{
  "assignment_boundary": [
    "x = E1,x \u00d7 E2,x the vector space structure as a direct product",
    "w = (w1 ,",
    "ST = S + Sb",
    "Xm = \u03b1m } , and P \u0010Q \u0011 j P j\u2208Jk |Jk | m q \u03b8 J1 ,",
    "M = 2",
    "uv = \u03d5m (hu , hv , huv , euv )",
    "ax + by = cz , J",
    "j = 1,",
    "std=[0",
    "L = \u03d5i",
    "n=0 and a \u2018fermionic\u2019 one, spanned by {f2n+1 (\u03b8)}\u221e",
    "i = 1,",
    "m = G\u2113m + G\u2113m is unaffected by the location of the source",
    "Vy = 3",
    "s = 23",
    "vac = 0 and Normal-Ordered Hamiltonian",
    "N = 20000, and 107 random samples for both cases",
    "d = 1 the map Dom(\u039b) \u220b \u03b1 7\u2192 \u039b(\u03b1) is continuous",
    "Ebw = max |E|, \u20137\u2013 (4",
    "ch = \u03b7fs \u03b7atm"
  ],
  "inequality_constraint": [
    "z \u2265 1, we define log\u03bd z recursively by log1 z = max{2, log z} and for \u03bd \u2265 2 by log\u03bd z = max{2, log\u03bd-1 z}",
    "v = 1 + \u03c3 + t, Z \u221e Z \u221e \u0010 2 \u00113\u03c3+1 3\u03c3 -\u03c0t/2 \u03c0(1+\u03c3)/2 (1 + \u03c3 + t) e dt = e v 3\u03c3 e-\u03c0v/2 dv \u2264 e\u03c0(1+\u03c3)/2 \u0393(3\u03c3 + 1)",
    "c < 0 and diverges on the real axis for c > 0",
    "N \u2265 \u03b5-1 , there is some x \u2208 (0, 2-N ] such that f (x) > xr-\u03b5",
    "n \u2265 2 and let \u03c3 = (\u03c31 ,",
    "B < 0",
    "i \u2264 c0 \u03b5total",
    "L < 0), h1 (\u03a3, L) = 0 (deg L \u2265 2g - 1)",
    "i \u2264 m - 1",
    "s > 1",
    "dy \u2264 16\u03b1 2 R 1-\u03b1 \u2225w\u03b4,R V\u03b5 \u22252L2 (R2 ) + C(U0 )",
    "n \u2264 M",
    "m < \u221e, then \u2225\u2206\u00b1 p \u2225p,p \u2264 D and \u2225\u2206H p \u2225p,p \u2264 2D for all 1 \u2264 p \u2264 \u221e",
    "N \u2265 2",
    "y\u2265y0 m1 ,m2 ,m3 ,m4 \u22651 m1 +m2 =m3 +m4 Applying the Cauchy-Schwarz inequality twice, we find that N+ (W1 , W2 ) is bounded by Z X dy \u03bb(m\u20321 )4 W1 (m1 y)W1 (m2 y)W1 (m3 y)W1 (m4 y) 2 |Ip (m)|",
    "h > 0",
    "v \u2264 u \u2264 \u00b52 v, one has the explicit formulas, see (2",
    "W < 1 inside the same Fourier window: e + (E : |E| < \u2206E) > W",
    "D \u2264 c1 + c1 p2-2n+ak + = c1 + c1 p 2-2n+ak + n-1 n-1 X X (ci+1 - ci )p-i + p2-n (ci+1 - ci ) i=1 n-1 X (ci+1 - ci )p -i +p 1-n i=1 n-1 X i=1 (ci+1 - ci ) + p 1-n (p - 1) i=1 n-1 X (ci+1 - ci )",
    "r = \u03bas + \u03c1, with 1 \u2264 \u03c1 \u2264 s - 1"
  ],
  "algebraic_generic": [
    "u+v=s u+v=s-1 where again, abusing notation, we use t here to denote the maps in F and G",
    "E = En (\u03b1, \u03b2, \u03b3), n = 0, 1, 2, 3 which may be real or complex, and which remain all \u03bb-independent",
