harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E_{a,b}^{(M)}E_{d,c}^{(N)}=\begin{cases}(-1)^{\bar E_{a,b}\bar E_{d,c}}E_{d,c}E_{a,b}-(q_b-q_b^{-1})^{-1}K_{c,b}E_{a,c}E_{d,b},M=N=1; \displaystyle \sum_{t=0}^{\min(M,N)}(-1)^tq_{b}^{\frac{-t(2N-3t-1)}{2}}(q_b-q_b^{-1})^{t}[t]_q!E_{d,c}^{(N-t)}E_{d,b}^{(t)}K_{c,b}^{t}E_{a,b}^{(M-t)}E_{a,c}^{(t)},.\end{cases}
M_{\kappa,t}(z) = (1-z) \frac{\kappa}{2i\pi}\int_{\gamma_{\kappa}}\frac{[(K_{2t}(y))^2-1]}{[t(K_{2t}(y))^2 + (2-t)][wK_{2t}(y) - \kappa]}\frac{dw}{wR(z,w)}.
a_h^+(r_h^+ u - r_h u, e_h + e_h^+) = a_h^+(r_h^+ u - u, e_h + e_h^+) + a_h(u - r_h u, e_h + e_h^+) &\;\;\;+ a_h^+(u - r_h u, e_h + e_h^+) - a_h(u - r_h u, e_h + e_h^+).
\Vert ( B_\varepsilon (\vartheta)-\widehat{\zeta}^*Q_0^\varepsilon )^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}\leqslant \Vert f\Vert^2_{L_\infty}\Vert (\widetilde{B}_\varepsilon (\vartheta)-\widehat{\zeta}^* I)^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}&\leqslant \Vert f\Vert^2_{L_\infty}c(\phi).
G(x)-v(x,c) = \frac{1}{r}E_x\left[\left(g(X_{T(r)}-cr)\right)\mathbb{I}_{[\overline{X}_{T(r)}\geq x^{\ast}(c)]}\right] &= E_x\left[ \int_0^{\infty}\sup_{0\leq u \leq t}(\kappa(X_u) -rc)^+ e^{-rt}\,dt \right].
\gamma_1\colon =-\limsup_{n\to\infty}\frac{\max_{Q\in\mathcal{Q}_n}\log_{b^{-1}}P_n(Q)}{n}, \gamma_2\colon =-\limsup_{n\to\infty}\frac{\min_{Q\in\mathcal{Q}_n}\log_{b^{-1}}P_n(Q)}{n}.
I_1 = (1 + \alpha) \int_{L \ge |y| > R}\frac{U (x, t) - U(x-y)}{|y|^{2+\alpha}}\frac{y}{|y|}dy &\quad+ \frac{U (x, t) -U(x-y)}{|y|^{1+\alpha}}\bigg|^L_R + \frac{U (x, t) -U(x-y)}{|y|^{1+\alpha}}\bigg|^{-R}_{-L}.
v_{2}(t,r) = \frac{-c_{b}r}{2\pi}\int_{0}^{\pi}\sin^{2}(\theta) d\theta \int_{0}^{\infty}d\xi \left(\chi_{\leq \frac{1}{4}}(\xi) \left(\frac{b-1}{\xi \log^{b}(\frac{1}{\xi})}+\frac{b(b-1)}{\xi \log^{b+1}(\frac{1}{\xi})}\right)+\psi(\xi)\right)\left(\frac{\sin(\xi t_{+})}{t_{+}^{2}}+ \frac{\sin(\xi t_{-})}{t_{-}^{2}}\right)
K_0={x_0^{d-4}\over (x_0^2 +{\bf x}^2)^{d-2}}dx^i dx^j -{x_0^{d-5}x^i \over (x_0^2 +{\bf x}^2)^{d-2}}dx^0 dx^j +{x_0^{d-5}x^j \over (x_0^2 +{\bf x}^2)^{d-2}}dx^0 dx^i,
\Delta^{I,J\cup\{k\}}\Delta^{I\cup\{i\},J\cup\{j,l\}}=\Delta^{I,J\cup\{j\}}\Delta^{I,J\cup\{l\}}+\Delta^{I\cup\{i\},J\cup\{j,k\}}\Delta^{I\cup \{i\}, J\cup\{k,l\}}.
\int_X\mathrm{Tr}\,e^{-t\widetilde\Box_{b,m}}(x,x)dv_X(x)\sim (\sum\limits^{p_1}_{s=1}e^{\frac{i2\pi(s-1)}{p_1}m})\big(t^{-n}c_{n}+t^{-n+1}c_{n-1}+t^{-n+2}c_{n-2}+\cdots\big) &\,\, +t^{-n+\frac{e}{2}}\tilde c_{n-\frac{e}{2}}+O(t^{-n+\frac{e}{2}+1}).
(1+\epsilon(0)) \Phi(-K)- (1+\epsilon(0)/2)\mathbf{u}^* \succeq (1+\epsilon(0))\big[1-C_2K^{-\omega_*}\big]\mathbf{u}^*- (1+\epsilon(0)/2)\mathbf{u}^* = \big[K_1\theta^{-\beta}/2-C_2K^{-\omega_*}(1+K_1\theta^{-\beta})\big]\mathbf{u}^* \succeq{\bf 0}
|\psi(x, t)|\lesssim \frac{t_0^{-\varepsilon}}{R^{a-2s}}\sum_{j=1}^k\frac{\mu_0^{\frac{n-2s}{2}+\sigma}(t)}{1+|y_j|^{a-2s}}+ e^{-\delta(t-t_0)}\|\psi_0\|_{L^{\infty}(\mathbb{R}^n)},
\Bigl(\sum\alpha_k\Bigr)\circ\beta=\sum_{n=0}^\infty\Bigl(\sum_{k=1}^\infty a_{k,n}\Bigr)\beta^n=\sum_{k=1}^\infty\Bigl(\sum_{n=0}^\infty a_{k,n}\beta^n\Bigr)=\sum\alpha_k(\beta).
