harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left|J_{1}(x)\right|\leq\left(\sup_{B_{R}\setminus N_{\delta}}f_{k}\right)\int_{B_{2R}(x)}\frac{1}{\left|x-y\right|}dy=CR^{2}\left(\sup_{B_{R}\setminus N_{\delta}}f_{k}\right)\to0,\quad\mbox{as}k\to+\infty.
\Delta \mathbf{b}^k - \Delta \widehat{\mathbf{b}}^k = (\mathbf{I}- \mathbf{B}^k)^{-1}\, (\mathbf{b}^k - \widehat{\mathbf{b}}^k) & = \frac{\lambda_k \, \left[1 - \Delta b^{S_{k+2}}_{k+1}\right]}{\alpha + \lambda_{k-1}+ \mu_k}\, (\mathbf{I}- \mathbf{B}^k)^{-1}\, \mathbf{e}^k.
g_{12}=g_{21}=0, g_{1 1 2}=g_{1 2 1}=g_{2 1 1}=0, g_{2 2 2}=0, g_{1112}=g_{1121}=g_{1211}=g_{2111}=0, g_{1222}=g_{2122}=g_{2212}=g_{2221}=0.
\partial\cdot N=\partial^{\mu}\eta_{\mu\nu}N^{\nu}=\partial_{\nu}N^{\nu}=(\frac{\partial_{t}}{c},-\vec{\nabla})\cdot(nc,n\vec{u})=\frac{\partial_{t}}{c}(nc)+\vec{\nabla}\cdot n\vec{u}=\partial_{t}n+\vec{\nabla}\cdot n\vec{u}=0
\tilde{\lambda}=:\frac{\sum_{j=1}^N|\lambda_j|}{2\pi},\tilde{d}_I(t):=\min_{1\leq j\leq N}\inf_{\alpha\in \mathbb{R}}|\alpha-\Phi(z_j(t), t)|,\tilde{d}_P(t)=\min_{j\neq k}|z_j(t)-z_k(t)|.
\mu_{1,c}(x) := \max\{x_2,x_3,c(x_1+x_2+x_3)\}, \mu_{2,c}(x) := \max\{x_1,x_3,c(x_1+x_2+x_3)\}, \mu_{3,c}(x) := \max\{x_1,x_2,c(x_1+x_2+x_3)\}.
A_{1}= C_{1},\A_{2}= C_{2}+ z t^{r}_{2}E, & B = ( - t_{1}+ c +\alpha z) C_{1}-\frac{t_{2}}{r+2}C_{2}-\frac{z(r+1)}{2(r+2)}D -\frac{zt_{2}^{r+1}}{r+2}E,
\mathcal{T}_z\circ \mathcal{T}_\zeta=\mathcal{T}_{z+\zeta}\ , \ \ \mathcal{T}_z^{-1}=\mathcal{T}^*_z=\mathcal{T}_{-z}\ , \ \ \mathcal{T}_0={\rm Id}_{L^2(\mathbb{R}^d)}\ .
4 x \left(54 x^2-20 x+1\right)Z +16 x \left(27 x^2-13 x+1\right)Z_{X}+24 x (2 x-1) (6 x-1)Z_{XX}+(4 x-1) \left(16 x^2-12 x+1\right)Z_{XXX}= 0.
g(\mathrm{div}^{\nabla}(T_{r}),X)=(-1)^{r}PX(S_{r})+g((\nabla_{E}A_{N})T_{r-1}E,X) +g(\mathrm{div}^{\nabla}(T_{r-1}),A_{N}X)+\sum_{i=1}^{n}\epsilon_{i}g((\nabla_{Z_{i}}A_{N})T_{r-1}Z_{i},X).
I_{2}( y , a , b , d ) = \frac{1}{a}h ( d ) \sum_{n = 1}^{\infty}\int d p \, p^{d - 3}\frac{1}{\sqrt{\vec{p}^{\, 2}+ m^{2}+ ( \frac{n \pi}{a})^{2}}}\frac{\cosh ( ( b - 2 y ) \sqrt{\vec{p}^{\, 2}+ m^{2}+ ( \frac{n \pi}{a})^{2}})}{\sinh ( b \sqrt{\vec{p}^{\, 2}+ m^{2}+ ( \frac{n \pi}{a})^{2}})}
S_{b}^{( 2 )}={\frac{1}{4 \pi}}\int d \tau \int_{0}^{2 \pi \alpha}d \sigma \Big [{\frac{1}{z_{0}^{2}}}( \partial z^{( 1 )})^{2}+{\frac{1}{z_{0}^{2}}}( \partial x_{4}^{( 1 )})^{2}+{\frac{3 ( z^{( 1 )})^{2}}{z_{0}^{4}}}( \dot{z_{0}}^{2}+ \dot{r}_{0}^{2})
\tilde{\zeta}(s)=\frac{V_q}{(4\pi)^{q/2}}\frac{\Gamma(s-q/2)}{\Gamma(s)}\Big\lbrace(\mu_c^2-\mu^2)^{q/2-s}+ \sum_{{\bf p}\ne{\bf 0}}(\sigma^d_{\bf p}+m^2-\mu^2)^{-s}\Big\rbrace\;,
{\cal K}_{{\bf C}^{n}}( z , \bar{z}) = N \, \mathrm{e x p}\left( \mathrm{i}\sum_{j = 1}^{n}\bar{z}^{j}z^{j}\right) \mathrm{d}z^{1}\wedge \ldots \wedge \mathrm{d}z^{n}\wedge \mathrm{d}\bar{z}^{1}\wedge \ldots \wedge \mathrm{d}\bar{z}^{n},
\sum_{i=1}^k \lambda_i \left( \frac{|\nabla \Phi_i(x)|^2}{2}- \frac{|x|^2}{2}\right) = \sum_{i=1}^k \lambda_i \left( \frac{|\nabla \Phi_i(x)|^2}{2}- \frac{|x|^2}{2}- \langle \nabla \Phi_i(x) - x, x \rangle \right) & = \frac{1}{2}\sum_{i=1}^k \lambda_i |\nabla \Phi_i(x) -x|^2.
