harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal{A}_{1}:=\{A_5 , A_6,A_7, L_2(7) , L_2(8), M_{22}\}, \mathcal{A}_{2}:=\{A_{8}, L_{3}(4), L_{2}(49), U_3(3), S_4(7), M_{11}\}, \mathcal{A}_{3}:=\{A_9, A_{10}, A_{11}, A_{12}, L_2(11), U_3(5),U_4(3),U_5(2), U_6(2), S_4(3), S_6(2), O_8^{+}(2), M_{12}, McL, HS, J_2\}.
\Omega_{b\nu}(az,bz)=\frac{\bar{I}_{\nu}^{(a)}(az)/\bar{I}_{\nu}^{(b)}(bz)}{\bar{K}_{\nu}^{(a)}(az)\bar{I}_{\nu}^{(b)}(bz)-\bar{K}_{\nu}^{(b)}(bz)\bar{I}_{\nu}^{(a)}(az)}.
V[52]=V[\begin{matrix}1\\ 3\\ 2\end{matrix}][\begin{matrix}2&3&1\end{matrix}]=V[52]^{*}=V^{*}[\begin{matrix}2\\ 3\\ 1\end{matrix}][\begin{matrix}1&3&2\end{matrix}]=[\begin{matrix}s&p^{*}&r^{*}\\p^{\prime*}&q&p^{\prime}\\r&p&s^{*}\end{matrix}]
E\{C_{i_1}\cdots C_{i_k}\}= \frac{m_{i_1}\cdots m_{i_k}}{q_{\phi}(n)}\sum_{h_1,\dots,h_k \ge 1}\phi^{h_1 + \cdots + h_k}q_{\phi}(n-h_1i_1 - \cdots - h_ki_k),
log|f(z_{0})|=\sum_{k=1}^{n}log|\frac{z_{0}-a_{k}}{1-\overline{a}_{k}z_{0}}|+\frac{1}{2\pi}\int_{0}^{2\pi}P_{r_{0}}(\varphi_{0}-\theta)log|f(e^{i\theta})|d\theta.
U_{j}( \underline{{p}}, \underline{{q}}) \equiv \frac{g m_{j}}{\sqrt{1 6 \pi^{3}}}\frac{1}{p^{+}\sqrt{q^{+}}}, V_{2 s}( \underline{{p}}, \underline{{q}}) \equiv \frac{g}{\sqrt{8 \pi^{3}}}\frac{\vec{\epsilon}_{2 s}^{*}\cdot \vec{p}_{\perp}}{p^{+}\sqrt{q^{+}}},
{\cal L}= -{1 \over 4}F^{\mu \nu}F_{\mu \nu}+{\kappa \over 4}\epsilon^{\alpha \beta \gamma}F_{\alpha \beta}A_{\gamma}+ i \bar{\psi}{\gamma}^{\mu}{\partial}_{\mu}\psi - m \bar{\psi}\psi + eA_{\mu}(J^{\mu}+ lG^{\mu}) +{1 \over 2}g (\bar{\psi}\psi)^2,
\sum_{a , b}\eta_{a b}\vartheta \left[ \begin{array}{c}{a}\\{b}\\ \end{array}\right] \vartheta \left[ \begin{array}{c}{a}\\{b + v_{5}}\\ \end{array}\right] \prod_{r = 3}^{4}\vartheta \left[ \begin{array}{c}{a + \frac{1}{2}}\\{b + v_{r}}\\ \end{array}\right] = 0
\check{\mathcal{H}}(\mathcal{L}_+;q,t)-\check{\mathcal{H}}(\mathcal{L}_-;q,t)=z^{2\epsilon}\check{\mathcal{H}}(\mathcal{L}_0;q,t) \check{\mathcal{H}}(\mathcal{L}^{+1};q,t)=t\check{\mathcal{H}}(\mathcal{L};q,t), check{\mathcal{H}}(\mathcal{L}^{-1};q,t)=t^{-1}\check{\mathcal{H}}(\mathcal{L};q,t)
(R^i_k)_{_{\mu F_0}}=\frac{1}{\mu^2 F_0^2}\big(2\frac{\partial G^i_{_{F_0}}}{\partial x^k}-\frac{\partial^2 G^i_{_{F_0}}}{\partial x^j \partial y^k}y^j +2G^j\frac{\partial^2 G^i_{_{F_0}}}{\partial y^j \partial y^k}- \frac{\partial G^i_{_{F_0}}}{\partial y^j}\frac{\partial G^j_{_{F_0}}}{\partial y^k}\big) &=\frac{1}{\mu^2}(R^i_k)_{_{F_0}}.
g_{j_2}h(J\setminus(D\cup J_1)) =g(I_1)g(I_2)h(J\setminus(D\cup J_1)) &=g(J_1\setminus I_1)g(J_2\setminus I_2)h(J\setminus(D\cup J_2))=g_{j_1}h(J\setminus(D\cup J_2)).
\frac{k \mu}{2 \pi}\widehat{L}\equiv - J^{-}( \theta ) ( L ( \theta ) - \partial_{\theta}J^{3}/ 2 ) , \frac{k \nu}{2 \pi}\widehat{\widetilde{L}}\equiv - \widetilde{J}^{+}( \theta ) ( \widetilde{L}( \theta ) + \partial_{\theta}\widetilde{J}^{3}/ 2 )
\left. b_0 \, \frac{d}{dt}\mbox{Cap}_{\mathcal{A}}(E_t) \right|_{t = 0}\, = (p-1) \int_{\mathbb{S}^{n-1}}(h_1(\xi)-h_0(\xi)) d\mu(\xi) = \left. b_1 \, \frac{d}{dt}\mbox{Cap}_{\mathcal{A}}(E_t) \right|_{t = 1}.