f_j = \begin{cases}2 \csc\left( \dfrac{\pi}{2j+4}\right) - 2\cot\left( \dfrac{\pi}{2j+2}\right), 2 \nmid j , 2 \cot\left( \dfrac{\pi}{2j+4}\right) - 2\csc\left( \dfrac{\pi}{2j+2}\right), 2 \mid j . \end{cases}
\varphi_t(x_\varepsilon,t_\varepsilon)\le J[u_\varepsilon^*,p_\varepsilon](x_\varepsilon,t_\varepsilon)+K_{(0,x_\varepsilon-\delta)}[\varphi,p_\varepsilon](x_\varepsilon,t_\varepsilon) &+K_{(x_\varepsilon-\delta,x_\varepsilon)}[u_\varepsilon^*,p_\varepsilon](x_\varepsilon,t_\varepsilon)+f(x_\varepsilon,t_\varepsilon),
\zeta_T(\nu|\beta)={\varrho^{-2\nu}\over \pi i}\int_{\mu}^{\infty}d\omega \int_{C_+}dz\ln\left(1-e^{i\beta z}\right)\left[{2\nu z \breve{\Phi}(\omega;z) \over(z^2+\omega^2)^{\nu+1}}-{\partial_z\breve{\Phi}(\omega;z) \over(z^2+\omega^2)^{\nu}}\right].
R_{\alpha,0}=\rho,~R_{\alpha, e_1}=R_{\alpha,2e_1}=0, R_{\alpha, me_1}-f_{(m-2)e_1}\theta_{11}=-\frac{1}{m}\theta_{11}(R_{\alpha, (m-2)e_1}-f_{(m-4)e_1}\theta_{11}).
\int_{\mathcal{Y}^{k+1}}g(y_{[n,n+k]})P^{\mu}(dy_{[n,n+k]}|Y_{[0,n-1]}) &=\int_{\mathcal{X}}\int_{\mathcal{Y}^{k+1}}g(y_{[n,n+k]})P(dy_{[n,n+k]}|X_{n}=x_{n})\pi_{n-}^{\mu}(dx_{n})
r(\mathbf{x},\mathbf{y}):=\begin{bmatrix}\mathbf{x}\mathbf{y}\end{bmatrix}- \Pi_{D}(\begin{bmatrix}\mathbf{x}\mathbf{y}\end{bmatrix}- \begin{bmatrix}H\mathbf{x}+\mathbf{g}-A^T\mathbf{y}A\mathbf{x}-\mathbf{b}\end{bmatrix}),
N_r(s, t) = \sum_{\substack{a,b r=s+t-a+b}}\left ( \binom{a}{2}+\binom{b}{2}\right )\binom{s}{a}\binom{r-s}{b}=\sum_{\substack{a,b r=s+t-a+b}}\binom{a}{2}\binom{s}{a}\binom{r-s}{b}+ \sum_{\substack{a,b r=s+t-a+b}}\binom{b}{2}\binom{s}{a}\binom{r-s}{b}.
p(x) \leq r(x)\leq p^*(x), \tilde{r}(x) = r(x)-\tilde{\epsilon}\textrm{for some}0 < \tilde{\epsilon}< p^--1, \tilde{\sigma}(x) = r(x)\frac{p(x)-1}{p(x)}, \tilde{\tau}(x) = p(x)\left( 1 - \frac{1}{\tilde{r}(x)}\right), \tilde{\delta}(x) = \tilde{r}(x) - 1.
\left( 1 - \frac{\mu+\tau}{\mu_{k_0+1}(\lambda) + \tau}\right) \sum_{k=1}^{k_0}\tilde{c}_k^2 \; - \; \left( 1 - \frac{\mu+\tau}{\mu_1(\lambda) + \tau}\right) \sum_{k=1}^{k_0}\tilde{c}_k^2 \; + \; \ell(\hat{u}).
\varphi_\lambda'/\varphi_\lambda(z(\lambda,\hat{X})) = \eta_1/\omega_1\int_1^{\hat{X}}\frac{dX}{2\sqrt{X(X-1)(X-\lambda)}}+\int_{1}^{\hat{X}}\frac{(X-\frac13(\lambda+1))dX}{2\sqrt{X(X-1)(X-\lambda)}}+\pi i/\omega_1.
\mathcal{I}_1^*(h,k) = J_{11}+ J_{21}+J_{23}+ J_{31}+J_{33}+ O\big( X^{-\frac{1}{2}+\varepsilon}Q^{\frac{5}{2}}(hk)^{\varepsilon}\big) + O\big( X^{\varepsilon}Q^{\frac{3}{2}+\varepsilon}(hk)^{\varepsilon}\big),
\alpha_\ell:=\frac{1}{2^{4(\ell-1)}}\sum_{k=1}^{\ell}\frac{[(2 \ell-2k)!]^3}{[( \ell-k)!]^4}\frac{[(2k-2)!]^3}{[(k-1)!]^4}\in\mathbb Z,\forall \ell\in\mathbb Z_{>0}.
\ell_2\left(\mu\otimes_\mathcal O \dfrac{\partial}{\partial x},\mu\otimes_\mathcal O \dfrac{\partial}{\partial y}\right):=\frac{\partial\varphi}{\partial x}\,\mu\otimes_\mathcal O \dfrac{\partial}{\partial y}-\frac{\partial\varphi}{\partial y}\,\mu\otimes_\mathcal O \dfrac{\partial}{\partial x}
\Pi_{1,a}:=S_1 E_{a,a}(S_1)^{-1}, \Pi_{2,a}:=\tilde S_2 E_{a,a}(\tilde S_2)^{-1}, L_{1}:=S_1\Lambda(S_1)^{-1}, L_{2}:=\tilde S_2\Lambda^{\top}(\tilde S_2)^{-1}, B_{1,j,a}:=\big(\Pi_{1,a}(L_{1})^j\big)_+, B_{2,j,a}:=\big(\Pi_{2,a}(L_{2})^j\big)_-.
