harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{d f}{d \sigma}( \sigma ( 0 ) ) = \pm \sqrt{\frac{\gamma}{1 6 2}}\frac{1}{k k_{1}}\bigg ( 1 - \bigg ( \frac{k_{1}}{k}\bigg )^{2}f^{2}( \sigma ( 0 ) ) \bigg )^{- 1 / 2}.
\begin{array}{l}{{n - 2 n_{1}- 1 - 2 ( n_{2}+ \cdots + n_{j}) - \alpha_{1 j - 1}^{( k - 1 )}\geq a_{1}^{( j )}\geq \cdots \geq a_{n_{j}- n_{j + 1}}^{( j )}\geq 0 ,}}\\ \end{array}
Y_{J_{2}, n_{2}}^{+}( 1 ) W_{J_{3}, n_{3}}^{+}( x ) = \frac{1}{1 - x}f_{J_{2}n_{2}, J_{3}n_{3}}^{J_{2}+ J_{3}- 1 , n_{2}+ n_{3}}Y_{J_{2}+ J_{3}- 1 , n_{2}+ n_{3}}^{+}( 1 ) + r e g u l a r ~ t e r m s
X_{V}\left(\{x^i,y^j\}_{TM}\right) -\{X_{V}(x^i),y^j\}_{TM}-\{x^i,X_{V}(y^j)\}_{TM}= &=X_{V}\left(\pi^{ij}({\bf x})\right)-\{x^i, v^j({\bf x})\}_{TM}=0 ,
H(x, y) = \frac{1}{\mathbf{N}_{n, 0}\left(x\mathbf{N}_{K/E}(x)^{-n}\det y\right)}F\left(x, y^{-1}\right) F(x, y) = \frac{1}{\mathbf{N}_{n, 0}\left(x^{-1}\mathbf{N}_{K/E}(x)^{n}\det y\right)}H\left(x, y^{-1}\right)
{\cal E}^{U ( n )}= \frac{1}{2}\, \frac{g_{S Y M}^{2}}{( 2 \pi )^{d - 2}\, | n + \theta m | \sqrt{\det ( G^{k l})}}\, G^{i j}( n_{i}+ \theta m_{i}) ( n_{j}+ \theta m_{j}) ,
\max_{j\in\{1,\dots,K\}}\frac{R_j(1-\frac{\prod_{k=1}^K\delta_k}{\delta_j})}{(1-\delta_j)\frac{\prod_{k=1}^K\delta_k}{\delta_j}(1-\prod_{k=1}^K\delta_k)}+\max_{\pi}\sum_{i=1}^K\frac{R_{\pi_i}}{1-\prod_{k=1}^i\delta_{\pi_k}}\leq 1 ,
\{J, H_{c}\}= \frac{1}{k}\int d^{2}\vec{x}d^{2}\vec{y}d^{2}\vec{z}\{G(\vec{x}-\vec{y})G(\vec{y}-\vec{z})-G(\vec{x}-\vec{z})G(\vec{y}-\vec{z})\}f_{abc}A^{a}_{2}(\vec{x})J_{2}^{b}(\vec{y})J_{\circ}^{c}(\vec{z})
\times \exp \left( - \frac{1}{2}\sum_{b , b^{\prime}= 1}^{N}\sum_{j = 1}^{n_{b}}\sum_{j^{\prime}= 1}^{n_{b^{\prime}}}\Big ( 1 - \delta_{b b^{\prime}}\delta_{j j^{\prime}}\Big ) \tilde{\tilde{V}}( x_{j}^{( b )}- x_{j^{\prime}}^{( b^{\prime})}) \right)
r_1(t;z)=-\frac{b(k)}{2ik a(k)}e^{-2i\eta^2 t}= r_{1}( z)e^{-2i\eta^2 t}, & r_2(t;z)=\frac{2ikb(k)}{a(k)}e^{-2i\eta^2 t}= r_{2}( z)e^{-2i\eta^2 t}.
X_{k}=\sum_{j=0}^{\infty}(j+1)^{-N}\varepsilon_{k-j}=\sum_{j=0}^{\infty}(j+1)^{-U^{*}DU}\varepsilon_{k-j}=\sum_{j=0}^{\infty}U^{*}\left( (j+1)^{-D}(U\varepsilon_{k-j})\right).
M_{4}= \left( \begin{array}{c c c c}{c_{1 1}}&{c_{1 3}}&{0}&{d_{1 3}}\\{c_{1 3}}&{c_{3 3}}&{- d_{1 3}}&{0}\\{0}&{- d_{1 3}}&{c_{1 1}}&{c_{1 3}}\\{d_{1 3}}&{0}&{c_{1 3}}&{c_{3 3}}\\ \end{array}\right)
E_{2}( r , a ) = \frac{( \varepsilon - 1 )^{2}}{1 6 \pi^{2}a^{4}}\, \frac{2 3}{6 0}\, \frac{( d^{4}- 1 0 d^{2}- 1 5 )}{( 1 + d )^{4}( 1 - d )^{4}}\, , d = \frac{r}{a}\, , r < a .
\renewcommand{\arraystretch}{1 . 4}T_{i}^{( 2 )}( s ) = \left( \begin{array}{c c}{{\rho_{i}/ s + O ( 1 )}}&{{O \left( s^{( k - n - 1 ) / 2}\right)}}\\{{O \left( s^{( k - n - 1 ) / 2}\right)}}&{{T_{i}^{( 1 )}( 1 ) + O ( s )}}\\ \end{array}\right) , \quad i = 1 , 2 , 3 .
{\bf h_{\varphi}}= \left( \begin{array}{c c}{{e^{\varphi / 4}}}&{{{\frac{1}{2}}\, \varphi_{w}e^{- \varphi / 4}}}\\{{0}}&{{e^{- \varphi / 4}}}\\ \end{array}\right) , \qquad{\bf \bar{h}_{\varphi}}= \left( \begin{array}{c c}{{e^{\varphi / 4}}}&{{0}}\\{{-{\frac{1}{2}}\, \varphi_{\bar{w}}e^{- \varphi / 4}}}&{{e^{- \varphi / 4}}}\\ \end{array}\right) .
