harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\operatorname{p}^{(OW)}_f[\pi,\eta]=\lim_{i\in I}\frac{P_{f_{KF_i}}(\eta_{KF_i})}{\theta(KF_i)}=\lim_{i\in I}\frac{P_{\sum_{F_i}f_{K}}((\eta_K)_{F_i})}{\theta(K)|F_i|}=\operatorname{dens}(\omega)\operatorname{p}^{(OW)}_{f_K}[\phi,\eta_K].
\lim_{n \to \infty}n^{s -2}T_2^{(n)}(1 + \gamma^{(n)}) = \sqrt{2}u^{\frac{s - 1}{2}}\int_{-\infty}^\infty e^{-s \tau}e^{- \sqrt{u}\cosh \tau}\cosh \tau d \tau = \sqrt{2}u^{\frac{s - 1}{2}}\left(K_{s + 1}(\sqrt{u}) + K_{s-1}(\sqrt{u})\right).
(R_{T,\left\lfloor Tr\right\rfloor -n}^{(1)})^{j}=((-\log T+\sum_{k=0}^{T}(k+d)^{-1})-\sum_{k=\left\lfloor Tr\right\rfloor -n}^{T}(k+d)^{-1})^{j}=(-\psi(d)-\sum_{k=\left\lfloor Tr\right\rfloor -n}^{T}(k+d)^{-1})^{j}+o(1).
d + t_0 w +q w^2 + 4 w g(K) - 2 w - 2 t_0 - 2 q w +2b - 2d &= -d - t_0 w - qw^2 + w^2 s_{\overline{K}}+4 w^2 \nu(\overline{K}) - 2 w^2 + 2 d + 2 t_0 w + 2 q w^2.
\left\|\tilde{s}_i^2 A^T \tilde{u}_i^A - \tilde{c}_i B^T B \tilde{g}_i\right\|_2 =\left\|A^T \tilde{u}_i^A - \tilde{c}_i Z^T Z \tilde{g}_i\right\|_2 =\left\|A^T \tilde{u}_i^A - \tilde{c}_i R^T R \tilde{g}_i\right\|_2 = \left\|\alpha_{k+1}R^T v_{k+1}e_{k+1}^T x_i\right\|_2
f_j =\{D_{\mathfrak{p}}\pi(u), (D_{\mathfrak{p}}\pi)^2(u), \ldots, v_i, D_{\mathfrak{p}}(\pi(v_i)),\ldots\}f'_k =\{D_{\mathfrak{p}}\pi(u), (D_{\mathfrak{p}}\pi)^2(u), \ldots\}
f^{L}(\vec x) =\sum_{i=0}^{n-1}(|x_{\sigma(i+1)}|-|x_{\sigma(i)}|)f(V_{\sigma(i)}^+,V_{\sigma(i)}^-) &=\sum_{i=0}^{n-1}\int_{|x_{\sigma(i)}|}^{|x_{\sigma(i+1)}|}f(V_{t}^+,V_{t}^-)dt &=\int_0^{\|x\|_\infty}f(V_t^+,V_t^-)dt.
{\mathcal K}({\nabla}^Q)\,=\, - \frac{1}{2\pi\sqrt{-1}}\left(\int_{(X\times M)/M}({\mathcal K}(\nabla^L)+\frac{1}{2}{\mathcal K}(\nabla^L)^2)\wedge (1+\frac{1}{2}p^*_1{\mathcal K}(\omega_X))\right)_2
\sum_{m=1}^N\alpha^m\|Q_m^{(1)}-Q_m^{(2)}\|_{L^1}&\le \Bigg(1-\left(\inf_{j\in\{1,2\}}|W_0^{(j)}|\right)^{-1}\sup_{j\in \{1,2\}}\sum_{m=1}^N\alpha^m\|W_m^{(j)}\|_{L^1}\Bigg)^{-1}&\quad\cdot |W_0^{(1)}W_0^{(2)}|^{-1}\sum_{m=0}^N\alpha^m\|W_m^{(1)}\|_{L^1}\sum_{n=0}^N\alpha^n\|W_n^{(1)}-W_n^{(2)}\|_{L^1}.
\lim_{u\to\infty}\sup_{(s,\tau)\neq (s',\tau'),|\tau-\tau(u)|, |\tau'-\tau(u)|, |s-s'|\leq \delta_u}\left|\frac{1-r_u(s,\tau, s',\tau')}{\frac{\sigma^2(u|s-s'+\tau-\tau'|)+\sigma^2(u|s-s'|)}{2\sigma^2(u\tau^*)}}-1\right|=0.
\begin{array}{r c l}{{}}&{{}}&{{G = 4 \partial C + 6 F^{1 \, m}B_{m}\, ,}}\\{{}}&{{}}&{{}}\\{{}}&{{}}&{{H_{m}= 3 \partial B_{m}+ 3 \epsilon_{m n p}F^{1 \, n}A^{2 \, p}\, ,}}\\{{}}&{{}}&{{}}\\{{}}&{{}}&{{F^{2 \, m}= 2 \partial A^{2 \, m}\, ,}}\\ \end{array}
\int\!{\cal D}\varphi\,\delta_\eta \phi{\delta S_0\over \delta \phi}e^{-(S_0+S_5)}= (\delta^{(0)}_\eta \phi+ \delta^{(1)}_\eta \phi){\delta S_0[\phi]\over \delta \phi}e^{-(S_0[\phi]+S^{(1)}[\phi])}
E_{KL}=\frac{1}{2}(\frac{\partial U_{K}}{\partial X_{L}}+\frac{\partial U_{L}}{\partial X_{K}}+\frac{\partial U_{M}}{\partial X_{K}}\frac{\partial U_{M}}{\partial X_{L}})\approx\frac{1}{2}(\frac{\partial U_{K}}{\partial X_{L}}+\frac{\partial U_{L}}{\partial X_{K}})
e^{-\frac{\Omega}{kT}}=\sum_{N_{1}=0}^{\infty}...\sum_{N_{s}=0}^{\infty}\int...\int\frac{1}{h^{n}C}e^{\frac{\mu_{1}N_{1}+...+\mu_{s}N_{s}-E}{kT}}dp_{1}...dq_{n}
e_2(|Q|;W_{\beta}) \geq \min\left\{\frac{1}{2}, \left[ \ln\left(\frac{\sqrt{2}|Q|^{1/2}}{a_{\beta}/\Xi_\beta}\exp\left(\frac{\mathcal{I}_0(1)}{\mathcal{I}_1(1)}\right)\right)\right]^{-1}\right\}&= \frac{1}{\zeta_2 + \left(-\ln\left(\frac{a_{\beta}/\Xi_{\beta}}{\sqrt{2}|Q|^{1/2}}\right)\right)_+},
\sum_{q\le P}\sum_{\substack{a=1 (a,q)=1}}^q\int_{\beta\in\mathcal{B}(q,a)\setminus[-\frac{1}{2qP},\frac{1}{2qP}]}\frac{S(q,a)^3}{q^4}\mathcal{I}(\beta,q)d\beta\ll X^2\log^3X.