S=\int d^{3}x \left[ -\left(\frac{\partial\phi^{A}}{\partial x^{i}}G_{AB}[\phi] \frac{\partial\phi^{B}}{\partial x^{j}}\right)+\frac{i}{2\kappa}\left| \frac{\partial\phi^{A}}{\partial x^{i}}\right| \sqrt{G[\phi]}R[\phi]\right],
\sum_{n=1}^N\lambda_n\left(\frac{A_n}{\Lambda_n}\right)^{-p}\leq \left(\frac{p+1}{p}\right)^q\sum_{n=1}^N\lambda_n\left(\frac{A_n}{\Lambda_n}\right)^{-p\mathrlap{+q}}a_n^{-q}- \frac{q}{p+1}\Lambda_N\left(\frac{A_N}{\Lambda_N}\right)^{-p}.
V = - \frac{1}{2}\phi^{2}+ g \kappa \phi^{3}+ \frac{1}{3 g^{2}\kappa^{2}\gamma^{2}}\left( 1 - 3 g^{2}\kappa^{2}\beta \gamma \phi^{2}- ( 1 - 2 g^{2}\kappa^{2}\beta \gamma \phi^{2})^{3 / 2}\right)
\frac{\partial\sigma_{t+1}}{\partial m}(z) =s_t(z)[1-s_t(z)]\frac{\partial\pi_{t}}{\partial m}(z)+c\left[C_{t}\left(z\right)\right]( 1-c[C_{t}(z)]) \left(1-\frac{\partial\pi_{t}}{\partial m}(z)\right) + (s_t(z)-c[C_{t}(z)])^{2}\cdot\frac{\partial\sigma_{t}}{\partial m}(z).
\inf_{(\rho,m)}\sup_{S\in H^1}\mathcal L(\rho,m,S) = \inf_{(\rho,m)}\{\mathcal A(\rho,m)+\mathbb I_{C_F(\rho^a,\rho^b)}(\rho,m)\}= \inf_{(\rho,m)\in C_F(\rho^0,\rho^1)}\{\mathcal A(\rho,m)\}.
\alpha^{\prime}m_{\mathrm{v e c}}^{2}\equiv - \alpha^{\prime}\sum_{\sigma , \rho = 0}^{p^{\prime}}G^{\sigma \rho}k_{\sigma}k_{\rho}= 0 ~ , \qquad \sum_{\sigma , \rho = 0}^{p^{\prime}}G^{\sigma \rho}k_{\sigma}\zeta_{\rho}( k ) = 0 ~ .
0 = \lambda_{C\,D}+ \lambda_{A\, D}+ \lambda_{A\,C}= \lambda_{C\,D}+ \lambda_{B\,D}+ \lambda_{B\,C}= \lambda_{A\,B}+ \lambda_{B\, C}+ \lambda_{A\,C}= \lambda_{A\,B}+\lambda_{B\,D}+\lambda_{A\,D}= \lambda_{A\,A}+\lambda_{A\,C}+\lambda_{A\,C}= \lambda_{C\,C}+ \lambda_{A\,C}+ \lambda_{A\,C}.
\begin{aligned}\partial_s g_{R,X_\ast,V_\ast}+ v\cdot \nabla_x g_{R,X_\ast,V_\ast}- \nabla (\phi*_x \rho[g_{R,X_\ast,V_\ast}]) \cdot \nabla_v \mu = -e_0\nabla \Phi((x-X_\ast+(R-s)V_\ast) \cdot \nabla_v \mu, g_{R,X_\ast,V_\ast}(0,\cdot) = 0.\end{aligned}
\int_{\Sigma}\int_{\Sigma}\frac{(w(x) - w(y) ) w(x) \psi(x)^2}{|x-y|^{N+s}}d x d y=\int_{\Sigma}A(x) w(x)^2 \psi(x)^2 dx=\int_{\Sigma}A(x) \phi(x)^2 dx .
A_{p-1}^{1\rightarrow m}(x) =1+\sum_{k>0}[D(1,p-1+k):D(m,p-1)]x^k &=1+\sum_{k>0}[D(1,p+k):D(m,p)]x^k-(1-\delta_{m,2})\sum_{k>0}[D(1,p-1+k):D(m,p+1)]x^k &= A_p^{1\rightarrow m}(x)-(1-\delta_{m,2})x^2A_{p+1}^{1\rightarrow m}(x).
H_{k,\beta}'= \frac{1}{sP_{k-1}L_k^\beta}\bigg(- \Big(\sum_{j=1}^{k-1}\frac1{P_j}+\frac{\beta}{P_k}\Big)\Big(1-\sum_{j=1}^{k-1}\frac1{P_j}-\frac{\beta}{P_k}\Big) + \sum_{j=1}^{k-1}\frac1{P_j}\Big(\sum_{i=1}^j\frac1{P_i}+\frac{\beta}{P_k}\Big) \bigg).
{{P}_{{\rm{DB}}}}= \left\{\frac{{E{\sum\limits_{m = 1}^{{M_{\rm{R}}}}{\sum\limits_{k = 1}^K{\sum\limits_{n = 1}^K{{\left|{\left[{{{\bf{\Xi}}_{{\rm{DB}},m}}}\right]_{k,k}^*{{\left[{{{\bf{\Xi}}_{{\rm{DB}},m}}}\right]}_{n,n}}}\right|}}}}}}}{{K{\sigma^2}}}<\gamma_{\rm th}\right\}.