\min_{x \in \mathbb R^{n \times q}}\sum_{i=1}^m \log\Big(\sum_{j=1}^q \exp\big(\sum_{l=1}^n A_{i,l}x_{l,j}\big)\Big) + \sum_{i=1}^n \sum_{j=1}^q x_{i,j}b_{i,j}+ \sum_{l=1}^n \sqrt{\sum_{j=1}^q x_{l,j}^2}
[a^{\dagger}a,a^{\dagger}b] = \lambda a^{\dagger}b, [bb^{\dagger},ba^{\dagger}] = \lambda' ba^{\dagger}, [a^{\dagger}b,b^{\dagger}a] = \mu a^{\dagger}a + \nu, [ba^{\dagger},ab^{\dagger}] = \mu'aa^{\dagger}+ \nu',
\tilde\Phi_k\left((x_1, v_1), \ldots, (x_{k}, v_{k}) \right) & =\bigg(l_{k}(x_1, \ldots, x_{k}), \sum_{i=1}^{k}(-1)^{|x_i|(|x_{i+1}|+\ldots+|x_{k}|)}\phi_{k-1}(x_1,\ldots, \widehat{x_i},\ldots, x_{k},v_i)\bigg),
|f(\varphi(h,x))-f(x)| =|a_i\varphi(h,x) + b_i- (a_n x + b_n)| \leq |\varphi(h,x)-x| \cdot |a_i| + |(a_i-a_n) x- (b_n-b_i)| & \leq \dfrac{\varepsilon}{4}\cdot u(x) \cdot |a_i| + |a_i-a_n| \cdot | x- j_i| .
x(P_1) = \, 4 t (1 + t) (-1 + 2 t) (-1 + t + t^2) (-1 + 2 t + t^2)^2 (-1 + 2 t + t^3) (-1 + t + t^2 + t^3)^2, x(P_2) = \, 4 t^4 (1 + t) (-1 + 2 t) (-1 + 2 t + t^2)^2 (-1 + 2 t + t^3) (-1 + 2 t - t^2 + t^3 + 2 t^4 + t^5), x(P_3) = \, 4 t^2 (-1 + 2 t) (-1 + 2 t + t^2)^2 (-1 + t + t^2 + t^3)^2 (-1 + 2 t - t^2 + t^3 + 2 t^4 + t^5).
S(X,Y)=\sum_{\alpha=1}^p\sum_{j=p+1}^nf^j(Y)\overline\omega^\alpha_j(X)f_\alpha\,\,\, \mbox{and}\,\,\, A_{f_\alpha}(X)=-\sum_{i=p+1}^{d_1}\overline\omega_\alpha^i(X)f_i.
\Delta_{\infty}(\lambda,b)=\delta_{\infty}^{2}(\lambda,b)\quad\mbox{with}\quad\delta_{\infty}(\lambda,b):=b\big[I_{1}(\lambda)K_{1}(\lambda)+I_{1}(\lambda b)K_{1}(\lambda b)\big]-(1+b^{2})I_{1}(\lambda b)K_{1}(\lambda).
\frac{2}{\frac{1}{M}+\frac{1}{N}}=\frac{2}{\kappa_{1}+\kappa_{2}}=\frac{1}{H}=\frac{2M}{1+\frac{(bN)^{2}}{a^{4}}}=\frac{2N}{\frac{a^{4}}{(bN)^{2}}+1}.
j_{\mu}^{m a t t e r}= \bar{\varphi}_{2}\partial_{\mu}\bar{\varphi}_{1}- ( \bar{\varphi}_{1}+ \mathrm{~ \frac ~ m e ~}) \partial_{\mu}\bar{\varphi}_{2}+ e A_{\mu}( \bar{\varphi}_{2}^{2}+ ( \bar{\varphi}_{1}+ \mathrm{~ \frac ~ m e ~})^{2}) + O ( \hbar )
\left( \hat{\epsilon}_{\hat{\mu}}{}^{\hat{a}}\right)=\left(\begin{array}{cc}e_{\mu}{}^{a}k^{-1}B_{\mu}& 0 k^{-1}\end{array}\right)\, ,\hspace{1cm}\left( \hat{\epsilon}_{\hat{a}}{}^{\hat{\mu}}\right)=\left(\begin{array}{cc}e_{a}{}^{\mu}-B_{a}& 0 k \end{array}\right)\, .