    "k=0 \u221e 8192 X \u2032\u2032 g27 (k) = 180224, g27 (k) = - 2 , \u03c0 (5",
    "t = 0 is given by d E[F (\u03a0t )] = dt Z X E[F (\u03a0t + \u03b4z ) - F (\u03a0t )] \u03bd(dz)",
    "d = pn predicts the existence of d+1 = 7 bases, only sets of 3 MUBs have been analytically constructed to date",
    "y = L(x) + c, we obtain DL-1 (a) DL-1 (b) g(x) = f (y + a + b) + f (y + a) + f (y + b) + f (y) = Da Db f (y)",
    "r = Q, Q f (r) = 1 - r \u0012 \u00132 = (r - Q)2",
    "S = 0) = \u03f50 \u00b7 (\u03f50 - \u03b42 ) \u00b7 (\u03f50 - \u03b43 )",
    "G = vs (U \u00b7)F , if p = q we obviously have \u2225G \u25e6 U -1 \u2225p = \u2225G\u2225p , since det(U ) = 1",
    "CIa = caI,i ei\u03d5I,i C-I = (CIa )* , i Eq",
    "i = A2i h2i-1 + B2i (y) y = u(y) (e\u22a4 m y) = u(y)",
    "a = 41, w = 3, so we are again in Case III",
    "a = 41 and let c = - 13 16",
    "k=0 k=0 (5",
    "higher = better; rank 1 = most profitable)",
    "s = sgn(\u03be1 )\u03c1f = 1/\u03be1 has a fixed sign",
    "i = M i Mni = C",
    "X = - f\u03b1 \u03b4v , \u03c9Y = - g\u03b1 \u03b4v \u03b1",
    "B = \u212620 = 10 Hz",
    "ij = 0 for i 6= j"
  ],
  "dirac_notation": [
    "ve = (-2\u2206 + v)f = 2 \u03b4|x|=b , b log( ab ) 1 and b \u2a7d \u03c1- 2",
    "lim = h \u03b1\u2192\u03b10 \u03b1\u03c0 c\u0303ll (\u03b1, \u03bb (\u03b1))\u2225u h j,\u03bb (\u03b1) \u2225 \u2225uk,\u03bbh (\u03b1) \u2225 l l l = P2 m,n=1 alm aln \u27e8\u03d5n,\u03bb0 , \u03d5m,\u03bb0 \u27e9 |all |2 ||\u03d5j,\u03bb0 || ||\u03d5k,\u03bb0 || ||\u03c8l,\u03bb0 ||2",
    "dx = 1, Rd Rd where \u03c1 = \u03c1(x) is a non\u2013negative infinitely differentiable function supported in the ball {x : |x| \u2a7d 1}",
    "L = {cI | c \u2208 C} \u21d0\u21d2 C = {cI | c \u2208 C}",
    "l = y\u0302i,l |P\u0302l (zi,T )] \u2243 P\u0302l (zi,T ), (k) (k) where y\u0302i,l = arg maxj Pj,l (zi,T )",
    "Fp = c / (a + c) = P(low confidence | correct)",
    "Gk = \u27e8a, b\u27e9 and such that the three subgroups of Gk of index 2 are H1 = \u27e8a, b2 , G\u2032k \u27e9 = \u27e8a, b2 \u27e9, H2 = \u27e8ab, b2 , G\u2032k \u27e9 = \u27e8ab, b2 \u27e9 and H3 = \u27e8a2 , b, G\u2032k \u27e9 = \u27e8a2 , b\u27e9",
    "L = L- log | det Df | acts on densities: R s for large n n (f* \u00b50 )(g) = g \u00b7 L h0 dm",
    "u=t+s-2t1 \u0001 -(t+s-2t1 )A xy Z \u221e ======== -uA du e |t-s| \u0012 -|t-s|A \u0013 e , = xy A xy (4",
    "d = \u03f1 ei\u03b4 for \u03f1 = |d|, becomes \u03bbWw = \u03f1 ei\u03b4 (\u03bb + Caw \u03f1)",
    "Ep = |FLOF F | \u00d7 b3 = 6 MeV",
    "W = 0, 4 , HomZ\u03bd2W (W \u2297 W, I) \u2243 C0|1 , for \u03bdW = 2, 6",
    "r = 1, and \u03c1(\u03b6a,r , \u03b6a,|a| ) = log |a| \u2192 \u221e as a \u2192 \u221e",