\mathrm{det}\left(\mathbf{A}\right)=\epsilon_{i_{1}\cdots i_{n}}A_{1i_{1}}\ldots A_{ni_{n}}=\epsilon_{i_{1}\cdots i_{n}}A_{i_{1}1}\ldots A_{i_{n}n}=\frac{1}{n!}\epsilon_{i_{1}\cdots i_{n}}\,\epsilon_{j_{1}\cdots j_{n}}A_{i_{1}j_{1}}\ldots A_{i_{n}j_{n}}
( M_{0 , \pi})_{A}^{B}= \frac{1}{2}\alpha_{0 , \pi}( \sigma_{1})_{A}^{B}\ , \quad ( \bar{M}_{0 , \pi})_{A}^{B}= \frac{1}{2}\alpha_{0 , \pi}\mathrm{i}( \sigma_{2})_{A}^{B}\ ,
f_{KP}(i, W') = \begin{cases}\max_{l \in \{0,1,\dots,\min\{d_i,\lfloor W'/w_i \rfloor\}\}}\{f_{KP}(i-1,W'- w_i l) + p_i l \}, i > 0 W' > 0, 0, . \end{cases}
\min\ (x_1-x_2 + x_3 - \bar x_1 + \bar x_2 - \bar x_3 + 1)/2 x_1- x_2 + x_3 - \bar x_1 + \bar x_2 - \bar x_3 \leq 0 x_i + \bar x_i =1, \, i=1,2,3 x_1,x_2,x_3,\bar x_1, \bar x_2, \bar x_3\ .
(a^\ast(\phi)\psi)_n(x_1, \ldots, x_n) := \frac{1}{\sqrt{n}}\sum^n_{j=1}\phi(x_j) \psi_{n-1}(x_1, \ldots, \widehat x_j, \ldots, x_n) (a(\bar\phi)\psi)_n(x_1, \ldots, x_n):= \sqrt{n+1}\int d x\ \bar\phi(x) \psi_{n+1}(x, x_1, \ldots, x_n)
\Omega_{i,j}=\begin{cases}10^{\xi_{k,j}/10}||X_{j}-X_{k}||^{-\alpha}i=k \displaystyle\frac{h{P}_{i}}{GP_{k}}10^{\xi_{i,j}/10}||X_{j}-X_{i}||^{-\alpha}i\in\mathcal{S}\end{cases}
G(t_0;\tilde{c})=\int_{\{\Psi<-t_0\}}|\tilde{F}|^2_h\tilde{c}(-\Psi)= \frac{G(T_1;c)}{\int_{T_1}^{+\infty}c(t)e^{-t}dt}\int_{t_0}^{+\infty}\tilde{c}(s)e^{-s}ds,
s^h :=s^{(m^2-1) \times m}= s([m,1])^{m-1}s([m-1,1])=s_1s_2s_3s_1 s^{hv}:=s^{(m^2-m-1) \times (m-1)}= s([m,1])^{m-2}s([m-1,1]) =s_1s_2s_1s_2s_3s_1,
e^{- 2 \phi ( x^{+}, x^{-})}= \frac{N}{4 8}\frac{\displaystyle \left( C_{3}^{+}+ \frac{\Lambda}{8}C_{4}^{-}\right) x^{+}+ \left( C_{3}^{-}+ \frac{\Lambda}{8}C_{4}^{+}\right) x^{-}}{\displaystyle 1 + \frac{\Lambda}{8}x^{+}x^{-}}+ a .
W_{\boldsymbol{\mathrm{T}}\circ\Theta^{(n,\sigma)}}=\frac{1}{n}C_{\boldsymbol{\mathrm{T}}\circ\Theta^{(n,\sigma)}}=\frac{1}{n}\left( \sum_{i,j=1}^{n}E_{ij}\otimes \boldsymbol{\mathrm{T}}\circ\Theta^{(n,\sigma)}(E_{ij})\right) &=\frac{1}{n}\left( \sum_{i,j=1}^{n}E_{ij}\otimes W_{ji}^{(n,\sigma)}\right),
W_{0}^{0}= | \xi |^{- 1}, \qquad W_{1}^{1}= | \xi |^{- 3}, \qquad W_{2}^{2}\approx \frac 1 4 | \xi |^{- 1}, \qquad W_{3}^{3}\approx \frac 9 4 | \xi |^{- 3}, \ldots
a*(b*c) = a*(bc + \pi_1(b,c)t + \pi_2(b,c)t^2+\dots) = abc + \pi_1(a,bc)t + \pi_2(a,bc)t^2 + \dots + a\pi_1(b,c)t + \pi_1(a,\pi_1(b,c))t^2 + \dots + a\pi_2(b,c)t^2 + \dots &= abc + \Big(a\pi_1(b,c) + \pi_1(a,bc)\Big)t + \Big(\pi_2(a,bc) + \pi_1(a,\pi_1(b,c)) + a\pi_2(b,c)\Big)t^2 + \dots,
q_i=q^{d_i},\, \, [n]_q!=\prod_{i=1}^n \frac{q^n-q^{-n}}{q-q^{-1}},\ \E_i^{(n)}=\frac{E_i^n}{[n]_{q_i}!}\ \ \ \F_i^{(n)}=\frac{F_i^n}{[n]_{q_i}!}.
f(m) \ast_R f(n) = (\underbrace{1_R+\dots+1_R}_{m}) \ast_R(\underbrace{1_R+\dots+1_R}_{n}) &= \underbrace{1_R \ast (\underbrace{1_R+\dots+1_R}_{n}) + \dots+1_R\ast (\underbrace{1_R+\dots+1_R}_{n})}_{m}&= \underbrace{1_R+\dots+1_R}_{mn}= f(m\ast n).