\P \big( |\widehat{d}_{j,k}| \neq 0 (j,k) \in \mathcal{J}_n^c (\gamma) \big) \leq \frac{2n^{1-\frac{1}{2}(\tau - \gamma)^2}}{(\tau-\gamma)\sqrt{\log n}}, \P \big( |\widehat{d}_{j,k}| = 0 (j,k) \in \mathcal{J}_n (2\tau) \big) \leq \frac{2n^{1-\frac{\tau^2}{2}}}{\tau \sqrt{\log n}},
I_{\alpha,\beta}:= \psi_{\alpha,\beta}= \{a_{\beta}\in G_{\beta}:\exists a_{\alpha}\in G_{\alpha}, a_{\alpha}\psi_{\alpha,\beta}=a_{\beta}\}, K_{\alpha,\beta}:= \psi_{\alpha,\beta}= \{a_{\alpha}\in G_{\alpha}:a_{\alpha}\psi_{\alpha,\beta}=e_{\beta}\},
H_1(t) =\varrho\min\left\{1; \left[\frac{h(t)}{c_{2,\infty}}\right]^{\alpha}\right\}= \varrho\min\left\{1; \frac{1}{c_{2,\infty}^{\alpha}}\left[\frac{1}{\|1/c_{2,0}^\alpha\|_{L^{\infty}(\Omega)}}+ \alpha(\alpha+1)k_3t \right]^{-1}\right\}
\begin{aligned}\pi_{1}^{-1}(o) =\left\{o\times[U,V,W_{1},0,0]\right\}\cup\left\{o\times[U,V,0,W_{2},0]\right\}\cup\left\{o\times[U,V,0,0,W_{3}]\right\}=:\mathbb{P}_{1}^{2}\cup\mathbb{P}_{2}^{2}\cup\mathbb{P}_{3}^{2}.\end{aligned}
\displaystyle X_{n+1,x}=\begin{cases}0, \mbox{if}x = x_n, E_x(S_n), \mbox{if}x \neq x_n \textrm{and}W_{n,x}\geq t^{\ast}, E_x(S_n + t^{\ast}) + t^{\ast}, \mbox{if}x \neq x_n \textrm{and}W_{n,x}< t^{\ast}.\end{cases}
\overline{\Phi}[r]=\frac{N^{2}}{2}\sum_{j=1}^{n}\sum_{k=1}^{n}\int_{0}^{1}ds\int_{0}^{1}ds^{\prime}\overline{\Phi}(|r_{j}(s)-r_{k}(s^{\prime})|)-\frac{1}{2}nN\overline{\Phi}(0),(2)
\left|\left|\frac{x_1u_1+y_1v_1}{\mathcal{N}(q_1)}-\frac{k}{\mathcal{N}(q_2)}\right|\right|=\left|\frac{x_1'u_1+y_1'v_1}{\mathcal{N}(q_1)}-\frac{k}{\mathcal{N}(q_2)}\right|
\rho_k-\rho_{k+1}\geq ||M_{\mathcal{S}_{k}}Y||^2-||M_{\mathcal{S}_{k}\cup\{i_{k+1}^*\}}Y||^2 - n\sum_{j=1}^{s_{k}+1}p_{\lambda_n}(|\hat{\beta}_j^*|) &\geq \frac{(X_{i_{k+1}^*}^\top M_{\mathcal{S}_{k}}Y)^2}{X_{i_{k+1}^*}^\top X_{i_{k+1}^*}}-n\lambda_n\sum_{j=1}^{s_{k}+1}|\hat{\beta}_j^*|,
Z = \beta^{N/2}\Gamma((N-1)/2)^{-1}{\LARGE\int_0^\infty}d\sigma{\LARGE\int_{-\infty}^\infty}d\sigma_1 ( 4 \beta^2 \sigma^2 + (2 \beta \sigma_1 + a )^2 + m^2 )^M \sigma^{N-2}exp[ -\beta (\sigma^2 + \sigma_1^2) ]
\sum_{j\le j_0}\psi_j2^{jr}\Vert\dot{\Delta}_j(T_f g)\Vert_{L^2}\le C \Big(\sup_{j'\le j_0+4}2^{j'(r-r_2)}\Vert \dot\Delta_{j'}g\Vert_{L^2}\Big) \sum_{j\le j_0}\sum_{j'\le j+4}2^{jr_2}\Vert \dot\Delta_{j'}f \Vert_{L^\infty}\psi_j.
f ( \nu ) = - i e^{2}\int \frac{d^{n}k}{{( 2 \pi )}^{n}}p_{\alpha}D_{F}^{\alpha \beta}( k ) p_{\beta}\int_{0}^{\nu}d \nu^{\prime}\int_{0}^{\nu^{\prime}}d \nu^{\prime \prime}e^{i \nu^{\prime \prime}( k . p )}.
\begin{aligned}(H_i-q)(H_i+q^{-1})=0, 1\leq i\leq r-1, H_iH_j=H_jH_i, |i-j|\geq 2, H_iH_{i+1}H_i=H_{i+1}H_iH_{i+1}, 1\leq i\leq r-2. \end{aligned}
\langle 1 \cdots M|\mathcal{B}^\prime(z_1)\cdots \mathcal{B}^\prime(z_N)|\overline{x_1}\cdots\overline{x_N}\rangle=\frac{(1+t^{-1}z_1^2)}{tz_1}\sum_{\overline{y}}\frac{z_1^{\overline{y}}-z_1^{-\overline{y}}}{z_1-z_1^{-1}}\langle \overline{y}|\mathcal{B}^\prime(z_2) \cdots \mathcal{B}^\prime(z_N)|\overline{x_1}\cdots\overline{x_N}\rangle,
[(N_b,0,0,0)\otimes(0,1,0,0)+(0,1,0,0)\otimes(N_b,0,0,0)] = [(1,0,0,0)\otimes(0,N_b,0,0)+(0,N_b,0,0)\otimes(1,0,0,0)] = [(1,0,0,0)\otimes(0,0,-N_a,0)+(0,0,-N_a,0)\otimes(1,0,0,0)] = -[(N_a,0,0,0)\otimes(0,0,1,0)+(0,0,1,0)\otimes(N_a,0,0,0)] = 0.
\stackrel{\wedge}{\Sigma}_{(2)}(p,m,\lambda ,\Lambda_0) =\frac{i\lambda^2}{192\pi^6}\int_0^1dx\int_0^1dy\frac y{\left[ yC\right]^{3/2}}\int d^3kd^3l\frac 1{\left[ k^2+l^2+Dp^2-m^2+i\epsilon \right]^3}
\langle E_\infty^k(\cdot, u), V \rangle = \int_0^\infty \widetilde{V}_0(y) y^{u-1}\frac{dy}{y}= \frac{\Gamma(s + \frac{k+1}{2}- 1)}{(4\pi)^{u + \frac{k+1}{2}- 1}}\sum_{n \geq 1}\frac{r_{2k+1}(n)r_1(n)}{n^{u + \frac{k+1}{2}- 1}}
y_{n}^{k+1}= y_{n}^k + \epsilon_k\left((I-J_{\gamma A})(x^k) - y_{N+1}^k\right); y_{i}^{k+1}= y_{i}^k + \epsilon_k \left(\frac{\gamma}{N}B_{i}( J_{\gamma A}(x^k)) - y_{i}^k\right).