    "k=0 k! 2 \u03c3(Dx , D\u03be , Dy , D\u03b7 ) \u0013k h a(x, \u03be)b(y, \u03b7) i |x=y,\u03be=\u03b7",
    "b=- i |\u03b1|2 , \u210f = 0",
    "S = 2000 Normalized feature distribution shift | fid=9, r = 0",
    "u=0 , Q5 = t 11 -1 (u)t 00 (u)|u=0 - Q32 , From one identi\ufb01es the auxiliary space A = {a, b}",
    "r = r- ) = -Q1 |r=r-",
    "a = 2 |DR| f (R) k a = |DR| f (R) (\u03c4\u0302 a + r\u0302a ) , (5",
    "Pe = Pe, so in this case \u03b2E = 1 and |\u03b2E | = |\u03b1E | = 1"
  ],
  "asymptotic_complexity": [
    "B = M\u2126 , 2 \u03c1= , 3 A= 3 r\u2126 , 2 we obtain M\u2126 (x) \u223c 3 r\u2126 e2x/3",
    "tor = GdR,K p K p Kp c (V ) \u2297Ep BdR , R\u03bdKp ,* (OBdR,log ) \u223c = GdR,K p where the last isomorphism is given by Proposition 3",
    "yi = f \u22c6 (qi )+ \u03b7i , \u03b7i \u223c N (0, \u03c3 2 I), with \u03c3 = 0",
    "I +W = i 4\u03c0 \u03ba -1 I + i 4\u03c0 \u03ba -1W 4\u03c0 M = i 4\u03c0 \u03ba -1 \u2211 (-i 4\u03c0)m \u03ba -mW m + O(\u03ba -M-2 ) as \u03ba \u2192 +\u221e",
    "zk \u2248 \u03b2k*",
    "n = 1, we have seen in the isomorphisms W1 \u223c = F1 which implies a morphism X \u223c = Y1 -\u2192 X1 \u223c = X 0",
    "takes \u22487 h 47 min per seed: SSI \u224812",
    "h \u2248 0 to compute the right hand side of the integral Ir is reasonable",
    "NLL \u2248 0",
    "N = 1 to \u223c 10-4 G0 at N = 9",
    "for \u2248 23",
    "Mi = (1 + O(\u03b520 ))\u03c1\u25e6 + \u03d6\u2020 (1, 1) + O(\u03b520 )Mi , M M M Pi = (11",
    "mt \u2248 170 GeV",
    "Rsh \u2248 8",
    "BC = 1 lies in Sp(2, R) \u223c SL(2, R) A linear canonical map \u03be = (q, p)\u22a4 7\u2192 \u03be \u2032 = S \u03be with S = ( C = \u0001D 0 1 and preserves \u03c9 = dp \u2227 dq",
    "d = N Fb (\u2206\u03d5 ) + \u03b4FLR\u03b7 has the perturbative expansion: \u03b4FLR\u03b7 = - g02 g03 g04 g05 H + H - H + H5 + O(g06 ), 2 3 4 2!(4!)2 3!(4!)3 4!(4!)4 5!(4!)5 36 (6",
    "arcsinh \u2248 5",
    "O = O1-1 O2 , we have 2\u03b3 \u220f \u2223xi - xj \u2223 \u03a3n (\u03b1, \u03b2, \u03b3) \u2236= \u222b O \u2208 O(2n)",
    "r = 1 - (r+ + r- )\u03bac + r+ O(\u03bac ), 3 c 3 2 \u03bac rc = 1 - (r+ + r- )\u03bac + r+ O(\u03ba2c )",
    "N \u2248 \u03b3N \u2297 \u03b3N"
  ],
  "set_transformation": [
    "pxy = 0 \u2208 L/L\u2032 , and so pxy \u2208 L\u2032 , which in turn implies p(x + py)(x + y) = p(x2 + (1 + p)xy + py 2 ) \u2208 L\u2032",
    "r = 0 and, for every j \u2208 Lacc , sets ci,j = 1/\u03b1r,j",
    "have = X q- P i\u2208I (\u03c3i +i+ 12 ) \u2113(wI w\u03c4 ) q wI w\u03c4 \u2208Wr = (A",
    "V = 1G\u03b8 (OV ) \u2297 1s\u03c3 (OV ) et f = fV \u2297 f V \u2208 Cc\u221e ((G\u03b8 \u00d7 s\u03c3 )(A))",