= \frac{3}{2}\times \frac{1}{2}\times \frac{4}{3}\times \frac{2}{3}\times \frac{5}{4}\times \frac{3}{4}\times \frac{6}{5}\times \frac{4}{5}\times \cdots \times ( \frac{2 0 1 4}{2 0 1 3}) \times \frac{2 0 1 2}{2 0 1 3}
d_1h_{i+1}+(-1)^{i}h_{i+1}d_1 =d_0h_i+(-1)^ih_id_0,\qquad 1\leq i\leq r-1, 0 =(-1)^rd_0h_r+h_rd_0, d_1h_1+h_1d_1 =g-f.
A_{1}:=\sup_{\eta_{k-1}<t<\eta_k}\biggl(\int_t^{\eta_k}v_0^{p}\biggr)^{1/p}\biggl(\int_{\eta_{k-1}}^t v_1^{-p'}\biggr)^{1/p'}\le\biggl(\int_{\eta_{k-1}}^{\eta_k}v_0^{p}\biggr)^{1/p}\biggl(\int_{\eta_{k-1}}^{\eta_{k}}v_1^{-p'}\biggr)^{1/p'}\le 1.
\left( \begin{array}{c c}{{f_{\alpha}^{I}}}&{{\bar{X}^{I}}}\\ \end{array}\right) = e^{K / 2}\left( \begin{matrix}{{- \frac{1}{z + \bar{z}}}}&{{1}}\\{{- \frac{i}{z + \bar{z}}}}&{{- i}}\\ \end{matrix}\right) \ ,
n_1\!\left(\sum_{w,h}\! k_{h,w}j_h\!+\!r_V b\!-\!j_p\!\right)\!\!=\!m\!\!\left(\!i_p\!-\!n_1\!-\!\sum_{w,h}k_{h,w}i_h \!-\!r_Va\!\right)\!+\!m n_1\left(\sum_{w,h}k_{h,w}+r_V n_2\right).
\begin{matrix}{\hat{\delta}F^{0 + a b}= 4 \bar{\cal N}^{0 \Lambda}F_{\Lambda}^{+ a b},}&{\hat{\delta}F_{i}^{+ a b}= 0 ,}\\{\hat{\delta}F^{0 - a b}= - 4{\cal N}^{0 \Lambda}F_{\Lambda}^{- a b},}&{\hat{\delta}F^{i - a b}= 0 .}\\ \end{matrix}
a_{ij}^1(y)= \begin{cases}a_{11}^0(y) + \frac{\delta \xi(y_1+y_2)}{r^0(y)}i=j=1, a_{22}^0(y) - \frac{\delta \xi(y_1+y_2)}{r^0(y)}i=j=2, a_{ij}^0(y) . \end{cases}
p_{x,y}(k)=4\left( xt-x+1\right)^{2}k^{2}+4(\allowbreak tx-yxt^{2}-xt^{2}-y+yt-t+yx)k +(t+y)^{2}+ty\left( yt-2t-2y\right) +4txy\left( t-1\right)
P\left(\mathsf{A},\frac{s_1+s_2}{2}\right) =\frac{1}{2}P(\mathsf{A},s_1)+\frac{1}{2}P(\mathsf{A},s_2) &=\frac{1}{2}\left(h(\mu_1)+s_1\Lambda(\mathsf{A},\mu_1)\right)+\frac{1}{2}\left(h(\mu_1)+s_2\Lambda(\mathsf{A},\mu_2)\right) &=h\left(\frac{1}{2}\left(\mu_1+\mu_2\right)\right)+ \Lambda\left(\mathsf{A},\frac{1}{2}\left(\mu_1+\mu_2\right)\right)
h[\epsilon] = -h_1[\epsilon]+h_1'[\epsilon] \lesssim{\sf R}_{\sf in}^{-\frac{1}{8}(2n-9)}(T-t)^{{\sf d}_1}\lambda^{-\frac{n+2}{2}}\sigma (1+|y|^2)^{-1-\frac{1}{16}(2n-9)}{\bf 1}_{|y|<2{\sf R}_{\sf in}}.
\sqrt{N}|\nabla U(q)| \geq \xi(q) \cdot \nabla U(q) = \sum_{k =1}^N A\alpha \xi_k(q) \cdot q_k |q_k|^{\alpha-2}+ \sum_{j <k}B \beta (\xi_j(q)-\xi_k(q)) \cdot \frac{q_k-q_j}{|q_k-q_j|^{\beta+2}}&\geq - D_2' |q|^{\alpha-1}+ B\beta |q_1 -q_2|^{-\beta-1}
< V_{j_{1}, m_{1}}( z_{1}) V_{j_{2}, m_{2}}( z_{2}) > = | z_{1}- z_{2}|^{- 4 \Delta_{1}}\left[ A ( j_{1}) \delta ( j_{1}+ j_{2}+ 1 ) + B ( j_{1}) \delta ( j_{1}- j_{2}) \right]
\frac{1}{|I|}\iint_{(-|I|,\, 0)\times I}\frac{|\tilde\mu(z)|^2}{-2{\rm Re}z}dxdy = \frac{1}{4|I|}\iint_{(-t,0)\times I}(2{\rm Re}z)^2 |Sh(z)|^2 dxdy &+ \frac{1}{|I|}\iint_{(-|I|,-t]\times I}\frac{|\mu_t (z + t)|^2}{(-2{\rm Re}z)}dxdy.
