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15292 lines
No EOL
507 KiB
JSON
15292 lines
No EOL
507 KiB
JSON
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]
|
|
],
|
|
"equations": [
|
|
{
|
|
"eq_number": 1,
|
|
"title": "Newton's Three Laws of Motion",
|
|
"domain_id": 1,
|
|
"significance": "Foundation of all classical mechanics; inertial frames; F=dp/dt; action=reaction"
|
|
},
|
|
{
|
|
"eq_number": 2,
|
|
"title": "Lagrangian Mechanics (Principle of Least Action)",
|
|
"domain_id": 1,
|
|
"significance": "Action S=\u222bL dt; \u03b4S=0 \u2192 Euler-Lagrange equations"
|
|
},
|
|
{
|
|
"eq_number": 3,
|
|
"title": "Hamiltonian Mechanics",
|
|
"domain_id": 1,
|
|
"significance": "Canonical eqs: q\u0307=\u2202H/\u2202p, \u1e57=\u2212\u2202H/\u2202q; symplectic structure"
|
|
},
|
|
{
|
|
"eq_number": 4,
|
|
"title": "Hamilton-Jacobi Equation",
|
|
"domain_id": 1,
|
|
"significance": "\u2202S/\u2202t + H(q,\u2202S/\u2202q,t)=0; bridges classical\u2192quantum"
|
|
},
|
|
{
|
|
"eq_number": 5,
|
|
"title": "Euler-Lagrange Equation",
|
|
"domain_id": 1,
|
|
"significance": "d/dt(\u2202L/\u2202q\u0307) \u2212 \u2202L/\u2202q = 0; from \u03b4S=0"
|
|
},
|
|
{
|
|
"eq_number": 6,
|
|
"title": "D'Alembert's Principle",
|
|
"domain_id": 1,
|
|
"significance": "Virtual work for dynamics: \u03a3(F_i\u2212\u1e57_i)\u00b7\u03b4r_i=0"
|
|
},
|
|
{
|
|
"eq_number": 7,
|
|
"title": "Euler's Rigid Body Rotation Equations",
|
|
"domain_id": 1,
|
|
"significance": "I\u00b7\u03c9\u0307 + \u03c9\u00d7(I\u00b7\u03c9) = \u03c4; angular momentum dynamics"
|
|
},
|
|
{
|
|
"eq_number": 8,
|
|
"title": "Conservation of Momentum",
|
|
"domain_id": 1,
|
|
"significance": "dP/dt = \u03a3F_ext; P constant when \u03a3F_ext=0"
|
|
},
|
|
{
|
|
"eq_number": 9,
|
|
"title": "Conservation of Angular Momentum",
|
|
"domain_id": 1,
|
|
"significance": "dL/dt = \u03c4_ext; L=I\u03c9 constant when \u03c4=0"
|
|
},
|
|
{
|
|
"eq_number": 10,
|
|
"title": "Conservation of Energy",
|
|
"domain_id": 1,
|
|
"significance": "dE/dt=0 for isolated system; time translation symmetry"
|
|
},
|
|
{
|
|
"eq_number": 11,
|
|
"title": "Work-Energy Theorem",
|
|
"domain_id": 1,
|
|
"significance": "W=\u0394KE; \u222bF\u00b7dr = \u00bdmv\u00b2_f \u2212 \u00bdmv\u00b2_i"
|
|
},
|
|
{
|
|
"eq_number": 12,
|
|
"title": "Impulse-Momentum Theorem",
|
|
"domain_id": 1,
|
|
"significance": "J=\u222bF dt=\u0394p"
|
|
},
|
|
{
|
|
"eq_number": 13,
|
|
"title": "Center of Mass Equation",
|
|
"domain_id": 1,
|
|
"significance": "MR\u0308_cm=\u03a3F_ext; COM moves like point particle"
|
|
},
|
|
{
|
|
"eq_number": 14,
|
|
"title": "Hooke's Law",
|
|
"domain_id": 18,
|
|
"significance": "F=\u2212kx; \u03c3=E\u03b5; linear elastic response"
|
|
},
|
|
{
|
|
"eq_number": 15,
|
|
"title": "Parallel Axis Theorem",
|
|
"domain_id": 1,
|
|
"significance": "I=I_cm+Md\u00b2"
|
|
},
|
|
{
|
|
"eq_number": 16,
|
|
"title": "Coriolis Force",
|
|
"domain_id": 1,
|
|
"significance": "F_cor=\u22122m \u03c9\u00d7v' (rotating frame)"
|
|
},
|
|
{
|
|
"eq_number": 17,
|
|
"title": "Centrifugal Force",
|
|
"domain_id": 1,
|
|
"significance": "F_cf=\u2212m \u03c9\u00d7(\u03c9\u00d7r) (rotating frame)"
|
|
},
|
|
{
|
|
"eq_number": 18,
|
|
"title": "Simple Harmonic Motion",
|
|
"domain_id": 1,
|
|
"significance": "x\u0308+\u03c9\u00b2x=0; x=A cos(\u03c9t+\u03c6); T=2\u03c0/\u03c9"
|
|
},
|
|
{
|
|
"eq_number": 19,
|
|
"title": "Damped Harmonic Oscillator",
|
|
"domain_id": 1,
|
|
"significance": "x\u0308+2\u03b2x\u0307+\u03c9\u2080\u00b2x=0; under/over/critically damped"
|
|
},
|
|
{
|
|
"eq_number": 20,
|
|
"title": "Forced Oscillator + Resonance",
|
|
"domain_id": 1,
|
|
"significance": "x\u0308+2\u03b2x\u0307+\u03c9\u2080\u00b2x=(F\u2080/m)cos \u03c9t; A=F\u2080/m/\u221a((\u03c9\u2080\u00b2\u2212\u03c9\u00b2)\u00b2+4\u03b2\u00b2\u03c9\u00b2)"
|
|
},
|
|
{
|
|
"eq_number": 21,
|
|
"title": "Coupled Oscillators (Normal Modes)",
|
|
"domain_id": 1,
|
|
"significance": "m\u1e8d\u2081=\u2212k x\u2081\u2212k'(x\u2081\u2212x\u2082); symmetric/antisymmetric modes"
|
|
},
|
|
{
|
|
"eq_number": 22,
|
|
"title": "Pendulum Equation",
|
|
"domain_id": 1,
|
|
"significance": "\u03b8\u0308+(g/L)sin \u03b8=0; small angle: \u03c9=\u221a(g/L)"
|
|
},
|
|
{
|
|
"eq_number": 23,
|
|
"title": "Kinematics (Constant Acceleration)",
|
|
"domain_id": 1,
|
|
"significance": "v=v\u2080+at, x=x\u2080+v\u2080t+\u00bdat\u00b2, v\u00b2=v\u2080\u00b2+2a\u0394x"
|
|
},
|
|
{
|
|
"eq_number": 24,
|
|
"title": "Universal Gravitation Law",
|
|
"domain_id": 2,
|
|
"significance": "F=\u2212G m\u2081m\u2082/r\u00b2 r\u0302"
|
|
},
|
|
{
|
|
"eq_number": 25,
|
|
"title": "Gravitational Potential Energy",
|
|
"domain_id": 2,
|
|
"significance": "U=\u2212GMm/r; F=\u2212\u2207U"
|
|
},
|
|
{
|
|
"eq_number": 26,
|
|
"title": "Kepler's First Law",
|
|
"domain_id": 2,
|
|
"significance": "Planetary orbits are ellipses with Sun at one focus"
|
|
},
|
|
{
|
|
"eq_number": 27,
|
|
"title": "Kepler's Second Law",
|
|
"domain_id": 2,
|
|
"significance": "Equal areas swept in equal times (areal velocity constant)"
|
|
},
|
|
{
|
|
"eq_number": 28,
|
|
"title": "Kepler's Third Law",
|
|
"domain_id": 2,
|
|
"significance": "T\u00b2\u221da\u00b3; T\u00b2=(4\u03c0\u00b2/GM)a\u00b3"
|
|
},
|
|
{
|
|
"eq_number": 29,
|
|
"title": "Escape Velocity",
|
|
"domain_id": 2,
|
|
"significance": "v_esc=\u221a(2GM/r)"
|
|
},
|
|
{
|
|
"eq_number": 30,
|
|
"title": "Orbital Velocity (Circular)",
|
|
"domain_id": 2,
|
|
"significance": "v_orb=\u221a(GM/r)"
|
|
},
|
|
{
|
|
"eq_number": 31,
|
|
"title": "Poisson Equation (Gravity)",
|
|
"domain_id": 2,
|
|
"significance": "\u2207\u00b2\u03a6=4\u03c0G\u03c1"
|
|
},
|
|
{
|
|
"eq_number": 32,
|
|
"title": "Tidal Force",
|
|
"domain_id": 2,
|
|
"significance": "F_tide\u22482GMm\u0394r/r\u00b3"
|
|
},
|
|
{
|
|
"eq_number": 33,
|
|
"title": "Gravitational Time Dilation (GR)",
|
|
"domain_id": 6,
|
|
"significance": "\u0394t'=\u0394t\u221a(1\u22122GM/rc\u00b2)"
|
|
},
|
|
{
|
|
"eq_number": 34,
|
|
"title": "Precession of Perihelion (GR)",
|
|
"domain_id": 6,
|
|
"significance": "\u0394\u03c6=6\u03c0GM/(a(1\u2212e\u00b2)c\u00b2) per orbit"
|
|
},
|
|
{
|
|
"eq_number": 35,
|
|
"title": "Lense-Thirring Precession (Frame Dragging)",
|
|
"domain_id": 6,
|
|
"significance": "\u03a9_LT=GJ/(2c\u00b2r\u00b3)(3(r\u0302\u00b7J\u0302)r\u0302\u2212J\u0302)"
|
|
},
|
|
{
|
|
"eq_number": 36,
|
|
"title": "Coulomb's Law",
|
|
"domain_id": 3,
|
|
"significance": "F=(1/4\u03c0\u03b5\u2080)q\u2081q\u2082/r\u00b2 r\u0302"
|
|
},
|
|
{
|
|
"eq_number": 37,
|
|
"title": "Lorentz Force Law",
|
|
"domain_id": 3,
|
|
"significance": "F=q(E+v\u00d7B)"
|
|
},
|
|
{
|
|
"eq_number": 38,
|
|
"title": "Maxwell's Equations (Differential)",
|
|
"domain_id": 3,
|
|
"significance": "\u2207\u00b7E=\u03c1/\u03b5\u2080, \u2207\u00b7B=0, \u2207\u00d7E=\u2212\u2202B/\u2202t, \u2207\u00d7B=\u03bc\u2080J+\u03bc\u2080\u03b5\u2080\u2202E/\u2202t"
|
|
},
|
|
{
|
|
"eq_number": 39,
|
|
"title": "Maxwell's Equations (Integral)",
|
|
"domain_id": 3,
|
|
"significance": "\u222eE\u00b7dA=Q/\u03b5\u2080, \u222eB\u00b7dA=0, \u222eE\u00b7dl=\u2212d\u03a6_B/dt, \u222eB\u00b7dl=\u03bc\u2080I+\u03bc\u2080\u03b5\u2080d\u03a6_E/dt"
|
|
},
|
|
{
|
|
"eq_number": 40,
|
|
"title": "Maxwell's Equations (Covariant / Tensor)",
|
|
"domain_id": 3,
|
|
"significance": "\u2202_\u03bcF^{\u03bc\u03bd}=\u03bc\u2080J^\u03bd; \u2202_\u03bcF\u0303^{\u03bc\u03bd}=0"
|
|
},
|
|
{
|
|
"eq_number": 41,
|
|
"title": "Scalar and Vector Potentials",
|
|
"domain_id": 3,
|
|
"significance": "B=\u2207\u00d7A; E=\u2212\u2207\u03c6\u2212\u2202A/\u2202t"
|
|
},
|
|
{
|
|
"eq_number": 42,
|
|
"title": "Gauge Invariance (U(1) in E&M)",
|
|
"domain_id": 3,
|
|
"significance": "A_\u03bc\u2192A_\u03bc+\u2202_\u03bc\u039b; E,B unchanged"
|
|
},
|
|
{
|
|
"eq_number": 43,
|
|
"title": "Biot-Savart Law",
|
|
"domain_id": 3,
|
|
"significance": "dB=(\u03bc\u2080/4\u03c0) I dl\u00d7r\u0302/r\u00b2"
|
|
},
|
|
{
|
|
"eq_number": 44,
|
|
"title": "Amp\u00e8re's Force Law (Wire)",
|
|
"domain_id": 3,
|
|
"significance": "dF=I dl\u00d7B"
|
|
},
|
|
{
|
|
"eq_number": 45,
|
|
"title": "Ohm's Law",
|
|
"domain_id": 3,
|
|
"significance": "V=IR; J=\u03c3E"
|
|
},
|
|
{
|
|
"eq_number": 46,
|
|
"title": "Kirchhoff's Current Law (KCL)",
|
|
"domain_id": 3,
|
|
"significance": "\u03a3I_in=\u03a3I_out at junction"
|
|
},
|
|
{
|
|
"eq_number": 47,
|
|
"title": "Kirchhoff's Voltage Law (KVL)",
|
|
"domain_id": 3,
|
|
"significance": "\u03a3V around closed loop=0"
|
|
},
|
|
{
|
|
"eq_number": 48,
|
|
"title": "Faraday's Law of Induction",
|
|
"domain_id": 3,
|
|
"significance": "\u03b5=\u2212d\u03a6_B/dt; induced EMF=\u2212flux change"
|
|
},
|
|
{
|
|
"eq_number": 49,
|
|
"title": "Lenz's Law",
|
|
"domain_id": 3,
|
|
"significance": "Induced current opposes flux change"
|
|
},
|
|
{
|
|
"eq_number": 50,
|
|
"title": "Poynting's Theorem",
|
|
"domain_id": 3,
|
|
"significance": "\u2202u/\u2202t+\u2207\u00b7S=\u2212J\u00b7E; S=(1/\u03bc\u2080)E\u00d7B"
|
|
},
|
|
{
|
|
"eq_number": 51,
|
|
"title": "Electromagnetic Wave Equation",
|
|
"domain_id": 3,
|
|
"significance": "\u25a1E=0; \u25a1B=0; c=1/\u221a(\u03bc\u2080\u03b5\u2080)"
|
|
},
|
|
{
|
|
"eq_number": 52,
|
|
"title": "EM Stress-Energy Tensor",
|
|
"domain_id": 3,
|
|
"significance": "T^{\u03bc\u03bd}=(1/\u03bc\u2080)[F^\u03bc_\u03b1 F^{\u03bd\u03b1}+\u00bcg^{\u03bc\u03bd}F\u00b2]"
|
|
},
|
|
{
|
|
"eq_number": 53,
|
|
"title": "Lienard-Wiechert Potentials",
|
|
"domain_id": 3,
|
|
"significance": "Retarded potentials for arbitrarily moving point charge"
|
|
},
|
|
{
|
|
"eq_number": 54,
|
|
"title": "Larmor Formula (Non-rel. Radiation)",
|
|
"domain_id": 3,
|
|
"significance": "P=q\u00b2 a\u00b2/(6\u03c0\u03b5\u2080 c\u00b3)"
|
|
},
|
|
{
|
|
"eq_number": 55,
|
|
"title": "Li\u00e9nard Formula (Relativistic Radiation)",
|
|
"domain_id": 3,
|
|
"significance": "P=(q\u00b2\u03b3\u2076/6\u03c0\u03b5\u2080c\u00b3)[a\u00b2\u2212(v\u00d7a)\u00b2/c\u00b2]"
|
|
},
|
|
{
|
|
"eq_number": 56,
|
|
"title": "Abraham-Lorentz Force (Radiation Reaction)",
|
|
"domain_id": 3,
|
|
"significance": "F_rad=(q\u00b2/6\u03c0\u03b5\u2080c\u00b3)d\u00b3r/dt\u00b3"
|
|
},
|
|
{
|
|
"eq_number": 57,
|
|
"title": "Coulomb Gauge",
|
|
"domain_id": 3,
|
|
"significance": "\u2207\u00b7A=0"
|
|
},
|
|
{
|
|
"eq_number": 58,
|
|
"title": "Lorenz Gauge",
|
|
"domain_id": 3,
|
|
"significance": "\u2202_\u03bc A^\u03bc=0"
|
|
},
|
|
{
|
|
"eq_number": 59,
|
|
"title": "RC Circuit Charging",
|
|
"domain_id": 3,
|
|
"significance": "q(t)=C \u03b5(1\u2212e^{\u2212t/RC}); \u03c4=RC"
|
|
},
|
|
{
|
|
"eq_number": 60,
|
|
"title": "RL Circuit Time Constant",
|
|
"domain_id": 3,
|
|
"significance": "I(t)=I\u2080 e^{\u2212tR/L}; \u03c4=L/R"
|
|
},
|
|
{
|
|
"eq_number": 61,
|
|
"title": "LC Oscillation",
|
|
"domain_id": 3,
|
|
"significance": "\u03c9\u2080=1/\u221a(LC); q\u0308+\u03c9\u2080\u00b2 q=0"
|
|
},
|
|
{
|
|
"eq_number": 62,
|
|
"title": "RLC Damped Oscillation",
|
|
"domain_id": 3,
|
|
"significance": "q\u0308+(R/L)q\u0307+(1/LC)q=0; \u03b3=R/(2L)"
|
|
},
|
|
{
|
|
"eq_number": 63,
|
|
"title": "Skin Effect",
|
|
"domain_id": 3,
|
|
"significance": "\u03b4=\u221a(2/\u03c9\u03bc\u03c3); penetration depth"
|
|
},
|
|
{
|
|
"eq_number": 64,
|
|
"title": "Dielectric Polarization (Linear)",
|
|
"domain_id": 3,
|
|
"significance": "D=\u03b5\u2080E+P=\u03b5_r \u03b5\u2080 E"
|
|
},
|
|
{
|
|
"eq_number": 65,
|
|
"title": "Magnetic Susceptibility",
|
|
"domain_id": 3,
|
|
"significance": "M=\u03c7_m H; B=\u03bc\u2080(H+M)=\u03bc_r \u03bc\u2080 H"
|
|
},
|
|
{
|
|
"eq_number": 66,
|
|
"title": "Zeroth Law of Thermodynamics",
|
|
"domain_id": 4,
|
|
"significance": "Thermal equilibrium is transitive; defines temperature"
|
|
},
|
|
{
|
|
"eq_number": 67,
|
|
"title": "First Law of Thermodynamics",
|
|
"domain_id": 4,
|
|
"significance": "dU=\u03b4Q\u2212\u03b4W; \u0394U=Q\u2212W"
|
|
},
|
|
{
|
|
"eq_number": 68,
|
|
"title": "Second Law of Thermodynamics",
|
|
"domain_id": 4,
|
|
"significance": "dS_total\u22650; entropy never decreases"
|
|
},
|
|
{
|
|
"eq_number": 69,
|
|
"title": "Third Law of Thermodynamics",
|
|
"domain_id": 4,
|
|
"significance": "S\u21920 as T\u21920 (perfect crystal)"
|
|
},
|
|
{
|
|
"eq_number": 70,
|
|
"title": "Ideal Gas Law",
|
|
"domain_id": 4,
|
|
"significance": "pV=nRT=N k_B T"
|
|
},
|
|
{
|
|
"eq_number": 71,
|
|
"title": "Van der Waals Equation of State",
|
|
"domain_id": 4,
|
|
"significance": "(p+an\u00b2/V\u00b2)(V\u2212nb)=nRT"
|
|
},
|
|
{
|
|
"eq_number": 72,
|
|
"title": "Kinetic Theory: Pressure",
|
|
"domain_id": 4,
|
|
"significance": "p=(1/3) N m \u27e8v\u00b2\u27e9/V"
|
|
},
|
|
{
|
|
"eq_number": 73,
|
|
"title": "Equipartition Theorem",
|
|
"domain_id": 4,
|
|
"significance": "\u27e8E\u27e9=f k_B T/2; C_V=(f/2)R"
|
|
},
|
|
{
|
|
"eq_number": 74,
|
|
"title": "Maxwell-Boltzmann Speed Distribution",
|
|
"domain_id": 4,
|
|
"significance": "f(v)=4\u03c0(m/2\u03c0k_B T)^{3/2} v\u00b2 e^{\u2212mv\u00b2/2kBT}"
|
|
},
|
|
{
|
|
"eq_number": 75,
|
|
"title": "Carnot Efficiency",
|
|
"domain_id": 4,
|
|
"significance": "\u03b7_max=1\u2212T_c/T_h"
|
|
},
|
|
{
|
|
"eq_number": 76,
|
|
"title": "Clausius-Clapeyron Relation",
|
|
"domain_id": 4,
|
|
"significance": "dP/dT=L/(T \u0394V) for phase coexistence"
|
|
},
|
|
{
|
|
"eq_number": 77,
|
|
"title": "Gibbs Phase Rule",
|
|
"domain_id": 4,
|
|
"significance": "F=C\u2212P+2"
|
|
},
|
|
{
|
|
"eq_number": 78,
|
|
"title": "Helmholtz Free Energy",
|
|
"domain_id": 4,
|
|
"significance": "F=U\u2212TS; \u0394F\u22640 at const T,V (spontaneous)"
|
|
},
|
|
{
|
|
"eq_number": 79,
|
|
"title": "Gibbs Free Energy",
|
|
"domain_id": 4,
|
|
"significance": "G=H\u2212TS; \u0394G\u22640 at const T,P (spontaneous)"
|
|
},
|
|
{
|
|
"eq_number": 80,
|
|
"title": "Enthalpy",
|
|
"domain_id": 4,
|
|
"significance": "H=U+pV; \u0394H=Q_p"
|
|
},
|
|
{
|
|
"eq_number": 81,
|
|
"title": "Maxwell Relations (Thermodynamics)",
|
|
"domain_id": 4,
|
|
"significance": "(\u2202T/\u2202V)_S=\u2212(\u2202p/\u2202S)_V; (\u2202T/\u2202p)_S=(\u2202V/\u2202S)_p; (\u2202S/\u2202V)_T=(\u2202p/\u2202T)_V; (\u2202S/\u2202p)_T=\u2212(\u2202V/\u2202T)_p"
|
|
},
|
|
{
|
|
"eq_number": 82,
|
|
"title": "TdS Equations",
|
|
"domain_id": 4,
|
|
"significance": "T dS=C_V dT+T(\u2202p/\u2202T)_V dV; T dS=C_p dT\u2212T(\u2202V/\u2202T)_p dp"
|
|
},
|
|
{
|
|
"eq_number": 83,
|
|
"title": "Specific Heat Relations (C_p\u2212C_V)",
|
|
"domain_id": 4,
|
|
"significance": "C_p\u2212C_V=\u2212T(\u2202V/\u2202T)_p\u00b2/(\u2202V/\u2202p)_T=TV\u03b1\u00b2/\u03ba_T"
|
|
},
|
|
{
|
|
"eq_number": 84,
|
|
"title": "Joule-Thomson Coefficient",
|
|
"domain_id": 4,
|
|
"significance": "\u03bc_JT=(\u2202T/\u2202p)_H=(V/C_p)(T\u03b1\u22121)"
|
|
},
|
|
{
|
|
"eq_number": 85,
|
|
"title": "Entropy of Mixing",
|
|
"domain_id": 4,
|
|
"significance": "\u0394S_mix=\u2212k_B(N\u2081 ln x\u2081+N\u2082 ln x\u2082)"
|
|
},
|
|
{
|
|
"eq_number": 86,
|
|
"title": "Planck's Blackbody Radiation Law",
|
|
"domain_id": 5,
|
|
"significance": "B_\u03bd=(2h\u03bd\u00b3/c\u00b2)/(e^{h\u03bd/kT}\u22121)"
|
|
},
|
|
{
|
|
"eq_number": 87,
|
|
"title": "Wien's Displacement Law",
|
|
"domain_id": 5,
|
|
"significance": "\u03bb_max T=2.898\u00d710\u207b\u00b3 m\u00b7K"
|
|
},
|
|
{
|
|
"eq_number": 88,
|
|
"title": "Stefan-Boltzmann Law",
|
|
"domain_id": 5,
|
|
"significance": "j*=\u03c3 T\u2074; \u03c3=2\u03c0\u2075k_B\u2074/(15h\u00b3c\u00b2)"
|
|
},
|
|
{
|
|
"eq_number": 89,
|
|
"title": "Photoelectric Effect Equation (Einstein)",
|
|
"domain_id": 5,
|
|
"significance": "K_max=h\u03bd\u2212\u03c6; photon quanta"
|
|
},
|
|
{
|
|
"eq_number": 90,
|
|
"title": "Einstein A and B Coefficients",
|
|
"domain_id": 5,
|
|
"significance": "A_21/B_21=8\u03c0h\u03bd\u00b3/c\u00b3; B_12/B_21=g\u2082/g\u2081"
|
|
},
|
|
{
|
|
"eq_number": 91,
|
|
"title": "Compton Scattering Formula",
|
|
"domain_id": 5,
|
|
"significance": "\u0394\u03bb=(h/m_e c)(1\u2212cos \u03b8); \u0394\u03bb_max\u22480.00486 nm"
|
|
},
|
|
{
|
|
"eq_number": 92,
|
|
"title": "de Broglie Wavelength",
|
|
"domain_id": 5,
|
|
"significance": "\u03bb=h/p=h/(\u03b3mv)"
|
|
},
|
|
{
|
|
"eq_number": 93,
|
|
"title": "Schr\u00f6dinger Equation (Time-Dependent)",
|
|
"domain_id": 5,
|
|
"significance": "i\u210f\u2202\u03c8/\u2202t=\u0124\u03c8"
|
|
},
|
|
{
|
|
"eq_number": 94,
|
|
"title": "Time-Independent Schr\u00f6dinger Equation",
|
|
"domain_id": 5,
|
|
"significance": "\u0124\u03c8=E\u03c8"
|
|
},
|
|
{
|
|
"eq_number": 95,
|
|
"title": "Born Rule (Probability Interpretation)",
|
|
"domain_id": 5,
|
|
"significance": "\u03c1(r,t)=|\u03c8(r,t)|\u00b2"
|
|
},
|
|
{
|
|
"eq_number": 96,
|
|
"title": "Probability Current (QM)",
|
|
"domain_id": 5,
|
|
"significance": "j=(\u210f/2mi)(\u03c8*\u2207\u03c8\u2212\u03c8\u2207\u03c8*); \u2202\u03c1/\u2202t+\u2207\u00b7j=0"
|
|
},
|
|
{
|
|
"eq_number": 97,
|
|
"title": "Canonical Commutation Relations",
|
|
"domain_id": 5,
|
|
"significance": "[x\u0302_i,p\u0302_j]=i\u210f\u03b4_{ij}"
|
|
},
|
|
{
|
|
"eq_number": 98,
|
|
"title": "Heisenberg Uncertainty Principle",
|
|
"domain_id": 5,
|
|
"significance": "\u0394x\u00b7\u0394p\u2265\u210f/2; \u0394E\u00b7\u0394t\u2265\u210f/2"
|
|
},
|
|
{
|
|
"eq_number": 99,
|
|
"title": "Harmonic Oscillator Energy Levels (QM)",
|
|
"domain_id": 5,
|
|
"significance": "E_n=\u210f\u03c9(n+\u00bd); \u00e2|n\u27e9=\u221an|n\u22121\u27e9, \u00e2\u2020|n\u27e9=\u221a(n+1)|n+1\u27e9"
|
|
},
|
|
{
|
|
"eq_number": 100,
|
|
"title": "Hydrogen Atom Energy Levels",
|
|
"domain_id": 5,
|
|
"significance": "E_n=\u2212R_y/n\u00b2; R_y=13.605693123 eV"
|
|
},
|
|
{
|
|
"eq_number": 101,
|
|
"title": "Angular Momentum Quantization",
|
|
"domain_id": 5,
|
|
"significance": "L\u00b2|l,m\u27e9=\u210f\u00b2 l(l+1); L_z|l,m\u27e9=\u210f m"
|
|
},
|
|
{
|
|
"eq_number": 102,
|
|
"title": "Spin-\u00bd Algebra (Pauli Matrices)",
|
|
"domain_id": 5,
|
|
"significance": "S=(\u210f/2)\u03c3; [\u03c3_i,\u03c3_j]=2i\u03b5_{ijk}\u03c3_k; {\u03c3_i,\u03c3_j}=2\u03b4_{ij}"
|
|
},
|
|
{
|
|
"eq_number": 103,
|
|
"title": "Spin-Orbit Coupling",
|
|
"domain_id": 5,
|
|
"significance": "H_SO=(1/2m\u00b2c\u00b2)(1/r)(dV/dr) L\u00b7S"
|
|
},
|
|
{
|
|
"eq_number": 104,
|
|
"title": "Dirac Equation",
|
|
"domain_id": 5,
|
|
"significance": "(i\u210f\u03b3^\u03bc\u2202_\u03bc\u2212mc)\u03c8=0"
|
|
},
|
|
{
|
|
"eq_number": 105,
|
|
"title": "Klein-Gordon Equation",
|
|
"domain_id": 5,
|
|
"significance": "(\u25a1+m\u00b2c\u00b2/\u210f\u00b2)\u03c6=0"
|
|
},
|
|
{
|
|
"eq_number": 106,
|
|
"title": "Fine Structure Formula (Hydrogen)",
|
|
"domain_id": 5,
|
|
"significance": "\u0394E_FS=(R_y \u03b1\u00b2/n\u00b3)[1/(j+\u00bd)\u22123/(4n)]"
|
|
},
|
|
{
|
|
"eq_number": 107,
|
|
"title": "Lamb Shift",
|
|
"domain_id": 5,
|
|
"significance": "\u0394E(2S\u22122P)\u22481057.8 MHz; QED vacuum effects"
|
|
},
|
|
{
|
|
"eq_number": 108,
|
|
"title": "Anomalous Magnetic Moment (Electron)",
|
|
"domain_id": 7,
|
|
"significance": "a_e=(g\u22122)/2\u22480.00115965218091; QED+EW+hadronic"
|
|
},
|
|
{
|
|
"eq_number": 109,
|
|
"title": "Pauli Exclusion Principle",
|
|
"domain_id": 5,
|
|
"significance": "No two identical fermions in same quantum state; \u03c8 antisymmetric"
|
|
},
|
|
{
|
|
"eq_number": 110,
|
|
"title": "Spin-Statistics Theorem",
|
|
"domain_id": 5,
|
|
"significance": "Half-int spin\u2192fermion (anticommutators); int\u2192boson (commutators)"
|
|
},
|
|
{
|
|
"eq_number": 111,
|
|
"title": "Fermi's Golden Rule",
|
|
"domain_id": 5,
|
|
"significance": "\u0393_{i\u2192f}=(2\u03c0/\u210f)|\u27e8f|V|i\u27e9|\u00b2 \u03c1(E_f)"
|
|
},
|
|
{
|
|
"eq_number": 112,
|
|
"title": "Time-Dependent Perturbation Theory (1st Order)",
|
|
"domain_id": 5,
|
|
"significance": "c_f(t)=\u2212(i/\u210f)\u222b\u2080\u1d57 \u27e8f|V(t')|i\u27e9 e^{i\u03c9_fi t'} dt'"
|
|
},
|
|
{
|
|
"eq_number": 113,
|
|
"title": "WKB Approximation",
|
|
"domain_id": 5,
|
|
"significance": "\u03c8\u223c(1/\u221ap)exp(\u00b1i\u222b p dx/\u210f); Bohr-Sommerfeld quantization"
|
|
},
|
|
{
|
|
"eq_number": 114,
|
|
"title": "Born Approximation (Scattering)",
|
|
"domain_id": 5,
|
|
"significance": "f(\u03b8,\u03c6)=\u2212(2m/\u210f\u00b2)(1/4\u03c0)\u222b e^{\u2212iq\u00b7r} V(r) d\u00b3r"
|
|
},
|
|
{
|
|
"eq_number": 115,
|
|
"title": "Partial Wave Expansion (Scattering)",
|
|
"domain_id": 5,
|
|
"significance": "f(\u03b8)=(1/k)\u03a3(2l+1)e^{i\u03b4_l} sin \u03b4_l P_l(cos \u03b8)"
|
|
},
|
|
{
|
|
"eq_number": 116,
|
|
"title": "Optical Theorem",
|
|
"domain_id": 5,
|
|
"significance": "Im f(0)=(k/4\u03c0)\u03c3_total"
|
|
},
|
|
{
|
|
"eq_number": 117,
|
|
"title": "Feynman Path Integral",
|
|
"domain_id": 5,
|
|
"significance": "\u27e8x_f,t_f|x_i,t_i\u27e9=\u222b D[x(t)] exp(iS[x]/\u210f)"
|
|
},
|
|
{
|
|
"eq_number": 118,
|
|
"title": "Von Neumann Equation",
|
|
"domain_id": 5,
|
|
"significance": "i\u210f \u2202\u03c1\u0302/\u2202t=[\u0124,\u03c1\u0302]"
|
|
},
|
|
{
|
|
"eq_number": 119,
|
|
"title": "Ehrenfest Theorem",
|
|
"domain_id": 5,
|
|
"significance": "d\u27e8A\u27e9/dt=(1/i\u210f)\u27e8[A,\u0124]\u27e9+\u27e8\u2202A/\u2202t\u27e9"
|
|
},
|
|
{
|
|
"eq_number": 120,
|
|
"title": "Bell's Inequality",
|
|
"domain_id": 5,
|
|
"significance": "|E(a,b)\u2212E(a,c)|\u22641+E(b,c)"
|
|
},
|
|
{
|
|
"eq_number": 121,
|
|
"title": "Lorentz Transformations (Boost)",
|
|
"domain_id": 6,
|
|
"significance": "x'=\u03b3(x\u2212vt); t'=\u03b3(t\u2212vx/c\u00b2); \u03b3=1/\u221a(1\u2212v\u00b2/c\u00b2)"
|
|
},
|
|
{
|
|
"eq_number": 122,
|
|
"title": "Minkowski Spacetime Interval",
|
|
"domain_id": 6,
|
|
"significance": "ds\u00b2=\u2212c\u00b2dt\u00b2+dx\u00b2+dy\u00b2+dz\u00b2=\u03b7_\u03bc\u03bd dx^\u03bc dx^\u03bd"
|
|
},
|
|
{
|
|
"eq_number": 123,
|
|
"title": "Time Dilation",
|
|
"domain_id": 6,
|
|
"significance": "\u0394t'=\u03b3\u0394t (moving clock runs slow)"
|
|
},
|
|
{
|
|
"eq_number": 124,
|
|
"title": "Length Contraction",
|
|
"domain_id": 6,
|
|
"significance": "L'=L/\u03b3 (moving object contracts)"
|
|
},
|
|
{
|
|
"eq_number": 125,
|
|
"title": "Relativistic Energy-Momentum Relation",
|
|
"domain_id": 6,
|
|
"significance": "E\u00b2=(pc)\u00b2+(mc\u00b2)\u00b2; E=\u03b3mc\u00b2; p=\u03b3mv"
|
|
},
|
|
{
|
|
"eq_number": 126,
|
|
"title": "Mass-Energy Equivalence",
|
|
"domain_id": 6,
|
|
"significance": "E=mc\u00b2; \u0394E=\u0394m c\u00b2"
|
|
},
|
|
{
|
|
"eq_number": 127,
|
|
"title": "Relativistic Doppler Effect",
|
|
"domain_id": 6,
|
|
"significance": "f_obs=f_s\u221a[(1+\u03b2)/(1\u2212\u03b2)] (longitudinal); transverse: f_obs=\u03b3f_s"
|
|
},
|
|
{
|
|
"eq_number": 128,
|
|
"title": "Relativistic Velocity Addition",
|
|
"domain_id": 6,
|
|
"significance": "u=(u'+v)/(1+u'v/c\u00b2)"
|
|
},
|
|
{
|
|
"eq_number": 129,
|
|
"title": "Einstein Field Equations (GR)",
|
|
"domain_id": 6,
|
|
"significance": "G_\u03bc\u03bd+\u039bg_\u03bc\u03bd=(8\u03c0G/c\u2074)T_\u03bc\u03bd"
|
|
},
|
|
{
|
|
"eq_number": 130,
|
|
"title": "Einstein-Hilbert Action",
|
|
"domain_id": 6,
|
|
"significance": "S=(c\u2074/16\u03c0G)\u222b d\u2074x\u221a(\u2212g)(R\u22122\u039b)+S_matter"
|
|
},
|
|
{
|
|
"eq_number": 131,
|
|
"title": "Schwarzschild Metric",
|
|
"domain_id": 6,
|
|
"significance": "ds\u00b2=\u2212(1\u2212r_s/r)c\u00b2dt\u00b2+dr\u00b2/(1\u2212r_s/r)+r\u00b2d\u03a9\u00b2; r_s=2GM/c\u00b2"
|
|
},
|
|
{
|
|
"eq_number": 132,
|
|
"title": "Kerr Metric (Rotating Black Hole)",
|
|
"domain_id": 6,
|
|
"significance": "Rotating axisymmetric vacuum solution; a=J/Mc"
|
|
},
|
|
{
|
|
"eq_number": 133,
|
|
"title": "FLRW Metric",
|
|
"domain_id": 6,
|
|
"significance": "ds\u00b2=\u2212c\u00b2dt\u00b2+a\u00b2(t)[dr\u00b2/(1\u2212kr\u00b2)+r\u00b2d\u03a9\u00b2]"
|
|
},
|
|
{
|
|
"eq_number": 134,
|
|
"title": "Geodesic Equation",
|
|
"domain_id": 6,
|
|
"significance": "d\u00b2x^\u03bc/d\u03c4\u00b2+\u0393^\u03bc_\u03b1\u03b2(dx^\u03b1/d\u03c4)(dx^\u03b2/d\u03c4)=0"
|
|
},
|
|
{
|
|
"eq_number": 135,
|
|
"title": "Gravitational Wave (TT Gauge)",
|
|
"domain_id": 6,
|
|
"significance": "h_\u03bc\u03bd^{TT} has only h_+,h_\u00d7 spatial transverse components"
|
|
},
|
|
{
|
|
"eq_number": 136,
|
|
"title": "Bekenstein-Hawking Black Hole Entropy",
|
|
"domain_id": 6,
|
|
"significance": "S_BH=k_B A/4\u2113_P\u00b2=k_B c\u00b3A/(4G\u210f)"
|
|
},
|
|
{
|
|
"eq_number": 137,
|
|
"title": "Hawking Temperature",
|
|
"domain_id": 6,
|
|
"significance": "T_H=\u210fc\u00b3/(8\u03c0GMk_B); T_H(M\u2299)\u22486.2\u00d710\u207b\u2078 K"
|
|
},
|
|
{
|
|
"eq_number": 138,
|
|
"title": "Black Hole Area Theorem (Hawking 1971)",
|
|
"domain_id": 6,
|
|
"significance": "dA/dt\u22650; horizon area never decreases"
|
|
},
|
|
{
|
|
"eq_number": 139,
|
|
"title": "Standard Model Lagrangian (Full)",
|
|
"domain_id": 7,
|
|
"significance": "\u2112_SM=\u2112_gauge+\u2112_fermion+\u2112_Higgs+\u2112_Yukawa; SU(3)\u00d7SU(2)\u00d7U(1)"
|
|
},
|
|
{
|
|
"eq_number": 140,
|
|
"title": "Yang-Mills Field Strength",
|
|
"domain_id": 7,
|
|
"significance": "F_\u03bc\u03bd^a=\u2202_\u03bcA_\u03bd^a\u2212\u2202_\u03bdA_\u03bc^a+gf^{abc}A_\u03bc^bA_\u03bd^c"
|
|
},
|
|
{
|
|
"eq_number": 141,
|
|
"title": "QED Lagrangian",
|
|
"domain_id": 7,
|
|
"significance": "\u2112_QED=\u03c8\u0304(i\u2202\u0338\u2212m)\u03c8\u2212e\u03c8\u0304\u03b3^\u03bc\u03c8A_\u03bc\u2212\u00bcF_\u03bc\u03bdF^{\u03bc\u03bd}"
|
|
},
|
|
{
|
|
"eq_number": 142,
|
|
"title": "QCD Lagrangian",
|
|
"domain_id": 7,
|
|
"significance": "\u2112_QCD=\u03a3\u03c8\u0304_f(iD\u0338\u2212m_f)\u03c8_f\u2212\u00bcG_a^{\u03bc\u03bd}G^a_{\u03bc\u03bd}"
|
|
},
|
|
{
|
|
"eq_number": 143,
|
|
"title": "Electroweak Symmetry Breaking",
|
|
"domain_id": 7,
|
|
"significance": "SU(2)_L\u00d7U(1)_Y\u2192U(1)_EM via Higgs VEV"
|
|
},
|
|
{
|
|
"eq_number": 144,
|
|
"title": "QCD Beta Function (1-loop)",
|
|
"domain_id": 7,
|
|
"significance": "\u03b2(\u03b1_s)=\u2212(b\u2080/2\u03c0)\u03b1_s\u00b2; b\u2080=11\u22122n_f/3"
|
|
},
|
|
{
|
|
"eq_number": 145,
|
|
"title": "DGLAP Evolution Equations",
|
|
"domain_id": 7,
|
|
"significance": "\u2202q/\u2202lnQ\u00b2=(\u03b1_s/2\u03c0)\u222b(dz/z)[P_qq q+P_qg g]; gluon evolution similarly"
|
|
},
|
|
{
|
|
"eq_number": 146,
|
|
"title": "CKM Matrix (Quark Mixing)",
|
|
"domain_id": 7,
|
|
"significance": "3\u00d73 unitary; 4 parameters (3 angles+1 CP phase)"
|
|
},
|
|
{
|
|
"eq_number": 147,
|
|
"title": "PMNS Matrix (Neutrino Mixing)",
|
|
"domain_id": 7,
|
|
"significance": "3\u00d73 leptonic mixing; \u03b8\u2081\u2082\u224833\u00b0,\u03b8\u2082\u2083\u224845\u00b0,\u03b8\u2081\u2083\u22488.5\u00b0"
|
|
},
|
|
{
|
|
"eq_number": 148,
|
|
"title": "Gell-Mann\u2013Oakes\u2013Renner Relation",
|
|
"domain_id": 7,
|
|
"significance": "m_\u03c0\u00b2=\u2212(m_u+m_d)\u27e8\u03c8\u0304\u03c8\u27e9/f_\u03c0\u00b2"
|
|
},
|
|
{
|
|
"eq_number": 149,
|
|
"title": "Higgs Mechanism (Mass Generation)",
|
|
"domain_id": 7,
|
|
"significance": "Scalar VEV v=246 GeV\u2192W,Z masses; fermion masses via Yukawa"
|
|
},
|
|
{
|
|
"eq_number": 150,
|
|
"title": "Weinberg Angle",
|
|
"domain_id": 7,
|
|
"significance": "sin\u00b2\u03b8_W=1\u2212M_W\u00b2/M_Z\u00b2; 0.23121\u00b10.00004"
|
|
},
|
|
{
|
|
"eq_number": 151,
|
|
"title": "Faddeev-Popov Gauge Fixing + Ghosts",
|
|
"domain_id": 7,
|
|
"significance": "Anticommuting scalar ghosts cancel unphysical gluon d.o.f."
|
|
},
|
|
{
|
|
"eq_number": 152,
|
|
"title": "BRST Symmetry",
|
|
"domain_id": 7,
|
|
"significance": "Residual global symmetry after gauge fixing"
|
|
},
|
|
{
|
|
"eq_number": 153,
|
|
"title": "Running Coupling (RGE, General)",
|
|
"domain_id": 7,
|
|
"significance": "\u03bc dg/d\u03bc=\u03b2(g); \u03bc d m/d\u03bc=\u03b3_m m"
|
|
},
|
|
{
|
|
"eq_number": 154,
|
|
"title": "Fermi's Theory (4-Fermion, Low-Energy EW)",
|
|
"domain_id": 7,
|
|
"significance": "\u2112_eff=\u2212(G_F/\u221a2) J_\u03bc^{CC} J^{CC\u2020\u03bc}"
|
|
},
|
|
{
|
|
"eq_number": 155,
|
|
"title": "Pati-Salam Model (SU(4)\u00d7SU(2)\u00d7SU(2))",
|
|
"domain_id": 7,
|
|
"significance": "Partial unification with lepton as 4th color"
|
|
},
|
|
{
|
|
"eq_number": 156,
|
|
"title": "Grand Unified Theories (GUTs)",
|
|
"domain_id": 7,
|
|
"significance": "SU(5), SO(10), E\u2086 unification at ~10\u00b9\u2076 GeV"
|
|
},
|
|
{
|
|
"eq_number": 157,
|
|
"title": "Axion (Peccei-Quinn Solution to Strong CP)",
|
|
"domain_id": 7,
|
|
"significance": "a\u2192\u03b3\u03b3; m_a~\u03bceV\u2212meV"
|
|
},
|
|
{
|
|
"eq_number": 158,
|
|
"title": "Muon g\u22122 Anomaly",
|
|
"domain_id": 7,
|
|
"significance": "a_\u03bc(exp) = 0.001165920705(148) (Fermilab final, June 2025); a_\u03bc(theory) = 0.00116592033(62) (lattice QCD white paper, May 2025). Now consistent; long-standing 4.2\u03c3 tension resolved."