|I''_3| \lesssim \ \| |D|^\frac12 R\|_{BMO}\|b_\alpha\|_{BMO}\sum_{j,k \leq l}2^{\frac{j-l}2+\frac{k-l}2}\| r_j\|_{\dot H^\frac12}\|w_k\|_{L^2}\lesssim \ \| |D|^\frac12 R\|_{BMO}\|b_\alpha\|_{BMO}\| r\|_{\dot H^\frac12}\|w\|_{L^2}.
\|a\|^4 =\|\Psi(a)\|^4 =\|\Psi(a)^*\Psi(a)\Psi(a)^*\Psi(a)\| \leq \|\Psi(a)^*\Psi(aa^*)\Psi(a)\| \leq \|\Psi(a)^*\|a\|^2h\Psi(a)\| \leq \|a\|^3\|h\Psi(a)\|.
G_{k,\ell}(R,M) \lesssim\begin{cases}R^{2\nu(k)-n+3}M^{-n}\min\Big\{1,\Big(\frac{M}{R}\Big)^\beta\Big\}\|b^0_{k,\ell}(\rho)\chi(\frac{\rho}M)\rho^{\frac{n-1}2}\|^2_{L^2},~R\lesssim 1; \min\Big\{1,\Big(\frac{M}{R}\Big)^\beta\Big\}R^{-(n-2)}M^{-n}\|b^0_{k,\ell}(\rho)\chi(\frac{\rho}M)\rho^{\frac{n-1}2}\|^2_{L^2},~R\gg1.\end{cases}
\|,\|_{Fun}:\mathcal{H}_{Fun}\to\mathbb{R},\Psi\mapsto \|\Psi\|^2_{Fun}=\,\langle \Psi |\Psi\rangle :=\,\sum^N_{k=1}\,\int_{TM^k_4}\,dvol_{\eta^k_4}\,|\alpha_k|^2.
{\left<T_{\sigma^-\sigma^-}^{\left(in\right)}\left(\sigma^-\right)\right>}_{in}=0-{1\over24}D_{\sigma^-}^S\left(y^-\right)={\lambda^2\over48}\left(1-{1\over{{\left(1+\lambda\Delta e^{\lambda\sigma^-}\right)}^2}}\right)\;,
2^{nd}\int \mu(B(x,2^{-n}))^q \,dx \le 2^{nd}\int \left(\sum_{I\in \mathcal{D}_{n,x}}\mu(I)\right)^q \,dx &\le O_{d,q}(1) 2^{nd}\int \sum_{I\in \mathcal{D}_{n,x}}\mu(I)^q \,dx &\le O_{d,q}(1) \sum_{I\in\mathcal{D}_n}\mu(I)^q = O_{d,q}(1) S_{n,q}\mu.
\hat{\mathbb{E}}_t[\eta X+Y] =\hat{\mathbb{E}}_t[\sum_{i=1}^n(\eta_i X_i+Y_i)I_{A_i}] & =\sum_{i=1}^n\hat{\mathbb{E}}_t[\eta_i X_i+Y_i]I_{A_i}&=\sum_{i=1}^n(\eta_i^+\hat{\mathbb{E}}_t[X_i]+\eta_i^-\hat{\mathbb{E}}_t[-X_i]+Y_i)I_{A_i}&=\eta^+\hat{\mathbb{E}}_t[X]+\eta^-\hat{\mathbb{E}}_t[-X]+Y.
\begin{aligned}\bold{d}=[\sigma^2_{R}(l_{x,1},l_{y,1}),...,\sigma^2_{R}(l_{x,n_R},l_{y,n_R})]^{T}, \widetilde{\bold{d}}=[\sigma^2_{S}(m_{x,1},m_{y,1}),...,\sigma^2_{S}(m_{x,n_S},m_{y,n_S})]^{T}. \end{aligned}
J_N = 1 + \sum_{d=1}^\infty \frac{z^{2d}}{d!d!}N^{2d}\sum_{\substack{\lambda \vdash d \ell(\lambda) \leq N}}s_\lambda(PQ) \frac{(\dim \mathsf{V}^\lambda)^2}{\dim \mathsf{W}_N^\lambda},
W(D) \ge W(H)+\sum_{i=n_0+1}^{n}\left(2i-1+(r-1)^2\right) &> 2\binom{n_0}{2}+ 2\sum_{i'=n_0}^{n-1}i' +(n-n_0)+(n-n_0)(r-1)^2 &\ge 2\binom{n}{2}+an_0+ (n-n_0)a &= 2\binom{n}{2}+an
\kappa(\epsilon) = \mu (I-e^{-\alpha}\tilde{P})^{-1}r + \epsilon e^{-\alpha}\mu(I-e^{-\alpha}\tilde{P})^{-1}\tilde{P}^{(1)}(I-e^{-\alpha}\tilde{P})^{-1}r + O(\epsilon^2(\log(1/\epsilon))^w)
|u(x,t)| \leq \frac{1}{2 \pi t}\int_{B(x,t)}\frac{|u_0(y)|}{[t^2 -|y-x|^2]^{1/2}}dy\;\;\; + \frac{1}{2\pi}\int_{B(x,t)}\frac{|u_1(y)| + |\nabla u_0 (y)|}{[t^2 -|y-x|^2]^{1/2}}dy + \frac{1}{2 \pi}\int_0^t \int_{B(x,t-s)}\frac{|F(y,s)|}{\left[(t-s)^2 -|y-x|^2\right]^{1/2}}dy ds = I_1+ I_2 + I_3.