    "M = 5, and vary only initial penalty \u03c10 \u2208 {10-4 , 10-3 , 10-2 , 10-1 } with \u03c10U U = \u03c10V V = \u03c10U V = \u03c10",
    "dj=0 aj X j \u2208 C[X] and \u03bd \u2208 C[G], we define P (\u03bd) := aj \u03bd *j , j=0 *0 *j where \u03bd := \u03b40 (the Dirac delta function) and \u03bd denotes the j-fold convolution power of \u0001 [ b P \u03bd",
    "A = I, it suffices to show that we can choose \u03bbk \u2192 \u221e such that B = B(E, \u03bbk ) = B2 (E, \u03bbk )B1 (E, \u03bbk ) satisfies B2 = -I",
    "i=1 Write Hi := C\u03b1i \u2297 Cni for all i \u2208 [k]",
    "k = Fq and \u03c3 = idK : K \u2192 K",
    "ce = c, ce\u2032 : 11X \u2032 \u2192 i*\u2032 i \u2032! A \u2032 [d \u2032 ] \u03f5A \u2032 [d \u2032 ] \u25e6 ce\u2032 = c \u2032",
    "i=1 \u0001 6 5 5 \u00015 We set \u21266 := \u21262 \u00d7 \u21263 \u2282 Z/2 3",
    "A = ac db \u2208 GL2 (Z) such that f (ax+by, cx+ dy) = g(x, y)",
    "L = (Lij ) \u2208 Mm (Z), Lii = fr(Li ), Lij = lk(Li , Lj ) (i \u0338= j)",
    "n = exp - c\u03b1 n 1 X pn (0, x) = N k\u2208\u039b \u001a W k \u03b1 2\u03c0i k\u00b7x \u03b1\u0001 e L + O ne-c\u03b1 W exp -c\u03b1 n L \u0013 \u0012 \u001b L 1 X W k \u03b1 2\u03c0i k\u00b7x \u03b1\u0001 e L + O ne-c\u03b1 W = exp -c\u03b1 n N k\u2208\u039b L \u0013 \u0012 (4",
    "C = Hp + 1 where H = max{ai,n : i \u2208 {0,",
    "J = \u03b1:J \u21a0 I (\u03b1) Here \u0237 (\u03b1) : X S,\u2192 X S-SdR , where -dR o n \u00d7S I (\u03b1) \u2032 \u2032 ( R ) : x \u2229 x = ; for \u03b1 ( i ) = \u0338 \u03b1 ( i ) X S( R ) = ( x ) \u2208 X i i i\u2208 I i S- dR dR \u00d7 I Now, since \u2294 is \u00e9tale (see, e",
    "t = y q - y + \u03b7 q z q - \u03b7z = u + \u03b7 q z q - \u03b7z, so \u03b7 q z q - \u03b7z \u2208 K(u, v, t)",
    "uv + vu = 0 , and the quaternion w = 12 (-1 + u + v + uv) \u2208 A",
    "s = 1 when d = 2 or s \u2208 ( 12 , min{ 72 - p, 23 }) when d = 3 (hence falling in the second regime covered by Proposition 3",
    "f = (fv )v\u2208C0 with O \u03d5f := fv \u2208 HC (l) \u2282 HC1"
  ],
  "logical_boolean": [
    "first = Y false (where Y = \u00acY[!]\u00ac denotes \u201cweak yesterday\u201d) is satisfied exactly at the first position of a finite trace",
    "K = C = 0, or K = C = 1\u2228 G",
    "Beff = 0 \u21d4 N \u00b7 B = + w\u2126 = 0 to compute an element of HN (\u2126) and solve the 2 (4",
    "w = X -u+Y \u21d2 u = X +Y -w, when u = X/2 \u21d2 w = X/2+Y , and when u = X - Y \u21d2 w = 2Y",
    "d = 2), this reduces to p 2 e-R(\u03b1) /2 = \u03b1 \u21d2 R(\u03b1) = 2 log(1/\u03b1)",
    "cL = N X k\u03b1 dim g\u03b1 \u03b1=1 k\u03b1 + h\u2228 \u03b1 , (3",