|
|
},
|
|
{
|
|
"eq_number": 159,
|
|
"title": "First Friedmann Equation",
|
|
"domain_id": 8,
|
|
"significance": "H\u00b2=(\u0227/a)\u00b2=8\u03c0G\u03c1/3\u2212kc\u00b2/a\u00b2+\u039bc\u00b2/3; H\u2080=67.4 km/s/Mpc"
|
|
},
|
|
{
|
|
"eq_number": 160,
|
|
"title": "Second Friedmann Equation",
|
|
"domain_id": 8,
|
|
"significance": "\u00e4/a=\u22124\u03c0G(\u03c1+3p/c\u00b2)/3+\u039bc\u00b2/3"
|
|
},
|
|
{
|
|
"eq_number": 161,
|
|
"title": "Cosmological Fluid Equation",
|
|
"domain_id": 8,
|
|
"significance": "\u03c1\u0307+3H(\u03c1+p/c\u00b2)=0"
|
|
},
|
|
{
|
|
"eq_number": 162,
|
|
"title": "Redshift Relation",
|
|
"domain_id": 8,
|
|
"significance": "1+z=a\u2080/a(t); \u03bb_obs=\u03bb_emit(1+z)"
|
|
},
|
|
{
|
|
"eq_number": 163,
|
|
"title": "Hubble-Lema\u00eetre Law",
|
|
"domain_id": 8,
|
|
"significance": "v=H\u2080 d (low z)"
|
|
},
|
|
{
|
|
"eq_number": 164,
|
|
"title": "CMB Blackbody Spectrum",
|
|
"domain_id": 8,
|
|
"significance": "T\u2080=2.72548\u00b10.00057 K; \u0394T/T\u2080<50 ppm"
|
|
},
|
|
{
|
|
"eq_number": 165,
|
|
"title": "BBN Primordial Element Abundances",
|
|
"domain_id": 8,
|
|
"significance": "Y_p=0.24709\u00b10.00025; D/H=(2.527\u00b10.030)\u00d710\u207b\u2075"
|
|
},
|
|
{
|
|
"eq_number": 166,
|
|
"title": "Sound Horizon at Recombination",
|
|
"domain_id": 8,
|
|
"significance": "r_s\u2248147 Mpc (comoving); BAO standard ruler"
|
|
},
|
|
{
|
|
"eq_number": 167,
|
|
"title": "Sachs-Wolfe Effect (CMB)",
|
|
"domain_id": 8,
|
|
"significance": "\u0394T/T=\u2212\u03a6/(3c\u00b2) at large angular scales"
|
|
},
|
|
{
|
|
"eq_number": 168,
|
|
"title": "Dark Energy Equation of State",
|
|
"domain_id": 8,
|
|
"significance": "w=p/\u03c1c\u00b2=\u22121.03\u00b10.03"
|
|
},
|
|
{
|
|
"eq_number": 169,
|
|
"title": "Deceleration Parameter",
|
|
"domain_id": 8,
|
|
"significance": "q\u2080=\u2212\u00e4a/\u0227\u00b2=\u22120.53\u00b10.02"
|
|
},
|
|
{
|
|
"eq_number": 170,
|
|
"title": "Matter Power Spectrum",
|
|
"domain_id": 8,
|
|
"significance": "P(k)~k^{n_s}; n_s=0.9649\u00b10.0042"
|
|
},
|
|
{
|
|
"eq_number": 171,
|
|
"title": "Cosmic Distance Ladder Relations",
|
|
"domain_id": 8,
|
|
"significance": "d_L=(1+z)\u03c7; \u03bc=5log\u2081\u2080(d_L/10pc)"
|
|
},
|
|
{
|
|
"eq_number": 172,
|
|
"title": "Inflation (Slow-Roll)",
|
|
"domain_id": 8,
|
|
"significance": "\u03b5=\u2212H\u0307/H\u00b2\u226a1; \u03b7\u226a1 scalar spectral index"
|
|
},
|
|
{
|
|
"eq_number": 173,
|
|
"title": "Hubble Tension",
|
|
"domain_id": 8,
|
|
"significance": "H\u2080(CMB)=67.4\u00b10.5 vs H\u2080(local)=73.0\u00b11.0 (5\u03c3)"
|
|
},
|
|
{
|
|
"eq_number": 174,
|
|
"title": "S\u2088 Tension",
|
|
"domain_id": 8,
|
|
"significance": "\u03c3\u2088(\u03a9_m/0.3)^{0.5}=0.832\u00b10.013 (CMB) vs ~0.76 (WL)"
|
|
},
|
|
{
|
|
"eq_number": 175,
|
|
"title": "Age of the Universe",
|
|
"domain_id": 8,
|
|
"significance": "t\u2080=13.797\u00b10.023 Gyr (Planck 2018)"
|
|
},
|
|
{
|
|
"eq_number": 176,
|
|
"title": "Navier-Stokes Equation (Incompressible)",
|
|
"domain_id": 9,
|
|
"significance": "\u2202v/\u2202t+(v\u00b7\u2207)v=\u2212(1/\u03c1)\u2207p+\u03bd\u2207\u00b2v+g; \u2207\u00b7v=0"
|
|
},
|
|
{
|
|
"eq_number": 177,
|
|
"title": "Continuity Equation (Fluid)",
|
|
"domain_id": 9,
|
|
"significance": "\u2202\u03c1/\u2202t+\u2207\u00b7(\u03c1v)=0"
|
|
},
|
|
{
|
|
"eq_number": 178,
|
|
"title": "Euler Equation (Inviscid)",
|
|
"domain_id": 9,
|
|
"significance": "\u2202v/\u2202t+(v\u00b7\u2207)v=\u2212(1/\u03c1)\u2207p+g (\u03bc=0 limit)"
|
|
},
|
|
{
|
|
"eq_number": 179,
|
|
"title": "Bernoulli's Equation",
|
|
"domain_id": 9,
|
|
"significance": "p+\u00bd\u03c1v\u00b2+\u03c1gz=constant (steady, incompressible, inviscid)"
|
|
},
|
|
{
|
|
"eq_number": 180,
|
|
"title": "Stokes Law (Drag on Sphere)",
|
|
"domain_id": 9,
|
|
"significance": "F_d=6\u03c0\u03bcRv (Re\u226a1)"
|
|
},
|
|
{
|
|
"eq_number": 181,
|
|
"title": "Poiseuille Flow (Hagen-Poiseuille)",
|
|
"domain_id": 9,
|
|
"significance": "Q=\u03c0GR\u2074/(8\u03bc); v_z(r)=(G/4\u03bc)(R\u00b2\u2212r\u00b2)"
|
|
},
|
|
{
|
|
"eq_number": 182,
|
|
"title": "Reynolds Number",
|
|
"domain_id": 9,
|
|
"significance": "Re=\u03c1UL/\u03bc; transition at Re~2300 (pipe)"
|
|
},
|
|
{
|
|
"eq_number": 183,
|
|
"title": "Kolmogorov Energy Spectrum",
|
|
"domain_id": 9,
|
|
"significance": "E(k)=C_K \u03b5^{2/3}k^{\u22125/3}; C_K\u22481.5"
|
|
},
|
|
{
|
|
"eq_number": 184,
|
|
"title": "Froude Number",
|
|
"domain_id": 9,
|
|
"significance": "Fr=v/\u221a(gL); wave/gravity scaling"
|
|
},
|
|
{
|
|
"eq_number": 185,
|
|
"title": "Mach Number",
|
|
"domain_id": 9,
|
|
"significance": "Ma=v/c_s; compressibility measure"
|
|
},
|
|
{
|
|
"eq_number": 186,
|
|
"title": "Kutta-Joukowski Theorem (Lift)",
|
|
"domain_id": 9,
|
|
"significance": "L'=\u03c1v\u0393 (lift per unit span)"
|
|
},
|
|
{
|
|
"eq_number": 187,
|
|
"title": "Torricelli's Law (Efflux Speed)",
|
|
"domain_id": 9,
|
|
"significance": "v=\u221a(2gh); speed of fluid from orifice"
|
|
},
|
|
{
|
|
"eq_number": 188,
|
|
"title": "Archimedes' Principle",
|
|
"domain_id": 9,
|
|
"significance": "F_b=\u03c1_fluid V_displaced g"
|
|
},
|
|
{
|
|
"eq_number": 189,
|
|
"title": "Surface Tension (Young-Laplace)",
|
|
"domain_id": 9,
|
|
"significance": "\u0394p=2\u03b3/R (spherical); \u0394p=\u03b3(1/R\u2081+1/R\u2082)"
|
|
},
|
|
{
|
|
"eq_number": 190,
|
|
"title": "Kelvin-Helmholtz Instability Condition",
|
|
"domain_id": 9,
|
|
"significance": "Instability when (\u03c1\u2081\u03c1\u2082/\u03c1\u2081+\u03c1\u2082)(v\u2081\u2212v\u2082)\u00b2>2\u221a(\u03c1\u2081\u03c1\u2082)g\u03b3"
|
|
},
|
|
{
|
|
"eq_number": 191,
|
|
"title": "Snell's Law of Refraction",
|
|
"domain_id": 10,
|
|
"significance": "n\u2081 sin\u03b8\u2081=n\u2082 sin\u03b8\u2082"
|
|
},
|
|
{
|
|
"eq_number": 192,
|
|
"title": "Thin Lens Equation",
|
|
"domain_id": 10,
|
|
"significance": "1/f=1/d_o+1/d_i"
|
|
},
|
|
{
|
|
"eq_number": 193,
|
|
"title": "Lens Maker's Formula",
|
|
"domain_id": 10,
|
|
"significance": "1/f=(n\u22121)(1/R\u2081\u22121/R\u2082)"
|
|
},
|
|
{
|
|
"eq_number": 194,
|
|
"title": "Magnification (Geometric Optics)",
|
|
"domain_id": 10,
|
|
"significance": "M=\u2212d_i/d_o=h_i/h_o"
|
|
},
|
|
{
|
|
"eq_number": 195,
|
|
"title": "Scalar Wave Equation (d'Alembert)",
|
|
"domain_id": 10,
|
|
"significance": "\u2202\u00b2u/\u2202t\u00b2=c\u00b2\u2207\u00b2u"
|
|
},
|
|
{
|
|
"eq_number": 196,
|
|
"title": "Young's Double-Slit Interference",
|
|
"domain_id": 10,
|
|
"significance": "\u0394y=\u03bbL/d (fringe spacing); d sin\u03b8=m\u03bb (maxima)"
|
|
},
|
|
{
|
|
"eq_number": 197,
|
|
"title": "Single-Slit Diffraction",
|
|
"domain_id": 10,
|
|
"significance": "I(\u03b8)=I\u2080[sin(\u03b2/2)/(\u03b2/2)]\u00b2; \u03b2=2\u03c0a sin\u03b8/\u03bb"
|
|
},
|
|
{
|
|
"eq_number": 198,
|
|
"title": "Grating Equation",
|
|
"domain_id": 10,
|
|
"significance": "d(sin\u03b8_i+sin\u03b8_m)=m\u03bb"
|
|
},
|
|
{
|
|
"eq_number": 199,
|
|
"title": "Bragg's Law (X-ray Diffraction)",
|
|
"domain_id": 10,
|
|
"significance": "n\u03bb=2d sin\u03b8"
|
|
},
|
|
{
|
|
"eq_number": 200,
|
|
"title": "Fresnel Equations (Amplitude Reflection/Transmission)",
|
|
"domain_id": 10,
|
|
"significance": "r_s=(n\u2081cos\u03b8_i\u2212n\u2082cos\u03b8_t)/(n\u2081cos\u03b8_i+n\u2082cos\u03b8_t); etc."
|
|
},
|
|
{
|
|
"eq_number": 201,
|
|
"title": "Brewster's Angle",
|
|
"domain_id": 10,
|
|
"significance": "\u03b8_B=arctan(n\u2082/n\u2081); reflected p-pol vanishes"
|
|
},
|
|
{
|
|
"eq_number": 202,
|
|
"title": "Malus's Law",
|
|
"domain_id": 10,
|
|
"significance": "I=I\u2080 cos\u00b2\u03b8"
|
|
},
|
|
{
|
|
"eq_number": 203,
|
|
"title": "Rayleigh Criterion (Resolution Limit)",
|
|
"domain_id": 10,
|
|
"significance": "\u03b8_min=1.22\u03bb/D"
|
|
},
|
|
{
|
|
"eq_number": 204,
|
|
"title": "Abbe Sine Condition",
|
|
"domain_id": 10,
|
|
"significance": "n y sin\u03b8=n' y' sin\u03b8'"
|
|
},
|
|
{
|
|
"eq_number": 205,
|
|
"title": "Numerical Aperture",
|
|
"domain_id": 10,
|
|
"significance": "NA=n sin\u03b8; resolution d=\u03bb/(2 NA)"
|
|
},
|
|
{
|
|
"eq_number": 206,
|
|
"title": "Fermat's Principle of Least Time",
|
|
"domain_id": 10,
|
|
"significance": "\u03b4\u222b n ds=0; light path minimizes optical path length"
|
|
},
|
|
{
|
|
"eq_number": 207,
|
|
"title": "Huygens-Fresnel Principle",
|
|
"domain_id": 10,
|
|
"significance": "Every point on wavefront = source of spherical wavelets"
|
|
},
|
|
{
|
|
"eq_number": 208,
|
|
"title": "Fabry-P\u00e9rot Etalon Transmission",
|
|
"domain_id": 10,
|
|
"significance": "T=T_max/[1+(2F/\u03c0)\u00b2 sin\u00b2(\u03b4/2)]"
|
|
},
|
|
{
|
|
"eq_number": 209,
|
|
"title": "Critical Angle (Total Internal Reflection)",
|
|
"domain_id": 10,
|
|
"significance": "\u03b8_c=arcsin(n\u2082/n\u2081)"
|
|
},
|
|
{
|
|
"eq_number": 210,
|
|
"title": "Fraunhofer vs Fresnel Diffraction Condition",
|
|
"domain_id": 10,
|
|
"significance": "Fresnel number F=a\u00b2/\u03bbz; F\u226a1\u2192Fraunhofer; F\u226b1\u2192Fresnel"
|
|
},
|
|
{
|
|
"eq_number": 211,
|
|
"title": "Speed of Sound",
|
|
"domain_id": 11,
|
|
"significance": "c=\u221a(K/\u03c1); c_air=\u221a(\u03b3RT/M)\u2248331.3+0.606\u00b7T_\u00b0C m/s"
|
|
},
|
|
{
|
|
"eq_number": 212,
|
|
"title": "Doppler Effect (Sound)",
|
|
"domain_id": 11,
|
|
"significance": "f'=f(c\u00b1v_o)/(c\u2213v_s)"
|
|
},
|
|
{
|
|
"eq_number": 213,
|
|
"title": "Standing Waves (String/Column)",
|
|
"domain_id": 11,
|
|
"significance": "\u03bb_n=2L/n (fixed-fixed/open-open); \u03bb_n=4L/n (fixed-free)"
|
|
},
|
|
{
|
|
"eq_number": 214,
|
|
"title": "Decibel Scale (SPL)",
|
|
"domain_id": 11,
|
|
"significance": "L_p=10 log\u2081\u2080(p\u00b2/p\u2080\u00b2) dB; p\u2080=20 \u03bcPa"
|
|
},
|
|
{
|
|
"eq_number": 215,
|
|
"title": "Shock Wave Rankine-Hugoniot Relations",
|
|
"domain_id": 11,
|
|
"significance": "Conservation eqs across shock: \u03c1\u2081v\u2081=\u03c1\u2082v\u2082; p\u2081+\u03c1\u2081v\u2081\u00b2=p\u2082+\u03c1\u2082v\u2082\u00b2; etc."
|
|
},
|
|
{
|
|
"eq_number": 216,
|
|
"title": "Beat Frequency",
|
|
"domain_id": 11,
|
|
"significance": "f_beat=|f\u2081\u2212f\u2082|"
|
|
},
|
|
{
|
|
"eq_number": 217,
|
|
"title": "Helmholtz Resonator Frequency",
|
|
"domain_id": 11,
|
|
"significance": "f=(c/2\u03c0)\u221a(A/VL_eff)"
|
|
},
|
|
{
|
|
"eq_number": 218,
|
|
"title": "Drude Model (Electrical Conductivity)",
|
|
"domain_id": 12,
|
|
"significance": "\u03c3=n e\u00b2\u03c4/m; J=\u03c3E"
|
|
},
|
|
{
|
|
"eq_number": 219,
|
|
"title": "Bloch's Theorem",
|
|
"domain_id": 12,
|
|
"significance": "\u03c8_k(r)=e^{ik\u00b7r} u_k(r); u_k periodic"
|
|
},
|
|
{
|
|
"eq_number": 220,
|
|
"title": "Kronig-Penney Model (1D Band Structure)",
|
|
"domain_id": 12,
|
|
"significance": "cos ka=cos \u03b1a+(P/\u03b1a)sin \u03b1a"
|
|
},
|
|
{
|
|
"eq_number": 221,
|
|
"title": "Fermi-Dirac Distribution",
|
|
"domain_id": 12,
|
|
"significance": "f(E)=1/[e^{(E\u2212\u03bc)/k_B T}+1]"
|
|
},
|
|
{
|
|
"eq_number": 222,
|
|
"title": "Free Electron Density of States",
|
|
"domain_id": 12,
|
|
"significance": "g(E)=(1/2\u03c0\u00b2)(2m/\u210f\u00b2)^{3/2}\u221aE"
|
|
},
|
|
{
|
|
"eq_number": 223,
|
|
"title": "Sommerfeld Model (Electron Heat Capacity)",
|
|
"domain_id": 12,
|
|
"significance": "C_V=(\u03c0\u00b2/2)n k_B(k_B T/E_F)"
|
|
},
|
|
{
|
|
"eq_number": 224,
|
|
"title": "BCS Theory (Superconductivity)",
|
|
"domain_id": 12,
|
|
"significance": "T_c=1.13\u0398_D e^{\u22121/N(0)V}; \u0394(T); Cooper pairs"
|
|
},
|
|
{
|
|
"eq_number": 225,
|
|
"title": "BCS Gap Equation at T=0",
|
|
"domain_id": 12,
|
|
"significance": "\u0394(0)=1.76 k_B T_c"
|
|
},
|
|
{
|
|
"eq_number": 226,
|
|
"title": "London Equations (Perfect Diamagnetism)",
|
|
"domain_id": 12,
|
|
"significance": "\u2202J_s/\u2202t=(n_s e\u00b2/m)E; \u2207\u00d7J_s=\u2212(n_s e\u00b2/m)B"
|
|
},
|
|
{
|
|
"eq_number": 227,
|
|
"title": "Josephson Effects (DC + AC)",
|
|
"domain_id": 12,
|
|
"significance": "I=I_c sin \u03c6 (DC); d\u03c6/dt=(2e/\u210f)V=(2\u03c0/\u03a6\u2080)V (AC)"
|
|
},
|
|
{
|
|
"eq_number": 228,
|
|
"title": "Curie's Law (Paramagnetism)",
|
|
"domain_id": 12,
|
|
"significance": "\u03c7=C/T; C=N\u03bc\u00b2/(3k_B)"
|
|
},
|
|
{
|
|
"eq_number": 229,
|
|
"title": "Curie-Weiss Law (Ferromagnetism)",
|
|
"domain_id": 12,
|
|
"significance": "\u03c7=C/(T\u2212T_c) above Curie point"
|
|
},
|
|
{
|
|
"eq_number": 230,
|
|
"title": "Heisenberg Exchange Interaction",
|
|
"domain_id": 12,
|
|
"significance": "H=\u2212J \u03a3_{\u27e8ij\u27e9} S_i\u00b7S_j"
|
|
},
|
|
{
|
|
"eq_number": 231,
|
|
"title": "Magnetic Hysteresis Loop",
|
|
"domain_id": 12,
|
|
"significance": "B\u2212H loop; remanence B_r, coercivity H_c"
|
|
},
|
|
{
|
|
"eq_number": 232,
|
|
"title": "Einstein Relation (Diffusion)",
|
|
"domain_id": 12,
|
|
"significance": "D=\u03bc k_B T/e"
|
|
},
|
|
{
|
|
"eq_number": 233,
|
|
"title": "Seebeck Effect (Thermoelectricity)",
|
|
"domain_id": 12,
|
|
"significance": "\u0394V=S \u0394T; S=\u2212(\u03c0\u00b2k_B\u00b2T/3e)(d ln \u03c3/dE)_{E=\u03bc}"
|
|
},
|
|
{
|
|
"eq_number": 234,
|
|
"title": "Hall Effect",
|
|
"domain_id": 12,
|
|
"significance": "V_H=(I B)/(n e d); R_H=1/(n e)"
|
|
},
|
|
{
|
|
"eq_number": 235,
|
|
"title": "Quantum Hall Effect (Integer)",
|
|
"domain_id": 12,
|
|
"significance": "R_H=h/(\u03bd e\u00b2); \u03bd integer; exact quantization"
|
|
},
|
|
{
|
|
"eq_number": 236,
|
|
"title": "Wiedemann-Franz Law",
|
|
"domain_id": 12,
|
|
"significance": "\u03ba/(\u03c3T)=L; L=(\u03c0\u00b2/3)(k_B/e)\u00b2\u22482.44\u00d710\u207b\u2078 W\u03a9/K\u00b2"
|
|
},
|
|
{
|
|
"eq_number": 237,
|
|
"title": "Debye Model (Lattice Heat Capacity)",
|
|
"domain_id": 12,
|
|
"significance": "C_V\u2248(12\u03c0\u2074/5) N k_B (T/\u0398_D)\u00b3 for T\u226a\u0398_D"
|
|
},
|
|
{
|
|
"eq_number": 238,
|
|
"title": "Mott Insulator Transition",
|
|
"domain_id": 12,
|
|
"significance": "U/t\u226bW\u2192Mott insulating gap; metal-insulator transition"
|
|
},
|
|
{
|
|
"eq_number": 239,
|
|
"title": "Density Functional Theory (Kohn-Sham Equations)",
|
|
"domain_id": 12,
|
|
"significance": "(\u2212\u00bd\u2207\u00b2+v_eff(r))\u03c6_i(r)=\u03b5_i \u03c6_i(r)"
|
|
},
|
|
{
|
|
"eq_number": 240,
|
|
"title": "Landau Fermi Liquid Theory",
|
|
"domain_id": 12,
|
|
"significance": "Quasiparticles with renormalized mass m*/m; same quantum numbers"
|
|
},
|
|
{
|
|
"eq_number": 241,
|
|
"title": "Radioactive Decay Law",
|
|
"domain_id": 13,
|
|
"significance": "N(t)=N\u2080 e^{\u2212\u03bbt}; T_{1/2}=ln 2/\u03bb; \u03c4=1/\u03bb"
|
|
},
|
|
{
|
|
"eq_number": 242,
|
|
"title": "Bethe-Weizs\u00e4cker (Semi-Empirical) Mass Formula",
|
|
"domain_id": 13,
|
|
"significance": "B=a_vA\u2212a_sA^{2/3}\u2212a_cZ\u00b2/A^{1/3}\u2212a_a(N\u2212Z)\u00b2/A+\u03b4(A,Z)"
|
|
},
|
|
{
|
|
"eq_number": 243,
|
|
"title": "Geiger-Nuttall Law (\u03b1-Decay)",
|
|
"domain_id": 13,
|
|
"significance": "log T_{1/2}=A+B/\u221aE_\u03b1"
|
|
},
|
|
{
|
|
"eq_number": 244,
|
|
"title": "Nuclear Shell Model (Magic Numbers)",
|
|
"domain_id": 13,
|
|
"significance": "Magic no: 2,8,20,28,50,82,126; spin-orbit coupling"
|
|
},
|
|
{
|
|
"eq_number": 245,
|
|
"title": "Q-Value of Nuclear Reaction",
|
|
"domain_id": 13,
|
|
"significance": "Q=(m_initial\u2212m_final)c\u00b2"
|
|
},
|
|
{
|
|
"eq_number": 246,
|
|
"title": "Neutrino Oscillation Probability",
|
|
"domain_id": 13,
|
|
"significance": "P(\u03bd_\u03b1\u2192\u03bd_\u03b2)=sin\u00b2(2\u03b8) sin\u00b2(\u0394m\u00b2 L/4E)"
|
|
},
|
|
{
|
|
"eq_number": 247,
|
|
"title": "Four-Factor Formula (Nuclear Reactor)",
|
|
"domain_id": 13,
|
|
"significance": "k_eff=\u03b7 \u03b5 p f; criticality when k_eff=1"
|
|
},
|
|
{
|
|
"eq_number": 248,
|
|
"title": "Rutherford Scattering Cross-Section",
|
|
"domain_id": 13,
|
|
"significance": "d\u03c3/d\u03a9=(Z\u2081Z\u2082e\u00b2/16\u03c0\u03b5\u2080E)\u00b2 csc\u2074(\u03b8/2)"
|
|
},
|
|
{
|
|
"eq_number": 249,
|
|
"title": "M\u00f6ssbauer Effect (Recoilless \u03b3 Emission)",
|
|
"domain_id": 13,
|
|
"significance": "Fraction f=exp(\u2212k\u00b2\u27e8x\u00b2\u27e9)"
|
|
},
|
|
{
|
|
"eq_number": 250,
|
|
"title": "Breit-Wigner Resonance (Nuclear Reactions)",
|
|
"domain_id": 13,
|
|
"significance": "\u03c3(E)=\u03c0\u019b\u00b2 g (\u0393_a \u0393_b)/[(E\u2212E_R)\u00b2+\u0393\u00b2/4]"
|
|
},
|
|
{
|
|
"eq_number": 251,
|
|
"title": "Lane-Emden Equation (Polytropic Stars)",
|
|
"domain_id": 14,
|
|
"significance": "(1/\u03be\u00b2)d(\u03be\u00b2 d\u03b8/d\u03be)/d\u03be=\u2212\u03b8^n"
|
|
},
|
|
{
|
|
"eq_number": 252,
|
|
"title": "Eddington Luminosity Limit",
|
|
"domain_id": 14,
|
|
"significance": "L_Edd=4\u03c0GM m_p c/\u03c3_T\u22481.3\u00d710\u00b3\u00b9(M/M\u2299) W"
|
|
},
|
|
{
|
|
"eq_number": 253,
|
|
"title": "Chandrasekhar Limit (White Dwarf)",
|
|
"domain_id": 14,
|
|
"significance": "M_Ch\u22481.44 M\u2299 (electron degeneracy pressure)"
|
|
},
|
|
{
|
|
"eq_number": 254,
|
|
"title": "TOV Limit (Neutron Star Maximum Mass)",
|
|
"domain_id": 14,
|
|
"significance": "M_max\u22482\u22123 M\u2299 (equation of state dependent)"
|
|
},
|
|
{
|
|
"eq_number": 255,
|
|
"title": "Hertzsprung-Russell Diagram + Main Sequence",
|
|
"domain_id": 14,
|
|
"significance": "L\u221dM^{3.5} (MS, M>0.5M\u2299); stellar radii, T_eff"
|
|
},
|
|
{
|
|
"eq_number": 256,
|
|
"title": "Mass-Luminosity Relation",
|
|
"domain_id": 14,
|
|
"significance": "L/L\u2299\u2248(M/M\u2299)^{3.5} (MS, intermediate mass)"
|
|
},
|
|
{
|
|
"eq_number": 257,
|
|
"title": "Virial Theorem (Astrophysics)",
|
|
"domain_id": 14,
|
|
"significance": "2\u27e8T\u27e9+\u27e8U\u27e9=0 for gravitational systems"
|
|
},
|
|
{
|
|
"eq_number": 258,
|
|
"title": "Jeans Instability Criterion (Star Formation)",
|
|
"domain_id": 14,
|
|
"significance": "\u03bb_J=c_s\u221a(\u03c0/G\u03c1); M_J\u221dc_s\u00b3/\u221a(G\u00b3\u03c1)"
|
|
},
|
|
{
|
|
"eq_number": 259,
|
|
"title": "Schwarzschild Criterion (Convection)",
|
|
"domain_id": 14,
|
|
"significance": "|dT/dr|_rad>|dT/dr|_ad\u2192convective instability"
|
|
},
|
|
{
|
|
"eq_number": 260,
|
|
"title": "pp Chain Energy Release",
|
|
"domain_id": 14,
|
|
"significance": "4p\u2192\u2074He+2e\u207a+2\u03bd_e+26.73 MeV"
|
|
},
|
|
{
|
|
"eq_number": 261,
|
|
"title": "CNO Cycle (Massive Stars)",
|
|
"domain_id": 14,
|
|
"significance": "C, N, O catalytic H fusion; dominant above ~1.3 M\u2299"
|
|
},
|
|
{
|
|
"eq_number": 262,
|
|
"title": "Triple-Alpha Process (Helium Burning)",
|
|
"domain_id": 14,
|
|
"significance": "3 \u2074He\u2192\u00b9\u00b2C+7.65 MeV (Hoyle resonance at 7.65 MeV)"
|
|
},
|
|
{
|
|
"eq_number": 263,
|
|
"title": "Core-Collapse Supernova Mechanism",
|
|
"domain_id": 14,
|
|
"significance": "Fe core infall\u2192neutrino burst\u2192explosion (delayed neutrino mechanism)"
|
|
},
|
|
{
|
|
"eq_number": 264,
|
|
"title": "Type Ia Supernova (Standardizable Candle)",
|
|
"domain_id": 14,
|
|
"significance": "Chandrasekhar mass WD thermonuclear detonation; Phillips rel."
|
|
},
|
|
{
|
|
"eq_number": 265,
|
|
"title": "Neutron Star Equation of State (Various)",
|
|
"domain_id": 14,
|
|
"significance": "p(\u03c1) from nuclear matter theory; constraints from NS masses"
|
|
},
|
|
{
|
|
"eq_number": 266,
|
|
"title": "Oppenheimer-Snyder Collapse (BH Formation)",
|
|
"domain_id": 14,
|
|
"significance": "Dust ball collapse\u2192BH; event horizon forms"
|
|
},
|
|
{
|
|
"eq_number": 267,
|
|
"title": "Pulsar Spin-Down",
|
|
"domain_id": 14,
|
|
"significance": "\u0116=\u2212I \u03c9 \u03c9\u0307; B_dipole\u22483.2\u00d710\u00b9\u2079\u221a(P \u1e56) G"
|
|
},
|
|
{
|
|
"eq_number": 268,
|
|
"title": "Olbers' Paradox Resolution",
|
|
"domain_id": 14,
|
|
"significance": "Dark night sky\u2192finite age+expanding universe"
|
|
},
|
|
{
|
|
"eq_number": 269,
|
|
"title": "Debye Length (Plasma Screening)",
|
|
"domain_id": 15,
|
|
"significance": "\u03bb_D=\u221a(\u03b5\u2080 k_B T/(n e\u00b2))"
|
|
},
|
|
{
|
|
"eq_number": 270,
|
|
"title": "Plasma Frequency",
|
|
"domain_id": 15,
|
|
"significance": "\u03c9_p=\u221a(n e\u00b2/(\u03b5\u2080 m_e))\u224856.4\u221an (rad/s)"
|
|
},
|
|
{
|
|
"eq_number": 271,
|
|
"title": "Alfv\u00e9n Wave Speed",
|
|
"domain_id": 15,
|
|
"significance": "v_A=B\u2080/\u221a(\u03bc\u2080\u03c1)"
|
|
},
|
|
{
|
|
"eq_number": 272,
|
|
"title": "MHD Induction Equation",
|
|
"domain_id": 15,
|
|
"significance": "\u2202B/\u2202t=\u2207\u00d7(v\u00d7B)+\u03b7\u2207\u00b2B"
|
|
},
|
|
{
|
|
"eq_number": 273,
|
|
"title": "Saha Ionization Equation",
|
|
"domain_id": 15,
|
|
"significance": "n_{i+1}n_e/n_i=(2/\u03bb\u00b3_deB)(U_{i+1}/U_i)e^{\u2212\u03c7/(k_B T)}"
|
|
},
|
|
{
|
|
"eq_number": 274,
|
|
"title": "Gyro-frequency (Larmor Frequency)",
|
|
"domain_id": 15,
|
|
"significance": "\u03c9_c=qB/m; r_L=v_\u22a5/\u03c9_c"
|
|
},
|
|
{
|
|
"eq_number": 275,
|
|
"title": "Beta Parameter (Plasma Confinement)",
|
|
"domain_id": 15,
|
|
"significance": "\u03b2=2\u03bc\u2080 p/B\u00b2"
|
|
},
|
|
{
|
|
"eq_number": 276,
|
|
"title": "Lawson Criterion (Fusion Ignition)",
|
|
"domain_id": 15,
|
|
"significance": "n T \u03c4_E>3\u00d710\u00b2\u00b9 keV\u00b7s/m\u00b3 (D-T)"
|
|
},
|
|
{
|
|
"eq_number": 277,
|
|
"title": "Noether's Theorem",
|
|
"domain_id": 16,
|
|
"significance": "Continuous symmetry \u21d4 conserved current/charge"
|
|
},
|
|
{
|
|
"eq_number": 278,
|
|
"title": "Stokes' Theorem",
|
|
"domain_id": 16,
|
|
"significance": "\u222b_S (\u2207\u00d7F)\u00b7dS=\u222e_C F\u00b7dl"
|
|
},
|
|
{
|
|
"eq_number": 279,
|
|
"title": "Gauss's Divergence Theorem",
|
|
"domain_id": 16,
|
|
"significance": "\u222b_V \u2207\u00b7F dV=\u222e_S F\u00b7dS"
|
|
},
|
|
{
|
|
"eq_number": 280,
|
|
"title": "Green's Theorem (2D)",
|
|
"domain_id": 16,
|
|
"significance": "\u222c(\u2202Q/\u2202x\u2212\u2202P/\u2202y)dxdy=\u222e Pdx+Qdy"
|
|
},
|
|
{
|
|
"eq_number": 281,
|
|
"title": "Fourier Transform",
|
|
"domain_id": 16,
|
|
"significance": "F(k)=\u222b f(x)e^{\u2212ikx}dx; f(x)=(1/2\u03c0)\u222b F(k)e^{ikx}dk"
|
|
},
|
|
{
|
|
"eq_number": 282,
|
|
"title": "Laplace's Equation",
|
|
"domain_id": 16,
|
|
"significance": "\u2207\u00b2\u03c6=0; harmonic functions"
|
|
},
|
|
{
|
|
"eq_number": 283,
|
|
"title": "Poisson's Equation",
|
|
"domain_id": 16,
|
|
"significance": "\u2207\u00b2\u03c6=\u2212f(x); fundamental PDE of physics"
|
|
},
|
|
{
|
|
"eq_number": 284,
|
|
"title": "Bessel's Equation",
|
|
"domain_id": 16,
|
|
"significance": "x\u00b2 y''+x y'+(x\u00b2\u2212n\u00b2)y=0"
|
|
},
|
|
{
|
|
"eq_number": 285,
|
|
"title": "Legendre's Equation",
|
|
"domain_id": 16,
|
|
"significance": "(1\u2212x\u00b2)y''\u22122xy'+n(n+1)y=0"
|
|
},
|
|
{
|
|
"eq_number": 286,
|
|
"title": "Hermite's Equation",
|
|
"domain_id": 16,
|
|
"significance": "y''\u22122xy'+2ny=0"
|
|
},
|
|
{
|
|
"eq_number": 287,
|
|
"title": "Associated Legendre Equation",
|
|
"domain_id": 16,
|
|
"significance": "(1\u2212x\u00b2)y''\u22122xy'+[n(n+1)\u2212m\u00b2/(1\u2212x\u00b2)]y=0"
|
|
},
|
|
{
|
|
"eq_number": 288,
|
|
"title": "Chebyshev Polynomials",
|
|
"domain_id": 16,
|
|
"significance": "T_n(cos\u03b8)=cos(n\u03b8); orthogonality"
|
|
},
|
|
{
|
|
"eq_number": 289,
|
|
"title": "Laguerre Polynomials",
|
|
"domain_id": 16,
|
|
"significance": "x y''+(1\u2212x)y'+n y=0"
|
|
},
|
|
{
|
|
"eq_number": 290,
|
|
"title": "Spherical Harmonics (Y_l^m)",
|
|
"domain_id": 16,
|
|
"significance": "Y_l^m(\u03b8,\u03c6)=\u221a((2l+1)(l\u2212m)!/4\u03c0(l+m)!) P_l^m(cos\u03b8) e^{im\u03c6}"
|
|
},
|
|
{
|
|
"eq_number": 291,
|
|
"title": "Gamma Function",
|
|
"domain_id": 16,
|
|
"significance": "\u0393(z)=\u222b\u2080^\u221e t^{z\u22121}e^{\u2212t}dt; \u0393(n+1)=n!"
|
|
},
|
|
{
|
|
"eq_number": 292,
|
|
"title": "Error Function",
|
|
"domain_id": 16,
|
|
"significance": "erf(x)=(2/\u221a\u03c0)\u222b\u2080^x e^{\u2212t\u00b2}dt"
|
|
},
|
|
{
|
|
"eq_number": 293,
|
|
"title": "Delta Function (Dirac)",
|
|
"domain_id": 16,
|
|
"significance": "\u222b \u03b4(x\u2212a)f(x)dx=f(a); \u222b \u03b4(x)dx=1"
|
|
},
|
|
{
|
|
"eq_number": 294,
|
|
"title": "Eigenvalue Equation",
|
|
"domain_id": 16,
|
|
"significance": "\u00c2v=\u03bbv"
|
|
},
|
|
{
|
|
"eq_number": 295,
|
|
"title": "Separation of Variables Method",
|
|
"domain_id": 16,
|
|
"significance": "\u03c8(x,y,z)=X(x)Y(y)Z(z); decouples PDEs"
|
|
},
|
|
{
|
|
"eq_number": 296,
|
|
"title": "Boltzmann Distribution",
|
|
"domain_id": 17,
|
|
"significance": "p_i=g_i e^{\u2212\u03b2E_i}/Z; \u03b2=1/k_B T"
|
|
},
|
|
{
|
|
"eq_number": 297,
|
|
"title": "Canonical Partition Function",
|
|
"domain_id": 17,
|
|
"significance": "Z=\u03a3 g_i e^{\u2212\u03b2E_i}; F=\u2212k_B T ln Z"
|
|
},
|
|
{
|
|
"eq_number": 298,
|
|
"title": "Grand Canonical Partition Function",
|
|
"domain_id": 17,
|
|
"significance": "\u039e=\u03a3_{N} \u03a3_{E} e^{\u2212\u03b2(E\u2212\u03bcN)}; \u03a9=\u2212k_B T ln \u039e"
|
|
},
|
|
{
|
|
"eq_number": 299,
|
|
"title": "Boltzmann Entropy Formula",
|
|
"domain_id": 17,
|
|
"significance": "S=k_B ln \u03a9"
|
|
},
|
|
{
|
|
"eq_number": 300,
|
|
"title": "Gibbs Entropy Formula",
|
|
"domain_id": 17,
|
|
"significance": "S=\u2212k_B \u03a3 p_i ln p_i"
|
|
},
|
|
{
|
|
"eq_number": 301,
|
|
"title": "Fluctuation-Dissipation Theorem",
|
|
"domain_id": 17,
|
|
"significance": "\u27e8x\u00b2\u27e9_\u03c9=(2k_B T/\u03c9) Im \u03c7(\u03c9)"
|
|
},
|
|
{
|
|
"eq_number": 302,
|
|
"title": "Einstein-Smoluchowski Relation (Diffusion)",
|
|
"domain_id": 17,
|
|
"significance": "\u27e8x\u00b2\u27e9=2Dt; D=\u03bc k_B T"
|
|
},
|
|
{
|
|
"eq_number": 303,
|
|
"title": "Jarzynski Equality",
|
|
"domain_id": 17,
|
|
"significance": "\u27e8e^{\u2212W/k_B T}\u27e9=e^{\u2212\u0394F/k_B T}"
|
|
},
|
|
{
|
|
"eq_number": 304,
|
|
"title": "Crooks Fluctuation Theorem",
|
|
"domain_id": 17,
|
|
"significance": "P_F(W)/P_R(\u2212W)=e^{(W\u2212\u0394F)/k_B T}"
|
|
},
|
|
{
|
|
"eq_number": 305,
|
|
"title": "Ising Model (1D/2D Exact Solution)",
|
|
"domain_id": 17,
|
|
"significance": "2D Onsager solution: T_c=2.269 J/k_B"
|
|
},
|
|
{
|
|
"eq_number": 306,
|
|
"title": "Central Limit Theorem (Statistical)",
|
|
"domain_id": 17,
|
|
"significance": "(1/n)\u03a3 X_i \u2192 N(\u03bc,\u03c3\u00b2/n)"
|
|
},
|
|
{
|
|
"eq_number": 307,
|
|
"title": "Bose-Einstein Condensation (T_c)",
|
|
"domain_id": 17,
|
|
"significance": "T_c=(2\u03c0\u210f\u00b2/m k_B)(n/\u03b6(3/2))^{2/3}"
|
|
},
|
|
{
|
|
"eq_number": 308,
|
|
"title": "Kramers-Kronig Relations (Dispersion)",
|
|
"domain_id": 17,
|
|
"significance": "Re \u03c7(\u03c9)=(1/\u03c0) P\u222b Im \u03c7(\u03c9')/(\u03c9'\u2212\u03c9)d\u03c9'"
|
|
},
|
|
{
|
|
"eq_number": 309,
|
|
"title": "Cauchy Stress Principle",
|
|
"domain_id": 18,
|
|
"significance": "t=\u03c3\u00b7n; traction vector=stress tensor\u00b7normal"
|
|
},
|
|
{
|
|
"eq_number": 310,
|
|
"title": "Generalized Hooke's Law (Linear Elasticity)",
|
|
"domain_id": 18,
|
|
"significance": "\u03c3_{ij}=C_{ijkl} \u03b5_{kl}; 21 independent elastic constants"
|
|
},
|
|
{
|
|
"eq_number": 311,
|
|
"title": "Infinitesimal Strain Tensor",
|
|
"domain_id": 18,
|
|
"significance": "\u03b5_{ij}=(1/2)(\u2202_j u_i+\u2202_i u_j)"
|
|
},
|
|
{
|
|
"eq_number": 312,
|
|
"title": "Young's Modulus / Elastic Modulus",
|
|
"domain_id": 18,
|
|
"significance": "E=\u03c3/\u03b5 (uniaxial); stress-strain ratio"
|
|
},
|
|
{
|
|
"eq_number": 313,
|
|
"title": "Shear Modulus",
|
|
"domain_id": 18,
|
|
"significance": "G=\u03c4/\u03b3; G=E/[2(1+\u03bd)] (isotropic)"
|
|
},
|
|
{
|
|
"eq_number": 314,
|
|
"title": "Bulk Modulus",
|
|
"domain_id": 18,
|
|
"significance": "K=\u2212V dp/dV; K=E/[3(1\u22122\u03bd)] (isotropic)"
|
|
},
|
|
{
|
|
"eq_number": 315,
|
|
"title": "Poisson's Ratio",
|
|
"domain_id": 18,
|
|
"significance": "\u03bd=\u2212\u03b5_transvers/\u03b5_axial; \u22121<\u03bd<0.5"
|
|
},
|
|
{
|
|
"eq_number": 316,
|
|
"title": "Euler-Bernoulli Beam Equation",
|
|
"domain_id": 18,
|
|
"significance": "EI d\u2074w/dx\u2074=q(x); deflection"
|
|
},
|
|
{
|
|
"eq_number": 317,
|
|
"title": "Timoshenko Beam Theory",
|
|
"domain_id": 18,
|
|
"significance": "Shear deformation included; more accurate for short beams"
|
|
},
|
|
{
|
|
"eq_number": 318,
|
|
"title": "Elastic Wave Speeds (P and S waves)",
|
|
"domain_id": 18,
|
|
"significance": "v_P=\u221a((K+4G/3)/\u03c1); v_S=\u221a(G/\u03c1)"
|
|
},
|
|
{
|
|
"eq_number": 319,
|
|
"title": "Creep / Viscoelastic Maxwell Model",
|
|
"domain_id": 18,
|
|
"significance": "d\u03b5/dt=(1/E) d\u03c3/dt + \u03c3/\u03b7"
|
|
},
|
|
{
|
|
"eq_number": 320,
|
|
"title": "Plastic Yield (Von Mises Criterion)",
|
|
"domain_id": 18,
|
|
"significance": "\u03c3_v=\u221a(\u00bd[(\u03c3\u2081\u2212\u03c3\u2082)\u00b2+(\u03c3\u2082\u2212\u03c3\u2083)\u00b2+(\u03c3\u2083\u2212\u03c3\u2081)\u00b2])\u2265\u03c3_y"
|
|
},
|
|
{
|
|
"eq_number": 321,
|
|
"title": "Shannon Entropy",
|
|
"domain_id": 19,
|
|
"significance": "H=\u2212\u03a3 p_i log\u2082 p_i (bits)"
|
|
},
|
|
{
|
|
"eq_number": 322,
|
|
"title": "Shannon-Hartley Channel Capacity",
|
|
"domain_id": 19,
|
|
"significance": "C=B log\u2082(1+S/N)"
|
|
},
|
|
{
|
|
"eq_number": 323,
|
|
"title": "Nyquist-Shannon Sampling Theorem",
|
|
"domain_id": 19,
|
|
"significance": "f_s\u22652 f_max to perfectly reconstruct"
|
|
},
|
|
{
|
|
"eq_number": 324,
|
|
"title": "Landauer's Principle",
|
|
"domain_id": 19,
|
|
"significance": "Erasure of 1 bit dissipates \u2265k_B T ln 2 heat"
|
|
},
|
|
{
|
|
"eq_number": 325,
|
|
"title": "Kolmogorov Complexity (Algorithmic Info)",
|
|
"domain_id": 19,
|
|
"significance": "K_U(x)=min{|p|:U(p)=x}"
|
|
},
|
|
{
|
|
"eq_number": 326,
|
|
"title": "Maximum Entropy Principle (Jaynes)",
|
|
"domain_id": 19,
|
|
"significance": "Maximize S subject to constraints\u2192least biased distribution"
|
|
},
|
|
{
|
|
"eq_number": 327,
|
|
"title": "Speed of Light Defines Meter",
|
|
"domain_id": 20,
|
|
"significance": "c=299792458 m/s EXACT"
|
|
},
|
|
{
|
|
"eq_number": 328,
|
|
"title": "Planck Constant Defines Kilogram",
|
|
"domain_id": 20,
|
|
"significance": "h=6.62607015e-34 J\u00b7s EXACT"
|
|
},
|
|
{
|
|
"eq_number": 329,
|
|
"title": "Elementary Charge Defines Ampere",
|
|
"domain_id": 20,
|
|
"significance": "e=1.602176634e-19 C EXACT"
|
|
},
|
|
{
|
|
"eq_number": 330,
|
|
"title": "Boltzmann Constant Defines Kelvin",
|
|
"domain_id": 20,
|
|
"significance": "k_B=1.380649e-23 J/K EXACT"
|
|
},
|
|
{
|
|
"eq_number": 331,
|
|
"title": "Avogadro Number Defines Mole",
|
|
"domain_id": 20,
|
|
"significance": "N_A=6.02214076e23 EXACT"
|
|
},
|
|
{
|
|
"eq_number": 332,
|
|
"title": "Josephson Voltage Standard",
|
|
"domain_id": 12,
|
|
"significance": "V=n f/K_J; K_J=2e/h=483597.9 GHz/V EXACT"
|
|
},
|
|
{
|
|
"eq_number": 333,
|
|
"title": "Quantum Hall Resistance Standard",
|
|
"domain_id": 12,
|
|
"significance": "R_H=h/(i e\u00b2); R_K=h/e\u00b2=25812.80745... \u03a9"
|
|
},
|
|
{
|
|
"eq_number": 334,
|
|
"title": "Bragg's Law (Generalized, Powder Diffraction)",
|
|
"domain_id": 22,
|
|
"significance": "n\u03bb = 2d sin \u03b8; foundation of all crystal structure determination"
|
|
},
|
|
{
|
|
"eq_number": 335,
|
|
"title": "Laue Equations (3D Diffraction Condition)",
|
|
"domain_id": 22,
|
|
"significance": "a\u00b7\u0394k=2\u03c0h, b\u00b7\u0394k=2\u03c0k, c\u00b7\u0394k=2\u03c0l; constructive interference in 3D lattice"
|
|
},
|
|
{
|
|
"eq_number": 336,
|
|
"title": "Structure Factor Equation",
|
|
"domain_id": 22,
|
|
"significance": "F_{hkl} = \u03a3_j f_j exp[2\u03c0i(hx_j+ky_j+lz_j)]; determines diffraction intensities"
|
|
},
|
|
{
|
|
"eq_number": 337,
|
|
"title": "Atomic Scattering Factor (X-ray Form Factor)",
|
|
"domain_id": 22,
|
|
"significance": "f(q) = \u222b \u03c1(r) exp(iq\u00b7r) d\u00b3r; Fourier transform of electron density"
|
|
},
|
|
{
|
|
"eq_number": 338,
|
|
"title": "Reciprocal Lattice Vector Definition",
|
|
"domain_id": 22,
|
|
"significance": "G = h a* + k b* + l c*; a*=(b\u00d7c)/V_cell, etc."
