harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S [ x^{\hat{i}}, x^{\alpha}] = \int d^{2}\xi \left[ \partial x^{\hat{i}}f_{\hat{i}\hat{j}}\bar{\partial}x^{\hat{j}}+ \partial x^{\hat{i}}f_{\hat{i}\beta}\bar{\partial}x^{\beta}+ \partial x^{\alpha}f_{\alpha \hat{j}}\bar{\partial}x^{\hat{j}}+ \partial x^{\alpha}f_{\alpha \beta}\bar{\partial}x^{\beta}\right] .
S_{n}(t)=\frac{1}{(n\sigma_0^2)^{1/2}}\left( \log \|M_{\lfloor tn \rfloor}\|-nt\lambda_1 + (nt-\lfloor nt \rfloor) (\log \|M_{\lfloor tn \rfloor +1}\| - \log\|M_{\lfloor tn \rfloor}\|) \right)
\prod_{\alpha\in\Phi^+}\xi^{\langle\alpha,\rho\rangle}-\xi^{-\langle\alpha,\rho\rangle}= \xi^{-2\langle\rho,\rho\rangle}\sum_{w\in\mathfrak{S}_{n+1}}(-1)^{l(w)}\xi^{\langle 2\rho,w\bullet 0\rangle}.
f_{T,\theta}(e_{\psi_n}g_{n_1,\cdots, n_s}(c_1,\cdots,c_s)e_{\psi_n})=\sum_{t\in T}f_{T}(e_{\psi_n}g_{n_1,\cdots, n_s}(c_1,\cdots,c_s)e_{\psi_n})(t)\chi(t).
|b_{n,r,s}| \le C\sum_{j, k=1}^n(j+r+|k+s|)^{-\beta}L(j+r+|k+s|) &\propto n\sum_{k=1}^n(r+|k+s|)^{-\beta}L(r+|k+s|) &\propto n^2(r+|s|)^{-\beta}L(r+|s|).
\tilde{Q}^{\epsilon_{+}}_{n+k}([E^{ss}/G],\overline{\beta})\rightarrow{Q}^{\epsilon_{+}}_{n+k}([E^{ss}/G],\overline{\beta}) \xrightarrow{i}{Q}^{\epsilon_{+}}_{n+k}(\tilde{\P},(d',d''-kd_0))\xrightarrow{b_{\epsilon_{+},k}}{Q}^{\epsilon_{+}}_{n}(\tilde{\P},(d',d''))\xrightarrow{c_{\epsilon_{+}}}{Q}^{0+}_{n}(\tilde{\P},(d',d'')).
G(z, \chi): = \prod_{p}\left(1 - \frac{\chi(p)}{p^{z}}\right)\left(1 - \frac{\chi(p)^{2}}{p^{2 z}}\right) \left(\sum_{\alpha = 0}^{\infty}\frac{a( p^{\alpha}) \chi( p^{\alpha})}{p^{\alpha z}}\right),
\ell (\mathbf x,\mathbf y) = (1+\delta)^{-(p^2qR+3q/4)}\varphi\{\mathbf y|\boldsymbol{\nu}_{J,R}\mathbf x,(1+\delta)^{-1/4}\mathbf{B}^{1}\}, u (\mathbf x,\mathbf y) = (1+\delta)^{p^2qR+3q/4}\varphi\{\mathbf y|\boldsymbol{\nu}_{J,R}\mathbf x, (1+\delta)\mathbf{B}^2\},
\Bigg\|Ue_i - \sum_{k=1}^{\lceil \theta^{-1}\rceil - 1}Ue_{j_k}\Bigg\| =\Bigg\|Ue_i - U \Bigg(\sum_{k=1}^{\lceil \theta^{-1}\rceil - 1}e_{j_k}\Bigg) \Bigg\| = \Bigg\|e_i - \sum_{k=1}^{\lceil \theta^{-1}\rceil - 1}e_{j_k}\Bigg\| = 1.
A' \ = \ 2^{n-1}\sum_{\omega = 0}^{a-1}2^{-\omega}(-1)^{\omega}I(\omega, 0) \sum_{\alpha = 0}^\omega \sum_{\ell = 0}^{a - 1 - \omega}\sum_{i= 0}^{a - \ell - \omega - 1}(-1)^i \binom{n}{\ell + \omega}\binom{\ell + \omega}{\alpha}\binom{n - \ell - \omega}{i}\binom{\ell + \omega - \alpha}{\omega - \alpha}.
\left\{\begin{aligned}(R_1,R_2,R_3):\R_1=\mathcal{C}(\bar{\alpha}P_1),\R_2=\mathcal{C}\left(\tfrac{a\alpha P_1}{1+a\bar{\alpha}P_1}\right), &R_3=\mathcal{C}(\beta P_2),\ \alpha\in[0,1],\ \beta\in[0,\min\{1,\tfrac{b}{1+aP_1}\}]\end{aligned}\right\}
\tau\left(\alpha_j(s),\beta_j(s);\xi_2^{(j)},\xi_1^{(j)}y_j\right)=(-1)^{\epsilon_j}|\xi_2^{(j)}|^{\alpha_j(s)+\beta_j(s)-1}\tau\left(\alpha_j(s),\beta_j(s);1,d^{(j)}y_j\right),
\operatorname{Res}_u(v) = (-\Delta\tilde u + \tilde u - \tilde f, v)_{\Omega_h \setminus \Omega}+ (\partial_{n_h}\tilde u - \tilde\tau, v)_{\Gamma_h}, \operatorname{Res}_g(v) = (v, -\Delta\tilde g + \tilde g)_{\Omega_h \setminus \Omega}+ (v, \partial_{n_h}\tilde g)_{\Gamma_h}.