\mathcal{T}=\mathcal{T}(\dot{Q},Q)=\frac{1}{2}\dot{Q}^{\mathrm{T}}\alpha\dot{Q}+\frac{1}{2}\dot{Q}^{\mathrm{T}}\theta Q,\qquad\mathcal{V}=\mathcal{V}(\dot{Q},Q)=\frac{1}{2}Q^{\mathrm{T}}\eta Q-\frac{1}{2}\dot{Q}^{\mathrm{T}}\theta Q
\frac{1}{6}\sum_{i=1}^{m}r^2_i(r^2_i -1)-\frac{15}{8}m_2-\frac{3}{4}m_3-\frac{1}{2}m_4 = \sum_{i=1}^{n}\lambda^2_i \frac{1}{6}\sum_{i=1}^{m}r^2_i(r^2_i -1)-\frac{15}{8}m_2-\frac{3}{4}m_3-\frac{1}{2}m_4 -\lambda^2_1 =\sum_{i=2}^{n}\lambda^2_i
\widetilde{v}_0 := \widetilde{\phi}(\widetilde{\beta}_0) = \widetilde{c}_5, \widetilde{v^\prime}_0 := \widetilde{\phi}(\widetilde{\beta^\prime}_0) = \widetilde{c}^\prime_5, \widetilde{w}_0 := \widetilde{\iota}\widetilde{\phi}(\widetilde{\beta}_0) = \widetilde{c}_1
G_1(s) =(n+p)\Big(p(P_q^{n,p-q}-P_{q-1}^{n+1,p-q})\Big)-2s\Big(\big(P_q^{n-1,p-q}\big)'-p\big(P_{q-1}^{n,p-q}\big)'\Big)-(n+1)\big(P_q^{n-1,p-q}-pP_{q-1}^{n,p-q}\big),
\begin{array}{c l l}{{( 3 , 6 ) \otimes ( 2 , \overline{{{4}}})}}&{{=}}&{{( 2 \oplus 4 , \, 4 \oplus \overline{{{2 0}}}) \, ,}}\\{{{}}}&{{{}}}&{{{}}}\\{{( 1 , 1 0 ) \otimes ( 2 , \overline{{{4}}})}}&{{=}}&{{( 2 , \, 4 \oplus 3 6 ) \, ,}}\\ \end{array}
\underset{\delta\rightarrow0}{\lim}\left|III_{\delta}\right|\leq2\underset{\overline{t}\in[0,T]}{\sup}\left|A_{-}(\overline{t},\overline{t},\cdot)\right|_{\mathrm{Lip}}(\tau-\sigma)=\underset{\overline{t}\in[0,T]}{\sup}\left|A(\overline{t},\cdot)-\widetilde{A}(\overline{t},\cdot)\right|_{\mathrm{Lip}}(\tau-\sigma).
\frac{(-1)^{\theta m(m-1)/2}q^{c(N, m, \theta)}}{\prod\limits_{r=1}^m{\prod\limits_{j=1}^{\theta(N - r+1)-1}{\big(z_r - q^{j-\theta}\big)}}}\cdot\prod_{r=1}^m{T_{q, z_r}^{\theta r - \theta + \tau_r^+ - \tau_r^-}}\left( \prod_{j=1}^{\theta N - 1}{\big(z_r - q^{\theta(N - 1) - j}\big)}\right).
\sup_{z \in D}|Pg(z)|\rho_D^{-1}(z) =: \lambda \Longrightarrow \sup_{z \in D}|Sg(z)|\rho_D^{-2}(z) \leq 64\lambda + \lambda^2/2; \lim_{z \to \partial D}|Pg(z)|\rho_D^{-1}(z) = 0 \Longrightarrow \lim_{z \to \partial D}|Sg(z)|\rho_D^{-2}(z) = 0.
[r, s, s] = [[r,s], s] = [r^{-1}s^{-1}rs, s] = [r^3srs,s] = [r^2, s] = r^2sr^2s = 1 \leq D_8\cap\langle s\rangle &[s, s, s] = [[s,s],s] = [1,s] = 1 \leq D_8\cap\langle s\rangle &[y, s, s] = [[y,s], s] = [y^{-1}s^{-1}ys, s] = [y^3sys,s] = [y^4s^2, s] = [1, s] = 1 \leq C_4 \cap \langle s\rangle
d\big(Z_2(G)/Z(G)\big) = d\big(Z(G)\big) d(G). d\big(\Omega_1(Z_2(G))\big) \ge d\big(Z(G)\big) d(G). d\bigg(\frac{Z(\Phi(G))\cap Z_3(G)}{Z(G)}\bigg) \ge d\big(\Omega_1(Z_2(G))\big)d(G)- d\big(\Omega_1(Z(G))\big){{d(G)}\choose{2}}.