Q(T, x) = \prod_{j = 1}^{8}\mathcal{H}(u_j, x) = (x-1)^5 \ \dfrac{x^2 - 4x + 1}{x-1}\ \dfrac{x^2 - 5x + 1}{x-1}\ \frac{x^5 - 11x^4 + 38x^3 - 42x^2 + 8x}{(x^2-4x+1)(x^2-5x+1)}= x(x-1)^3 (x^4 - 11x^3 + 38x^2 - 42x+8) = x(x-1)^3 (x-4) (x^3 - 7x^2 + 10x - 2) .
(0_S, a\mapsto 0_A)*(o,\delta) = \big(0_S\cdot o, a\mapsto 0_A\cdot [o,\delta] + i(0_S)\cdot \delta(a)\big) = \big(0_S\cdot o, a\mapsto 0_A\cdot [o,\delta] + 0_A\cdot \delta(a)\big) = \big(0_S, a\mapsto 0_A\big)
\displaystyle{\prod_{i=1}^{d^{'}}}\big{[}X_{e+\delta_{i}}\big{]}^{r}\times\big{[}F|(\displaystyle{\bigcup_{1\leq i<\theta}}X_{e+\delta_d+i})\big{]}^{r,f^{'}}\subset\big{[}F|(\displaystyle{\bigcup_{1\leq i<\theta}}X_{e+i})\big{]}^{r,f}.
\big |g_{ij}(x,t)-g_{ij}(0,0) \big | = |g_{ij}(x,t)-\delta_{ij}| \leq \sup | \nabla_{(x,t)}g_{ij}(x,t)| \Vert (x,t)\Vert +O(t^2) \leq Cr'+O(t^2),
{\bf F}_\epsilon(t)+\int_0^t\int_\Omega \Big(\mu|\nabla u_\epsilon|^2+\widetilde\mu|\nabla\cdot u_\epsilon|^2+|\Delta d_\epsilon+\frac{1}{\epsilon^2}(1-|d_\epsilon|^2)d_\epsilon|^2\Big)\le{\bf F_\epsilon}(0)={\bf E}(0).
\mathcal{H}(...) = \sum_{n_{2}\in \mathbb{Z}}^{\infty}\,V\left(\frac{n_2}{N_0/n^2_1}\right) \,\mathcal{C}^{+}(n^2_1n_2,m;q)\,\overline{\mathcal{C}^{+}(n^2_1n_2,m;q)}&\times\,\,\, \mathcal{J}^{+}_1(n^2_1n_2,m,q)\,\overline{\mathcal{J}^{+}_1(n^2_1n_2,m,q)}.
(1+x)^{n}=(\begin{matrix}n\\ 0\end{matrix})x^{0}+(\begin{matrix}n\\ 1\end{matrix})x^{1}+(\begin{matrix}n\\ 2\end{matrix})x^{2}+\cdot\cdot\cdot+(\begin{matrix}n\\n-1\end{matrix})x^{n-1}+(\begin{matrix}n\\n\end{matrix})x^{n},
n\Re(\tau)-j_0\Re(\varpi(\tau)) \leq n\Re(\tau)+\frac{j_0}{\alpha}\left(\Re(\tau)-A_R\Re(\tau)^{2\mu}+A_I\Im(\tau)^{2\mu}\right) & \leq -nc_\star \Im(\tau)^{2\mu}+ \left(\frac{j_0}{\alpha}+n\right)\tau_p -\frac{j_0}{\alpha}A_R\tau_p^{2\mu}.
\begin{aligned}\left\lceil\frac{2InM-(I+1)Il}{(nM-1)M}\right\rceil = \left\lceil\frac{2(p_1-2)ewv\frac{p_1-1}{e}-(p_1-1)(p_1-2)(wv-w+\frac{ew}{r})}{(ewv\frac{p_1-1}{e}-1)\frac{p_1-1}{e}}\right\rceil = \left\lceil e-\frac{wve+\frac{(p_1-2)we(e-r)}{r}-e}{wv(p_1-1)-1}\right\rceil.\end{aligned}
m(Q) = \frac{8 \Omega^3}{3}\ \frac{\ell}{2 \pi}\frac{\pi}{2}\int_\zeta r(z) dz &= \frac{8 \Omega^3}{3}\frac{\ell}{4 \pi}\int_\zeta \pi r(z) dz &= \frac{8 \Omega^3}{3}\frac{\mathcal{A}}{4 \pi},
\begin{aligned}\int_X |\nabla \sigma|^2 \,d\mu_g \le \int_X \big[ -2|\nabla |\sigma||^2 - R_g |\sigma|^2 + (\sqrt{8}iF_A^+, |\sigma|\sigma) + \frac{1}{2}(D_A \Phi, D_A(|\Phi|^2 \Phi)) \big] \,d\mu_g\end{aligned}
\sum_{n=0}^{\infty}\frac{(\frac12-4k)_n (\frac12-2k)_n}{(1+2k)_n(1)_n}(3-2\sqrt 2)^{2n}\left[ 16(3\sqrt 2 -4) n + 16(13-9 \sqrt 2) k + 4(5\sqrt 2 -7) \right] = C_2 (3\sqrt 2-4)^{2k}\frac{2^{9k}}{3^{3k}}\, \frac{(1)_k \left(\frac12\right)_k}{\left(\frac{1}{12}\right)_k (\frac{5}{12})_k},
U(d_1,d_2) U(e_1,e_2) =\begin{pmatrix}1 d_1+e_1 d_2+e_2 -d_1 d_2-d_1 e_2-d_2 e_1-e_1 e_2 0 1 0 -d_2-e_2 0 0 1 -d_1-e_1 0 0 0 1\end{pmatrix}.