F_{\tiny \mathrm{d i v}}^{C}[ g , \beta , \mu ] = -{\frac{\eta}{4 8 \kappa \beta^{2}}}\int_{\Sigma}\left[ b f_{1}+ a \left( p_{1}{\frac{4 \pi^{2}}{\kappa^{2}\beta^{2}}}{\cal P}+ p_{2}R + p_{3}{\cal Q}\right) \right] .
dc^{T}-cd^{T}=[\begin{matrix}0&c_{2}d_{1}-c_{1}d_{2}&c_{3}d_{1}-c_{1}d_{3}\\c_{1}d_{2}-c_{2}d_{1}&0&c_{3}d_{2}-c_{2}d_{3}\\c_{1}d_{3}-c_{3}d_{1}&c_{2}d_{3}-c_{3}d_{2}&0\end{matrix}]
\langle 0 |a_{p}(6{\textstyle \sum{}_{4}}){1 \over E-g\sum_{2}}(6{\textstyle \sum_{4}}){1 \over E-g\sum_{2}}(6{\textstyle \sum_{4}})a^{\dagger}_{p}|0\rangle ={54 \over pg^{2}}I\hbar^{5},
P\left( \frac{1}{n}\sum_{i=0}^{n-1}E[X_{i+1}-X_i \mid \mathcal{F}^X_i ]^2 \geq \epsilon \right) &\leq P( I_n^X \leq - n^{3/4}) + P( S_n^X \geq n^{3/4}) + P\left( \sum_{|x| \leq n^{3/4}}\bar\delta_{x}\geq \epsilon n \right) &\leq 2P( S_n^X \geq n^{3/4}) + \frac{E[\bar\delta_{0}] + 2n^{3/4}E[\bar\delta_{1}]}{\epsilon n},
|-\int_{t}^{t+\frac{1}{2}}\frac{ds}{(s-t)}\int_{0}^{s-t}\rho d\rho \frac{\lambda''(s)}{r}\partial_{r}\left(\frac{-1-\rho^{2}+r^{2}}{\sqrt{(-1-\rho^{2}+r^{2})^{2}+4r^{2}}}\right)| &\leq \frac{C}{t^{2}\log^{b+1}(t)}
\log T(f)\ =\ \Phi(\log f)+ \sum_{n=1}^\infty \frac{1}{n!}L_n\left(\Phi\left(\frac{f'}{f}\right),\ldots,\Phi\left(\frac{f^{(n)}}{f}\right)\right)\epsilon^n &=\ \Phi(\log f)+ \sum_{n=1}^\infty \frac{1}{n!}\Phi\left(L_n\left(\frac{f'}{f},\ldots,\frac{f^{(n)}}{f}\right)\right)\epsilon^n.
T_2r_{12}(a,b) =T_2(\lambda_a(b),\lambda^{-1}_{\lambda_a(b)}( (a\circ b)^{-1}a(a\circ b)))=(\lambda_a(b),(a\circ b)^{-1}a(a\circ b)), =(\lambda_a(b),\lambda_a(b)^{-1}a\lambda_a(b))=s_{12}(a,\lambda_a(b))=s_{12}T_2(a,b)
\P\left(V_n\right. \left.=2r_{1,n}, \, \tau_n=2r_n, \, L_n=k_n,\, X^{\left(n\right)}_1=j_n|X_{0}^{\left(n\right)}=0\right) &=\left(\sum\limits_{i=0}^{k_n}\P(A_{r_n, r_{1,n}, k_n, i})\right)\P\left(B_{n, r_n, j_n}|X_{t_{2r_n}}^{\left(n\right)}=0\right)
S = \int dx^{10}\sqrt{g}\bigg{\{}e^{-2 \phi}\bigg{(}R -4 (\partial \phi)^2 \bigg{)}+{1\over 2}(\partial C_{[0]})^2 +{1\over 4 \cdot 3!}(F_{[3]})^2 +{1\over 4 \cdot 5!}(F_{[5]})^2 +{1\over 4 \cdot 7!}(F_{[7]})^2 \bigg{\}}\, .
\left|\|Ax-b\|_2-\|Ay-b\|_2\right|\le\|Ax-Ay\|_2=\left\|\sum_{i\in[n]}(x_i-y_i)a_i\right\|_2\le\sum_{i\in[n]}|x_i-y_i|\left\|a_i\right\|_2\le\max_{j\in[n]}\left\|a_j\right\|_2\|x-y\|_1.
K_{N}( \lambda , \lambda^{\prime}) ~ = ~ \sqrt{\lambda \lambda^{\prime}}\prod_{f = 1}^{\alpha}\sqrt{\lambda^{2}+ m_{f}^{2}}\sqrt{{\lambda^{\prime}}^{2}+ m_{f}^{2}}e^{- \frac{N}{2}( V ( \lambda^{2}) + V ({\lambda^{\prime}}^{2}) )}\frac{1}{N}\sum_{j = 0}^{N - 1}\frac{Q_{j}( \lambda ) Q_{j}( \lambda^{\prime})}{r_{j}}~ ,
- \ln Z_{1}= \int_{0}^{\infty}\frac{d s}{s}\left( \mathrm{T r}e^{s D^{2}}- \frac 1 4 \mathrm{T r}e^{s ( D^{2}{\bf 1}_{1 6}+ i \Gamma . F / 2 )}- \frac 1 2 \mathrm{S t r}e^{s \Delta_{2}}\right) ,
\Delta{{\mathbf{x}}^{T}}\Delta \mathbf{s}=\left( 1-\nu \right)\left( \Delta{{\mathbf{y}}^{T}}\mathbf{Ax}-\Delta{{\mathbf{x}}^{T}}\left({{\mathbf{A}}^{T}}\mathbf{y}+\mathbf{C}\mathbf{x}+\mathbf{s}\right)\right).