\int_{\mathcal F(t)}\rho_f(\partial_t u + u \cdot \nabla u ) \cdot (\partial_t u + \hat{\Lambda}\cdot \nabla u - u \cdot \nabla \hat{\Lambda})= \int_{\mathcal F(t)}{\rm div}\sigma \cdot (\partial_t u + \hat{\Lambda}\cdot \nabla u - u \cdot \nabla \hat{\Lambda})\,.
\sum\limits_{n = 1}^\infty{\frac{{\sigma_{z_1,\beta}(n)\sigma_{z_2,\beta}(n)}}{{{n^s}}}}= \frac{{\zeta (s)\zeta (\beta (s - z_1))\zeta (\beta (s - z_1))\zeta (\beta (s - z_1 - z_2))}}{{\zeta (\beta (2s - z_1 - z_2))}}
\|P_N a -a \| \leq \|P_N (a - a_j)\| + \|P_N a_j - a_j\| + \|a_j-a\| &\leq (1+c) \| a - a_j \| + \max_{i=1, \ldots, m}\|P_N a_i - a_i\| &\leq (1+c) \delta + \max_{i=1, \ldots, m}\|P_N a_i - a_i\|,
\partial D_{n+m}=\partial(D^{n}\times D^{m})-(\partial D^{n}\times \partial D^{m}) &= (\partial D^{n}\times D^{m}) \cup (D^{n}\times \partial D^{m}) - (\partial D^{n}\times \partial D^{m}) &= \left( \partial D^{n}\times \mathrm{Int}(D^{m}) \right) \coprod \left( \mathrm{Int}(D^{n}) \times \partial D^{m}\right)
I_{1}= \frac{1}{1 6 \pi^{2}}\int \frac{d^{4}k}{( 2 \pi )^{4}}\frac{d^{4}l}{( 2 \pi )^{4}}\frac{1}{\Big ( 1 + k^{2 n}/ \Lambda^{2 n}\Big ) \Big ( 1 + l^{2 n}/ \Lambda^{2 n}\Big ) k^{4}l^{4}}.
m_l\log\log x + O(1) = \sum_{p\leq x}\frac{|a_L(p)|^2}{p}= \sum_{p\leq x}\frac{|a_{\zeta^{m_L}}(p)+a_{L^*}(p)|^2}{p}&=\sum_{p\leq x}\frac{|a_{\zeta^{m_L}}(p)|^2}{p}+ \sum_{p\leq x}\frac{|a_{L^*}(p)|^2}{p}+ O(1) &=m_L\log\log x + \sum_{p\leq x}\frac{|a_{L^*}(p)|^2}{p}+ O(1).
h_2 =\frac{k\int^{x}_{C_4}( C_2 F_{1}(C_1/2,1/(8\xi)) + C_3 G_{1}(C_1/2,1/(8\xi)) )d\xi}{2x( C_2 F_{1}(C_1/2,1/(8x)) + C_3 G_{1}(C_1/2,1/(8x)) )}.
\| \nabla S_k(x,y) \|_{2}^2 \le \sum_{j=k}^n \frac{1}{(x_j^o - \delta_n)^2}+ \frac{1}{(y_j^o - \delta_n)^2}\le 8 n \sum_{j=k}^n \frac{1}{j}\le 16 n \log (n/k),
\bigl\{\mathbf{p}\bigl(\tfrac{\partial_1a}{a},\dots,\tfrac{\partial_ma}{a}\bigr) \ \big| \a\in A\bigr\}\neq \{0\},\intertext{and such that there exists an element $f\in K$ satisfying}\mathbf{p}(\partial_1u,\dots,\partial_mu)-\mathbf{q}(\partial_1r_1,\dots,\partial_mr_1)=\delta_xf.
-\langle\frac{dP_{b}}{dt}\rangle=\frac{192\pi G^{5/3}m_{1}m_{2}(m_{1}+m_{2})^{-1/3}}{5c^{5}(1-e^{2})^{7/2}}(1+\frac{73}{24}e^{2}+\frac{37}{96}e^{4})(\frac{P_{b}}{2\pi})^{-5/3}
K(x,t)-\frac{e^{-1/4t}}{8\pi t}\mathbb I &=\frac{e^{-|x|^2/4t}-e^{-1/4t}}{8\pi t}\mathbb I+\left(\frac{e^{-|x|^2/4t}}{8\pi t}-\frac{1-e^{-|x|^2/4t}}{2\pi |x|^2}\right)\left(\mathbb I-\frac{2x\otimes x}{|x|^2}\right)=:\frac{A+B}{8\pi}.
G_N(x, r) ={}\oint_{\Sigma}g_N(t; x, r) \frac{dt}{2\pi i}, \g_N(t; x, r) = e^{\frac{N}{M + 1}\big( f_{M, N}(q_{M}(0)) -f_{M, N}(t) + \frac{(x + r - C)(t - q_M(0))}{\rho_{M, N}}\big)},
{\cal C}\Gamma_{a}= \left( \begin{matrix}{c \tilde{\gamma}_{a}}&{0}\\{0}&{c^{T}\gamma_{a}}\\ \end{matrix}\right) \otimes \Omega \ ; \qquad{\cal C}\Gamma_{a^{\prime}}= \left( \begin{matrix}{0}&{- c}\\{c^{T}}&{0}\\ \end{matrix}\right) \otimes \Omega \gamma_{a^{\prime}}\ .