    "Lij = bri \u2200 i and N X Lij = bcj \u2200 j i=1 j=1 with r denoting row and c denoting column",
    "LLM = R\u03c1 \u03bb \u2227 \u03ba\u03c1 \u03bb = R\u03c1 \u03bb[\u00b5\u03bd] \u03ba\u03c1 \u03bb [\u03b1\u03b2] dx\u00b5 \u2227 dx\u03bd \u2227 dx\u03b1 \u2227 dx\u03b2",
    "dx =\u21d2 -c2s \u00b5\u2032s,\u03b8 d2 u du := - \u03b2 = \u03bb u with \u03b2 s s dx2 dx \u03c1s,\u03b8 (A",
    "R = R1 \u03c8 R + 2\u03c0Z and the action becomes R2 S= 4\u03c0 Z 1 d\u03d5L \u2227 *d\u03d5L + 4\u03c0 \u0393L Z Z i d\u03c8 \u2227 *d\u03c8 + 2\u03c0 \u0393R R (\u03d5L - R I 1 R \u03c8 )d\u03d5",
    "max =\u21d2 1 1 (1 - cos \u03b8) - K\u0303dM (1 - cos \u03b8)(1 + cos \u03b8) - (1 - cos \u03b8) = 0",
    "s = 0 \u21d2 s = s(z)",
    "dH = 0 , 1 d(e-2\u03d5 *10 H) - F0 \u2227 *10 F2 - F2 \u2227 *10 F410 - F410 \u2227 F410 = 0 , 2 -2\u03d5 10 IIB: d(e *10 H) - F1 \u2227 *10 F3 - F3 \u2227 *10 F5 = 0",
    "y = d\u03b1y\u2228 , H\u03b1\u03b7 = d\u03b1\u03b7\u2228 , we get the equality l\u03b1 \u00b7 akH ,\u03b1y = ares \u25a1 k,\u03b1\u03b7 by Lemma 9",
    "n = (pd k)ps + 1 = wps + 1 and we have (x n + ym - 1)(-x n + ym + 1) = 0 \u21d0\u21d2 y2m - x2n + 2x n - 1 - 0 s s s \u21d0\u21d2 (yl ) p y - (x2w ) p x2 + 2(x k ) p x - 1 = 0",
    "xr = 0, we have 0 = \u03b9Rx (\u03c4x \u2227 \u03c9xr ) = \u03c9xr which is a contradiction, and thus we must have \u03c4x \u2227 \u03c9xr \u0338= 0",
    "m = 0 \u21d2 \u03c1m = \u03c1m,0 e-3N (3",
    "ax = ay = az = a =\u21d2 bx = by = bz = b Under this condition, the equations of motion simplify to dui 1 = - \u03f5ijk uj Bk d\u03c4 \u03b3 which is formally identical to the Lorentz equation for a particle moving in a uniform effective magnetic field Bi",
    "Ltop = F a \u2227 F a = d(Aa \u2227 dAa + f abc Aa \u2227 Ab \u2227 Ac )",
    "j = Ri jkl ek \u2227 el = dei + \u0393i j \u2227 ej"
  ],
  "matrix_tensor": [
    "k=1 \uf8f4 \uf8f4 \uf8f4 \uf8f3(1 - e\u00b12\u03c0iz ) 12 \u03b8 = \u00b1 \u03c0 , 2 which (B",
    "C=- \u221a 2 , 2 m - \u03c92 \uf8ee B=\uf8f0 m sech mx \u03c9 !2 \uf8f9 - 1\uf8fb C",
    "j=1 \u03c0 \uf8f1 \uf8fc -2m-1 \u221e 2 + \u03bb2 \uf8f21 \uf8fd X \u03bb p j j \u0010 \u0011 \u0001 = (-\u03b2)-m \u03b6p (2m + 1) + \u00b7 \uf8f32 p p + \u03c01 + \u03bb2j \u03c3 \u03bbj \u03b2 e2\u03b2\u03bbj - 1 \uf8fe j=1 \u03c0 -2 2m m+1 X j=0 (1,p) j (-1) (1,p) B2j B2m-2j+2 (2j)!(2m - 2j + 2)! \u03b1m+1-j \u03b2 j , (1",
    "BE = 8\u03bb - 8\u03bb3 , \uf8f4 \uf8f4 \uf8f4 \uf8f2BF = 4(i - 1)\u03bb2 , \uf8f4 DE = -16i + 16\u03bb4 , \uf8f4 \uf8f4 \uf8f3 DF = 8i\u03bb - 8i\u03bb3",
    "D = 2, z z ! D (\u03c8R (z), \u03c8I (z)) = \u0014 \u0015 \u0014 \u0015 \uf8f4 1 2 1 \uf8f4 \uf8f4 \u221a Re , , if D = 7 or D = 11",