\int_{-1}^1\frac{d\left(|\Phi_n|^2\right)}{dx}w_{\alpha,\beta+1}\,dx = \int_{-1}^1 P_n^{(\alpha,\beta-1)}\cdot 2\Re\dfrac{d\Phi_n}{dx}\cdot w_{\alpha,\beta+1}\,dx + 2\Re \mu \int_{-1}^1\left|\dfrac{d\Phi_n}{dx}\right|^2w_{\alpha,\beta+1}\, dx.
\mathcal{L}(U_{\mu}^{-}(x_{\delta})) =(a+(\chi_2\lambda_2V_{2}-\chi_1\lambda_1V_1)(x;u)-(b+\chi_2\mu_2-\chi_1\mu_1)U_{\mu}^-(x_\delta))U_{\mu}^-(x_\delta) & \geq (a -\frac{a\tilde M}{b+\chi_2\mu_2-\chi_1\mu_1-M}-(b+\chi_2\mu_2-\chi_1\mu_1)U_{\mu}^{-}(x_{\delta}))U_{\mu}^{-}(x_{\delta})
cov(X,Y)=\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{1}{2}(x_{i}-x_{j})(y_{i}-y_{j})=\frac{1}{n^{2}}\sum_{i}\sum_{j>i}(x_{i}-x_{j})(y_{i}-y_{j}).
\min\left|\left|I-\mathcal{K}_{i}^{(z)}P_{i}^{(z)}\right|\right|_{F}^{2}= \min\sum_{j=1}^{n}\left|\left|e^{(j)}-\mathcal{K}_{i}^{(z)}p^{(j)}_{i}\right|\right|^{2}_2,
A_{N}= \frac{{\cal N}_{N}}{2 \pi^{3}( 2 \pi^{2})}\int_{0}^{1}d z \frac{z}{\sqrt{z}\sqrt{1 - z}}\int_{0}^{z^{2}}\frac{d y}{y}\left( \frac{- \ln ( y )}{4 \pi^{2}}\right)^{{\textstyle{\frac{k + 1 - D}{2}}}}e^{{\textstyle 1 6 l y}}
\Big(\dot{\Lambda}^{\frac{d}{p}-1}e^{(t - \tau)\Delta}\mathbb{P}\nabla .\big(u(\tau) \otimes v(\tau)\big)\Big)_j =\frac{1}{(t -\tau)^{\frac{\{\frac{d}{p}\}+ d+1}{2}}}\sum_{l, k = 1}^d K_{l, k, j}\Big(\frac{.}{\sqrt{t - \tau}}\Big)*\Big(\dot{\Lambda}^{[\frac{d}{p}]-1}\big(u_l (\tau)v_k(\tau)\big)\Big),
\left. \begin{array}{l l}\mu_E \nu_1- \nu_E \mu_1 - (y_2-y_3) \Lambda = 0 \mu_E\nu_2 - \nu_E\mu_2 - (y_3-y_1) \Lambda =0 \mu_E\nu_3 - \nu_E\mu_3 - (y_3-y_1) \Lambda = 0. \end{array}\right.
\partial_\beta\tilde{V}(\rho,\beta)=\frac{16\left(\left(\frac{(\rho\beta)\sinh(\rho\beta)-2(\cosh(\rho\beta)-1)}{(\rho\beta)^3}\right)\rho^3+\rho\sinh(\rho\beta)\right)}{\left(\left(\frac{\cosh(\rho\beta)-1}{(\rho\beta)^2}\right)\rho^2+\cosh(\rho\beta)+1\right)^3}.
q_{b}( a , r ) = - \frac{1}{2 \pi a^{3}r^{n}S_{D}\sigma^{D - 1}}\sum_{l = 0}^{\infty}D_{l}\int_{m a}^{\infty}d z \, z^{3}\frac{\bar{K}_{\nu_{l}}( z )}{\bar{I}_{\nu_{l}}( z )}
\begin{array}{c}{\Gamma^{M}\Gamma^{N}+ \Gamma^{N}\Gamma^{M}= 2 \eta^{M N}, \qquad \eta^{M N}= ( + , - , - , \dots ) ,}\\{( \Gamma^{M})^{*}= - \Gamma^{M}, \qquad ( \Gamma^{0})^{\mathrm{T}}= - \Gamma^{0}, \qquad ( \Gamma^{i})^{\mathrm{T}}= \Gamma^{i}, \quad i = 1 , \dots , 9 .}\\ \end{array}
R_{-2m}=(-1)^m \frac{\zeta(2m+1) y^{2m}}{2^{2m+1}\pi^{2m}}, R_0=- \frac{1}{2}\zeta(2m+1), R_1=(-1)^{m}\frac{(2\pi)^{2m +2}B_{2m+2}}{2 (2m+2)! y}, \sum_{i=0}^m R_{-(2i+1)}=(-1)^{m+1}\sum_{i=0}^{m}\frac{(-1)^i B_{2i+2}B_{2m-2i}(2\pi)^{2m-2i}y^{2i+1}}{2(2i+2)! (2m-2i)!}.