\limsup_{N \to \infty}\left| \frac{1}{N}\sum_{n=1}^{N}F_{1, h}(T^nx)F_{2, h}(T^{2n}x) \right|^2 & \leq \frac{C}{K}+ \frac{C}{(K+1)^2}\sum_{k=1}^K (K - k+1) \left( \limsup_{N \to \infty}\frac{1}{N}\sum_{n=1}^{N}\left(F_{1, h}\cdot F_{1, h}\circ T^k \right)(T^nx) \left(F_{2, h}\cdot F_{2, h}\circ T^{2k}\right)(T^{2n}x) \right).
i e \left( 2 \pi \right)^4 \delta(p^\prime - p - q )\Gamma^\mu_{abc}(q,p) =\int dx dy dz e^{i \, (p^\prime z - p y - q x)}{\delta^3 \Gamma\over \delta \bar{\psi}_c (z) \delta\psi_b(y) \delta A^\mu_a(x)}\, .
L_1 = \langle x_{\alpha_2}, x_{\alpha_1 + \alpha_2}\rangle, L_2 = \langle x_{\alpha_1}, x_{\alpha_1 + \alpha_2}\rangle, L_3 = \langle x_{\alpha_1}+ x_{\alpha_2}, x_{\alpha_1 + \alpha_2}\rangle.
\lim_{n\rightarrow \infty}\exp \left(-\frac{\lambda^2}2 n^{2\alpha+1}\int_{t- \delta}^{\eta_n(t)}\psi_{n,1}^2(s,\eta_n(t)) ds \right) = \exp\left( -\frac{\lambda^2}2 \kappa_1^2 \right)
B(F) =\max\left( \max_{1}\|h_{n}\star_{r}^{l}h_{n}\|,\max_{2}\|h_{n}\star_{r}^{l}h_{m}\| ,\max_{n=1,\dots ,k}\|h_{n}\|^{2}\right) B'(F) =\max( | 1-\sigma^{2}|,B(F),B(F)^{3/2})
\begin{array}{rcll}{\cal C}_+ = \sigma_2\otimes\sigma_1\otimes \sigma_2 \otimes \sigma_1\otimes \ldots\qquad \eta =+1{\cal C}_- = \sigma_1\otimes \sigma_2 \otimes \sigma_1\otimes\sigma_2\otimes \ldots \propto{\cal C}_+ \Gamma_*\qquad \eta =-1 \,.\end{array}
M_\phi(f)(x)=\begin{cases}e^{-x^2}, \mbox{$x\in[-\frac{\sqrt{2}}2,\frac{\sqrt{2}}2]$,}\frac1{\sqrt{2e}}\frac1x, \mbox{$x\in(-\infty,-\frac{\sqrt{2}}2)\bigcup(\frac{\sqrt{2}}2,+\infty)$.}\end{cases}
L_z \left| n, \, j \, j_z \right> = \left( \frac{n-j}{2}- \frac{n+j}{4}\right)\left| n, \, j \, j_z \right>, L^2 \left| n, \, j \, j_z \right> = \frac{n+j}{4}\left(\frac{n+j}{4}+ 1\right)\left| n, \, j \, j_z \right>.
{\gamma_{ap}}= \frac{{{{\left\|{{{\bf{h}}_{{s_0},a{p_0}}}}\right\|}^2}{{\left|{{X_{{s_0},a{p_0}}}}\right|}^{- \alpha}}}}{{\underbrace{{I_{s,ap}}+{{I_{ap,ap}}}}_{I{n_{ap}}}+{{{\delta^2}}\mathord{\left/{\vphantom{{{\delta^2}}{{P_s}}}}\right. \kern-\nulldelimiterspace}{{P_s}}}}},
\zeta^{\prime}\left( 0 | A \right) = \frac{1}{4 \pi}\int \left[ d^{2}z \right] t r^{\prime}[ k_{1}\left( z \right) ] \qquad , \qquad \left[ \frac{d}{d s}\zeta^{\prime}\left( s | A \right) \right]_{s = 0}= \gamma \zeta^{\prime}\left( 0 | A \right)
\frac{z_re^{x_r}}{1-z_re^{x_r}}\prod_{j=1}^{r-1}\frac{z_je^{\sum_{\nu=j}^{r-2}x_\nu+(x_{r-1}+x_r)}}{1-z_je^{\sum_{\nu=j}^{r-2}x_\nu+(x_{r-1}+x_{r})}},
\lambda_{A}^{i}= \epsilon^{A B}\mathcal{C}\left( \overline{{\lambda}}^{i B}\right)^{T}\quad ; \quad \overline{{\lambda}}^{i B}= \left( \lambda_{B}^{i}\right)^{\dagger}\gamma_{0}\quad ; \quad A , B = 1 , \dots , 2 .
V_{0}= \textstyle{\left[ -{\frac{1}{2}}\left( -{\frac{1}{2}}-{\frac{1}{2}}-{\frac{1}{2}}\right)^{3}| | -{\frac{1}{2}}\left( -{\frac{1}{2}}-{\frac{1}{2}}-{\frac{1}{2}}\right)^{3}\right]}.