    "j = \uf8f4 \uf8f4 \uf8f3\u00b5 j , if xi, j = NaN where \u00b5 j = N1 PN i=1 xi, j is the mean of feature j",
    "d = 1, \uf8f4 \uf8f4 \uf8f4 \uf8f2C , if d \u0338= 1 is a square or d = -432, 3 (3",
    "j = pi\u03b1 , b j = pi\u03b1 , \uf8f4 \uf8f4 if \u039e if \u039e j j \uf8f2\u03b1j \uf8f20 b j = idL2 , , b j = idL2 , \u03a3j,j = 0 \u03a3j+d,j+d = 0 if \u039e if \u039e \uf8f4 \uf8f4 \uf8f3 \uf8f3 2 tanh(\u03d1j /2) if b \u039ej = R\u03d1",
    "Am = = Hence, we have that X\u2032 \uf8f1 m-t (m - t + 1)! \uf8f4 \uf8f4 \uf8f2(-1) if r = m - t + 1, (-1)m-t (m - t)! if r = m - t, \uf8f4 \uf8f4 \uf8f3 0 otherwise",
    "j = L, \uf8f2{b2 }, \u2032 \u2032 A\u0302(i, j) = {b1 , b2 , b2 }, if i = 2, j = 3,",
    "x = \uf8f0y \uf8fb = \uf8f0 Y \uf8fb + \uf8f0v(t, X, Z)\uf8fb",
    "p=1 m Q \u0393(\u03bdpj ) j=1 dEp m Q 2 P i=1 n Q \u0393(\u00b5pi + Ep Mpi ) \u0393(\u03bdpj + Ep Npj ) #(U+ U- )Ep # \u0393(Ep + 1) j=1 \uf8ee n \uf8f9 m Q Q \u0393(\u03bdpj ) \u0393(\u00b5pi + Ep Mpi ) Z\u221e \uf8ef \uf8fa 1 i=1 \uf8ef j=1 (m,n) Ep \uf8fa (U+ U- )#",
    "D = 20 and a root j of PD (j), we have \uf8f1 \uf8f4 \uf8f22 if p \u2261 0, 2, 3 (mod 5) and p \u0338= 13, mj = 4 if p = 13, \uf8f4 \uf8f3 0 otherwise",
    "p = 5, \uf8f4 \uf8f4 \uf8f4 12 if p = 3, \uf8f4 \uf8f4 \uf8f4 \uf8f324 if p = 2",
    "L = \uf8f0( U\u00b5j ) [U\u00b5\u2113 (1 - U0 )-1 V0 + Vz \u03b4\u00b5\u2113 ,z ]\uf8fb - \uf8f0( U\u00b5j ) (1 - U0 )-1 V0 \uf8fb , (D33) j=1 1,1 j=1 1,1 where \u2113 is again the position of the right-most non-trivial Pauli matrix",
    "C = 1 - , 2314 \uf8f4 \uf8f4 C \uf8f4 \uf8f4 \uf8f4 1 \uf8f4 \uf8f4 C3124 = , \uf8f4 \uf8f4 1-C \uf8f4 \uf8f4 \uf8f4 \uf8f4 C \uf8f4 \uf8f3C3214 =",
    "i=1 using the maximum-likelihood estimator: \uf8f9-1 \uf8ee c,t n X c,t \u03b1\u0302MLE = 1 + nc,t \uf8f0 \u03b5c,t i \uf8fb ln c,t \u03b5 min i=1 , where nc,t is the number of observations above the lower threshold \u03b5c,t min , chosen by the Clauset et al",
    "k=0 m=0 ( )= \u03b9 \u03bbs -k -1 X \u221e \u03c9 k(m+1) \u00b5m \u03c3 -k C \uf8f4 T k=0 \u00b5 - \u03c9 \u03bb \uf8f4 1 TX 12 \uf8f4 2 \uf8f4 \uf8f4 - \uf8f3 T k=0 m=0 \u03bbm+1 for s = \u221e, (4",
    "j = \u03bbj0 ,1 - 1, \u03bbj0 ,3 , \u03bbj0 ,2 + 1 \uf8f4 \u0001 \uf8f3 \u03bbj0 -1,3 + (p - 1) + nj0 -1 , \u03bbj0 -1,2 - nj0 -1 , \u03bbj0 -1,1 - (p - 1) if j \u0338= j0 , j0 - 1 if j = j0 if j = j0 - 1",
    "g = \uf8f0v1 v2 v3 \uf8fb = \uf8f01 0"
  ],
  "differential_calculus": [
    "m = m 2 X \u03f5:[m]\u2192{\u00b7,*} Recall that circular elements satisfy \u0011 \u0010 \u03f5(i) \u03f5(j) \u0338= 0 \u03ba2 cip , ciq (ei\u03b8 )\u2206(\u03f5) X Y \u03c0\u2208N C2\u03f5 (mn) V \u2208\u03c0 V ={ip ,iq } \u0011 \u0010 \u03f5(i) \u03f5(j) \u03ba2 cip , ciq",