\lim_{j\to-\infty}\frac{(-1)^j(x_{j+1}-x_j)}{|j|^t\!\cdot\!|r_2|^j}=(-1)^t(r_2-1)b, & \ \lim_{j\to\infty}\frac{(-1)^j(x_{j+1}-x_j)}{j^s\!\cdot\!|r_1|^j}=(r_1-1)a.
z_{[ r ]}^{( n )}\: T_{[ r ]}^{( n )}\; = \; - 4 \: ( n \: T_{[ r ]}^{( n + 1 )}+ T_{\{r \}}^{( n + 2 )}) \: + \:{\mathrm{( s m o o t h ~ f u n c t i o n s )}}\; .
D^2 (p+r) = D^2 (p) + \frac{1}{2}r^{\alpha\beta}\left [ \frac{\partial}{\partial r^{\alpha\beta}}D^2(r)\right]_{r=p}+ \frac{1}{2}\left[ \frac{1}{2}\frac{\partial^2}{\partial r_{\alpha\beta}\partial r^{\alpha\beta}}D^2 (r) \right]_{r=p}.
\int_{\mathbb R^2}\sum_{k\in \Bbb Z}2^{2k}\vert \psi_{j,k}*f(y_1,y_2,y_3)\vert^2 w(R_{2^{-j},2^{-k}}(y_1,y_2,y_3) ){dy_2\,dy_3}&=\int_{\mathbb R^2}\sum_{k\in \Bbb Z}2^{2k}\big\vert \big(f\ast_1\psi^{(1)}_{j}(y_1,\cdot,\cdot)\big) \ast_{2,3}\psi^{(2)}_{j,k}(y_2,y_3)\big\vert^2 w_{y_1,2^{-j}}(R_{2^{-k},2^{-j-k}}(y_2,y_3) ){dy_2\,dy_3}.
{{\boldsymbol{x}}_k}= \left[{\begin{array}{*{20}{c}}{{{\boldsymbol{I}}_2}}{\Delta T{{\boldsymbol{I}}_2}}{\boldsymbol{0}}{{{\boldsymbol{I}}_2}}\end{array}}\right]{{\boldsymbol{x}}_{k - 1}}+{{\boldsymbol{q}}_{k - 1}},
+\cos\psi\cot\eta\cdot\left(\frac{F[p,q](\frac{i}{\mu_{0}})w_{2}[p,q](\frac{i}{\mu_{0}})w_{3}[p,q](\frac{i}{\mu_{0}})}{w_{1}[p,q](\frac{i}{\mu_{0}})}\right)^{-\frac{1}{2}}\gamma^{0}\gamma^{1}\gamma^{0}\partial_{\psi}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
\Phi_2 (z_1 , z_2)=\langle (\tilde{m})^{\infty}|u_{kj}(z_1)u_{il}(z_2)| (m)^0\rangle = (z_1 z_2)^{\Delta}(T_2 \chi_1^+ (z)+T_1 \chi_2^+ (z))
H=\begin{pmatrix}y_{1,1}y_{1,2}y_{1,3}y_{2,1}y_{2,2}y_{2,3}\sum_{j=1}^N s^j_3(x)\, m^j_1 \sum_{j=1}^N s^j_3(x)\, m^j_2 1 +\sum_{j=1}^N s^j_3(x)\, m^j_3\end{pmatrix}.
S_{a_{1},b_{1}}^{a_{2},b_{2},q_{1}+q_{2}}\left( p_{1}+p_{2},l\right)=\sum_{m=0}^{p_{2}+q_{2}}S_{a_{1},b_{1}}^{a_{2},b_{2},q_{2}}\left(p_{2},m\right) S_{a_{1},a_{1}m+b_{1}}^{a_{2},a_{2}m+b_{2},q_{1}}\left(p_{1},l-m\right) .
I_{(Y_-,D_{Y,-})}(y,z)=ze^{t_-/z}\sum_{d\in\mathbb K_{-}}\tilde y^{d}\left(\prod_{i\in M_0}\frac{\prod_{a\leq 0,\langle a\rangle=\langle D_i\cdot d\rangle}(\bar{D}_i+az)}{\prod_{a \leq D_i\cdot d,\langle a\rangle=\langle D_i\cdot d\rangle}(\bar{D}_i+az)}\right) \left(\prod_{i=1}^k\prod_{0<a\leq v_i\cdot d}(v_i+az)\right)\textbf{1}_{[-d]}I_{D_{Y,-},d},
MO(f)(z) \geq \left(\frac{1}{2}\int_{D(z,r)}\int_{D(z,r)}|f(u)-f(v)|^2 |k_z(u)|^2 |k_z(v)|^2 dV(u)dV(v)\right)^{1/2}&\geq \left(\frac{C^2}{2|D(z,r)|^2}\int_{D(z,r)}\int_{D(z,r)}|f(u)-f(v)|^2 dV(u) dV(v)\right)^{1/2}&= C MO_r(f)(z).
D_{\psi_{i}}= \sum_{N = 0}^{+ \infty}{\left( \frac{\Gamma_{i}e^{-{\frac{\hbar \omega_{i}}{k T}}}}{1 - \left( 1 - \Gamma_{i}\right) e^{-{\frac{\hbar \omega_{i}}{k T}}}}\right)}^{N}\left| \rho_{i}^{N}\right> \left< \rho_{i}^{N}\right| .