\kappa_{n}^{2}=\zeta_{n}^{2}sin\left(\sqrt{CL\left(\omega_{n}^{2}\right)^{2}-\frac{CR^{2}}{4L}}x_{0}\right)+\eta_{n}^{2}cos\left(\sqrt{CL\left(\omega_{n}^{2}\right)^{2}-\frac{CR^{2}}{4L}}x_{0}\right).
v_n(x) =(1-x) v_{n-1}(x) + \int_0^x \max \{v_{n-1}(x), 1+ v_{n-1}(x-y) \}\, dy & = (1-x) \widetilde{v}_{n-1}(x) + \int_0^x \max \{\widetilde{v}_{n-1}(x), 1+ \widetilde{v}_{n-1}(x-y) \}\, dy & = (1-x) \widetilde{v}_{n-1}(x) +\int_0^x \max \{\widetilde{v}_{n-1}(x), 1+ \widetilde{v}_{n-1}(y) \}\, dy = \widetilde{v}_{n}(x),
H_{0}= \frac{1}{2}\int d^{3}x \left( B_{a}^{i j}F_{i j}^{a}- A_{i}^{a}A_{a}^{i}+ \left( \partial^{i}B_{0 i}^{a}\right) \left( \partial_{j}B_{a}^{0 j}\right) \right) \equiv \int d^{3}x h .
H_{a,b,c}(u,v;m+2) = (\tilde d_m + u\partial_u +v\partial_v) H_{a,b,c}(u,v;m), H_{a,b+1,c}(u,v;m) = b^{-1}\partial_v H_{a,b,c}(u,v;m), H_{a,b,c+1}(u,v;m) = c^{-1}\partial_u H_{a,b,c}(u,v;m) .
\langle t, t\rangle_n=\sum\limits_{j,k}\alpha_j^*\alpha_k \otimes \sum\limits_p d^*_{pj}d_{pk}=\sum\limits_p\left(\sum\limits_j \alpha_j\otimes d_{pj}\right)^*\left(\sum\limits_k \alpha_k \otimes d_{pk}\right) \geq 0.
2 \iint G_{h/2}\ast \partial_t^{h}u^h \cdot \left(\xi \cdot \nabla \right) G_{h/2}\ast u^h \,dx\,dt &=\frac1h \iint \!\left(\nabla \cdot \xi\right) ( 1-u^h \cdot G_h\ast u^h) \,dx\,dt -2 \sum_{i,j}\iint \!\partial_i \xi_j \,\partial_i G_{h/2}\ast u^h \cdot \partial_j G_{h/2}\ast u^h \,dx\,dt +o(1).
\displaystyle \sigma^n(t,x,y) = (\sigma(t,x,y) \mathbf{1}(|x|\le n)+ \sigma(0,0,0) \mathbf{1}(|x| > n))*\phi_n(x), & \displaystyle b^n(t,x,y) = b(t,x,y) \mathbf{1}(|x|\le n).
\Psi_{\Upsilon,M}(-2\delta-w_2) \Psi_{\Upsilon,M}(w_2) = \frac{M^{-2\delta}+M^{-2\delta(1-\Upsilon)}-M^{-\Upsilon w_2-2\delta}-M^{\Upsilon w_2-2\delta(1-\Upsilon)}}{w_2^2(2\delta+w_2)^2(\Upsilon\log M)^2}.
\sum_{n\ge 1, n\neq k}\frac{1}{\alpha^4_{\nu,n}-\alpha^4_{\nu,k}}=\frac{1}{8\alpha^4_{\nu,k}}\left[2\nu+3-\frac{\alpha^2_{\nu,k}+2\nu-1}{\alpha^2_{\nu,k}-2\nu+1}\right]-\frac{1}{2\alpha^2_{\nu,k}}\sum_{n\ge 1}\frac{1}{\alpha^2_{\nu,n}+\alpha^2_{\nu,k}},
\mu(c) =\frac{(1+c^2)(1+c)^2}{K_1(1-b)}\left(1+\frac{\Gamma K}{(K+c)^2}\right), \lambda(c) =\frac{1}{T(c)}\left(\frac{\Gamma c}{K+c}-\frac{(b+c)(1+c)}{1-b}\left(1+\frac{\Gamma K}{(K+c)^2}\right)\right).
\begin{matrix}{\bar{C}_{z z}^{\prime \prime}= - b_{z z},}&{C^{+ \prime}= c^{z},}\\{\bar{\Gamma}_{+ z}^{\prime \prime}= - i \beta_{+ z},}&{\Gamma^{+ \prime}= - i \gamma^{+},}\\{\bar{\Gamma}_{- z}^{\prime \prime}= \beta_{- z},}&{\Gamma^{- \prime}= - \gamma^{-},}\\ \end{matrix}
{\bf E}^{\tau p}={\frac{1}{2}}\left( \begin{array}{c c}{{a_{1}^{\tau p}}^{\dagger}}&{{a_{2}^{\tau p}}^{\dagger}}\\ \end{array}\right){\bf \sigma}\left( \begin{array}{c}{a_{1}^{\tau p}}\\{a_{2}^{\tau p}}\\ \end{array}\right)
\big | \big | \tilde{\delta}_{~ v}^{u}\big | \big | = \left( \begin{array}{l l}{0_{[ N - 2 ] \times [ N - 2 ]}}{0_{[ N - 2 ] \times [ 2 ]}}{0_{[ 2 ] \times [ N - 2 ]}}{1_{[ 2 ] \times [ 2 ]}}\end{array}\right) ~ ~ ,
B_n = \bigoplus_{k \ge 0}\bigoplus_{\substack{\ell_1, \dots, \ell_k \ge 1, m \ge 0, \ell_1 + \cdots + \ell_k + m + k = n}}A_{\ell_1}\otimes_S \cdots \otimes_S A_{\ell_k}\otimes_S G_m \otimes_S R.