    "j = e(\u2206j ) and l(\u2206j ) = l(\u2206ij ) imply that i\u2228 \u25a1 s = j = ij by (7",
    "i=1 We start by focusing on the case i = 3 where, as we shall see, the operator A\u03f5,3 (\u2206) turns out to be defined also for \u03f5 = 0 and \u2206 \u2208 B(\u03a3), possibly unbounded with unbounded 71 \u2206 \u2206 \u2206 complement",
    "i = N/2, hSy i = hSz i = 0, and h\u2206Sz i = h\u2206Sy i = N /2",
    "t = \u0001 \u2113 W0\u2113 + \u2206Wt\u2113 (x) x\u2113-1 = W0\u2113 x\u2113-1 + B \u2113 Ex,t A\u2113 x\u2113-1 , t t t \u2113 \u2206Wt\u2113 (x) = B \u2113 Ex,t A\u2113",
    "F = 1 , 0] - -1 -[G0 ] F + \u2206* G = 0",
    "j = 100, the precision loss of \u2206loss = 13",
    "H = \u03bd\u2206\u0398H , subject to the same initial datum \u0398(\u00b7, 0) = \u0398H (\u00b7, 0) = \u03980",
    "Y = 14 \u0010 1 1 X + Y \u0011 i h h i (X-Y )2 1 1 1 1 - 14 XY (X+Y ) , we get E X+Y = 2 E Rn - 4 \u2206n , and hence h E 1 i Rn+1 = 1 + pdiamond h 1 i 1 + pdiamond E - \u2206n",
    "m=0 p=1 where \u0012 \u0013 \u0012 \u0013 \u221a \u03b3 \u03c0 (-1)m 4\u03bb m+2p \u0393(\u2206 + m + p) 2 C p Ap , Ap,m = - 8\u03bb m!p! \u03c0 \u0393(\u2206) \u0012 \u0013 \u0012 \u0013 \u221a \u03b3 \u03c0 (-1)m 4\u03bb m+2p \u0393(\u2206 + m + p) 2 Bp,m = - Cp B p",
    "a = 0, the principal symbols satisfy F0r (\u03c9\u0303) = 27M 2 2 \u03c9\u0303 , \u2206 F0\u03b8 (\u2113\u0303) = \u2113\u03032 (cf",
    "HN = N X i=1 -\u2206i + V ext (xi ) + X VN (xi , xj , xk ), (1",
    "s = Ps \u0393c (Vs ) and Ran \u2206s = Sols (M), showing that it descends to a bijective map \u2206s : T Ss (M) \u2192 Sols (M)",
    "k=0 \u03b1k,J \u03c4 J-2k \u0012 2\u03c4 p \u0001 d-1 \u0013 \u2206\u2032 -J-2\u2206+2k+d-1 2 K d-2+\u2206\u2032 -J-2\u2206+2k (p\u03c4 ) , 2 \u0001 d (-1)k 2d-2+J-2k \u03c0 2 -1 \u0393 2 \u0393(J + 1)\u0393 d2 + J - k - 1 \u0001, \u03b1k,J = \u0393(k + 1)\u0393(d + J - 2)\u0393(J - 2k + 1)\u0393 21 (J - 2k - \u2206\u2032 + 2\u2206) (2",
    "Vi = FPGI (Mi , \u03c0i\u221e ) such that Ii \u2286 \u2206, Mi \u2208 Oalgi and \u03c0i\u221e \u2208 B Ii i for i = 0, 1",
    "D = {u \u2208 H 1 (R2 ) : \u2206u \u2208 L2 (R2 )} = H 2 (R2 )",
    "J = 4, the self-consistent equations are easily solved, and we obtain \u03bb = 4J/\u03c0, \u2206c = 2 2/\u03c0, and \u2206s = 0",
    "s = 0, rel b = -babs \u03b6\u2206abs + \u03b6\u2206rel = -bq (X) q (X) - bq (X)",
    "x = Set \u22061 (x) := Q1 (x) - p2 , \u22060 (x) := Q0 (x) - p2",
    "L = 1 the equation has only one solution z1 = z = 4 (1 + \u03b1) \u2206"
  ],
  "sum_prod_operators": [