N_{w,G}^{\chi}=\langle N_{w,G},\chi\rangle=\frac{1}{|G|}\sum_{g\in G}N_{w,G}(g)\overline{\chi}(g)=\frac{1}{|G|}\sum_{(g_{1},\cdots,g_{r})\in G^{r}}\overline{\chi}(w(g_{1},\cdots,g_{r})),
{f_{17}}={f_{18}}- \frac{1}{2}\left({{f_3}-{f_4}}\right) - \frac{1}{2}\left({{f_7}-{f_8}}\right) + \frac{1}{2}\left({{f_9}-{f_{10}}}\right) - \frac{1}{8}\left({2{F_y}-{F_z}}\right) - \frac{1}{6}\rho{u_z}.
\begin{array}{ccl}\vspace{.2cm}\omega^2(x)= -{\rm Tr}\Big[\Big(W'(\hat \Phi)+[\hat X,\hat Y]\Big)\frac1{x-\hat\Phi}\Big] \ , \omega\Big(x+i\frac m2\Big)\omega\Big(x-i\frac m2\Big)={\rm Tr}\left(\hat Y\hat X\frac1{x-\hat\Phi-i\frac m2}-\hat X\hat Y\frac1{x-\hat\Phi+i\frac m2}\right)\ . \end{array}
Z_{n}^{2}(\boldsymbol{\beta}_{0}) =\frac{1}{n}\sum\limits_{i=1}^{n}\rho_{1}\left( \frac{u_{i}}{s_{n}}\right) +\frac{\iota_n}{n}\sum\limits_{j=1}^{s}\frac{\vert \beta_{0,j}\vert^{t}}{\vert \widehat{\beta}_{2,j}\vert^{\varsigma}}.
\begin{cases}dY_t = \left[ \mathrm{Diag}(Y_t) - Y_t Y_t^T \right] \Gamma^T d B_t + D^T Y_t dt + \left[ \mathrm{Diag}(Y_t) - Y_t Y_t^T \right]\left( a - \Sigma Y_t - b(S_t Y_t)\right)dt dS_t = S_t \left( a - b(S_t Y_t)\right)^T Y_t dt + S_t Y_t^T \Gamma^T d B_t,\end{cases}
\kappa^* = \arg \max_{\kappa \in \mathcal{P}(\mathcal{M})}\sum_{u \in \tau(\kappa)}\cfrac{M-W_u^k+1}{1-p_u}&= \arg \max_{C \in \mathbf{C}}\sum_{v_{um}\in C}\cfrac{M-W_u^k+1}{1-p_u},
\lim\limits_{n\to+\infty}\dfrac{n F_n(x)}{(32\ln n)^{\frac{1}{2}}}=1,\ \lim\limits_{n\to+\infty}n F_n(x)=+\infty,\ \lim\limits_{n\to+\infty}n^{\gamma}F_n(x)=0,\ \forall\ \gamma<1.
\overline{D}+ \widehat{(s^{t}{s'}^{(1-t)})}= \overline{D}+ t\widehat{(s)}+ (1-t) \widehat{(s')}& =t(\overline{D}+ \widehat{(s)}) + (1-t)(\overline{D}+ \widehat{(s')}) \geqslant 0,
\mbox{tr}_{\mbox{\tiny ZERO}\atop\mbox{\tiny MODES}}= 2 V_{p+1}\int \frac{d^{p+1}k}{\left( 2\pi\right)^{p+1}}\, e^{-2\pi t\alpha^\prime k^2 -\frac{t y^2}{2\pi \alpha^\prime}}= 2V_{p+1}\,\left( 8\pi^2 \alpha^\prime t\right)^{-\frac{p+1}{2}}\; e^{-\frac{t y^2}{2\pi\alpha^\prime}},
( \xi , \eta ) = \frac{c}{2 \pi}\int ( \xi^{a}\nabla_{a}\nabla_{b}\nabla_{c}\eta^{c}) d x^{b}+ \frac{h}{2 \pi}\int ( \xi^{a}D_{a b}\nabla_{c}\eta^{c}) d x^{b}- ( \xi \leftrightarrow \eta ) .
\sigma(\{\chi_{0}\}) = \int f\cdot Pf \,d\mu = \int Pf\cdot Pf \,d\mu \geq \Bigl(\int Pf \,d\mu\Bigr)^{2}t\mapsto t^{2}= \Bigl( \int f \,d\mu\Bigr)^{2}P1_{X}= 1_{X}= \mu(D)^{2}f=1_{D},
{\cal L}_{I}= \frac{2}{9}\left[ \left\{\phantom{|}_{a b}^{\mu \nu}\right\}\Omega_{\mu}{}^{a c d e}\Omega_{\nu}{}^{b c d e}- \frac{1}{4 8}\left\{\phantom{|}_{a b c d e}^{\mu \nu \alpha \beta \gamma}\right\}\Omega_{\mu}{}^{a b c d}E_{\nu \alpha \beta \gamma}{}^{e}\right]
T^{H_t,H,H_t}_{f^{[2]}}(W+t(V_k-W),V_k-W) &=f(H_t)(W+t(V_k-W))(H-iI)^{-1}(V_k-W)(H_t-iI)^{-1}&\quad-T_{(fu)^{[1]}}^{H_t,H}\big((W+t(V_k-W))(H-iI)^{-1}\big)(V_k-W)(H_t-iI)^{-1}&\quad-T_{(fu)^{[1]}}^{H_t,H_t}\big((W+t(V_k-W))(H-iI)^{-1}(V_k-W)(H_t-iI)^{-1}\big) &\quad+T_{(fu^2)^{[2]}}^{H_t,H,H_t}\big((W+t(V_k-W))(H-iI)^{-1},(V_k-W)(H_t-iI)^{-1}\big)
t^{\sigma}\|e^{-tH^{\beta}}u_0\|_{L^a}\leq{C}t^{\sigma}t^{-\frac d{2\beta}(\frac{2\beta}{d(m-1)}-\frac 1a)}\|u_0\|_{L^{\frac{d(m-1)}{2\beta}}}={C}\|u_0\|_{L^{\frac{d(m-1)}{2\beta}}}\leq{C}\|u_0\|_{\exp L^{p}},
\sup_{i\geq 1}\sup_{x\in K}\left| x^z \cdot \prod_{j=1}^{\infty}{\frac{\big(q^{-z+\nu_j^{(i)}}t^j; q\big)_{\infty}}{\big(q^{-z+\nu_j^{(i)}}t^{j-1}; q\big)_{\infty}}}\right| \leq c\cdot a^r, \textrm{if}z = r \pm \frac{\pi\sqrt{-1}}{\ln{q}}.