D^{b a}= 2 ({\frac{1}{\eta + B}})^{a b}- \eta^{a b}\, \, \, \, , \, \, \, \, D^{i j}= - \delta^{i j}\, \, \, \, , \, \, \, \,{D^{\mu}}_{\alpha}D^{\nu \alpha}= \eta^{\mu \nu},
I_1= \int dx \left[ \alpha \Omega^{1/2}e^{-\psi_1}\alpha' \Omega'-\frac{3}{4}\alpha^2 \Omega^{3/2}e^{-\psi_1}\psi_1^{\prime 2}-\frac{\mu l^2}{2}\Omega^{-1}e^{\psi_1}\right] ,
P_{j}^{\alpha}(t,x) = 2^{j\alpha/p}\int_{{\mathbb R}^d}e^{2\pi ix\cdot\xi}\widehat{\Psi}(2^{-j}\xi) e^{-t|\xi|^\alpha}d\xi &= 2^{j\alpha/p}2^{jd}\int_{{\mathbb R}^d}e^{2\pi i2^{j}x\cdot\xi}\widehat{\Psi}(\xi) e^{-t2^{j\alpha}|\xi|^\alpha}d\xi.
y_{r_1,\ldots,r_k}=\left(\prod_{i=1}^k\mu(r_i)\varphi(r_i)\right)\sum_{\substack{d_1,\ldots,d_k r_i\mid d_i\forall i}}\frac{\lambda_{d_1,\ldots,d_k}}{\prod_{i=1}^kd_i},
\omega=\left(\{\alpha^+_i(\omega)\},\{\alpha^+_i(\omega)\},\gamma_1(\omega),\gamma_2(\omega)\right)\mapsto\left(-\alpha_1^-(\omega),-\alpha_2^-(\omega),\cdots, \alpha_2^+(\omega),\alpha_1^+(\omega)\right)\in Conf(\mathbb{R}^*).
\frac{d^{2}f}{d x^{2}}- \frac{( 5 x^{3}+ 1 )}{2 x ( 1 - x^{3})}\frac{d f}{d x}- \frac{l ( l + 6 )}{4 x^{2}( 1 - x^{3})}f + \frac{\lambda^{2}}{( 1 - x^{3})^{2}}\left( 1 + \frac{\rho_{0}^{6}}{( \omega R )^{6}x^{3}}\right) f = 0 \, ,
4x^{3-k}t^{1-\alpha}\eta_{2xt}-4(k+c)x^{2-k}\eta_{2xx}+4k(k+c)x^{1-k}\eta_{2x}-m(k+1)x^3(-(1-\alpha)t^{-1-\alpha}\tau &+(1-\alpha)t^{-\alpha}\tau_t-t^{1-\alpha}\tau_{tt})+2mkx^2t^{1-\alpha}\sigma_{1t}+2mk(k+c)\sigma_1=0.
\sum_{k=1}^{n-2}(n-1)p^{2n-k-5}= (n-1) \left(p^{2n-6}+ p^{2n-7}+ \ldots + p^{n-3}\right) &= (n-1) p^{n-3}\sum_{i=0}^{n-3}p^i &= (n-1)p^{n-3}\frac{1-p^{n-2}}{1-p}.
\Theta\left(f_{1},f_{2},T\right)=\int_{\mathbb{R}^{nd}}f_{2}\left(Q\left(v_{1}\right)-Q\left(v_{2}\right),\dots,Q\left(v_{n-1}\right)-Q\left(v_{n}\right)\right)f_{1}\left(T^{-1}\left\Vert v\right\Vert \right)dv.
H^{(6+3)}_{3B,3}(\tau),H^{(6+3)}_{3C,3}(\tau),H^{(6+3)}_{2B,3}(\tau),H^{(6+3)}_{2C,3}(\tau),H^{(6+3)}_{4B,3}(\tau),H^{(6+3)}_{6B,3}(\tau),H^{(6+3)}_{6C,3}(\tau):=0.
S L ( 2 ,{\bf R})_{S}: \ \ \left( \begin{matrix}{{\vec{\hat{\alpha}}}}\\{{\vec{\hat{\beta}}}}\\ \end{matrix}\right) \to \left( \begin{matrix}{{a}}&{{c}}\\{{b}}&{{d}}\\ \end{matrix}\right) \left( \begin{matrix}{{\vec{\hat{\alpha}}}}\\{{\vec{\hat{\beta}}}}\\ \end{matrix}\right) .
U_K = \sup_{n \in \mathbb{N}}\sum_{k=K}^\infty k B^n_k(T) = \sup_{n \in \mathbb{N}}\sum_{k=K}^\infty k \left( p_k - \zeta^n_k(T) \right) \le \sum_{k=K}^\infty k p_k, & \sup_{n \in \mathbb{N}}\sum_{k=K}^\infty k \|B^n_k\|_\infty \le \sup_{n \in \mathbb{N}}\sum_{k=K}^\infty k \|\zeta^n_k\|_\infty = \sum_{k=K}^\infty k p_k.
\mathrm{R H S}= \frac{- 4}{\sqrt{( 2 \pi )^{3}k}}( - 1 )^{\alpha}e^{- i [ \alpha ] \theta}\sin^{2}\pi \alpha \oint d z \frac{e^{- i \theta}}{( z + e^{\epsilon}) ( z - e^{- i \theta}e^{- \epsilon})}
g = \left( \begin{array}{c c}{1}&{X}\\{}&{}\\{0}&{1}\\ \end{array}\right) \left( \begin{array}{c c}{\exp ( \frac 1 2 \Phi )}&{0}\\{}&{}\\{0}&{\exp ( - \frac 1 2 \Phi )}\\ \end{array}\right) \left( \begin{array}{c c}{1}&{0}\\{}&{}\\{Y}&{1}\\ \end{array}\right) ,
\Delta S_{M}^{\mathrm{d i v}}( \lambda = 0 ) = \frac{2}{( 4 - d ) 6 4 \pi^{2}}\int d^{4}x \sqrt{- g}\left[ M_{\mu \nu \rho \sigma}M^{\mu \nu \rho \sigma}+ 2 M_{\mu \nu \rho \sigma}\left( P^{\rho \sigma \mu \nu}+ \frac{1}{6}R \delta^{\rho ( \mu}\delta^{\sigma \nu )}\right) \right] .