M^{(1)}_{1,\phi}(\infty)={8\Psi^2(\varphi)}\begin{pmatrix}2 -3\cr0 0\end{pmatrix},{M^{(2)}_{1,\phi}(\infty)=16\Psi^2(\varphi)\begin{pmatrix}1 -3\cr0 1\end{pmatrix},M^{(3)}_{1,\phi}(\infty)=0.}
\tilde{R}= \begin{pmatrix}\tilde{g}_{n-k+1}\tilde{g}_{n-k+2}\vdots \tilde{g}_{n}\end{pmatrix}\/; \hat{R}= \begin{pmatrix}\hat{g}_{n-k+1}\hat{g}_{n-k+2}\vdots \hat{g}_{n}\end{pmatrix}\/; E_{n-k+1}^\zeta = \begin{pmatrix}e_{n-k+1}(\zeta) \vdots \vdots e_n(\zeta) \end{pmatrix}
{\cal H}={1\over2}P^2+m^2_L\left(\sum_{\alpha_i\in\Pi\cap\Delta_L}e^{\alpha_i\cdot Q}+e^{\alpha_0\cdot Q}\right)+ m^2_S\sum_{\alpha_i\in\Pi\cap\Delta_S}e^{2\alpha_i\cdot Q},
\begin{array}{l l l}{{A_{0}^{+ 3}| h , \ell_{3}^{+}, \ell_{3}^{-}, u \rangle}}&{{=}}&{{\ell_{3}^{+}| h , \ell_{3}^{+}, \ell_{3}^{-}, u \rangle}}\\{{\}}&{{\}}\\{{A_{0}^{- 3}| h , \ell_{3}^{+}, \ell_{3}^{-}, u \rangle}}&{{=}}&{{\ell_{3}^{-}| h , \ell_{3}^{+}, \ell_{3}^{-}, u \rangle}}\\ \end{array}
G_{n}^{(1)}=P_{n,0}P_{n,1}^{R}P_{n,2}P_{n,3}^{R}\cdot\cdot\cdot andG_{n}^{(2)}=P_{n,0}^{R}P_{n,1}P_{n,2}^{R}P_{n,3}\cdot\cdot\cdot
\frac{1}{J_n}\sum_{j=1}^{J_n}\sum_{x^n\in \mathsf{A}^n}E(x^n\mid j) \mathrm{tr}\left(\overline{W}_{s''}^{\otimes n}\left( \rho_{x^n}\right)D_{j}^n\right)\geq 1 - (\frac{3}{\tau_n})^{2{d'}^4}2^{-n^{1/16}\beta}\geq 1 - 2^{-\frac{1}{2}n^{1/16}\beta}
\widetilde\eta_i(x)-b_i(x/N) = \widetilde\eta_i(x)(1-b_i(x/N)) -b_i(x/N)(1-\widetilde\eta_i(x)) & = \sum\limits_{0\le j \neq i\le 3}\Big( \widetilde\eta_i(x) b_j(x/N) - b_i(x/N) \widetilde\eta_j(x/N) \Big) .
(\omega(g)\phi,\phi)= ~\|\phi\|^2 (\bar{\xi}\xi')^{\nu_2+\frac{3}{2}}(\bar{\eta}\eta'\bar{\zeta}\zeta')^{\nu_1+\frac{3}{2}}(\gamma\bar{\gamma}')^{\alpha+\frac{3}{2}}(\cosh t)^{-\nu_2-\alpha-3}&\times ( \cos \theta \cos \theta' + \cosh^{-1}t\zeta\bar{\zeta}'\sin \theta \sin \theta')^r.
\lim_{\zeta\to+0}F(\zeta_0+\zeta) =\lim_{\zeta\to+0}\int_Ug(\zeta_0+\zeta,w)dV_w =\int_U\lim_{\zeta\to+0}g(\zeta_0+\zeta,w)dV_w=\int_Ug(\zeta_0,w)dV_w=F(\zeta_0),
\partial^L \varphi(x):=\left\{v\in \Re^n: \exists (x^k, v^k ) \to (x, v)\{\rm with}\ \lim_{y\to x^k}\frac{f(y)-f(x^k)-\langle v^k, y-x^k\rangle}{\|y-x^k\|}\geq0\ \forall k\right\}.
E^{\nu}\left[\left.\frac{d\mu}{d\nu}(X_{0})\right|\bigcap_{n \geq 0}\mathcal{F}_{0,\infty}^{\mathcal{Y}}\vee \mathcal{F}_{n,\infty}^{\mathcal{X}}\right] =E^{\nu}\left[\left.\frac{d\mu}{d\nu}(X_{0})\right|{F}_{0,\infty}^{\mathcal{Y}}\right|
h_{g;k_{1},...,k_{n}}=\frac{m!}{\#Aut(k_{1},...,k_{n})}\prod_{i=1}^{n}\frac{k_{i}^{k_{i}}}{k_{i}!}\int_{\overline{M}_{g,n}}\frac{c(E^{*})}{(1-k_{1}\psi_{1})\cdot\cdot\cdot(1-k_{n}\psi_{n})}.
\overline{y}_n^{*}:\Gamma(X'_n,\varphi_n^{*}(\mathcal{F}_{n}))\xrightarrow{\sim}\Gamma(\overline{S}_n, \overline{y}_n^*(\varphi_n^*(\mathcal{F}_{n})))=\mathbb{V}_{n}(\mathcal{F}).
\mathrm{div}\big((1+\dot{S}_m a)\dot{\Delta}_j\nabla\Pi\big)= \mathrm{div}\dot{\Delta}_j\big(f+\dot{E}_m+\nabla \dot{S}_m a\cdot\nabla u+\dot{S}_m a\Delta u\big)-\mathrm{div}([\dot{\Delta}_j,\dot{S}_m a]\nabla\Pi).
{\cal L}=- \frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^{2}(A_{\mu}+\partial_{\mu}\theta)(A^{\mu}+\partial^{\mu}\theta)+A^{\mu}\partial_\mu B-\partial_{\mu}\bar{\cal C}\partial^{\mu}{\cal C}-\frac{1}{2}\alpha B^{2}.
y^{2}zw - x^3z + \Bigl(\frac{I_{4}}{12}w+z\Bigr)xzw-\Bigl(\frac{I_{10}}{4}w^{2}+\frac{I_{2}}{24}z^{2}\Bigr)w^{2}-\Bigl(\frac{I_{2}I_{4}-3I_{6}}{108}\Bigr)zw^{3}=0.
Z_a(x):=\sum_{\zeta(\rho)=0, \rho=\beta+i\gamma \atop a<\Re \rho=\beta, |\gamma|<x}\frac{x^\beta}{|\rho|}, Z(x):= \sum_{\zeta(\rho)=0, \rho=\beta+i\gamma \atop \theta<\Re \rho=\beta, |\gamma|<x}\frac{x^\beta}{|\rho|}.