\begin{array}{lcl}\Delta X_{\pm (i,j)}= X_{\pm (i,j)}\otimes e^{-hH_{j;n-1}/2}+ e^{-hH_{i;n-1}/2}\otimes X_{\pm (i,j)}+ + (1-e^{\pm h})\sum_{k=i+1}^{j-1}(X_{\pm (i,k)}\otimes X_{\pm (k,j)}).\end{array}
\frac{N_p}{D_p}\leq\frac{-\frac{2+\gamma}{4}(1+o_R(1)+o_p(1))}{\frac{1}{2+\gamma}(1+o_R(1))}=-\frac{(2+\gamma)^2}{4}(1+o_R(1)+o_p(1))\overset{\eqref{varie}}{=}-\frac{\ell^2+2}{2}(1+o_R(1)+o_p(1))
\begin{aligned}\|D^r\tilde{f}_P\|_p = \frac{\|D^r g\|_p}{h^{d+r}}\left(h^d\sum_{i=1}^Sp_i^p\right)^{\frac{1}{p}}&\le \frac{\|D^r g\|_pR^{\frac{d}{p}}}{2h^{d+r}}\left(\frac{1}{S}\sum_{i=1}^Sp_i^p\right)^{\frac{1}{p}}&= \frac{c_0^{\frac{d}{p}}\|D^r g\|_p}{2h^{r-s}}\cdot n\ln n\left(\frac{1}{S}\sum_{i=1}^Sp_i^p\right)^{\frac{1}{p}}\end{aligned}
\textbf{p}^{(\ell)}_k= p_k^{1/3}\left[b^{(\ell)}_{i_\ell}(t_k)\right]_{i_\ell=1}^{n_\ell}\in \mathbb{R}^{n_\ell}b^{(\ell)}_{i_\ell}(t_k)= \int_{\mathbb{R}}\psi^{(\ell)}_{i_\ell}(x_\ell) e^{- t_k^2 x^2_\ell}\mathrm{d}x_\ell,
\Theta_{[ 1 / 2 ]}\, = \, \left[ \begin{array}{c c c}{{0}}&{{{}}}&{{- 1}}\\{{1}}&{{{}}}&{{0}}\\ \end{array}\right] \, , \, \, \Theta_{[ 1 ]}\, = \, \left[ \begin{array}{c c c c c}{{0}}&{{{}}}&{{0}}&{{{}}}&{{1}}\\{{0}}&{{{}}}&{{- 1}}&{{{}}}&{{0}}\\{{1}}&{{{}}}&{{0}}&{{{}}}&{{0}}\\ \end{array}\right] \quad .
(\lambda)_{\upsilon}:=\frac{\Gamma(\lambda+\upsilon)}{\Gamma(\lambda)}=\begin{cases}1 , \quad~~~~~~~~~~~~~ (\upsilon=0~;~\lambda\in\mathbb{C}\backslash\{0\}) \lambda(\lambda+1)...(\lambda+n-1), (\upsilon=n\in\mathbb{N}~;~\lambda\in\mathbb{C}). \end{cases}
(1+\tilde{\delta})\tilde{a}_{m-1}^{n-1}\tilde{a}_{m+1}^{n}=\tilde{\delta}\tilde{a}_{m+1}^{n-1}\tilde{a}_{m-1}^n +\tilde{a}_{m}^{n}\tilde{a}_m^{n-1}.
\sup_{0<|h|<\frac{1}{2}}\left\|\frac{\delta_h \psi}{|h|^\alpha}\right\|_{L^p(\mathbb{R}^N)}\le \frac{1}{2}\,\sup_{0<|h|<\frac{1}{2}}\left\|\frac{\delta^2_h \psi}{|h|^\alpha}\right\|_{L^p(\mathbb{R}^N)}+4^\alpha\,\left\|\psi\right\|_{L^p(\mathbb{R}^N)}+2^{\alpha-1}\,\sup_{0<|h'|<\frac{1}{2}}\left\|\frac{\delta_{h'}\psi}{|h'|^\alpha}\right\|_{L^p(\mathbb{R}^N)}.
D^{c}( x_{1}, t ) \equiv - \frac{1}{( 2 \pi )^{2}}\int_{- \infty}^{\infty}d q_{0}\int_{- \infty}^{\infty}d q_{1}\frac{1}{q_{0}^{2}- q_{1}^{2}+ i \epsilon}e^{- \frac{i}{\hbar}c q_{0}t}e^{\frac{i}{\hbar}q_{1}x_{1}}
\begin{cases}\operatorname{supp}(\nabla \phi) \cap \{x_{n+1}>0\}\subset \mathbb B_3^+ \setminus \mathbb B_{2}^+ |\operatorname{div}(x_{n+1}^a \nabla \phi)| \leq C x_{n+1}^a\ \mathbf 1_{\mathbb B_3^+ \setminus \mathbb B_{2}^+}.\end{cases}
\lvert \Gamma_{l}-\Gamma_{k}\rvert \le \sum_{i=k}^{l-1}\lvert \Gamma_{i+1}-\Gamma_{i}\rvert\le \sum_{i=k}^{l-1}\tau^{n/2+1}\tau^{\beta i}\mu &\le \tau^{n/2+1}\mu\sum_{i=k}^{\infty}\tau^{\beta i}=\tau^{n/2+1}\frac{\tau^{\beta k}}{1-\tau^{\beta}}\mu.