M_1M_1^T=\begin{bmatrix}1 \frac{1+\sqrt{d}}{2}\end{bmatrix}\begin{bmatrix}1 \frac{1+\sqrt{d}}{2}\end{bmatrix}=\begin{bmatrix}1 \frac{1+\sqrt{d}}{2}\frac{1+\sqrt{d}}{2}\frac{1+d+2\sqrt{d}}{4}\end{bmatrix}.
\hat{\Gamma}_{4}^{a^{\prime}a^{\prime}b^{\prime}b^{\prime}}( p_{1}, p_{2}, - p_{3}, - p_{4}) = [ V ( p_{1}) \otimes V ( p_{2}) ]^{a^{\prime}a^{\prime}a a}{\Gamma}_{4}^{a a b b}( p_{1}, p_{2}, - p_{3}, - p_{4}) [ V^{T}( - p_{3}) \otimes V^{T}( - p_{4}) ]^{b b b^{\prime}b^{\prime}}.
(x_-,x_+) = \bigsqcup_{1\le b\le b(k)}J^{(k)}(b) \bigsqcup_{\ell >k}\bigsqcup_{\varepsilon = \pm}\bigsqcup_{1\le b\le b_\varepsilon (\ell)}J_\varepsilon^{(\ell)}(b) \;\;\; \hbox{mod.}\, 0\, .
\begin{array}{r c l}{{{A^{\prime}}^{a}T^{a}}}&{{=}}&{{U A^{a}T^{a}U^{- 1}\ \ \ , \ \ \{\pi^{\prime}}^{a}T^{a}= U \pi^{a}T^{a}U^{- 1}\ \ \ ,}}\\{{{\phi^{\prime}}^{a}T^{a}}}&{{=}}&{{U \phi^{a}T^{a}U^{- 1}\, + \, \frac{i}{g}U \frac{d}{d t}U^{- 1}\ \ \ .}}\\ \end{array}
\sum_{k=1}^{\infty}\zeta\left( k+1\right) \left( r+k\beta\mid\alpha\right)_{n}x^{k}& \quad=-\left( r\mid\alpha\right)_{n}\left( \psi\left( 1-x\right)+\gamma\right) +\sum_{k=1}^{n}S\left( n,k;\alpha,\beta,r\right)k!\zeta\left( k+1,1-x\right) \left( \beta x\right)^{k}.
\alpha_2 = (-xX_1w_{12}-X_2w_{22}-X_3w_{32}+xX_2\tilde \varphi) dx + (\partial_xw_{12}+X_3w_{22}-X_2w_{32}-xX_3\tilde \varphi) x^{-1}\theta^1 + (-X_3w_{12}+\partial_xw_{22}+xX_1w_{32}-\tilde \varphi) \theta^2 + (X_2w_{12}-xX_1w_{22}+\partial_xw_{32}) \theta^3 .
R^{(\gamma),0}_1\varphi=\int_0^\infty \mathrm{e}^{-t}{}_hP_t^{(\gamma)}\varphi dt=\frac{u_{\lambda_\gamma+1}\ast \psi_\gamma}{\psi_\gamma}=\frac{u_{\lambda_\gamma}-u_{\lambda_\gamma+1}}{u_{\lambda_\gamma}}=1-w_{1,\lambda_\gamma}.
+ \frac{1}{2}({\gamma}^0 \otimes{\gamma}^0) (e_1{\phi}^{\rm self}_{(1)}+ e_2{\phi}^{\rm self}_{(2)}) - \frac{1}{2}e_1 b ({\gamma}^1\otimes{\gamma}^0 ) - \frac{1}{2}e_2 b ({\gamma}^0 \otimes{\gamma}^1 )
L^-=q\left(\begin{array}{ccc}q^{-h_2}0 0 (q-q^{-1})\alpha^- q^{-h_1-h_2}0 (q^{-1}-q)b^- (q^{-1}-q)\beta^- (-1)^Fq^{-h_1-2h_2}\end{array}\right)
\frac{\binom{K-2}{s-1}}{\binom{K-2}{s-1}+\sum_{i=0}^{K-1-s}\binom{K-1}{s+i}(N-1)^iN}\geq \frac{\binom{K-1}{s-1}}{\binom{K-1}{s-1}+\sum_{i=0}^{K-s}\binom{K}{s+i}(N-1)^iN}.
\hat{x_{4}}=H_{4}^{T}y_{4}=\frac{1}{2}[\begin{matrix}1&1&\sqrt{2}&0\\ 1&1&-\sqrt{2}&0\\ 1&-1&0&\sqrt{2}\\ 1&-1&0&-\sqrt{2}\end{matrix}][\begin{matrix}5\\ -2\\ -1/\sqrt{2}\\ -1/\sqrt{2}\end{matrix}]=[\begin{matrix}1\\ 2\\ 3\\ 4\end{matrix}]
\mathcal{F}_{(w)}(B;\ell) := \frac{\tilde{\mathcal{B}}_{(w)}^{(2)}(B;\ell)}{\mathcal{B}_{(w)}^{(2)}(B;\ell)}= \prod_{p > 2 \atop p \nmid \ell}\frac{\Big( 1+ \sum_{n=1}^\infty \frac{\tau_B(p^{2n})}{p^n}\Big(\frac{p^w}{p^w+1}\Big) \Big)}{\Big( 1+ \sum_{n=1}^\infty \frac{\tau_B(p^{2n})}{p^n}\Big)}\prod_{p | \ell}\frac{p^w}{p^w+1}.
