harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{\Omega}\vert \nabla{y_2}_*\vert^{p - 2}\nabla{y_2}_*\cdot \nabla wdz + \int_{\Omega}\eta\vert \nabla{y_2}_*\vert^{q - 2}\nabla{y_2}_*\cdot \nabla wdz \geq \int_{\Omega}a_2{y_2}_*^{- \nu}wdz + \lambda(\kappa_2 + 1)\int_{\Omega}\vert{y_1}_*\vert^{\kappa_1 + 1}\vert{y_2}_*\vert^{\kappa_2}w dz.
\left\Vert \widehat{P}_{n}^{(\alpha,\beta)}\right\Vert_{\widehat{w}_{\alpha,\beta}}^{2}=\left( \frac{2^{\alpha+\beta+1}(\alpha+n)(\beta+n)}{4(\alpha+n+1)(\beta+n-1)(\alpha+\beta+2n-1)}\right) \left( \frac{\Gamma(\alpha+n)\Gamma(\beta+n)}{\Gamma(n)\Gamma(\alpha+\beta+n)}\right)\quad(n\in\mathbb{N}).
(-\triangle_{g})^{\alpha}u(x) = \int_{\mathbb{R}^{d}}g\bigg{(}\frac{|u(x)-u(y)|}{|x-y|^{\alpha}}\bigg{)}\frac{u(x)-u(y)}{|u(x)-u(y)|}\frac{dy}{|x-y|^{d+\alpha}},
\int_{\mathbb{R}^{3}}w_{0}\varphi dx+\int^{\infty}_{0}\int_{\mathbb{R}^{3}}\left(a(\partial_{t}\varphi+\Delta\varphi+u^{c,\gamma}\cdot\nabla\varphi)-(a\cdot\nabla)u^{c,\gamma}\varphi\right)dxdt=0.
\begin{cases}2\pi \partial_t x(t,y,\eta)=-\nabla_\xi a ( x(t,y,\eta),\xi(t,y,\eta)) 2\pi \partial_t\xi(t,y,\eta)=\nabla_x a (x(t,y,\eta),\xi(t,y,\eta)) x(0,y,\eta)=y, \xi(0,y,\eta)=\eta.\end{cases}
lim_{x\rightarrow\infty}\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=lim_{x\rightarrow\infty}\frac{e^{2x}+1}{e^{2x}-1}=lim_{x\rightarrow\infty}\frac{2e^{2x}}{2e^{2x}}=1.
df_{z}(w) ~ = \delta (z-w) \nabla\left(\frac{1}{\delta-|w|}\right)^{T}- \frac{\delta}{\delta -|w|}Id \crcr ~ = \delta (z-w) \frac{1}{(\delta - |w|)^{2}}\left(\frac{w}{|w|}\right)^{T}- \frac{\delta}{\delta -|w|}Id.
\sum_{i = j}^{3}i S_{i j}^{F}({\vec{x}}-{\vec{x^{\prime}}}) = \int{\frac{\mathrm{d}^{3}{\vec{k}}}{( 2 \pi )^{3}}}{{\frac{e^{i{\vec{k}}. ({\vec{x}}-{\vec{x^{\prime}})}}}{\omega_{k}}}}.
Y^{n}(t) \doteq X_{0}^{n}(0)+\sum_{k=0}^{\infty}\frac{k-2}{n}\int_{[0,t]\times \lbrack 0,1]}{{1}}_{[0,r_{k}(\boldsymbol{X}^{n}(s-)))}(y)\,N_{k}^{n}(ds\,dy), \eta^{n}(t) \doteq \sum_{k=0}^{\infty}\frac{2}{n}\int_{[0,t]\times\lbrack 0,1]}{{1}}_{\{X_{0}^{n}(s-)<0\}}{{1}}_{[0,r_{k}(\boldsymbol{X}^{n}(s-)))}(y)\,N_{k}^{n}(ds\,dy).
h(x) =\sum_{j=0}^{i-1}\int_{t_j}^{t_{j+1}}\eta(t)dt+\int_{t_i}^{f(x)}\eta(t)dt &\geq \sum_{j=0}^{i-1}\frac{(t_{j+1}-t_j)^2}{4}+\frac{(f(x)-t_i)^2}{4}.
\begin{array}{l}{{\left[ M_{i_{1}j_{1}}( x_{1}^{-}, y_{1}^{-}; x^{+}) , M_{i_{2}j_{2}}( x_{2}^{-}, y_{2}^{-}; x^{+}) \right]}}\\{{{}~ ~ ={\frac{1}{2 N}}~ \delta_{j_{1}i_{2}}\delta ( y_{1}^{-}- x_{2}^{-}) M_{i_{1}j_{2}}( x_{1}^{-}, y_{2}^{-}; x^{+})}}\\{{{}~ ~ -{\frac{1}{2 N}}~ \delta_{j_{2}i_{1}}\delta ( y_{2}^{-}- x_{1}^{-}) ~ M_{i_{2}j_{1}}( x_{2}^{-}, y_{1}^{-}; x^{+})}}\\ \end{array}
\frac{1}{n^2+1}= \frac{\imath}{2}\left( \frac{1}{n+\imath}- \frac{1}{n-\imath}\right) \frac{1}{(n^2+1)^2}= -\frac{1}{4}\left(\frac{\imath}{(n-\imath)}- \frac{\imath}{(n+\imath)}+ \frac{1}{(n+\imath)^2}+ \frac{1}{(n-\imath)^2}\right),
\|\chi_1(H-E)^{-1}\chi_2\|= \|\chi_1e^{t\langle \cdot-x_0\rangle_\epsilon}(H_{t,\epsilon}-E)^{-1}e^{-t\langle \cdot-x_0\rangle_\epsilon}\chi_2\| \leqslant \|\chi_1 e^{t\langle \cdot -x_0\rangle_\epsilon}\|\,\|(H_{t,\epsilon}-E)^{-1}\|\,\|e^{-t\langle \cdot -x_0\rangle_\epsilon}\chi_2\|.