\|y\|^{2}_{\pi}=\sum_{n=1}^{N}\bar y_{n}\;\pi_{n}\; y_{n}=\sum_{n=1}^{N}\frac{\bar y_{n}}{\varphi_{n}}\;y_{n}\pi_{n}\varphi_{n}& =\sum_{n=1}^{N}\frac{\bar y_{n}}{\varphi_{n}}\left(\sum_{q=1}^{n}y_{q}\pi_q\varphi_{q}-\sum_{q=1}^{n-1}y_{q}\pi_q\varphi_{q}\right)
\Theta_1^\beta(f_0) := \Big\{f\in \Theta : \big| f(x)-f_0(x) \big| \leq C \Big(\frac{\log n}{n}\Big)^{\frac{\beta}{\beta+1}}+ C \Big(\frac{\log n}{n}f_0(x)\Big)^{\frac{\beta}{2\beta+1}}, \ \forall x\in[0,1]\Big\}
\frac{1}{\lambda_{r}}\sigma_{0}-\frac{2}{\pi}\sigma_{0}+\frac{1}{\pi r}\frac{\displaystyle \Gamma\left(\frac{3}{2}+i m r\right) \Gamma\left(\frac{3}{2}-i m r\right)}{\displaystyle \Gamma\left(1+i m r\right) \Gamma\left(1-i m r\right)}=0\, ,
\int_{0}^{\infty}{\frac{\operatorname{s i n}( b \, u ) d u}{\mathrm{e}^{u}- 1}}= \sum_{n = 1}^{\infty}\int_{0}^{\infty}\operatorname{s i n}( b \, u ) \, \mathrm{e}^{- n \, u}d u
|\int_{B^+_\rho(x_0)}\eta(x_0-y)\overline{u}(t,y)\,dy-\int_{B^+_\rho(x_0)}\eta(x_0-y)\overline{u}(s,y)\,dy| &=|\int_{B^+_\rho(x_0)}\eta(x_0-y)u(t,y)\,dy-\int_{B^+_\rho(x_0)}\eta(x_0-y)u(s,y)\,dy| &=|\int_s^t \int_{B^+_\rho(x_0)}\eta(x_0-y)u_t(\tau,y)\,dy\,d\tau|.
Q_n(X_1\times X_2,f,d,\epsilon) \leq\sum_{(x,y)\in E\times F}(1/\epsilon)^{S_nf(x,y)}&=\sum_{x\in E}(1/\epsilon)^{S_nf_1(x)}\cdot \sum_{y\in F}(1/\epsilon)^{S_nf_2(y)},
\int_{e}^{w}\int_{t}^{t+\Delta t}(\rho c\frac{\partial T}{\partial t}dt)dV=\int_{t}^{t+\Delta t}[(kA\frac{\partial T}{\partial x})_{e}-(kA\frac{\partial T}{\partial x})_{w}]dt+\int_{t}^{t+\Delta t}\overline{S}\Delta Vdt
\sum_{k=1}^{+\infty}\dfrac{\cos (kx)}{k^{2}+\alpha^{2}}=\dfrac{\pi}{2\alpha}\dfrac{\cosh(\alpha(\pi -x))}{\sinh(\alpha \pi)}-\dfrac{1}{2\alpha^{2}}\ \ \ \ \ \mathrm{for}\ \ 0\leq x\leq 2\pi
J V_g(v,\lambda) = J (U_g v, U^*_{\tau(g)}\lambda) = (\Phi^{-1}U^*_{\tau(g)}\lambda, \Phi U_{r^2}U_g v) = (U_{\tau(g)}\Phi^{-1}\lambda, \Phi U_{\tau^2(g)}U_{r^2}v) = (U_{\tau(g)}\Phi^{-1}\lambda, U^*_{\tau^2(g)}\Phi U_{r^2}v) = V_{\tau(g)}(\Phi^{-1}\lambda,\Phi U_{r^2}v) = V_{\tau(g)}J(v,\lambda).
\theta = \left( \begin{matrix}{{\theta^{\alpha}}}\\{{\theta^{\alpha^{\prime}}}}\\ \end{matrix}\right) \ , \qquad \kappa = \left( \begin{matrix}{{\kappa^{\alpha}}}\\{{\kappa^{\alpha^{\prime}}}}\\ \end{matrix}\right) \ , \qquad \epsilon = \left( \begin{matrix}{{\epsilon^{\alpha}}}\\{{\epsilon^{\alpha^{\prime}}}}\\ \end{matrix}\right)
p_{\mathrm{CW}}(\lambda)=-\frac{\lambda^{s}}{n\Delta(\{\sigma\})\prod_{j=1}^n\sigma_j^{s+1}\Gamma(j+s)}&\times\begin{vmatrix}0 [\lambda^{k-1}]_{k=1,...,n}[e^{-\sigma_j^{-1}\lambda}]_{j=1,...,n}[\Gamma(k+s)\sigma_j^{k+s}]_{\substack{j=1,...,n k=1,...,n}}\end{vmatrix},
\sum_{n= X_1}^{X_2}a_n b_n = \sum_{n=X_1}^{X_2}a_n ( B(n) + R(n) - B(n-1) - R(n-1)) = \sum_{n = X_1}^{X_2}a_n( B(n) - B(n-1) ) + \sum_{n=X_1}^{X_2}a_n ( R(n)- R(n-1) ).