\int_I \frac{1}{dist^2(\gamma(t),\rho(A))}|\gamma^{\prime}(t)|dt = 2\int_0^{\infty}\frac{1}{t^2+(\frac{g}{2})^2}dt = \left[\frac{4}{g}\arctan\left(\frac{2t}{g}\right)\right]_0^{\infty}= \frac{2\pi}{g}
\frac{1}{d_j (A)}= \underset{{\mathcal{M}\subset \mathbb{C}^{2n}}\atop{\dim \mathcal{M}=2n-j+1}}{\min}\,\,\,\underset{{x \in \mathcal{M}}\atop{\langle x, Ax \rangle = 1}}{\max}\,\,\, \langle x, i J x \rangle,
\frac{d}{d\epsilon}K(\epsilon)E(\sqrt{1-\epsilon^{2}})=\frac{1}{\epsilon(1-\epsilon^{2})}[E(\epsilon)E(\sqrt{1-\epsilon^{2}})-K(\epsilon)E(\sqrt{1-\epsilon^{2}})+\epsilon^{2}K(\epsilon)K(\sqrt{1-\epsilon^{2}})]
{P^{\textrm{proposed}}_{{\mathop{\textrm{int}}}}}= \prod\limits_{i = 1}^N{\Pr \left({\frac{{N_0 + |h_{is}|^2 P_i}}{{N_0 + |h_{ie}|^2 P_i}}< 1}\right)}&= \prod\limits_{i = 1}^N{\Pr \left({|{h_{is}}{|^2}< |{h_{ie}}{|^2}}\right)}&=\prod\limits_{i = 1}^N{P_{{\mathop{\textrm{int}}}}^i},
\begin{array}{l}\det \langle L_{1}(u,p_{1},\bar{p}_{1}) \rangle \cdot \det\langle L_{2}(u,p_{2},\bar{p}_{2}) \rangle = \vspace{-2mm}\qquad \qquad \qquad = \det \langle L_{1}(u,p_{3},\bar{p}_{3})\rangle \cdot \det \langle L_{2}(u,p_{4},\bar{p}_{4}) \rangle.\end{array}
\overline{S}_n^{\boldsymbol{\mu},\sigma}-\sum_{k=n+1}^{d}C_{k}^{(d), \sigma}(\mathbf{P}_{\!\sigma}) \cdot\bigg( \int\! \log \boldsymbol{\mu}_k^{\sigma}\,\mathrm{d}\mathbf{p}_{\sigma_k}+ \sum_{\ell=1}^{n}\chi_{\ell}^{\sigma}(\mathbf{p}_{\sigma_k})\big( \overline{S}_{\ell}^{\boldsymbol{\mu},\sigma}-\overline{S}_{\ell-1}^{\boldsymbol{\mu},\sigma}\big) \bigg),
|\tilde{P}_{q_n-1}(\theta_{m_n+\ell q_n+1})-\tilde{P}_{q_n-1}(\theta_{m_n-q_n+1})| \leq \|F_{q_n-1}(\theta_{m_n+\ell q_n+1})-F_{q_n-1}(\theta_{m_n-q_n+1})\| \leq \sum_{j=0}^{q_n-2}\| F_{q_n-j-2}(\theta_{m_n+\ell q_n+2+j})\|\cdot \| F(\theta_{m_n-q_n+1+j})-F(\theta_{m_n+\ell q_n+1+j})\|\cdot \| F_{j}(\theta_{m_n-q_n+1}) \|,
I(f,g) \geqslant \int_0^\infty \int_0^\infty \bigg(\lambda \int_0^{(\max\{u(a), v(b)\}/2\omega_n)^{1/n}}c^{\lambda-1}J(a,b,c) dc \bigg) da db & \geqslant \frac{(2\omega_n)^{-\lambda/n}}2 \int_0^\infty \int_0^\infty u(a) v(b) (\max\{u(a), v(b)\})^{\lambda /n}da db.
d_{\mathfrak{g}_1}(\iota_a \omega_{1,1})=\rho_2(a)\circ \omega_{2,0}, d_{\mathfrak{g}_2}(\iota_x \omega_{1,1})=\rho_1(x)\circ \omega_{0,2},\forall a\in \mathfrak{g}_2,x\in \mathfrak{g}_1,
\mathcal{C}_{ij}\simeq \Omega_{X}|_{D_{ij}}\mathcal{C}_{ij}^l \simeq \omega_l|_{D_{ij}}, l=i,j \omega_X|_{D_i}\simeq \mathcal{C}_i\ \otimes \omega_i, \det(\mathcal{C}_{ij}) \simeq \mathcal{C}_{ij}^l \otimes \mathcal{C}_l|_{D_{ij}}l=i,j
\iint_{\max(|x-y|, |x-z|) > \varepsilon}|K(x,y,z)-K_{\Phi}(x,y,z)| |f(y)| |g(z)|\,d\mu(y)\,d\mu(z) &\lesssim \sum_{j=1}^3 \varepsilon^{j\beta}\iint_{\max(|x-y|, |x-z|) > \varepsilon}\frac{|f(y)| |g(z)|}{(|x-y|+|x-z|)^{2m+j\beta}}\,d\mu(y) \,d\mu(z)\lesssim M_{\mu}f(x) M_{\mu}g(x),
{\mathcal Z}^{reg}_A[J] \,=\, \int{\mathcal D}{\mathcal A}_\mu \, \,\exp \left\{i \int d^3x [-\frac{\xi^2}{4}{\mathcal F}_{\mu\nu}{\mathcal F}^{\mu\nu}\, +\, \frac{\kappa}{2}\epsilon^{\mu\nu\lambda}{\mathcal A}_\mu \partial_\nu{\mathcal A}_\lambda +J^\mu{\mathcal A}_\mu ]\right\}\,,
(u_{n+1}+u_n)^{p+1}= \sum\limits_{j=0}^{p+1}\binom{p+1}{j}u_{n+1}^j u_n^{p+1-j}, (u_{n+1,r}+u_n)^{p+1}= \sum\limits_{j=0}^{p+1}\binom{p+1}{j}u_{n+1,r}^j u_n^{p+1-j},