C_{p,g}^{(\ell)}=\sum_{j=1}^{p}(-1)^{j-1}\sum_{t=0}^{\lfloor\frac{p-j}{2}\rfloor}\binom{n-p+j+2t}{t}\sum_{\beta=0}^{p-j-2t}2^{p-j-2t-\beta}\binom{n-\ell}{\beta}\binom{\ell}{p-j-2t-\beta}\sum_{i=0}^{j-1}\binom{\beta}{p+i-j-t-g}.
- q^{5}{\cal N}_{i p}C_{i a}C^{v a}\left. \partial^{v}\right.^{*}x^{s}= q^{4}C_{i a}C_{p \delta}\hat{\cal R}_{a l}^{p \delta}{\cal N}_{s l}{\cal N}_{i p}- q^{4}\left. \hat{\cal R}^{- 1}\right._{l d}^{i k}{\cal N}_{k n}{\cal N}_{s l}{\cal N}_{i p}{\cal N}_{t d}x^{n}C_{d b}C^{m b}{\cal N}_{d t}\left. \partial^{m}\right.^{*}~ .
g \mapsto F_g, \;\;\; F_g(f) = \mathbb E \sum_{t\in \mathcal T}\langle \Delta g_t,\Delta f_t\rangle \stackrel{(*)}= \mathbb E \langle g_{\infty}, f_{\infty}\rangle, \|F_g\|_{(\mathcal A^{\mathcal T}_{p,q})^{*}}\eqsim_{p,q}\|g\|_{\mathcal A^{\mathcal T}_{p',q'}}.
{\frac{m}{2 \pi}}\cosh \theta = \rho_{0}( \theta ) + \widetilde{\rho}_{0}( \theta ) - \sum_{k = 0}^{3}\int_{- \infty}^{\infty}\Psi_{0 k}( \theta - \theta^{\prime}) \rho_{k}( \theta^{\prime}) d \theta^{\prime}
\tilde{J}_{\mu}^{( \pm )}\equiv \int d^{3}x \, j_{\mu}^{( \pm )}( x ) = \sum_{{\bf k}}\frac{k_{\mu}}{\omega_{{\bf k}}}( a_{{\bf k}}^{\dagger}a_{{\bf k}}\pm b_{{\bf k}}^{\dagger}b_{{\bf k}}) .
\sum_{j=1}^{i+1}E_{2}(V(d w), B_{j}) \leq 2 E_{2}(V(d w), B_{1}) + \frac{2c_{4}\lambda^{\frac{p}{2}}}{\sigma^{n}}\sum_{j=1}^{i}\omega\left( R_{j}\right)\leq \frac{\sigma^{n}\lambda^{\frac{p}{2}}}{100}.
G_2(s_j,v_j;x)=\int_{x}^{x+2s_j}\frac{f(t)dt}{(x+s_j-t)^2+v_j^2}\leq \int_{x}^{x+2s_j}\frac{f(t)dt}{k^2s_j^2}=\frac{1}{k^2s_j^2}\int_{x}^{x+2s_j}f(t)dt.
\int_0^r \left[(-\phi''-\frac{z}{2}\phi'+\frac{z}{2}e_\sigma\psi')\cdot (z^{2}\phi) +(-\psi''-\frac{z}{2}\psi'+\frac{z}{2}e_\sigma\phi')\cdot (z^{2}\psi)\right]{\rm{d}}z = \int_0^r \left( |z(\phi',\psi')|^2+\frac{3}{4}|z(\phi,\psi)|^2+2z(\phi'\phi+\psi'\psi)- \frac{3 e_\sigma+ze_\sigma'}{2}z^{2}\phi\psi\right).
(\eta_1, \dots, \eta_n)=(\xi'_1,\dots, \xi'_n)\begin{pmatrix}X \cdots 0 0 \cdots 0 \vdots \ddots \vdots \vdots \ddots \vdots 0 \cdots X 1 \cdots 0 0 \cdots 0 1 \cdots 0 \vdots \ddots \vdots \vdots \ddots \vdots 0 \cdots 0 0 \cdots 1 \end{pmatrix}=(\xi'_1,\dots, \xi'_n) e^ih^i(X).
|U_3(x)| \leq C\int_{|y|>2|x|}\left(\frac{|x|}{|y|}+\frac{1}{|a||y|^2}\right)|f(y)|\,dy &\leq C(|x|^{-1}+|a|^{-1}|x|^{-3})\int_{|y|>2|x|}|y||f(y)|\,dy=o(|x|^{-1})
\left(\ell_{2}'\right)^{3}= \frac{k_3\ell_{2}^{4}}{k_0-k_2 \ell_{2}^{2}}l_{1}= k_2\ell_2+k_1+k_0/\ell_2 b_{25}^6= \frac{c_{45}^3( k_0-k_2 \ell_{2}^{2})}{k_3\ell_{2}^{4}}b_{13}= \,b_{15}=0 b_{23}= (k_4\ell_2-k_0)b_{25}b_{12}^2= \frac{c_{12}b_{25}^2\ell_2^2}{c_{45}( k_0-k_2 \ell_{2}^{2})}
\begin{array}{r c l}{\delta_{i}x^{\alpha}}&{= 0}&{= \displaystyle \frac{1}{i \hbar}[ x^{\alpha}, L_{i}^{\mu}\circ p_{\mu}]}\\{\delta_{i}x^{\mu}}&{= L_{i}^{\mu}}&{= \displaystyle \frac{1}{i \hbar}[ x^{\mu}, L_{i}^{\mu}\circ p_{\mu}] .}\\ \end{array}
\delta S [ \theta , \bar{A}_{+}, \bar{\Pi}^{\perp -}] = \frac{k}{8 \pi}\int_{\partial \cal M}\partial_{+}\theta \partial_{-}\theta + \partial_{-}\theta \left( \bar{A}_{+}- \mathrm{\frac{8 \pi}{k}}\bar{\Pi}^{\perp -}\right) .