S_{1^{j}}( z ) = \sum_{{\scriptstyle k_{1}+ 2 k_{2}+ \cdots + m k_{m}= j}\atop{\scriptstyle k_{1}\geq 0 , \cdots , k_{m}\geq 0}}( - 1 )^{k_{2}+ k_{4}+ \cdots + k_{2 \left[ \frac m 2 \right]}}\frac 1{k_{1}! \cdots k_{m}!}z_{1}^{k_{1}}\cdots z_{m}^{k_{m}}
\frac{1}{u_{1}}+\frac{1}{u_{2}}+\frac{1}{u_{3}}+\cdot\cdot\cdot+\frac{1}{u_{n}}=\frac{1}{u_{1}-}\frac{u_{1}^{2}}{u_{1}+u_{2}-}\frac{u_{2}^{2}}{u_{2}+u_{3}-}\cdot\cdot\cdot\frac{u_{n-1}^{2}}{u_{n-1}+u_{n}},
\|f\|_{M_m^{r,s}(\mathbb{R}^d)}= \|m V_gf\|_{L_{x,\omega}^{r,s}(\mathbb{R}^{2d})}= \left(\int_{\mathbb{R}^d}\left(\int_{\mathbb{R}^d}|V_g f(x, \omega) m (x, \omega) |^r d x\right)^{\frac{s}{r}}d \omega\right)^{\frac{1}{s}}< \infty.
F(1,y) =\sum_{j=0}^{2N}f_j y^j = f_0+\ldots+f_{r-1}y^{r-1}+y^r\sum_{j=r}^{2N}f_jy^{j-r}&=y^r(1+ y^4)(1-y^4)^{2\lfloor \frac{N-2}{4}\rfloor}\cdot \sum_{h=0}^{\lfloor \frac{N-2}{4}\rfloor}\epsilon_{\lfloor\frac{N-2}{4}\rfloor -h}(y^{4}(1-y^4)^{-2})^h.
\begin{array}{rcl}[X^+_+,X^+_-]=0~, & k_\pm X^+_\pm k_\pm^{-1}=q_\pm X^+_\pm~,~~~~k_\pm X^+_\mp k_\pm^{-1}=-X^+_\mp~, & X^+_\pm X^-_\pm - q_\pm X^-_\pm X^+_\pm=\displaystyle\frac{1-k^{-2}_\pm}{1-q^{-1}_\pm}~, & X^+_\pm X^-_\mp-q_\pm X^-_\mp X^+_\pm=0~.\end{array}
\bigl(\mathcal{S}_N L_{M}\bigr)(n\,|\, b) - \bigl(\mathcal{S}_NL_{M}\bigr)(0\,|\, b) = \sum\limits_{i=1}^k (-1)^i \binom{n}{i}\bigl(\mathcal{S}_N L_{M-i}\bigr)(0\,|\, b) + +(-1)^{k+1}\sum\limits_{i_1=0}^{n-1}\sum\limits_{i_2=0}^{i_1-1}\cdots \sum\limits_{i_{k+1}=0}^{i_{k}-1}\bigl(\mathcal{S}_NL_{M-k-1}\bigr)(i_{k+1}\,|\, b).
\psi_S(x) =(x+1)^{n-3}\,\Big[x^3-(s_1+t_2-1)\,x^2-(2s_1+2t_2+1)\,x+3s_1+3t_2-1\Big] &=(x+1)^{n-3}\,(x-1)\,\Big[x^2-(s_1+t_2-2)\,x-3s_1-3t_2+1\Big] &=(x+1)^{n-3}\,(x-1)\,\Big[x^2-(n-4)\,x-3n+7\Big],
\beta^{(2)}_1=\lim_m\frac{\sum_{n=3}^m\frac{1-c_{2n}}{1+c_{1n}}}{\sum_{n=3}^m\frac{1-c_{1n}}{1+c_{1n}}},\quad\beta^{(3)}_1=\lim_m\frac{\sum_{n=3}^m\frac{(1-c_{1n})(1-c_{2n})}{1+c_{1n}}}{\sum_{n=3}^m\frac{1-c_{1n}}{1+c_{1n}}},
{\cal H}( p ) = - 2 i \varepsilon^{\mu \nu \rho}\varepsilon^{\alpha \beta \lambda}g_{\nu \alpha}\frac{\mu^{\epsilon}}{( 2 \pi )^{d}}\int{\cal D}q \frac{1}{( p + q )^{2}q^{2}}I_{\beta \mu \lambda \rho}( q ) ,
{F}^V_{{\rm dR},1}\Big(\frac{t}{\tau},-\frac{1}{\tau}\Big) =2^{N/2}\tau^{(\dim M -\dim V+1)/2}e^{\pi\sqrt{-1}nt^2/\tau}{F}^V_{{\rm dR},2}(t,\tau),{F}^V_{{\rm dR},3}\Big(\frac{t}{\tau},-\frac{1}{\tau}\Big) =\tau^{(\dim M -\dim V+1)/2}e^{\pi\sqrt{-1}nt^2/\tau}{F}^V_{{\rm dR},3}(t,\tau).
\nu((\alpha, \beta),(s, t),Z)=\frac{1}{r}\log \Phi(\alpha+ s,\beta+ t, Z)+(1-\frac{1}{r})\log \Phi(\alpha +\frac{1}{1-r}s, \beta +\frac{1}{1-r}t ,Z)-\log \Phi(\alpha,\beta,Z),
{\frac{\ddot{\hat{c}}}{\hat{c}}}+ 3{\frac{\dot{\hat{a}}_{0}}{\hat{a}_{0}}}{\frac{\dot{\hat{c}}}{\hat{c}}}= -{\frac{2}{\omega}}\zeta \left({\frac{\hat{c}}{c}}\right)^{2}\hat{\varrho}_{m}.
\gamma_s^{-\frac12}d(u_s^n-H_s^n) = \gamma_s^{-\frac12}\left( \rho_s + \mu_s + \frac12 \eta_s^2 - \frac38 \sigma_s^2 \right) u_s^n ds - \gamma_s^{-\frac12}\left( \frac12 \mu_s - \frac18 \sigma_s^2 \right) H_s^n ds & + \gamma_s^{-\frac12}\frac12 \sigma_s (u_s^n-H_s^n) dW_s^1 + \gamma_s^{-\frac12}Z_s dv_s^n .