\mathcal{Y}= \big\{w \in \mathcal{L}\smallsetminus \mathrm{Z}(\mathcal{L}) \mid \mathrm{C}_\mathcal{L}\big( \mathrm{C}_\mathcal{L}(w) \big) = \mathcal{F}w + \mathrm{Z}(\mathcal{L}) \big\}
\exp(t\,\omega_{0}(U)) = \sup\left\{|\mu|\,;\,\mu \in \mathfrak{S}(U(t))\right\}, \exp(t\,\omega_{\mathrm{ess}}(U)) = r_{\mathrm{ess}}(U(t)) = \sup\left\{|\mu|\,;\,\mu \in \mathfrak{S}_{\mathrm{ess}}(U(t))\right\}
\psi(\mbox{\boldmath$r$}) = \psi_{inc}(\mbox{\boldmath$r$})+\frac{2m}{\hbar^{2}}\int G(|\mbox{\boldmath$r-r'$}|) v(\mbox{\boldmath$r'$})\psi(\mbox{\boldmath$r'$})d\mbox{\boldmath$r'$}\,,
\Psi_{( h ; 1 )}( z ) \Psi_{( h^{\prime}; 1 )}( 0 ) = \ldots + \frac{C_{( h , h^{\prime}; 2 ) \iota}}{D_{\iota \iota^{*}}}z^{h_{\iota}- h - h^{\prime}}\chi_{\iota}( 0 ) + \ldots ,
\nabla_i\nabla_jR^2= 2\nabla_iR\nabla_jR+2R\nabla_i\nabla_jR,\frac{\alpha'}{4}\nabla_i\nabla_jR^3=\frac{3}{2}\alpha'R\nabla_iR\nabla_jR+\frac{3}{4}\alpha'R^2\nabla_i\nabla_jR.
G_{p o l a r}( x , y ; k ) = - 4 i \pi \left( S ( x ) \gamma^{5}\right) \otimes \left( S ( y ) \gamma^{5}\right) \frac{\sin \left[ k x / 2 \right] \sin \left[ k y / 2 \right]}{( k^{2}- e^{2}/ \pi + i \epsilon )},
\psi ( \tau ) = \sqrt{\frac{1}{\pi \alpha}}\sum_{n = 1}^{\infty}\left( \psi_{n}^{( 1 )}\cos{\frac{n \tau}{\alpha}}+ \psi_{n}^{( 2 )}\sin{\frac{n \tau}{\alpha}}\right) +{\frac{\psi_{0}}{\sqrt{2 \pi \alpha}}}
- \log x \int_{1}^{\infty}x^{t}\left( \log t+ B_{X}+ O\left( \dfrac{1}{t}\right) \right) dt = I_{1}+ I_{2}+ O(I_{3}), I_{1}= -\log x \int_{1}^{\infty}x^{t}\log t \, dt, I_{2}= -B_{X}\cdot \log x \int_{1}^{\infty}x^{t}dt = B_{X}\cdot x, I_{3}= - \log x \int_{1}^{\infty}\dfrac{x^t}{t}dt.
f(x,y)=\begin{cases}x ,\;x\geq x^*, \frac{1}{2}e^{cx^*}(x^*-\frac{1}{c})e^{-cx}+\frac{1}{2}e^{-cx^*}(x^*+\frac{1}{c})e^{cx},\;-x^*< x<x^*, y\geq0 0 ,\;x\leq-x^*,\end{cases}
\frac{\int_0^t \frac{1}{\sigma(s)}\,ds}{\log \sigma(t)}=\frac{\int_0^{T(M)}\frac{1}{\sigma(s)}\,ds}{\log \sigma(t)}+ \frac{\int_{T(M)}^t \frac{1}{\sigma(s)}\,ds}{\log \sigma(t)}&\leq\frac{\int_0^{T(M)}\frac{1}{\sigma(s)}\,ds}{\log t}+\frac{1}{M}\cdot \frac{\log t - \log T(M)}{\log t}.
\overline{H}\left(\bigwedge_{a \in \mathcal{A}}G_a \right) = \sum_{a \in \mathcal{A}}\overline{H}(G_a) = \sum_{a \in \mathcal{A}}H_\kappa(G_a), \overline{H}\left(\bigsqcup_{a \in \mathcal{A}}^{P_A}G_a \right) = \sum_{a \in \mathcal{A}}P_A(a) \overline{H}(G_a) = \sum_{a \in \mathcal{A}}P_A(a) H_\kappa(G_a).
A_{11}= -(1+e^{h\sigma})\sigma bd - (1-e^{h\sigma})\sigma a_\sigma \;, A_{12}= (1+e^{h\sigma})bd + (1-e^{h\sigma}) a_\sigma \;, A_{21}= bd+a_{2\sigma}+e^{h\sigma}\left((bd-a_\sigma)(h\sigma+1)+\sigma\right) \;, A_{22}= 1-e^{h\sigma}(h(bd-a_\sigma)+1) \;,
(1-\|p\|_\infty) \cdot p_i \leq \pi_S(i) \leq 2 \cdot p_i, \textstyle{\sum_{k}}\,p_k = S, \pi_{S-1}(i) \leq \pi_S(i), \P_S( \{i,j\}\subseteq I) \leq \pi_S(i) \cdot \pi_S(j), i \neq j .
\bar{q}_{\beta}(n,t)=\sum_{\Omega^{n}_{k}}\prod_{j=1}^{k}\rho^{x_{j}}\binom{r+x_{j}-1}{x_{j}}\left(\frac{\lambda t^{\beta}(1-\rho)^{r}}{1-(1-\rho)^{r}}\right)^{k}E_{\beta, k\beta+1}^{k+1}\left(-\lambda t^{\beta}\right).
w_{\alpha}^{(i)}(0, t) = \dfrac{1}{2\, \mathrm{v}_i \, h_i^2}\Big(\boldsymbol{d_{\alpha}}(t) - \mathrm{v}_1 \, h_1^2 \, w_{\alpha}^{(1)}(0, t)\Big), \ \t \in [0, T], i\in \{2, 3\}.