\mathcal{D}_{f_{0}}= &\left\{f\in C^{\infty}(I\times (-1,1),\mathbb{R}^{2}): f(s,0)=f_{0}, \frac{\partial f}{\partial s}(s,\varepsilon )=1, \forall s\in I, \forall \varepsilon\in(-1,1) \right\}
{\cal L}_{\mathrm{B I}}= - \sqrt{- \mathrm{d e t}\big ( \eta_{\mu \nu}+ F_{\mu \nu}+ \bar{\chi}\Gamma_{\mu}\partial_{\nu}\chi +{\textstyle{\frac{1}{4}}}\bar{\chi}\Gamma^{a}\partial_{\mu}\chi \bar{\chi}\Gamma_{a}\partial_{\nu}\chi \big )}.
\sum_{P\in Q^{(5,3)}}X_{P}= X_{1+2+3}+ X_{1+2+4}+ X_{1+2+5}+ X_{1+3+4}+ X_{1+3+5}&+ X_{1+4+5}+ X_{2+3+4}+ X_{2+3+5}+ X_{2+4+5}+ X_{3+4+5}.
\frac{1}{s^{N-2\alpha+1}}\int_{\{y=0\}\cap \mathcal{B}_s(x)}f_n\,dX+\frac{d}{ds}\Bigl(\frac{1}{s^{N-2\alpha+1}}\int_{\{y>0\}\cap \partial \mathcal{B}_s(z)}y^{1-2\alpha}w_n \,dS\Bigr)=0,
[y^{p^r}_{12}, y^{p^s}_{23}]=[x^{p^r}_{12}, x^{p^s}_{23}]=x^{p^r}_{12}(1- x^{-p^r}_{12}x^{p^s}_{23}x^{p^r}_{12}x^{-p^s}_{23})x^{p^s}_{23}.
B_{\bar{0}}(\theta) =y^{(2)}_{1,2}(\theta)+i =3 \omega^{-1}w_E[\varphi(E\omega^{-1}) , \varphi(E\omega^{-5})] \frac{d^2}{dE^2}\varphi(E \omega^3) +i, B_{0}(\theta) =y^{(1)}_{1,2}(\theta) -i =-3\omega w_E[\varphi(E\omega) , \varphi(E\omega^{5})] \frac{d^2}{dE^2}\varphi(E \omega^{-3}) -i.
\mathbf{C}_{\mathbf{s}\mathbf{s}_\mathcal{Q}}=\sqrt{\frac{2}{\pi}}\mathbf{K}\mathbf{C}_{\mathbf{s}},\mbox{where}\mathbf{K}=\left(\mathbf{C}_{\mathbf{s}}\right)^{-\frac{1}{2}}.
\left[{\cal D}^\mu \cdot \Gamma \right]_{{\rm flat}}=x^\nu \left[{\Theta}_\nu^{~\mu}\left( x\right) \cdot \Gamma \right]_{{\rm flat}}-x^\nu \left[w_\nu^{~\mu}\left( x\right) \cdot \Gamma \right]_{{\rm flat}}-\left[\Lambda^{\mu}\left( x\right) \cdot \Gamma \right]_{{\rm flat}}\,\,,
G^1=\frac{-x^1(y^1)^2(y^2)^2}{\left[ 2(y^1)^2+(x^1)^2(y^2)^2\right]}, G^2=\frac{2(y^1)^3(y^2)}{x^1\left[ 2(y^1)^2+(x^1)^2(y^2)^2\right]}.
S=-\frac{1}{2}\int d^2xdy\,\Bigl(i\bar\psi\gamma^\mu\frac{\partial\psi}{\partial x^\mu}-\frac{1}{2}\bar\psi\gamma_5\frac{\partial\psi}{\partial y}+\frac{\partial\phi}{\partial x^\mu}\frac{\partial\phi}{\partial x_\mu}-\frac{1}{4}\frac{\partial\phi}{\partial y}\frac{\partial\phi}{\partial y}+F^2\Bigr)
\psi^*\left(\begin{array}{c c}a b c d \end{array}\right) = \left(\begin{array}{c c c}a^{2 \, p^{e_1}}0 b^{2 \, p^{e_1}}a^{p^{e_1}}\, c^{p^{e_1}}1 b^{p^{e_1}}\, d^{p^{e_1}}c^{2 \, p^{e_1}}0 d^{2 \, p^{e_1}}\end{array}\right) .
\langle(\Pi_z{}^{a}\gamma_a{\cal C})\rangle \,\langle(\Pi_z{}^{a}\gamma_a{\cal C})^\dagger \rangle =- |Z|^2{\rm I}\ , \qquad \langle(\gamma_{\hat \mu}{\cal C}\;\Pi_z{}^{\hat\mu})\rangle \,\langle(\gamma_{\hat \mu}{\cal C}\; \Pi_z{}^{\hat \mu})^\dagger \rangle =|Z|^2{\rm I}\ ,
\small\mathbf{y}_{i,j,t}= \begin{cases}\mathbf{H}_{i,j,t}\mathbf{x}_{i,j,t}+ \mathbf{n}_{i,j,t}, (i,j) \in \mathbb{S}_b, b \notin \mathbb{I}, \mathbf{H}_{i,j,t}\mathbf{x}_{i,j,t}+ \mathbf{H_I}_{i,j,t}\mathbf{x_I}_{i,j,t}+ \mathbf{n}_{i,j,t}, (i,j) \in \mathbb{S}_b, b\in \mathbb{I},\end{cases}
\mathbb{E}\sum_{x \in V_{\sigma}(\mathbf{X}_0)}f(\mathbf{X}_0, x) =\int_0^\infty \mathbb{E}\sum_{\substack{x \in V_{\sigma}(\mathbf{X}_0(\mathbf{T}_{\sigma_1}\cup\{z_1\})) x \geq \{\sigma_1 z_1\}}}f(\mathbf{X}_0(\mathbf{T}_{\sigma_1}\cup\{z_1\}), x) \,dz_1.