{\cal D}_{--}\psi_{+ + \dot{q}}^{~ ~ ~ -}={\cal D}_{+ +}\psi_{-- \dot{q}}^{~ ~ ~ -}+ 4 i \psi_{+ + \dot{p}}^{~ ~ ~ -}\psi_{-- \dot{p}}^{~ ~ ~ -}\psi_{-- \dot{q}}^{~ ~ ~ -}
\mathbb{E}\left[\left(\tilde{Z}_\beta^{(n)}\right)^k\right] = n^{k-\frac{\beta k}{2}}(1-n^{1-\beta})^{-k}(1-n^{\beta-1})^k \_2F_1[-k,n^{\beta}+1;n+2;n^{1-\beta}]
S^{loc}f(x) \leq \int_{\mathbb{R}^d}\sum_j \chi_{B_j}(x_1)|\chi_{N_2}(x_1,y_1)-\chi_{4B_j}(y_1)|\int_{Y}|S(x,y)||f(y)|\,d\mu(y_2)\,dy_1 &+\sum_j \chi_{B_j}(x_1)|S(f(y)\chi_{4B_j}(y_1))(x)|:= S_2(f)+S_1(f),
\begin{aligned}\int_{K A^{-t}R^*}\mu \ast |\mathcal F(\varphi_{R}\circ A^{-1})|(y) dy \lesssim |\det A| \prod^d_{k=1}\lambda_{k}\sum^{\infty}_{j=1}2^{-Mj}\iint \chi_{K A^{-t}R^*}(y) \chi_{2^{j}A^{-t}R^*}(y-u) d\mu(u) dy. \end{aligned}
H ( t , M , \kappa_{\tau}) = \rho^{\tau [ \frac{d - \frac{m}{2}}{2}+ 1 ]}\nonumber H ( \frac{t}{\rho^{2 \tau}}, \frac{M}{\rho^{2 \tau [ \frac{d - \frac{m}{2}}{2}- 1 ]}}, \frac{\kappa_{\tau}}{\rho}) .
P_{-1,0}(\mathfrak{v})=\left( \begin{array}{c}0 v^{-}_{-1,0}r^{\frac{1}{2}}\end{array}\right),\mbox{}P_{1,0}(\mathfrak{v})=\left( \begin{array}{c}v^{+}_{1,0}r^{\frac{1}{2}}0\end{array}\right).
( W_t,\widetilde W_t)_{t\in [0,t_{i-1}]}= \psi(( W_t,\widetilde B_t)_{t\in [0,t_{i-1}]}), (V_{t}, \widetilde V_{t})_{t\in [0,t_i-t_{i-1}]}= \varphi( (W_t-W_{t_{i-1}},\widetilde B_t)_{t\in [t_{i-1},t_i]})
\varepsilon_k^{-1}((0,\beta)) = \{x: \varepsilon_{k}(x) \in (0,\beta)\}= \bigcap_{\beta\leq r \leq 1}w(\cdot, r)^{-1}([0, k)) = \bigcap_{\beta\leq r \leq 1, r\in \mathbb Q}w(\cdot, r)^{-1}([0, k)).
\frac{d}{dx}\mathbb X(x,t) = B(x,t) \sigma_2 C(x,t), \dfrac{\partial}{\partial t}\mathbb X(x,t) = i A B(x,t) \sigma_2 C(x,t) - i B(x,t)\sigma_2 C(x,t) A_\zeta + iB(x,t) \gamma C(x,t).
L^{+}=L_{1}\cup L_{2}\cup L_{3}& = \left(D^{k}(a_{1})^{+}\cap E_{1}\cap D_{b_{1}}(\tfrac{1}{2})\right)\cup\left(D^{k}(a_{1})^{+}\cap E_{2}\cap D_{b_{2}}(\tfrac{1}{2})\right) & = D^{k}(a_{1})^{+}\cap\left[\left(E_{1}\cap D_{b_{1}}(\tfrac{1}{2})\right)\cup\left(E_{2}\cap D_{b_{2}}(\tfrac{1}{2})\right)\right]
\mathbb{P}\left(Y_1 \leq y\right)\sim y\frac{\delta}{1-\delta},y\rightarrow 0 &A_1(N)=\mathbb{P}\left(Y_1 \leq \theta_N\right) \sim \frac{K\log 2}{N}\frac{\delta}{1-\delta},~N\rightarrow \infty .
\frac{\partial}{\partial x_\ell}{\cal L}^{-1}(\mbox{e}^{- \lambda^2\|\cdot\|^2}) (x) = - \frac{x_\ell}{2 \|x\|^2}\left( \frac{\sqrt{\pi}}{2 \|\lambda^2 x\|}\mbox{erf}(\|\lambda x\|) - \mbox{e}^{- \lambda^2 \|x\|^2}\right), \ell=1,\ldots,d.
\psi_v\cdot g_u =(\psi\cdot g_u)_v-(\psi\cdot g_{uv}) =(\phi\cdot f_u)_v-\frac1v g_v\cdot g_{uv}=(\phi\cdot f_u)_v-\frac1{2v}(g_v\cdot g_{v})_{u}=(\phi\cdot f_u)_v-\frac1{2v}(f_v\cdot f_{v})_u =(\phi\cdot f_u)_v-\frac1{v}f_v\cdot f_{uv}=\phi_v\cdot f_u.