\Omega_{n}^{\pm}(\lambda,b) :=\frac{B_{n}(\lambda,b)\pm\sqrt{\Delta_{n}(\lambda,b)}}{2b}&=\frac{1-b^{2}}{2b}\Lambda_{1}(\lambda,b)+\frac{1}{2}\Big(\Omega_{n}(\lambda)-\Omega_{n}(\lambda b)\Big) &\quad\pm\frac{1}{2b}\sqrt{\Big(b\big[\Omega_{n}(\lambda)+\Omega_{n}(\lambda b)\big]-(1+b^{2})\Lambda_{1}(\lambda,b)\Big)^{2}-4b^{2}\Lambda_{n}^{2}(\lambda,b)}.
(u_1:u_2:u_3:u_4:u_5:u_6)= \left(\frac{x_1}{\lambda_3}:\frac{x_0}{\lambda_3}:\frac{y_1}{\lambda_2}:\frac{y_0}{\lambda_2}:\frac{z_1}{\lambda_1}:\frac{z_0}{\lambda_1}\right), \left( \frac{x_1}{x_0}, \frac{y_1}{y_0},\frac{z_1}{z_0}\right) = \left( \frac{u_1}{u_2}, \frac{u_3}{u_4},\frac{u_5}{u_6}\right),
z=h_2^{-1}\left(\frac{x-h_2(y)}{1-y}\right),y\leq 1 &s=\alpha\star\left(\frac{y}{2}+(1-y)z\right) &r(x)=-\frac{h_2^\prime(x)}{h_2^{\prime\prime}(x)}=x(1-x)\ln \frac{1-x}{x}\ 0\leq x\leq \frac{1}{2}&h_2^\prime(x)=\frac{1}{\ln 2}\ln\frac{1-x}{x}\ 0\leq x\leq \frac{1}{2}&h_2^{\prime\prime}(x)=-\frac{1}{\ln 2}\frac{1}{x(1-x)}\ 0\leq x\leq \frac{1}{2}.
r_0=r_0(\xi)=-\frac{|\xi|^2}{2(1+|\xi|^2)}+\frac{|\xi|^2 \sqrt{3+4|\xi|^2}}{2(1+|\xi|^2)}i=\varphi(\xi)+\phi(\xi)i, r_1=r_1(\xi)=-\frac{|\xi|^2}{2(1+|\xi|^2)}-\frac{|\xi|^2 \sqrt{3+4|\xi|^2}}{2(1+|\xi|^2)}i=\varphi(\xi)-\phi(\xi)i.
\sum_{k =0}^\infty |T((1+k)^{-\frac{s}{2}}h_{k})|^2 = \sum_{k=2n}^\infty (1+k)^{-s}\left|\sum_{x\in X_n}\omega(x) h_{k}(x)\right|^2 = \sum_{x,y\in X_n}\omega(x)\omega(y) \sum_{k=2n}^\infty (1+k)^{-s}h_{k}(x)h_{k}(y)\, .
\begin{array}{ccccc}ds^2_{brane}\stackrel{r\to \infty}{\approx}\underbrace{dx^m \otimes dx^n \,\eta_{m n}}- \underbrace{dr^2 +r^2 \, ds^2_{X}(y)}\null \null M_{p+1}\null C\left (X^{D-p-2}\right)\,,\end{array}
\sum_{i_1 +\cdot+i_r =k}\sum_{\sigma\in \mathfrak{S}(i_1,\ldots,i_r )}\epsilon(\sigma)\frac{1}{r!}\prod_{j=1}^r\Phi_{i_j-1}(x_{\sigma(i_{1}+\cdots+i_{j-1}+1)},\ldots, x_{\sigma(i_{1}+\cdots+i_j)}).
\begin{aligned}\displaystyle Q^{k,s}_d = \operatorname{Shrink}\left(\frac{\lambda}{\lambda +r}M^k_d + \frac{r}{\lambda +r}(v^{s-1}- b^{s-1}), \frac{1}{(\lambda+r)\eta}\right). v^s = Q^{k,s}_d + b^{s-1}- \frac{1}{n}\left({\bf 1}^T ( Q^{k,s}_d + b^{s-1}) - N\right). b^{s}= b^{s-1}+ Q^{k,s}_d - v^s.\end{aligned}
\begin{aligned}U_N(\vec u) \sim U_N(\vec u^\pm) + \frac{1}{2N}\sum_{k=1}^N \lambda^\pm_k (b^\pm_k)^2 \vec \psi^{(k)\pm}\cdot \vec \psi^{(k)\pm}\textrm{as}\ \vec u \to \vec u^\pm = E_N^\pm + \frac{1}{2}\sum_{k=1}^N \lambda_k^\pm (b^\pm_k)^2 .\end{aligned}
T ={\frac{\eta \sinh ( \beta - r_{-}\Delta \phi_{n}) \pm [ ( \cosh ( \beta - r_{-}\Delta \phi_{n}) - \eta X \cosh r_{+}\Delta \phi_{n})^{2}+ \sinh^{2}r_{+}\Delta \phi_{n}]^{1 / 2}}{\cosh r_{+}\Delta \phi_{n}}}.
D_0 =\prod_{2l+1\in A_1}\langle d_{2l+1}\rangle\times\prod_{2^{n-2}-(2l+1)\in B_1}\langle d_{2^{n-2}-(2l+1)}\rangle= &=\prod_{2l+1\in A_1}\langle d_{2l+1}\rangle\times\prod_{2^{n-2}-(2l+1)\in B_1}\langle d_{2l+1}q(1,2l+1)\rangle= &=\prod_{2l+1\in A_1}\langle d_{2l+1}\rangle\times\prod_{2^{n-2}-(2l+1)\in B_1}\langle q(1,2l+1)\rangle.