|
|
},
|
|
{
|
|
"eq_number": 339,
|
|
"title": "Brillouin Zone Boundaries",
|
|
"domain_id": 22,
|
|
"significance": "2 k\u00b7G = |G|\u00b2; electron wave diffraction condition at BZ boundaries"
|
|
},
|
|
{
|
|
"eq_number": 340,
|
|
"title": "Ewald Sphere Construction",
|
|
"domain_id": 22,
|
|
"significance": "|k| = |k'| = 2\u03c0/\u03bb; \u0394k = G falls on sphere \u2192 diffraction"
|
|
},
|
|
{
|
|
"eq_number": 341,
|
|
"title": "Patterson Function (Interatomic Vectors)",
|
|
"domain_id": 22,
|
|
"significance": "P(u,v,w) = \u222b |F_{hkl}|\u00b2 exp[\u22122\u03c0i(hu+kv+lw)] d*h d*k d*l"
|
|
},
|
|
{
|
|
"eq_number": 342,
|
|
"title": "Debye-Waller Factor (Thermal Motion)",
|
|
"domain_id": 22,
|
|
"significance": "f_T(q) = f\u2080(q) exp(\u2212\u00bd\u27e8(u\u00b7q)\u00b2\u27e9); B = 8\u03c0\u00b2\u27e8u\u00b2\u27e9"
|
|
},
|
|
{
|
|
"eq_number": 343,
|
|
"title": "Space Group Symmetry Operations",
|
|
"domain_id": 22,
|
|
"significance": "230 space groups in 3D; {R|t} r = R r + t"
|
|
},
|
|
{
|
|
"eq_number": 344,
|
|
"title": "Interplanar Spacing (Cubic Systems)",
|
|
"domain_id": 22,
|
|
"significance": "1/d\u00b2 = (h\u00b2+k\u00b2+l\u00b2)/a\u00b2 (cubic); general: depends on lattice parameters"
|
|
},
|
|
{
|
|
"eq_number": 345,
|
|
"title": "Scherrer Equation (Crystallite Size)",
|
|
"domain_id": 22,
|
|
"significance": "D = K \u03bb / (\u03b2 cos \u03b8); K\u22480.9; \u03b2=FWHM in radians"
|
|
},
|
|
{
|
|
"eq_number": 346,
|
|
"title": "Williamson-Hall Analysis (Size + Strain)",
|
|
"domain_id": 22,
|
|
"significance": "\u03b2 cos \u03b8 = K\u03bb/D + 4\u03b5 sin \u03b8; separates size and microstrain broadening"
|
|
},
|
|
{
|
|
"eq_number": 347,
|
|
"title": "True Stress \u2014 True Strain Definition",
|
|
"domain_id": 21,
|
|
"significance": "\u03c3_true = F/A_inst; \u03b5_true = ln(L/L\u2080) = ln(1+\u03b5_eng)"
|
|
},
|
|
{
|
|
"eq_number": 348,
|
|
"title": "Hollomon Equation (Work Hardening)",
|
|
"domain_id": 21,
|
|
"significance": "\u03c3 = K \u03b5^n; n = strain hardening exponent; K = strength coefficient"
|
|
},
|
|
{
|
|
"eq_number": 349,
|
|
"title": "Hall-Petch Relationship (Grain Size Strengthening)",
|
|
"domain_id": 21,
|
|
"significance": "\u03c3_y = \u03c3\u2080 + k_y / \u221ad; d = grain diameter"
|
|
},
|
|
{
|
|
"eq_number": 350,
|
|
"title": "Orowan Equation (Precipitation Strengthening)",
|
|
"domain_id": 21,
|
|
"significance": "\u0394\u03c4 = G b / L; L = interparticle spacing; b = Burgers vector"
|
|
},
|
|
{
|
|
"eq_number": 351,
|
|
"title": "Schmid's Law (Critical Resolved Shear Stress)",
|
|
"domain_id": 21,
|
|
"significance": "\u03c4_CRSS = \u03c3_y cos \u03c6 cos \u03bb; m = cos \u03c6 cos \u03bb (Schmid factor)"
|
|
},
|
|
{
|
|
"eq_number": 352,
|
|
"title": "Taylor Equation (Dislocation Strengthening)",
|
|
"domain_id": 21,
|
|
"significance": "\u03c4 = \u03b1 G b \u221a\u03c1; \u03c1 = dislocation density; \u03b1\u22480.2\u20130.5"
|
|
},
|
|
{
|
|
"eq_number": 353,
|
|
"title": "Petch-Forwood Hardness-Yield Strength Relation",
|
|
"domain_id": 21,
|
|
"significance": "H \u2248 3 \u03c3_y (metals); Vickers/Brinell \u2248 3 \u00d7 yield"
|
|
},
|
|
{
|
|
"eq_number": 354,
|
|
"title": "Griffith Criterion (Brittle Fracture)",
|
|
"domain_id": 21,
|
|
"significance": "\u03c3_f = \u221a(2E\u03b3_s / \u03c0a); critical stress for crack propagation"
|
|
},
|
|
{
|
|
"eq_number": 355,
|
|
"title": "Stress Intensity Factor (LEFM, Mode I)",
|
|
"domain_id": 21,
|
|
"significance": "K_I = Y \u03c3 \u221a(\u03c0a); fracture when K_I \u2265 K_Ic"
|
|
},
|
|
{
|
|
"eq_number": 356,
|
|
"title": "J-Integral (Elastic-Plastic Fracture)",
|
|
"domain_id": 21,
|
|
"significance": "J = \u222b_\u0393 (W dy \u2212 T_i \u2202u_i/\u2202x ds); path-independent energy release rate"
|
|
},
|
|
{
|
|
"eq_number": 357,
|
|
"title": "Paris' Law (Fatigue Crack Growth)",
|
|
"domain_id": 21,
|
|
"significance": "da/dN = C (\u0394K)^m; C, m material constants; m\u22482\u20134 for metals"
|
|
},
|
|
{
|
|
"eq_number": 358,
|
|
"title": "Basquin Equation (High-Cycle Fatigue)",
|
|
"domain_id": 21,
|
|
"significance": "\u03c3_a = \u03c3_f' (2N_f)^b; b\u2248\u22120.05 to \u22120.12 for metals"
|
|
},
|
|
{
|
|
"eq_number": 359,
|
|
"title": "Coffin-Manson Relation (Low-Cycle Fatigue)",
|
|
"domain_id": 21,
|
|
"significance": "\u0394\u03b5_p/2 = \u03b5_f' (2N_f)^c; c\u2248\u22120.5 to \u22120.7"
|
|
},
|
|
{
|
|
"eq_number": 360,
|
|
"title": "Norton-Bailey Creep Law",
|
|
"domain_id": 21,
|
|
"significance": "\u03b5_cr = A \u03c3^n t^m (primary creep); d\u03b5_cr/dt = B \u03c3^n (secondary)"
|
|
},
|
|
{
|
|
"eq_number": 361,
|
|
"title": "Larson-Miller Parameter (Creep Rupture)",
|
|
"domain_id": 21,
|
|
"significance": "P = T (C + log t_r); C\u224820; T in K, t_r in hours"
|
|
},
|
|
{
|
|
"eq_number": 362,
|
|
"title": "Mohr-Coulomb Failure Criterion",
|
|
"domain_id": 21,
|
|
"significance": "\u03c4 = c + \u03c3_n tan \u03c6; c=cohesion, \u03c6=internal friction angle"
|
|
},
|
|
{
|
|
"eq_number": 363,
|
|
"title": "Drucker-Prager Yield Criterion",
|
|
"domain_id": 21,
|
|
"significance": "\u221aJ\u2082 + \u03b1 I\u2081 = k; pressure-dependent yielding"
|
|
},
|
|
{
|
|
"eq_number": 364,
|
|
"title": "Weibull Distribution (Brittle Failure Statistics)",
|
|
"domain_id": 21,
|
|
"significance": "P_f = 1 \u2212 exp[\u2212(\u03c3/\u03c3\u2080)^m]; m = Weibull modulus"
|
|
},
|
|
{
|
|
"eq_number": 365,
|
|
"title": "Stoney Equation (Thin Film Stress)",
|
|
"domain_id": 21,
|
|
"significance": "\u03c3_f = E_s h_s\u00b2 \u03ba / [6(1\u2212\u03bd_s) h_f]; substrate curvature \u2192 film stress"
|
|
},
|
|
{
|
|
"eq_number": 366,
|
|
"title": "Debye Specific Heat Model (Full)",
|
|
"domain_id": 21,
|
|
"significance": "C_V = 9 N k_B (T/\u0398_D)\u00b3 \u222b\u2080^{\u0398_D/T} x\u2074 e^x / (e^x\u22121)\u00b2 dx"
|
|
},
|
|
{
|
|
"eq_number": 367,
|
|
"title": "Dulong-Petit Law",
|
|
"domain_id": 21,
|
|
"significance": "C_V = 3R \u2248 24.94 J/(mol\u00b7K) at high T (classical limit of Debye)"
|
|
},
|
|
{
|
|
"eq_number": 368,
|
|
"title": "Einstein Heat Capacity Model",
|
|
"domain_id": 21,
|
|
"significance": "C_V = 3 N k_B (\u0398_E/T)\u00b2 e^{\u0398_E/T} / (e^{\u0398_E/T}\u22121)\u00b2"
|
|
},
|
|
{
|
|
"eq_number": 369,
|
|
"title": "Wiedemann-Franz Law (Electronic Thermal Conductivity)",
|
|
"domain_id": 21,
|
|
"significance": "\u03ba_e / (\u03c3 T) = L; L = (\u03c0\u00b2/3)(k_B/e)\u00b2 \u2248 2.44\u00d710\u207b\u2078 W \u03a9/K\u00b2"
|
|
},
|
|
{
|
|
"eq_number": 370,
|
|
"title": "Debye-Callaway Model (Lattice Thermal Conductivity)",
|
|
"domain_id": 21,
|
|
"significance": "\u03ba_l = (k_B/2\u03c0\u00b2v)(k_B T/\u210f)\u00b3 \u222b\u2080^{\u0398_D/T} \u03c4_c x\u2074 e^x / (e^x\u22121)\u00b2 dx"
|
|
},
|
|
{
|
|
"eq_number": 371,
|
|
"title": "Thermal Expansion Coefficient (Gr\u00fcneisen Relation)",
|
|
"domain_id": 21,
|
|
"significance": "\u03b1 = \u03b3 C_V / (3 B V); \u03b3 = Gr\u00fcneisen parameter; B = bulk modulus"
|
|
},
|
|
{
|
|
"eq_number": 372,
|
|
"title": "Gr\u00fcneisen Equation of State (Solids)",
|
|
"domain_id": 21,
|
|
"significance": "P(V) = \u2212dU\u2080/dV + \u03b3 U_th/V; \u03b3 = Gr\u00fcneisen parameter"
|
|
},
|
|
{
|
|
"eq_number": 373,
|
|
"title": "Lindemann Melting Criterion",
|
|
"domain_id": 21,
|
|
"significance": "T_m \u2248 C \u03b8_D\u00b2 M V^{2/3}; C depends on crystal structure"
|
|
},
|
|
{
|
|
"eq_number": 374,
|
|
"title": "Stefan-Boltzmann Radiative Heat Transfer (Between Surfaces)",
|
|
"domain_id": 21,
|
|
"significance": "q = \u03b5_eff \u03c3 (T\u2081\u2074\u2212T\u2082\u2074); view factor + emissivity correction"
|
|
},
|
|
{
|
|
"eq_number": 375,
|
|
"title": "Complex Dielectric Constant",
|
|
"domain_id": 21,
|
|
"significance": "\u03b5* = \u03b5' \u2212 i \u03b5''; tan \u03b4 = \u03b5''/\u03b5'; loss tangent"
|
|
},
|
|
{
|
|
"eq_number": 376,
|
|
"title": "Clausius-Mossotti Relation (Polarizability)",
|
|
"domain_id": 21,
|
|
"significance": "(\u03b5_r\u22121)/(\u03b5_r+2) = N \u03b1 / (3 \u03b5\u2080); links macro/micro dielectric properties"
|
|
},
|
|
{
|
|
"eq_number": 377,
|
|
"title": "Debye Relaxation (Dipole Response)",
|
|
"domain_id": 21,
|
|
"significance": "\u03b5*(\u03c9) = \u03b5_\u221e + (\u03b5_s\u2212\u03b5_\u221e) / (1 + i \u03c9 \u03c4)"
|
|
},
|
|
{
|
|
"eq_number": 378,
|
|
"title": "Cole-Cole Relaxation (Distributed)",
|
|
"domain_id": 21,
|
|
"significance": "\u03b5*(\u03c9) = \u03b5_\u221e + (\u03b5_s\u2212\u03b5_\u221e) / [1 + (i \u03c9 \u03c4)^{1\u2212\u03b1}]"
|
|
},
|
|
{
|
|
"eq_number": 379,
|
|
"title": "Havriliak-Negami Relaxation",
|
|
"domain_id": 21,
|
|
"significance": "\u03b5*(\u03c9) = \u03b5_\u221e + (\u03b5_s\u2212\u03b5_\u221e) / [1 + (i \u03c9 \u03c4)^\u03b1]^\u03b2"
|
|
},
|
|
{
|
|
"eq_number": 380,
|
|
"title": "Curie-Weiss Law for Ferroelectrics (Above T_c)",
|
|
"domain_id": 21,
|
|
"significance": "\u03b5_r = C / (T \u2212 T_c); C = Curie constant"
|
|
},
|
|
{
|
|
"eq_number": 381,
|
|
"title": "Piezoelectric Constitutive Equations",
|
|
"domain_id": 21,
|
|
"significance": "S = s^E T + d^t E; D = d T + \u03b5^T E (strain-charge form)"
|
|
},
|
|
{
|
|
"eq_number": 382,
|
|
"title": "Pyroelectric Coefficient",
|
|
"domain_id": 21,
|
|
"significance": "p = dP_s/dT; \u0394Q = p A \u0394T"
|
|
},
|
|
{
|
|
"eq_number": 383,
|
|
"title": "Fowler-Nordheim Tunneling (Field Emission)",
|
|
"domain_id": 21,
|
|
"significance": "J = (A/\u03c6)(\u03b2E)\u00b2 exp(\u2212B \u03c6^{3/2} / \u03b2E); A,B constants"
|
|
},
|
|
{
|
|
"eq_number": 384,
|
|
"title": "Poole-Frenkel Conduction (Insulators)",
|
|
"domain_id": 21,
|
|
"significance": "\u03c3 = \u03c3\u2080 exp[\u2212q(\u03c6_B\u2212\u221a(qE/\u03c0\u03b5))/k_B T]"
|
|
},
|
|
{
|
|
"eq_number": 385,
|
|
"title": "Varistor I-V Characteristic (Nonlinear)",
|
|
"domain_id": 21,
|
|
"significance": "I = k V^\u03b1; \u03b1 >> 1 (ZnO varistors \u03b1\u224820\u2013100)"
|
|
},
|
|
{
|
|
"eq_number": 386,
|
|
"title": "Percolation Threshold (Conductivity)",
|
|
"domain_id": 21,
|
|
"significance": "\u03c3 = \u03c3\u2080 (p \u2212 p_c)^t; p = volume fraction; p_c = percolation threshold"
|
|
},
|
|
{
|
|
"eq_number": 387,
|
|
"title": "Intrinsic Carrier Concentration (Semiconductors)",
|
|
"domain_id": 23,
|
|
"significance": "n_i = \u221a(N_c N_v) exp(\u2212E_g / 2 k_B T); N_c = 2(2\u03c0 m_e* k_B T/h\u00b2)^{3/2}"
|
|
},
|
|
{
|
|
"eq_number": 388,
|
|
"title": "Fermi Level in Doped Semiconductors",
|
|
"domain_id": 23,
|
|
"significance": "n-type: E_F = E_c \u2212 k_B T ln(N_c/N_d); p-type: E_F = E_v + k_B T ln(N_v/N_a)"
|
|
},
|
|
{
|
|
"eq_number": 389,
|
|
"title": "Mass Action Law (Semiconductors)",
|
|
"domain_id": 23,
|
|
"significance": "n p = n_i\u00b2; product constant at fixed T"
|
|
},
|
|
{
|
|
"eq_number": 390,
|
|
"title": "Shockley Diode Equation (Ideal)",
|
|
"domain_id": 23,
|
|
"significance": "I = I_s [exp(q V / n k_B T) \u2212 1]; I_s = reverse saturation current"
|
|
},
|
|
{
|
|
"eq_number": 391,
|
|
"title": "Built-in Potential (p-n Junction)",
|
|
"domain_id": 23,
|
|
"significance": "V_bi = (k_B T / q) ln(N_a N_d / n_i\u00b2)"
|
|
},
|
|
{
|
|
"eq_number": 392,
|
|
"title": "Depletion Width (p-n Junction)",
|
|
"domain_id": 23,
|
|
"significance": "W = \u221a[2\u03b5_s (V_bi\u2212V)(1/N_a+1/N_d)/q]"
|
|
},
|
|
{
|
|
"eq_number": 393,
|
|
"title": "MOS Capacitor Threshold Voltage",
|
|
"domain_id": 23,
|
|
"significance": "V_th = V_FB + 2\u03c6_F + \u221a(4\u03b5_s q N_a \u03c6_F)/C_ox"
|
|
},
|
|
{
|
|
"eq_number": 394,
|
|
"title": "MOSFET Drain Current (Saturation, Long Channel)",
|
|
"domain_id": 23,
|
|
"significance": "I_D = (\u03bc_n C_ox W / 2L) (V_GS \u2212 V_th)\u00b2"
|
|
},
|
|
{
|
|
"eq_number": 395,
|
|
"title": "Subthreshold Swing (MOSFET)",
|
|
"domain_id": 23,
|
|
"significance": "SS = (k_B T/q) ln(10) (1 + C_dep/C_ox); ideal: 60 mV/decade at 300K"
|
|
},
|
|
{
|
|
"eq_number": 396,
|
|
"title": "Avalanche Breakdown (Impact Ionization)",
|
|
"domain_id": 23,
|
|
"significance": "M = 1 / [1 \u2212 (V/V_BR)^n]; n\u22483\u20136"
|
|
},
|
|
{
|
|
"eq_number": 397,
|
|
"title": "Quantum Confinement Energy (Particle in a Box)",
|
|
"domain_id": 23,
|
|
"significance": "E_n = n\u00b2 \u03c0\u00b2 \u210f\u00b2 / (2 m* L\u00b2); blue shift with decreasing size"
|
|
},
|
|
{
|
|
"eq_number": 398,
|
|
"title": "Brus Equation (Semiconductor Nanocrystal Band Gap)",
|
|
"domain_id": 23,
|
|
"significance": "E_g(R) = E_g(bulk) + \u210f\u00b2\u03c0\u00b2/(2\u03bc R\u00b2) \u2212 1.8e\u00b2/(\u03b5_r R); \u03bc = reduced exciton mass"
|
|
},
|
|
{
|
|
"eq_number": 399,
|
|
"title": "Kane's k\u00b7p Band Model (Non-Parabolicity)",
|
|
"domain_id": 23,
|
|
"significance": "E(1+\u03b1E) = \u210f\u00b2 k\u00b2 / (2 m*); \u03b1 = 1/E_g; non-parabolic correction"
|
|
},
|
|
{
|
|
"eq_number": 400,
|
|
"title": "Mott Transition (Doped Semiconductor)",
|
|
"domain_id": 23,
|
|
"significance": "n_c^{1/3} a_B* \u2248 0.25; insulator-metal transition at critical doping"
|
|
},
|
|
{
|
|
"eq_number": 401,
|
|
"title": "Anderson Localization (Disordered Materials)",
|
|
"domain_id": 23,
|
|
"significance": "W/V > W_c \u2192 localized states; mobility edge at E_c"
|
|
},
|
|
{
|
|
"eq_number": 402,
|
|
"title": "Tauc Plot (Band Gap from Absorption)",
|
|
"domain_id": 23,
|
|
"significance": "(\u03b1 h \u03bd)^{1/r} = A (h\u03bd \u2212 E_g); r=\u00bd for direct, r=2 for indirect"
|
|
},
|
|
{
|
|
"eq_number": 403,
|
|
"title": "Stoner Criterion (Itinerant Ferromagnetism)",
|
|
"domain_id": 21,
|
|
"significance": "N(E_F) I > 1; spontaneous magnetization when DOS \u00d7 exchange exceeds unity"
|
|
},
|
|
{
|
|
"eq_number": 404,
|
|
"title": "Stoner-Wohlfarth Model (Single-Domain Particle)",
|
|
"domain_id": 21,
|
|
"significance": "E = K V sin\u00b2\u03b8 \u2212 \u03bc\u2080 M_s H V cos(\u03c6\u2212\u03b8); hysteresis from anisotropy+Zeeman"
|
|
},
|
|
{
|
|
"eq_number": 405,
|
|
"title": "N\u00e9el Temperature (Antiferromagnetism)",
|
|
"domain_id": 21,
|
|
"significance": "T_N = (2J S(S+1)/3k_B) z (from mean-field); sublattice ordering temperature"
|
|
},
|
|
{
|
|
"eq_number": 406,
|
|
"title": "Curie Temperature (Mean-Field Ferromagnetism)",
|
|
"domain_id": 21,
|
|
"significance": "T_c = (2J S(S+1)/3k_B) z; z = coordination number"
|
|
},
|
|
{
|
|
"eq_number": 407,
|
|
"title": "Bloch T^{3/2} Law (Magnetization at Low T)",
|
|
"domain_id": 21,
|
|
"significance": "M_s(T) = M_s(0) [1 \u2212 (T/T_c)^{3/2}] (3D Heisenberg)"
|
|
},
|
|
{
|
|
"eq_number": 408,
|
|
"title": "Landau-Lifshitz-Gilbert Equation (Magnetization Dynamics)",
|
|
"domain_id": 21,
|
|
"significance": "dM/dt = \u2212\u03b3 M \u00d7 H_eff + (\u03b1/M_s) M \u00d7 dM/dt"
|
|
},
|
|
{
|
|
"eq_number": 409,
|
|
"title": "Brown's Paradox (Domain Wall Motion)",
|
|
"domain_id": 21,
|
|
"significance": "v = (\u03b3 \u0394 / \u03b1)(H \u2212 H_c); soft magnetic materials"
|
|
},
|
|
{
|
|
"eq_number": 410,
|
|
"title": "Magnetostriction (Joule Magnetostriction)",
|
|
"domain_id": 21,
|
|
"significance": "\u0394L/L = (3/2) \u03bb_s (cos\u00b2\u03b8 \u2212 1/3); \u03bb_s = saturation magnetostriction"
|
|
},
|
|
{
|
|
"eq_number": 411,
|
|
"title": "Giant Magnetoresistance (GMR, CIP)",
|
|
"domain_id": 21,
|
|
"significance": "\u0394R/R = (R_AP\u2212R_P)/R_P; spin-dependent scattering at interfaces"
|
|
},
|
|
{
|
|
"eq_number": 412,
|
|
"title": "Tunneling Magnetoresistance (TMR, Julliere Model)",
|
|
"domain_id": 21,
|
|
"significance": "TMR = (R_AP\u2212R_P)/R_P = 2P\u2081P\u2082/(1\u2212P\u2081P\u2082); P = spin polarization"
|
|
},
|
|
{
|
|
"eq_number": 413,
|
|
"title": "RKKY Interaction (Indirect Exchange)",
|
|
"domain_id": 21,
|
|
"significance": "J(R) \u221d cos(2k_F R) / R\u00b3; oscillatory coupling through conduction electrons"
|
|
},
|
|
{
|
|
"eq_number": 414,
|
|
"title": "Superexchange (Anderson-Goodenough-Kanamori Rules)",
|
|
"domain_id": 21,
|
|
"significance": "J_ij \u221d \u2212b\u00b2/U (for 180\u00b0 cation-anion-cation); sign depends on orbital filling"
|
|
},
|
|
{
|
|
"eq_number": 415,
|
|
"title": "Complex Refractive Index (General)",
|
|
"domain_id": 21,
|
|
"significance": "\u00f1 = n + i \u03ba; I(z) = I\u2080 exp(\u2212\u03b1 z); \u03b1 = 4\u03c0\u03ba/\u03bb"
|
|
},
|
|
{
|
|
"eq_number": 416,
|
|
"title": "Kramers-Kronig Relations (Optical Constants)",
|
|
"domain_id": 21,
|
|
"significance": "n(\u03c9)\u22121 = (2/\u03c0) P \u222b\u2080^\u221e \u03c9' \u03ba(\u03c9')/(\u03c9'\u00b2\u2212\u03c9\u00b2) d\u03c9'; causality \u2192 dispersion relations"
|
|
},
|
|
{
|
|
"eq_number": 417,
|
|
"title": "Tauc-Lorentz Model (Amorphous Semiconductor Optics)",
|
|
"domain_id": 21,
|
|
"significance": "\u03b5_2(E) = [A E\u2080 C (E\u2212E_g)\u00b2] / [(E\u00b2\u2212E\u2080\u00b2)\u00b2 + C\u00b2 E\u00b2] E for E>E_g; 0 otherwise"
|
|
},
|
|
{
|
|
"eq_number": 418,
|
|
"title": "Sellmeier Equation (Refractive Index Dispersion)",
|
|
"domain_id": 21,
|
|
"significance": "n\u00b2(\u03bb) = 1 + \u03a3_i A_i \u03bb\u00b2 / (\u03bb\u00b2 \u2212 \u03bb_i\u00b2); empirical fit for transparent regions"
|
|
},
|
|
{
|
|
"eq_number": 419,
|
|
"title": "Cauchy Equation (Refractive Index Fit)",
|
|
"domain_id": 21,
|
|
"significance": "n(\u03bb) = A + B/\u03bb\u00b2 + C/\u03bb\u2074; empirical for transparent region"
|
|
},
|
|
{
|
|
"eq_number": 420,
|
|
"title": "Urbach Tail (Absorption Edge)",
|
|
"domain_id": 21,
|
|
"significance": "\u03b1(E) = \u03b1\u2080 exp[\u03c3 (E\u2212E\u2080) / k_B T]; exponential absorption below band edge"
|
|
},
|
|
{
|
|
"eq_number": 421,
|
|
"title": "Beer-Lambert Law (Absorption)",
|
|
"domain_id": 3,
|
|
"significance": "A = log\u2081\u2080(I\u2080/I) = \u03b5 c L; absorbance proportional to concentration and path"
|
|
},
|
|
{
|
|
"eq_number": 422,
|
|
"title": "Kubelka-Munk Theory (Diffuse Reflectance)",
|
|
"domain_id": 21,
|
|
"significance": "F(R_\u221e) = (1\u2212R_\u221e)\u00b2/(2R_\u221e) = K/S \u221d \u03b1; for thick opaque scattering media"
|
|
},
|
|
{
|
|
"eq_number": 423,
|
|
"title": "Fresnel Loss at Normal Incidence",
|
|
"domain_id": 21,
|
|
"significance": "R = [(n\u2081\u2212n\u2082)/(n\u2081+n\u2082)]\u00b2; reflection coefficient at normal incidence"
|
|
},
|
|
{
|
|
"eq_number": 424,
|
|
"title": "Drude Model for Free-Carrier Absorption",
|
|
"domain_id": 21,
|
|
"significance": "\u03b5(\u03c9) = \u03b5_\u221e \u2212 \u03c9_p\u00b2/(\u03c9\u00b2 + i \u03c9/\u03c4); \u03c9_p = \u221a(n e\u00b2/\u03b5\u2080 m*)"
|
|
},
|
|
{
|
|
"eq_number": 425,
|
|
"title": "Forster Resonance Energy Transfer (FRET) Efficiency",
|
|
"domain_id": 21,
|
|
"significance": "E = 1 / [1 + (r/R\u2080)\u2076]; R\u2080 = F\u00f6rster radius (~1\u201310 nm)"
|
|
},
|
|
{
|
|
"eq_number": 426,
|
|
"title": "Stokes Shift (Luminescence)",
|
|
"domain_id": 21,
|
|
"significance": "\u0394E = E_abs \u2212 E_em > 0; from vibrational relaxation"
|
|
},
|
|
{
|
|
"eq_number": 427,
|
|
"title": "Dexter Energy Transfer (Exchange)",
|
|
"domain_id": 21,
|
|
"significance": "k_ET \u221d exp(\u22122r/L); short-range (\u22721 nm) electron exchange"
|
|
},
|
|
{
|
|
"eq_number": 428,
|
|
"title": "Vickers Hardness Definition",
|
|
"domain_id": 21,
|
|
"significance": "HV = 1.854 F / d\u00b2; F in kgf, d = average diagonal (mm)"
|
|
},
|
|
{
|
|
"eq_number": 429,
|
|
"title": "Brinell Hardness",
|
|
"domain_id": 21,
|
|
"significance": "HB = 2F / [\u03c0 D (D \u2212 \u221a(D\u00b2\u2212d\u00b2))]; D = ball diameter"
|
|
},
|
|
{
|
|
"eq_number": 430,
|
|
"title": "Rockwell Hardness (Indirect)",
|
|
"domain_id": 21,
|
|
"significance": "HR = N \u2212 h/s; h = penetration depth; N,s depend on scale"
|
|
},
|
|
{
|
|
"eq_number": 431,
|
|
"title": "Knoop Hardness (Thin Films / Brittle)",
|
|
"domain_id": 21,
|
|
"significance": "HK = 14.229 F / d\u2081\u00b2; long diagonal; shallow penetration"
|
|
},
|
|
{
|
|
"eq_number": 432,
|
|
"title": "Nanoindentation (Oliver-Pharr Method)",
|
|
"domain_id": 21,
|
|
"significance": "H = P_max/A; E_r = \u221a\u03c0 S/(2\u03b2\u221aA); S = dP/dh at unload"
|
|
},
|
|
{
|
|
"eq_number": 433,
|
|
"title": "Charpy Impact Toughness",
|
|
"domain_id": 21,
|
|
"significance": "KV = m g (h_initial \u2212 h_final); energy absorbed in fracture (J)"
|
|
},
|
|
{
|
|
"eq_number": 434,
|
|
"title": "Izod Impact Test",
|
|
"domain_id": 21,
|
|
"significance": "Similar to Charpy; energy absorbed per unit width (J/m)"
|
|
},
|
|
{
|
|
"eq_number": 435,
|
|
"title": "Rubber Elasticity (Gaussian Chain, Affine)",
|
|
"domain_id": 24,
|
|
"significance": "\u03c3_true = n k_B T (\u03bb \u2212 1/\u03bb\u00b2); n = crosslink density; \u03bb = extension ratio"
|
|
},
|
|
{
|
|
"eq_number": 436,
|
|
"title": "Mooney-Rivlin Equation (Hyperelastic)",
|
|
"domain_id": 24,
|
|
"significance": "W = C\u2081\u2080(I\u2081\u22123) + C\u2080\u2081(I\u2082\u22123); I\u2081,I\u2082 = invariants of Cauchy-Green tensor"
|
|
},
|
|
{
|
|
"eq_number": 437,
|
|
"title": "Flory-Huggins Theory (Polymer Solution Free Energy)",
|
|
"domain_id": 24,
|
|
"significance": "\u0394G_mix/k_B T = n\u2081 ln \u03c6\u2081 + n\u2082 ln \u03c6\u2082 + \u03c7 n\u2081 \u03c6\u2082; \u03c7 = Flory interaction parameter"
|
|
},
|
|
{
|
|
"eq_number": 438,
|
|
"title": "Williams-Landel-Ferry (WLF) Equation",
|
|
"domain_id": 24,
|
|
"significance": "log a_T = \u2212C\u2081 (T\u2212T_ref) / (C\u2082 + T\u2212T_ref); time-temperature superposition"
|
|
},
|
|
{
|
|
"eq_number": 439,
|
|
"title": "Arrhenius Viscosity (Above Glass Transition)",
|
|
"domain_id": 24,
|
|
"significance": "\u03b7(T) = \u03b7\u2080 exp(E_a / R T) (simple) or Vogel-Fulcher-Tammann: \u03b7 = \u03b7\u2080 exp[B/(T\u2212T\u2080)]"
|
|
},
|
|
{
|
|
"eq_number": 440,
|
|
"title": "Rouse Model (Unentangled Polymer Dynamics)",
|
|
"domain_id": 24,
|
|
"significance": "\u03c4_R = \u03b6 N\u00b2 b\u00b2 / (3\u03c0\u00b2 k_B T); longest relaxation time of unentangled chain"
|
|
},
|
|
{
|
|
"eq_number": 441,
|
|
"title": "Reptation Model (de Gennes, Entangled Dynamics)",
|
|
"domain_id": 24,
|
|
"significance": "\u03c4_rep \u221d N\u00b3; D_rep \u221d N\u207b\u00b2; disentanglement time; Nobel 1991"
|
|
},
|
|
{
|
|
"eq_number": 442,
|
|
"title": "Entanglement Molecular Weight",
|
|
"domain_id": 24,
|
|
"significance": "M_e = \u03c1 R T / G_N\u2070; from plateau modulus G_N\u2070"
|
|
},
|
|
{
|
|
"eq_number": 443,
|
|
"title": "Flory-Fox Equation (T_g vs Molecular Weight)",
|
|
"domain_id": 24,
|
|
"significance": "T_g = T_g\u221e \u2212 K_F / M_n; T_g increases with MW to asymptotic limit"
|
|
},
|
|
{
|
|
"eq_number": 444,
|
|
"title": "Cahn-Hilliard Equation (Spinodal Decomposition)",
|
|
"domain_id": 24,
|
|
"significance": "\u2202c/\u2202t = M \u2207\u00b2[\u2202f/\u2202c \u2212 2\u03ba \u2207\u00b2c]; diffusion modulated by gradient energy"
|
|
},
|
|
{
|
|
"eq_number": 445,
|
|
"title": "Avrami Equation (Crystallization Kinetics)",
|
|
"domain_id": 27,
|
|
"significance": "X(t) = 1 \u2212 exp(\u2212k t^n); n = Avrami exponent (dimensionality + nucleation mode)"
|
|
},
|
|
{
|
|
"eq_number": 446,
|
|
"title": "Lauritzen-Hoffman Theory (Polymer Crystal Growth)",
|
|
"domain_id": 27,
|
|
"significance": "G = G\u2080 exp[\u2212U*/R(T\u2212T_\u221e)] exp[\u2212K_g / (T \u0394T f)]; secondary nucleation"
|
|
},
|
|
{
|
|
"eq_number": 447,
|
|
"title": "Young's Equation (Contact Angle)",
|
|
"domain_id": 25,
|
|
"significance": "\u03b3_sv = \u03b3_sl + \u03b3_lv cos \u03b8; balance of interfacial tensions"
|
|
},
|
|
{
|
|
"eq_number": 448,
|
|
"title": "Wenzel Equation (Rough Surface Wetting)",
|
|
"domain_id": 25,
|
|
"significance": "cos \u03b8* = r cos \u03b8; r = actual/projected area > 1; roughness amplifies wetting"
|
|
},
|
|
{
|
|
"eq_number": 449,
|
|
"title": "Cassie-Baxter Equation (Composite/Heterogeneous Wetting)",
|
|
"domain_id": 25,
|
|
"significance": "cos \u03b8* = f\u2081 cos \u03b8\u2081 + f\u2082 cos \u03b8\u2082; f\u2081+f\u2082=1; trapped air \u2192 superhydrophobic"
|
|
},
|
|
{
|
|
"eq_number": 450,
|
|
"title": "Laplace Pressure (Curved Interface)",
|
|
"domain_id": 9,
|
|
"significance": "\u0394P = \u03b3 (1/R\u2081 + 1/R\u2082); pressure inside curved surface"
|
|
},
|
|
{
|
|
"eq_number": 451,
|
|
"title": "Kelvin Equation (Capillary Condensation)",
|
|
"domain_id": 25,
|
|
"significance": "ln(P/P\u2080) = \u22122\u03b3 V_m / (r R T); condensation in pores below saturation"
|
|
},
|
|
{
|
|
"eq_number": 452,
|
|
"title": "Langmuir Adsorption Isotherm (Monolayer)",
|
|
"domain_id": 25,
|
|
"significance": "\u03b8 = K P / (1 + K P); \u03b8 = fractional coverage; K = adsorption equilibrium constant"
|
|
},
|
|
{
|
|
"eq_number": 453,
|
|
"title": "BET Isotherm (Brunauer-Emmett-Teller, Multilayer)",
|
|
"domain_id": 25,
|
|
"significance": "P/[V(P\u2080\u2212P)] = 1/(V_m C) + (C\u22121)P/(V_m C P\u2080); surface area from multilayer adsorption"
|
|
},
|
|
{
|
|
"eq_number": 454,
|
|
"title": "Freundlich Isotherm (Heterogeneous Surfaces)",
|
|
"domain_id": 25,
|
|
"significance": "q = K_F P^{1/n}; empirical; heterogeneous adsorption"
|
|
},
|
|
{
|
|
"eq_number": 455,
|
|
"title": "Gibbs Adsorption Equation",
|
|
"domain_id": 25,
|
|
"significance": "d\u03b3 = \u2212\u03a3 \u0393_i d\u03bc_i; \u0393_i = surface excess concentration"
|
|
},
|
|
{
|
|
"eq_number": 456,
|
|
"title": "Amontons-Coulomb Friction Law (Dry Friction)",
|
|
"domain_id": 18,
|
|
"significance": "F_f \u2264 \u03bc_s N (static); F_f = \u03bc_k N (kinetic); \u03bc_k < \u03bc_s"
|
|
},
|
|
{
|
|
"eq_number": 457,
|
|
"title": "Archard's Law (Adhesive Wear)",
|
|
"domain_id": 25,
|
|
"significance": "V = k F s / H; k = wear coefficient; H = hardness"
|
|
},
|
|
{
|
|
"eq_number": 458,
|
|
"title": "Hamaker Constant (Van der Waals Between Surfaces)",
|
|
"domain_id": 25,
|
|
"significance": "A = \u03c0\u00b2 C \u03c1\u2081 \u03c1\u2082; F_vdW/A = \u2212A / (6\u03c0 d\u00b3) (flat surfaces)"
|
|
},
|
|
{
|
|
"eq_number": 459,
|
|
"title": "DLVO Theory (Colloid Stability)",
|
|
"domain_id": 25,
|
|
"significance": "V_total(d) = V_vdW + V_edl; van der Waals + electric double-layer"
|
|
},
|
|
{
|
|
"eq_number": 460,
|
|
"title": "Zeta Potential (Smoluchowski Equation)",
|
|
"domain_id": 25,
|
|
"significance": "\u03b6 = \u03b7 \u03bc_e / \u03b5; \u03bc_e = electrophoretic mobility; \u03b7 = viscosity"
|
|
},
|
|
{
|
|
"eq_number": 461,
|
|
"title": "Derjaguin Approximation (Force Between Curved Surfaces)",
|
|
"domain_id": 25,
|
|
"significance": "F_sphere(d) = 2\u03c0R W_flat(d); relates sphere-sphere to flat-plate energy"
|
|
},
|
|
{
|
|
"eq_number": 462,
|
|
"title": "Johnson-Kendall-Roberts (JKR) Adhesion Model",
|
|
"domain_id": 25,
|
|
"significance": "a\u00b3 = (R/K)[F + 3\u03c0W_ad R + \u221a(6\u03c0W_adRF + (3\u03c0W_adR)\u00b2)]; elastic + adhesion contact"
|
|
},
|
|
{
|
|
"eq_number": 463,
|
|
"title": "Derjaguin-Muller-Toporov (DMT) Model",
|
|
"domain_id": 25,
|
|
"significance": "a\u00b3 = (R/K)[F + 2\u03c0W_ad R]; adhesion without distortion of contact profile"
|
|
},
|
|
{
|
|
"eq_number": 464,
|
|
"title": "Fick's First Law (Steady-State Diffusion)",
|
|
"domain_id": 21,
|
|
"significance": "J = \u2212D \u2202c/\u2202x; flux proportional to concentration gradient"
|
|
},
|
|
{
|
|
"eq_number": 465,
|
|
"title": "Fick's Second Law (Time-Dependent Diffusion)",
|
|
"domain_id": 21,
|
|
"significance": "\u2202c/\u2202t = D \u2202\u00b2c/\u2202x\u00b2; for constant D; general: \u2202c/\u2202t = \u2202/\u2202x(D \u2202c/\u2202x)"
|
|
},
|
|
{
|
|
"eq_number": 466,
|
|
"title": "Diffusion Solutions (Common)",
|
|
"domain_id": 21,
|
|
"significance": "Thin film: c(x,t) = (M/\u221a(4\u03c0Dt)) exp(\u2212x\u00b2/4Dt); Error function: c = C\u2080 erfc(x/\u221a(4Dt))"
|
|
},
|
|
{
|
|
"eq_number": 467,
|
|
"title": "Arrhenius Diffusion Coefficient",
|
|
"domain_id": 21,
|
|
"significance": "D = D\u2080 exp(\u2212E_a / k_B T); thermally activated diffusion"
|
|
},
|
|
{
|
|
"eq_number": 468,
|
|
"title": "Darken Equations (Interdiffusion / Kirkendall Effect)",
|
|
"domain_id": 21,
|
|
"significance": "D\u0303 = (X_B D_A + X_A D_B) \u03a6; \u03a6 = thermodynamic factor including non-ideality"
|
|
},
|
|
{
|
|
"eq_number": 469,
|
|
"title": "Nernst-Planck Equation (Ion Transport)",
|
|
"domain_id": 21,
|
|
"significance": "J_i = \u2212D_i \u2207c_i \u2212 (z_i F/RT)D_i c_i \u2207\u03c6 + c_i v; diffusion + migration + convection"
|
|
},
|
|
{
|
|
"eq_number": 470,
|
|
"title": "Stokes-Einstein Relation (Diffusion of Spheres)",
|
|
"domain_id": 21,
|
|
"significance": "D = k_B T / (6\u03c0 \u03b7 r); hydrodynamic radius from diffusion"
|
|
},
|
|
{
|
|
"eq_number": 471,
|
|
"title": "Tracer Diffusion Correlation Factor",
|
|
"domain_id": 21,
|
|
"significance": "D* = f D_rand; f = correlation factor; f<1 for vacancy mechanism"
|
|
},
|
|
{
|
|
"eq_number": 472,
|
|
"title": "Gibbs-Thomson Effect (Curvature Depression of Melting/Equilibrium Point)",
|
|
"domain_id": 27,
|
|
"significance": "T_m(r) = T_m(\u221e)(1 \u2212 2\u03b3_sl / (\u03c1_s \u0394H_f r)); small particles melt at lower T"
|
|
},
|
|
{
|
|
"eq_number": 473,
|
|
"title": "Classical Nucleation Theory (Homogeneous)",
|
|
"domain_id": 27,
|
|
"significance": "\u0394G = (4\u03c0/3)r\u00b3 \u0394G_v + 4\u03c0r\u00b2 \u03b3; r* = \u22122\u03b3/\u0394G_v; \u0394G* = 16\u03c0\u03b3\u00b3/(3\u0394G_v\u00b2)"
|
|
},
|
|
{
|
|
"eq_number": 474,
|
|
"title": "Johnson-Mehl-Avrami-Kolmogorov (JMAK) Equation",
|
|
"domain_id": 27,
|
|
"significance": "f = 1 \u2212 exp[\u2212(kt)^n]; n depends on nucleation+growth dimensionality"
|
|
},
|
|
{
|
|
"eq_number": 475,
|
|
"title": "Turnbull's Nucleation Rate (Steady-State)",
|
|
"domain_id": 27,
|
|
"significance": "I = N_v (k_B T/h) exp[\u2212(\u0394G*+\u0394G_a)/k_B T]; includes kinetic barrier"
|
|
},
|
|
{
|
|
"eq_number": 476,
|
|
"title": "Lever Rule (Phase Diagram Tie Line)",
|
|
"domain_id": 27,
|
|
"significance": "f_\u03b1 = (C\u2080\u2212C_\u03b2)/(C_\u03b1\u2212C_\u03b2); f_\u03b2 = (C_\u03b1\u2212C\u2080)/(C_\u03b1\u2212C_\u03b2)"
|
|
},
|
|
{
|
|
"eq_number": 477,
|
|
"title": "Gibbs-Thomson-Freundlich (Ostwald Ripening / LSW Theory)",
|
|
"domain_id": 27,
|
|
"significance": "\u27e8r\u27e9\u00b3 \u2212 \u27e8r\u2080\u27e9\u00b3 = k t; k \u221d \u03b3 D c_\u221e V_m\u00b2/(R T); coarsening of precipitates"
|
|
},
|
|
{
|
|
"eq_number": 478,
|
|
"title": "Darken-Gurry Plot (Solubility Limits)",
|
|
"domain_id": 21,
|
|
"significance": "Extensive solubility when |\u0394R_atom|<15% and |\u0394\u03c7|<0.4 (electronegativity difference)"
|
|
},
|
|
{
|
|
"eq_number": 479,
|
|
"title": "Hume-Rothery Rules (Alloy Formation)",
|
|
"domain_id": 21,
|
|
"significance": "(1) Size <15% (2) Similar electronegativity (3) Same valence (4) Same crystal structure"
|
|
},
|
|
{
|
|
"eq_number": 480,
|
|
"title": "Vegard's Law (Lattice Parameter in Solid Solutions)",
|
|
"domain_id": 21,
|
|
"significance": "a_AB = x_A a_A + x_B a_B; linear interpolation; deviations = non-ideal mixing"
|
|
},
|
|
{
|
|
"eq_number": 481,
|
|
"title": "Rule of Mixtures (Composite Modulus, Isostrain)",
|
|
"domain_id": 21,
|
|
"significance": "E_c = E_f V_f + E_m V_m (Voigt bound, upper); 1/E_c = V_f/E_f + V_m/E_m (Reuss bound, lower)"
|
|
},
|
|
{
|
|
"eq_number": 482,
|
|
"title": "Hashin-Shtrikman Bounds (Composite Moduli)",
|
|
"domain_id": 21,
|
|
"significance": "Tighter bounds than Voigt-Reuss; K_lower = K_m + V_f/[1/(K_f\u2212K_m)+3V_m/(3K_m+4G_m)]; etc."