\mathcal{I}:=\int^1_0\mathbb{L}'(\mathbf{U}^a+{\mathbf V}_k+\tau({\mathbf V}_{k+\frac{1}{2}}-{\mathbf V}_k),\Psi^a+\Psi_k+\tau(\Psi_{k+\frac{1}{2}}-\Psi_k))({\mathbf V}_{k+\frac{1}{2}}-{\mathbf V}_k,\Psi_{k+\frac{1}{2}}-\Psi_k)d\tau.
k_2^\nu T_{\rho \sigma \mu \nu}^{abcd}(p_1,p_2;k_1,k_2)=-ig^2e^\rho(p_1)e^\sigma (p_2)[k_{1\mu}S_{\rho \sigma}^{abcd}+\widetilde{G}_{\rho\sigma \mu}^{abcd}]
F(x^{t+1})-F(x)+\frac{\alpha_t^2}{2\gamma_t}\|x-z^{t+1}\|^2\overset{\eqref{F-ineq}}{\leq}\ (1-\alpha_t)\left(F(x^t)-F(x)\right)+\frac{\alpha_{t}^2(1-\beta_t)}{2\gamma_{t}}\|x-z^t\|^2 \overset{\eqref{alpha-eqn}}{=}\ (1-\alpha_t)\left(F(x^t)-F(x)+\frac{\alpha_{t-1}^2}{2\gamma_{t-1}}\|x-z^t\|^2\right).
K_w(z) = \frac{E_+(z)F_+(z)F_+(w)^*E_+(w)^*-E_-(z)F_-(z)F_-(w)^*E_-(w)^*}{\rho_w(z)}& = E_+(z)K_w^\mathfrak{F}(z)E_+(w)^*+F_-(z)K_w^\mathfrak{E}(z)F_-(w)^*.
\left|\int_{M}a(x)\left(f_{\varepsilon}(W_{\delta_{\varepsilon}(t),\eta}+\Phi_{\delta_{\varepsilon}(t),\eta})-f_{\varepsilon}(W_{\delta_{\varepsilon}(t),\eta})-f_{\varepsilon}'(W_{\delta_{\varepsilon}(t),\eta})\Phi_{\delta_{\varepsilon}(t),\eta}\right)Z_{\delta_{\varepsilon}(t),\eta}^{k}d\mu_{g}\right|=o(|\varepsilon|).
M = \frac{2P_3(1)}{L}\sideset{}{^+}\sum_{\chi (q)}\epsilon(\chi) \sum_{b \leq y_2}\frac{\Lambda(b) \overline{\chi}(b) P_2[b]}{b^{\frac{1}{2}}}\sum_n \frac{\overline{\chi}(n)}{n^{\frac{1}{2}}}V \left( \frac{n}{q}\right).
L_2^{-1}=(1-T) (1+T) (2T^2 - T + 1), L_3^{-1}=(1+T) (1-T^2), L_{11}^{-1}=( 1+T)^2 (11 T^2 - 4 T + 1), L_{37}^{-1}=(1-T)(37^2 T^4 + 148 T^3 + 14 T^2 + 4 T + 1),
\overline{\nabla}h_{\Omega}(x)=\sum_{i}h_{i}e_{i}+hx, F_{i}(x)=\sum_{j}(h_{ij}+h\delta_{ij})e_{j},F_{ij}(x)=\sum_{k}(h_{ijk}+h_{k}\delta_{ij})e_{k}-(h_{ij}+h\delta_{ij})x.
K^{\alpha \gamma}|_{\beta \delta}(x,y) = K^{\gamma \alpha}|_{\delta \beta}(x,y)= \overline{K^{\beta \delta}|_{\alpha \gamma}(y,x)}K^\alpha_{\;\;\,\gamma}|_\beta^{\;\;\,\delta}(x,y) = K^\delta_{\;\;\,\beta}|_\gamma^{\;\;\,\alpha}(y,x)= \overline{K^\gamma_{\;\;\,\alpha}|_\delta^{\;\;\,\beta}(x,y)}\:.
\frac{1}{n}\sum_{k\in N}e_{k,j+n-2}^n+\frac{1}{n}e^{n-1}_{i,j}+\frac{1}{n}\left(1-\frac{1}{n}\right)e_{i,j}^{n-2}+\frac{1}{n}\sum_{\tau=1}^{n-3}e_{i,j+\tau}^{n-\tau-2}
e_{i}={\bf 1}\otimes \cdots \otimes \left( \begin{array}{c c c c}{0}&{0}&{0}&{0}\\{0}&{q^{- 1}}&{- 1}&{0}\\{0}&{- 1}&{q}&{0}\\{0}&{0}&{0}&{0}\\ \end{array}\right) \otimes \cdots \otimes{\bf 1}\in{\bf C}^{\otimes 2 l},
E=\partial x_{1}+\partial x_{3},\;\; Z_{1}=\cos \varphi_{3}\partial x_{2}-\sin \varphi_{2}\sin \varphi_{3}\partial x_{5}, Z_{2}=\cos \varphi_{3}\partial x_{4}-\cos \varphi_{2}\sin \varphi_{3}\partial x_{5}\;\;\mbox{and}\;\; N=\frac{1}{2}(-\partial x_{1}+\partial x_{3}).
g_t(\alpha(x),\alpha(y))=\alpha\circ g_t(x,y), g_t(\alpha(x),g_t(y,z))-g_t(g_t(x,y),\alpha(z))=(-1)^{|x||y|}\{g_t(\alpha(y),g_t(x,z))-g_t(g_t(y,x),\alpha(z))\}, g_t(g_t(x,y),\alpha(z))=(-1)^{|y||z|}g_t(g_t(x,z),\alpha(y)).
(\iota\otimes (\varphi^*\circ S))\Delta(\sigma_{-\mathrm i /p}(x^*)) =(R\otimes (\varphi^*\circ \tau_{-\mathrm i/2}))(R\otimes R)\Delta(\sigma_{-\mathrm i /p}(x^*)) &= ( (\varphi^*\circ \tau_{-\mathrm i/2})\otimes R)\Delta (R\circ \sigma_{-\mathrm i /p}(x^*)) .