= \int dt d\theta d\phi dt^{\prime}d\theta^{\prime}d\phi^{\prime}\frac{\sin\theta\sin\theta^{\prime}\Phi_{-}(t,\theta,\phi)\Phi_{-}(t^{\prime},\theta^{\prime},\phi^{\prime})}{[\cosh(\frac{t-t^{\prime}}{a}) -(\sin \theta \sin\theta^{\prime}\cos (\phi-\phi^{\prime}) + \cos \theta \cos \theta^{\prime})]^{h}}.
R^{\widetilde\nabla}(e_1,e_2)\phi = -\frac12 K \omega\cdot\phi -dA(e_1,e_2)\phi -d\beta(e_1,e_2)\omega\cdot\phi &+2(\det A)\omega\cdot\phi - 2 \beta(e_2)J(A(e_1))\phi + 2 \beta(e_1)J(A(e_2))\phi.
\Big\vert\frac{1}{N^{s+1}}\int\limits_{\mathbf{w}\in \mathbb{R}^{s+1}}\prod\limits_{j=1}^d g_j (\psi_j(\mathbf{w}) + a_j) F(\mathbf{w})\, d\mathbf{w}\Big\vert \ll_C \rho^{-\Omega(1)}\sigma_F^{-1}.
=\frac{x^5(1-2x+x^2-x^3)}{(1-x)^6(1-2x)^2}+\frac{x^6(1-x-x^2)}{(1-x)^5(1-2x)^3}+\frac{x^5(2-x)}{(1-x)^3(1-2x)(1-3x+x^2)}&\quad+\frac{x^6}{(1-x)^2(1-2x)^2(1-3x+x^2)}+\frac{x^3}{(1-2x)^3}&=\frac{x^3(1-9x+37x^2-91x^3+142x^4-141x^5+90x^6-36x^7+6x^8)}{(1-x)^6(1-2x)^3(1-3x+x^2)}.
\min \sum_{j=1}^K y_j, \sum_{i=1}^n w_i x_{ij}\leq W y_j, j = 1,2,\ldots,K, & \sum_{j=1}^K x_{ij}\geq d_i, \forall i \in I, & y_j \in \{0, 1 \}, j = 1,2,\ldots,K, & x_{ij}\in \mathbb{Z}_+, \forall i \in I, j=1,2,...,K.
\frac{2^{16N-16}F(N-1,N-1)}{2N^2\binom{2N}{N}^2}&\quad=\frac{3\cdot 2^{4 N-5}\left(12 N^2-16 N+5\right) \binom{2 N-2}{N-1}\binom{3 N-3}{N-1}\binom{6 N-6}{3 N-3}}{N^2 \binom{2 N}{N}^2}&=\frac{3\cdot 2^{4N-7}(6N-5)!(N-1)!}{(2N-1)!(2N-2)!(3N-3)!}\in \mathbb{Z},
{\cal L}^F = 2iv_0:\!\psi_+^{\dagger}\partial_+\psi_+ + \psi_-^{\dagger}\partial_-\psi_-\!:+v_1J_+J_- + iv_2 \left( \psi_+\psi_--\psi_-^{\dagger}\psi_+^{\dagger}\right).
A_k^{(s,t)}:=\left(\begin{array}{cccc}q_1^{(s,t)}1 q_1^{(s,t)}e_{1}^{(s,t)}q_2^{(s,t)}+e_1^{(s,t)}\ddots &\ddots \ddots 1 & q_{k-1}^{(s,t)}e_{k-1}^{(s,t)}q_k^{(s,t)}+e_{k-1}^{(s,t)}\end{array}\right).
\lambda_1(\omega) = \inf \left\{\frac{\int_0^D u'(x)^2\omega(x)\,\textrm{d}x}{\int_0^D u(x)^2\omega(x)\,\textrm{d}x}:u \in H^1(0,D),\,\int_0^D u(x)\omega(x)\textrm{d}x = 0 \right\},
V_\varphi\widetilde{g}(x,\xi) =\int_{\mathbb{R}^d}g(Ct)e^{-2\pi i\xi \cdot t}\overline{\varphi(t-x)}dt\asymp \int_{\mathbb{R}^d}g(s)e^{-2\pi i((C^{-1})^T\xi)\cdot s}\overline{\varphi(C^{-1}(s-Cx))}ds &=V_{\widetilde{\varphi}}g(B(x,\xi)),
\delta \int d^4x \mathcal{L}=\int d^4x\partial_\mu\left(\frac 12 \{\partial^\mu\Phi,\delta_0\Phi\}_\star+\delta x^\mu\mathcal{L}+\frac{\lambda}{12}\frac{\sinh(\Delta)}{\Delta}(\Phi\star \Phi~,~\tilde{\partial}^\mu[\Phi,\delta_0\Phi]_\star)\right).
Q^{\mathcal{O}| 2 s + \tau , 2 t + \tau^{\prime}}= Q^{\mathcal{O}_{\tau , \tau^{\prime}}^{\mu , \nu}| 2 s + \tau , 2 t + \tau^{\prime}}= Q_{2 s + \tau , 2 t + \tau^{\prime}}^{\mu , \nu}\sim g_{2 s + \tau , 2 t + \tau^{\prime}}^{\tau^{\prime}- 1 - \nu , \tau + 1 - \mu}\, \, \det \mathcal{A}_{2 s + \tau , 2 t + \tau^{\prime}}^{\mu , \nu}\, \,
\phi = \int_0^{\tau_n \wedge (T-\frac1n)}\frac{1}{T-s}\; ds + \int_{\tau_n \wedge (T-\frac1n)}^{\tau_n}n\; ds &=\log\left(\frac{T}{T-\tau_n \wedge (T-\frac1n)}\right) + n\left(\tau_n-\tau_n \wedge \left(T-\frac1n\right)\right).