\Psi_1'(0)= 2\cdot\frac{d}{dx}\bigg({}_2F_1\bigg[\begin{matrix}-a -a-x &1\end{matrix}\bigg|\,z\bigg]_{p-1}\bigg)\bigg|_{x=0}= 2\cdot\frac{d}{dx}\bigg((1-z)^a{}_2F_1\bigg[\begin{matrix}-a 1+a+x &1\end{matrix}\bigg|\,\frac z{z-1}\bigg]_{p-1}\bigg)\bigg|_{x=0}
e_k (id:\ell^n_p(\mathbb{C})\rightarrow \ell^n_q(\mathbb{C}))\sim \begin{cases}1 1\leq k\leq \log_2(2n), \displaystyle \biggl(\frac{\log_2(1+2n/k)}{k}\biggr)^{\frac{1}{p}-\frac{1}{q}}\log_2(2n)\leq k\leq 2n, begin{align*}10pt]\displaystyle 2^{-\frac{k-1}{2n}}n^{\frac{1}{q}-\frac{1}{p}}2n\leq k.\end{cases}
\frac{\partial J_{MCCC}}{\partial w^{*}}= \frac{\partial E_{DY}[G^{C}_{\sigma\,\sqrt{2}}(e)]}{\partial w^{*}}= E_{DY}\left [ G^{C}_{\sigma\,\sqrt{2}}(e)\frac{\partial (ee^*)}{\partial w^{*}}\right ] = \textbf{0}
\sum_{n\geq0}U_{2k,2a-1}(n)q^n= \frac{(-q^2;q)_\infty(q^{2a},q^{4k-2a},q^{4k};q^{4k})_\infty}{(q^2;q^2)_\infty}&+\frac{xq(-q^2;q)_\infty(q^{2a-2},q^{4k-2a+2},q^{4k};q^{4k})_\infty}{(q^2;q^2)_\infty}
M_{1}(\mathcal{NC}(G)) =(q-1)(q^{3}-q-1)((q-1)(q^{3}-q-1)-1)^{2}~~~~-4((q-1)(q^{3}-q-1)-1)\dfrac{q(q-1)}{2}( q^{4}- 2q^{3}- q^{2}+ 2q + 1 ) ~~~~~+ q(q-1)(q^{6}-4q^{5}+4q^{4}+2q^{3}-4q^{2}+q-1) = (q-1)(q^{11}-2q^{10}-4q^{9}+9q^{8}+5q^{7}-15q^{6}+q^{5}+7q^{4}-2q^{3}+q^{2}-q)
\nu_2(t_5(4n+j))=4\left\lceil\frac{\nu_2(n+1)}{2}\right\rceil-(\nu_2(n+1)\pmod{2}),j\in\{0,1,2,3\}, \nu_2(t_9(8n+j))=5\left\lceil\frac{\nu_2(n+1)}{2}\right\rceil-2(\nu_2(n+1)\pmod{2}),j\in\{0,1,...,7\}.
\sum_{\nu_{1},\nu_{2}}(\begin{matrix}\mu_{1}&\mu_{2}&\gamma\\ \nu_{1}&\nu_{2}&\nu\end{matrix})(\begin{matrix}\mu_{1}&\mu_{2}&\gamma^{\prime}\\ \nu_{1}&\nu_{2}&\nu^{\prime}\end{matrix})=\delta_{\nu\nu^{\prime}}\delta_{\gamma,\gamma^{\prime}}
\Gamma[\phi] = \sum_n \int\frac{d^4p_1}{(2\pi)^4}\cdots\frac{d^4p_n}{(2\pi)^4}(2\pi)^4\delta^4\left(\sum_ip_i\right)\Gamma^{(n)}(\{p_i\})\hat\phi(p_1)\cdots\hat\phi(p_n)
\phi =\left( H\otimes \Delta \right) \left( \alpha^{-1}\right) \cdot \left(1_{H}\otimes \alpha^{-1}\right) \cdot \left( \alpha \otimes 1_{H}\right)\cdot \left( \Delta \otimes H\right) \left( \alpha \right) , \left( \varepsilon_{H}\otimes H\right) \left( \alpha \right) =\lambda,\left( H\otimes \varepsilon_{H}\right) \left( \alpha \right) =\rho .
iv_3 = i \frac 12 a_1 2! 1!\left( X_1+X_2 + X_3 \right) + i \frac 12 1! 2! \left( X_{1+2}+ X_{1+3}+ X_{2+3}\right) &= 2ia_1 \left( X_1 + X_2 + X_3 \right) .
\phi_{- 2 , 1}( \tau , z ) = - \frac{2 E_{6}( \tau ) \theta_{0 , 1}( \tau , z )}{\Delta ( \tau )}= - \frac{2}{q}+ 9 6 0 - 2 \sum_{( \gamma , \gamma ) = 2}\zeta^{\gamma}+ O ( q ) \, ,
\sigma_{n}(C_{n})\,\,\mbox{{\rm cov}}(S_{n},U_{n})=(1-\alpha_{n})^{1/2}\int_{C_{n}}{\bf E\,}\left[ \left| T_{\eta}(x)\right|\,U_{n}\,(1-\alpha_{n})^{-1/2}\right] \sqrt{k_{n}(x)}\,dx
\lambda_{M/K}^l(x) =\min\{\phi_{L/K}^{j,m}(\phi_{M/L}^k(x)):(j,k)\in \Omega_l\}&=\min\{\tilde{\phi}_{L/K}^{j,m}(\tilde{\phi}_{M/L}^k(x)):(j,k)\in \Omega_{l_0}0\le l_0\le l\}.
P_1 \le \sum_{k \ge 1}\Big| \int \gamma_k(y) S(x,y)\,d(\mu - c_{Q^{(k+s_0)}}\mathcal{H}^n_{\mid L_{Q^{(k+s_0)}}})(y)\Big | &+ \sum_{k \ge 1}\Big| \int \gamma_k(y) S(x,y)\,d(c_{Q^{(k+s_0)}}\mathcal{H}^n_{\mid L_{Q^{(k+s_0)}}}- c_Q \mathcal{H}^n_{\mid L_Q})(y)\Big | = J_1 + J_2
h_1(t) = \begin{cases}O\left(\frac{Nd\gamma^2_t}{\sigma^2_t}+ Nd\gamma_t\sigma_t^2\right), \mbox{if $j=1$}, &O({N^2d^2\gamma^2_t}+ Nd\gamma_t\sigma_t^2), \mbox{if $j=2$}.\end{cases}
y^{2}\partial_{y}y^{- 2}\partial_{y}H +{\frac{1}{y^{2}}}\left( f^{-{\frac{3}{2}}}( \mu ) \partial_{\mu}f^{{\frac{3}{2}}}( \mu ) \partial_{\mu}- m^{2}f \right) H = - \omega^{2}H \ .
tL_x - s \ell_x = t \sum_{j=1}^{s-t}L_{k_j}- s \sum_{j=1}^{s-t}\ell_{k_j}= \sum_{j=1}^{s-t}(t L_{k_j}- s \ell_{k_j}) \equiv \sum_{j=1}^{s-t}r_j \equiv (s-t)r \equiv 0 \! \! \! \pmod{s-t}.