|
|
},
|
|
{
|
|
"eq_number": 483,
|
|
"title": "Halpin-Tsai Equations (Short Fiber Composites)",
|
|
"domain_id": 21,
|
|
"significance": "E/E_m = (1 + \u03be \u03b7 V_f)/(1 \u2212 \u03b7 V_f); \u03b7 = (E_f/E_m\u22121)/(E_f/E_m+\u03be); \u03be = shape factor"
|
|
},
|
|
{
|
|
"eq_number": 484,
|
|
"title": "Porosity-Young's Modulus Relation (Empirical)",
|
|
"domain_id": 21,
|
|
"significance": "E = E\u2080 (1 \u2212 P)^n or exp(\u2212bP); P = porosity fraction; n\u22482\u20134"
|
|
},
|
|
{
|
|
"eq_number": 485,
|
|
"title": "Gibson-Ashby Model (Cellular Solids / Foams)",
|
|
"domain_id": 21,
|
|
"significance": "E*/E_s = C (\u03c1*/\u03c1_s)^n; n=2 open cell; n=3 closed cell; \u03c3*/\u03c3_ys \u221d (\u03c1*/\u03c1_s)^{3/2}"
|
|
},
|
|
{
|
|
"eq_number": 486,
|
|
"title": "Eshelby Inclusion Problem (Stress in Ellipsoidal Inclusion)",
|
|
"domain_id": 21,
|
|
"significance": "\u03b5^T = S \u03b5*; S = Eshelby tensor (depends on inclusion shape + matrix Poisson ratio)"
|
|
},
|
|
{
|
|
"eq_number": 487,
|
|
"title": "Moss-Burstein Shift (Doped Semiconductor Absorption Edge)",
|
|
"domain_id": 21,
|
|
"significance": "\u0394E_g = (\u210f\u00b2/2m*)(3\u03c0\u00b2n)^{2/3}; Fermi filling blocks lowest transitions"
|
|
},
|
|
{
|
|
"eq_number": 488,
|
|
"title": "Franz-Keldysh Effect (Electro-Absorption)",
|
|
"domain_id": 21,
|
|
"significance": "\u03b1(E,F) \u221d exp[\u2212(E_g\u2212E)^{3/2} / e\u210fF]; band edge shift in electric field"
|
|
},
|
|
{
|
|
"eq_number": 489,
|
|
"title": "Pockels Effect (Linear Electro-Optic)",
|
|
"domain_id": 21,
|
|
"significance": "\u0394(1/n\u00b2)_i = r_ij E_j; r_ij = linear electro-optic coefficients"
|
|
},
|
|
{
|
|
"eq_number": 490,
|
|
"title": "Kerr Effect (Quadratic Electro-Optic)",
|
|
"domain_id": 21,
|
|
"significance": "\u0394n = K \u03bb E\u00b2; quadratic field dependence"
|
|
},
|
|
{
|
|
"eq_number": 491,
|
|
"title": "Photoconductivity (Rose Model)",
|
|
"domain_id": 21,
|
|
"significance": "\u0394\u03c3 = e \u03bc \u03c4 G L / d; G=generation rate, \u03c4=lifetime; gain = \u03c4/t_transit"
|
|
},
|
|
{
|
|
"eq_number": 492,
|
|
"title": "Shockley-Read-Hall Recombination Rate",
|
|
"domain_id": 23,
|
|
"significance": "U = (np\u2212n_i\u00b2) / [\u03c4_p (n+n\u2081) + \u03c4_n (p+p\u2081)]; trap-assisted recombination"
|
|
},
|
|
{
|
|
"eq_number": 493,
|
|
"title": "Auger Recombination Rate",
|
|
"domain_id": 23,
|
|
"significance": "U_Auger = C_n n\u00b2p + C_p n p\u00b2; three-particle non-radiative recombination"
|
|
},
|
|
{
|
|
"eq_number": 494,
|
|
"title": "BCS Energy Gap at T=0",
|
|
"domain_id": 12,
|
|
"significance": "\u0394(0) = 1.764 k_B T_c; universal BCS ratio"
|
|
},
|
|
{
|
|
"eq_number": 495,
|
|
"title": "Ginzburg-Landau Coherence Length",
|
|
"domain_id": 12,
|
|
"significance": "\u03be(T) = \u03be(0) / \u221a(1\u2212T/T_c); \u03be(0) = \u221a(\u210f\u00b2/2m*|\u03b1|); spatial variation of order parameter"
|
|
},
|
|
{
|
|
"eq_number": 496,
|
|
"title": "Ginzburg-Landau Penetration Depth",
|
|
"domain_id": 12,
|
|
"significance": "\u03bb(T) = \u03bb(0)/\u221a(1\u2212T/T_c); magnetic field penetration into superconductor"
|
|
},
|
|
{
|
|
"eq_number": 497,
|
|
"title": "Ginzburg-Landau Parameter (\u03ba)",
|
|
"domain_id": 12,
|
|
"significance": "\u03ba = \u03bb/\u03be; \u03ba < 1/\u221a2 \u2192 Type I; \u03ba > 1/\u221a2 \u2192 Type II"
|
|
},
|
|
{
|
|
"eq_number": 498,
|
|
"title": "Abrikosov Vortex Lattice (Lower/Upper Critical Fields)",
|
|
"domain_id": 12,
|
|
"significance": "H_c1 = H_c ln \u03ba/(\u221a2 \u03ba); H_c2 = \u221a2 \u03ba H_c; vortex state between"
|
|
},
|
|
{
|
|
"eq_number": 499,
|
|
"title": "Flux Pinning (Bean Critical State Model)",
|
|
"domain_id": 12,
|
|
"significance": "J_c = constant; \u2207\u00d7B = \u03bc\u2080 J_c; critical state penetration profile"
|
|
},
|
|
{
|
|
"eq_number": 500,
|
|
"title": "Little-Parks Effect (Fluxoid Quantization)",
|
|
"domain_id": 12,
|
|
"significance": "T_c oscillates with flux through cylinder; period = \u03a6\u2080 = h/2e"
|
|
},
|
|
{
|
|
"eq_number": 501,
|
|
"title": "Andreev Reflection",
|
|
"domain_id": 12,
|
|
"significance": "e\u207b \u2192 NS interface reflects as h\u207a; retroreflection; sub-gap conductance enhancement"
|
|
},
|
|
{
|
|
"eq_number": 502,
|
|
"title": "Nernst Equation (Electrode Potential)",
|
|
"domain_id": 21,
|
|
"significance": "E = E\u2070 \u2212 (RT/nF) ln Q; E\u2070 = standard reduction potential"
|
|
},
|
|
{
|
|
"eq_number": 503,
|
|
"title": "Butler-Volmer Equation (Electrode Kinetics)",
|
|
"domain_id": 21,
|
|
"significance": "j = j\u2080 [exp(\u03b1_a F \u03b7/RT) \u2212 exp(\u2212\u03b1_c F \u03b7/RT)]; \u03b7 = overpotential"
|
|
},
|
|
{
|
|
"eq_number": 504,
|
|
"title": "Tafel Equation (High Overpotential Limit)",
|
|
"domain_id": 21,
|
|
"significance": "\u03b7 = a + b log |j|; b = 2.303 RT/(\u03b1 nF) \u2248 120 mV/decade (\u03b1=0.5 at 298K)"
|
|
},
|
|
{
|
|
"eq_number": 505,
|
|
"title": "Randles-Sevcik Equation (Cyclic Voltammetry Peak Current)",
|
|
"domain_id": 21,
|
|
"significance": "i_p = 0.4463 n F A C \u221a(n F v D/RT); reversible: i_p \u221d \u221av"
|
|
},
|
|
{
|
|
"eq_number": 506,
|
|
"title": "Cottrell Equation (Chronoamperometry)",
|
|
"domain_id": 21,
|
|
"significance": "i(t) = n F A C \u221a(D) / \u221a(\u03c0 t); diffusion-limited current decay"
|
|
},
|
|
{
|
|
"eq_number": 507,
|
|
"title": "Faraday's Laws of Electrolysis",
|
|
"domain_id": 3,
|
|
"significance": "m = (Q M)/(n F); mass deposited proportional to charge; Q=It"
|
|
},
|
|
{
|
|
"eq_number": 508,
|
|
"title": "Wagner Number (Current Distribution Uniformity)",
|
|
"domain_id": 21,
|
|
"significance": "Wa = \u03ba (d\u03b7/dj) / L; Wa \u226b 1 \u2192 uniform; Wa \u226a 1 \u2192 non-uniform"
|
|
},
|
|
{
|
|
"eq_number": 509,
|
|
"title": "Zener Anelasticity (Standard Linear Solid)",
|
|
"domain_id": 21,
|
|
"significance": "\u03b5 = \u03c3/E_R + (\u03c3/E_U\u2212\u03c3/E_R) (1\u2212e^{\u2212t/\u03c4}); relaxation strength \u0394 = (E_U\u2212E_R)/\u221a(E_U E_R)"
|
|
},
|
|
{
|
|
"eq_number": 510,
|
|
"title": "Debye Peak (Internal Friction, Point Defect Relaxation)",
|
|
"domain_id": 21,
|
|
"significance": "tan \u03b4 = \u0394 \u03c9 \u03c4 / (1 + \u03c9\u00b2 \u03c4\u00b2); \u03c4 = \u03c4\u2080 exp(E_a/k_B T); peak at \u03c9\u03c4=1"
|
|
},
|
|
{
|
|
"eq_number": 511,
|
|
"title": "Bordoni Peak (Dislocation Relaxation)",
|
|
"domain_id": 21,
|
|
"significance": "kink-pair formation on dislocations; tan \u03b4 peak with E_a~0.1\u20130.2 eV"
|
|
},
|
|
{
|
|
"eq_number": 512,
|
|
"title": "Granato-L\u00fccke Theory (Dislocation Damping)",
|
|
"domain_id": 21,
|
|
"significance": "\u03b5_d = (\u039b L\u00b2 \u03c3)/(6 G) (amplitude-independent); breakaway at high amplitude"
|
|
},
|
|
{
|
|
"eq_number": 513,
|
|
"title": "Einstein Viscosity Equation (Rigid Sphere Suspension, Dilute)",
|
|
"domain_id": 26,
|
|
"significance": "\u03b7 = \u03b7_s (1 + 2.5 \u03c6); \u03c6 = volume fraction; dilute limit \u03c6\u226a1"
|
|
},
|
|
{
|
|
"eq_number": 514,
|
|
"title": "Krieger-Dougherty Equation (Concentrated Suspension)",
|
|
"domain_id": 26,
|
|
"significance": "\u03b7 = \u03b7_s (1 \u2212 \u03c6/\u03c6_m)^{\u2212[\u03b7]\u03c6_m}; \u03c6_m = maximum packing; [\u03b7]\u22482.5"
|
|
},
|
|
{
|
|
"eq_number": 515,
|
|
"title": "Frank-Oseen Free Energy (Liquid Crystal Elastic)",
|
|
"domain_id": 26,
|
|
"significance": "F = \u00bd[K\u2081(\u2207\u00b7n)\u00b2 + K\u2082(n\u00b7\u2207\u00d7n)\u00b2 + K\u2083(n\u00d7\u2207\u00d7n)\u00b2]; splay, twist, bend"
|
|
},
|
|
{
|
|
"eq_number": 516,
|
|
"title": "Frederiks Transition Threshold (Liquid Crystal)",
|
|
"domain_id": 26,
|
|
"significance": "E_c = (\u03c0/d) \u221a(K/\u03b5\u2080\u0394\u03b5); voltage for director reorientation"
|
|
},
|
|
{
|
|
"eq_number": 517,
|
|
"title": "Rayleigh Instability (Liquid Jet Breakup)",
|
|
"domain_id": 26,
|
|
"significance": "\u03bb_max = 9.016 r\u2080; fastest growing wavelength \u2192 uniform droplet formation"
|
|
},
|
|
{
|
|
"eq_number": 518,
|
|
"title": "Plateau-Rayleigh Instability for Liquid Threads",
|
|
"domain_id": 26,
|
|
"significance": "Cylindrical liquid thread unstable for \u03bb > 2\u03c0r; surface-tension-driven breakup"
|
|
},
|
|
{
|
|
"eq_number": 519,
|
|
"title": "Kissinger Equation (DSC/DTA Peak Kinetics)",
|
|
"domain_id": 21,
|
|
"significance": "ln(\u03b2/T_p\u00b2) = \u2212E_a/(R T_p) + ln(A R/E_a); \u03b2 = heating rate; T_p = peak temperature"
|
|
},
|
|
{
|
|
"eq_number": 520,
|
|
"title": "Ozawa-Flynn-Wall Equation (Isoconversional Kinetics)",
|
|
"domain_id": 21,
|
|
"significance": "log \u03b2 = const \u2212 0.4567 E_a/(R T); model-free kinetic analysis"
|
|
},
|
|
{
|
|
"eq_number": 521,
|
|
"title": "Tammann Nucleation Diagram (Nucleation vs Growth Rate)",
|
|
"domain_id": 27,
|
|
"significance": "Nucleation rate I(T) and growth rate U(T) bell-shaped; overlap \u2192 crystallization window"
|
|
},
|
|
{
|
|
"eq_number": 522,
|
|
"title": "Time-Temperature-Transformation (TTT) Diagram Equation",
|
|
"domain_id": 27,
|
|
"significance": "\u03c4(T) \u221d exp(\u0394G*/k_B T + E_a/k_B T); C-curve shape; nose at intermediate T"
|
|
},
|
|
{
|
|
"eq_number": 523,
|
|
"title": "Thornton Structure Zone Model (Thin Film Growth)",
|
|
"domain_id": 21,
|
|
"significance": "T/T_m vs Ar pressure \u2192 Zone 1 (porous), Zone T (dense fibrous), Zone 2 (columnar), Zone 3 (recrystallized)"
|
|
},
|
|
{
|
|
"eq_number": 524,
|
|
"title": "Herring Scaling Laws (Sintering Kinetics)",
|
|
"domain_id": 21,
|
|
"significance": "(\u0394L/L\u2080)^n \u221d t; n=1 viscous flow; n=2 volume diffusion; n=3 grain boundary diffusion; n=5 surface diffusion"
|
|
},
|
|
{
|
|
"eq_number": 525,
|
|
"title": "Pilling-Bedworth Ratio (Oxide Protectiveness)",
|
|
"domain_id": 21,
|
|
"significance": "PBR = V_oxide / V_metal consumed; 1 < PBR < 2 \u2192 protective; PBR > 2 \u2192 spallation; PBR < 1 \u2192 porous"
|
|
},
|
|
{
|
|
"eq_number": 526,
|
|
"title": "Ellingham Diagram (Oxide Thermodynamic Stability)",
|
|
"domain_id": 21,
|
|
"significance": "\u0394G\u2070 = RT ln p_O\u2082; line slope = \u2212\u0394S\u2070; lower line \u2192 more stable oxide"
|
|
},
|
|
{
|
|
"eq_number": 527,
|
|
"title": "Mott-Gurney Law (Space-Charge-Limited Current)",
|
|
"domain_id": 21,
|
|
"significance": "J = (9/8) \u03b5 \u03bc V\u00b2 / L\u00b3; trap-free SCLC; Child's law for solids"
|
|
},
|
|
{
|
|
"eq_number": 528,
|
|
"title": "Richardson-Dushman Equation (Thermionic Emission)",
|
|
"domain_id": 21,
|
|
"significance": "J = A_R T\u00b2 exp(\u2212\u03c6/k_B T); A_R = 4\u03c0 m e k_B\u00b2/h\u00b3 \u2248 1.20\u00d710\u2076 A/(m\u00b2K\u00b2)"
|
|
},
|
|
{
|
|
"eq_number": 529,
|
|
"title": "Schottky Barrier Height (Metal-Semiconductor)",
|
|
"domain_id": 23,
|
|
"significance": "\u03c6_Bn = \u03c6_m \u2212 \u03c7_s; \u03c6_Bp = E_g/q + \u03c7_s \u2212 \u03c6_m (ideal, no interface states)"
|
|
},
|
|
{
|
|
"eq_number": 530,
|
|
"title": "Spicer's Unified Defect Model (Fermi Level Pinning at Interfaces)",
|
|
"domain_id": 23,
|
|
"significance": "E_F pinned by deep native defects at interface; independent of metal work function"
|
|
},
|
|
{
|
|
"eq_number": 531,
|
|
"title": "Wolff's Law (Bone Remodeling, Mechanical Adaptation)",
|
|
"domain_id": 21,
|
|
"significance": "Bone density distribution adapts to principal stress trajectories; \u03c3_ij \u2192 \u03c1_ij"
|
|
},
|
|
{
|
|
"eq_number": 532,
|
|
"title": "Fung's Quasi-Linear Viscoelasticity (Soft Tissue)",
|
|
"domain_id": 21,
|
|
"significance": "\u03c3(t) = \u222b\u2080\u1d57 G(t\u2212\u03c4) \u2202\u03c3_e(\u03b5)/\u2202\u03b5 \u00b7 \u2202\u03b5/\u2202\u03c4 d\u03c4; separable elastic + relaxation"
|
|
},
|
|
{
|
|
"eq_number": 533,
|
|
"title": "Ogden Hyperelastic Model (Biological Tissue)",
|
|
"domain_id": 21,
|
|
"significance": "W = \u03a3 (\u03bc_k/\u03b1_k) (\u03bb\u2081^{\u03b1_k} + \u03bb\u2082^{\u03b1_k} + \u03bb\u2083^{\u03b1_k} \u2212 3); principal stretches; fits large deformations"
|
|
},
|
|
{
|
|
"eq_number": 534,
|
|
"title": "Matthiessen's Rule (Electrical Resistivity Additivity)",
|
|
"domain_id": 21,
|
|
"significance": "\u03c1_total = \u03c1_thermal + \u03c1_impurity + \u03c1_deformation; independent contributions sum"
|
|
},
|
|
{
|
|
"eq_number": 535,
|
|
"title": "Nordheim's Rule (Alloy Resistivity)",
|
|
"domain_id": 21,
|
|
"significance": "\u03c1_alloy = \u03c1_pure + C x(1\u2212x); x = atomic fraction; max at x=0.5 for disordered binary"
|
|
},
|
|
{
|
|
"eq_number": 536,
|
|
"title": "Miedema's Rules (Alloy Formation Enthalpy)",
|
|
"domain_id": 21,
|
|
"significance": "\u0394H_form = f(\u0394\u03c6*, \u0394n_ws^{1/3}); work function + electron density mismatch \u2192 semi-empirical model"
|
|
},
|
|
{
|
|
"eq_number": 537,
|
|
"title": "K\u00f6hler's Rule (Magnetoresistance Scaling)",
|
|
"domain_id": 21,
|
|
"significance": "\u0394\u03c1(B)/\u03c1(0) = F[B/\u03c1(0)]; Kohler plot universal for given material"
|
|
},
|
|
{
|
|
"eq_number": 538,
|
|
"title": "Zener Breakdown (Band-to-Band Tunneling)",
|
|
"domain_id": 23,
|
|
"significance": "D = exp[\u22124\u221a(2m*) E_g^{3/2}/(3 e \u210f E)]; tunneling probability through forbidden gap"
|
|
},
|
|
{
|
|
"eq_number": 539,
|
|
"title": "Klemens Model (Thermal Boundary Resistance / Kapitza)",
|
|
"domain_id": 21,
|
|
"significance": "R_K = 4 / (\u03c1 c v \u03b6); acoustic mismatch model; acoustic impedance mismatch \u2192 resistance"
|
|
},
|
|
{
|
|
"eq_number": 540,
|
|
"title": "Diffuse Mismatch Model (Thermal Boundary Resistance)",
|
|
"domain_id": 21,
|
|
"significance": "R_K from transmission probability of phonons regardless of mode; rough interfaces"
|
|
},
|
|
{
|
|
"eq_number": 541,
|
|
"title": "Seismic Wave Equation (Elastic)",
|
|
"domain_id": 28,
|
|
"significance": "\u03c1 \u2202\u00b2u/\u2202t\u00b2 = (\u03bb+\u03bc)\u2207(\u2207\u00b7u) + \u03bc\u2207\u00b2u; vector elastic wave equation; P and S waves"
|
|
},
|
|
{
|
|
"eq_number": 542,
|
|
"title": "Snell's Law (Seismic Refraction at Interfaces)",
|
|
"domain_id": 28,
|
|
"significance": "sin i\u2081/v\u2081 = sin i\u2082/v\u2082 = p (ray parameter); seismic ray tracing through layered Earth"
|
|
},
|
|
{
|
|
"eq_number": 543,
|
|
"title": "Gutenberg-Richter Magnitude-Energy Relation",
|
|
"domain_id": 28,
|
|
"significance": "log\u2081\u2080 E = 4.8 + 1.5 M (E in Joules); log\u2081\u2080 E = 5.24 + 1.44 M (later refinement)"
|
|
},
|
|
{
|
|
"eq_number": 544,
|
|
"title": "Omori's Law (Aftershock Decay)",
|
|
"domain_id": 28,
|
|
"significance": "n(t) = K / (c + t)^p; p\u22481 (often 0.9\u20131.4); K,c constants; t=time after mainshock"
|
|
},
|
|
{
|
|
"eq_number": 545,
|
|
"title": "Frequency-Magnitude Distribution (Gutenberg-Richter Law)",
|
|
"domain_id": 28,
|
|
"significance": "log\u2081\u2080 N(\u2265M) = a \u2212 b M; b\u22481 (global average); N=cumulative number of earthquakes"
|
|
},
|
|
{
|
|
"eq_number": 546,
|
|
"title": "Bullen's Compressibility-Pressure Hypothesis (Earth Interior)",
|
|
"domain_id": 28,
|
|
"significance": "K = a + bP; Earth core compressibility varies linearly with pressure"
|
|
},
|
|
{
|
|
"eq_number": 547,
|
|
"title": "Love Wave Dispersion Equation (Surface Waves)",
|
|
"domain_id": 28,
|
|
"significance": "tan(\u03ba\u03b2\u2081H) = (\u03bc\u2082\u03b2\u2082)/(\u03bc\u2081\u03b2\u2081); Love wave existence condition in layer over half-space"
|
|
},
|
|
{
|
|
"eq_number": 548,
|
|
"title": "Airy Isostasy (Crustal Compensation Model)",
|
|
"domain_id": 28,
|
|
"significance": "Mountain root thickness = h \u03c1_c/(\u03c1_m\u2212\u03c1_c); h=elevation; \u03c1_c,\u03c1_m=crust/mantle density"
|
|
},
|
|
{
|
|
"eq_number": 549,
|
|
"title": "Free-Air Gravity Anomaly",
|
|
"domain_id": 28,
|
|
"significance": "\u0394g_FA = g_obs \u2212 g_theoretical(\u03bb) + \u03b4g_FAC; \u03b4g_FAC=0.3086h mGal (free-air correction, h in m)"
|
|
},
|
|
{
|
|
"eq_number": 550,
|
|
"title": "Bouguer Gravity Anomaly (Complete)",
|
|
"domain_id": 28,
|
|
"significance": "\u0394g_B = g_obs \u2212 g_theo(\u03bb) + 0.3086h \u2212 0.0419\u03c1h + \u03b4g_terrain; \u03c1=2.67 g/cm\u00b3 typical"
|
|
},
|
|
{
|
|
"eq_number": 551,
|
|
"title": "Geodetic Reference System (GRS80/WGS84 Ellipsoid)",
|
|
"domain_id": 28,
|
|
"significance": "a=6378137m, f=1/298.257223563 (WGS84); meridian radius M=a(1\u2212e\u00b2)/(1\u2212e\u00b2sin\u00b2\u03c6)^{3/2}"
|
|
},
|
|
{
|
|
"eq_number": 552,
|
|
"title": "Geoid Undulation (Stokes' Formula)",
|
|
"domain_id": 28,
|
|
"significance": "N = (R/4\u03c0\u03b3)\u222b\u222b \u0394g S(\u03c8) d\u03c3; S(\u03c8) = Stokes function; \u0394g=gravity anomaly; \u03c8=angular distance"
|
|
},
|
|
{
|
|
"eq_number": 553,
|
|
"title": "Plate Motion on a Sphere (Euler Pole Rotation)",
|
|
"domain_id": 28,
|
|
"significance": "v = \u03c9 \u00d7 R; v=linear velocity, \u03c9=angular velocity vector, R=Earth radius vector"
|
|
},
|
|
{
|
|
"eq_number": 554,
|
|
"title": "Geomagnetic Secular Variation (IGRF Model)",
|
|
"domain_id": 28,
|
|
"significance": "B(r,\u03b8,\u03c6,t) = \u2212\u2207[a \u03a3(g_n^m cos m\u03c6 + h_n^m sin m\u03c6)(a/r)^{n+1} P_n^m(cos \u03b8)]"
|
|
},
|
|
{
|
|
"eq_number": 555,
|
|
"title": "Curie Temperature Isotherm (Magnetic Crustal Thickness)",
|
|
"domain_id": 28,
|
|
"significance": "Magnetic minerals become paramagnetic above ~580\u00b0C (magnetite Curie point); ~20-30 km depth"
|
|
},
|
|
{
|
|
"eq_number": 556,
|
|
"title": "Darcy's Law (Groundwater Flow in Porous Media)",
|
|
"domain_id": 31,
|
|
"significance": "Q = \u2212K A (dh/dl); v_Darcy = Q/A = \u2212K \u2207h; K = hydraulic conductivity"
|
|
},
|
|
{
|
|
"eq_number": 557,
|
|
"title": "Dupuit-Forchheimer Assumption (Unconfined Aquifer)",
|
|
"domain_id": 31,
|
|
"significance": "Q = \u2212K h (dh/dx); h=saturated thickness; flow lines approximately horizontal"
|
|
},
|
|
{
|
|
"eq_number": 558,
|
|
"title": "Theis Solution (Well Hydraulics, Confined Aquifer)",
|
|
"domain_id": 31,
|
|
"significance": "s(r,t) = Q/(4\u03c0T) \u222b_u^\u221e (e^{-x}/x) dx; u = r\u00b2S/(4Tt); T=transmissivity, S=storativity"
|
|
},
|
|
{
|
|
"eq_number": 559,
|
|
"title": "Manning Equation (Open Channel Flow)",
|
|
"domain_id": 31,
|
|
"significance": "v = (1/n) R_h^{2/3} S^{1/2}; n=Manning roughness; R_h=hydraulic radius; S=slope"
|
|
},
|
|
{
|
|
"eq_number": 560,
|
|
"title": "Rational Method (Peak Runoff Estimation)",
|
|
"domain_id": 31,
|
|
"significance": "Q_peak = C i A; C=runoff coefficient (0-1); i=rainfall intensity; A=watershed area"
|
|
},
|
|
{
|
|
"eq_number": 561,
|
|
"title": "Richards Equation (Unsaturated Flow in Soils)",
|
|
"domain_id": 31,
|
|
"significance": "\u2202\u03b8/\u2202t = \u2207\u00b7[K(\u03b8) \u2207(\u03c8+z)]; \u03b8=moisture content; \u03c8=pressure head; K(\u03b8)=hydraulic conductivity"
|
|
},
|
|
{
|
|
"eq_number": 562,
|
|
"title": "Horton Infiltration Model",
|
|
"domain_id": 31,
|
|
"significance": "f(t) = f_c + (f\u2080\u2212f_c) e^{\u2212kt}; infiltration capacity decays exponentially"
|
|
},
|
|
{
|
|
"eq_number": 563,
|
|
"title": "Penman-Monteith Evapotranspiration Equation",
|
|
"domain_id": 31,
|
|
"significance": "ET = [\u0394(R_n\u2212G) + \u03c1_a c_p (e_s\u2212e_a)/r_a] / [\u0394 + \u03b3(1+r_s/r_a)]"
|
|
},
|
|
{
|
|
"eq_number": 564,
|
|
"title": "Stomatal Conductance (Jarvis Model)",
|
|
"domain_id": 31,
|
|
"significance": "g_s = g_s_max \u00b7 f\u2081(PAR) \u00b7 f\u2082(T) \u00b7 f\u2083(VPD) \u00b7 f\u2084(CO\u2082) \u00b7 f\u2085(\u03c8_leaf); multiplicative stress functions"
|
|
},
|
|
{
|
|
"eq_number": 565,
|
|
"title": "Linear Wave Theory (Airy Wave, Dispersion Relation)",
|
|
"domain_id": 30,
|
|
"significance": "\u03c9\u00b2 = gk tanh(kh); deep water (kh\u226b1): \u03c9\u00b2=gk, c=g/\u03c9; shallow water (kh\u226a1): \u03c9\u00b2=ghk\u00b2, c=\u221a(gh)"
|
|
},
|
|
{
|
|
"eq_number": 566,
|
|
"title": "Stokes Drift (Mass Transport Under Waves)",
|
|
"domain_id": 30,
|
|
"significance": "U_s = \u00bd a\u00b2 \u03c9 k e^{2kz}; net Lagrangian drift under progressive waves; ~O(\u03b5\u00b2)"
|
|
},
|
|
{
|
|
"eq_number": 567,
|
|
"title": "Significant Wave Height (Sverdrup-Munk-Bretschneider)",
|
|
"domain_id": 30,
|
|
"significance": "H_s = H_{1/3} \u2248 4\u221am\u2080; m\u2080 = \u222b S(f) df (zeroth spectral moment)"
|
|
},
|
|
{
|
|
"eq_number": 568,
|
|
"title": "Tide-Generating Potential (Equilibrium Theory)",
|
|
"domain_id": 30,
|
|
"significance": "V_tide = \u2212(3/2) GM_\u2299 R\u00b2/r\u00b3 (cos\u00b2\u03b8 \u2212 1/3); Laplace's tidal equations govern dynamic response"
|
|
},
|
|
{
|
|
"eq_number": 569,
|
|
"title": "Geostrophic Balance (Ocean Currents)",
|
|
"domain_id": 30,
|
|
"significance": "f v = (1/\u03c1) \u2202p/\u2202x; f u = \u2212(1/\u03c1) \u2202p/\u2202y; f=2\u03a9 sin \u03c6 (Coriolis parameter); large-scale flow"
|
|
},
|
|
{
|
|
"eq_number": 570,
|
|
"title": "Ekman Transport (Wind-Driven Surface Layer)",
|
|
"domain_id": 30,
|
|
"significance": "M_E = \u03c4_wind / (\u03c1 f); net transport 90\u00b0 to right of wind (NH); left (SH)"
|
|
},
|
|
{
|
|
"eq_number": 571,
|
|
"title": "Thermohaline Circulation (Stommel-Arons Model)",
|
|
"domain_id": 30,
|
|
"significance": "Balance of advection, diffusion, and sources/sinks of heat and salt; deep ocean circulation"
|
|
},
|
|
{
|
|
"eq_number": 572,
|
|
"title": "Sverdrup Balance (Wind-Driven Gyre Circulation)",
|
|
"domain_id": 30,
|
|
"significance": "\u03b2 v = f \u2202w/\u2202z + curl_z(\u03c4)/(\u03c1); \u03b2=df/dy; meridional transport from wind stress curl"
|
|
},
|
|
{
|
|
"eq_number": 573,
|
|
"title": "Munk's Western Boundary Current Theory",
|
|
"domain_id": 30,
|
|
"significance": "A_H \u2207\u2074\u03c8 \u2212 \u03b2 \u2202\u03c8/\u2202x = \u2212curl_z(\u03c4)/\u03c1; lateral friction balances \u03b2-effect; Gulf Stream width"
|
|
},
|
|
{
|
|
"eq_number": 574,
|
|
"title": "Sonar Equation (Active, Monostatic)",
|
|
"domain_id": 48,
|
|
"significance": "SL \u2212 2TL + TS = NL \u2212 DI + DT; SL=source level, TL=transmission loss, TS=target strength, NL=noise, DI=directivity index, DT=detection threshold"
|
|
},
|
|
{
|
|
"eq_number": 575,
|
|
"title": "Sound Speed in Seawater (UNESCO/IES-80/CTD)",
|
|
"domain_id": 48,
|
|
"significance": "c(S,T,P) = 1449.2 + 4.6T \u2212 0.055T\u00b2 + 0.00029T\u00b3 + (1.34\u22120.010T)(S\u221235) + 0.016z"
|
|
},
|
|
{
|
|
"eq_number": 576,
|
|
"title": "Acoustic Doppler Current Profiler (ADCP) Principle",
|
|
"domain_id": 48,
|
|
"significance": "v_radial = (c \u0394f)/(2 f\u2080); Doppler shift from scatterers moving with water; 4 beams \u2192 3D velocity"
|
|
},
|
|
{
|
|
"eq_number": 577,
|
|
"title": "Hydrostatic Equation (Atmospheric)",
|
|
"domain_id": 29,
|
|
"significance": "dp/dz = \u2212\u03c1 g; pressure decreases exponentially with height in isothermal atmosphere"
|
|
},
|
|
{
|
|
"eq_number": 578,
|
|
"title": "Ideal Gas Law (Moist Air, Virtual Temperature)",
|
|
"domain_id": 29,
|
|
"significance": "p = \u03c1 R_d T_v; T_v = T (1 + 0.608 q); q=specific humidity; virtual temperature correction"
|
|
},
|
|
{
|
|
"eq_number": 579,
|
|
"title": "Potential Temperature (Adiabatic Reference)",
|
|
"domain_id": 29,
|
|
"significance": "\u03b8 = T (p\u2080/p)^{R_d/c_p}; R_d/c_p \u2248 0.286; conserved under adiabatic vertical displacement"
|
|
},
|
|
{
|
|
"eq_number": 580,
|
|
"title": "Brunt-V\u00e4is\u00e4l\u00e4 Frequency (Atmospheric Stability)",
|
|
"domain_id": 29,
|
|
"significance": "N\u00b2 = (g/\u03b8) d\u03b8/dz; buoyancy oscillation frequency; N\u00b2>0 \u2192 stable oscillation"
|
|
},
|
|
{
|
|
"eq_number": 581,
|
|
"title": "Geostrophic Wind (Pressure Gradient + Coriolis Balance)",
|
|
"domain_id": 29,
|
|
"significance": "u_g = \u2212(1/\u03c1f) \u2202p/\u2202y; v_g = (1/\u03c1f) \u2202p/\u2202x; wind parallel to isobars; f=2\u03a9 sin \u03c6"
|
|
},
|
|
{
|
|
"eq_number": 582,
|
|
"title": "Thermal Wind Equation (Vertical Wind Shear)",
|
|
"domain_id": 29,
|
|
"significance": "\u2202u_g/\u2202z = \u2212(g/fT) \u2202T/\u2202y; \u2202v_g/\u2202z = (g/fT) \u2202T/\u2202x; temperature gradient \u2192 wind shear"
|
|
},
|
|
{
|
|
"eq_number": 583,
|
|
"title": "Rossby Number (Inertial vs Coriolis)",
|
|
"domain_id": 29,
|
|
"significance": "Ro = U/(f L); Ro \u226a 1 \u2192 geostrophic; Ro ~ 1 \u2192 gradient wind; Ro \u226b 1 \u2192 cyclostrophic"
|
|
},
|
|
{
|
|
"eq_number": 584,
|
|
"title": "Rossby Wave Phase Speed (Planetary Waves)",
|
|
"domain_id": 29,
|
|
"significance": "c = \u016b \u2212 \u03b2/k\u00b2; \u03b2 = 2\u03a9 cos \u03c6 / R; westward phase speed relative to mean flow"
|
|
},
|
|
{
|
|
"eq_number": 585,
|
|
"title": "Clausius-Clapeyron (Water Vapor Saturation Pressure)",
|
|
"domain_id": 29,
|
|
"significance": "de_s/dT = L e_s/(R_v T\u00b2); saturated vapor pressure increases ~7%/K near surface"
|
|
},
|
|
{
|
|
"eq_number": 586,
|
|
"title": "Schwarzschild Equation (Radiative Transfer, No Scattering)",
|
|
"domain_id": 29,
|
|
"significance": "dI_\u03bd/ds = \u2212k_\u03bd \u03c1 I_\u03bd + k_\u03bd \u03c1 B_\u03bd(T); absorption + thermal emission along path"
|
|
},
|
|
{
|
|
"eq_number": 587,
|
|
"title": "K\u00f6hler Theory (Cloud Droplet Activation)",
|
|
"domain_id": 29,
|
|
"significance": "S\u22121 = A/r \u2212 B/r\u00b3; A=Kelvin term (curvature); B=solute term (Raoult effect); critical supersaturation at r*"
|
|
},
|
|
{
|
|
"eq_number": 588,
|
|
"title": "Terminal Fall Speed of Cloud/Rain Drops (Stokes/Davies)",
|
|
"domain_id": 29,
|
|
"significance": "v_t = k \u00b7 r\u00b2 (cloud droplets, r<40\u03bcm, Stokes regime); v_t = k' \u00b7 r^{0.5} (rain, r>0.6mm, turbulent)"
|
|
},
|
|
{
|
|
"eq_number": 589,
|
|
"title": "Mie Scattering (Atmospheric Aerosols, Clouds)",
|
|
"domain_id": 29,
|
|
"significance": "Q_ext, Q_sca, Q_abs as function of size parameter x=2\u03c0r/\u03bb and refractive index m"
|
|
},
|
|
{
|
|
"eq_number": 590,
|
|
"title": "Rayleigh Scattering (Molecular Atmosphere)",
|
|
"domain_id": 29,
|
|
"significance": "I_sca \u221d 1/\u03bb\u2074; cross-section \u03c3_R \u221d 1/\u03bb\u2074; blue sky, red sunsets; polarization patterns"
|
|
},
|
|
{
|
|
"eq_number": 591,
|
|
"title": "Lightning Return Stroke Current (Heidler Function / Bruce-Golde)",
|
|
"domain_id": 29,
|
|
"significance": "i(t) = (I\u2080/\u03b7)((t/\u03c4\u2081)^n/(1+(t/\u03c4\u2081)^n)) exp(\u2212t/\u03c4\u2082); I\u2080\u224810-200 kA; \u03c4\u2081\u22481-2\u03bcs, \u03c4\u2082\u224810-100\u03bcs"
|
|
},
|
|
{
|
|
"eq_number": 592,
|
|
"title": "Primitive Equations (Numerical Weather Prediction)",
|
|
"domain_id": 29,
|
|
"significance": "Conservation of momentum (3D), mass (continuity), energy (thermodynamic), moisture, and ideal gas law"
|
|
},
|
|
{
|
|
"eq_number": 593,
|
|
"title": "Nernst Equation (Membrane Equilibrium Potential)",
|
|
"domain_id": 32,
|
|
"significance": "E_ion = (RT/zF) ln([ion]_out/[ion]_in); equilibrium potential for single ion species"
|
|
},
|
|
{
|
|
"eq_number": 594,
|
|
"title": "Goldman-Hodgkin-Katz (GHK) Voltage Equation",
|
|
"domain_id": 32,
|
|
"significance": "V_m = (RT/F) ln[(P_K[K]_o+P_Na[Na]_o+P_Cl[Cl]_i)/(P_K[K]_i+P_Na[Na]_i+P_Cl[Cl]_o)]"
|
|
},
|
|
{
|
|
"eq_number": 595,
|
|
"title": "Hodgkin-Huxley Equations (Action Potential)",
|
|
"domain_id": 32,
|
|
"significance": "C_m dV/dt = \u2212g_K n\u2074(V\u2212E_K) \u2212 g_Na m\u00b3h(V\u2212E_Na) \u2212 g_L(V\u2212E_L) + I_stim"
|
|
},
|
|
{
|
|
"eq_number": 596,
|
|
"title": "FitzHugh-Nagumo Model (Simplified Excitable Dynamics)",
|
|
"domain_id": 32,
|
|
"significance": "dv/dt = v \u2212 v\u00b3/3 \u2212 w + I; dw/dt = \u03b5(v + a \u2212 bw); 2-variable reduction of HH"
|
|
},
|
|
{
|
|
"eq_number": 597,
|
|
"title": "Cable Equation (Neuronal Dendrite/Axon)",
|
|
"domain_id": 32,
|
|
"significance": "\u03bb\u00b2 \u2202\u00b2V/\u2202x\u00b2 = \u03c4_m \u2202V/\u2202t + V; \u03bb=\u221a(r_m/r_i); \u03c4_m=r_m c_m; passive spread along membrane"
|
|
},
|
|
{
|
|
"eq_number": 598,
|
|
"title": "Einstein-Smoluchowski Relation (Molecular Motor Stalling Force)",
|
|
"domain_id": 32,
|
|
"significance": "F_stall = k_B T / \u03b4; \u03b4=step size (~8 nm for kinesin); ~6 pN stall force"
|
|
},
|
|
{
|
|
"eq_number": 599,
|
|
"title": "Bell's Model (Bond Rupture Under Force)",
|
|
"domain_id": 32,
|
|
"significance": "k_off(F) = k\u2080 exp(F \u03b3/k_B T); \u03b3=reactive compliance (~0.1-0.5 nm); slip bond kinetics"
|
|
},
|
|
{
|
|
"eq_number": 600,
|
|
"title": "Hill's Equation (Muscle Force-Velocity Relation)",
|
|
"domain_id": 32,
|
|
"significance": "(F + a)(v + b) = (F\u2080 + a)b; hyperbola; v_max = F\u2080 b/a; a,b constants"
|
|
},
|
|
{
|
|
"eq_number": 601,
|
|
"title": "Huxley Sliding Filament Model (Cross-Bridge Dynamics)",
|
|
"domain_id": 32,
|
|
"significance": "\u2202n(x,t)/\u2202t = f(x)[1\u2212n(x,t)] \u2212 g(x) n(x,t); n=attached cross-bridge probability; x=distortion"
|
|
},
|
|
{
|
|
"eq_number": 602,
|
|
"title": "Monod-Wyman-Changeux (MWC) Model (Allosteric Transitions)",
|
|
"domain_id": 32,
|
|
"significance": "L = [T\u2080]/[R\u2080]; Y = \u03b1(1+\u03b1)^{n-1}/(L + (1+\u03b1)^n); \u03b1=[S]/K_R; cooperative ligand binding"
|
|
},
|
|
{
|
|
"eq_number": 603,
|
|
"title": "Michaelis-Menten Enzyme Kinetics",
|
|
"domain_id": 33,
|
|
"significance": "v = V_max [S] / (K_m + [S]); K_m = (k_{\u22121}+k_cat)/k\u2081; V_max = k_cat [E]_total"
|
|
},
|
|
{
|
|
"eq_number": 604,
|
|
"title": "Transition State Theory (Eyring Equation)",
|
|
"domain_id": 33,
|
|
"significance": "k = (k_B T/h) exp(\u2212\u0394G\u2021/RT) = (k_B T/h) exp(\u0394S\u2021/R) exp(\u2212\u0394H\u2021/RT)"
|
|
},
|
|
{
|
|
"eq_number": 605,
|
|
"title": "Arrhenius Equation (Chemical Reaction Rate)",
|
|
"domain_id": 33,
|
|
"significance": "k = A exp(\u2212E_a/RT); log\u2081\u2080(k\u2082/k\u2081) = (E_a/2.303R)(1/T\u2081\u22121/T\u2082); activation energy from T-dependence"
|
|
},
|
|
{
|
|
"eq_number": 606,
|
|
"title": "Marcus Theory (Electron Transfer Rate)",
|
|
"domain_id": 33,
|
|
"significance": "k_ET = (2\u03c0/\u210f) H_AB\u00b2 (1/\u221a(4\u03c0\u03bb k_B T)) exp[\u2212(\u0394G\u2070+\u03bb)\u00b2/(4\u03bb k_B T)]; Nobel 1992"
|
|
},
|
|
{
|
|
"eq_number": 607,
|
|
"title": "Butler-Volmer Equation (Electrode Kinetics)",
|
|
"domain_id": 33,
|
|
"significance": "j = j\u2080[exp(\u03b1_a F \u03b7/RT) \u2212 exp(\u2212\u03b1_c F \u03b7/RT)]; \u03b7=overpotential; \u03b1_a+\u03b1_c\u22481"
|
|
},
|
|
{
|
|
"eq_number": 608,
|
|
"title": "Beer-Lambert Law (Spectroscopy)",
|
|
"domain_id": 34,
|
|
"significance": "A = \u2212log\u2081\u2080(T) = \u03b5 c L; absorbance, molar absorptivity, concentration, path length"
|
|
},
|
|
{
|
|
"eq_number": 609,
|
|
"title": "F\u00f6rster Resonance Energy Transfer (FRET) Efficiency",
|
|
"domain_id": 32,
|
|
"significance": "E = R\u2080\u2076/(R\u2080\u2076+r\u2076); R\u2080\u2076 \u221d \u03ba\u00b2 \u03a6_D J(\u03bb)/n\u2074; R\u2080~1-10 nm"
|
|
},
|
|
{
|
|
"eq_number": 610,
|
|
"title": "Einstein Rate Equations (Laser Dynamics)",
|
|
"domain_id": 34,
|
|
"significance": "dN\u2082/dt = R_p \u2212 B\u2082\u2081 \u03c1(\u03bd) N\u2082 \u2212 A\u2082\u2081 N\u2082; d\u03c6/dt = B\u2082\u2081 \u03c1(\u03bd)(N\u2082\u2212N\u2081) c' \u2212 \u03c6/\u03c4_c"
|
|
},
|
|
{
|
|
"eq_number": 611,
|
|
"title": "Laser Threshold Condition",
|
|
"domain_id": 34,
|
|
"significance": "g_th = \u03b1_int + (1/2L) ln(1/R\u2081R\u2082); gain must overcome internal loss + mirror transmission"
|
|
},
|
|
{
|
|
"eq_number": 612,
|
|
"title": "Schawlow-Townes Linewidth (Fundamental Laser Linewidth)",
|
|
"domain_id": 34,
|
|
"significance": "\u0394\u03bd = (\u03c0 h\u03bd (\u0394\u03bd_c)\u00b2)/P_out; quantum-limited linewidth; narrower with higher power"
|
|
},
|
|
{
|
|
"eq_number": 613,
|
|
"title": "Mode-Locking Condition (fs/ps Pulses)",
|
|
"domain_id": 34,
|
|
"significance": "T_R = 2L/c (round-trip time); f_rep = 1/T_R; N locked modes \u2192 \u03c4_p = T_R/N \u221d 1/\u0394\u03bd_gain"
|
|
},
|
|
{
|
|
"eq_number": 614,
|
|
"title": "Nonlinear Polarization (\u03c7^{(n)} Expansion)",
|
|
"domain_id": 34,
|
|
"significance": "P_i = \u03b5\u2080[\u03c7^{(1)}_{ij} E_j + \u03c7^{(2)}_{ijk} E_j E_k + \u03c7^{(3)}_{ijkl} E_j E_k E_l + ...]"