    "j=1 Since WAn acts by permutations, WAn (\u03c91 ) = {ei - 1 n+1 \u2211 ej , n + 1 j=1 j = 1 ,",
    "UKIM = exp (-i(J\u03c3z \u2297\u03c3z + \u2211 ha (1\u2297\u03c3a +\u03c3a \u22971))), a=x,z with J = 1, hx = 0",
    "l=n ) \u2192 C , 13 N,e Bn,t \u2236 CN \u2192 \u21132 ({l}\u221e l=n ) by \u221e (AN,e n,t [x])j = \u2211 dj (2l, t)x(l), l=n N N,e Bn,t [u](l) = -i \u2211 dj (2l, t)uj , (3",
    "in=1 U1q (Kn,i ) \u220fin=1 Uq (Kn,i ) \u220fin=1 Kq (k i ) 0 0 U1q (Kn ) Uq ( K n ) Kq ( k ) 0",
    "i = N -1 \u2211Nj=1 \u03b4 j2 (T, T ), where N is the number of particles in the ensemble, which in (1+1)-dimensional space \u03b4 j2 (T, T ) is de\ufb01ned by? I",
    "TK = Tmin ] (ker Tmax \u00d7 {0}) = {(u, f ) \u2208 Tmax : u(b) = u(a)} \u222b = {(u, f ) \u2208 Tmax : U (\u00b7, 0)* wf = 0}",
    "i = \u03b7i \u2a7e 1/\u03b2) r1 \u2a7dv 24\u03b4 \u2211 (2\u03b6 \u03b2)r1 \u2211 (2\u03b6 \u03b2)r2 \u00b7 EB({i},r1 ) \u00b7 e j r2 =0 r1 \u2a7e1 \u2a7dv 48\u03b4 \u2211 (2\u03b6 \u03b2)r \u00b7 EB({i},r) \u00b7 e j",
    "F = \u2211\u221e n=1 f (n)x \u2208 KJxK be a k-Mahler series",
    "dx = dx \u0393(s - 1) 0 \u0393(s - 1) 0 x 2m 1 - x2 z 2 m=1 m \u222b \u221e (-2 log x)s-2 x2m 4 \u2211 (2z)2m 1 (2m) (s - 1) dx",
    "k = \u2211 Q(y) 1 - \u220f P(Yj \u0338= y) y\u2208L ! j =1 \u0010 \u0011 = \u2211 Q(y) 1 - (1 - Q(y))k",
    "m = P\u03bd,\u00b5,s \u220f m\u2032 \u2260m 15 Counting these GT labels reproduces the irrep dimension, \u03bd1 \u03bd2 \u00b51 #GT(\u03bd) = \u2211 \u2211 \u2211 (2s + 1) = dim V\u03bd , \u00b51 =\u03bd2 \u00b52 =-\u03bd2 s=\u2223\u00b52 \u2223 in agreement with Eq",
    "Ei = \u2211(-1)\u2223J\u2223 DJ J B B B = i - d\u00b5 i + \u22ef, i ByJ By By,\u00b5 i = 1,",
    "Ht = - \u2211 p j log p j , p j = softmax WU ht crit , j j (\u2113 ) where WU is the unembedding matrix",
    "H = \u2211nj=1 Hj and each operator Hj is supported on sites { j, j + 1,",
    "i = f (t), where f (t) = \u2211 ait i",
    "j=1 n v13,2 (x) = -16\u03c0 2 \u2211 \u03b1 j \u03b4 (x - y j ), (3",
    "k = 4 can be generated by a single additional generator added to the matchgate commutant: \u2126{(x{1234} = 1, x\u2205 = 2n - 1)} = \u2211(\u03b3j )\u22974",
    "Br = {(Pv )v\u2208\u2126k \u2208 V (Ak ) : \u2211 invv (\u03b1(Pv )) = 0, \u2200\u03b1 \u2208 Br(V )} v\u2208\u2126k is the so-called Brauer\u2013Manin set, and (2",
    "x = \u220fi\u221e=1 pi i , pi \u2208 P, \u03b1i \u2208 N0 , the explicit solution is ld( x ) = \u2211i\u221e=1 \u03b1pi",
    "n=0 k=0 \u03bb \u221e \u221e \u221e n tn tn (\u03b3,Y ) (\u03b3,Y ) = \u2211 Dk,\u03bb \u2211 SY2,\u03bb (n, k) = \u2211 \u2211 Dk,\u03bb SY2,\u03bb (n, k)"
  ]
}