\begin{array}{c}{{I_{\nu}\left( - y - i x \right) = e^{\displaystyle{- \imath \pi \nu}}I_{\nu}\left( y + i x \right)}}\\{{I_{\nu}\left( y + i x \right) = \left( I_{\nu}\left( y - i x \right) \right)^{*}.}}\\ \end{array}
R^\nabla_{X,Y}Z = R^W_{X,Y}Z - \frac{1}{2}(dJ\theta)(X,Y)JZ-\frac{1}{2}(d\theta)(X,Y)Z -\frac{1}{2}\left(D^W_X\left(D^W_YJ\right)-D^W_Y\left(D^W_XJ\right)-D^W_{[X,Y]}J \right)JZ +\frac{1}{4}\left(\left(D^W_XJ\right)\left(D^W_YJ\right) -\left(D^W_YJ\right)\left(D^W_XJ\right) \right)Z.
\sum_{k=0}^{n-1}\frac{(a_1/q;q^d)_k (a_2/q;q^d)_k \cdots (a_d/q;q^d)_k q^{dk}}{(a_1q^d;q^d)_k (a_2 q^d;q^d)_k \dotsb(a_dq^d;q^d)_k}\equiv 0 \pmod{\Phi_n(q)}
\int_{- 1}^{x_{n,j}^{(\alpha )}}{{P_{B,n}}f(x)\,dx}= \sum\limits_{i = 0}^n{f_{n,i}^{(\alpha )}{\mkern 1mu}\int_{- 1}^{x_{n,j}^{(\alpha )}}{{\cal L}_{B,n,i}^{(\alpha )}(x)\,dx}}, j = 0, \ldots, n.
\begin{aligned}u\left(\xi a+(1-\xi)b\right)w\left(\xi a+(1-\xi)b\right) +u\left((1-\xi)a+tb\right)w\left((1-\xi)a+\xi b\right) & \leq (2\xi^2-2 \xi+1) \left[u(a)w(a)+ u(b)w(b)\right] & +2t(1-\xi)\left[u(a)w(b)+u(b)w(a)\right].\end{aligned}
\int_{t_l}^{t_{l+1}}|\tilde G(\mu^{N,M}(t),z^{N,M}(t))dt - \tilde G( \mu^{N,M}(t_l), z^{N,M}(t))|dt\leq 2M\Delta(\epsilon) \ \ \{\rm if}\ \ \ \t_l\in\cup_{i=1}^k ( \mathcal{T}_i - \delta(\epsilon) , \mathcal{T}_i+ \delta (\epsilon)).
\Big\|\langle M\rangle_n- (n a_n^2\sigma^2 +v_n ) \Big\|_t \leq c_1 \Big\|\sum_{k=1}^{n-1}(\frac{a_kT_k}{k})^2\Big\|_t \leq c_2\sum_{k=1}^{n-1}\frac{1}{k^2}\|a_kT_k\|_{2t}^2.
\|f^{=d}\|_q \leq \gamma' \rho^{-d}= \gamma \left( 99\cdot2 \, r \, r' q \, (r')^{-2/q}\sqrt{\frac{\log(1 / \gamma)}{d}}\right)^d = \gamma \left( 99\cdot2\cdot\sqrt{2}\, r \, q \, (r')^{-2/q}\right)^d.
-\frac{(e^Ly-e^{-L}x)^2}{e^{2L}-e^{-2L}}+\frac{(e^Ly-e^{-L}x-(e^{2L}-e^{-2L})\frac r{\sqrt2})^2}{e^{2L}-e^{-2L}}=\frac{(e^{2L}-e^{-2L})r^2}2-\sqrt2r(e^Ly-e^{-L}x)
ce^{(2+\beta)S}\frac{1}{\Delta(\epsilon)}\sum_{k=j'}^{j''}\delta_{k}+\frac{P}{\delta(\epsilon)}7\vartheta\leq ce^{(2+\beta)S}\frac{P}{\Delta(\epsilon)}+\frac{P}{\delta(\epsilon)}7\vartheta<<|\log\epsilon|^{2},
e_{rs}=\frac{1}{2}(\frac{\partial u_{r}}{\partial x_{s}}+\frac{\partial u_{s}}{\partial x_{r}}-\frac{\partial u_{k}}{\partial x_{r}}\frac{\partial u_{k}}{\partial x_{s}})\approx\frac{1}{2}(\frac{\partial u_{r}}{\partial x_{s}}+\frac{\partial u_{s}}{\partial x_{r}})
\mathbf{M}= [\mathbf{W}^H \mathbf{h}_1, \mathbf{W}^H \mathbf{g}_1]\begin{bmatrix}\frac{1}{k_1}+ \frac{P_{\scriptscriptstyle{5}}|h_{21}|^{2}}{k_2}0 0 \frac{P_{\scriptscriptstyle{8}}|h_{31}|^{2}}{k_3}\end{bmatrix}\begin{bmatrix}\mathbf{h}^H_1 \mathbf{W}\mathbf{g}^H_1 \mathbf{W}\end{bmatrix}\mathbf{P}^2_{\scriptscriptstyle{A}}
n\frac{\partial\langle v_{j}\rangle}{\partial t}+\sum_{i}n\langle v_{i}\rangle\frac{\partial\langle v_{j}\rangle}{\partial x_{i}}=-n\frac{\partial\Phi}{\partial x_{j}}-\sum_{i}\frac{\partial(n\sigma_{ij}^{2})}{\partial x_{i}}(j=1,2,3.)