|
|
},
|
|
{
|
|
"eq_number": 615,
|
|
"title": "Phase-Matching Condition (Nonlinear Optics)",
|
|
"domain_id": 34,
|
|
"significance": "\u0394k = k_3 \u2212 k_2 \u2212 k_1 = 0 for SHG; n(2\u03c9)=n(\u03c9) required; birefringent or QPM"
|
|
},
|
|
{
|
|
"eq_number": 616,
|
|
"title": "Nonlinear Schr\u00f6dinger Equation (Optical Solitons in Fibers)",
|
|
"domain_id": 34,
|
|
"significance": "i \u2202A/\u2202z \u2212 (\u03b2\u2082/2)\u2202\u00b2A/\u2202t\u00b2 + \u03b3|A|\u00b2A = 0; balance GVD (\u03b2\u2082) and Kerr nonlinearity (\u03b3)"
|
|
},
|
|
{
|
|
"eq_number": 617,
|
|
"title": "Kramers-Kronig Relations (Optical Dispersion)",
|
|
"domain_id": 34,
|
|
"significance": "n(\u03c9)\u22121 = (2/\u03c0)P\u222b\u2080^\u221e \u03c9'\u03ba(\u03c9')/(\u03c9'\u00b2\u2212\u03c9\u00b2)d\u03c9'; causality \u2192 real and imaginary parts of \u03c7 linked"
|
|
},
|
|
{
|
|
"eq_number": 618,
|
|
"title": "Rate Equations for Semiconductor Lasers",
|
|
"domain_id": 34,
|
|
"significance": "dN/dt = \u03b7_i I/qV \u2212 R(N) \u2212 v_g g(N) N_ph; dN_ph/dt = \u0393 v_g g(N) N_ph \u2212 N_ph/\u03c4_ph + \u03b2_sp R_sp"
|
|
},
|
|
{
|
|
"eq_number": 619,
|
|
"title": "Master Equation for Mode-Locked Lasers (Haus)",
|
|
"domain_id": 34,
|
|
"significance": "\u0394A = (g\u2212l + jD) A + (g/\u03a9_g\u00b2 + jD_g) \u2202\u00b2A/\u2202t\u00b2 + (\u03b3\u2212j\u03b4)|A|\u00b2A; Haus master equation"
|
|
},
|
|
{
|
|
"eq_number": 620,
|
|
"title": "Coupled-Mode Theory (Waveguides, Gratings, Resonators)",
|
|
"domain_id": 34,
|
|
"significance": "da_\u03bc/dz = \u2212j \u03a3_\u03ba K_\u03bc\u03ba a_\u03ba exp[j(\u03b2_\u03ba\u2212\u03b2_\u03bc)z]; coupling between waveguide/grating modes"
|
|
},
|
|
{
|
|
"eq_number": 621,
|
|
"title": "Zeeman Effect (Normal + Anomalous)",
|
|
"domain_id": 35,
|
|
"significance": "\u0394E = \u03bc_B g_J m_J B; g_J = 1 + [J(J+1)+S(S+1)\u2212L(L+1)]/[2J(J+1)]; Land\u00e9 g-factor"
|
|
},
|
|
{
|
|
"eq_number": 622,
|
|
"title": "Stark Effect (Linear + Quadratic)",
|
|
"domain_id": 35,
|
|
"significance": "Linear: \u0394E = 3ea\u2080 n (n\u2081\u2212n\u2082) E / 2 (Hydrogen); Quadratic: \u0394E = \u2212\u00bd \u03b1 E\u00b2 (general)"
|
|
},
|
|
{
|
|
"eq_number": 623,
|
|
"title": "Hyperfine Structure (Fermi Contact Interaction)",
|
|
"domain_id": 35,
|
|
"significance": "\u0394E_HFS = (A/2) [F(F+1) \u2212 I(I+1) \u2212 J(J+1)]; A \u221d \u03bc_B \u03bc_N \u27e81/r\u00b3\u27e9 |\u03c8(0)|\u00b2"
|
|
},
|
|
{
|
|
"eq_number": 624,
|
|
"title": "Born-Oppenheimer Approximation (Molecular Hamiltonian Separation)",
|
|
"domain_id": 35,
|
|
"significance": "\u03a8(r,R) \u2248 \u03c8_e(r;R) \u03c7_N(R); electronic Schr\u00f6dinger eq at fixed nuclear geometry; then nuclear motion"
|
|
},
|
|
{
|
|
"eq_number": 625,
|
|
"title": "Franck-Condon Principle (Vibrational Transition Intensities)",
|
|
"domain_id": 35,
|
|
"significance": "I_v'v'' \u221d |\u222b \u03c8_v'* \u03c8_v'' dR|\u00b2; vertical transitions; overlap of vibrational wavefunctions"
|
|
},
|
|
{
|
|
"eq_number": 626,
|
|
"title": "Molecular Rotational Spectroscopy (Rigid Rotor)",
|
|
"domain_id": 35,
|
|
"significance": "E_J = B J(J+1); B = \u210f\u00b2/(2I); I = \u03bc R\u00b2; \u0394J = \u00b11 selection rule \u2192 2B spacing"
|
|
},
|
|
{
|
|
"eq_number": 627,
|
|
"title": "Molecular Vibrational Spectroscopy (Harmonic)",
|
|
"domain_id": 35,
|
|
"significance": "E_v = \u210f\u03c9(v+\u00bd); \u03c9 = \u221a(k/\u03bc); fundamental transition \u03bd\u2080 = \u03c9/(2\u03c0c)"
|
|
},
|
|
{
|
|
"eq_number": 628,
|
|
"title": "Morse Potential (Anharmonic Diatomic)",
|
|
"domain_id": 35,
|
|
"significance": "V(r) = D_e [1 \u2212 e^{\u2212a(r\u2212r_e)}]\u00b2; analytical eigenvalues E_v = \u210f\u03c9(v+\u00bd)\u2212\u210f\u03c9x_e(v+\u00bd)\u00b2"
|
|
},
|
|
{
|
|
"eq_number": 629,
|
|
"title": "Rydberg Formula (Atomic Series Limits)",
|
|
"domain_id": 35,
|
|
"significance": "1/\u03bb = R\u221e/(n\u2081+\u03b4\u2081)\u00b2 \u2212 R\u221e/(n\u2082+\u03b4\u2082)\u00b2; R\u221e=10973731.568157 m\u207b\u00b9; \u03b4=quantum defect"
|
|
},
|
|
{
|
|
"eq_number": 630,
|
|
"title": "Racah Algebra (Angular Momentum Coupling in Complex Atoms)",
|
|
"domain_id": 35,
|
|
"significance": "Wigner 3-j, 6-j, 9-j symbols; recoupling coefficients; matrix elements of tensor operators"
|
|
},
|
|
{
|
|
"eq_number": 631,
|
|
"title": "Newtonian Constitutive Equation (Viscous Fluid)",
|
|
"domain_id": 36,
|
|
"significance": "\u03c4 = \u03bc \u03b3\u0307; \u03c3 = \u2212p I + 2 \u03bc D; D = strain-rate tensor; \u03c4 \u221d shear rate linearly"
|
|
},
|
|
{
|
|
"eq_number": 632,
|
|
"title": "Power-Law Fluid (Ostwald-de Waele Model)",
|
|
"domain_id": 36,
|
|
"significance": "\u03c4 = K \u03b3\u0307^n; \u03b7_app = K \u03b3\u0307^{n-1}; n<1 \u2192 shear-thinning (pseudoplastic); n>1 \u2192 shear-thickening; n=1 \u2192 Newtonian"
|
|
},
|
|
{
|
|
"eq_number": 633,
|
|
"title": "Bingham Plastic (Yield Stress Fluid)",
|
|
"domain_id": 36,
|
|
"significance": "\u03c4 = \u03c4_y + \u03bc_p \u03b3\u0307 for \u03c4 > \u03c4_y; no flow for \u03c4 < \u03c4_y"
|
|
},
|
|
{
|
|
"eq_number": 634,
|
|
"title": "Herschel-Bulkley Model (Yield + Power-Law)",
|
|
"domain_id": 36,
|
|
"significance": "\u03c4 = \u03c4_y + K \u03b3\u0307^n for \u03c4 > \u03c4_y"
|
|
},
|
|
{
|
|
"eq_number": 635,
|
|
"title": "Carreau-Yasuda Model (Shear-Thinning With Zero/Infinite Limits)",
|
|
"domain_id": 36,
|
|
"significance": "\u03b7(\u03b3\u0307) = \u03b7_\u221e + (\u03b7\u2080\u2212\u03b7_\u221e)[1 + (\u03bb \u03b3\u0307)^a]^{(n-1)/a}"
|
|
},
|
|
{
|
|
"eq_number": 636,
|
|
"title": "Maxwell Viscoelastic Model (Liquid)",
|
|
"domain_id": 36,
|
|
"significance": "d\u03b5/dt = (1/E) d\u03c3/dt + \u03c3/\u03b7; relaxation time \u03c4 = \u03b7/E; elastic at short times, viscous at long"
|
|
},
|
|
{
|
|
"eq_number": 637,
|
|
"title": "Kelvin-Voigt Model (Viscoelastic Solid)",
|
|
"domain_id": 36,
|
|
"significance": "\u03c3 = E \u03b5 + \u03b7 d\u03b5/dt; retardation time = \u03b7/E; creep compliance J(t)=[1\u2212e^{\u2212t/\u03c4}]/E"
|
|
},
|
|
{
|
|
"eq_number": 638,
|
|
"title": "Generalized Maxwell / Wiechert Model (Multiple Relaxation Times)",
|
|
"domain_id": 36,
|
|
"significance": "G(t) = G_\u221e + \u03a3_i G_i exp(\u2212t/\u03c4_i); relaxation spectrum H(\u03c4); Prony series"
|
|
},
|
|
{
|
|
"eq_number": 639,
|
|
"title": "Cox-Merz Rule (Steady vs Dynamic Viscosity Equivalence)",
|
|
"domain_id": 36,
|
|
"significance": "\u03b7(\u03b3\u0307) \u2248 |\u03b7*(\u03c9)| when \u03b3\u0307 = \u03c9; empirical equivalence for many polymer melts/solutions"
|
|
},
|
|
{
|
|
"eq_number": 640,
|
|
"title": "Trouton Ratio (Extensional/Shear Viscosity Ratio)",
|
|
"domain_id": 36,
|
|
"significance": "Tr = \u03b7_E / \u03b7; Newtonian: Tr=3; viscoelastic: Tr\u226b3; strain-hardening in extension"
|
|
},
|
|
{
|
|
"eq_number": 641,
|
|
"title": "Amontons-Coulomb Friction Laws",
|
|
"domain_id": 37,
|
|
"significance": "F_f = \u03bc N (macroscopic); independent of apparent contact area and sliding speed (approximately)"
|
|
},
|
|
{
|
|
"eq_number": 642,
|
|
"title": "Archard's Law (Adhesive Wear)",
|
|
"domain_id": 37,
|
|
"significance": "V = k F s / H; k=wear coefficient (~10^{-2} to 10^{-7}); softer material hardness controls"
|
|
},
|
|
{
|
|
"eq_number": 643,
|
|
"title": "Stribeck Curve (Lubrication Regimes)",
|
|
"domain_id": 37,
|
|
"significance": "\u03bc = f(\u03b7 N/p, roughness, geometry); boundary \u2192 mixed \u2192 EHL \u2192 hydrodynamic as speed increases"
|
|
},
|
|
{
|
|
"eq_number": 644,
|
|
"title": "Reynolds Equation (Thin-Film Lubrication)",
|
|
"domain_id": 37,
|
|
"significance": "\u2202/\u2202x[(h\u00b3/\u03b7)\u2202p/\u2202x] + \u2202/\u2202y[(h\u00b3/\u03b7)\u2202p/\u2202y] = 6(U\u2202h/\u2202x + 2\u2202h/\u2202t)"
|
|
},
|
|
{
|
|
"eq_number": 645,
|
|
"title": "Hertzian Contact (Elastic Contact Between Curved Surfaces)",
|
|
"domain_id": 18,
|
|
"significance": "a = (3FR/4E*)^{1/3}; p_max = 3F/(2\u03c0a\u00b2); E* = [(1\u2212\u03bd\u2081\u00b2)/E\u2081+(1\u2212\u03bd\u2082\u00b2)/E\u2082]\u207b\u00b9"
|
|
},
|
|
{
|
|
"eq_number": 646,
|
|
"title": "Elastohydrodynamic Lubrication (EHL) Film Thickness (Hamrock-Dowson)",
|
|
"domain_id": 37,
|
|
"significance": "h_min/R_x = 3.63 U\u2070\u00b7\u2076\u2078 G\u2070\u00b7\u2074\u2079 W\u207b\u2070\u00b7\u2070\u2077\u00b3 (1\u2212e^{\u22120.68k}); U=\u03b7\u2080u/E'R_x, G=\u03b1E', W=w/E'R_x\u00b2"
|
|
},
|
|
{
|
|
"eq_number": 647,
|
|
"title": "Janssen Effect (Pressure Saturation in Silos)",
|
|
"domain_id": 38,
|
|
"significance": "p(z) = (\u03c1 g D / 4 \u03bc_w K) [1 \u2212 exp(\u22124 \u03bc_w K z/D)]; pressure saturates at finite depth"
|
|
},
|
|
{
|
|
"eq_number": 648,
|
|
"title": "Coulomb Yield Criterion (Granular Failure)",
|
|
"domain_id": 38,
|
|
"significance": "\u03c4 = \u03c3 tan \u03c6 + c; \u03c6=internal friction angle (~25-45\u00b0 for sands); c=cohesion (0 for dry sand)"
|
|
},
|
|
{
|
|
"eq_number": 649,
|
|
"title": "Angle of Repose (Granular Pile)",
|
|
"domain_id": 38,
|
|
"significance": "tan \u03c6_r = H_max / R; \u03c6_r \u2248 \u03c6 (internal friction angle); ~30-40\u00b0 for most granular materials"
|
|
},
|
|
{
|
|
"eq_number": 650,
|
|
"title": "Brazil Nut Effect (Granular Convection/Segregation)",
|
|
"domain_id": 38,
|
|
"significance": "Larger particles rise during vibration or shaking due to percolation + convection"
|
|
},
|
|
{
|
|
"eq_number": 651,
|
|
"title": "Bagnold Scaling (Granular Flow Rheology - Inertial)",
|
|
"domain_id": 38,
|
|
"significance": "\u03c4 = a (\u03c1_p d\u00b2) \u03b3\u0307\u00b2 (inertial regime); Bagnold number Ba = \u03c1_p d\u00b2 \u03b3\u0307/\u03b7_f; Ba>450 \u2192 grain inertia dominates"
|
|
},
|
|
{
|
|
"eq_number": 652,
|
|
"title": "\u03bc(I) Rheology (Inertial Number Scaling for Dense Granular Flow)",
|
|
"domain_id": 38,
|
|
"significance": "\u03bc(I) = \u03bc_s + (\u03bc\u2082\u2212\u03bc_s)/(1+I\u2080/I); I = \u03b3\u0307 d/\u221a(p/\u03c1_p); dimensionless inertial number"
|
|
},
|
|
{
|
|
"eq_number": 653,
|
|
"title": "Coulomb Blockade Condition (Single-Electron Transistor)",
|
|
"domain_id": 39,
|
|
"significance": "E_c = e\u00b2/(2C_\u03a3) > k_B T; charging energy must exceed thermal energy for CB to be observed"
|
|
},
|
|
{
|
|
"eq_number": 654,
|
|
"title": "Landauer Formula (Ballistic Conductance)",
|
|
"domain_id": 39,
|
|
"significance": "G = (2e\u00b2/h) \u03a3 T_n; G\u2080 = 2e\u00b2/h \u2248 77.5 \u03bcS (~12.9 k\u03a9); quantized conductance"
|
|
},
|
|
{
|
|
"eq_number": 655,
|
|
"title": "Kondo Effect (Resistance Minimum in Dilute Magnetic Alloys)",
|
|
"domain_id": 39,
|
|
"significance": "R \u221d \u2212ln(T) below Kondo temperature T_K; magnetic impurity spin screened by conduction electrons"
|
|
},
|
|
{
|
|
"eq_number": 656,
|
|
"title": "2D Electron Gas Density of States (Constant)",
|
|
"domain_id": 39,
|
|
"significance": "g_{2D}(E) = m*/(\u03c0\u210f\u00b2) = constant; independent of energy"
|
|
},
|
|
{
|
|
"eq_number": 657,
|
|
"title": "Graphene Dirac Dispersion (Massless 2D Fermions)",
|
|
"domain_id": 39,
|
|
"significance": "E = \u00b1 v_F |k|; v_F \u2248 10\u2076 m/s; linear dispersion near Dirac points K,K'"
|
|
},
|
|
{
|
|
"eq_number": 658,
|
|
"title": "Quantum Confinement (Infinite Well \u2014 Nanowire/Quantum Well)",
|
|
"domain_id": 39,
|
|
"significance": "E_n = n\u00b2\u03c0\u00b2\u210f\u00b2/(2m*L\u00b2); 1D wire; 2D well adds E_{n_x,n_y} terms; 0D dot adds all three"
|
|
},
|
|
{
|
|
"eq_number": 659,
|
|
"title": "Casimir Force (Between Ideal Plates, Nanoscale)",
|
|
"domain_id": 39,
|
|
"significance": "F/A = \u2212\u03c0\u00b2\u210fc/(240 d\u2074); d=separation; attractive; zero-point EM fluctuations"
|
|
},
|
|
{
|
|
"eq_number": 660,
|
|
"title": "DLVO Theory (Colloidal Nanoparticle Stability)",
|
|
"domain_id": 39,
|
|
"significance": "V_total(d) = V_vdW + V_EDL; van der Waals attraction + electric double-layer repulsion"
|
|
},
|
|
{
|
|
"eq_number": 661,
|
|
"title": "Single Qubit State (Bloch Sphere)",
|
|
"domain_id": 40,
|
|
"significance": "|\u03c8\u27e9 = cos(\u03b8/2)|0\u27e9 + e^{i\u03c6} sin(\u03b8/2)|1\u27e9; pure state on Bloch sphere surface"
|
|
},
|
|
{
|
|
"eq_number": 662,
|
|
"title": "Bell States (Maximally Entangled Two-Qubit States)",
|
|
"domain_id": 40,
|
|
"significance": "|\u03a6\u207a\u27e9=(|00\u27e9+|11\u27e9)/\u221a2; |\u03a6\u207b\u27e9=(|00\u27e9\u2212|11\u27e9)/\u221a2; |\u03a8\u207a\u27e9=(|01\u27e9+|10\u27e9)/\u221a2; |\u03a8\u207b\u27e9=(|01\u27e9\u2212|10\u27e9)/\u221a2"
|
|
},
|
|
{
|
|
"eq_number": 663,
|
|
"title": "No-Cloning Theorem",
|
|
"domain_id": 40,
|
|
"significance": "An unknown quantum state cannot be copied perfectly; U|\u03c8\u27e9|0\u27e9 \u2260 |\u03c8\u27e9|\u03c8\u27e9 for all |\u03c8\u27e9"
|
|
},
|
|
{
|
|
"eq_number": 664,
|
|
"title": "Holevo Bound (Classical Information From Qubit)",
|
|
"domain_id": 40,
|
|
"significance": "\u03c7 \u2264 S(\u03c1) \u2212 \u03a3 p_i S(\u03c1_i); at most 1 classical bit extractable per qubit"
|
|
},
|
|
{
|
|
"eq_number": 665,
|
|
"title": "Deutsch-Jozsa Algorithm Speedup",
|
|
"domain_id": 40,
|
|
"significance": "Single-query solution to balanced/constant problem; first clear quantum advantage"
|
|
},
|
|
{
|
|
"eq_number": 666,
|
|
"title": "Grover's Search Algorithm (Quadratic Speedup)",
|
|
"domain_id": 40,
|
|
"significance": "O(\u221aN) quantum search via amplitude amplification; ~\u03c0\u221aN/4 Grover iterations"
|
|
},
|
|
{
|
|
"eq_number": 667,
|
|
"title": "Shor's Factoring Algorithm",
|
|
"domain_id": 40,
|
|
"significance": "Quantum period-finding via QFT \u2192 polynomial-time integer factorization; O((log N)\u00b3); exponentially faster than classical best known"
|
|
},
|
|
{
|
|
"eq_number": 668,
|
|
"title": "Concatenated Quantum Error Correction Threshold Theorem",
|
|
"domain_id": 40,
|
|
"significance": "If gate error < p_th (~10\u207b\u00b2 to 10\u207b\u2074 depending on code), errors can be arbitrarily suppressed"
|
|
},
|
|
{
|
|
"eq_number": 669,
|
|
"title": "Lorenz Equations (Deterministic Chaos)",
|
|
"domain_id": 41,
|
|
"significance": "\u1e8b = \u03c3(y\u2212x); \u1e8f = x(\u03c1\u2212z)\u2212y; \u017c = xy\u2212\u03b2 z; \u03c3=10, \u03b2=8/3, \u03c1=28 \u2192 strange attractor"
|
|
},
|
|
{
|
|
"eq_number": 670,
|
|
"title": "Logistic Map (Period-Doubling Route to Chaos)",
|
|
"domain_id": 41,
|
|
"significance": "x_{n+1} = r x_n (1\u2212x_n); period-doubling bifurcations; chaos at r\u22483.57; Feigenbaum universality"
|
|
},
|
|
{
|
|
"eq_number": 671,
|
|
"title": "Lyapunov Exponent (Chaos Diagnostic)",
|
|
"domain_id": 41,
|
|
"significance": "\u03bb = lim_{t\u2192\u221e} (1/t) ln |\u03b4x(t)/\u03b4x(0)|; \u03bb>0 \u2192 chaos; \u03bb<0 \u2192 stable; \u03bb=0 \u2192 marginal"
|
|
},
|
|
{
|
|
"eq_number": 672,
|
|
"title": "KAM Theorem (Kolmogorov-Arnold-Moser)",
|
|
"domain_id": 41,
|
|
"significance": "Most invariant tori survive small perturbations if frequency ratio is sufficiently irrational"
|
|
},
|
|
{
|
|
"eq_number": 673,
|
|
"title": "Kuramoto Model (Synchronization of Coupled Oscillators)",
|
|
"domain_id": 41,
|
|
"significance": "\u03b8\u0307_i = \u03c9_i + (K/N) \u03a3_j sin(\u03b8_j\u2212\u03b8_i); K>K_c \u2192 phase transition to global synchronization"
|
|
},
|
|
{
|
|
"eq_number": 674,
|
|
"title": "Mandelbrot Set (Fractal Geometry)",
|
|
"domain_id": 41,
|
|
"significance": "z_{n+1} = z_n\u00b2 + c; bounded orbits \u2192 c \u2208 Mandelbrot set; fractal boundary with infinite complexity"
|
|
},
|
|
{
|
|
"eq_number": 675,
|
|
"title": "Bloch Equations (NMR/MRI Signal)",
|
|
"domain_id": 42,
|
|
"significance": "dM/dt = \u03b3 M \u00d7 B \u2212 (M_x i\u0302+M_y j\u0302)/T\u2082 \u2212 (M_z\u2212M\u2080)k\u0302/T\u2081; relaxation toward equilibrium"
|
|
},
|
|
{
|
|
"eq_number": 676,
|
|
"title": "Larmor Frequency (NMR Precession)",
|
|
"domain_id": 42,
|
|
"significance": "\u03c9\u2080 = \u03b3 B\u2080; \u03b3_H/2\u03c0 = 42.577 MHz/T; proton Larmor frequency in clinical MRI"
|
|
},
|
|
{
|
|
"eq_number": 677,
|
|
"title": "Beer-Lambert Law (X-ray/\u03b3-ray Attenuation, CT)",
|
|
"domain_id": 42,
|
|
"significance": "I = I\u2080 e^{\u2212\u03bcx}; \u03bc/p = mass attenuation coefficient; CT: \u03bc(x,y) \u2192 Hounsfield units"
|
|
},
|
|
{
|
|
"eq_number": 678,
|
|
"title": "Radon Transform (CT Image Reconstruction)",
|
|
"domain_id": 42,
|
|
"significance": "p(s,\u03b8) = \u222b_{-\u221e}^\u221e f(s cos\u03b8\u2212t sin\u03b8, s sin\u03b8+t cos\u03b8) dt; projection \u2192 2D image via filtered backprojection"
|
|
},
|
|
{
|
|
"eq_number": 679,
|
|
"title": "Ultrasound Wave Equation (Medical Imaging)",
|
|
"domain_id": 42,
|
|
"significance": "\u2202\u00b2p/\u2202t\u00b2 = c\u00b2\u2207\u00b2p; reflection at tissue interfaces (acoustic impedance mismatch Z=\u03c1c)"
|
|
},
|
|
{
|
|
"eq_number": 680,
|
|
"title": "Attenuation of Ultrasound in Tissue",
|
|
"domain_id": 42,
|
|
"significance": "I(x) = I\u2080 e^{\u2212\u03b1 f^n x}; n\u22481 for most soft tissues; \u03b1\u22480.5-1 dB/(cm\u00b7MHz)"
|
|
},
|
|
{
|
|
"eq_number": 681,
|
|
"title": "Linear-Quadratic (LQ) Model (Radiation Therapy Cell Survival)",
|
|
"domain_id": 43,
|
|
"significance": "S = exp(\u2212\u03b1D \u2212 \u03b2D\u00b2); \u03b1/\u03b2 ratio ~3 Gy for late-responding (CNS, spinal); ~10 Gy for early-responding/acutely responding; D=total dose"
|
|
},
|
|
{
|
|
"eq_number": 682,
|
|
"title": "Bragg Peak (Proton/Ion Beam Depth-Dose)",
|
|
"domain_id": 43,
|
|
"significance": "dE/dx peaks sharply at end of range (Bragg peak); R \u221d E^{1.7\u22121.8}; sharp distal falloff"
|
|
},
|
|
{
|
|
"eq_number": 683,
|
|
"title": "Bethe-Bloch Formula (Stopping Power for Charged Particles)",
|
|
"domain_id": 43,
|
|
"significance": "\u2212\u27e8dE/dx\u27e9 = K z\u00b2 (Z/A)(1/\u03b2\u00b2)[\u00bd ln(2m_e c\u00b2\u03b2\u00b2\u03b3\u00b2T_max/I\u00b2) \u2212 \u03b2\u00b2 \u2212 \u03b4(\u03b2\u03b3)/2]"
|
|
},
|
|
{
|
|
"eq_number": 684,
|
|
"title": "Dosimetry: Cavity Theory (Bragg-Gray / Spencer-Attix)",
|
|
"domain_id": 43,
|
|
"significance": "D_med = (S\u0304/\u03c1)_med^wall \u00b7 D_wall; dose to medium from dose measured in wall/gas cavity"
|
|
},
|
|
{
|
|
"eq_number": 685,
|
|
"title": "MIRD Formalism (Internal Dosimetry)",
|
|
"domain_id": 43,
|
|
"significance": "D\u0304(r_T\u2190r_S) = \u00c3_S \u03a3_i \u0394_i \u03c6_i(r_T\u2190r_S); \u00c3_S=cumulated activity; \u0394_i=mean energy per transition; \u03c6=fraction absorbed"
|
|
},
|
|
{
|
|
"eq_number": 686,
|
|
"title": "Shockley-Queisser Limit (Single-Junction Solar Cell Efficiency)",
|
|
"domain_id": 44,
|
|
"significance": "\u03b7_max \u2248 33.7% for E_g=1.34 eV under AM1.5 spectrum (non-concentrated); detailed balance limit"
|
|
},
|
|
{
|
|
"eq_number": 687,
|
|
"title": "Solar Cell I-V Characteristic (One-Diode Model)",
|
|
"domain_id": 44,
|
|
"significance": "I = I_ph \u2212 I\u2080 [exp(q(V+IR_s)/nk_B T)\u22121] \u2212 (V+IR_s)/R_sh"
|
|
},
|
|
{
|
|
"eq_number": 688,
|
|
"title": "Fill Factor (Solar Cell)",
|
|
"domain_id": 44,
|
|
"significance": "FF = (V_mpp I_mpp)/(V_oc I_sc); \u03b7 = P_max/P_in = FF \u00b7 V_oc \u00b7 J_sc / P_in"
|
|
},
|
|
{
|
|
"eq_number": 689,
|
|
"title": "Betz Limit (Wind Turbine Maximum Efficiency)",
|
|
"domain_id": 44,
|
|
"significance": "C_p_max = 16/27 \u2248 59.3%; maximum fraction of kinetic power extractable from wind"
|
|
},
|
|
{
|
|
"eq_number": 690,
|
|
"title": "Rankine Cycle Efficiency (Steam Power Plant)",
|
|
"domain_id": 44,
|
|
"significance": "\u03b7 = (W_turbine \u2212 W_pump)/Q_in \u2248 1 \u2212 T_c/T_h (ideal Carnot upper bound, real ~30-45%)"
|
|
},
|
|
{
|
|
"eq_number": 691,
|
|
"title": "Brayton Cycle Efficiency (Gas Turbine / Jet Engine)",
|
|
"domain_id": 44,
|
|
"significance": "\u03b7 = 1 \u2212 1/r_p^{(\u03b3\u22121)/\u03b3}; r_p = compressor pressure ratio; \u03b3 = c_p/c_v"
|
|
},
|
|
{
|
|
"eq_number": 692,
|
|
"title": "Nernst Equation (Fuel Cell Open-Circuit Voltage)",
|
|
"domain_id": 44,
|
|
"significance": "E_rev = \u2212\u0394G/(nF); H\u2082/O\u2082 Fuel Cell: E\u2070 = 1.229 V at 25\u00b0C; actual: E = E_rev \u2212 \u03b7_act \u2212 \u03b7_ohm \u2212 \u03b7_conc"
|
|
},
|
|
{
|
|
"eq_number": 693,
|
|
"title": "Parker Spiral (Interplanetary Magnetic Field)",
|
|
"domain_id": 45,
|
|
"significance": "B_r \u221d 1/r\u00b2; B_\u03c6 = \u2212B_r (\u03a9 r sin \u03b8)/v_SW; Archimedean spiral angle; v_SW\u2248400 km/s (slow), ~750 km/s (fast)"
|
|
},
|
|
{
|
|
"eq_number": 694,
|
|
"title": "Chapman-Ferraro Model (Magnetopause Standoff Distance)",
|
|
"domain_id": 45,
|
|
"significance": "Balance of solar wind dynamic pressure with Earth's magnetic pressure: R_MP ~ 10 R_E (subsolar)"
|
|
},
|
|
{
|
|
"eq_number": 695,
|
|
"title": "Alfv\u00e9n Mach Number (Solar Wind \u2014 Magnetosphere Coupling)",
|
|
"domain_id": 45,
|
|
"significance": "M_A = v_SW / v_A; M_A typical solar wind ~5-10; super-Alfv\u00e9nic flow \u2192 bow shock forms"
|
|
},
|
|
{
|
|
"eq_number": 696,
|
|
"title": "Dungey Cycle (Magnetospheric Convection via Reconnection)",
|
|
"domain_id": 45,
|
|
"significance": "Open flux transport from dayside reconnection \u2192 tail lobe \u2192 nightside reconnection \u2192 return flow (2-cell convection)"
|
|
},
|
|
{
|
|
"eq_number": 697,
|
|
"title": "St\u00f8rmer Theory (Charged Particle Motion in Dipole Field)",
|
|
"domain_id": 45,
|
|
"significance": "Allowed/forbidden zones for cosmic ray access; rigidity cutoff P_c = 59.6 cos\u2074 \u03bb / r\u00b2 (GV, dipole approx.)"
|
|
},
|
|
{
|
|
"eq_number": 698,
|
|
"title": "Radiation Belt Diffusion Equation (Fokker-Planck Approach)",
|
|
"domain_id": 45,
|
|
"significance": "\u2202f/\u2202t = L\u00b2 \u2202/\u2202L (D_LL L^{-2} \u2202f/\u2202L) + radial diffusion + sources (CRAND, injections) + losses (wave-particle, atmospheric)"
|
|
},
|
|
{
|
|
"eq_number": 699,
|
|
"title": "Auroral Electron Acceleration (Knight Relation)",
|
|
"domain_id": 45,
|
|
"significance": "j_\u2225 = K (V \u2212 V_c); K = field-aligned conductance; V_c = critical voltage; parallel potential drop above aurora"
|
|
},
|
|
{
|
|
"eq_number": 700,
|
|
"title": "Chapman-Jouguet (CJ) Detonation Theory",
|
|
"domain_id": 46,
|
|
"significance": "Detonation products at sonic condition relative to shock front (M=1); Rayleigh line tangent to Hugoniot at CJ point"
|
|
},
|
|
{
|
|
"eq_number": 701,
|
|
"title": "ZND Model (Zeldovich-Von Neumann-D\u00f6ring Structure)",
|
|
"domain_id": 46,
|
|
"significance": "Lead shock \u2192 von Neumann spike (induction zone, no reaction) \u2192 reaction zone \u2192 CJ plane"
|
|
},
|
|
{
|
|
"eq_number": 702,
|
|
"title": "Rankine-Hugoniot Relations (General Shock Jump Conditions)",
|
|
"domain_id": 46,
|
|
"significance": "\u03c1\u2081 u\u2081 = \u03c1\u2082 u\u2082; p\u2081+\u03c1\u2081u\u2081\u00b2 = p\u2082+\u03c1\u2082u\u2082\u00b2; h\u2081+\u00bdu\u2081\u00b2 = h\u2082+\u00bdu\u2082\u00b2; conservation across any shock or detonation front"
|
|
},
|
|
{
|
|
"eq_number": 703,
|
|
"title": "Mie-Gr\u00fcneisen Equation of State (Solids Under Shock)",
|
|
"domain_id": 46,
|
|
"significance": "p(V,E) = p_ref(V) + (\u03b3(V)/V)[E \u2212 E_ref(V)]; \u03b3(V)/V = Gr\u00fcneisen parameter / volume"
|
|
},
|
|
{
|
|
"eq_number": 704,
|
|
"title": "Hopkinson-Cranz (Cube-Root) Blast Scaling Law",
|
|
"domain_id": 46,
|
|
"significance": "R\u2081/R\u2082 = (W\u2081/W\u2082)^{1/3} at equal overpressure; scaled distance Z = R/W^{1/3}"
|
|
},
|
|
{
|
|
"eq_number": 705,
|
|
"title": "Taylor-Sedov Blast Wave (Point Explosion, Self-Similar Solution)",
|
|
"domain_id": 46,
|
|
"significance": "R(t) = \u03be\u2080 (E/\u03c1\u2080)^{1/5} t^{2/5} (strong shock, spherical); nuclear fireball radius"
|
|
},
|
|
{
|
|
"eq_number": 706,
|
|
"title": "Veselago's Left-Handed Material Condition",
|
|
"domain_id": 47,
|
|
"significance": "\u03b5 < 0 and \u03bc < 0 simultaneously \u2192 \u00f1 < 0 (negative index); reversed Snell's law, reversed Doppler, reversed Cherenkov"
|
|
},
|
|
{
|
|
"eq_number": 707,
|
|
"title": "Pendry's Perfect Lens (Subwavelength Imaging)",
|
|
"domain_id": 47,
|
|
"significance": "n = \u22121, \u03bc = \u03b5 = \u22121 \u2192 amplifies evanescent waves \u2192 subwavelength resolution; no diffraction limit"
|
|
},
|
|
{
|
|
"eq_number": 708,
|
|
"title": "Transformation Optics (Cloaking / Invisibility)",
|
|
"domain_id": 47,
|
|
"significance": "g'^{\u03bc\u03bd} = \u039b^\u03bc_\u03b1 \u039b^\u03bd_\u03b2 g^{\u03b1\u03b2} where \u039b relates virtual to physical space; coordinate transformation \u2192 anisotropic \u03b5,\u03bc"
|
|
},
|
|
{
|
|
"eq_number": 709,
|
|
"title": "Effective Medium Theory (Maxwell Garnett / Bruggeman)",
|
|
"domain_id": 47,
|
|
"significance": "MG: (\u03b5_eff\u2212\u03b5_h)/(\u03b5_eff+2\u03b5_h) = f (\u03b5_i\u2212\u03b5_h)/(\u03b5_i+2\u03b5_h); Bruggeman: f(\u03b5_i\u2212\u03b5_eff)/(\u03b5_i+2\u03b5_eff) + (1\u2212f)(\u03b5_h\u2212\u03b5_eff)/(\u03b5_h+2\u03b5_eff)=0"
|
|
},
|
|
{
|
|
"eq_number": 710,
|
|
"title": "NTU-Effectiveness Method (Heat Exchanger Design)",
|
|
"domain_id": 49,
|
|
"significance": "\u03b5 = Q/Q_max; NTU = UA/C_min; \u03b5 = f(NTU, C_r, flow arrangement); C_r = C_min/C_max"
|
|
},
|
|
{
|
|
"eq_number": 711,
|
|
"title": "Natural Frequency of a Cantilever Beam",
|
|
"domain_id": 49,
|
|
"significance": "f_n = (\u03b2_n L)\u00b2/(2\u03c0 L\u00b2) \u221a(EI/\u03c1A); \u03b2\u2081L=1.875, \u03b2\u2082L=4.694, \u03b2\u2083L=7.855 for cantilever"
|
|
},
|
|
{
|
|
"eq_number": 712,
|
|
"title": "PID Control Law (Feedback Control)",
|
|
"domain_id": 49,
|
|
"significance": "u(t) = K_p e(t) + K_i \u222b\u2080\u1d57 e(\u03c4)d\u03c4 + K_d de/dt; e(t)=setpoint \u2212 measurement"
|
|
},
|
|
{
|
|
"eq_number": 713,
|
|
"title": "Nyquist Stability Criterion",
|
|
"domain_id": 49,
|
|
"significance": "N = Z \u2212 P; encirclements of \u22121 point determine closed-loop stability from open-loop transfer function"
|
|
},
|
|
{
|
|
"eq_number": 714,
|
|
"title": "Young-Laplace Equation (Capillary Pressure, Droplets/Bubbles)",
|
|
"domain_id": 25,
|
|
"significance": "\u0394p = \u03b3 (1/R\u2081 + 1/R\u2082); \u0394p = 2\u03b3/R (spherical); \u0394p = 4\u03b3/R for soap bubble (2 interfaces)"
|
|
},
|
|
{
|
|
"eq_number": 715,
|
|
"title": "Kelvin Equation (Curvature + Vapor Pressure)",
|
|
"domain_id": 25,
|
|
"significance": "ln(P/P_sat) = 2\u03b3 V_m/(r R T); concave meniscus (r<0) \u2192 condensation below P_sat"
|
|
},
|
|
{
|
|
"eq_number": 716,
|
|
"title": "Washburn Equation (Capillary Rise Dynamics)",
|
|
"domain_id": 25,
|
|
"significance": "h(t) = \u221a(\u03b3 R cos \u03b8 t/(2\u03b7)); Lucas-Washburn; \u221at dependence for capillary imbibition"
|
|
},
|
|
{
|
|
"eq_number": 717,
|
|
"title": "Poiseuille Law (Microfluidic Channel Flow)",
|
|
"domain_id": 36,
|
|
"significance": "Q = (\u0394p w h\u00b3)/(12 \u03b7 L) [1 \u2212 0.63 h/w] for rectangular channel (h\u226aw); ~h\u00b3 dependence"
|
|
},
|
|
{
|
|
"eq_number": 718,
|
|
"title": "Stribeck Curve \u2014 Empirical Friction-Speed-Load Relation",
|
|
"domain_id": 37,
|
|
"significance": "\u03bc = \u03bc_b + (\u03bc_h\u2212\u03bc_b) / [1 + (\u03b7 N/p)^m]; boundary \u2192 mixed \u2192 hydrodynamic transition"
|
|
},
|
|
{
|
|
"eq_number": 719,
|
|
"title": "Zener-Hollomon Parameter (Hot Deformation)",
|
|
"domain_id": 21,
|
|
"significance": "Z = \u03b5\u0307 exp(Q/RT); flow stress \u03c3 = f(Z); Z unifies temperature and strain-rate effects"
|
|
},
|
|
{
|
|
"eq_number": 720,
|
|
"title": "Ashby Deformation Mechanism Maps",
|
|
"domain_id": 21,
|
|
"significance": "\u03c3/G vs T/T_m plot with boundaries between plasticity, power-law creep, diffusional flow, etc."
|
|
},
|
|
{
|
|
"eq_number": 721,
|
|
"title": "Tabor Parameter (Indentation Representative Strain)",
|
|
"domain_id": 21,
|
|
"significance": "\u03b5_rep \u2248 0.2 tan \u03b2; \u03b2 = indenter angle; uniaxial stress-strain relation from hardness"
|
|
},
|
|
{
|
|
"eq_number": 722,
|
|
"title": "Lode Parameter (Stress Triaxiality for Ductile Fracture)",
|
|
"domain_id": 21,
|
|
"significance": "\u03b7 = \u03c3_m/\u03c3_v; \u03c3_m=hydrostatic stress; \u03c3_v=von Mises stress; \u03b7>1/3 needed for void growth"
|
|
},
|
|
{
|
|
"eq_number": 723,
|
|
"title": "Dislocation Density Evolution (Kocks-Mecking Model)",
|
|
"domain_id": 21,
|
|
"significance": "d\u03c1/d\u03b5 = k\u2081 \u221a\u03c1 \u2212 k\u2082 \u03c1; storage (hardening) vs dynamic recovery (annihilation); Stage II\u2192III"
|
|
},
|
|
{
|
|
"eq_number": 724,
|
|
"title": "Bauschinger Effect (Kinematic Hardening)",
|
|
"domain_id": 21,
|
|
"significance": "Yield stress in reverse loading lower than forward loading; dislocation back-stress accumulation"
|
|
},
|
|
{
|
|
"eq_number": 725,
|
|
"title": "Schottky Diode I-V (Thermionic Emission Model)",
|
|
"domain_id": 23,
|
|
"significance": "J = A* T\u00b2 exp(\u2212q\u03c6_Bn/k_B T) [exp(qV/k_B T)\u22121]; A* = Richardson constant modified for effective mass"
|
|
},
|
|
{
|
|
"eq_number": 726,
|
|
"title": "Solar Cell Quantum Efficiency (External/Internal)",
|
|
"domain_id": 44,
|
|
"significance": "EQE(\u03bb) = (J_sc(\u03bb)/q) / \u03a6(\u03bb); IQE(\u03bb) = EQE(\u03bb) / (1\u2212R(\u03bb)\u2212T(\u03bb))"
|
|
},
|
|
{
|
|
"eq_number": 727,
|
|
"title": "Detailed Balance Limit (Tandem/Multijunction Solar Cells)",
|
|
"domain_id": 44,
|
|
"significance": "\u03b7_max \u2192 45.7% (2-junction), 51.3% (3-junction), 68.2% (infinite junctions) at 1 sun; ~86.8% at max concentration"
|
|
},
|
|
{
|
|
"eq_number": 728,
|
|
"title": "Tandem Solar Cell Current-Matching Condition",
|
|
"domain_id": 44,
|
|
"significance": "J_sc_top = J_sc_bottom (series-connected); otherwise limited by lower subcell current"
|
|
},
|
|
{
|
|
"eq_number": 729,
|
|
"title": "Thermoelectric Figure of Merit (ZT)",
|
|
"domain_id": 44,
|
|
"significance": "ZT = S\u00b2 \u03c3 T / \u03ba; S=Seebeck coefficient; \u03c3=electrical conductivity; \u03ba=thermal conductivity"
|
|
},
|
|
{
|
|
"eq_number": 730,
|
|
"title": "Seebeck Effect (Thermoelectric Voltage)",
|
|
"domain_id": 44,
|
|
"significance": "\u0394V = \u2212\u222b S(T) dT; V_oc = S \u0394T for small \u0394T; S\u221dk_B/e (\u224886 \u03bcV/K per k_B/e)"
|
|
},
|
|
{
|
|
"eq_number": 731,
|
|
"title": "Peltier Effect (Thermoelectric Heat Pumping)",
|
|
"domain_id": 44,
|
|
"significance": "Q\u0307 = \u03a0 I; \u03a0 = S T (Kelvin relation); heat absorbed/released at junction"
|
|
},
|
|
{
|
|
"eq_number": 732,
|
|
"title": "Electrochemical Overpotential Components (Fuel Cell/Battery)",
|
|
"domain_id": 44,
|
|
"significance": "V_cell = E_rev \u2212 \u03b7_act \u2212 \u03b7_ohm \u2212 \u03b7_conc; \u03b7_act from Butler-Volmer; \u03b7_ohm=IR; \u03b7_conc=(RT/nF)ln(1\u2212j/j_L)"
|
|
},
|
|
{
|
|
"eq_number": 733,
|
|
"title": "Peukert's Law (Battery Capacity vs Discharge Rate)",
|
|
"domain_id": 44,
|
|
"significance": "C = I^k t (k>1 for lead-acid, k\u22481.1-1.3); capacity decreases at higher discharge rates"
|
|
},
|
|
{
|
|
"eq_number": 734,
|
|
"title": "Ragone Plot (Energy vs Power Density)",
|
|
"domain_id": 44,
|
|
"significance": "Specific energy (Wh/kg) vs specific power (W/kg); batteries, fuel cells, capacitors, flywheels occupy different regions"
|
|
},
|
|
{
|
|
"eq_number": 735,
|
|
"title": "Magnetorheological Fluid (Bingham Plastic with Field-Dependent Yield)",
|
|
"domain_id": 36,
|
|
"significance": "\u03c4 = \u03c4_y(B) + \u03b7 \u03b3\u0307; yield stress controllable via applied magnetic field; ~50-100 kPa max"
|
|
},
|
|
{
|
|
"eq_number": 736,
|
|
"title": "Electrorheological Fluid (Field-Dependent Viscosity)",
|
|
"domain_id": 36,
|
|
"significance": "\u03b7(E) = \u03b7\u2080 + \u03b1 E^n; viscosity increases with electric field; Winslow effect"
|
|
},
|
|
{
|
|
"eq_number": 737,
|
|
"title": "Phononic Crystal Band Gap (Bragg Scattering of Sound)",
|
|
"domain_id": 47,
|
|
"significance": "\u03a9(k+G)=\u03a9(k); periodic elastic constants \u2192 band gaps for acoustic/elastic waves"
|
|
},
|
|
{
|
|
"eq_number": 738,
|
|
"title": "NSM Semantic Primes Explication",
|
|
"domain_id": 51,
|
|
"significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
|
|
},
|
|
{
|
|
"eq_number": 738,
|
|
"title": "NSM Semantic Primes Explication",
|
|
"domain_id": 51,
|
|
"significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
|
|
},
|
|
{
|
|
"eq_number": 738,
|
|
"title": "NSM Semantic Primes Explication",
|
|
"domain_id": 51,
|
|
"significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
|
|
},
|
|
{
|
|
"eq_number": 738,
|
|
"title": "NSM Semantic Primes Explication",
|
|
"domain_id": 51,
|
|
"significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
|
|
},
|
|
{
|
|
"eq_number": 738,
|
|
"title": "NSM Semantic Primes Explication",
|
|
"domain_id": 51,
|
|
"significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
|
|
},
|
|
{
|
|
"eq_number": 738,
|
|
"title": "NSM Semantic Primes Explication",
|
|
"domain_id": 51,
|
|
"significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
|
|
},
|
|
{
|
|
"eq_number": 738,
|
|
"title": "NSM Semantic Primes Explication",
|
|
"domain_id": 51,
|
|
"significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
|
|
},
|
|
{
|
|
"eq_number": 738,
|
|
"title": "NSM Semantic Primes Explication",
|
|
"domain_id": 51,
|
|
"significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
|
|
},
|
|
{
|
|
"eq_number": 738,
|
|
"title": "NSM Semantic Primes Explication",
|
|
"domain_id": 51,
|
|
"significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
|
|
},
|
|
{
|
|
"eq_number": 738,
|
|
"title": "NSM Semantic Primes Explication",
|
|
"domain_id": 51,
|
|
"significance": "Universal semantic decomposition using 64 irreducible primes. Foundation for language compression and cognitive load analysis."