\sum_{j=0}^{a_i-1}( 1-\frac{i\pi_q(i)-ij}{q^i}) \le 2a_i q^{-\lceil \frac{i}{2}\rceil}+ \frac{i}{q^i}\frac{a_i(a_i-1)}{2}\le 4a_i^2 i q^{-\lceil \frac{i}{2}\rceil}.
v_{4}(t,r) = \frac{-1}{2\pi}\int_{t}^{\infty}ds \int_{0}^{s-t}\frac{\rho d\rho}{\sqrt{(s-t)^{2}-\rho^{2}}}\int_{0}^{2\pi}d\theta \frac{v_{4,c}(s,\sqrt{r^{2}+2 r \rho \cos(\theta) + \rho^{2}})}{\sqrt{r^{2}+2 r \rho \cos(\theta) + \rho^{2}}}\left(r+\rho \cos(\theta)\right)
d s^{2}= ( H_{1}H_{5})^{- \frac{1}{2}}\left[ - d t^{2}+ d x^{2}+ H_{1}\sum_{i = 6}^{9}d x_{i}^{2}\right] + ( H_{1}H_{5})^{\frac{1}{2}}[ d r^{2}+ r^{2}d \Omega_{3}^{2}] ,
f(g(xy))+(f(xy))g + \alpha_1 \big(f(g(xy)) + f((xy)g)\big) + \alpha_2 \big(g(f(xy)) + g((xy)f)\big) = (f(gx))y+((fx)g)y + \alpha_1 \big((f(gx))y + (f(xg))y\big) + \alpha_2 \big((g(fx))y + (g(xf))y\big) - (f(gy))x+((fy)g)x + \alpha_1 \big((f(gy))x + (f(yg))x\big) + \alpha_2 \big((g(fy))x + (g(yf))x\big).
\gamma(\sigma \times \sigma^\prime, \psi) = \sum_{\substack{h \in U_m(\mathfrak{f}) \backslash G_m(\mathfrak{f}) x \in M_{m,n-m-1}(\mathfrak{f})}}J_{\sigma, \psi}\begin{pmatrix}1 I h x\end{pmatrix}J_{\sigma^\prime, \psi^{-1}}(h).
\kappa_n=\frac{n^4 (a^2 - 4 b - n^2)}{4 (2 n - 1) (2 n + 1)}=-\frac{n^4}{16},\lambda_n=\frac{2 n^2 + 2 n + 1 + 4 b - a^2}{4}=\frac{8 n^2 +8 n + 3}{16}.
\left\vert t_{v_{i}}\zeta-s_{v_{i}}\right\vert= t_{v_{i}}^{-\omega(v_{i})}, \lim_{i\to\infty}\omega(v_{i})=\infty, &\limsup_{i\to\infty}\frac{\log t_{v_{i+1}}}{\log t_{v_{i}+1}}<\infty.
P_{\mu}^{5}( k ) = i \int \frac{d^{2}p}{( 2 \pi )^{2}}\frac{\mathrm{t r}\{\gamma_{\mu}\left[ ( p \! \! \! \slash + k \! \! \! \slash ) + m \right] \gamma_{5}( p \! \! \! \slash + m ) \}}{\left[ ( p + k )^{2}- m^{2}\right] ( p^{2}- m^{2})}.
\mathrm{Tr}(\mathcal{W}_p(a_0\otimes\cdots\otimes a_p)De^{-s^2D^2}) = \mathrm{Tr}(\Gamma a_0\left(\prod_{k=1}^{p-1}[F,a_k]\right)\delta(a_p)De^{-s^2D^2}) = \mathrm{Tr}(\Gamma a_0\left(\prod_{k=1}^{p-1}[F,a_k]\right)\delta(a_p)F|D|e^{-s^2D^2}).
{\frac{\mu^{4}}{1 6}}V{\frac{d^{3}\vec{k}}{( 2 \pi )^{2}}}{\frac{1}{| \vec{k}|^{4}}}{\frac{( k_{y}/ H_{5})^{2}}{\mathrm{s i n h}^{2}( k_{y}\pi / H_{5})}}.
\eta^{24}\left(i\frac{\mathbf K(\sqrt{1-\lambda})}{\mathbf K(\sqrt{\lambda})}\right) =\left[\frac{2\mathbf K(\sqrt{\lambda})}{\pi}\right]^{12}\frac{\lambda^2(1-\lambda)^2}{256},\eta^{24}\left(2i\frac{\mathbf K(\sqrt{1-\lambda})}{\mathbf K(\sqrt{\lambda})}\right) =\left[\frac{2\mathbf K(\sqrt{\lambda})}{\pi}\right]^{12}\frac{\lambda^{4}(1-\lambda)}{65536},
(*) \leq \sum_{k=2}^{p+1}\sum_{j=0}^{p+1-k}\binom{p+1}{k}\binom{p+1-k}{j}2^{k}\gamma_{n}^{2j+k}\left\| \xi_{n+1}\right\|^{k}\left\| Z_{n}- m \right\|^{k}V_{n}^{p+1-k-j}& \leq \sum_{k=2}^{p+1}\sum_{j=0}^{p+1-k}\binom{p+1}{k}\binom{p+1-k}{j}2^{2k}\gamma_{n}^{2j+k}\left\| Z_{n}- m \right\|^{k}V_{n}^{p+1-k-j}
\psi_k(z) = -\rho \sum_{m=1}^N \frac{1}{2\pi i}\int_{L_m}\overline{n^2(t)}\;\overline{\psi_m(t)}E_1(t-z) \textrm{d}t+C,\;z \in D_k \;(k=1,2,\ldots, N),
A_5 :=\div (\bar m) \big( h'(\rho) - h'(\bar \rho) - h''(\bar \rho)( \rho-\bar \rho) \big)\, , A_6 := \big( h'( \rho) - h'(\bar \rho)\big) \big( \div(m) - \div(\bar m) \big)\, .
ds^2 = -\frac{t^2}{(t^2 + r_-^2)(t^2-r_+^2)}dt^2 +t^2 (d\phi +\frac{r_+ r_-}{t^2}dr)^2 +\frac{(t^2 + r_-^2)(t^2-r_+^2)}{t^2}dr^2 .