F_N(2) =\frac{1}{N}\sum^{N-1}_{n=1}\frac{1-\cos2\phi_n}{(1-\cos\phi_n)^2}&=\frac{1}{2N}\sum^{N-1}_{n=1}\frac{\sin^2\phi_n}{\sin^4\phi_n/2}=\frac{2}{N}\sum^{N-1}_{n=1}\frac{\cos^2\phi_n/2}{\sin^2\phi_n/2}&=\frac{4N}{3}-2+\frac{2}{3N}
a_{4 , 4}= r a_{4 , 4}^{\mathrm{\tiny{s c a l}}}+{\frac{1}{4 !}}\int_{\Sigma_{r}}\bar{h}^{1 / 2}d^{3}x \left[{\frac{1}{1 6}}\mathrm{T r}( \bar{\gamma}^{i}\bar{\gamma}^{j}\bar{F}_{i j})^{4}-{\frac{r}{2}}( \bar{F}^{i j}\bar{F}_{i j})^{2}\right] ,
N_{M_2\cap T}\left(m\right) = \frac{1}{2}\bigg(\sum_{a\mid t}\chi_{4D}(a)a^{-1}\bigg)L(2,\chi_{4D})^{-1}&\times (2\pi)^2\cdot\Gamma(2)^{-1}\cdot t\cdot |3^4|^{-1/2}\prod_{p \mid 2D}\alpha_p(t,S_2),
\sum_{v \in V}\prod_{i=1}^k x_i(v)+\frac 1k \sum_{i=1}^k \sum_{v \in V}(1-x_i(v))^k \geq 2^{1-k}N+\delta_0 |B_1|\geq (2^{1-k}+\delta_0 \varepsilon_0)N.
(T_3,T_4)=\begin{cases}(\mathrm{A}(3,r+1)^2) r=2,\ \mathfrak{M}\in\mathcal{T}^2\langle 243,8\rangle r=1, (1^3,\mathrm{A}(3,r+1)) r=2,\ \mathfrak{M}\in\mathcal{T}^2\langle 243,6\rangle, ((1^3)^2) r=2,\ \mathfrak{M}\in\mathcal{T}^2\langle 243,3\rangle r\ge 3.\end{cases}
\begin{array}{rcl}\Pi\left(t_{1}\right) = \mathbb{E}_{t_{1}}^{\mathbb{\tilde{P}}}\left[\Pi\left(t_{2}\right)\right]-\lambda^{-1}_{t_{1}}\left[\Pi\left(t_{2}\right),\sum_{p\in P}C_{p}\left[k_{p},F_{p}\left(t\right),G\left(t\right)\right]-\left(\sum_{c\in C}s_{c}p_{c}\right)D\right]\end{array}
\begin{aligned}\|u\|_{C^{k,2+\alpha}_{WF}([T_0,T]\times \bar S_{n,m})}\leq C\left(\|g\|_{C^{k,\alpha}_{WF}([0,T]\times\bar S_{n,m})}+ \|f\|_{C(\bar S_{n,m})}\right).\end{aligned}
R(\kappa) =\sum_{j=-1}^{\beta-4}\kappa^jG_j+\mathcal O(\kappa^{\beta-3}), G_j =G_{0,j}-\sum_{\genfrac{}{}{0pt}{}{j_1\ge 0,j_2\ge -1,j_3\ge 0}{j_1+j_2+j_3=j}}G_{0,j_1}vC_{j_2}v^*G_{0,j_3},
ds^2_{deS}= -{\ell^2\over 4}\left({d\rho\over \rho}\right)^2 +{1\over \rho}\, d\vec x\cdot d\vec x = -{\ell^2\over 4}{d\rho^2\over \rho^2}+{1\over \rho}\,\delta_{ab}dx^a dx^b\,.
\alpha_{\mathfrak{n}}(q^{\ast}hq,s,\chi)=\frac{\prod_{i=1}^{m-r}L_{\mathfrak{n}}(2s-4m+2r+2i+1,\chi^2)}{\prod_{i=0}^{m-1}L_{\mathfrak{n}}(2s-2i,\chi^2)}\prod_{p}P_{h,q,p}(\chi^{\ast}(p)p^{-s}),
[T(\psi_1 *_G \psi_2)](k') = |H| \cdot \sum_{g\in G}\psi_1(k_g h_g) \psi_2(h_g^{-1}k_g^{-1}k') begin{align*}5 pt] = |H| \cdot \sum_{g\in G}\psi_1(k_g) \psi_2(k_g^{-1}k').
\begin{aligned}f (0) = f(a) = f (1) = 0 a \in (0,1) & f(x) < 0 x \in (0,a) \, , f(x) > 0 x \in (a,1) & f^\prime (0) < 0 , f^\prime (a) > 0, f^\prime (1) < 0\, . \end{aligned}
\sum_{\substack{j=3 j\equiv_2 1}}^\infty |b_j|< \frac{2\pi_{p'}}{\pi^2(p-1)}\left[2+\frac{\pi^2}{2}(p-2)\right]\left[\left(1-\frac{1}{2^{p'}}\right) \zeta(p')-1\right].
\hat{p}(z_m|\boldsymbol{y},\hat{p}_m,\tau_m^p,\boldsymbol{\theta})=\frac{p(y_m|z_m,\boldsymbol{\theta})\mathcal{N}(z_m|\hat{p}_m,\tau_m^p)}{\int_zp(y_m|z_m,\boldsymbol{\theta})\mathcal{N}(z_m|\hat{p}_m,\tau_m^p)}
\delta{F}=-\displaystyle\int_{t_0}^{t_1}\displaystyle\sum_{k=1}^N\left(E_k\tilde{L}\delta{x}_k- \dfrac{\partial\tilde{L}}{\partial\dot{{x_k}}}\left(\delta\dfrac{d{x_k}}{dt}-\dfrac{d}{dt}\delta{x_k}\right)\right)dt,
\omega Z_{i ( 2 )}\rightarrow Z_{i ( 2 )}, \ \, o m e g a \tilde{B}_{i ( 1 2 )}^{1}\rightarrow \tilde{B}_{i ( 1 2 )}^{1}, \ \ \omega \tilde{\psi}_{i ( 1 2 )}^{1}\rightarrow \tilde{\psi}_{i ( 1 2 )}^{1}( i = 1 , 2 , 3 ) ,
S = \int \frac{1}{2}\epsilon^{\mu\nu\lambda\rho}\left( B^a_{\mu\nu}(x)R^a_{\lambda\rho}(x) +\underbrace{\phi^{ab}(x)B^a_{\mu\nu}(x)B^b_{\lambda\rho}(x)}_{{\rm constraint\: term}}\right).