|
|
},
|
|
{
|
|
"eq_number": 739,
|
|
"title": "Cognitive Load Matrix (Invariant-Enhanced)",
|
|
"domain_id": 51,
|
|
"significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 739,
|
|
"title": "Cognitive Load Matrix (Invariant-Enhanced)",
|
|
"domain_id": 51,
|
|
"significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 739,
|
|
"title": "Cognitive Load Matrix (Invariant-Enhanced)",
|
|
"domain_id": 51,
|
|
"significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 739,
|
|
"title": "Cognitive Load Matrix (Invariant-Enhanced)",
|
|
"domain_id": 51,
|
|
"significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 739,
|
|
"title": "Cognitive Load Matrix (Invariant-Enhanced)",
|
|
"domain_id": 51,
|
|
"significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 739,
|
|
"title": "Cognitive Load Matrix (Invariant-Enhanced)",
|
|
"domain_id": 51,
|
|
"significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 739,
|
|
"title": "Cognitive Load Matrix (Invariant-Enhanced)",
|
|
"domain_id": 51,
|
|
"significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 739,
|
|
"title": "Cognitive Load Matrix (Invariant-Enhanced)",
|
|
"domain_id": 51,
|
|
"significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 739,
|
|
"title": "Cognitive Load Matrix (Invariant-Enhanced)",
|
|
"domain_id": 51,
|
|
"significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 739,
|
|
"title": "Cognitive Load Matrix (Invariant-Enhanced)",
|
|
"domain_id": 51,
|
|
"significance": "8-dimensional cognitive load model with invariant preservation. Critical for assessing processing overhead in semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 740,
|
|
"title": "Evolutionary Operator (Universal)",
|
|
"domain_id": 51,
|
|
"significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
|
|
},
|
|
{
|
|
"eq_number": 740,
|
|
"title": "Evolutionary Operator (Universal)",
|
|
"domain_id": 51,
|
|
"significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
|
|
},
|
|
{
|
|
"eq_number": 740,
|
|
"title": "Evolutionary Operator (Universal)",
|
|
"domain_id": 51,
|
|
"significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
|
|
},
|
|
{
|
|
"eq_number": 740,
|
|
"title": "Evolutionary Operator (Universal)",
|
|
"domain_id": 51,
|
|
"significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
|
|
},
|
|
{
|
|
"eq_number": 740,
|
|
"title": "Evolutionary Operator (Universal)",
|
|
"domain_id": 51,
|
|
"significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
|
|
},
|
|
{
|
|
"eq_number": 740,
|
|
"title": "Evolutionary Operator (Universal)",
|
|
"domain_id": 51,
|
|
"significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
|
|
},
|
|
{
|
|
"eq_number": 740,
|
|
"title": "Evolutionary Operator (Universal)",
|
|
"domain_id": 51,
|
|
"significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
|
|
},
|
|
{
|
|
"eq_number": 740,
|
|
"title": "Evolutionary Operator (Universal)",
|
|
"domain_id": 51,
|
|
"significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
|
|
},
|
|
{
|
|
"eq_number": 740,
|
|
"title": "Evolutionary Operator (Universal)",
|
|
"domain_id": 51,
|
|
"significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
|
|
},
|
|
{
|
|
"eq_number": 740,
|
|
"title": "Evolutionary Operator (Universal)",
|
|
"domain_id": 51,
|
|
"significance": "Conserved operator frozen across 120 Myr evolution. Model for compression operators that remain stable across contexts."
|
|
},
|
|
{
|
|
"eq_number": 741,
|
|
"title": "Hutter Prize Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
|
|
},
|
|
{
|
|
"eq_number": 741,
|
|
"title": "Hutter Prize Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
|
|
},
|
|
{
|
|
"eq_number": 741,
|
|
"title": "Hutter Prize Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
|
|
},
|
|
{
|
|
"eq_number": 741,
|
|
"title": "Hutter Prize Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
|
|
},
|
|
{
|
|
"eq_number": 741,
|
|
"title": "Hutter Prize Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
|
|
},
|
|
{
|
|
"eq_number": 741,
|
|
"title": "Hutter Prize Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
|
|
},
|
|
{
|
|
"eq_number": 741,
|
|
"title": "Hutter Prize Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
|
|
},
|
|
{
|
|
"eq_number": 741,
|
|
"title": "Hutter Prize Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
|
|
},
|
|
{
|
|
"eq_number": 741,
|
|
"title": "Hutter Prize Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
|
|
},
|
|
{
|
|
"eq_number": 741,
|
|
"title": "Hutter Prize Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Weighted compression metric with decoder and resource penalties. Foundation for compression efficiency optimization."
|
|
},
|
|
{
|
|
"eq_number": 742,
|
|
"title": "Semantic Compression Operator",
|
|
"domain_id": 51,
|
|
"significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
|
|
},
|
|
{
|
|
"eq_number": 742,
|
|
"title": "Semantic Compression Operator",
|
|
"domain_id": 51,
|
|
"significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
|
|
},
|
|
{
|
|
"eq_number": 742,
|
|
"title": "Semantic Compression Operator",
|
|
"domain_id": 51,
|
|
"significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
|
|
},
|
|
{
|
|
"eq_number": 742,
|
|
"title": "Semantic Compression Operator",
|
|
"domain_id": 51,
|
|
"significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
|
|
},
|
|
{
|
|
"eq_number": 742,
|
|
"title": "Semantic Compression Operator",
|
|
"domain_id": 51,
|
|
"significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
|
|
},
|
|
{
|
|
"eq_number": 742,
|
|
"title": "Semantic Compression Operator",
|
|
"domain_id": 51,
|
|
"significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
|
|
},
|
|
{
|
|
"eq_number": 742,
|
|
"title": "Semantic Compression Operator",
|
|
"domain_id": 51,
|
|
"significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
|
|
},
|
|
{
|
|
"eq_number": 742,
|
|
"title": "Semantic Compression Operator",
|
|
"domain_id": 51,
|
|
"significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
|
|
},
|
|
{
|
|
"eq_number": 742,
|
|
"title": "Semantic Compression Operator",
|
|
"domain_id": 51,
|
|
"significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
|
|
},
|
|
{
|
|
"eq_number": 742,
|
|
"title": "Semantic Compression Operator",
|
|
"domain_id": 51,
|
|
"significance": "Language-specific compression operator using NSM primes as conserved basis. Core of semantic-aware compression."
|
|
},
|
|
{
|
|
"eq_number": 743,
|
|
"title": "Prime-to-Byte Mapping",
|
|
"domain_id": 51,
|
|
"significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
|
|
},
|
|
{
|
|
"eq_number": 743,
|
|
"title": "Prime-to-Byte Mapping",
|
|
"domain_id": 51,
|
|
"significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
|
|
},
|
|
{
|
|
"eq_number": 743,
|
|
"title": "Prime-to-Byte Mapping",
|
|
"domain_id": 51,
|
|
"significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
|
|
},
|
|
{
|
|
"eq_number": 743,
|
|
"title": "Prime-to-Byte Mapping",
|
|
"domain_id": 51,
|
|
"significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
|
|
},
|
|
{
|
|
"eq_number": 743,
|
|
"title": "Prime-to-Byte Mapping",
|
|
"domain_id": 51,
|
|
"significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
|
|
},
|
|
{
|
|
"eq_number": 743,
|
|
"title": "Prime-to-Byte Mapping",
|
|
"domain_id": 51,
|
|
"significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
|
|
},
|
|
{
|
|
"eq_number": 743,
|
|
"title": "Prime-to-Byte Mapping",
|
|
"domain_id": 51,
|
|
"significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
|
|
},
|
|
{
|
|
"eq_number": 743,
|
|
"title": "Prime-to-Byte Mapping",
|
|
"domain_id": 51,
|
|
"significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
|
|
},
|
|
{
|
|
"eq_number": 743,
|
|
"title": "Prime-to-Byte Mapping",
|
|
"domain_id": 51,
|
|
"significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
|
|
},
|
|
{
|
|
"eq_number": 743,
|
|
"title": "Prime-to-Byte Mapping",
|
|
"domain_id": 51,
|
|
"significance": "Maps semantic primes to compression primitives. Enables semantic-aware byte-level optimization."
|
|
},
|
|
{
|
|
"eq_number": 744,
|
|
"title": "Context as Cognitive Load Function",
|
|
"domain_id": 51,
|
|
"significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
|
|
},
|
|
{
|
|
"eq_number": 744,
|
|
"title": "Context as Cognitive Load Function",
|
|
"domain_id": 51,
|
|
"significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
|
|
},
|
|
{
|
|
"eq_number": 744,
|
|
"title": "Context as Cognitive Load Function",
|
|
"domain_id": 51,
|
|
"significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
|
|
},
|
|
{
|
|
"eq_number": 744,
|
|
"title": "Context as Cognitive Load Function",
|
|
"domain_id": 51,
|
|
"significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
|
|
},
|
|
{
|
|
"eq_number": 744,
|
|
"title": "Context as Cognitive Load Function",
|
|
"domain_id": 51,
|
|
"significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
|
|
},
|
|
{
|
|
"eq_number": 744,
|
|
"title": "Context as Cognitive Load Function",
|
|
"domain_id": 51,
|
|
"significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
|
|
},
|
|
{
|
|
"eq_number": 744,
|
|
"title": "Context as Cognitive Load Function",
|
|
"domain_id": 51,
|
|
"significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
|
|
},
|
|
{
|
|
"eq_number": 744,
|
|
"title": "Context as Cognitive Load Function",
|
|
"domain_id": 51,
|
|
"significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
|
|
},
|
|
{
|
|
"eq_number": 744,
|
|
"title": "Context as Cognitive Load Function",
|
|
"domain_id": 51,
|
|
"significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
|
|
},
|
|
{
|
|
"eq_number": 744,
|
|
"title": "Context as Cognitive Load Function",
|
|
"domain_id": 51,
|
|
"significance": "Cognitive load determines regulatory state for compression. Links processing overhead to prime activation."
|
|
},
|
|
{
|
|
"eq_number": 745,
|
|
"title": "Gap Adaptation Equation",
|
|
"domain_id": 51,
|
|
"significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
|
|
},
|
|
{
|
|
"eq_number": 745,
|
|
"title": "Gap Adaptation Equation",
|
|
"domain_id": 51,
|
|
"significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
|
|
},
|
|
{
|
|
"eq_number": 745,
|
|
"title": "Gap Adaptation Equation",
|
|
"domain_id": 51,
|
|
"significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
|
|
},
|
|
{
|
|
"eq_number": 745,
|
|
"title": "Gap Adaptation Equation",
|
|
"domain_id": 51,
|
|
"significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
|
|
},
|
|
{
|
|
"eq_number": 745,
|
|
"title": "Gap Adaptation Equation",
|
|
"domain_id": 51,
|
|
"significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
|
|
},
|
|
{
|
|
"eq_number": 745,
|
|
"title": "Gap Adaptation Equation",
|
|
"domain_id": 51,
|
|
"significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
|
|
},
|
|
{
|
|
"eq_number": 745,
|
|
"title": "Gap Adaptation Equation",
|
|
"domain_id": 51,
|
|
"significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
|
|
},
|
|
{
|
|
"eq_number": 745,
|
|
"title": "Gap Adaptation Equation",
|
|
"domain_id": 51,
|
|
"significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
|
|
},
|
|
{
|
|
"eq_number": 745,
|
|
"title": "Gap Adaptation Equation",
|
|
"domain_id": 51,
|
|
"significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
|
|
},
|
|
{
|
|
"eq_number": 745,
|
|
"title": "Gap Adaptation Equation",
|
|
"domain_id": 51,
|
|
"significance": "Evolutionary fracking principle: gap width controls coupling strength. Adaptive prime filtering based on load."
|
|
},
|
|
{
|
|
"eq_number": 746,
|
|
"title": "Unified Semantic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 746,
|
|
"title": "Unified Semantic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 746,
|
|
"title": "Unified Semantic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 746,
|
|
"title": "Unified Semantic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 746,
|
|
"title": "Unified Semantic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 746,
|
|
"title": "Unified Semantic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 746,
|
|
"title": "Unified Semantic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 746,
|
|
"title": "Unified Semantic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 746,
|
|
"title": "Unified Semantic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 746,
|
|
"title": "Unified Semantic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Complete integration of primes, cognitive load, and gap adaptation. Master equation for semantic compression."
|
|
},
|
|
{
|
|
"eq_number": 747,
|
|
"title": "Gap-Dependent Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
|
|
},
|
|
{
|
|
"eq_number": 747,
|
|
"title": "Gap-Dependent Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
|
|
},
|
|
{
|
|
"eq_number": 747,
|
|
"title": "Gap-Dependent Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
|
|
},
|
|
{
|
|
"eq_number": 747,
|
|
"title": "Gap-Dependent Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
|
|
},
|
|
{
|
|
"eq_number": 747,
|
|
"title": "Gap-Dependent Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
|
|
},
|
|
{
|
|
"eq_number": 747,
|
|
"title": "Gap-Dependent Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
|
|
},
|
|
{
|
|
"eq_number": 747,
|
|
"title": "Gap-Dependent Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
|
|
},
|
|
{
|
|
"eq_number": 747,
|
|
"title": "Gap-Dependent Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
|
|
},
|
|
{
|
|
"eq_number": 747,
|
|
"title": "Gap-Dependent Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
|
|
},
|
|
{
|
|
"eq_number": 747,
|
|
"title": "Gap-Dependent Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Threshold function for prime activation based on gap width. Implements stress response in compression."
|
|
},
|
|
{
|
|
"eq_number": 748,
|
|
"title": "Prime Compression Matrix",
|
|
"domain_id": 51,
|
|
"significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
|
|
},
|
|
{
|
|
"eq_number": 748,
|
|
"title": "Prime Compression Matrix",
|
|
"domain_id": 51,
|
|
"significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
|
|
},
|
|
{
|
|
"eq_number": 748,
|
|
"title": "Prime Compression Matrix",
|
|
"domain_id": 51,
|
|
"significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
|
|
},
|
|
{
|
|
"eq_number": 748,
|
|
"title": "Prime Compression Matrix",
|
|
"domain_id": 51,
|
|
"significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
|
|
},
|
|
{
|
|
"eq_number": 748,
|
|
"title": "Prime Compression Matrix",
|
|
"domain_id": 51,
|
|
"significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
|
|
},
|
|
{
|
|
"eq_number": 748,
|
|
"title": "Prime Compression Matrix",
|
|
"domain_id": 51,
|
|
"significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
|
|
},
|
|
{
|
|
"eq_number": 748,
|
|
"title": "Prime Compression Matrix",
|
|
"domain_id": 51,
|
|
"significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
|
|
},
|
|
{
|
|
"eq_number": 748,
|
|
"title": "Prime Compression Matrix",
|
|
"domain_id": 51,
|
|
"significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
|
|
},
|
|
{
|
|
"eq_number": 748,
|
|
"title": "Prime Compression Matrix",
|
|
"domain_id": 51,
|
|
"significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
|
|
},
|
|
{
|
|
"eq_number": 748,
|
|
"title": "Prime Compression Matrix",
|
|
"domain_id": 51,
|
|
"significance": "64\u00d764 matrix of prime weights and cross-correlations. Encodes conserved topology of semantic relationships."
|
|
},
|
|
{
|
|
"eq_number": 749,
|
|
"title": "Matrix-Vector Compression",
|
|
"domain_id": 51,
|
|
"significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
|
|
},
|
|
{
|
|
"eq_number": 749,
|
|
"title": "Matrix-Vector Compression",
|
|
"domain_id": 51,
|
|
"significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
|
|
},
|
|
{
|
|
"eq_number": 749,
|
|
"title": "Matrix-Vector Compression",
|
|
"domain_id": 51,
|
|
"significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
|
|
},
|
|
{
|
|
"eq_number": 749,
|
|
"title": "Matrix-Vector Compression",
|
|
"domain_id": 51,
|
|
"significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
|
|
},
|
|
{
|
|
"eq_number": 749,
|
|
"title": "Matrix-Vector Compression",
|
|
"domain_id": 51,
|
|
"significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
|
|
},
|
|
{
|
|
"eq_number": 749,
|
|
"title": "Matrix-Vector Compression",
|
|
"domain_id": 51,
|
|
"significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
|
|
},
|
|
{
|
|
"eq_number": 749,
|
|
"title": "Matrix-Vector Compression",
|
|
"domain_id": 51,
|
|
"significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
|
|
},
|
|
{
|
|
"eq_number": 749,
|
|
"title": "Matrix-Vector Compression",
|
|
"domain_id": 51,
|
|
"significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
|
|
},
|
|
{
|
|
"eq_number": 749,
|
|
"title": "Matrix-Vector Compression",
|
|
"domain_id": 51,
|
|
"significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
|
|
},
|
|
{
|
|
"eq_number": 749,
|
|
"title": "Matrix-Vector Compression",
|
|
"domain_id": 51,
|
|
"significance": "Linear algebra formulation of semantic compression with gap modulation. Efficient implementation target."
|
|
},
|
|
{
|
|
"eq_number": 750,
|
|
"title": "Invariant Load with Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
|
|
},
|
|
{
|
|
"eq_number": 750,
|
|
"title": "Invariant Load with Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
|
|
},
|
|
{
|
|
"eq_number": 750,
|
|
"title": "Invariant Load with Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
|
|
},
|
|
{
|
|
"eq_number": 750,
|
|
"title": "Invariant Load with Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
|
|
},
|
|
{
|
|
"eq_number": 750,
|
|
"title": "Invariant Load with Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
|
|
},
|
|
{
|
|
"eq_number": 750,
|
|
"title": "Invariant Load with Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
|
|
},
|
|
{
|
|
"eq_number": 750,
|
|
"title": "Invariant Load with Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
|
|
},
|
|
{
|
|
"eq_number": 750,
|
|
"title": "Invariant Load with Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
|
|
},
|
|
{
|
|
"eq_number": 750,
|
|
"title": "Invariant Load with Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
|
|
},
|
|
{
|
|
"eq_number": 750,
|
|
"title": "Invariant Load with Prime Activation",
|
|
"domain_id": 51,
|
|
"significance": "Modified invariant load that only counts active primes. Reduces penalty under high-stress conditions."
|
|
},
|
|
{
|
|
"eq_number": 751,
|
|
"title": "Gap Threshold Function",
|
|
"domain_id": 51,
|
|
"significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
|
|
},
|
|
{
|
|
"eq_number": 751,
|
|
"title": "Gap Threshold Function",
|
|
"domain_id": 51,
|
|
"significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
|
|
},
|
|
{
|
|
"eq_number": 751,
|
|
"title": "Gap Threshold Function",
|
|
"domain_id": 51,
|
|
"significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
|
|
},
|
|
{
|
|
"eq_number": 751,
|
|
"title": "Gap Threshold Function",
|
|
"domain_id": 51,
|
|
"significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
|
|
},
|
|
{
|
|
"eq_number": 751,
|
|
"title": "Gap Threshold Function",
|
|
"domain_id": 51,
|
|
"significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
|
|
},
|
|
{
|
|
"eq_number": 751,
|
|
"title": "Gap Threshold Function",
|
|
"domain_id": 51,
|
|
"significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
|
|
},
|
|
{
|
|
"eq_number": 751,
|
|
"title": "Gap Threshold Function",
|
|
"domain_id": 51,
|
|
"significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
|
|
},
|
|
{
|
|
"eq_number": 751,
|
|
"title": "Gap Threshold Function",
|
|
"domain_id": 51,
|
|
"significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
|
|
},
|
|
{
|
|
"eq_number": 751,
|
|
"title": "Gap Threshold Function",
|
|
"domain_id": 51,
|
|
"significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
|
|
},
|
|
{
|
|
"eq_number": 751,
|
|
"title": "Gap Threshold Function",
|
|
"domain_id": 51,
|
|
"significance": "Piecewise threshold mapping gap width to severity cutoff. Implements discrete stress response levels."
|
|
},
|
|
{
|
|
"eq_number": 752,
|
|
"title": "Hutter Prize Penalty with Invariants",
|
|
"domain_id": 51,
|
|
"significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
|
|
},
|
|
{
|
|
"eq_number": 752,
|
|
"title": "Hutter Prize Penalty with Invariants",
|
|
"domain_id": 51,
|
|
"significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
|
|
},
|
|
{
|
|
"eq_number": 752,
|
|
"title": "Hutter Prize Penalty with Invariants",
|
|
"domain_id": 51,
|
|
"significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
|
|
},
|
|
{
|
|
"eq_number": 752,
|
|
"title": "Hutter Prize Penalty with Invariants",
|
|
"domain_id": 51,
|
|
"significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
|
|
},
|
|
{
|
|
"eq_number": 752,
|
|
"title": "Hutter Prize Penalty with Invariants",
|
|
"domain_id": 51,
|
|
"significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
|
|
},
|
|
{
|
|
"eq_number": 752,
|
|
"title": "Hutter Prize Penalty with Invariants",
|
|
"domain_id": 51,
|
|
"significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
|
|
},
|
|
{
|
|
"eq_number": 752,
|
|
"title": "Hutter Prize Penalty with Invariants",
|
|
"domain_id": 51,
|
|
"significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
|
|
},
|
|
{
|
|
"eq_number": 752,
|
|
"title": "Hutter Prize Penalty with Invariants",
|
|
"domain_id": 51,
|
|
"significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
|
|
},
|
|
{
|
|
"eq_number": 752,
|
|
"title": "Hutter Prize Penalty with Invariants",
|
|
"domain_id": 51,
|
|
"significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
|
|
},
|
|
{
|
|
"eq_number": 752,
|
|
"title": "Hutter Prize Penalty with Invariants",
|
|
"domain_id": 51,
|
|
"significance": "Extended Hutter Prize penalty including invariant preservation cost. Tradeoff between compression and semantic fidelity."
|
|
},
|
|
{
|
|
"eq_number": 753,
|
|
"title": "Gap Adaptation Dynamics",
|
|
"domain_id": 51,
|
|
"significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
|
|
},
|
|
{
|
|
"eq_number": 753,
|
|
"title": "Gap Adaptation Dynamics",
|
|
"domain_id": 51,
|
|
"significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
|
|
},
|
|
{
|
|
"eq_number": 753,
|
|
"title": "Gap Adaptation Dynamics",
|
|
"domain_id": 51,
|
|
"significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
|
|
},
|
|
{
|
|
"eq_number": 753,
|
|
"title": "Gap Adaptation Dynamics",
|
|
"domain_id": 51,
|
|
"significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
|
|
},
|
|
{
|
|
"eq_number": 753,
|
|
"title": "Gap Adaptation Dynamics",
|
|
"domain_id": 51,
|
|
"significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
|
|
},
|
|
{
|
|
"eq_number": 753,
|
|
"title": "Gap Adaptation Dynamics",
|
|
"domain_id": 51,
|
|
"significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
|
|
},
|
|
{
|
|
"eq_number": 753,
|
|
"title": "Gap Adaptation Dynamics",
|
|
"domain_id": 51,
|
|
"significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
|
|
},
|
|
{
|
|
"eq_number": 753,
|
|
"title": "Gap Adaptation Dynamics",
|
|
"domain_id": 51,
|
|
"significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
|
|
},
|
|
{
|
|
"eq_number": 753,
|
|
"title": "Gap Adaptation Dynamics",
|
|
"domain_id": 51,
|
|
"significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
|
|
},
|
|
{
|
|
"eq_number": 753,
|
|
"title": "Gap Adaptation Dynamics",
|
|
"domain_id": 51,
|
|
"significance": "Gradient descent dynamics for gap adaptation. Ensures convergence to optimal load-balanced state."
|
|
},
|
|
{
|
|
"eq_number": 754,
|
|
"title": "Prime Conservation Theorem (Spectral Entropy Bound)",
|
|
"domain_id": 51,
|
|
"significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
|
|
},
|
|
{
|
|
"eq_number": 754,
|
|
"title": "Prime Conservation Theorem (Spectral Entropy Bound)",
|
|
"domain_id": 51,
|
|
"significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
|
|
},
|
|
{
|
|
"eq_number": 754,
|
|
"title": "Prime Conservation Theorem (Spectral Entropy Bound)",
|
|
"domain_id": 51,
|
|
"significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
|
|
},
|
|
{
|
|
"eq_number": 754,
|
|
"title": "Prime Conservation Theorem (Spectral Entropy Bound)",
|
|
"domain_id": 51,
|
|
"significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
|
|
},
|
|
{
|
|
"eq_number": 754,
|
|
"title": "Prime Conservation Theorem (Spectral Entropy Bound)",
|
|
"domain_id": 51,
|
|
"significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
|
|
},
|
|
{
|
|
"eq_number": 754,
|
|
"title": "Prime Conservation Theorem (Spectral Entropy Bound)",
|
|
"domain_id": 51,
|
|
"significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
|
|
},
|
|
{
|
|
"eq_number": 754,
|
|
"title": "Prime Conservation Theorem (Spectral Entropy Bound)",
|
|
"domain_id": 51,
|
|
"significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
|
|
},
|
|
{
|
|
"eq_number": 754,
|
|
"title": "Prime Conservation Theorem (Spectral Entropy Bound)",
|
|
"domain_id": 51,
|
|
"significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
|
|
},
|
|
{
|
|
"eq_number": 754,
|
|
"title": "Prime Conservation Theorem (Spectral Entropy Bound)",
|
|
"domain_id": 51,
|
|
"significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
|
|
},
|
|
{
|
|
"eq_number": 754,
|
|
"title": "Prime Conservation Theorem (Spectral Entropy Bound)",
|
|
"domain_id": 51,
|
|
"significance": "Compression ratio bounded by spectral entropy of prime activation under conserved operator."
|
|
},
|
|
{
|
|
"eq_number": 755,
|
|
"title": "Invariant Preservation Theorem",
|
|
"domain_id": 51,
|
|
"significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
|
|
},
|
|
{
|
|
"eq_number": 755,
|
|
"title": "Invariant Preservation Theorem",
|
|
"domain_id": 51,
|
|
"significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
|
|
},
|
|
{
|
|
"eq_number": 755,
|
|
"title": "Invariant Preservation Theorem",
|
|
"domain_id": 51,
|
|
"significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
|
|
},
|
|
{
|
|
"eq_number": 755,
|
|
"title": "Invariant Preservation Theorem",
|
|
"domain_id": 51,
|
|
"significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
|
|
},
|
|
{
|
|
"eq_number": 755,
|
|
"title": "Invariant Preservation Theorem",
|
|
"domain_id": 51,
|
|
"significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
|
|
},
|
|
{
|
|
"eq_number": 755,
|
|
"title": "Invariant Preservation Theorem",
|
|
"domain_id": 51,
|
|
"significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
|
|
},
|
|
{
|
|
"eq_number": 755,
|
|
"title": "Invariant Preservation Theorem",
|
|
"domain_id": 51,
|
|
"significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
|
|
},
|
|
{
|
|
"eq_number": 755,
|
|
"title": "Invariant Preservation Theorem",
|
|
"domain_id": 51,
|
|
"significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
|
|
},
|
|
{
|
|
"eq_number": 755,
|
|
"title": "Invariant Preservation Theorem",
|
|
"domain_id": 51,
|
|
"significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
|
|
},
|
|
{
|
|
"eq_number": 755,
|
|
"title": "Invariant Preservation Theorem",
|
|
"domain_id": 51,
|
|
"significance": "Critical invariants (severity=\u221e) preserved regardless of gap width. Hard constraint on compression."
|
|
},
|
|
{
|
|
"eq_number": 756,
|
|
"title": "Matrix Evolution Learning Rule",
|
|
"domain_id": 51,
|
|
"significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
|
|
},
|
|
{
|
|
"eq_number": 756,
|
|
"title": "Matrix Evolution Learning Rule",
|
|
"domain_id": 51,
|
|
"significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
|
|
},
|
|
{
|
|
"eq_number": 756,
|
|
"title": "Matrix Evolution Learning Rule",
|
|
"domain_id": 51,
|
|
"significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
|
|
},
|
|
{
|
|
"eq_number": 756,
|
|
"title": "Matrix Evolution Learning Rule",
|
|
"domain_id": 51,
|
|
"significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
|
|
},
|
|
{
|
|
"eq_number": 756,
|
|
"title": "Matrix Evolution Learning Rule",
|
|
"domain_id": 51,
|
|
"significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
|
|
},
|
|
{
|
|
"eq_number": 756,
|
|
"title": "Matrix Evolution Learning Rule",
|
|
"domain_id": 51,
|
|
"significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
|
|
},
|
|
{
|
|
"eq_number": 756,
|
|
"title": "Matrix Evolution Learning Rule",
|
|
"domain_id": 51,
|
|
"significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
|
|
},
|
|
{
|
|
"eq_number": 756,
|
|
"title": "Matrix Evolution Learning Rule",
|
|
"domain_id": 51,
|
|
"significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
|
|
},
|
|
{
|
|
"eq_number": 756,
|
|
"title": "Matrix Evolution Learning Rule",
|
|
"domain_id": 51,
|
|
"significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
|
|
},
|
|
{
|
|
"eq_number": 756,
|
|
"title": "Matrix Evolution Learning Rule",
|
|
"domain_id": 51,
|
|
"significance": "Gradient-based learning of prime matrix while conserving topology. Analogous to evolutionary mutation."
|
|
},
|
|
{
|
|
"eq_number": 757,
|
|
"title": "Cross-Linguistic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
|
|
},
|
|
{
|
|
"eq_number": 757,
|
|
"title": "Cross-Linguistic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
|
|
},
|
|
{
|
|
"eq_number": 757,
|
|
"title": "Cross-Linguistic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
|
|
},
|
|
{
|
|
"eq_number": 757,
|
|
"title": "Cross-Linguistic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
|
|
},
|
|
{
|
|
"eq_number": 757,
|
|
"title": "Cross-Linguistic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
|
|
},
|
|
{
|
|
"eq_number": 757,
|
|
"title": "Cross-Linguistic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
|
|
},
|
|
{
|
|
"eq_number": 757,
|
|
"title": "Cross-Linguistic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
|
|
},
|
|
{
|
|
"eq_number": 757,
|
|
"title": "Cross-Linguistic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
|
|
},
|
|
{
|
|
"eq_number": 757,
|
|
"title": "Cross-Linguistic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
|
|
},
|
|
{
|
|
"eq_number": 757,
|
|
"title": "Cross-Linguistic Compression Equation",
|
|
"domain_id": 51,
|
|
"significance": "Unified compression across languages with language-specific gap functions. Enables transfer learning."
|
|
},
|
|
{
|
|
"eq_number": 758,
|
|
"title": "Language-Specific Gap Function",
|
|
"domain_id": 51,
|
|
"significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
|
|
},
|
|
{
|
|
"eq_number": 758,
|
|
"title": "Language-Specific Gap Function",
|
|
"domain_id": 51,
|
|
"significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
|
|
},
|
|
{
|
|
"eq_number": 758,
|
|
"title": "Language-Specific Gap Function",
|
|
"domain_id": 51,
|
|
"significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
|
|
},
|
|
{
|
|
"eq_number": 758,
|
|
"title": "Language-Specific Gap Function",
|
|
"domain_id": 51,
|
|
"significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
|
|
},
|
|
{
|
|
"eq_number": 758,
|
|
"title": "Language-Specific Gap Function",
|
|
"domain_id": 51,
|
|
"significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
|
|
},
|
|
{
|
|
"eq_number": 758,
|
|
"title": "Language-Specific Gap Function",
|
|
"domain_id": 51,
|
|
"significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
|
|
},
|
|
{
|
|
"eq_number": 758,
|
|
"title": "Language-Specific Gap Function",
|
|
"domain_id": 51,
|
|
"significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
|
|
},
|
|
{
|
|
"eq_number": 758,
|
|
"title": "Language-Specific Gap Function",
|
|
"domain_id": 51,
|
|
"significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
|
|
},
|
|
{
|
|
"eq_number": 758,
|
|
"title": "Language-Specific Gap Function",
|
|
"domain_id": 51,
|
|
"significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
|
|
},
|
|
{
|
|
"eq_number": 758,
|
|
"title": "Language-Specific Gap Function",
|
|
"domain_id": 51,
|
|
"significance": "Gap function parameterized by language complexity. Accounts for morphological and syntactic differences."
|
|
},
|
|
{
|
|
"eq_number": 759,
|
|
"title": "0-AVMR: Square Shell Identity",
|
|
"domain_id": 51,
|
|
"significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
|
|
},
|
|
{
|
|
"eq_number": 759,
|
|
"title": "0-AVMR: Square Shell Identity",
|
|
"domain_id": 51,
|
|
"significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
|
|
},
|
|
{
|
|
"eq_number": 759,
|
|
"title": "0-AVMR: Square Shell Identity",
|
|
"domain_id": 51,
|
|
"significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
|
|
},
|
|
{
|
|
"eq_number": 759,
|
|
"title": "0-AVMR: Square Shell Identity",
|
|
"domain_id": 51,
|
|
"significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
|
|
},
|
|
{
|
|
"eq_number": 759,
|
|
"title": "0-AVMR: Square Shell Identity",
|
|
"domain_id": 51,
|
|
"significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
|
|
},
|
|
{
|
|
"eq_number": 759,
|
|
"title": "0-AVMR: Square Shell Identity",
|
|
"domain_id": 51,
|
|
"significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
|
|
},
|
|
{
|
|
"eq_number": 759,
|
|
"title": "0-AVMR: Square Shell Identity",
|
|
"domain_id": 51,
|
|
"significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
|
|
},
|
|
{
|
|
"eq_number": 759,
|
|
"title": "0-AVMR: Square Shell Identity",
|
|
"domain_id": 51,
|
|
"significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
|
|
},
|
|
{
|
|
"eq_number": 759,
|
|
"title": "0-AVMR: Square Shell Identity",
|
|
"domain_id": 51,
|
|
"significance": "Foundational partition of natural numbers into discrete shells indexed by k = floor(sqrt(n)). Basis for hierarchical vector aggregation."
|
|
},
|
|
{
|
|
"eq_number": 760,
|
|
"title": "0-AVMR: Tip Coordinate Map",
|
|
"domain_id": 51,
|
|
"significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
|
|
},
|
|
{
|
|
"eq_number": 760,
|
|
"title": "0-AVMR: Tip Coordinate Map",
|
|
"domain_id": 51,
|
|
"significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
|
|
},
|
|
{
|
|
"eq_number": 760,
|
|
"title": "0-AVMR: Tip Coordinate Map",
|
|
"domain_id": 51,
|
|
"significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
|
|
},
|
|
{
|
|
"eq_number": 760,
|
|
"title": "0-AVMR: Tip Coordinate Map",
|
|
"domain_id": 51,
|
|
"significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
|
|
},
|
|
{
|
|
"eq_number": 760,
|
|
"title": "0-AVMR: Tip Coordinate Map",
|
|
"domain_id": 51,
|
|
"significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
|
|
},
|
|
{
|
|
"eq_number": 760,
|
|
"title": "0-AVMR: Tip Coordinate Map",
|
|
"domain_id": 51,
|
|
"significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
|
|
},
|
|
{
|
|
"eq_number": 760,
|
|
"title": "0-AVMR: Tip Coordinate Map",
|
|
"domain_id": 51,
|
|
"significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
|
|
},
|
|
{
|
|
"eq_number": 760,
|
|
"title": "0-AVMR: Tip Coordinate Map",
|
|
"domain_id": 51,
|
|
"significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
|
|
},
|
|
{
|
|
"eq_number": 760,
|
|
"title": "0-AVMR: Tip Coordinate Map",
|
|
"domain_id": 51,
|
|
"significance": "Injective coordinate system on each shell mapping (a,b) to (product, difference). Enables vector aggregation with discriminant invariant."
|
|
},
|
|
{
|
|
"eq_number": 761,
|
|
"title": "0-AVMR: Interaction Score",
|
|
"domain_id": 51,
|
|
"significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
|
|
},
|
|
{
|
|
"eq_number": 761,
|
|
"title": "0-AVMR: Interaction Score",
|
|
"domain_id": 51,
|
|
"significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
|
|
},
|
|
{
|
|
"eq_number": 761,
|
|
"title": "0-AVMR: Interaction Score",
|
|
"domain_id": 51,
|
|
"significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
|
|
},
|
|
{
|
|
"eq_number": 761,
|
|
"title": "0-AVMR: Interaction Score",
|
|
"domain_id": 51,
|
|
"significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
|
|
},
|
|
{
|
|
"eq_number": 761,
|
|
"title": "0-AVMR: Interaction Score",
|
|
"domain_id": 51,
|
|
"significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
|
|
},
|
|
{
|
|
"eq_number": 761,
|
|
"title": "0-AVMR: Interaction Score",
|
|
"domain_id": 51,
|
|
"significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
|
|
},
|
|
{
|
|
"eq_number": 761,
|
|
"title": "0-AVMR: Interaction Score",
|
|
"domain_id": 51,
|
|
"significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
|
|
},
|
|
{
|
|
"eq_number": 761,
|
|
"title": "0-AVMR: Interaction Score",
|
|
"domain_id": 51,
|
|
"significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
|
|
},
|
|
{
|
|
"eq_number": 761,
|
|
"title": "0-AVMR: Interaction Score",
|
|
"domain_id": 51,
|
|
"significance": "Additive decomposition of interaction into mass, polarity, and spectral components. Basis for vector interaction terms."
|
|
},
|
|
{
|
|
"eq_number": 762,
|
|
"title": "0-AVMR: Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
|
|
},
|
|
{
|
|
"eq_number": 762,
|
|
"title": "0-AVMR: Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
|
|
},
|
|
{
|
|
"eq_number": 762,
|
|
"title": "0-AVMR: Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
|
|
},
|
|
{
|
|
"eq_number": 762,
|
|
"title": "0-AVMR: Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
|
|
},
|
|
{
|
|
"eq_number": 762,
|
|
"title": "0-AVMR: Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
|
|
},
|
|
{
|
|
"eq_number": 762,
|
|
"title": "0-AVMR: Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
|
|
},
|
|
{
|
|
"eq_number": 762,
|
|
"title": "0-AVMR: Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
|
|
},
|
|
{
|
|
"eq_number": 762,
|
|
"title": "0-AVMR: Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
|
|
},
|
|
{
|
|
"eq_number": 762,
|
|
"title": "0-AVMR: Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Composition mapping temporal-color encoding to genetic codon space. Theoretical bridge between temporal patterns and biological encoding."
|
|
},
|
|
{
|
|
"eq_number": 763,
|
|
"title": "0-AVMR: Genetic Entropy Bound",
|
|
"domain_id": 51,
|
|
"significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
|
|
},
|
|
{
|
|
"eq_number": 763,
|
|
"title": "0-AVMR: Genetic Entropy Bound",
|
|
"domain_id": 51,
|
|
"significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
|
|
},
|
|
{
|
|
"eq_number": 763,
|
|
"title": "0-AVMR: Genetic Entropy Bound",
|
|
"domain_id": 51,
|
|
"significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
|
|
},
|
|
{
|
|
"eq_number": 763,
|
|
"title": "0-AVMR: Genetic Entropy Bound",
|
|
"domain_id": 51,
|
|
"significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
|
|
},
|
|
{
|
|
"eq_number": 763,
|
|
"title": "0-AVMR: Genetic Entropy Bound",
|
|
"domain_id": 51,
|
|
"significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
|
|
},
|
|
{
|
|
"eq_number": 763,
|
|
"title": "0-AVMR: Genetic Entropy Bound",
|
|
"domain_id": 51,
|
|
"significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
|
|
},
|
|
{
|
|
"eq_number": 763,
|
|
"title": "0-AVMR: Genetic Entropy Bound",
|
|
"domain_id": 51,
|
|
"significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
|
|
},
|
|
{
|
|
"eq_number": 763,
|
|
"title": "0-AVMR: Genetic Entropy Bound",
|
|
"domain_id": 51,
|
|
"significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
|
|
},
|
|
{
|
|
"eq_number": 763,
|
|
"title": "0-AVMR: Genetic Entropy Bound",
|
|
"domain_id": 51,
|
|
"significance": "Information capacity bound for genetic coding system. H \u2248 4.2 bits bounded by log2(64) = 6 bits (codon space)."
|
|
},
|
|
{
|
|
"eq_number": 764,
|
|
"title": "0-AMMR: Shell Partition of Computation",
|
|
"domain_id": 51,
|
|
"significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
|
|
},
|
|
{
|
|
"eq_number": 764,
|
|
"title": "0-AMMR: Shell Partition of Computation",
|
|
"domain_id": 51,
|
|
"significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
|
|
},
|
|
{
|
|
"eq_number": 764,
|
|
"title": "0-AMMR: Shell Partition of Computation",
|
|
"domain_id": 51,
|
|
"significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
|
|
},
|
|
{
|
|
"eq_number": 764,
|
|
"title": "0-AMMR: Shell Partition of Computation",
|
|
"domain_id": 51,
|
|
"significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
|
|
},
|
|
{
|
|
"eq_number": 764,
|
|
"title": "0-AMMR: Shell Partition of Computation",
|
|
"domain_id": 51,
|
|
"significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
|
|
},
|
|
{
|
|
"eq_number": 764,
|
|
"title": "0-AMMR: Shell Partition of Computation",
|
|
"domain_id": 51,
|
|
"significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
|
|
},
|
|
{
|
|
"eq_number": 764,
|
|
"title": "0-AMMR: Shell Partition of Computation",
|
|
"domain_id": 51,
|
|
"significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
|
|
},
|
|
{
|
|
"eq_number": 764,
|
|
"title": "0-AMMR: Shell Partition of Computation",
|
|
"domain_id": 51,
|
|
"significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
|
|
},
|
|
{
|
|
"eq_number": 764,
|
|
"title": "0-AMMR: Shell Partition of Computation",
|
|
"domain_id": 51,
|
|
"significance": "Maps any computation (via G\u00f6del encoding) to discrete shell structure. Provides hierarchical organization for ENE operations."
|
|
},
|
|
{
|
|
"eq_number": 765,
|
|
"title": "0-AMMR: Shell Coordinate System",
|
|
"domain_id": 51,
|
|
"significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
|
|
},
|
|
{
|
|
"eq_number": 765,
|
|
"title": "0-AMMR: Shell Coordinate System",
|
|
"domain_id": 51,
|
|
"significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
|
|
},
|
|
{
|
|
"eq_number": 765,
|
|
"title": "0-AMMR: Shell Coordinate System",
|
|
"domain_id": 51,
|
|
"significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
|
|
},
|
|
{
|
|
"eq_number": 765,
|
|
"title": "0-AMMR: Shell Coordinate System",
|
|
"domain_id": 51,
|
|
"significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
|
|
},
|
|
{
|
|
"eq_number": 765,
|
|
"title": "0-AMMR: Shell Coordinate System",
|
|
"domain_id": 51,
|
|
"significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
|
|
},
|
|
{
|
|
"eq_number": 765,
|
|
"title": "0-AMMR: Shell Coordinate System",
|
|
"domain_id": 51,
|
|
"significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
|
|
},
|
|
{
|
|
"eq_number": 765,
|
|
"title": "0-AMMR: Shell Coordinate System",
|
|
"domain_id": 51,
|
|
"significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
|
|
},
|
|
{
|
|
"eq_number": 765,
|
|
"title": "0-AMMR: Shell Coordinate System",
|
|
"domain_id": 51,
|
|
"significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
|
|
},
|
|
{
|
|
"eq_number": 765,
|
|
"title": "0-AMMR: Shell Coordinate System",
|
|
"domain_id": 51,
|
|
"significance": "Tip map as injective coordinate system on each shell. Enables unique addressing within computational shells."
|
|
},
|
|
{
|
|
"eq_number": 766,
|
|
"title": "0-AMMR: Additive Shell Interaction",
|
|
"domain_id": 51,
|
|
"significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
|
|
},
|
|
{
|
|
"eq_number": 766,
|
|
"title": "0-AMMR: Additive Shell Interaction",
|
|
"domain_id": 51,
|
|
"significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
|
|
},
|
|
{
|
|
"eq_number": 766,
|
|
"title": "0-AMMR: Additive Shell Interaction",
|
|
"domain_id": 51,
|
|
"significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
|
|
},
|
|
{
|
|
"eq_number": 766,
|
|
"title": "0-AMMR: Additive Shell Interaction",
|
|
"domain_id": 51,
|
|
"significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
|
|
},
|
|
{
|
|
"eq_number": 766,
|
|
"title": "0-AMMR: Additive Shell Interaction",
|
|
"domain_id": 51,
|
|
"significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
|
|
},
|
|
{
|
|
"eq_number": 766,
|
|
"title": "0-AMMR: Additive Shell Interaction",
|
|
"domain_id": 51,
|
|
"significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
|
|
},
|
|
{
|
|
"eq_number": 766,
|
|
"title": "0-AMMR: Additive Shell Interaction",
|
|
"domain_id": 51,
|
|
"significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
|
|
},
|
|
{
|
|
"eq_number": 766,
|
|
"title": "0-AMMR: Additive Shell Interaction",
|
|
"domain_id": 51,
|
|
"significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
|
|
},
|
|
{
|
|
"eq_number": 766,
|
|
"title": "0-AMMR: Additive Shell Interaction",
|
|
"domain_id": 51,
|
|
"significance": "Interactions between shells decompose additively. Basis for multi-shell computation in ENE."
|
|
},
|
|
{
|
|
"eq_number": 767,
|
|
"title": "0-AMMR: Temporal-Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
|
|
},
|
|
{
|
|
"eq_number": 767,
|
|
"title": "0-AMMR: Temporal-Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
|
|
},
|
|
{
|
|
"eq_number": 767,
|
|
"title": "0-AMMR: Temporal-Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
|
|
},
|
|
{
|
|
"eq_number": 767,
|
|
"title": "0-AMMR: Temporal-Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
|
|
},
|
|
{
|
|
"eq_number": 767,
|
|
"title": "0-AMMR: Temporal-Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
|
|
},
|
|
{
|
|
"eq_number": 767,
|
|
"title": "0-AMMR: Temporal-Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
|
|
},
|
|
{
|
|
"eq_number": 767,
|
|
"title": "0-AMMR: Temporal-Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
|
|
},
|
|
{
|
|
"eq_number": 767,
|
|
"title": "0-AMMR: Temporal-Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
|
|
},
|
|
{
|
|
"eq_number": 767,
|
|
"title": "0-AMMR: Temporal-Genetic Transduction",
|
|
"domain_id": 51,
|
|
"significance": "Temporal patterns transduced to genetic encoding. Potential for time-aware semantic compression in ENE."