\theta_1 = \alpha_1 \nabla_1+ \alpha_2 \nabla_2+\alpha_3 \nabla_3 +\alpha_4 \nabla_4, \theta_2 = \beta_1 \nabla_1+ \beta_2 \nabla_2 +\beta_3 \nabla_3, \theta_3 = \gamma_1 \nabla_1+ \gamma_2 \nabla_2.
\sigma_i(a,b) = \sum_{k=0}^{b-a}t_i^{a+k}t_{i+1}^{b-k}= t_i^{a}t_{i+1}^{b}+ t_i^{a+1}t_{i+1}^{b-1}+ \cdots + t_i^{b}t_{i+1}^{a}.
q_k f(\{r_j\})=(1-\alpha)D_{\alpha,k}f(\{r_j\}), D_{\alpha,k}f(\{r_j\})\doteq\frac{f(r_1,\dots,\alpha r_k,\dots,r_N)-f(r_1,\dots,r_k,\dots,r_N)}{\alpha r_k-r_k}.
U_{2m+1}^{(2)}=U_{m}U_{m+1}&=U_{m}\left( pU_{m}-qU_{m-1}\right) &=pU_{m}U_{m}-q\left( pU_{m-1}-qU_{m-2}\right) U_{m-1}&=pU_{2m}^{(2)}-pqU_{2m-2}^{(2)}+q^{2}U_{2m-3}^{(2)}.
U^{- 1}( z_{1}, \dots , z_{N}) = P \exp \left[ -{\frac{1}{2 \pi \kappa}}\int_{\Gamma}\sum_{\alpha}d \zeta^{\alpha}\sum_{\beta \not = \alpha}\hat{Q}_{\alpha}^{a}\hat{Q}_{\beta}^{a}{\frac{1}{\zeta_{\alpha}- \zeta_{\beta}}}\right] ,
E_{\rm vac}= -\frac{1}{2}\sum_j(2G_j+1)|\epsilon_j| - \int_0^\infty \frac{dk}{2\pi}\,\sqrt{k^2+m_f^2}\,\frac{d}{dk}\,\overline{\delta}(k) + E^{(1,2)}+ E^{(3,4)}\, ,
\chi_P(P_a,P_b) = \left| \sum_{i=1}^k kP_a(i)P_b(i) - 1 \right| &= \left| \sum_{i=1}^k k \frac{1}{k}\left( 1 + \epsilon v_{a,i}\right) \frac{1}{k}\left( 1 + \epsilon v_{b,i}\right) - 1 \right| &= \left| \sum_{i=1}^k \frac{1}{k}\left( 1 + \epsilon v_{a,i}+ \epsilon v_{b,i}+ \epsilon^2 v_{a,i}v_{b,i}\right) - 1 \right| &= 0,
\frac{(\begin{matrix}z+k\\j\end{matrix})}{(\begin{matrix}k\\j\end{matrix})}\rightarrow(1-\frac{j}{k})^{-z}and\frac{(\begin{matrix}j\\j-k\end{matrix})}{(\begin{matrix}j-z\\j-k\end{matrix})}\rightarrow(\frac{j}{k})^{z}
\mathcal{F}=\int\bigg{[}\varepsilon_y^2\psi_B+\varepsilon^2\zeta_B-\frac{2}{p+1}\big{(}|\varepsilon+Q_b|^{p+1}-Q_b^{p+1}-(p+1)\varepsilon Q_b^p\big{)}\psi_B\bigg{]}.
\lbrack\lbrack\even,even \ ]]_{\star}=even \quad\lbrack\odd,odd\ ]_{\star}=odd \lbrack\lbrack\odd,odd \ ]]_{\star}=even \quad\lbrack\even,even\ ]_{\star}=odd \lbrack\lbrack\even,odd \ ]]_{\star}=odd \quad\lbrack\even,odd\ ]_{\star}=even
\hat{G}_{M N}= \left( \begin{matrix}{e^{a \varphi}g_{{\mu}{\nu}}+ G_{{m}{n}}A_{{\mu}}^{( 1 ) m}A_{{\nu}}^{( 1 ) n}}&{A_{{\mu}}^{( 1 ) m}G_{{m}{n}}}\\{A_{{\nu}}^{( 1 ) n}G_{{m}{n}}}&{G_{{m}{n}}}\\ \end{matrix}\right) ,
\sum_{i=1}^n(\gamma-\mu_i)^{-1}-\frac{n}{\lambda^{1/2}(1+\lambda^{1/2})}= O\left(\left(1+|\sigma_n|^{1/2}\right)n^{2/3}\right), \sum_{i=1}^n(\gamma-\mu_i)^{-2}=O(n^{4/3}).
\Xi = \begin{cases}M^{\nu+1},\; \; M \cdot \prod_{i = 1}^{\nu}M_i, \; \; M^2 + \sum_{i=2}^{\nu_\mathrm{max}}(n_\mathrm{NB, i}+ 1)^{\nu_i+1}, \; \; .\end{cases}
\Pi_{\mu_{1}\mu_{2}}^{\mathrm{U V}}= \left( \frac{1 3}{3}- \xi \right) \frac{g^{2}C_{G}}{\left( 4 \pi \right)^{2}}\left( \frac{1}{2 \epsilon}+ \mathrm{l n}\left( M \right) \right) \left( \eta_{\mu_{1}\mu_{2}}k^{2}- k_{\mu_{1}}k_{\mu_{2}}\right) ,
M(x,Y) = \min_{|y| \leq 2Y}\sum_{p \leq x}\frac{1 - h(p) + h(p) - \Re( h(p)\chi(p) p^{iy})}{p}&= \sum_{p \leq x}\frac{1 - h(p)}{p}+ \min_{|y| \leq 2Y}\sum_{p \leq x}\frac{h(p)(1 - \Re(\chi(p) p^{iy}))}{p}.