\begin{aligned}w_{i,j}=\, (s_{\sigma_{i-1,j}}s_{\sigma_{i-1,j}-1}\cdots s_{\widetilde{m}_{i-1,j}+1}) &(s_{\sigma_{i-1,j}+1}s_{\sigma_{i-1,j}}\cdots s_{\widetilde{m}_{i-1,j}+2})\cdot\cdots\cdot &(s_{\sigma_{i-1,j}+m_{i,j}-1}s_{\sigma_{i-1,j}+m_{i,j}-2}\cdots s_{\widetilde{m}_{i,j}})\end{aligned}
T_{a}^{L}= \left( \begin{array}{c c c}{{N_{a}}}&{{E_{a}^{\dagger}}}&{{F_{a}^{\dagger}}}\\{{E_{a}}}&{{P_{a}}}&{{G_{a}^{\dagger}}}\\{{F_{a}}}&{{G_{a}}}&{{b_{a}}}\\ \end{array}\right) ~ , ~ ~ v = \left( \begin{array}{c}{{u}}\\{{w}}\\{{z}}\\ \end{array}\right) ~ ,
\Phi_k(A)=\mathcal{F}_k \circ \mathcal{F}_{k-1}\circ \dots \circ \mathcal{F}_1(A),\ \ Psi_k(A)=\mathcal{F}_1 \circ \mathcal{F}_{2}\circ \dots \circ \mathcal{F}_k(A),\ \k \in \mathbb{N},
\mathrm{det}_N(z_j^{N-k+1}-z_j^{-N+k-1})=(-1)^N \prod_{j=1}^N z_j^{j-1-N}(1-z_j^2)\prod_{1 \le j <k \le N}(1-z_j z_k)(1-z_j z_k^{-1}),
\begin{array}{ccc}(b-c)(a^2+b^2-c^2-d^2) = 0 (b+c)(a^2+b^2-ad+bc) = 0 d(a^2+b^2-ad+bc)^2 = 0 a(c^2+d^2-ad+cb) = 0 (b+c)(c^2+d^2-ad+bc) = 0 ac+bd = 0 a+d \neq 0 \end{array}.
|\mu(T^{k(h_n+c_n)}(I_{n,a}) \cap B) - \mu(I_{n,a})\mu(B)| \leq \int \limits_{I_{n,a}}\Big{|}\frac{1}{r_n+1}\sum \limits_{i=0}^{r_n}\chi_{B}\circ T^{-ki-\frac{1}{2}k(k-1)}-\mu(B)\Big{|}d\mu + \frac{2k+2}{r_n+1}\mu(I_{n}).
\nu\Big(a^*_{p_1}(t) a_{q_1}(t) a^*_{p_2}(s) a_{q_2}(s)\Big) = \delta(q_1 - p_1)\delta(q_2 - p_2)f_0 (p_1) f_0(p_2) & + \delta(q_1 - p_2 )\delta(q_2 - p_1)e^{i(t-s) (E_{p_1}- E_{p_2})}f_0(p_1) \widetilde f_0 (p_2 ) \ .
\gamma^{0}= \left( \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\right) , \hspace *{0 . 5 c m}\gamma^{1}= \left( \begin{array}{c c}{0}&{- 1}\\{1}&{0}\\ \end{array}\right) , \hspace *{0 . 5 c m}\gamma^{5}= \left( \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right) .
\int\limits_{\mathbb R^d}a (z) \mu (\xi, \xi -z ) (\varkappa^{ij}_2 (\xi-z) - \varkappa^{ij}_2 (\xi) ) dz + b^i \varkappa_1^j(\xi) + \int\limits_{\mathbb R^d}a (z) \mu (\xi, \xi -z ) \big(\frac12 z^i z^j - z^i \varkappa^{j}_1 (\xi-z) \big) dz\ = \ \Theta^{ij}.
G ({\hat{x}},{\hat{y}}) \; = \; \int_{{\hat{z}}}{\hat{\Delta}}_{\varphi}({\hat{x}}, \hat{z}){\bar{\hat{\chi}}}({\hat{z}}){\hat{\mathcal D}}({\hat{z}},{\hat{y}}){\hat{\chi}}({\hat{y}}) \; .
H_{1,2}\leq 2^{q-1}\int_0^{2\pi}\int_{\frac{1}{2}}^{\gamma}\frac{dr d\theta}{|\gamma-r|^\tau|re^{i\theta}-1|^q}=2^{q-1}\sum_{j=1}^4\int_{\frac{(j-1)\pi}{2}}^{\frac{j\pi}{2}}\int_{\frac{1}{2}}^{\gamma}\frac{dr d\theta}{|\gamma-r|^\tau|re^{i\theta}-1|^q}=2^{q-1}\sum_{j=1}^4 H_{1,2}^j.
F = (x_0-a'_1y_1+\ldots-a'_my_m)x_1^{a_1}\cdots x_n^{a_n}+ (y_0-b''_1x_1+\ldots-b''_nx_n)y_1^{b_1}\cdots y_m^{b_m}- \sum_{i=1}^{r_1}(\beta_{i,1}y_1 + \ldots + \beta_{i,m}y_m)^d - \sum_{j=1}^{r_2}(\gamma_{j,1}x_1 + \ldots + \gamma_{j,n}x_n)^d.