\ast_{2,5}= 36 (k+4) b_4 \mathrel{\phantom{}}\bigg( (6k^2+136k+285)a_2b_3^3-8(6k^2+56k+125)a_2b_2b_3 &\mathrel{\phantom{}}\mathrel{\phantom{=}}+ \, 48(k^2+11k+30)a_2b_1b_4^2 +16(k^2+6k)a_0b_3b_4^2 &\mathrel{\phantom{}}\mathrel{\phantom{=}}- \, 2(8k^2+58k+105) a_1b_3^2b_4 + 16(2k^2+17k+35)a_1b_2b_4^2 \bigg)
b_3 = \gamma_2,c_3-b_2 = \gamma_1,c_2 = -\gamma_0, \alpha_2 = \frac{2c_3 - b_2}{b_3}, \alpha_1 = \frac{c_3^2-b_1-b_3c_2-b_2c_3}{b_3^2}, \alpha_0 = \frac{-c_1-c_3c_2}{b_3^2}.
\{f_1, f_2\}\star f_3 = f_1\star (f_2\star f_3 )-(-1)^{(m_2-1)m_3}f_1\star (f_3 \circ f_2)-(-1)^{(m_1-1)(m_2+m_3-1)}(f_2\star f_3 )\circ f_1- &(-1)^{(m_1-1)(m_2-1)}f_2\star (f_1\star f_3 )+(-1)^{(m_1-1)m_2}f_2\star (f_3 \circ f_1)+ &(-1)^{(m_3-1)(m_2-1)}(f_1\star f_3 )\circ f_2.
\ell_3^+ = 3 \times 3 + 2 = 11 \;, \ell_3^- = 3 \times 3 + 1 = 10 \;, m_3^+ = 3 \times 5 + 3 = 18 \;, m_3^- = 3 \times 5 + 2 = 17 \;, n_3^+ = 3 \times 7 + 4 = 25 \;, n_3^- = 3 \times 7 + 3 = 24 \;.
H(x,y) := \int{d^4 p \over (2\pi)^4}\left({\chi \over p^2+M_1^2}-{\chi \over p^2+M_2^2}\right){1 \over p^2}{{-1 \over \kappa}+{p^2 \over M^2}\left(1 +{p^2 \over \tilde M^2}\right) \over 1 +{p^2 \over M'{}^2}}e^{ip(x-y)}.
\mu^b(p(l)=k, \xi_1=x_1, \ldots \xi_k=x_k, \xi_{k+1}=x_{k+1},\tau_1\in dt^1, \ldots , \tau_{k+1}\in dt^{k+1}) &=1_{\{x_1=x_{k+1}\}}Q_{x_2}^{x_1}\cdots Q_{x_1}^{x_k}\frac{1}{t^1+\cdots+t^{k+1}}(-L^{x_1}_{x_1})e^{L_{x_1}^{x_1}t^1}\cdots (-L^{x_k}_{x_k})e^{L_{x_k}^{x_k}t^k}e^{L_{x_{k+1}}^{x_{k+1}}t^{k+1}}\,dt^1\cdots dt^{k+1}.
\Bigg |{\atop{\mathrm{d e t}\atop{\scriptstyle ( i , j )}}}\Bigg ( \frac{1}{w_{i}- z_{j}}\Bigg ) \Bigg | = \Bigg | \frac{\prod_{1 \leq i < j \leq n}( w_{i}- w_{j}) ( z_{i}- z_{j})}{\prod_{i , j = 1}^{n}( w_{i}- z_{j})}\Bigg | ,
c ( l_{0}+ 1 , l_{1}; l_{0}, l_{1}) = c ( l_{0}, l_{1}; l_{0}+ 1 , l_{1}) = c_{0}\sqrt{\frac{( k_{1}- l_{0}- 1 ) ( k_{1}+ l_{0})}{( l_{1}- l_{0}) ( l_{1}- l_{0}- 1 ) ( l_{1}+ l_{0}) ( l_{1}+ l_{0}+ 1 )}},
\Delta_{[N+2k]\times\underline{[k]}\times\underline{[2]}}f_{\underline{[k]}\times\underline{[k]}}\bar{\Delta}_{\underline{[2]}\times\underline{[k]}\times[N+2k]}= 1_{[N+2k]\times[N+2k]}- U_{[N+2k]\times\underline{[N]}}\bar{U}_{\underline{[N]}\times[N+2k]}.
\Bigl (\sum_i n_i[X \xleftarrow{f_i}M_i \xrightarrow{g_i}Y] \Bigr) \circ \Bigl (\sum_j m_j[Y \xleftarrow{h_j}N_j \xrightarrow{k_j}Z]\Bigr) & :=\sum_{i,j}n_im_j[(X \xleftarrow{f_i}M_i \xrightarrow{g_i}Y) \circ (Y \xleftarrow{h_j}N_j \xrightarrow{k_j}Z)].