\lim_{b \uparrow \infty}\frac{\overline{\overline{W}}^{(q+r)}(b)}{I_a^{(q,r)}(b)}= \lim_{b \uparrow \infty}\frac{\overline{\overline{W}}^{(q+r)}(b)}{W^{(q+r)}(b)}\lim_{b \uparrow \infty}\frac{W^{(q+r)}(b)}{I_a^{(q,r)}(b)}= \frac{1}{\Phi(q+r)}\frac{W^{(q)}(-a)}{Z^{(q)\prime}(-a, \Phi(q+r))}.
\vec{\Gamma}_{\nu}\partial_{\mu}\vec{J}^{\mu}= - \vec{\Gamma}_{\nu}\vec{q}\, \frac{d \Lambda}{d q_{0}}= - \partial_{\nu}\alpha \sin \alpha \, \frac{d \Lambda}{d q_{0}}= \partial_{\nu}q_{0}\, \frac{d \Lambda}{d q_{0}}= \partial_{\nu}\Lambda .
Z(\Lambda) = \{\mathbf n \in \mathbb Z^N \;|\: \Lambda \cdot \mathbf n = 0\}, \begin{aligned}V(\Lambda) = \{L \in \mathbb R^N \;|\: Z(\Lambda)\subset Z(L)\}, V_+(\Lambda) = V(\Lambda) \cap (\mathbb R_+^\ast)^N, W(\Lambda) = \{L \in \mathbb R^N \;|\: Z(\Lambda)=Z(L)\}, W_+(\Lambda) = W(\Lambda) \cap (\mathbb R_+^\ast)^N.\end{aligned}
K^{\alpha}_{R,m}(z,w) = \frac{1}{\pi R^{2}}\left\{\sum\limits_{n=0}^{m-1}\frac{\Gamma(\alpha+2+n)}{\Gamma(n+1)\Gamma(\alpha+1)}\frac{(z\overline{w})^{n}}{R^{2n}}+ \sum\limits_{n=m}^{+\infty}\frac{\Gamma(n-m+1)\Gamma(\alpha+2+n-m)}{\Gamma^{2}(n+1)\Gamma(\alpha+1)}\frac{(z\overline{w})^{n}}{R^{2(n-m)}}\right\}.
P_{out}(R_{E},\phi_{2}) \!\!= \!\! &(1\!-\!e^{\!-\!\frac{\rho_1|h_1|^{2}}{\rho_e}})P_{out,1}(R_{E},\phi_{2})+e^{-\frac{\rho_1|h_1|^{2}}{\rho_e}}P_{out,2}(R_{E},\phi_{2}),
\frac{1}{2}(\|E_1 \tilde v_k\|^2 + \|E_2 \tilde w_k\|^2) = \|E_1v_k\| \|E_2 w_k\| \leq \frac{1}{2}(\|E_1 v_k\|^2 + \|E_2 w_k\|^2).
\psi' \left(\prod_{i=1}^{n_1}((\right.c_{1i},\ldots,c_{ki}),2)\prod_{i=0}^{n_2}((c_{1i}',\ldots,c_{ki}'),+1)\left.\prod_{i=1}^{n_3}((c_{1i}'',\ldots,c_{ki}''),2)\right) = &\sum_{i=1}^{n_1}(c_{1i}z_1^{i-n_1-1},\ldots,c_{ki}z_k^{i-n_1-1})+\sum_{i=0}^{n_2}(c_{1i}'z_{1}^{i},\ldots,c_{ki}'z_{k}^{i})+\sum_{i=1}^{n_3}(c_{1i}''z_1^{i+n_3},\ldots,c_{ki}''z_k^{i+n_3}),
f(u)E_{i}+f(u+v)E_{i+1}+f(v)E_{i}+f(u)f(v)\delta E_{i}+f(u)f(u+v)f(v)E_{i}= f(v)E_{i+1}+f(u+v)E_{i}+f(u)E_{i+1}+f(v)f(u)\delta E_{i+1}+ f(v)f(u+v)f(u)E_{i+1}
\sum_{n = - n_{0}}^{\infty}\frac{a_{n}^{( 1 )}e^{- n \pi i / 2}}{\left( x - x_{0}\right)^{n}}= \sum_{n = - n_{0}}^{\infty}\frac{( - 1 )^{n}a_{n}^{( 2 )}e^{- n \pi i / 2}}{\left( x - x_{0}\right)^{n}},
\int_0^r \left[(\phi''+\frac{z}{2}\phi'-\frac{z}{2}e_\sigma\psi')\cdot z\phi'+(\psi''+\frac{z}{2}\psi'-\frac{z}{2}e_\sigma\phi')\cdot z\psi'\right]{\rm{d}}z =\frac{1}{2}\int_0^r |z(\phi',\psi')|^2+\int_0^r z(\phi'\phi''+\psi'\psi'')-\int_0^r z^2e_\sigma\phi'\psi'.
\sum_{T \leq \lvert t_j \rvert \leq 2T}\rho_j(h) \langle \mu_j, V \rangle = \sum_{T \leq \lvert t_j \rvert \leq 2T}\rho_j(h) \langle \mu_j, \theta^{2k+1}\overline{\theta}y^{\frac{k+1}{2}}\rangle \ll T^{3k + 8 + \epsilon}.
\mathbf{R}_{\boldsymbol{\epsilon}}=\mathbb{E}\left\{\left(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}\right)\left(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}\right)^\top\right\}=\mathbb{E}\left\{\boldsymbol{\epsilon}\boldsymbol{\epsilon}^\top\right\}~.