\partial_{r}\Phi_{i}( r , \theta ) |_{r = r_{h}}= H_{1}( r = r_{h}, \theta ) = \partial_{r}H_{j}( r , \theta ) |_{r = r_{h}}= 0 \ , \i = 1 , 2 , 3 , 4 \ , \j = 2 , 3 , 4
\int_{X}|\tilde{F}-(1-b(\Psi))fF^{1+\delta}|_{h}^2e^{v(\Psi)-\delta \max\{\psi+T,2\log|F|\}}c(-v(\Psi)) \le \bigg(\sup_{X}e^{-u(-v(\Psi))}\bigg)\int_{X}\frac{1}{B}\mathbb{I}_{\{-t_0-B<\Psi<-t_0\}}|fF|^2_h<+\infty.
288V^{2}=|\begin{matrix}0&1&1&1&1\\ 1&0&d_{12}^{2}&d_{13}^{2}&d_{14}^{2}\\ 1&d_{21}^{2}&0&d_{23}^{2}&d_{24}^{2}\\ 1&d_{31}^{2}&d_{32}^{2}&0&d_{34}^{2}\\ 1&d_{41}^{2}&d_{42}^{2}&d_{43}^{2}&0\\ \end{matrix}|
\|f(t)-f(t')\|_{L^2}= \|S(t)(f(t)-f(t'))\|_{L^2}&\lesssim \|u\|_{L^2((t',t), L^4_x)}\|v_0\|_{L^2}+ \|u\|_{L^2((t',t), L^4_x)}^3 \to 0,t,t'\to \infty
\displaystyle \left\langle\!\!\left\langle p,\sum_{i=1}^r (\hat{b}^i-\xi^i)-\sum_{j=1}^s\hat{a}^j\right\rangle\!\!\right\rangle= \int_0^T\left[\sum_{i=1}^r (\hat{b}_{h_0}^{i}(t)-\xi_{h_0}^{i}(t))-\sum_{j=1}^s\hat{a}_{h_0}^{j}(t)\right]dt\leq0.
s = \epsilon^{1/\tilde{p}}\nabla f(x_{k+1}) \left(A_k\frac{(N^2-1)^{\frac{\tilde p-2}{2 \tilde p-2}}}{2N}\right)^{\frac{\tilde p-1}{\tilde p}}u = (x_{k+1}- x^\ast) \epsilon^{-\frac{1}{\tilde p}}\left(A_k\frac{(N^2-1)^{\frac{\tilde p-2}{2 \tilde p-2}}}{2N}\right)^{-\frac{\tilde p-1}{\tilde p}}\alpha_k \frac{\tilde p-1}{\tilde p}.
c= \frac{2 \left(1-r^2\right)^{d_{N}}(m_2 (1-r)+m_1 (1+r) - 2m_1 m_2 s_{N_n})}{m_1 m_2 \left((1+r)^{d_{N}+1}-(1-r)^{d_{N}+1}\right)}
1.\ \ \F(n,k,l) =\{n-l\choose k-l}\ \ \ \ \ \ \ \ \ \ \ \ \ 0\le l\le k. 2. \F(n,k,-l) =\ f(k,l){n\choose k}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\le l\le k, 3. \F(n,k,-l) =\ g\Big(n,k-\frac{l+1}2,\frac{l+1}2\Big)+f(k,l-1){n\choose k}\ \ \ 0\le l\le k.
\begin{array}[c]{c}(\frac{\partial}{\partial t}-\Delta_{B})\log|X|_{\omega}^{2}=\frac{1}{|X|_{\omega}^{2}}\left( -g^{Ti\overline{j}}g_{k\overline{l}}^{T}(\partial_{i}^{k}X)(\overline{\partial_{j}^{l}X})+\frac{\left\vert\bigtriangledown^{T}|X|_{\omega}^{2}\right\vert_{\omega}^{2}}{|X|_{\omega}^{2}}\right) \leq0.\end{array}
\chi^{a b}= \sum_{n = 1}^{\infty}\left( - 1 \right)^{n}\, \left( \Lambda^{n}\right)^{a b}+ W^{a c}\sum_{n = 0}^{\infty}\left( - 1 \right)^{n}\, \left( \Lambda^{n}\right)^{c b}\, ,
\langle J^{'}(u),v\rangle =\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}g\left( \frac{u(x)-u(y)}{\vert x-y\vert^{\alpha}}\right)\frac{v(x)-v(y)}{\vert x-y\vert^{\alpha+d}}dxdy+\int_{\mathbb{R}^{d}}g(u)v\dx -\int_{\mathbb{R}^{d}}K(x)f(x,u)v\dx,
\widehat a_{k-1,k}= \widehat \delta_k + \frac{\widehat \beta_k}{\widehat\alpha_{k-1}}= -\frac{\delta_{n-k}}{\delta_{n-k+1}}\left(\theta_{n-k+1}+ \alpha_{n-k+2}\frac{\beta_{n-k+1}}{\beta_{n-k+2}}\right) + \frac{\delta_{n-k}\alpha_{n-k+2}\beta_{n-k+1}}{\beta_{n-k+2}\delta_{n-k+1}}.