\begin{aligned}\lim_{s \searrow 0}f(s) = 0 \qquad \lim_{s \searrow 0}f_s(s) = 1 \lim_{s \searrow 0}g(s) > 0 \qquad \lim_{s \searrow 0}g_s(s) = 0 \lim_{s \searrow 0}h_s(s) = 0 \end{aligned}
\aligned\tilde{U}(x) =S^{\frac{(N-\mu)(2-N)}{4(N-\mu+2)}}C(N,\mu)^{\frac{2-N}{2(N-\mu+2)}}U(x) &=S^{\frac{(N-\mu)(2-N)}{4(N-\mu+2)}}C(N,\mu)^{\frac{2-N}{2(N-\mu+2)}}\frac{[N(N-2)]^{\frac{N-2}{4}}}{(1+|x|^{2})^{\frac{N-2}{2}}}\endaligned
K_{4}= \frac{4 C}{\pi}\mathcal{I}\left(\frac{P_{S}\Omega_{S,R}\: g_{QAM}}{N_{0}P_{L_{S,R}}\: m_{S,R}}, m_{S, R}; \frac{\pi}{2}\right) - \frac{4C^{2}}{\pi}\mathcal{I}\left(\frac{P_{S}\Omega_{S,R}\: g_{QAM}}{N_{0}\: P_{L_{S,R}}m_{S,R}}, m_{S, R}; \frac{\pi}{4}\right)
\frac{1}{2\pi\mathbf{i}}\int_{\Gamma_{\mu}}e^{zt}z^{\alpha-2}(z^{\alpha}+\lambda_{k}^{s})^{-1}dz=\sum_{j=-L}^{L}\omega_{j}e^{z_{j}t}z_{j}^{\alpha-2}(z_{j}^{\alpha}+\lambda_{k}^{s})^{-1}+\mathcal{E}_{k}(t)
G_{p+2,q}^{m,n+1}(\begin{matrix}\alpha,a_{p},\alpha^{\prime}\\b_{q}\end{matrix}|z)=(-1)^{\alpha^{\prime}-\alpha}G_{p+2,q}^{m,n+1}(\begin{matrix}\alpha^{\prime},a_{p},\alpha\\b_{q}\end{matrix}|z),n\le p,\alpha^{\prime}-\alpha\in\mathbb{Z},
\mathbb{T}_{\varrho}(M,\{\mathbf{h}_p^{M}\}_{p=0}^3)=\frac{\mathbb{T}_{{\psi_{_1}}}(M_1,\{\mathbf{h}_p^{M_1}\}_{p=0}^{3}) \;\mathbb{T}_{{\psi_{_2}}}(M_2,\{\mathbf{h}_p^{M_2}\}_{p=0}^{3})}{\mathbb{T}_{{\varrho_{|_{\mathbb{D}^2}}}}(\mathbb{D}^2,\{\mathbf{h}_0^{\mathbb{D}^2}\})}.
\mathcal{L}_{1}^{\mu}[\beta b]\nabla_X\psi = - \frac{1}{\beta}\sinh{(\beta b(X) \sqrt{\mu}|\mathrm{D}|)}\mathrm{sech}(\sqrt{\mu}|\mathrm{D}|)\dfrac{1}{\sqrt{\mu}|\mathrm{D}|}\nabla_X\psi.
a = \inf\left\{x\le x_{o}:\u\left(\xi\right)< v\left( \xi\right) \xi \in\left]x,x_{o}\right]\right\}, b = \sup\left\{x\ge x_{o}:\u\left(\xi\right)\le v\left( \xi\right) \xi \in\left[x_{o},x\right]\right\}.
M_{a,1}:\,(r_1,y_1,z,\epsilon_1) = (r_1,\left(\frac{2(\vert \beta \vert+\mathcal O(z))}{\phi^{[k]}(\vert b\vert + \mathcal O(z))(-z)}\right)^{1/k}+ \mathcal O(r_1+\epsilon_1),z,\epsilon_1).
\bigwedge_{i=1}^m \left(\nu\delta^{\tau}\odot^\flat n_i\delta^{t_i}\right) = \bigwedge_{i=j}^m \left(\nu\delta^{\tau}\odot^\flat n_i\delta^{t_i}\right) = \bigwedge_{i=j}^m \left(\nu-n_i\right)\delta^\tau = (\nu-n_j)\delta^\tau.
\frac{1}{\sqrt{\det\left(\hat{{\cal G}}_{\parallel}^{(0)}\right)}}=\frac{1}{\sqrt{\det\left(\hat{{\cal G}}_{\perp}^{(0)}\right)}}=\exp\left(-\beta_T\int_{-\infty}^{\infty}dk\,\rho_0(k)\Phi(k^2;\beta_T,\mu^2)\right)
u_{ik}:= (-1)^{i-k-l}\binom{n}{k}^{-2}\frac{(i+k+l+\sigma)_{n+1-k-l}(\beta+2k+1)_{i-k-l}}{(2i+\sigma)(i-k-l)!(k+l-n)_{i-k-l}(\alpha+l+i+1-k)_{n-i}}, begin{align*}1ex] u_{ih}:= -u_{i,h-1}\frac{h^2(l+h-n-1)(n+l+\alpha+1-h)}{(h-n-1)^2(h-k)(h+k+\beta)}(h=k+1,k+2,\ldots,n-l);
\int_{0}^{z}<\zeta>^{2n+1}d\zeta=\frac{1}{2n+2}<z>^{2n+2}, \int_{0}^{z}<\zeta>^{2n}d\zeta=\frac{1}{2n+1}[<z>^{2n+1}+\frac{1}{2^{2n}}(z-<z>)].
\mathcal{P}'_{n+1}(x)=\dfrac{\zeta+n}{2\zeta+n}\left(x-\dfrac{\theta}{\zeta+n}\right)\mathcal{P}'_n(x)-\dfrac{n}{4(2\zeta+n)}(x^2+\omega^2)\mathcal{P}'_{n-1}(x), n \geq 1.
_{t_{0}}^{ABC}D_{t}^{\alpha}\left[ u(t)-u^*-u^*\ln\dfrac{u(t)}{u^*}\right] \leq \left( 1-\dfrac{u^*}{u(t)}\right)\:_{t_{0}}^{ABC}D_{t}^{\alpha}u(t),_{t_{0}}^{CF}D_{t}^{\alpha}\left[ u(t)-u^*-u^*\ln\dfrac{u(t)}{u^*}\right] \leq \left( 1-\dfrac{u^*}{u(t)}\right)\:_{t_{0}}^{CF}D_{t}^{\alpha}u(t).