|
|
},
|
|
{
|
|
"eq_number": 768,
|
|
"title": "0-AMMR: Information Capacity Bound",
|
|
"domain_id": 51,
|
|
"significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
|
|
},
|
|
{
|
|
"eq_number": 768,
|
|
"title": "0-AMMR: Information Capacity Bound",
|
|
"domain_id": 51,
|
|
"significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
|
|
},
|
|
{
|
|
"eq_number": 768,
|
|
"title": "0-AMMR: Information Capacity Bound",
|
|
"domain_id": 51,
|
|
"significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
|
|
},
|
|
{
|
|
"eq_number": 768,
|
|
"title": "0-AMMR: Information Capacity Bound",
|
|
"domain_id": 51,
|
|
"significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
|
|
},
|
|
{
|
|
"eq_number": 768,
|
|
"title": "0-AMMR: Information Capacity Bound",
|
|
"domain_id": 51,
|
|
"significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
|
|
},
|
|
{
|
|
"eq_number": 768,
|
|
"title": "0-AMMR: Information Capacity Bound",
|
|
"domain_id": 51,
|
|
"significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
|
|
},
|
|
{
|
|
"eq_number": 768,
|
|
"title": "0-AMMR: Information Capacity Bound",
|
|
"domain_id": 51,
|
|
"significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
|
|
},
|
|
{
|
|
"eq_number": 768,
|
|
"title": "0-AMMR: Information Capacity Bound",
|
|
"domain_id": 51,
|
|
"significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
|
|
},
|
|
{
|
|
"eq_number": 768,
|
|
"title": "0-AMMR: Information Capacity Bound",
|
|
"domain_id": 51,
|
|
"significance": "Genetic entropy bounds system information capacity. Provides theoretical limit for ENE compression."
|
|
},
|
|
{
|
|
"eq_number": 769,
|
|
"title": "0-AMMR: RG Flow Shell Preservation",
|
|
"domain_id": 51,
|
|
"significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
|
|
},
|
|
{
|
|
"eq_number": 769,
|
|
"title": "0-AMMR: RG Flow Shell Preservation",
|
|
"domain_id": 51,
|
|
"significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
|
|
},
|
|
{
|
|
"eq_number": 769,
|
|
"title": "0-AMMR: RG Flow Shell Preservation",
|
|
"domain_id": 51,
|
|
"significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
|
|
},
|
|
{
|
|
"eq_number": 769,
|
|
"title": "0-AMMR: RG Flow Shell Preservation",
|
|
"domain_id": 51,
|
|
"significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
|
|
},
|
|
{
|
|
"eq_number": 769,
|
|
"title": "0-AMMR: RG Flow Shell Preservation",
|
|
"domain_id": 51,
|
|
"significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
|
|
},
|
|
{
|
|
"eq_number": 769,
|
|
"title": "0-AMMR: RG Flow Shell Preservation",
|
|
"domain_id": 51,
|
|
"significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
|
|
},
|
|
{
|
|
"eq_number": 769,
|
|
"title": "0-AMMR: RG Flow Shell Preservation",
|
|
"domain_id": 51,
|
|
"significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
|
|
},
|
|
{
|
|
"eq_number": 769,
|
|
"title": "0-AMMR: RG Flow Shell Preservation",
|
|
"domain_id": 51,
|
|
"significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
|
|
},
|
|
{
|
|
"eq_number": 769,
|
|
"title": "0-AMMR: RG Flow Shell Preservation",
|
|
"domain_id": 51,
|
|
"significance": "Renormalization group flow preserves shell structure under scale transformations. Critical for ENE scale-invariant operations."
|
|
},
|
|
{
|
|
"eq_number": 770,
|
|
"title": "Graph Laplacian Spectral Decomposition",
|
|
"domain_id": 51,
|
|
"significance": "Spectral decomposition of graph Laplacian for graph-native computation. Enables eigenvector-based graph operations."
|
|
},
|
|
{
|
|
"eq_number": 770,
|
|
"title": "Graph Laplacian Spectral Decomposition",
|
|
"domain_id": 51,
|
|
"significance": "Spectral decomposition of graph Laplacian for graph-native computation. Enables eigenvector-based graph operations."
|
|
},
|
|
{
|
|
"eq_number": 770,
|
|
"title": "Graph Laplacian Spectral Decomposition",
|
|
"domain_id": 51,
|
|
"significance": "Spectral decomposition of graph Laplacian for graph-native computation. Enables eigenvector-based graph operations."
|
|
},
|
|
{
|
|
"eq_number": 770,
|
|
"title": "Graph Laplacian Spectral Decomposition",
|
|
"domain_id": 51,
|
|
"significance": "Spectral decomposition of graph Laplacian for graph-native computation. Enables eigenvector-based graph operations."
|
|
},
|
|
{
|
|
"eq_number": 770,
|
|
"title": "Graph Laplacian Spectral Decomposition",
|
|
"domain_id": 51,
|
|
"significance": "Spectral decomposition of graph Laplacian for graph-native computation. Enables eigenvector-based graph operations."
|
|
},
|
|
{
|
|
"eq_number": 770,
|
|
"title": "Graph Laplacian Spectral Decomposition",
|
|
"domain_id": 51,
|
|
"significance": "Spectral decomposition of graph Laplacian for graph-native computation. Enables eigenvector-based graph operations."
|
|
},
|
|
{
|
|
"eq_number": 770,
|
|
"title": "Graph Laplacian Spectral Decomposition",
|
|
"domain_id": 51,
|
|
"significance": "Spectral decomposition of graph Laplacian for graph-native computation. Enables eigenvector-based graph operations."
|
|
},
|
|
{
|
|
"eq_number": 770,
|
|
"title": "Graph Laplacian Spectral Decomposition",
|
|
"domain_id": 51,
|
|
"significance": "Spectral decomposition of graph Laplacian for graph-native computation. Enables eigenvector-based graph operations."
|
|
},
|
|
{
|
|
"eq_number": 771,
|
|
"title": "Graph Attention Mechanism",
|
|
"domain_id": 51,
|
|
"significance": "Attention-based message passing for graph neural networks. Enables context-aware graph operations."
|
|
},
|
|
{
|
|
"eq_number": 771,
|
|
"title": "Graph Attention Mechanism",
|
|
"domain_id": 51,
|
|
"significance": "Attention-based message passing for graph neural networks. Enables context-aware graph operations."
|
|
},
|
|
{
|
|
"eq_number": 771,
|
|
"title": "Graph Attention Mechanism",
|
|
"domain_id": 51,
|
|
"significance": "Attention-based message passing for graph neural networks. Enables context-aware graph operations."
|
|
},
|
|
{
|
|
"eq_number": 771,
|
|
"title": "Graph Attention Mechanism",
|
|
"domain_id": 51,
|
|
"significance": "Attention-based message passing for graph neural networks. Enables context-aware graph operations."
|
|
},
|
|
{
|
|
"eq_number": 771,
|
|
"title": "Graph Attention Mechanism",
|
|
"domain_id": 51,
|
|
"significance": "Attention-based message passing for graph neural networks. Enables context-aware graph operations."
|
|
},
|
|
{
|
|
"eq_number": 771,
|
|
"title": "Graph Attention Mechanism",
|
|
"domain_id": 51,
|
|
"significance": "Attention-based message passing for graph neural networks. Enables context-aware graph operations."
|
|
},
|
|
{
|
|
"eq_number": 771,
|
|
"title": "Graph Attention Mechanism",
|
|
"domain_id": 51,
|
|
"significance": "Attention-based message passing for graph neural networks. Enables context-aware graph operations."
|
|
},
|
|
{
|
|
"eq_number": 771,
|
|
"title": "Graph Attention Mechanism",
|
|
"domain_id": 51,
|
|
"significance": "Attention-based message passing for graph neural networks. Enables context-aware graph operations."
|
|
},
|
|
{
|
|
"eq_number": 772,
|
|
"title": "Graph Convolution",
|
|
"domain_id": 51,
|
|
"significance": "Spectral graph convolution for graph-native processing. Enables convolution operations on graph-structured data."
|
|
},
|
|
{
|
|
"eq_number": 772,
|
|
"title": "Graph Convolution",
|
|
"domain_id": 51,
|
|
"significance": "Spectral graph convolution for graph-native processing. Enables convolution operations on graph-structured data."
|
|
},
|
|
{
|
|
"eq_number": 772,
|
|
"title": "Graph Convolution",
|
|
"domain_id": 51,
|
|
"significance": "Spectral graph convolution for graph-native processing. Enables convolution operations on graph-structured data."
|
|
},
|
|
{
|
|
"eq_number": 772,
|
|
"title": "Graph Convolution",
|
|
"domain_id": 51,
|
|
"significance": "Spectral graph convolution for graph-native processing. Enables convolution operations on graph-structured data."
|
|
},
|
|
{
|
|
"eq_number": 772,
|
|
"title": "Graph Convolution",
|
|
"domain_id": 51,
|
|
"significance": "Spectral graph convolution for graph-native processing. Enables convolution operations on graph-structured data."
|
|
},
|
|
{
|
|
"eq_number": 772,
|
|
"title": "Graph Convolution",
|
|
"domain_id": 51,
|
|
"significance": "Spectral graph convolution for graph-native processing. Enables convolution operations on graph-structured data."
|
|
},
|
|
{
|
|
"eq_number": 772,
|
|
"title": "Graph Convolution",
|
|
"domain_id": 51,
|
|
"significance": "Spectral graph convolution for graph-native processing. Enables convolution operations on graph-structured data."
|
|
},
|
|
{
|
|
"eq_number": 772,
|
|
"title": "Graph Convolution",
|
|
"domain_id": 51,
|
|
"significance": "Spectral graph convolution for graph-native processing. Enables convolution operations on graph-structured data."
|
|
},
|
|
{
|
|
"eq_number": 773,
|
|
"title": "WGSL Vector Swizzle Operation",
|
|
"domain_id": 51,
|
|
"significance": "GPU vector component swizzling for efficient parallel processing. Enables flexible vector manipulation on GPU."
|
|
},
|
|
{
|
|
"eq_number": 773,
|
|
"title": "WGSL Vector Swizzle Operation",
|
|
"domain_id": 51,
|
|
"significance": "GPU vector component swizzling for efficient parallel processing. Enables flexible vector manipulation on GPU."
|
|
},
|
|
{
|
|
"eq_number": 773,
|
|
"title": "WGSL Vector Swizzle Operation",
|
|
"domain_id": 51,
|
|
"significance": "GPU vector component swizzling for efficient parallel processing. Enables flexible vector manipulation on GPU."
|
|
},
|
|
{
|
|
"eq_number": 773,
|
|
"title": "WGSL Vector Swizzle Operation",
|
|
"domain_id": 51,
|
|
"significance": "GPU vector component swizzling for efficient parallel processing. Enables flexible vector manipulation on GPU."
|
|
},
|
|
{
|
|
"eq_number": 773,
|
|
"title": "WGSL Vector Swizzle Operation",
|
|
"domain_id": 51,
|
|
"significance": "GPU vector component swizzling for efficient parallel processing. Enables flexible vector manipulation on GPU."
|
|
},
|
|
{
|
|
"eq_number": 773,
|
|
"title": "WGSL Vector Swizzle Operation",
|
|
"domain_id": 51,
|
|
"significance": "GPU vector component swizzling for efficient parallel processing. Enables flexible vector manipulation on GPU."
|
|
},
|
|
{
|
|
"eq_number": 773,
|
|
"title": "WGSL Vector Swizzle Operation",
|
|
"domain_id": 51,
|
|
"significance": "GPU vector component swizzling for efficient parallel processing. Enables flexible vector manipulation on GPU."
|
|
},
|
|
{
|
|
"eq_number": 773,
|
|
"title": "WGSL Vector Swizzle Operation",
|
|
"domain_id": 51,
|
|
"significance": "GPU vector component swizzling for efficient parallel processing. Enables flexible vector manipulation on GPU."
|
|
},
|
|
{
|
|
"eq_number": 774,
|
|
"title": "WGSL Workgroup Synchronization",
|
|
"domain_id": 51,
|
|
"significance": "Barrier synchronization for GPU workgroups. Ensures correct parallel execution order."
|
|
},
|
|
{
|
|
"eq_number": 774,
|
|
"title": "WGSL Workgroup Synchronization",
|
|
"domain_id": 51,
|
|
"significance": "Barrier synchronization for GPU workgroups. Ensures correct parallel execution order."
|
|
},
|
|
{
|
|
"eq_number": 774,
|
|
"title": "WGSL Workgroup Synchronization",
|
|
"domain_id": 51,
|
|
"significance": "Barrier synchronization for GPU workgroups. Ensures correct parallel execution order."
|
|
},
|
|
{
|
|
"eq_number": 774,
|
|
"title": "WGSL Workgroup Synchronization",
|
|
"domain_id": 51,
|
|
"significance": "Barrier synchronization for GPU workgroups. Ensures correct parallel execution order."
|
|
},
|
|
{
|
|
"eq_number": 774,
|
|
"title": "WGSL Workgroup Synchronization",
|
|
"domain_id": 51,
|
|
"significance": "Barrier synchronization for GPU workgroups. Ensures correct parallel execution order."
|
|
},
|
|
{
|
|
"eq_number": 774,
|
|
"title": "WGSL Workgroup Synchronization",
|
|
"domain_id": 51,
|
|
"significance": "Barrier synchronization for GPU workgroups. Ensures correct parallel execution order."
|
|
},
|
|
{
|
|
"eq_number": 774,
|
|
"title": "WGSL Workgroup Synchronization",
|
|
"domain_id": 51,
|
|
"significance": "Barrier synchronization for GPU workgroups. Ensures correct parallel execution order."
|
|
},
|
|
{
|
|
"eq_number": 774,
|
|
"title": "WGSL Workgroup Synchronization",
|
|
"domain_id": 51,
|
|
"significance": "Barrier synchronization for GPU workgroups. Ensures correct parallel execution order."
|
|
},
|
|
{
|
|
"eq_number": 775,
|
|
"title": "WGSL Shared Memory Reduction",
|
|
"domain_id": 51,
|
|
"significance": "Parallel reduction in GPU shared memory. Enables efficient aggregation across workgroup."
|
|
},
|
|
{
|
|
"eq_number": 775,
|
|
"title": "WGSL Shared Memory Reduction",
|
|
"domain_id": 51,
|
|
"significance": "Parallel reduction in GPU shared memory. Enables efficient aggregation across workgroup."
|
|
},
|
|
{
|
|
"eq_number": 775,
|
|
"title": "WGSL Shared Memory Reduction",
|
|
"domain_id": 51,
|
|
"significance": "Parallel reduction in GPU shared memory. Enables efficient aggregation across workgroup."
|
|
},
|
|
{
|
|
"eq_number": 775,
|
|
"title": "WGSL Shared Memory Reduction",
|
|
"domain_id": 51,
|
|
"significance": "Parallel reduction in GPU shared memory. Enables efficient aggregation across workgroup."
|
|
},
|
|
{
|
|
"eq_number": 775,
|
|
"title": "WGSL Shared Memory Reduction",
|
|
"domain_id": 51,
|
|
"significance": "Parallel reduction in GPU shared memory. Enables efficient aggregation across workgroup."
|
|
},
|
|
{
|
|
"eq_number": 775,
|
|
"title": "WGSL Shared Memory Reduction",
|
|
"domain_id": 51,
|
|
"significance": "Parallel reduction in GPU shared memory. Enables efficient aggregation across workgroup."
|
|
},
|
|
{
|
|
"eq_number": 775,
|
|
"title": "WGSL Shared Memory Reduction",
|
|
"domain_id": 51,
|
|
"significance": "Parallel reduction in GPU shared memory. Enables efficient aggregation across workgroup."
|
|
},
|
|
{
|
|
"eq_number": 775,
|
|
"title": "WGSL Shared Memory Reduction",
|
|
"domain_id": 51,
|
|
"significance": "Parallel reduction in GPU shared memory. Enables efficient aggregation across workgroup."
|
|
},
|
|
{
|
|
"eq_number": 776,
|
|
"title": "Vector Append Operation",
|
|
"domain_id": 51,
|
|
"significance": "Dynamic vector appending for incremental processing. Enables streaming vector operations."
|
|
},
|
|
{
|
|
"eq_number": 776,
|
|
"title": "Vector Append Operation",
|
|
"domain_id": 51,
|
|
"significance": "Dynamic vector appending for incremental processing. Enables streaming vector operations."
|
|
},
|
|
{
|
|
"eq_number": 776,
|
|
"title": "Vector Append Operation",
|
|
"domain_id": 51,
|
|
"significance": "Dynamic vector appending for incremental processing. Enables streaming vector operations."
|
|
},
|
|
{
|
|
"eq_number": 776,
|
|
"title": "Vector Append Operation",
|
|
"domain_id": 51,
|
|
"significance": "Dynamic vector appending for incremental processing. Enables streaming vector operations."
|
|
},
|
|
{
|
|
"eq_number": 776,
|
|
"title": "Vector Append Operation",
|
|
"domain_id": 51,
|
|
"significance": "Dynamic vector appending for incremental processing. Enables streaming vector operations."
|
|
},
|
|
{
|
|
"eq_number": 776,
|
|
"title": "Vector Append Operation",
|
|
"domain_id": 51,
|
|
"significance": "Dynamic vector appending for incremental processing. Enables streaming vector operations."
|
|
},
|
|
{
|
|
"eq_number": 776,
|
|
"title": "Vector Append Operation",
|
|
"domain_id": 51,
|
|
"significance": "Dynamic vector appending for incremental processing. Enables streaming vector operations."
|
|
},
|
|
{
|
|
"eq_number": 776,
|
|
"title": "Vector Append Operation",
|
|
"domain_id": 51,
|
|
"significance": "Dynamic vector appending for incremental processing. Enables streaming vector operations."
|
|
},
|
|
{
|
|
"eq_number": 777,
|
|
"title": "Vector Concatenation",
|
|
"domain_id": 51,
|
|
"significance": "Efficient vector concatenation for batch processing. Enables combining multiple vectors."
|
|
},
|
|
{
|
|
"eq_number": 777,
|
|
"title": "Vector Concatenation",
|
|
"domain_id": 51,
|
|
"significance": "Efficient vector concatenation for batch processing. Enables combining multiple vectors."
|
|
},
|
|
{
|
|
"eq_number": 777,
|
|
"title": "Vector Concatenation",
|
|
"domain_id": 51,
|
|
"significance": "Efficient vector concatenation for batch processing. Enables combining multiple vectors."
|
|
},
|
|
{
|
|
"eq_number": 777,
|
|
"title": "Vector Concatenation",
|
|
"domain_id": 51,
|
|
"significance": "Efficient vector concatenation for batch processing. Enables combining multiple vectors."
|
|
},
|
|
{
|
|
"eq_number": 777,
|
|
"title": "Vector Concatenation",
|
|
"domain_id": 51,
|
|
"significance": "Efficient vector concatenation for batch processing. Enables combining multiple vectors."
|
|
},
|
|
{
|
|
"eq_number": 777,
|
|
"title": "Vector Concatenation",
|
|
"domain_id": 51,
|
|
"significance": "Efficient vector concatenation for batch processing. Enables combining multiple vectors."
|
|
},
|
|
{
|
|
"eq_number": 777,
|
|
"title": "Vector Concatenation",
|
|
"domain_id": 51,
|
|
"significance": "Efficient vector concatenation for batch processing. Enables combining multiple vectors."
|
|
},
|
|
{
|
|
"eq_number": 777,
|
|
"title": "Vector Concatenation",
|
|
"domain_id": 51,
|
|
"significance": "Efficient vector concatenation for batch processing. Enables combining multiple vectors."
|
|
},
|
|
{
|
|
"eq_number": 778,
|
|
"title": "Vector Append with Capacity Growth",
|
|
"domain_id": 51,
|
|
"significance": "Amortized O(1) append with geometric capacity growth. Optimizes memory allocation."
|
|
},
|
|
{
|
|
"eq_number": 778,
|
|
"title": "Vector Append with Capacity Growth",
|
|
"domain_id": 51,
|
|
"significance": "Amortized O(1) append with geometric capacity growth. Optimizes memory allocation."
|
|
},
|
|
{
|
|
"eq_number": 778,
|
|
"title": "Vector Append with Capacity Growth",
|
|
"domain_id": 51,
|
|
"significance": "Amortized O(1) append with geometric capacity growth. Optimizes memory allocation."
|
|
},
|
|
{
|
|
"eq_number": 778,
|
|
"title": "Vector Append with Capacity Growth",
|
|
"domain_id": 51,
|
|
"significance": "Amortized O(1) append with geometric capacity growth. Optimizes memory allocation."
|
|
},
|
|
{
|
|
"eq_number": 778,
|
|
"title": "Vector Append with Capacity Growth",
|
|
"domain_id": 51,
|
|
"significance": "Amortized O(1) append with geometric capacity growth. Optimizes memory allocation."
|
|
},
|
|
{
|
|
"eq_number": 778,
|
|
"title": "Vector Append with Capacity Growth",
|
|
"domain_id": 51,
|
|
"significance": "Amortized O(1) append with geometric capacity growth. Optimizes memory allocation."
|
|
},
|
|
{
|
|
"eq_number": 778,
|
|
"title": "Vector Append with Capacity Growth",
|
|
"domain_id": 51,
|
|
"significance": "Amortized O(1) append with geometric capacity growth. Optimizes memory allocation."
|
|
},
|
|
{
|
|
"eq_number": 778,
|
|
"title": "Vector Append with Capacity Growth",
|
|
"domain_id": 51,
|
|
"significance": "Amortized O(1) append with geometric capacity growth. Optimizes memory allocation."
|
|
},
|
|
{
|
|
"eq_number": 779,
|
|
"title": "Graph Vector Append",
|
|
"domain_id": 51,
|
|
"significance": "Appending vectors to graph nodes for incremental graph updates. Enables dynamic graph processing."
|
|
},
|
|
{
|
|
"eq_number": 779,
|
|
"title": "Graph Vector Append",
|
|
"domain_id": 51,
|
|
"significance": "Appending vectors to graph nodes for incremental graph updates. Enables dynamic graph processing."
|
|
},
|
|
{
|
|
"eq_number": 779,
|
|
"title": "Graph Vector Append",
|
|
"domain_id": 51,
|
|
"significance": "Appending vectors to graph nodes for incremental graph updates. Enables dynamic graph processing."
|
|
},
|
|
{
|
|
"eq_number": 779,
|
|
"title": "Graph Vector Append",
|
|
"domain_id": 51,
|
|
"significance": "Appending vectors to graph nodes for incremental graph updates. Enables dynamic graph processing."
|
|
},
|
|
{
|
|
"eq_number": 779,
|
|
"title": "Graph Vector Append",
|
|
"domain_id": 51,
|
|
"significance": "Appending vectors to graph nodes for incremental graph updates. Enables dynamic graph processing."
|
|
},
|
|
{
|
|
"eq_number": 779,
|
|
"title": "Graph Vector Append",
|
|
"domain_id": 51,
|
|
"significance": "Appending vectors to graph nodes for incremental graph updates. Enables dynamic graph processing."
|
|
},
|
|
{
|
|
"eq_number": 779,
|
|
"title": "Graph Vector Append",
|
|
"domain_id": 51,
|
|
"significance": "Appending vectors to graph nodes for incremental graph updates. Enables dynamic graph processing."
|
|
},
|
|
{
|
|
"eq_number": 779,
|
|
"title": "Graph Vector Append",
|
|
"domain_id": 51,
|
|
"significance": "Appending vectors to graph nodes for incremental graph updates. Enables dynamic graph processing."
|
|
},
|
|
{
|
|
"eq_number": 780,
|
|
"title": "HNSW Hierarchical Navigable Small World",
|
|
"domain_id": 51,
|
|
"significance": "Graph-based approximate nearest neighbor search. Combines probability skip list with navigable small world graphs for fast vector similarity search."
|
|
},
|
|
{
|
|
"eq_number": 780,
|
|
"title": "HNSW Hierarchical Navigable Small World",
|
|
"domain_id": 51,
|
|
"significance": "Graph-based approximate nearest neighbor search. Combines probability skip list with navigable small world graphs for fast vector similarity search."
|
|
},
|
|
{
|
|
"eq_number": 780,
|
|
"title": "HNSW Hierarchical Navigable Small World",
|
|
"domain_id": 51,
|
|
"significance": "Graph-based approximate nearest neighbor search. Combines probability skip list with navigable small world graphs for fast vector similarity search."
|
|
},
|
|
{
|
|
"eq_number": 780,
|
|
"title": "HNSW Hierarchical Navigable Small World",
|
|
"domain_id": 51,
|
|
"significance": "Graph-based approximate nearest neighbor search. Combines probability skip list with navigable small world graphs for fast vector similarity search."
|
|
},
|
|
{
|
|
"eq_number": 780,
|
|
"title": "HNSW Hierarchical Navigable Small World",
|
|
"domain_id": 51,
|
|
"significance": "Graph-based approximate nearest neighbor search. Combines probability skip list with navigable small world graphs for fast vector similarity search."
|
|
},
|
|
{
|
|
"eq_number": 780,
|
|
"title": "HNSW Hierarchical Navigable Small World",
|
|
"domain_id": 51,
|
|
"significance": "Graph-based approximate nearest neighbor search. Combines probability skip list with navigable small world graphs for fast vector similarity search."
|
|
},
|
|
{
|
|
"eq_number": 780,
|
|
"title": "HNSW Hierarchical Navigable Small World",
|
|
"domain_id": 51,
|
|
"significance": "Graph-based approximate nearest neighbor search. Combines probability skip list with navigable small world graphs for fast vector similarity search."
|
|
},
|
|
{
|
|
"eq_number": 781,
|
|
"title": "Approximate Nearest Neighbor Search",
|
|
"domain_id": 51,
|
|
"significance": "Efficient vector search with slight accuracy penalty for massive speedup. Uses HNSW for O(log N) search complexity."
|
|
},
|
|
{
|
|
"eq_number": 781,
|
|
"title": "Approximate Nearest Neighbor Search",
|
|
"domain_id": 51,
|
|
"significance": "Efficient vector search with slight accuracy penalty for massive speedup. Uses HNSW for O(log N) search complexity."
|
|
},
|
|
{
|
|
"eq_number": 781,
|
|
"title": "Approximate Nearest Neighbor Search",
|
|
"domain_id": 51,
|
|
"significance": "Efficient vector search with slight accuracy penalty for massive speedup. Uses HNSW for O(log N) search complexity."
|
|
},
|
|
{
|
|
"eq_number": 781,
|
|
"title": "Approximate Nearest Neighbor Search",
|
|
"domain_id": 51,
|
|
"significance": "Efficient vector search with slight accuracy penalty for massive speedup. Uses HNSW for O(log N) search complexity."
|
|
},
|
|
{
|
|
"eq_number": 781,
|
|
"title": "Approximate Nearest Neighbor Search",
|
|
"domain_id": 51,
|
|
"significance": "Efficient vector search with slight accuracy penalty for massive speedup. Uses HNSW for O(log N) search complexity."
|
|
},
|
|
{
|
|
"eq_number": 781,
|
|
"title": "Approximate Nearest Neighbor Search",
|
|
"domain_id": 51,
|
|
"significance": "Efficient vector search with slight accuracy penalty for massive speedup. Uses HNSW for O(log N) search complexity."
|
|
},
|
|
{
|
|
"eq_number": 781,
|
|
"title": "Approximate Nearest Neighbor Search",
|
|
"domain_id": 51,
|
|
"significance": "Efficient vector search with slight accuracy penalty for massive speedup. Uses HNSW for O(log N) search complexity."
|
|
},
|
|
{
|
|
"eq_number": 782,
|
|
"title": "Proximity Graph Edge Probability",
|
|
"domain_id": 51,
|
|
"significance": "Probability of edge creation based on vector proximity in HNSW. Controls graph connectivity."
|
|
},
|
|
{
|
|
"eq_number": 782,
|
|
"title": "Proximity Graph Edge Probability",
|
|
"domain_id": 51,
|
|
"significance": "Probability of edge creation based on vector proximity in HNSW. Controls graph connectivity."
|
|
},
|
|
{
|
|
"eq_number": 782,
|
|
"title": "Proximity Graph Edge Probability",
|
|
"domain_id": 51,
|
|
"significance": "Probability of edge creation based on vector proximity in HNSW. Controls graph connectivity."
|
|
},
|
|
{
|
|
"eq_number": 782,
|
|
"title": "Proximity Graph Edge Probability",
|
|
"domain_id": 51,
|
|
"significance": "Probability of edge creation based on vector proximity in HNSW. Controls graph connectivity."
|
|
},
|
|
{
|
|
"eq_number": 782,
|
|
"title": "Proximity Graph Edge Probability",
|
|
"domain_id": 51,
|
|
"significance": "Probability of edge creation based on vector proximity in HNSW. Controls graph connectivity."
|
|
},
|
|
{
|
|
"eq_number": 782,
|
|
"title": "Proximity Graph Edge Probability",
|
|
"domain_id": 51,
|
|
"significance": "Probability of edge creation based on vector proximity in HNSW. Controls graph connectivity."
|
|
},
|
|
{
|
|
"eq_number": 782,
|
|
"title": "Proximity Graph Edge Probability",
|
|
"domain_id": 51,
|
|
"significance": "Probability of edge creation based on vector proximity in HNSW. Controls graph connectivity."
|
|
},
|
|
{
|
|
"eq_number": 783,
|
|
"title": "Property Graph Traversal",
|
|
"domain_id": 51,
|
|
"significance": "Efficient traversal of property graphs with nodes, edges, and properties. Basis for graph query languages like Cypher and GSQL."
|
|
},
|
|
{
|
|
"eq_number": 783,
|
|
"title": "Property Graph Traversal",
|
|
"domain_id": 51,
|
|
"significance": "Efficient traversal of property graphs with nodes, edges, and properties. Basis for graph query languages like Cypher and GSQL."
|
|
},
|
|
{
|
|
"eq_number": 783,
|
|
"title": "Property Graph Traversal",
|
|
"domain_id": 51,
|
|
"significance": "Efficient traversal of property graphs with nodes, edges, and properties. Basis for graph query languages like Cypher and GSQL."
|
|
},
|
|
{
|
|
"eq_number": 783,
|
|
"title": "Property Graph Traversal",
|
|
"domain_id": 51,
|
|
"significance": "Efficient traversal of property graphs with nodes, edges, and properties. Basis for graph query languages like Cypher and GSQL."
|
|
},
|
|
{
|
|
"eq_number": 783,
|
|
"title": "Property Graph Traversal",
|
|
"domain_id": 51,
|
|
"significance": "Efficient traversal of property graphs with nodes, edges, and properties. Basis for graph query languages like Cypher and GSQL."
|
|
},
|
|
{
|
|
"eq_number": 783,
|
|
"title": "Property Graph Traversal",
|
|
"domain_id": 51,
|
|
"significance": "Efficient traversal of property graphs with nodes, edges, and properties. Basis for graph query languages like Cypher and GSQL."
|
|
},
|
|
{
|
|
"eq_number": 783,
|
|
"title": "Property Graph Traversal",
|
|
"domain_id": 51,
|
|
"significance": "Efficient traversal of property graphs with nodes, edges, and properties. Basis for graph query languages like Cypher and GSQL."
|
|
},
|
|
{
|
|
"eq_number": 784,
|
|
"title": "Graph Pattern Matching",
|
|
"domain_id": 51,
|
|
"significance": "Pattern-based query in graph databases. Enables complex relationship queries like Cypher's MATCH clause."
|
|
},
|
|
{
|
|
"eq_number": 784,
|
|
"title": "Graph Pattern Matching",
|
|
"domain_id": 51,
|
|
"significance": "Pattern-based query in graph databases. Enables complex relationship queries like Cypher's MATCH clause."
|
|
},
|
|
{
|
|
"eq_number": 784,
|
|
"title": "Graph Pattern Matching",
|
|
"domain_id": 51,
|
|
"significance": "Pattern-based query in graph databases. Enables complex relationship queries like Cypher's MATCH clause."
|
|
},
|
|
{
|
|
"eq_number": 784,
|
|
"title": "Graph Pattern Matching",
|
|
"domain_id": 51,
|
|
"significance": "Pattern-based query in graph databases. Enables complex relationship queries like Cypher's MATCH clause."
|
|
},
|
|
{
|
|
"eq_number": 784,
|
|
"title": "Graph Pattern Matching",
|
|
"domain_id": 51,
|
|
"significance": "Pattern-based query in graph databases. Enables complex relationship queries like Cypher's MATCH clause."
|
|
},
|
|
{
|
|
"eq_number": 784,
|
|
"title": "Graph Pattern Matching",
|
|
"domain_id": 51,
|
|
"significance": "Pattern-based query in graph databases. Enables complex relationship queries like Cypher's MATCH clause."
|
|
},
|
|
{
|
|
"eq_number": 784,
|
|
"title": "Graph Pattern Matching",
|
|
"domain_id": 51,
|
|
"significance": "Pattern-based query in graph databases. Enables complex relationship queries like Cypher's MATCH clause."
|
|
},
|
|
{
|
|
"eq_number": 785,
|
|
"title": "Multi-Model Query Integration",
|
|
"domain_id": 51,
|
|
"significance": "Unified querying across document, key-value, and graph models. Enables ArangoDB-style multi-model databases."
|
|
},
|
|
{
|
|
"eq_number": 785,
|
|
"title": "Multi-Model Query Integration",
|
|
"domain_id": 51,
|
|
"significance": "Unified querying across document, key-value, and graph models. Enables ArangoDB-style multi-model databases."
|
|
},
|
|
{
|
|
"eq_number": 785,
|
|
"title": "Multi-Model Query Integration",
|
|
"domain_id": 51,
|
|
"significance": "Unified querying across document, key-value, and graph models. Enables ArangoDB-style multi-model databases."
|
|
},
|
|
{
|
|
"eq_number": 785,
|
|
"title": "Multi-Model Query Integration",
|
|
"domain_id": 51,
|
|
"significance": "Unified querying across document, key-value, and graph models. Enables ArangoDB-style multi-model databases."
|
|
},
|
|
{
|
|
"eq_number": 785,
|
|
"title": "Multi-Model Query Integration",
|
|
"domain_id": 51,
|
|
"significance": "Unified querying across document, key-value, and graph models. Enables ArangoDB-style multi-model databases."
|
|
},
|
|
{
|
|
"eq_number": 785,
|
|
"title": "Multi-Model Query Integration",
|
|
"domain_id": 51,
|
|
"significance": "Unified querying across document, key-value, and graph models. Enables ArangoDB-style multi-model databases."
|
|
},
|
|
{
|
|
"eq_number": 785,
|
|
"title": "Multi-Model Query Integration",
|
|
"domain_id": 51,
|
|
"significance": "Unified querying across document, key-value, and graph models. Enables ArangoDB-style multi-model databases."
|
|
},
|
|
{
|
|
"eq_number": 786,
|
|
"title": "Parallel Graph Processing",
|
|
"domain_id": 51,
|
|
"significance": "Native parallel graph engine for real-time analytics. TigerGraph-style parallel processing for massive graphs."
|
|
},
|
|
{
|
|
"eq_number": 786,
|
|
"title": "Parallel Graph Processing",
|
|
"domain_id": 51,
|
|
"significance": "Native parallel graph engine for real-time analytics. TigerGraph-style parallel processing for massive graphs."
|
|
},
|
|
{
|
|
"eq_number": 786,
|
|
"title": "Parallel Graph Processing",
|
|
"domain_id": 51,
|
|
"significance": "Native parallel graph engine for real-time analytics. TigerGraph-style parallel processing for massive graphs."
|
|
},
|
|
{
|
|
"eq_number": 786,
|
|
"title": "Parallel Graph Processing",
|
|
"domain_id": 51,
|
|
"significance": "Native parallel graph engine for real-time analytics. TigerGraph-style parallel processing for massive graphs."
|
|
},
|
|
{
|
|
"eq_number": 786,
|
|
"title": "Parallel Graph Processing",
|
|
"domain_id": 51,
|
|
"significance": "Native parallel graph engine for real-time analytics. TigerGraph-style parallel processing for massive graphs."
|
|
},
|
|
{
|
|
"eq_number": 786,
|
|
"title": "Parallel Graph Processing",
|
|
"domain_id": 51,
|
|
"significance": "Native parallel graph engine for real-time analytics. TigerGraph-style parallel processing for massive graphs."
|
|
},
|
|
{
|
|
"eq_number": 786,
|
|
"title": "Parallel Graph Processing",
|
|
"domain_id": 51,
|
|
"significance": "Native parallel graph engine for real-time analytics. TigerGraph-style parallel processing for massive graphs."
|
|
},
|
|
{
|
|
"eq_number": 787,
|
|
"title": "Shockwave Alignment and Relaxation",
|
|
"domain_id": 51,
|
|
"significance": "Quasi-charged cells align under shockwave, propagate charge symmetrically, then dissipate and relax. Four-phase cycle: anisotropic \u2192 shock_aligned \u2192 discharge \u2192 relaxed."
|
|
},
|
|
{
|
|
"eq_number": 787,
|
|
"title": "Shockwave Alignment and Relaxation",
|
|
"domain_id": 51,
|
|
"significance": "Quasi-charged cells align under shockwave, propagate charge symmetrically, then dissipate and relax. Four-phase cycle: anisotropic \u2192 shock_aligned \u2192 discharge \u2192 relaxed."
|
|
},
|
|
{
|
|
"eq_number": 787,
|
|
"title": "Shockwave Alignment and Relaxation",
|
|
"domain_id": 51,
|
|
"significance": "Quasi-charged cells align under shockwave, propagate charge symmetrically, then dissipate and relax. Four-phase cycle: anisotropic \u2192 shock_aligned \u2192 discharge \u2192 relaxed."
|
|
},
|
|
{
|
|
"eq_number": 787,
|
|
"title": "Shockwave Alignment and Relaxation",
|
|
"domain_id": 51,
|
|
"significance": "Quasi-charged cells align under shockwave, propagate charge symmetrically, then dissipate and relax. Four-phase cycle: anisotropic \u2192 shock_aligned \u2192 discharge \u2192 relaxed."
|
|
},
|
|
{
|
|
"eq_number": 787,
|
|
"title": "Shockwave Alignment and Relaxation",
|
|
"domain_id": 51,
|
|
"significance": "Quasi-charged cells align under shockwave, propagate charge symmetrically, then dissipate and relax. Four-phase cycle: anisotropic \u2192 shock_aligned \u2192 discharge \u2192 relaxed."
|
|
},
|
|
{
|
|
"eq_number": 787,
|
|
"title": "Shockwave Alignment and Relaxation",
|
|
"domain_id": 51,
|
|
"significance": "Quasi-charged cells align under shockwave, propagate charge symmetrically, then dissipate and relax. Four-phase cycle: anisotropic \u2192 shock_aligned \u2192 discharge \u2192 relaxed."
|
|
},
|
|
{
|
|
"eq_number": 788,
|
|
"title": "Phonon Force Law",
|
|
"domain_id": 51,
|
|
"significance": "Phonon correlation structure for self-healing. Force decays exponentially with Manhattan distance and oscillates with coherence period."
|
|
},
|
|
{
|
|
"eq_number": 788,
|
|
"title": "Phonon Force Law",
|
|
"domain_id": 51,
|
|
"significance": "Phonon correlation structure for self-healing. Force decays exponentially with Manhattan distance and oscillates with coherence period."
|
|
},
|
|
{
|
|
"eq_number": 788,
|
|
"title": "Phonon Force Law",
|
|
"domain_id": 51,
|
|
"significance": "Phonon correlation structure for self-healing. Force decays exponentially with Manhattan distance and oscillates with coherence period."
|
|
},
|
|
{
|
|
"eq_number": 788,
|
|
"title": "Phonon Force Law",
|
|
"domain_id": 51,
|
|
"significance": "Phonon correlation structure for self-healing. Force decays exponentially with Manhattan distance and oscillates with coherence period."
|
|
},
|
|
{
|
|
"eq_number": 788,
|
|
"title": "Phonon Force Law",
|
|
"domain_id": 51,
|
|
"significance": "Phonon correlation structure for self-healing. Force decays exponentially with Manhattan distance and oscillates with coherence period."
|
|
},
|
|
{
|
|
"eq_number": 788,
|
|
"title": "Phonon Force Law",
|
|
"domain_id": 51,
|
|
"significance": "Phonon correlation structure for self-healing. Force decays exponentially with Manhattan distance and oscillates with coherence period."
|
|
},
|
|
{
|
|
"eq_number": 789,
|
|
"title": "Cartesian Phonon Prime Integration",
|
|
"domain_id": 51,
|
|
"significance": "256\u00d7256 Cartesian coordinate space with 16-bit fixed addressing and Manhattan distance metric for hardware-efficient phonon transport."
|
|
},
|
|
{
|
|
"eq_number": 789,
|
|
"title": "Cartesian Phonon Prime Integration",
|
|
"domain_id": 51,
|
|
"significance": "256\u00d7256 Cartesian coordinate space with 16-bit fixed addressing and Manhattan distance metric for hardware-efficient phonon transport."
|
|
},
|
|
{
|
|
"eq_number": 789,
|
|
"title": "Cartesian Phonon Prime Integration",
|
|
"domain_id": 51,
|
|
"significance": "256\u00d7256 Cartesian coordinate space with 16-bit fixed addressing and Manhattan distance metric for hardware-efficient phonon transport."
|
|
},
|
|
{
|
|
"eq_number": 789,
|
|
"title": "Cartesian Phonon Prime Integration",
|
|
"domain_id": 51,
|
|
"significance": "256\u00d7256 Cartesian coordinate space with 16-bit fixed addressing and Manhattan distance metric for hardware-efficient phonon transport."
|
|
},
|
|
{
|
|
"eq_number": 789,
|
|
"title": "Cartesian Phonon Prime Integration",
|
|
"domain_id": 51,
|
|
"significance": "256\u00d7256 Cartesian coordinate space with 16-bit fixed addressing and Manhattan distance metric for hardware-efficient phonon transport."
|
|
},
|
|
{
|
|
"eq_number": 789,
|
|
"title": "Cartesian Phonon Prime Integration",
|
|
"domain_id": 51,
|
|
"significance": "256\u00d7256 Cartesian coordinate space with 16-bit fixed addressing and Manhattan distance metric for hardware-efficient phonon transport."
|
|
},
|
|
{
|
|
"eq_number": 790,
|
|
"title": "Phonon Load Dissipation",
|
|
"domain_id": 51,
|
|
"significance": "Phonon energy dissipates through discrete steps. Full dissipation returns cell to relaxed zero-load state."
|
|
},
|
|
{
|
|
"eq_number": 790,
|
|
"title": "Phonon Load Dissipation",
|
|
"domain_id": 51,
|
|
"significance": "Phonon energy dissipates through discrete steps. Full dissipation returns cell to relaxed zero-load state."
|
|
},
|
|
{
|
|
"eq_number": 790,
|
|
"title": "Phonon Load Dissipation",
|
|
"domain_id": 51,
|
|
"significance": "Phonon energy dissipates through discrete steps. Full dissipation returns cell to relaxed zero-load state."
|
|
},
|
|
{
|
|
"eq_number": 790,
|
|
"title": "Phonon Load Dissipation",
|
|
"domain_id": 51,
|
|
"significance": "Phonon energy dissipates through discrete steps. Full dissipation returns cell to relaxed zero-load state."
|
|
},
|
|
{
|
|
"eq_number": 790,
|
|
"title": "Phonon Load Dissipation",
|
|
"domain_id": 51,
|
|
"significance": "Phonon energy dissipates through discrete steps. Full dissipation returns cell to relaxed zero-load state."
|
|
},
|
|
{
|
|
"eq_number": 790,
|
|
"title": "Phonon Load Dissipation",
|
|
"domain_id": 51,
|
|
"significance": "Phonon energy dissipates through discrete steps. Full dissipation returns cell to relaxed zero-load state."
|
|
},
|
|
{
|
|
"eq_number": 791,
|
|
"title": "Shock Aligned Contact Energy",
|
|
"domain_id": 51,
|
|
"significance": "Positive aligned charge and contact coupling produce positive contact energy during shock-forced propagation."
|
|
},
|
|
{
|
|
"eq_number": 791,
|
|
"title": "Shock Aligned Contact Energy",
|
|
"domain_id": 51,
|
|
"significance": "Positive aligned charge and contact coupling produce positive contact energy during shock-forced propagation."
|
|
},
|
|
{
|
|
"eq_number": 791,
|
|
"title": "Shock Aligned Contact Energy",
|
|
"domain_id": 51,
|
|
"significance": "Positive aligned charge and contact coupling produce positive contact energy during shock-forced propagation."
|
|
},
|
|
{
|
|
"eq_number": 791,
|
|
"title": "Shock Aligned Contact Energy",
|
|
"domain_id": 51,
|
|
"significance": "Positive aligned charge and contact coupling produce positive contact energy during shock-forced propagation."
|
|
},
|
|
{
|
|
"eq_number": 791,
|
|
"title": "Shock Aligned Contact Energy",
|
|
"domain_id": 51,
|
|
"significance": "Positive aligned charge and contact coupling produce positive contact energy during shock-forced propagation."
|
|
},
|
|
{
|
|
"eq_number": 791,
|
|
"title": "Shock Aligned Contact Energy",
|
|
"domain_id": 51,
|
|
"significance": "Positive aligned charge and contact coupling produce positive contact energy during shock-forced propagation."
|
|
},
|
|
{
|
|
"eq_number": 792,
|
|
"title": "Photonic Spectral Witness",
|
|
"domain_id": 51,
|
|
"significance": "Spectral amplitudes encoded into optical mode amplitudes. Photon-count distribution recovers scalar observable \u03a9[u]."
|
|
},
|
|
{
|
|
"eq_number": 792,
|
|
"title": "Photonic Spectral Witness",
|
|
"domain_id": 51,
|
|
"significance": "Spectral amplitudes encoded into optical mode amplitudes. Photon-count distribution recovers scalar observable \u03a9[u]."
|
|
},
|
|
{
|
|
"eq_number": 792,
|
|
"title": "Photonic Spectral Witness",
|
|
"domain_id": 51,
|
|
"significance": "Spectral amplitudes encoded into optical mode amplitudes. Photon-count distribution recovers scalar observable \u03a9[u]."
|
|
},
|
|
{
|
|
"eq_number": 792,
|
|
"title": "Photonic Spectral Witness",
|
|
"domain_id": 51,
|
|
"significance": "Spectral amplitudes encoded into optical mode amplitudes. Photon-count distribution recovers scalar observable \u03a9[u]."
|
|
},
|
|
{
|
|
"eq_number": 792,
|
|
"title": "Photonic Spectral Witness",
|
|
"domain_id": 51,
|
|
"significance": "Spectral amplitudes encoded into optical mode amplitudes. Photon-count distribution recovers scalar observable \u03a9[u]."