C^{k_1,\ldots,k_n}_{m_1,\ldots,m_n}:= \sum_{l_i=0 \atop i=1,\ldots,n}^{k_i}\left(\prod_{i=1}^n\binom{k_i}{l_i}(-1)^{l_i+1}(l_1+\cdots+l_i+1)^{m_i-1}\right).
\begin{gathered}X \mapsto \frac{a(Y)X + b(Y)}{c(Y)X + d(Y)}, \Y \mapsto \frac{\alpha Y + \beta}{\gamma Y + \delta}\begin{bmatrix}a(Y) b(Y) c(Y) d(Y)\end{bmatrix}\in PGL_2(k(Y)),\ \begin{bmatrix}\alpha \beta \gamma \delta\end{bmatrix}\in PGL_2(k).\end{gathered}
\Gamma_{i j}^{i}= \frac{1}{h_{i}}\frac{\partial h_{i}}{\partial u_{j}}, \qquad \Gamma_{i i}^{j}= ( 2 \delta_{i j}- 1 ) \frac{h_{i}}{h_{j}^{2}}\frac{\partial h_{i}}{\partial u_{j}}.
\begin{array}{l}{\psi_{a}( z ) \psi^{b}( w ) = \frac{\delta_{a}^{b}}{( z - w )}+ \cdots}\\{\}\\{\psi_{a}( z ) \psi_{b}( w ) = \psi^{a}( z ) \psi^{b}( w ) = 0 + \cdots}\\ \end{array}
u_{0\theta}= \sum_{n=-\infty}^{\infty}inu_{0n}(X,T)e^{in\theta}u_{0X}= \sum_{n=-\infty}^{\infty}u_{0nX}(X,T)e^{in\theta}u_{0\theta\theta}= \sum_{n=-\infty}^{\infty}-n^{2}u_{0n}(X,T)e^{in\theta}.
(\ldots, x_1, x_0)(w_n) \Phi = (\ldots, x_1, x_0p_n,z_1, z_2, \ldots, z_k) \delta \circ (s_n)\Psi = (\ldots, x_1, x_0)f_n \circ ((s_n)\Psi)^{-1}\circ (s_n) \Psi = (\ldots, x_1, x_0)f_n.
E_C^{(2)\;reg}=-\frac{M_0}{4\sqrt{\alpha}}-\frac{M_0^2 R}{8\pi\alpha}\Gamma(-1)-\frac{M_0}{2\pi\sqrt{\alpha}}\sum\limits_{n=1}^{\infty}n^{-1}K_1\left(\frac{2n M_0 R}{\sqrt{\alpha}}\right).
H_0 =\frac{1}{36}\left(\begin{array}{cccc}33 12 15 18 12 6 6 6 15 6 7 8 18 6 8 12 \end{array}\right)T =\frac{1}{30}\left( \begin{array}{cccc}-47 -16 -11 -6 -16 22 12 2 -11 12 31/3 -4/3 -6 2 -4/3 -4/3 \end{array}\right)
F^{1}_{r{\tilde{\theta}}}= -\,\frac{1}{2\lambda}\,{\dot{a}}~~,~~ F^{2}_{r{\tilde{\varphi}}}= \frac{1}{2\lambda}\,{\sin{\widetilde{\theta}}\,{\dot{a}}}~~,~~ F^{3}_{{\tilde{\theta}}{\tilde{\varphi}}}= \frac{1}{2\lambda}\,{\sin{\widetilde{\theta}}}\left(1 - a^2 \right)~~.
\begin{aligned}f(t,s)= F_{\boldsymbol{\omega}\boldsymbol{\eta}}\big(L^{-1}_{\boldsymbol{\omega}}(t),K^{-1}_{\boldsymbol{\eta}}(s),f\big(L^{-1}_{\boldsymbol{\omega}}(t),K^{-1}_{\boldsymbol{\eta}}(s)\big)\big), ~~~~~\forall~ (t,s) \in L_{\boldsymbol{\omega}}(SG') \times K_{\boldsymbol{\eta}}(SG''), \end{aligned}
\mathcal{N}_1\left[\underline{\phi}\right](\xi)= \int_{\mathbb R}J_i(\xi-y)\underline{\phi}(y)dy-\underline{\phi}(\xi) begin{align*}0.2cm] \geq 1-\frac{1-b}{2}e^{\eta \xi}\int_{\mathbb R}J_1(y)e^{-\eta y}dy-\left(1-\displaystyle\frac{1-b}{2}e^{\eta \xi}\right) begin{align*}0.2cm] = \frac{1-b}{2}e^{\eta \xi}\left(1-\int_{\mathbb R}J_1(y)e^{-\eta y}dy\right).
I_\lambda(\omega) =I_\lambda(\omega)-\frac{1}{2}I'_\lambda(\omega)\omega &=\frac{\lambda}{4}\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2dx+\left(\frac{1}{2}-\frac{1}{2_\alpha^*}\right)\displaystyle\int_{\mathbb{R}^N}b(x)|\omega(x,0)|^{2_\alpha^*}dx &<0.
V(r,\alpha_{1},...,\alpha_{n})=\sum\limits_{l=1}^{n}\sum\limits_{k=1,\textrm{}k\neq l}^{n}m_{l}m_{k}G\left(2r\sin{\left(\frac{1}{2}|\alpha_{l}-\alpha_{k}|\right)}\right)-\sum\limits_{l=1}^{n}m_{l}A^{2}r^{2}
D^{n}(x^{-1}x)|_{y=1}=0 =\sum_{j=0}^n{n \choose j}D^j(x^{-1})|_{y=1}D^{n-j}(x)|_{y=1}begin{align*}3pt] =\sum_{j=0}^n{n \choose j}(-1)^{j}x^{-1}(1-x^2)^{{\lfloor j/2\rfloor}}D^{n-j}(x)|_{y=1}.
\Vert f \Vert'_{\ell^{p,\infty}}:=\sup_{E\subset \mathbb{Z}^n,\,1\leq \mu(E)<\infty}\mu(E)^{\frac{1}{p}-\frac{1}{r}}(\sum_{n'\in E}|f(n')|^r)^{\frac{1}{r}}
Z_{g Y M_{2}}( M , G ) = \sum_{R}\, ( \dim R )^{2 - 2 p}\, e^{- \sum_{r = 1}^{n}\, g_{r}A C_{r}( R ) \, + \, \sum_{\{k_{i}\}}\, a ( k_{1}, k_{2}, \ldots ) \prod_{s}\, ( C_{s}( R ) )^{k_{s}}}
\langle \partial_{z}\Phi ( z ) \partial_{z}\Phi ( 0 ) \rangle = \lim_{\alpha \rightarrow 0}{\frac{1}{\alpha^{2}}}\partial_{x}\partial_{y}\langle e^{i \alpha \Phi ( x )}e^{- i \alpha \Phi ( y )}\rangle |_{{x = z}\atop{y = 0}}.