{\cal S}_E=\frac{\pi}{4}\tau\nu \sum_{n=0}n^2 X_{n0}^2+ \frac{\pi}{2}\tau \sum_{n=0}\sum_{k=1}\left ( \rho \left ( \frac{2\pi k}{\tau}\right )^2 + n^2\nu \right ) |X_{nk}|^2,
\left( r!\sum_{j=0}^{r}\frac{\binom{b}{r-j}}{j!}x^{j}\frac{d^{j}}{dx^{j}}\right)^{p}=\sum_{k=0}^{rp}S_{1,b,r}\left( p,k\right) x^{k}\frac{d^{k}}{dx^{k}}.
X_0 = \begin{pmatrix}0 0 0 \ldots 0 *[1ex]1 0 0 \ldots 0 0 1 0 \ddots \vdots \vdots \vdots \ddots \ddots 0 0 0 \ldots 1 0\end{pmatrix}\ , Y_0 = \begin{pmatrix}0 1-n 0 \ldots 0 *[1ex]0 0 2-n \ldots 0 0 0 0 \ddots \vdots \vdots \vdots \ddots \ddots -1 0 0 \ldots 0 0\end{pmatrix}\ ,
-\theta_{1}\overline{\pi}_{1}\overline{p}\xi_{e_{k}a}(n) = -\theta_{1}\left(\overline{p}\xi_{e_{k}a}(n)+\operatorname{ker}(\alpha)\right)=-\alpha \overline{p}\xi_{e_{k}a}(n) -\theta_{2}^{-1}(\overline{\gamma}'(\xi_{e_{k}a}(n))) =-\theta_{2}^{-1}(c_{k}\beta \alpha \xi_{e_{k}a}(n))=-c_{k}\overline{\pi}_{3}\alpha \xi_{e_{k}a}(n)
\int_{\Sigma_R}\frac{h(Y)-h(X)}{|X-Y|^{n+1+s}}dY=\int_{B_R(0)}\frac{h(g(t)) - h(g(0))}{(g(t)^2+|t|^2)^{\frac{n+1+s}{2}}}\sqrt{1+|D g(t)|^2}\,dt
\vert B_{m}l_{2}^{m}(f(l_{1}/l_{2})-P_{m}(l_{1}/l_{2}))\vert \leq \left\vert B_{m}l_{2}^{m}\sum_{j=m+1}^{\infty}c_{j}\left(\frac{l_{1}}{l_{2}}\right)^{j}\right\vert \leq (B_{2}l_{2}^{m})^{-m}
\lbrace \left\|u(t)\right\|_{\alpha/2}^2 \rbrace = \frac{1}{2 \nu (T_2-T_1)}(|u(T_1)|^2-|u(T_2)|^2) \geq \frac{1}{2 \nu (T_2-T_1)}\Big(\frac{1}{2}D^{-2}-\sigma^{-2}T_2^{-2}\Big) \geq \frac{D^{-2}}{8 (T_2-T_1)}\nu^{-1},
\mathbb{E}_{\mathbf{H}}\{\Vert\hat{\mathbf{x}}_{lmmse}-\mathbf{x}\Vert^2\}&=m\sum_{k=0}^{l}\frac{\left(-1\right)^k}{\sigma_x^{2k}}\mu_{\mathbf{\Lambda}}\left(-k-1\right)+o\left(\sigma_x^{-2r}\right)
f^{n, k}(t, y, z) : =\mathbb{I}_{\{t\leq \sigma_n \}}\inf_{y^\prime, z^\prime}\big \{f^+(t, y^\prime, z^\prime) + n|y-y^\prime|+n|\lambda_t(z-z^\prime)| \big\}&-\mathbb{I}_{\{t\leq \sigma_k \}}\inf_{y^\prime, z^\prime}\big\{f^-(t, y^\prime, z^\prime) + k|y-y^\prime|+k|\lambda_t(z-z^\prime)| \big\},
\mathcal{\hat{I}}_{l}\left(\hat{\mathbf{\Sigma}}_{1:L}\right)=\textrm{log}\left|\mathbf{I}+\mathbf{H}_{l,l}^{\dagger}\hat{\mathbf{\Sigma}}_{l}\mathbf{H}_{l,l}\hat{\mathbf{\Omega}}_{l}^{-1}\right|.
u_t =p_1(u^2)_x+ \left( h^2u(h^2 q_0 u_{xx}+ p_3 u)_{xx}\right)_x+ \frac{(q_0-q_1+q_2)}{2}h^4\left( u_{xx}^2\right)_x+ \quad\h^2 \left(u_{x}(\beta p_3 u-h^2 (q_0-q_1) u_{xx})_x\right)_x.
T^{- 1}= \left( \begin{array}{c c}{a^{- 1}+ a^{- 1}b d^{- 1}c a^{- 1}}&{- a^{- 1}b d^{- 1}}\\{- d^{- 1}c a^{- 1}}&{d^{- 1}+ d^{- 1}c a^{- 1}b d^{- 1}}\\ \end{array}\right)
S_\vartheta = \prod_{n>0}\exp\left(-\cos 2\vartheta\; a_n \bar a_n\right)\prod_{n>0}(\sin 2 \vartheta)^{a_na_{-n}+ \bar a_n \bar a_{-n}}\prod_{n>0}\exp \left(\cos 2\vartheta\; a_{-n}\bar a_{-n}\right)
\mu_j > b :=\max\left\{\frac{2\kappa_{\mu g}}{\epsilon'}, \frac{16(\kappa_{ef}+\kappa_{efs})}{\eta_1 \theta_{fcd}\epsilon'}, \frac{2\kappa_{J_m}^2}{\epsilon'}, \frac{2\eta_2}{\epsilon'}, \lambda\,\mu_{\min}\right\}.