[a-yb,x] h + [y, D(x)] h = [ah,x] - [ybh,x] + [y, D(x)] h &= [D(\hat y), x] -[yD(h),x] + [\hat y,D(x)] - y[h, D(x)] &= [D(\hat y), x] + [\hat y, D(x)] - [y,x]D(h) - y[D(h),x] - y[h,D(x)] = D([\hat y,x]) - D(h) - yD([h,x]) = 0.
I_{\eta \, par}(0,n;r) = 2\pi i (-1)^{k/2}\sum_{j=0}^{k/2-1}\frac{(2\pi i n)^{j}}{j!}\binom{k-2-j}{k/2-1}\left[\frac{e^{-2\pi i n r}}{(2r)^{k-1-j}}+ \frac{e^{2\pi i n r}}{(-2r)^{k-1-j}}\right].
D^{ac}= \sum_{Y\in \mathcal{I}_X^G}\left(1-\sum_{\alpha\in \Phi_{Q^u}\cup \Phi_s^+}\alpha\circ\mathcal{P}(\mu_Y)\right)Y & +\sum_{D\in\mathcal{D}}\left(m_D-\sum_{\alpha\in \Phi_{Q^u}\cup \Phi_s^+}\alpha\circ\mathcal{P}(\rho(D))\right)\overline{D}.
h_{0}(z) =z\exp\left(\sqrt{\frac{69}{68}}\int_{0}^{z}\frac{x^2 +\sqrt{1+x^4}-1}{x}dx\right) &=z +\frac{\sqrt{69}}{4\sqrt{17}}z^3 +\frac{1}{4}\left(\frac{69}{136}+\frac{\sqrt{69}}{4\sqrt{17}}\right)z^5 +\cdots,.
h_v (z-b,D(b,z)) = h_{E_1 ,v}(f_1 (z)-\infty )-h_{E_1 ,v}(f_1 (b)-\infty )-h_{E_2 ,v}(f_2 (z)-\infty ) &\quad+ h_{E_2 ,v}(f_2 (b)-\infty ) +\chi (y(f_1 (z))y(f_2 (b))/y(f_1 (b))y(f_2 (z))).
\sum_{\Gamma t(l)<t}\nabla f(y_{\Gamma t(l)})\Gamma=\frac{\Gamma n_t}{t}\left[\sum_{\Gamma t(l)<t}\nabla f(y_{\Gamma t(l)})\frac{t}{n_t}\right]\simeq \frac{\Gamma n_t}{t}\int_{0}^{t}\nabla f(y_t)dt\simeq \frac{1}{\Delta t}\int_{0}^{t}\nabla f(y_t)dt.
\begin{cases}u_t - A_1u =h(t,x) \chi_{\omega}(x), \ (t,x) \in (0,T) \times (x_0,1), u(t,1)=(au_x)(t,x_0)=0, \t \in (0,T), u(0,x)=u_0(x)_{|_{(x_0,1]}},\end{cases}
\mu_{1 2}= -{\frac{1}{6 \pi^{2}}}{\frac{\pi^{D / 2}}{2^{D}}}\left[ -{\frac{( D - 3 ) ( D - 1 )}{D}}\Gamma \left( 2 -{\frac{D}{2}}\right) \zeta ( 2 - D ) +{\frac{\pi^{3 / 2 - D}}{2}}( D + 1 ) \Gamma \left({\frac{D + 1}{2}}\right) \zeta ( D + 1 ) \right] \,{\frac{\hbar L^{D - 1}}{a^{D}}}.
\left\lVert \pi\left(\rho^m\right)-\pi\left(\rho^n\right)\right\rVert_{{\mathcal{B}}(E)}= \left\lVert \left(A_\pi^\mu\right)^m \left(I_E-\left(A_\pi^\mu\right)^{n-m}\right)\right\rVert &\le \left\lVert A_\pi^\mu\vert_{E_\pi}\right\rVert^m \left\lVert I_E-\left(A_\pi^\mu\right)^{n-m}\right\rVert &\le 2 \left\lVert A_\pi^\mu\vert_{E_\pi}\right\rVert^m
\pi^{G_{2,s}}_{2,\ell}|_K = \sum_{\substack{d'-\ell-4 \ge e'\ge 0 begin{align*}.2ex] e'\equiv d'-\ell-4\pmod{2}}}\sum_{k'=0}^{e'}\gamma^{}_{e'}\otimes \gamma_{2d'+e'-2k'}^{}
F(\beta,\epsilon) =\lim_{N\rightarrow \infty}\limits \frac{1}{N}\mathbb{E}\log \mathcal{Z}_N=\lim_{N\rightarrow \infty}\limits \frac{1}{\gamma N}\mathbb{E}\log \mathcal{Z}_N^{\gamma}\leq \lim_{N\rightarrow \infty}\limits \frac{1}{\gamma N}\log \mathbb{E}\mathcal{Z}_N^{\gamma}
F_{1}= \int_{- 1}^{1}d y y e^{1 /{\tilde{g}}( y - 1 )}\left[ 1 - \frac{1}{6}( y - 1 ) - \frac{1}{6 \tilde{g}}( y - 1 )^{2}\right] = \tilde{g}+ \frac{1}{6}{\tilde{g}}^{2}+ . . .
\ln \frac{w_{2}}{w_{1}}( e^{\frac{2}{3}t_{1}+ \frac{1}{3}t_{2}}) + 2 \pi i = r \left[ \ln \frac{w_{3}}{w_{2}}( e^{\frac{2}{3}t_{1}+ \frac{1}{3}t_{2}}) - 2 \pi i \right] , ~ r \in{\bf R}.