\big|\partial_{\bf x}^{\gamma}V_{\alpha}({\bf x})\big| \le \sum_{j\in Q_{\alpha}}\frac{Z\sqrt{2}\gamma_j!}{|x_j|}\Big(\frac{8}{|x_j|}\Big)^{|\gamma_j|}&\ +\sum_{j\in Q_{\alpha}}\sum_{k=1,k\neq j}^{N}\frac{\sqrt{2}(\gamma_j!+\gamma_k!)}{|x_j-x_k|}\Big(\frac{8}{|x_j-x_k|}\Big)^{|\gamma_j|+|\gamma_k|}\,.
d(\varphi_{m,m}(x),f(x))\le \sum_{i=n_0}^{m-1}d(\varphi_{i+1,m}(x),\varphi_{i,m}(x))+d(\varphi_{n_0,m}(x),f_{n_0}(x))+d(f_{n_0}(x),f(x))\le \le\sum_{i=n_0}^{m-1}\frac{1}{2^i}+\frac{\varepsilon}{3}+\frac{\varepsilon}{3}<\frac{1}{2^{n_0-1}}+\frac{\varepsilon}{3}+\frac{\varepsilon}{3}<\varepsilon
\sum_{n=0}^{\infty}\phi_{n+1,\lambda}(x)\frac{t^{n}}{n!}=\frac{d}{dt}\sum_{n=0}^{\infty}\phi_{n,\lambda}(x)\frac{t^{n}}{n!}&=\frac{d}{dt}e^{x(e_{\lambda}(t)-1)}=xe_{\lambda}^{1-\lambda}(t)e^{x(e_{\lambda}(t)-1)}&=\sum_{n=0}^{\infty}\bigg(x\sum_{k=0}^{n}\binom{n}{k}(1-\lambda)_{n-k,\lambda}\phi_{k,\lambda}(x)\bigg)\frac{t^{n}}{n!}.
\begin{array}{c}{( q^{-{\frac{\sigma^{3}}{2}}}\otimes \sigma^{+}+ \sigma^{+}\otimes q^{\frac{\sigma^{3}}{2}}) \downarrow \otimes \downarrow =}{= q^{1 / 2}\downarrow \otimes \uparrow + q^{- 1 / 2}\uparrow \otimes \downarrow}\end{array}
\sum_{\chi \in X_N}\chi(a)\psi(b) \alpha_{\chi,\psi}^{g,h}= \sum_{\chi \in X_N}\chi(a) \alpha_{\chi,\psi}^{g,\delta}+ \sum_{\chi \in X_N}\psi(b)\alpha_{\chi,\psi}^{\delta,h}.
{\Gamma^{\prime}}_{A^{\mu}\varphi_{1}\varphi_{2}}^{( n )}( p , p_{1}, p_{2}) = \Gamma_{A^{\mu}\varphi_{1}\varphi_{2}}^{( n )}( p , p_{1}, p_{2}) - i e ( p_{1}^{\mu}- p_{2}^{\mu}) \delta z_{e}^{( n )}( \xi )
\lim_{\epsilon\to +0}\left[K_{\mu\nu}\right]_{x^\perp =-\epsilon}^{x^\perp =\epsilon}=\left.\frac{1}{m_4^3}\left(T_{\mu\nu}-\frac{1}{d-1}g_{\mu\nu}g^{\alpha\beta}T_{\alpha\beta}\right)\right|_{x^\perp =0}
\frac{{\cal E}( g , M )}{V}= M^{4}( c_{0 0}+ c_{0 1}g^{2}+ c_{0 2}g^{4}+ . . . ) + M^{3}\Lambda ( c_{1 0}+ c_{1 1}g^{2}+ . . . ) + M^{2}\Lambda^{2}c_{2 0}+ . . . + M \Lambda^{3}c_{3 0}+ . . .
[dx]_{n}=4\pi\delta(1-\sum_{i=1}^{n}x_{i})\prod_{i=1}^{n}\frac{dx_{i}}{4\pi\sqrt{x_{i}}},[d^{2}k_{\perp}]_{n}=4\pi^{2}\delta(\sum_{i=1}^{n}\vec{k}_{\perp i})\prod_{i=1}^{n}\frac{d^{2}k_{\perp i}}{4\pi^{2}}.
h(t)(g(h(t))-\log t) > \frac{g(h(t))}{g(\lambda h(t))}\frac{t}{\lambda}+h(t)\log t\left(\frac{g(h(t))}{g(\lambda h(t))}-1\right) &\ge\frac{g(h(t))}{g(\lambda h(t))}\frac{t}{\lambda}+ct\left(\frac{g(h(t))}{g(\lambda h(t))}-1\right),
H = \tanh(c_2\tau - c_3) = \tanh(d_2 \tau - d_3) &=\tanh \left(\frac{2 c_3}{\tau_f}K \tau_f - c_3 \right) &= \tanh \left(c_3(2K-1) \right) K =\frac{1}{2}\left(\frac{\tanh^{-1}H}{c_3}+1\right)
(\mathbb{E}\times P^{(i,n)}_t)(A_t^{(i,n)}\ |\ |Z^*(t(i+1)/n)|=\lfloor t^{d+\varepsilon}\rfloor+j) \geq (\mathbb{E}\times P^{(i,n)}_t)(A_t^{(i,n)}\ |\ |Z^*(t(i+1)/n)|=\lfloor t^{d+\varepsilon}\rfloor+(j+1))
\langle \varphi^{2}(x)\rangle_{b}\approx -\frac{\Gamma^{-2}(\nu_0+1)}{2^{2\nu_0}\pi a^{D-1}S_D\sigma^{D-1}}\left(\frac{r}{a}\right)^{2\nu_0-n}\int_{ma}^{\infty}dz\frac{z^{2\nu_0+1}}{\sqrt{z^2-m^2a^2}}\frac{\bar K_{\nu_0}(z)}{\bar I_{\nu_0}(z)},\quad r\to 0,
H=\sum_{i=1}^{4}\left( -\frac{\partial^{2}}{\partial x_{i}^{2}}+\omega^{2}x_{i}^{2}\right) +\lambda \sum_{i<j}^3 \frac{1}{(x_{i}-x_{j})^{2}}+\frac{g}{(x_1+x_2+x_3- 3 x_4)^2}+\frac{\mu}{\sum_{i=1}^{4}x_{i}^{2}}
F_1 :=-\theta (\partial_z\alpha) (\nabla_H\pi) -(\Delta_H\theta)(\partial_z\alpha)v -(\Delta_H\theta)\alpha(\partial_z v), F_2 :=-2(\nabla_H\theta)\alpha\cdot(\nabla_Hv) -2\theta(\partial_z\alpha)(\partial_zv) - \theta(\partial_z^2\alpha)v.