\frac{\dot{m}_{0}}{\dot{m}_{01}}=\sqrt{\frac{T_{01}}{T_{0}}}\sqrt{\frac{\epsilon_{0}^{2}(1-\epsilon_{c})^{2}-(\epsilon_{2}-\epsilon_{c}\epsilon_{0})^{2}}{\epsilon_{01}^{2}(1-\epsilon_{c})^{2}-(\epsilon_{21}-\epsilon_{c}\epsilon_{01})^{2}}}
Z = \int{\cal D}h_{\mu \nu}{\cal D}h_{\mu \nu}^{\prime}{\cal D}\theta_{\mu}{\cal D}b_{\mu \nu}{\cal D}c_{\mu \nu}{\cal D}\bar{c}_{\mu \nu}{\cal D}b_{\mu}{\cal D}c_{\mu}{\cal D}\bar{c}_{\mu}\exp i \int d^{4}x \left[ L_{h}+ L_{\mathrm{B}}\right] .
c(J/L)=\left|\left(\frac{H_1(\Upsilon_C,\mathbb{Z})^\vee}{2H_1(\Upsilon_C,\mathbb{Z})}\right)^{G_k}\right| \quad\textup{and}w(J/L)=(-1)^{\textup{rk}H_1(\Upsilon_C,\mathbb{Z})^{G_k}}.
\Delta \tilde{u}+ \mu \frac{\tilde{u}}{\delta^2}- \tilde{u}^p \le - \epsilon(2 \beta_- + \epsilon-1)M\delta^{\beta_- +\epsilon-2}+ MK K_1 \delta^{\beta_--1}= - M\delta^{\beta_- +\epsilon-2}\left[\epsilon (\epsilon-(1-2 \beta_-)) - KK_1 \delta^{1 - \epsilon}\right].
\dfrac{1}{(k-1)!}\delta_{i_1i_2\cdots i_k}^{j_1j_2\cdots j_k}(wv_1^t)^{i_1}_{j_1}(wv_2^t)^{i_2}_{j_2}(A_1)^{i_3}_{j_3}\cdots (A_{k-2})^{i_k}_{j_k}= 0.
h(\alpha,\tau,\psi_{R},\upsilon_{R},\psi_{T},\upsilon_{T})=\sum_{p=1}^{P}\gamma_{p}\delta(\alpha-\alpha_{p})\delta(\tau-\tau_{p})\cdot\delta(\psi_{R}-\psi_{R_{p}})\delta(\upsilon_{R}-\upsilon_{R_{p}})\cdot\delta(\psi_{T}-\psi_{T_{p}})\delta(\upsilon_{T}-\upsilon_{T_{p}})
\Xi =\frac{vN^{2\gamma-2}}{d\log^7N}\sum_{P<p\leq 2P}(\log p)\sum_{z=1}^{2Z-1}\theta_{z}(p^c) &=\frac{vN^{2\gamma-2}}{d\log^7N}\left(\sum_{P<p\leq 2P}(\log p)-\sum_{P<p\leq 2P}(\log p)\theta_{0}(p^c)\right) &\ll \frac{vN^{3\gamma-2}}{d\log^7N},
\nu_p (b'_{\lambda,\lambda}) = \nu_p (J_{\ell}^v (\lambda)|_{v=\theta}) - \nu_p (z_{\lambda}) = \sum_{s\ge 0}m_{p^s}(\lambda) (r+s+\gamma) - \sum_{s\ge 0}(sm_{p^s}(\lambda) + \nu_p (m_{p^s}(\lambda)) ) = (\gamma+r) \ell(\lambda) - \nu_p (x_{\lambda}).
Y_{m+1}= \iint_{Q_{m+1}}(v-k_{m+1})_+^p\,dv_0\,dt \le \iint_{Q_m}(v-k_{m+1})_+^p\phi_m^2\,dv_0\,dt \le \left(\iint_{Q_m}[(v-k_m)_+^p\phi_m^2]^{\frac{n+2}{n}}\,dv_0\,dt\right)^{\frac{n}{n+2}}|E_m|^{\frac{2}{n+2}}
\sigma^2 = \lim_{n\to\infty}\frac{1}{n}\mathbb{E}\left[ \big( \log \langle f, G_n x \rangle - n \lambda \big)^2 \right] = \lim_{n\to\infty}\frac{1}{n}\mathbb{E}\left[ \big( \log \rho(G_n) - n \lambda \big)^2 \right],
\partial_{\vec{z}}^{2}H_{F}+ H_{3}\partial_{\vec{x}}^{2}H_{F}= 0 , \ \ \ \partial_{\vec{z}}^{2}H_{1}+ H_{3}\partial_{\vec{x}}^{2}H_{1}= 0 , \ \ \ \partial_{\vec{z}}^{2}H_{3}= 0 .
\sqrt{\frac{\pi}{\log q^{-2}}}\sum_{n=0}^{\infty}\frac{\left(uv\right)^{n}}{\left(q;q\right)_{n}}=\sum_{j,k=0}^{\infty}q^{\left(j+k\right)/2}u^{j}v^{k}\int_{-\infty}^{\infty}S_{j}\left(q^{2\alpha-1/2};q\right)S_{k}\left(q^{2\alpha-1/2};q\right)q^{2\alpha^{2}}d\alpha,
(I-T_k)^{-1}=(I-S_k+\widetilde{T}_k)^{-1}=(I+(I-S_k)^{-1}\widetilde{T}_k)^{-1}(I-S_k)^{-1}&= \sum_{n=0}^{\infty}(-(I-S_k)^{-1}\widetilde{T}_k)^{n}(I-S_k)^{-1}.
\mathbf{b}_R(\psi_1,\theta_1) = \frac{1}{\sqrt{N}}\mathbf{b}(\phi_1,\vartheta_1), \mathbf{b}_T(\psi_{2,k,p},\theta_{2,k,p}) = \frac{1}{\sqrt{N}}\mathbf{b}(\phi_{2,k,p},\vartheta_{2,k,p})