\mathrm{Tr}(\mathrm{Tr}(\alpha^i \beta^j \gamma_k \omega_l)) = \mathrm{Tr}(\beta^j \omega_l \mathrm{Tr}(\alpha^i \gamma_k)) = \begin{cases}0, i \not= k \mathrm{Tr}(\beta^j \omega_l), i = k \end{cases}= \begin{cases}0, j \not= l k \not= i 1, j = l k = i\end{cases}
\mathbb{E}_{P,Q}\Big[\frac{\hat{N}_0}{n}\Big] = \frac{e^\alpha}{e^\alpha+1}\mathbb{P}(Y_1 = 0)+\frac{1}{e^\alpha+1}\mathbb{P}(Y_1 = 1) = \frac{1}{e^\alpha+1}\Big\{1+(e^\alpha-1)P(A)\Big\}.
d\mathbf{Y}_t = \mathbf{F}_t^{\ast}d\tilde{\mathbf{A}}_t^{\ast}+\mathbf{F}_t d\tilde{\mathbf{A}}_t+ \mathbf{G}_t d\tilde{\mathbf{\lambda}}_t, d\tilde{\mathbf{\lambda}}_t = \left(d\tilde{\mathbf{\Lambda}}_t\right).
[[g]] \ni \left(\begin{array}{cc|c}1 0 0 0 1 0 \hline-v_1 0 I_{n-2}\end{array}\right) g^{(1)}= \left(\begin{array}{cc|c}1 0 0 0 1 0 \hline0 v_2 I_{n-2}\end{array}\right) =: g^{(2)}.
a_{2k+2}\geq \tfrac{\gamma}{1-\gamma}o_{k+1}-e_{k}+1 = \tfrac{\gamma}{1-\gamma}\left(a_{2k+1}+o_{k}\right)-e_{k}+1 &\geq \tfrac{\gamma}{1-\gamma}\left(\tfrac{\beta}{1-\beta}e_{k}+1\right)-e_{k}&=\left(\tfrac{\gamma\beta}{\left(1-\gamma\right)\left(1-\beta\right)}-1\right)e_{k}+\tfrac{\gamma}{1-\gamma}
\lim_{t\rightarrow \infty}(b_t)^{1/t}=\lim_{t\rightarrow\infty}(h_t(\alpha_t))^{-1/t}=\lim_{t\rightarrow\infty}\{\alpha_t\exp[(t-1)u(\alpha_t)]\}^{-1/t}=\lim_{t\rightarrow\infty}\exp[-u(\alpha_t)]=m_1(\mu).
E[S_{n}^{2k+1}]=0;E[S_{n}^{2k}]=\frac{(\begin{matrix}n\\k\end{matrix})\frac{(2k)!}{2^{k}}}{n^{k}}=\frac{n(n-1)\cdot\cdot\cdot(n-k+1)}{n^{k}}(2k-1)!!
\langle \psi|\chi\rangle =\, \sum^N_{k=1}\int_{\varphi^{-1}_{k*}(T_xM_4)}d^4z_k \exp(-\,\imath \vartheta_{\psi k}(\varphi^{-1}_{k}(x),z_k))\cdot &\cdot \exp(\,\imath \vartheta_{\chi k}(\varphi^{-1}_{k}(x),z_k))\,n_{\psi k}(\varphi^{-1}_{k}(x),z_k) \,n_{\chi k}(\varphi^{-1}_{k}(x),z_k)
lim_{x\rightarrow3^{+}}\frac{\sqrt{x-3}}{x^{2}-9}=lim_{x\rightarrow3^{+}}\frac{\sqrt{x-3}}{(x+3)(x-3)}=lim_{x\rightarrow3^{+}}\frac{1}{(x+3)\sqrt{x-3}}=+\infty
p_t(z) = (8\pi)^{-d}e^{-dt}\sum_{k=0}^\infty (e^{-2t})^k L_k^{d-1}\Big( \frac{|z|^2}{2}\Big) e^{-\frac{|z|^2}{4}}= (8\pi)^{-d}e^{-dt}(1-e^{-2t})^{-d}\exp\Big( - \frac{1+e^{-2t}}{1-e^{-2t}}\frac{|z|^2}{4}\Big) & = (16\pi \sinh t)^{-d}\exp\Big(-\frac 1 4 \coth(t)|z|^2 \Big).
c_{n,\rho,q}^{(k)}(z) =\rho^n\underbrace{\int_0^1\cdots\int_0^1}_k\left(\frac{x_1\cdots x_k-z}{\rho}\right)_n d_q x_1\cdots d_q x_k &=\rho^n n!\underbrace{\int_0^1\cdots\int_0^1}_k\binom{(x_1\cdots x_k-z)/\rho}{n}d_q x_1\cdots d_q x_k\,,
|\Lambda_p| = -\frac{1}{2}\left(\sum_{i = 1}^{p - 1}b_i\right)^2 + \frac{p}{2}\sum_{i = 1}^{p - 1}b_i^2 + \sum_{i = 1}^{p - 1}\left(i - \frac{p - 1}{2}\right)b_i
I = \int_{\Omega}\left( \frac{1}{2}\epsilon_{\mu \nu \lambda}B_{\mu}^{*}( \partial_{\nu}- i g a_{\nu}) B_{\lambda}+ M B_{\mu}^{*}B^{\mu}+ \frac{1}{2}\epsilon_{\mu \nu \lambda}a_{\mu}\partial_{\nu}a_{\lambda}\right) .
F_{a}\to{\cal F}_{a}= \partial V_{a}- i \frac e{\sin \theta_{W}}\Bigl ( ( \hat{\Phi}+ \zeta \mathrm{v})^{\dagger}\frac{\tau_{a}^{T}( \hat{G})}2 ( \Phi + \mathrm{v}) - ( \Phi + \mathrm{v})^{\dagger}\frac{\tau_{a}^{T}( \hat{G})}2 ( \hat{\Phi}+ \zeta \mathrm{v}) \Bigr ) . \nonumber