\frac{1}{2\pi^2}\int_{^{(3)}_{\texttt{a}}}f(X_1,X_2) \frac{\sqrt{\rho_{\texttt{a}}(X_1,X_2)}}{O_{\texttt{a}}(X_1,X_2;q)}X_1X_2 &= \sum_{\substack{l_1, l_2\geq 0 l_1+l_2\leq m}}f\bigl(\boldsymbol{X}^{(m,3)}_{\texttt{a};l_1\omega_1+l_2\omega_2}\bigr) \hat{\Delta}^{(m,3)}_{\texttt{a};l_1\omega_1+l_2\omega_2},
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{\left(\tau,v_{s+k+1},\omega_{k+1}\right) \in \mathcal{B}^-}d\omega_{k+1}dv_{s+k+1}d\tau \leq & \leq C_d \left( s+k \right) T R^d \left[\frac{y}{\eta T}+ \left(\frac{\eta}{R}\right)^d + \theta^{d-1}\right]\end{aligned}
|\mathbb{A}^{i+1}_s-(1+\varrho^{i+1})\mathbb{A}^i_s|=|(d\mathcal{M}^{i+1})(\mathcal{M}^{i+1})^{-1}-(d\mathcal{M}^{i})(\mathcal{M}^{i})^{-1}-\varrho^{i+1}(d\mathcal{M}^{i})(\mathcal{M}^{i})^{-1}|.
A^{-1}_{\Lambda}=\left(\begin{array}{ccccc}\mathrm{Re}\left( \lambda^{\vee}_1 \right) \mathrm{Re}\left( \lambda^{\vee}_{2d}\right) \mathrm{Im}\left( \lambda^{\vee}_1 \right) \mathrm{Im}\left( \lambda^{\vee}_{2d}\right) \end{array}\right).
(K\circ v_j) f_0^{(r)}+ \sum_{q=1}^r \binom{r}{q}\sum c_{m_1, \ldots, m_q}\frac{K^{(M_q)}\circ v_j}{F^{M_q}}f_0^{(r-q)}\prod_{s=1}^q \big(f_0^{(s-1)}\big)^{m_s},
\begin{aligned}\mathop{\max}\limits_{\{{\mu}_k\}_{k = 0}^{K - 1}}\ \frac{1}{{K}}\sum\limits_{k = 0}^{K - 1}{{{\log}_2}}\left({1 +{\mu_k}{{\left\|{{{{\bf{\bar e}}}_k}}\right\|}^2}}\right){\rm{s.t.}}\ \ \sum\limits_{k = 0}^{K - 1}{{\mu_k}}\le KP. \end{aligned}
{\cal{A}}^{\,++\,class}_{nonrelat}\,=\,i\frac{e^2}{m \theta}\frac{\vec{s}\wedge\vec{q}}{\vec{q}^2}\,+\,\frac{e^4}{4 \pi m \theta^2}\ln\Biggl(\frac{-\Lambda^2}{\vec{p}^2}\Biggr)
X_{i,j}(x,x_{1})\left|J\right\rangle \simeq \sum_{n=0}^{|J|}\sum_{\substack{J' \subseteq J |{J'}|=n}}\mathcal{G}^{n}(i,J,{J'})W_{n+i+1,n+j+1}(x)\left|J\setminus{J'}\right\rangle \,,
e_{2}\psi = \left( \begin{array}{c}{{-{\psi_{2}}}}\\{{{\psi_{1}}}}\\{{-{\psi_{4}^{*}}}}\\{{{\psi_{3}^{*}}}}\\ \end{array}\right) \quad o r \quad e_{3}\mathbf{)}e_{1}\psi = \left( \begin{array}{c}{{{\psi_{2}}}}\\{{{\psi_{1}}}}\\{{{\psi_{4}^{*}}}}\\{{-{\psi_{3}^{*}}}}\\ \end{array}\right)
\nu [ A ] \; = \; \lim_{V_{4}\rightarrow \infty}\; \frac{g^{2}}{8 \pi^{2}}\; \int_{V_{4}}\; d^{4}x \; \partial_{\mu}K_{\mu}\; = \; \lim_{V_{4}\rightarrow \infty}\; \frac{g^{2}}{8 \pi^{2}}\; \int_{\partial V_{4}}\; d^{3}\sigma \; \hat{n}_{\mu}K_{\mu}\; .
\sum_{I:J \in \varphi(I)}\lambda^{|I|-|J|}f(I,J) = \sum_{I:J \in \varphi(I)}\frac{1}{(1+\lambda)^{|\tilde{I}_s|}}\le \sum_{I:J \in \varphi(I)}\frac{1}{(1+\lambda)^{|\gamma(I)|/4}},