|
|
},
|
|
{
|
|
"eq_number": 792,
|
|
"title": "Photonic Spectral Witness",
|
|
"domain_id": 51,
|
|
"significance": "Spectral amplitudes encoded into optical mode amplitudes. Photon-count distribution recovers scalar observable \u03a9[u]."
|
|
},
|
|
{
|
|
"eq_number": 793,
|
|
"title": "Pair-Bonded Shockwave Propagation",
|
|
"domain_id": 51,
|
|
"significance": "Two quasi-charged cells form temporary bond during shock alignment, enabling symmetric charge transfer before relaxation."
|
|
},
|
|
{
|
|
"eq_number": 793,
|
|
"title": "Pair-Bonded Shockwave Propagation",
|
|
"domain_id": 51,
|
|
"significance": "Two quasi-charged cells form temporary bond during shock alignment, enabling symmetric charge transfer before relaxation."
|
|
},
|
|
{
|
|
"eq_number": 793,
|
|
"title": "Pair-Bonded Shockwave Propagation",
|
|
"domain_id": 51,
|
|
"significance": "Two quasi-charged cells form temporary bond during shock alignment, enabling symmetric charge transfer before relaxation."
|
|
},
|
|
{
|
|
"eq_number": 793,
|
|
"title": "Pair-Bonded Shockwave Propagation",
|
|
"domain_id": 51,
|
|
"significance": "Two quasi-charged cells form temporary bond during shock alignment, enabling symmetric charge transfer before relaxation."
|
|
},
|
|
{
|
|
"eq_number": 793,
|
|
"title": "Pair-Bonded Shockwave Propagation",
|
|
"domain_id": 51,
|
|
"significance": "Two quasi-charged cells form temporary bond during shock alignment, enabling symmetric charge transfer before relaxation."
|
|
},
|
|
{
|
|
"eq_number": 793,
|
|
"title": "Pair-Bonded Shockwave Propagation",
|
|
"domain_id": 51,
|
|
"significance": "Two quasi-charged cells form temporary bond during shock alignment, enabling symmetric charge transfer before relaxation."
|
|
},
|
|
{
|
|
"eq_number": 794,
|
|
"title": "Phonon-Mediated Information Transport",
|
|
"domain_id": 51,
|
|
"significance": "Information encoded in phonon packets propagates through lattice. Lossy transport preserves spectral structure not exact state."
|
|
},
|
|
{
|
|
"eq_number": 794,
|
|
"title": "Phonon-Mediated Information Transport",
|
|
"domain_id": 51,
|
|
"significance": "Information encoded in phonon packets propagates through lattice. Lossy transport preserves spectral structure not exact state."
|
|
},
|
|
{
|
|
"eq_number": 794,
|
|
"title": "Phonon-Mediated Information Transport",
|
|
"domain_id": 51,
|
|
"significance": "Information encoded in phonon packets propagates through lattice. Lossy transport preserves spectral structure not exact state."
|
|
},
|
|
{
|
|
"eq_number": 794,
|
|
"title": "Phonon-Mediated Information Transport",
|
|
"domain_id": 51,
|
|
"significance": "Information encoded in phonon packets propagates through lattice. Lossy transport preserves spectral structure not exact state."
|
|
},
|
|
{
|
|
"eq_number": 794,
|
|
"title": "Phonon-Mediated Information Transport",
|
|
"domain_id": 51,
|
|
"significance": "Information encoded in phonon packets propagates through lattice. Lossy transport preserves spectral structure not exact state."
|
|
},
|
|
{
|
|
"eq_number": 794,
|
|
"title": "Phonon-Mediated Information Transport",
|
|
"domain_id": 51,
|
|
"significance": "Information encoded in phonon packets propagates through lattice. Lossy transport preserves spectral structure not exact state."
|
|
},
|
|
{
|
|
"eq_number": 795,
|
|
"title": "\u0394\u03c6\u03b3K\u03bb Compression Law",
|
|
"domain_id": 51,
|
|
"significance": "Compression-domain instance of GCCL with separate fields for transform pressure (\u03b3) and cost paid (K). Corrected from \u0394\u03c6\u03b3\u03bb which overloaded \u03b3."
|
|
},
|
|
{
|
|
"eq_number": 795,
|
|
"title": "\u0394\u03c6\u03b3K\u03bb Compression Law",
|
|
"domain_id": 51,
|
|
"significance": "Compression-domain instance of GCCL with separate fields for transform pressure (\u03b3) and cost paid (K). Corrected from \u0394\u03c6\u03b3\u03bb which overloaded \u03b3."
|
|
},
|
|
{
|
|
"eq_number": 795,
|
|
"title": "\u0394\u03c6\u03b3K\u03bb Compression Law",
|
|
"domain_id": 51,
|
|
"significance": "Compression-domain instance of GCCL with separate fields for transform pressure (\u03b3) and cost paid (K). Corrected from \u0394\u03c6\u03b3\u03bb which overloaded \u03b3."
|
|
},
|
|
{
|
|
"eq_number": 795,
|
|
"title": "\u0394\u03c6\u03b3K\u03bb Compression Law",
|
|
"domain_id": 51,
|
|
"significance": "Compression-domain instance of GCCL with separate fields for transform pressure (\u03b3) and cost paid (K). Corrected from \u0394\u03c6\u03b3\u03bb which overloaded \u03b3."
|
|
},
|
|
{
|
|
"eq_number": 795,
|
|
"title": "\u0394\u03c6\u03b3K\u03bb Compression Law",
|
|
"domain_id": 51,
|
|
"significance": "Compression-domain instance of GCCL with separate fields for transform pressure (\u03b3) and cost paid (K). Corrected from \u0394\u03c6\u03b3\u03bb which overloaded \u03b3."
|
|
},
|
|
{
|
|
"eq_number": 796,
|
|
"title": "Goxel Scalar Sub-Manifold",
|
|
"domain_id": 51,
|
|
"significance": "N-space shape inhabiting geometric volume, expressed as bounded scalar sub-manifold. Admitted only through declared projection, audit, and receipt gates."
|
|
},
|
|
{
|
|
"eq_number": 796,
|
|
"title": "Goxel Scalar Sub-Manifold",
|
|
"domain_id": 51,
|
|
"significance": "N-space shape inhabiting geometric volume, expressed as bounded scalar sub-manifold. Admitted only through declared projection, audit, and receipt gates."
|
|
},
|
|
{
|
|
"eq_number": 796,
|
|
"title": "Goxel Scalar Sub-Manifold",
|
|
"domain_id": 51,
|
|
"significance": "N-space shape inhabiting geometric volume, expressed as bounded scalar sub-manifold. Admitted only through declared projection, audit, and receipt gates."
|
|
},
|
|
{
|
|
"eq_number": 796,
|
|
"title": "Goxel Scalar Sub-Manifold",
|
|
"domain_id": 51,
|
|
"significance": "N-space shape inhabiting geometric volume, expressed as bounded scalar sub-manifold. Admitted only through declared projection, audit, and receipt gates."
|
|
},
|
|
{
|
|
"eq_number": 796,
|
|
"title": "Goxel Scalar Sub-Manifold",
|
|
"domain_id": 51,
|
|
"significance": "N-space shape inhabiting geometric volume, expressed as bounded scalar sub-manifold. Admitted only through declared projection, audit, and receipt gates."
|
|
},
|
|
{
|
|
"eq_number": 797,
|
|
"title": "Model Genome Encoding",
|
|
"domain_id": 51,
|
|
"significance": "Compact generative encoding of model family with codon\u2192gene\u2192chromosome\u2192genome\u2192phenotype hierarchy."
|
|
},
|
|
{
|
|
"eq_number": 797,
|
|
"title": "Model Genome Encoding",
|
|
"domain_id": 51,
|
|
"significance": "Compact generative encoding of model family with codon\u2192gene\u2192chromosome\u2192genome\u2192phenotype hierarchy."
|
|
},
|
|
{
|
|
"eq_number": 797,
|
|
"title": "Model Genome Encoding",
|
|
"domain_id": 51,
|
|
"significance": "Compact generative encoding of model family with codon\u2192gene\u2192chromosome\u2192genome\u2192phenotype hierarchy."
|
|
},
|
|
{
|
|
"eq_number": 797,
|
|
"title": "Model Genome Encoding",
|
|
"domain_id": 51,
|
|
"significance": "Compact generative encoding of model family with codon\u2192gene\u2192chromosome\u2192genome\u2192phenotype hierarchy."
|
|
},
|
|
{
|
|
"eq_number": 797,
|
|
"title": "Model Genome Encoding",
|
|
"domain_id": 51,
|
|
"significance": "Compact generative encoding of model family with codon\u2192gene\u2192chromosome\u2192genome\u2192phenotype hierarchy."
|
|
},
|
|
{
|
|
"eq_number": 798,
|
|
"title": "Kinetic Operation Token (KOT)",
|
|
"domain_id": 51,
|
|
"significance": "Accounting layer for action cost. Every transformation pays and leaves a trace. Prevents free transformations."
|
|
},
|
|
{
|
|
"eq_number": 798,
|
|
"title": "Kinetic Operation Token (KOT)",
|
|
"domain_id": 51,
|
|
"significance": "Accounting layer for action cost. Every transformation pays and leaves a trace. Prevents free transformations."
|
|
},
|
|
{
|
|
"eq_number": 798,
|
|
"title": "Kinetic Operation Token (KOT)",
|
|
"domain_id": 51,
|
|
"significance": "Accounting layer for action cost. Every transformation pays and leaves a trace. Prevents free transformations."
|
|
},
|
|
{
|
|
"eq_number": 798,
|
|
"title": "Kinetic Operation Token (KOT)",
|
|
"domain_id": 51,
|
|
"significance": "Accounting layer for action cost. Every transformation pays and leaves a trace. Prevents free transformations."
|
|
},
|
|
{
|
|
"eq_number": 798,
|
|
"title": "Kinetic Operation Token (KOT)",
|
|
"domain_id": 51,
|
|
"significance": "Accounting layer for action cost. Every transformation pays and leaves a trace. Prevents free transformations."
|
|
},
|
|
{
|
|
"eq_number": 799,
|
|
"title": "Bounded Lawful Surface",
|
|
"domain_id": 51,
|
|
"significance": "Set of transitions and phenotypes that can be expressed, replayed, checked, budgeted, and receipted under declared constraints."
|
|
},
|
|
{
|
|
"eq_number": 799,
|
|
"title": "Bounded Lawful Surface",
|
|
"domain_id": 51,
|
|
"significance": "Set of transitions and phenotypes that can be expressed, replayed, checked, budgeted, and receipted under declared constraints."
|
|
},
|
|
{
|
|
"eq_number": 799,
|
|
"title": "Bounded Lawful Surface",
|
|
"domain_id": 51,
|
|
"significance": "Set of transitions and phenotypes that can be expressed, replayed, checked, budgeted, and receipted under declared constraints."
|
|
},
|
|
{
|
|
"eq_number": 799,
|
|
"title": "Bounded Lawful Surface",
|
|
"domain_id": 51,
|
|
"significance": "Set of transitions and phenotypes that can be expressed, replayed, checked, budgeted, and receipted under declared constraints."
|
|
},
|
|
{
|
|
"eq_number": 799,
|
|
"title": "Bounded Lawful Surface",
|
|
"domain_id": 51,
|
|
"significance": "Set of transitions and phenotypes that can be expressed, replayed, checked, budgeted, and receipted under declared constraints."
|
|
},
|
|
{
|
|
"eq_number": 800,
|
|
"title": "Genotype-Phenotype Split",
|
|
"domain_id": 51,
|
|
"significance": "Separation of internal encoding from outward expression. Prevents projection from being mistaken for source object."
|
|
},
|
|
{
|
|
"eq_number": 800,
|
|
"title": "Genotype-Phenotype Split",
|
|
"domain_id": 51,
|
|
"significance": "Separation of internal encoding from outward expression. Prevents projection from being mistaken for source object."
|
|
},
|
|
{
|
|
"eq_number": 800,
|
|
"title": "Genotype-Phenotype Split",
|
|
"domain_id": 51,
|
|
"significance": "Separation of internal encoding from outward expression. Prevents projection from being mistaken for source object."
|
|
},
|
|
{
|
|
"eq_number": 800,
|
|
"title": "Genotype-Phenotype Split",
|
|
"domain_id": 51,
|
|
"significance": "Separation of internal encoding from outward expression. Prevents projection from being mistaken for source object."
|
|
},
|
|
{
|
|
"eq_number": 800,
|
|
"title": "Genotype-Phenotype Split",
|
|
"domain_id": 51,
|
|
"significance": "Separation of internal encoding from outward expression. Prevents projection from being mistaken for source object."
|
|
},
|
|
{
|
|
"eq_number": 801,
|
|
"title": "Mixture Primitive Combination",
|
|
"domain_id": 51,
|
|
"significance": "Multiple coding families (DNA, codons, proteins, ambiguity, etc.) can be mixed only under explicit decoder, residual, KOT, scale, projection, and receipt rules."
|
|
},
|
|
{
|
|
"eq_number": 801,
|
|
"title": "Mixture Primitive Combination",
|
|
"domain_id": 51,
|
|
"significance": "Multiple coding families (DNA, codons, proteins, ambiguity, etc.) can be mixed only under explicit decoder, residual, KOT, scale, projection, and receipt rules."
|
|
},
|
|
{
|
|
"eq_number": 801,
|
|
"title": "Mixture Primitive Combination",
|
|
"domain_id": 51,
|
|
"significance": "Multiple coding families (DNA, codons, proteins, ambiguity, etc.) can be mixed only under explicit decoder, residual, KOT, scale, projection, and receipt rules."
|
|
},
|
|
{
|
|
"eq_number": 801,
|
|
"title": "Mixture Primitive Combination",
|
|
"domain_id": 51,
|
|
"significance": "Multiple coding families (DNA, codons, proteins, ambiguity, etc.) can be mixed only under explicit decoder, residual, KOT, scale, projection, and receipt rules."
|
|
},
|
|
{
|
|
"eq_number": 801,
|
|
"title": "Mixture Primitive Combination",
|
|
"domain_id": 51,
|
|
"significance": "Multiple coding families (DNA, codons, proteins, ambiguity, etc.) can be mixed only under explicit decoder, residual, KOT, scale, projection, and receipt rules."
|
|
},
|
|
{
|
|
"eq_number": 802,
|
|
"title": "Layered Mountain Model",
|
|
"domain_id": 51,
|
|
"significance": "GCCL sits over layered state mountains: NUVMAP (address), AVMR (vector evolution), AMMR (commit history), O-AMMR (orthogonal projection), GCCL-Rep (transition rope)."
|
|
},
|
|
{
|
|
"eq_number": 802,
|
|
"title": "Layered Mountain Model",
|
|
"domain_id": 51,
|
|
"significance": "GCCL sits over layered state mountains: NUVMAP (address), AVMR (vector evolution), AMMR (commit history), O-AMMR (orthogonal projection), GCCL-Rep (transition rope)."
|
|
},
|
|
{
|
|
"eq_number": 802,
|
|
"title": "Layered Mountain Model",
|
|
"domain_id": 51,
|
|
"significance": "GCCL sits over layered state mountains: NUVMAP (address), AVMR (vector evolution), AMMR (commit history), O-AMMR (orthogonal projection), GCCL-Rep (transition rope)."
|
|
},
|
|
{
|
|
"eq_number": 802,
|
|
"title": "Layered Mountain Model",
|
|
"domain_id": 51,
|
|
"significance": "GCCL sits over layered state mountains: NUVMAP (address), AVMR (vector evolution), AMMR (commit history), O-AMMR (orthogonal projection), GCCL-Rep (transition rope)."
|
|
},
|
|
{
|
|
"eq_number": 802,
|
|
"title": "Layered Mountain Model",
|
|
"domain_id": 51,
|
|
"significance": "GCCL sits over layered state mountains: NUVMAP (address), AVMR (vector evolution), AMMR (commit history), O-AMMR (orthogonal projection), GCCL-Rep (transition rope)."
|
|
},
|
|
{
|
|
"eq_number": 803,
|
|
"title": "Wavefront Emission",
|
|
"domain_id": 51,
|
|
"significance": "State changes emit wavefronts that propagate through resonant field with amplitude, frequency, phase, position, and decay."
|
|
},
|
|
{
|
|
"eq_number": 803,
|
|
"title": "Wavefront Emission",
|
|
"domain_id": 51,
|
|
"significance": "State changes emit wavefronts that propagate through resonant field with amplitude, frequency, phase, position, and decay."
|
|
},
|
|
{
|
|
"eq_number": 803,
|
|
"title": "Wavefront Emission",
|
|
"domain_id": 51,
|
|
"significance": "State changes emit wavefronts that propagate through resonant field with amplitude, frequency, phase, position, and decay."
|
|
},
|
|
{
|
|
"eq_number": 803,
|
|
"title": "Wavefront Emission",
|
|
"domain_id": 51,
|
|
"significance": "State changes emit wavefronts that propagate through resonant field with amplitude, frequency, phase, position, and decay."
|
|
},
|
|
{
|
|
"eq_number": 804,
|
|
"title": "MOIM Behavioral Fingerprint",
|
|
"domain_id": 51,
|
|
"significance": "Objects become behavioral points across identity, conservation, transformation, scaling, and dynamics axes."
|
|
},
|
|
{
|
|
"eq_number": 804,
|
|
"title": "MOIM Behavioral Fingerprint",
|
|
"domain_id": 51,
|
|
"significance": "Objects become behavioral points across identity, conservation, transformation, scaling, and dynamics axes."
|
|
},
|
|
{
|
|
"eq_number": 804,
|
|
"title": "MOIM Behavioral Fingerprint",
|
|
"domain_id": 51,
|
|
"significance": "Objects become behavioral points across identity, conservation, transformation, scaling, and dynamics axes."
|
|
},
|
|
{
|
|
"eq_number": 804,
|
|
"title": "MOIM Behavioral Fingerprint",
|
|
"domain_id": 51,
|
|
"significance": "Objects become behavioral points across identity, conservation, transformation, scaling, and dynamics axes."
|
|
},
|
|
{
|
|
"eq_number": 805,
|
|
"title": "Universal Binding Manifold",
|
|
"domain_id": 51,
|
|
"significance": "Binding affinity surface for conceptual relationships with energy-based binding strength."
|
|
},
|
|
{
|
|
"eq_number": 805,
|
|
"title": "Universal Binding Manifold",
|
|
"domain_id": 51,
|
|
"significance": "Binding affinity surface for conceptual relationships with energy-based binding strength."
|
|
},
|
|
{
|
|
"eq_number": 805,
|
|
"title": "Universal Binding Manifold",
|
|
"domain_id": 51,
|
|
"significance": "Binding affinity surface for conceptual relationships with energy-based binding strength."
|
|
},
|
|
{
|
|
"eq_number": 805,
|
|
"title": "Universal Binding Manifold",
|
|
"domain_id": 51,
|
|
"significance": "Binding affinity surface for conceptual relationships with energy-based binding strength."
|
|
},
|
|
{
|
|
"eq_number": 806,
|
|
"title": "Info Bottleneck Principle",
|
|
"domain_id": 51,
|
|
"significance": "Optimal neural compression: minimize mutual information with input while maximizing with output."
|
|
},
|
|
{
|
|
"eq_number": 806,
|
|
"title": "Info Bottleneck Principle",
|
|
"domain_id": 51,
|
|
"significance": "Optimal neural compression: minimize mutual information with input while maximizing with output."
|
|
},
|
|
{
|
|
"eq_number": 806,
|
|
"title": "Info Bottleneck Principle",
|
|
"domain_id": 51,
|
|
"significance": "Optimal neural compression: minimize mutual information with input while maximizing with output."
|
|
},
|
|
{
|
|
"eq_number": 806,
|
|
"title": "Info Bottleneck Principle",
|
|
"domain_id": 51,
|
|
"significance": "Optimal neural compression: minimize mutual information with input while maximizing with output."
|
|
},
|
|
{
|
|
"eq_number": 807,
|
|
"title": "Free Energy Principle",
|
|
"domain_id": 51,
|
|
"significance": "Variational self-organization invariant: systems minimize free energy by minimizing surprise."
|
|
},
|
|
{
|
|
"eq_number": 807,
|
|
"title": "Free Energy Principle",
|
|
"domain_id": 51,
|
|
"significance": "Variational self-organization invariant: systems minimize free energy by minimizing surprise."
|
|
},
|
|
{
|
|
"eq_number": 807,
|
|
"title": "Free Energy Principle",
|
|
"domain_id": 51,
|
|
"significance": "Variational self-organization invariant: systems minimize free energy by minimizing surprise."
|
|
},
|
|
{
|
|
"eq_number": 807,
|
|
"title": "Free Energy Principle",
|
|
"domain_id": 51,
|
|
"significance": "Variational self-organization invariant: systems minimize free energy by minimizing surprise."
|
|
},
|
|
{
|
|
"eq_number": 808,
|
|
"title": "Predictive Coding",
|
|
"domain_id": 51,
|
|
"significance": "Hierarchical prediction error update: predictions drive learning and inference."
|
|
},
|
|
{
|
|
"eq_number": 808,
|
|
"title": "Predictive Coding",
|
|
"domain_id": 51,
|
|
"significance": "Hierarchical prediction error update: predictions drive learning and inference."
|
|
},
|
|
{
|
|
"eq_number": 808,
|
|
"title": "Predictive Coding",
|
|
"domain_id": 51,
|
|
"significance": "Hierarchical prediction error update: predictions drive learning and inference."
|
|
},
|
|
{
|
|
"eq_number": 808,
|
|
"title": "Predictive Coding",
|
|
"domain_id": 51,
|
|
"significance": "Hierarchical prediction error update: predictions drive learning and inference."
|
|
},
|
|
{
|
|
"eq_number": 809,
|
|
"title": "Onsager Reciprocity",
|
|
"domain_id": 51,
|
|
"significance": "Coupled transport symmetry law: cross-coupling coefficients are symmetric."
|
|
},
|
|
{
|
|
"eq_number": 809,
|
|
"title": "Onsager Reciprocity",
|
|
"domain_id": 51,
|
|
"significance": "Coupled transport symmetry law: cross-coupling coefficients are symmetric."
|
|
},
|
|
{
|
|
"eq_number": 809,
|
|
"title": "Onsager Reciprocity",
|
|
"domain_id": 51,
|
|
"significance": "Coupled transport symmetry law: cross-coupling coefficients are symmetric."
|
|
},
|
|
{
|
|
"eq_number": 809,
|
|
"title": "Onsager Reciprocity",
|
|
"domain_id": 51,
|
|
"significance": "Coupled transport symmetry law: cross-coupling coefficients are symmetric."
|
|
},
|
|
{
|
|
"eq_number": 810,
|
|
"title": "Jarzynski Equality",
|
|
"domain_id": 51,
|
|
"significance": "Non-equilibrium work-extraction relation connects work fluctuations to free energy difference."
|
|
},
|
|
{
|
|
"eq_number": 810,
|
|
"title": "Jarzynski Equality",
|
|
"domain_id": 51,
|
|
"significance": "Non-equilibrium work-extraction relation connects work fluctuations to free energy difference."
|
|
},
|
|
{
|
|
"eq_number": 810,
|
|
"title": "Jarzynski Equality",
|
|
"domain_id": 51,
|
|
"significance": "Non-equilibrium work-extraction relation connects work fluctuations to free energy difference."
|
|
},
|
|
{
|
|
"eq_number": 810,
|
|
"title": "Jarzynski Equality",
|
|
"domain_id": 51,
|
|
"significance": "Non-equilibrium work-extraction relation connects work fluctuations to free energy difference."
|
|
},
|
|
{
|
|
"eq_number": 811,
|
|
"title": "DNA Linking Number",
|
|
"domain_id": 51,
|
|
"significance": "Topological constraint on circular DNA: linking number equals twist plus writhe."
|
|
},
|
|
{
|
|
"eq_number": 811,
|
|
"title": "DNA Linking Number",
|
|
"domain_id": 51,
|
|
"significance": "Topological constraint on circular DNA: linking number equals twist plus writhe."
|
|
},
|
|
{
|
|
"eq_number": 811,
|
|
"title": "DNA Linking Number",
|
|
"domain_id": 51,
|
|
"significance": "Topological constraint on circular DNA: linking number equals twist plus writhe."
|
|
},
|
|
{
|
|
"eq_number": 811,
|
|
"title": "DNA Linking Number",
|
|
"domain_id": 51,
|
|
"significance": "Topological constraint on circular DNA: linking number equals twist plus writhe."
|
|
},
|
|
{
|
|
"eq_number": 812,
|
|
"title": "Cavity Persistence",
|
|
"domain_id": 51,
|
|
"significance": "Topological information processing metric: persistence of topological features."
|
|
},
|
|
{
|
|
"eq_number": 812,
|
|
"title": "Cavity Persistence",
|
|
"domain_id": 51,
|
|
"significance": "Topological information processing metric: persistence of topological features."
|
|
},
|
|
{
|
|
"eq_number": 812,
|
|
"title": "Cavity Persistence",
|
|
"domain_id": 51,
|
|
"significance": "Topological information processing metric: persistence of topological features."
|
|
},
|
|
{
|
|
"eq_number": 812,
|
|
"title": "Cavity Persistence",
|
|
"domain_id": 51,
|
|
"significance": "Topological information processing metric: persistence of topological features."
|
|
},
|
|
{
|
|
"eq_number": 813,
|
|
"title": "Hill Regulation",
|
|
"domain_id": 51,
|
|
"significance": "Nonlinear saturation feedback: sigmoid functions used throughout OTOM."
|
|
},
|
|
{
|
|
"eq_number": 813,
|
|
"title": "Hill Regulation",
|
|
"domain_id": 51,
|
|
"significance": "Nonlinear saturation feedback: sigmoid functions used throughout OTOM."
|
|
},
|
|
{
|
|
"eq_number": 813,
|
|
"title": "Hill Regulation",
|
|
"domain_id": 51,
|
|
"significance": "Nonlinear saturation feedback: sigmoid functions used throughout OTOM."
|
|
},
|
|
{
|
|
"eq_number": 813,
|
|
"title": "Hill Regulation",
|
|
"domain_id": 51,
|
|
"significance": "Nonlinear saturation feedback: sigmoid functions used throughout OTOM."
|
|
},
|
|
{
|
|
"eq_number": 814,
|
|
"title": "Wilson-Cowan Equations",
|
|
"domain_id": 51,
|
|
"significance": "Mean-field neural population dynamics for cognitive load modeling."
|
|
},
|
|
{
|
|
"eq_number": 814,
|
|
"title": "Wilson-Cowan Equations",
|
|
"domain_id": 51,
|
|
"significance": "Mean-field neural population dynamics for cognitive load modeling."
|
|
},
|
|
{
|
|
"eq_number": 814,
|
|
"title": "Wilson-Cowan Equations",
|
|
"domain_id": 51,
|
|
"significance": "Mean-field neural population dynamics for cognitive load modeling."
|
|
},
|
|
{
|
|
"eq_number": 814,
|
|
"title": "Wilson-Cowan Equations",
|
|
"domain_id": 51,
|
|
"significance": "Mean-field neural population dynamics for cognitive load modeling."
|
|
},
|
|
{
|
|
"eq_number": 815,
|
|
"title": "Turing Morphogenesis",
|
|
"domain_id": 51,
|
|
"significance": "Spontaneous symmetry breaking for pattern formation on manifolds."
|
|
},
|
|
{
|
|
"eq_number": 815,
|
|
"title": "Turing Morphogenesis",
|
|
"domain_id": 51,
|
|
"significance": "Spontaneous symmetry breaking for pattern formation on manifolds."
|
|
},
|
|
{
|
|
"eq_number": 815,
|
|
"title": "Turing Morphogenesis",
|
|
"domain_id": 51,
|
|
"significance": "Spontaneous symmetry breaking for pattern formation on manifolds."
|
|
},
|
|
{
|
|
"eq_number": 815,
|
|
"title": "Turing Morphogenesis",
|
|
"domain_id": 51,
|
|
"significance": "Spontaneous symmetry breaking for pattern formation on manifolds."
|
|
},
|
|
{
|
|
"eq_number": 816,
|
|
"title": "Mass Number Admissibility Gate",
|
|
"domain_id": 51,
|
|
"significance": "Three-layer Mass Number structure: Admissible (A), Residual (R), Boundary (\u03b5). Core rule: A \u2264 threshold * (R + \u03b5)."
|
|
},
|
|
{
|
|
"eq_number": 816,
|
|
"title": "Mass Number Admissibility Gate",
|
|
"domain_id": 51,
|
|
"significance": "Three-layer Mass Number structure: Admissible (A), Residual (R), Boundary (\u03b5). Core rule: A \u2264 threshold * (R + \u03b5)."
|
|
},
|
|
{
|
|
"eq_number": 816,
|
|
"title": "Mass Number Admissibility Gate",
|
|
"domain_id": 51,
|
|
"significance": "Three-layer Mass Number structure: Admissible (A), Residual (R), Boundary (\u03b5). Core rule: A \u2264 threshold * (R + \u03b5)."
|
|
},
|
|
{
|
|
"eq_number": 817,
|
|
"title": "Admissible Reduction Packet",
|
|
"domain_id": 51,
|
|
"significance": "Layer 1 of Mass Number: records concrete reduction achieved by modeling move. Must be grounded in surface feature/invariant."
|
|
},
|
|
{
|
|
"eq_number": 817,
|
|
"title": "Admissible Reduction Packet",
|
|
"domain_id": 51,
|
|
"significance": "Layer 1 of Mass Number: records concrete reduction achieved by modeling move. Must be grounded in surface feature/invariant."
|
|
},
|
|
{
|
|
"eq_number": 817,
|
|
"title": "Admissible Reduction Packet",
|
|
"domain_id": 51,
|
|
"significance": "Layer 1 of Mass Number: records concrete reduction achieved by modeling move. Must be grounded in surface feature/invariant."
|
|
},
|
|
{
|
|
"eq_number": 818,
|
|
"title": "Residual Risk Receipt",
|
|
"domain_id": 51,
|
|
"significance": "Layer 2 of Mass Number: records what remains unreduced after move. Must be inspectable and bounded."
|
|
},
|
|
{
|
|
"eq_number": 818,
|
|
"title": "Residual Risk Receipt",
|
|
"domain_id": 51,
|
|
"significance": "Layer 2 of Mass Number: records what remains unreduced after move. Must be inspectable and bounded."
|
|
},
|
|
{
|
|
"eq_number": 818,
|
|
"title": "Residual Risk Receipt",
|
|
"domain_id": 51,
|
|
"significance": "Layer 2 of Mass Number: records what remains unreduced after move. Must be inspectable and bounded."
|
|
},
|
|
{
|
|
"eq_number": 819,
|
|
"title": "Boundary Marker (\u03b5 Guard)",
|
|
"domain_id": 51,
|
|
"significance": "Layer 3 of Mass Number: ensures denominator never zero. Carries threshold for admissibility decisions."
|
|
},
|
|
{
|
|
"eq_number": 819,
|
|
"title": "Boundary Marker (\u03b5 Guard)",
|
|
"domain_id": 51,
|
|
"significance": "Layer 3 of Mass Number: ensures denominator never zero. Carries threshold for admissibility decisions."
|
|
},
|
|
{
|
|
"eq_number": 819,
|
|
"title": "Boundary Marker (\u03b5 Guard)",
|
|
"domain_id": 51,
|
|
"significance": "Layer 3 of Mass Number: ensures denominator never zero. Carries threshold for admissibility decisions."
|
|
},
|
|
{
|
|
"eq_number": 820,
|
|
"title": "NaNMass Doctrine",
|
|
"domain_id": 51,
|
|
"significance": "Apparent infinity is diagnostic, not destination. NaNMass means coordinate system failed to close mass."
|
|
},
|
|
{
|
|
"eq_number": 820,
|
|
"title": "NaNMass Doctrine",
|
|
"domain_id": 51,
|
|
"significance": "Apparent infinity is diagnostic, not destination. NaNMass means coordinate system failed to close mass."
|
|
},
|
|
{
|
|
"eq_number": 820,
|
|
"title": "NaNMass Doctrine",
|
|
"domain_id": 51,
|
|
"significance": "Apparent infinity is diagnostic, not destination. NaNMass means coordinate system failed to close mass."
|
|
},
|
|
{
|
|
"eq_number": 821,
|
|
"title": "Closure Path to Metric",
|
|
"domain_id": 51,
|
|
"significance": "Mass becomes distance only through admissibility closure. Raw mass \u2192 pseudometric \u2192 zero-distance quotient \u2192 metric."
|
|
},
|
|
{
|
|
"eq_number": 821,
|
|
"title": "Closure Path to Metric",
|
|
"domain_id": 51,
|
|
"significance": "Mass becomes distance only through admissibility closure. Raw mass \u2192 pseudometric \u2192 zero-distance quotient \u2192 metric."
|
|
},
|
|
{
|
|
"eq_number": 821,
|
|
"title": "Closure Path to Metric",
|
|
"domain_id": 51,
|
|
"significance": "Mass becomes distance only through admissibility closure. Raw mass \u2192 pseudometric \u2192 zero-distance quotient \u2192 metric."
|
|
},
|
|
{
|
|
"eq_number": 822,
|
|
"title": "Erd\u0151s Forced-Pattern Model",
|
|
"domain_id": 51,
|
|
"significance": "If system is large enough, disorder cannot remain pure. Organized substructure must appear."
|
|
},
|
|
{
|
|
"eq_number": 822,
|
|
"title": "Erd\u0151s Forced-Pattern Model",
|
|
"domain_id": 51,
|
|
"significance": "If system is large enough, disorder cannot remain pure. Organized substructure must appear."
|
|
},
|
|
{
|
|
"eq_number": 822,
|
|
"title": "Erd\u0151s Forced-Pattern Model",
|
|
"domain_id": 51,
|
|
"significance": "If system is large enough, disorder cannot remain pure. Organized substructure must appear."
|
|
},
|
|
{
|
|
"eq_number": 823,
|
|
"title": "General-Position Convexity Forcing",
|
|
"domain_id": 51,
|
|
"significance": "Points in general position: when does convex n-gon become unavoidable?"
|
|
},
|
|
{
|
|
"eq_number": 823,
|
|
"title": "General-Position Convexity Forcing",
|
|
"domain_id": 51,
|
|
"significance": "Points in general position: when does convex n-gon become unavoidable?"
|
|
},
|
|
{
|
|
"eq_number": 823,
|
|
"title": "General-Position Convexity Forcing",
|
|
"domain_id": 51,
|
|
"significance": "Points in general position: when does convex n-gon become unavoidable?"
|
|
},
|
|
{
|
|
"eq_number": 824,
|
|
"title": "Cup-Cap Monotonicity",
|
|
"domain_id": 51,
|
|
"significance": "Geometry converted to ordered subsequences. Convexity becomes pattern of slope changes."
|
|
},
|
|
{
|
|
"eq_number": 824,
|
|
"title": "Cup-Cap Monotonicity",
|
|
"domain_id": 51,
|
|
"significance": "Geometry converted to ordered subsequences. Convexity becomes pattern of slope changes."
|
|
},
|
|
{
|
|
"eq_number": 824,
|
|
"title": "Cup-Cap Monotonicity",
|
|
"domain_id": 51,
|
|
"significance": "Geometry converted to ordered subsequences. Convexity becomes pattern of slope changes."
|
|
},
|
|
{
|
|
"eq_number": 825,
|
|
"title": "Probabilistic Existence Method",
|
|
"domain_id": 51,
|
|
"significance": "Do not construct directly. Show random object avoids bad event with positive probability."
|
|
},
|
|
{
|
|
"eq_number": 825,
|
|
"title": "Probabilistic Existence Method",
|
|
"domain_id": 51,
|
|
"significance": "Do not construct directly. Show random object avoids bad event with positive probability."
|
|
},
|
|
{
|
|
"eq_number": 825,
|
|
"title": "Probabilistic Existence Method",
|
|
"domain_id": 51,
|
|
"significance": "Do not construct directly. Show random object avoids bad event with positive probability."
|
|
},
|
|
{
|
|
"eq_number": 826,
|
|
"title": "Extremal Density Threshold",
|
|
"domain_id": 51,
|
|
"significance": "Maximum possible density before forbidden structure is forced."
|
|
},
|
|
{
|
|
"eq_number": 826,
|
|
"title": "Extremal Density Threshold",
|
|
"domain_id": 51,
|
|
"significance": "Maximum possible density before forbidden structure is forced."
|
|
},
|
|
{
|
|
"eq_number": 826,
|
|
"title": "Extremal Density Threshold",
|
|
"domain_id": 51,
|
|
"significance": "Maximum possible density before forbidden structure is forced."
|
|
},
|
|
{
|
|
"eq_number": 827,
|
|
"title": "Sidon Additive Collision",
|
|
"domain_id": 51,
|
|
"significance": "Integers as collision surfaces. Forbidden equality becomes overlap in additive address space."
|
|
},
|
|
{
|
|
"eq_number": 827,
|
|
"title": "Sidon Additive Collision",
|
|
"domain_id": 51,
|
|
"significance": "Integers as collision surfaces. Forbidden equality becomes overlap in additive address space."
|
|
},
|
|
{
|
|
"eq_number": 827,
|
|
"title": "Sidon Additive Collision",
|
|
"domain_id": 51,
|
|
"significance": "Integers as collision surfaces. Forbidden equality becomes overlap in additive address space."
|
|
},
|
|
{
|
|
"eq_number": 828,
|
|
"title": "Order-Type Signature Function",
|
|
"domain_id": 51,
|
|
"significance": "Coordinates discarded. Only orientation signatures kept for convexity encoding."
|
|
},
|
|
{
|
|
"eq_number": 828,
|
|
"title": "Order-Type Signature Function",
|
|
"domain_id": 51,
|
|
"significance": "Coordinates discarded. Only orientation signatures kept for convexity encoding."
|
|
},
|
|
{
|
|
"eq_number": 828,
|
|
"title": "Order-Type Signature Function",
|
|
"domain_id": 51,
|
|
"significance": "Coordinates discarded. Only orientation signatures kept for convexity encoding."
|
|
},
|
|
{
|
|
"eq_number": 829,
|
|
"title": "Strain121 Temperature Limit",
|
|
"domain_id": 51,
|
|
"significance": "Absolute biological temperature limit: 122\u00b0C (395K) protein denaturation wall."
|
|
},
|
|
{
|
|
"eq_number": 829,
|
|
"title": "Strain121 Temperature Limit",
|
|
"domain_id": 51,
|
|
"significance": "Absolute biological temperature limit: 122\u00b0C (395K) protein denaturation wall."
|
|
},
|
|
{
|
|
"eq_number": 830,
|
|
"title": "Diatom Stiffness Limit",
|
|
"domain_id": 51,
|
|
"significance": "Silica shells approach inorganic material limits. \u03ba_T \u2248 2.7\u00d710^-11 Pa^-1."
|
|
},
|
|
{
|
|
"eq_number": 830,
|
|
"title": "Diatom Stiffness Limit",
|
|
"domain_id": 51,
|
|
"significance": "Silica shells approach inorganic material limits. \u03ba_T \u2248 2.7\u00d710^-11 Pa^-1."
|
|
},
|
|
{
|
|
"eq_number": 831,
|
|
"title": "Vibrio Natriegens Replication Speed",
|
|
"domain_id": 51,
|
|
"significance": "Absolute biological replication speed limit: 10-15 minute doubling time."
|
|
},
|
|
{
|
|
"eq_number": 831,
|
|
"title": "Vibrio Natriegens Replication Speed",
|
|
"domain_id": 51,
|
|
"significance": "Absolute biological replication speed limit: 10-15 minute doubling time."
|
|
},
|
|
{
|
|
"eq_number": 832,
|
|
"title": "Pyrococcus Pressure-Volume Work",
|
|
"domain_id": 51,
|
|
"significance": "P\u00b7\u0394V > kT prevents protein unfolding. Obligate piezophile stability condition."
|
|
},
|
|
{
|
|
"eq_number": 832,
|
|
"title": "Pyrococcus Pressure-Volume Work",
|
|
"domain_id": 51,
|
|
"significance": "P\u00b7\u0394V > kT prevents protein unfolding. Obligate piezophile stability condition."
|
|
},
|
|
{
|
|
"eq_number": 833,
|
|
"title": "Desulforudis Energy Flux",
|
|
"domain_id": 51,
|
|
"significance": "Deep biosphere champion: 10^-15 W/cell energy flux, 1000-year division time."
|
|
},
|
|
{
|
|
"eq_number": 833,
|
|
"title": "Desulforudis Energy Flux",
|
|
"domain_id": 51,
|
|
"significance": "Deep biosphere champion: 10^-15 W/cell energy flux, 1000-year division time."
|
|
},
|
|
{
|
|
"eq_number": 834,
|
|
"title": "Landauer Limit",
|
|
"domain_id": 51,
|
|
"significance": "Minimum energy per bit erasure: E = kT ln(2)."
|
|
},
|
|
{
|
|
"eq_number": 834,
|
|
"title": "Landauer Limit",
|
|
"domain_id": 51,
|
|
"significance": "Minimum energy per bit erasure: E = kT ln(2)."
|
|
},
|
|
{
|
|
"eq_number": 835,
|
|
"title": "Resonant Cavity Q-Factor Limit",
|
|
"domain_id": 51,
|
|
"significance": "Material damping prevents infinite Q. Q_max \u2248 100 for biological tissue."
|
|
},
|
|
{
|
|
"eq_number": 835,
|
|
"title": "Resonant Cavity Q-Factor Limit",
|
|
"domain_id": 51,
|
|
"significance": "Material damping prevents infinite Q. Q_max \u2248 100 for biological tissue."
|
|
},
|
|
{
|
|
"eq_number": 836,
|
|
"title": "Turing Pattern Growth Limit",
|
|
"domain_id": 51,
|
|
"significance": "Finite nutrient flux prevents infinite growth in reaction-diffusion systems."
|
|
},
|
|
{
|
|
"eq_number": 836,
|
|
"title": "Turing Pattern Growth Limit",
|
|
"domain_id": 51,
|
|
"significance": "Finite nutrient flux prevents infinite growth in reaction-diffusion systems."
|
|
},
|
|
{
|
|
"eq_number": 837,
|
|
"title": "Navier-Stokes Blow-up Rejection",
|
|
"domain_id": 51,
|
|
"significance": "Evolutionary rejection of blow-up: infinite vorticity, zero compressibility, zero viscosity, infinite energy."
|
|
},
|
|
{
|
|
"eq_number": 837,
|
|
"title": "Navier-Stokes Blow-up Rejection",
|
|
"domain_id": 51,
|
|
"significance": "Evolutionary rejection of blow-up: infinite vorticity, zero compressibility, zero viscosity, infinite energy."
|
|
},
|
|
{
|
|
"eq_number": 838,
|
|
"title": "Thermococcus Pressure Adaptability",
|
|
"domain_id": 51,
|
|
"significance": "Widest pressure-range organism: 1 atm to 130 MPa adaptive flexibility."
|
|
},
|
|
{
|
|
"eq_number": 838,
|
|
"title": "Thermococcus Pressure Adaptability",
|
|
"domain_id": 51,
|
|
"significance": "Widest pressure-range organism: 1 atm to 130 MPa adaptive flexibility."
|
|
},
|
|
{
|
|
"eq_number": 839,
|
|
"title": "Thermus Moderate Thermophily",
|
|
"domain_id": 51,
|
|
"significance": "Moderate thermophile: 50-80\u00b0C (Taq polymerase source)."
|
|
},
|
|
{
|
|
"eq_number": 839,
|
|
"title": "Thermus Moderate Thermophily",
|
|
"domain_id": 51,
|
|
"significance": "Moderate thermophile: 50-80\u00b0C (Taq polymerase source)."
|
|
},
|
|
{
|
|
"eq_number": 840,
|
|
"title": "E. Coli Replication Reference",
|
|
"domain_id": 51,
|
|
"significance": "Baseline replication efficiency: 20 minutes optimal doubling, 4.6M bp genome."
|
|
},
|
|
{
|
|
"eq_number": 840,
|
|
"title": "E. Coli Replication Reference",
|
|
"domain_id": 51,
|
|
"significance": "Baseline replication efficiency: 20 minutes optimal doubling, 4.6M bp genome."
|
|
},
|
|
{
|
|
"eq_number": 841,
|
|
"title": "Rotational Phase Encoding",
|
|
"domain_id": 51,
|
|
"significance": "4-bit \u03c0 field encodes 16 rotational states (22.5\u00b0 resolution) for geometric information flow."
|
|
},
|
|
{
|
|
"eq_number": 842,
|
|
"title": "Chiral Alignment Coupling",
|
|
"domain_id": 51,
|
|
"significance": "Alignment strength between rotational states: A = cos(\u0394\u03b8). Determines information flow."
|
|
},
|
|
{
|
|
"eq_number": 843,
|
|
"title": "Manifold Blit Equation",
|
|
"domain_id": 51,
|
|
"significance": "Hardware-accelerated manifold update: M_{k+1} = Quant_LLM( J_DAG[ M_k \u2295 (\u03a8_q \u2297 R_RT) ] )."
|
|
},
|
|
{
|
|
"eq_number": 844,
|
|
"title": "Blitter Accumulation",
|
|
"domain_id": 51,
|
|
"significance": "Saturating bitwise accumulation: saturate(M_k + \u03b4) for discrete Picard integral."
|
|
},
|
|
{
|
|
"eq_number": 845,
|
|
"title": "Quantum Walk Amplitude",
|
|
"domain_id": 51,
|
|
"significance": "Discrete diffusion for quadratic convergence: A_{t+1} = (A_t \u2297 K) / 4."
|
|
},
|
|
{
|
|
"eq_number": 846,
|
|
"title": "Anisotropic Torsion Flow",
|
|
"domain_id": 51,
|
|
"significance": "\u2202_t \u03d5 = \u2207_i(M^ij \u2207_j \u03b4F/\u03b4\u03d5) - \u03c3 \u2202\u03d5/\u2202I_lock for manifold evolution."
|
|
},
|
|
{
|
|
"eq_number": 847,
|
|
"title": "Interlocking Energy",
|
|
"domain_id": 51,
|
|
"significance": "I_lock = w(1 - cos(k\u00b7frustration)) for recursive deposition snagging."
|
|
},
|
|
{
|
|
"eq_number": 848,
|
|
"title": "Spike Sync TVI",
|
|
"domain_id": 51,
|
|
"significance": "Temporal Variant Index for spike trains: coarse-grained timing/rate/pattern/collapse."
|
|
},
|
|
{
|
|
"eq_number": 849,
|
|
"title": "Coarse-Graining Rule",
|
|
"domain_id": 51,
|
|
"significance": "Quantize time into bins: t_bin = floor(t / \u0394t) for jitter tolerance."
|
|
},
|
|
{
|
|
"eq_number": 850,
|
|
"title": "Soliton Phase Singularity",
|
|
"domain_id": 51,
|
|
"significance": "Phase winding number +1 around soliton center: topological charge = vortex."
|
|
}
|
|
]
|
|
} |