\sigma^2(c) =V(c, c)+2\sum_{k=1}^{36}\sum_{j=0}^\infty\frac1{6^{k+36j}}V\bigl(\langle 2^{k+36j}c\rangle, \langle 3^{k+36j}c\rangle\bigr) &=V(c, c)+\frac{2\cdot 6^{36}}{6^{36}-1}\sum_{k=1}^{36}\frac1{6^{k}}V\bigl(\langle 2^{k}c\rangle, \langle 3^{k}c\rangle\bigr) &=\frac{1222685807050467558645782547163492}{4561296506512477081905890170847375}.
S [ A_{\mu}, h ] = - \frac{1}{2 g^{2}}\int d x^{3}\, \mathrm{t r}[ F_{\mu \nu}\, F^{\mu \nu}] + \int d x^{3}\left( \mathrm{t r}( D_{\mu}h )^{2}+ \frac{\lambda}{4}( 2 \mathrm{t r}h^{\dagger}h - v^{2})^{2}\right) \, \, \, ,
\varphi_i=\left(\frac{z_iA_i(z_{i+1},z_{i+2},\ldots,z_{n-1})+B_i(z_{i+1},z_{i+2},\ldots,z_{n-1})}{z_iC_i(z_{i+1},z_{i+2},\ldots,z_{n-1})+D_i(z_{i+1},z_{i+2},\ldots,z_{n-1})}\right)\in\mathrm{PGL}(2,\mathbb{C}(z_{i+1},z_{i+2},\ldots,z_{n-1}))
\sum_{i=1}^{n}v_{i}^{q+1}a_{i}^{qj+l}&=\sum_{i=1}^{\frac{q-1}{a}}\omega^{ai(q+1)}\sum_{s=0}^{\frac{q-1}{2}}(\omega^{[2s(q-1)+a(2i+1)](qj+l)}+\omega^{[2s(q-1)+a(2i+2)](qj+l)}) &=\sum_{i=1}^{\frac{q-1}{a}}\omega^{ai(q+1)}(\omega^{a(2i+1)(qj+l)}+\omega^{a(2i+2)(qj+l)})\sum_{s=0}^{\frac{q-1}{2}}\omega^{2s(q-1)(qj+l)}.
B^{[ A_{\mu}\, ] \, 1}\, , \, \, B^{[ c \, ] \, 1}\, , \, \, B^{[ \overline{{c}}\, ] \, 1}\, , \, \, B^{[ G \, ] \, 1}\, \, ; \, \, \lambda^{[ A_{\mu}\, ] \, 1}\, , \, \, \lambda^{[ c \, ] \, 1}\, , \, \, \lambda^{[ \overline{{c}}\, ] \, 1}\, , \, \, \lambda^{[ G \, ] \, 1}\, \, . \nonumber
l(\varepsilon)=\left({\textstyle\bigcup\nolimits_{j=1}^{s}}\gamma(t_{j},\varepsilon)\right) \cup\left({\textstyle\bigcup\nolimits_{j=1}^{s-1}}(t_{j}+\varepsilon,t_{j+1}-\varepsilon)\right) \cup\lbrack-h,t_{1}-\varepsilon)\cup(t_{s}+\varepsilon,2\pi-h].
{\cal G}_{z 4}\left(\rho,z\right)={\displaystyle\frac{1}{4\pi A{\cal C}_{66}}\sum_{l=1}^{4}{\frac{\Lambda_{\rm{c4}}\left(a=-A_l\right)}{\sqrt{A_l\rho^2+z^2}\prod_{j=1,(j\neq l)}^{4}{\left(A_j-A_l\right)}}}}
{\cal O}_{\pm}^{a}( \{x \}) : = ( - 1 )^{\frac{N ( N - 1 )}{2}}\left[ \frac{1}{N !}\sum_{\pi}\mathrm{s i g n}( \pi ) \prod_{b = 1}^{N}\overline{{{\psi}}}_{\stackrel{{\scriptstyle 1}}{{\scriptstyle 2}}}^{( \pi ( b ) )}( x^{( b )}) \right]
S[\, H\ , B\,]= \int d^4x\left[\,{1\over 12}H_{\lambda\mu\nu}\, H^{\lambda\mu\nu}-{1\over 6}H^{\lambda\mu\nu}\partial_{[\, \lambda}B_{\mu\nu\,]}+{m^2\over 4}B_{\mu\nu}\, B^{\mu\nu}\,\right]
{\cal L}= - \frac{1}{4}\tilde{F}_{\mu \nu}\tilde{F}_{\mu \nu}+ \partial_{\mu}\psi \partial_{\mu}\psi + q^{2}\psi^{2}\tilde{A}_{\mu}\tilde{A}_{\mu}- \mu^{2}\psi^{2}- \frac{\lambda}{4}\psi^{4}
\tilde{t}=T(t), \tilde{x}_k= \sqrt{\dot{T}(t)}A_{kl}x_l+\beta_k(t), u(x,t)=\exp[F(x,t)] \tilde{u}(\tilde{x},\tilde{t}), F =\frac{{\ddot{T}}}{\dot{T}}\frac{|x|^2}{8}+ \frac{\dot{\beta}_k}{2\sqrt{\dot{T}}}A_{kl}x_l+f(t)
W = \left| \begin{array}{c c c c}{{0}}&{{i C_{-}}}&{{0}}&{{C_{-}}}\\{{0}}&{{- C_{+}}}&{{0}}&{{i C_{+}}}\\{{i C_{-}}}&{{0}}&{{C_{-}}}&{{0}}\\{{i C_{+}}}&{{0}}&{{C_{+}}}&{{0}}\\ \end{array}\right| ; \quad C_{\mp}\equiv \frac 1 2 ( C \mp 1 ) , \quad C \Psi \equiv \Psi^{*}, \quad C \mathcal{E}\equiv \mathcal{E}^{*}.