\left( \bigg| \frac{1}{\sqrt{k}l}\underset{i=1}{\overset{k}{\sum}}\left( \underset{j=1}{\overset{l}{\sum}}(B_i^j - \frac{1}{2}) \right)^2 - \sqrt{k}/4 \bigg| \leq c_{\alpha,n}\right) \leq \frac{1/2+16\eta_{p,l,k}/\sqrt{k}}{\eta_{p,l,k}^2}.
d_{S(G,t+1)}(x^{t+1},y^{t+1}) = d_{S(G,t+1)}(v_0^{t+1},v_0v_1^t)+\sum_{i=0}^{r-2}d_{S(G,t+1)}(v_{i+1}v_i^t,v_{i+1}v_{i+2}^t) + &+d_{S(G,t+1)}(v_rv_{r-1}^t,v_r^{t+1})+r.
\mathcal{H}^s_{\infty}(F) \ll \liminf_{N\to\infty}\sum_{n=N}^{\infty}\beta_1^n\beta_2^n\beta_2^{-n(1-(1+\theta_1)\log_{\beta_2}\beta_1)}\beta_1^{-n(1+\theta_{1})s}&=\liminf_{N\to\infty}\sum_{n=N}^{\infty}\beta_2^{n\left(2+\theta_1-(1+\theta_{1})s\right)\log_{\beta_2}\beta_1}&=0.
<\sum_{n=0}^{\infty}a_nZ^n,\sum_{n=0}^{\infty}b_nZ^n>_{H^2_\beta(\mathbb{U}_{\mathbb{B}\mathbb{C}})}= \sum_{n=0}^\infty{a_n\beta(n)\;.\;(b_n\beta(n))}^{\dagger_3}=\sum_{n=0}^\infty a_n(b_n)^{\dagger_3}\beta(n)^2.
\begin{array}{r c l}{Y}&{\sim}&{1 \qquad \mathrm{f o r}\quad d_{2 i - 1}= 0 \nonumber}\\{}&{\sim}&{\exp \left( - \frac{1}{2 \pi \sqrt{3}}\sum_{i}R_{2 i - 1}^{2}\right) \mathrm{f o r}\quad d_{2 i - 1}= \pm 1 , \pm 2}\\ \end{array}
c_jx_1X_2p_{j}+ X_2 r_j = c_jx_1\left( \frac{\partial}{\partial{x_2}}p_{j}+ \sum_{k>2}p_{k}\frac{\partial}{\partial{x_k}}p_{j}\right)+\frac{\partial}{\partial{x_2}}r_j+ \sum_{\ell>2}p_{\ell}\frac{\partial}{\partial{x_\ell}}r_j=0.
\Gamma^0=\frac{36}{\pi^2 d^6}\int_{\Omega}\int_{\Omega}\left\langle[\mathbf{P}]\cdot\Upsilon^0(x, z)\cdot \Lambda(x), \Pi(z)\right\rangle\,dx\,dz-\frac{36}{\pi^2 d^6}\sum_{m=1}^{\aleph}\int_{\mathbf{B}_m}\int_{\mathbf{B}_m}\left\langle[P_{D_m}]\cdot\Upsilon^0(x, z)\cdot Q_m, V_m\right\rangle\,dx\,dz,
- \frac{\hbar^{2}}{2A}\widetilde{\bf D}= - \frac{\hbar^{2}}{2A}\sum_{a}\frac{\partial^{2}}{\partial (q^{a})^{2}}+\widetilde{\mathbf{V}},\ \widetilde{\mathbf V}=-\frac{\hbar^{2}}{2A}\frac{1}{P_{\lambda}{}^{2}}+ \frac{\hbar^{2}}{4A}\frac{1}{P_{\lambda}}\sum_{a}\left(\frac{\partial P_{\lambda}}{\partial q^{a}}\right)^{2}
{\mathcal M}= \left( \begin{array}{c}\frac 12 (M+M^{*}) \frac 1{2i}(M-M^{*})\end{array}\right) , \quad{\mathcal Z}= \left( \begin{array}{c}\frac 12 (Z+Z^{*}) \frac 1{2i}(Z-Z^{*}) \end{array}\right).
\lambda_{k}^{\mathbf{i}}(\mathbf{s},\mathbf{s'}) = \alpha_{k-1}(S_{k-1}=\mathbf{s'})\gamma_{\mathbf{i}}(\mathbf{\overline{y}}_{k},\mathbf{s'},\mathbf{s})\beta_{k}(S_{k}=\mathbf{s}).
\lambda_1^2 = \sum_{y\sim x}\sum_{z\sim y}\mathbf{v}_z = \sum_{y\sim x}\sum_{\substack{z\sim y z\in N(x)}}\mathbf{v}_z + \sum_{y\sim x}\sum_{\substack{z\sim y z\not\in N(x)}}\mathbf{v}_z \leq 2 e\left(N(x)\right) + e\left(N(x), V(G) \setminus N(x), \right)
Tr\left( \exp\left(\frac{1}{4}t\Delta_{\nu}\right)\right) \simeq \frac{1}{(4\pi t)^{n}}\sum\limits_{j=0}^{+\infty}b_{j}^{\left( \nu ,n\right)}\t^{j},\text{\ \}t\searrow 0^{+}
u_n(X_{t})=\int_{t}^{\zeta}f_n(X_r,u_n(X_r))\,dr+\int_{t}^{\zeta}dA_{r}^{\mu_n}+\int_{t}^{\zeta}dA_{r}^{\nu_n}+\int_{t}^{\zeta}dM^n_{r},\,\, 0\le t\le\zeta,