harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
V_{Q}^{\prime}\, = \, \frac{{\hbar}^{2}}{8 M}\left\{R + \frac{( \partial_{a}f ) ( \partial^{a}f )}{4 M}\left[ 3 g_{\phantom{0}, 0}^{i j}\, g_{i j , 0}-{\left( g^{i j}\, g_{i j , 0}\right)}^{2}\right] \right\}.
R_p^{s,u}= \left[\log_2\left(1+\frac{\mathbf{w}_1^H\mathbf{h}_{pp}\mathbf{h}_{pp}^H\mathbf{w}_1}{\sigma^2+\mathbf{w}_2^H\mathbf{h}_{sp}\mathbf{h}_{sp}^H\mathbf{w}_2}\right)\right. &\left.-\log_2\left(1+\frac{\mathbf{w}_1^H\mathbf{h}_{ps}^H\mathbf{h}_{ps}\mathbf{w}_1}{\sigma^2}\right)\right]^+,
g(x)= \begin{cases}x, x<Q^{-1}(-1) \, \, x>1, T_1(x) x\in [-1,1], h T_{-1}^{-1}h^{-1}(x), x\in Q^{-1}([-1,T_{-1}(1)], , x\in [Q^{-1}(T_{-1}(1)), -1]. \end{cases}
X_j(t) = R_\omega(t) \, Q\Big(t + \frac{2\pi j}{n}\Big), 0\leq j\leq n-1, R_\omega(t)=\begin{bmatrix}\cos(\omega t) -\sin(\omega t) 0 \sin(\omega t) \cos(\omega t) 0 0 0 1\end{bmatrix},
U^*( \hat{x}_2 (1-UU^*) \hat{x}_4 - \hat{x}_4 (1-UU^*) \hat{x}_2 )U = U^*( \hat{x}_1 (1-UU^*) \hat{x}_3 - \hat{x}_3 (1-UU^*) \hat{x}_1 )U,
Q_{3,\vec{v}_{s,t,f,0}^{(1)}}= \left(\left(\begin{array}{rrr}-s 0 1 1 0 0 -t 1 0\end{array}\right), \left(\begin{array}{rrr}0 1 0 0 t 1 1 s 0\end{array}\right), \left(\begin{array}{rrr}1 0 0 0 1 0 0 0 1\end{array}\right)\right).
r(\nu,\theta) =2\cos(( d+1)\nu)\cos(( d+1)\theta) + \ldots + 2\cos( m\nu)\cos( m\theta)\,, = \frac 12 \big[\mathbf D_{m}(\nu-\theta)+\mathbf D_{m}(\nu+\theta)-\mathbf D_{d}(\nu-\theta)-\mathbf D_{d}(\nu+\theta) \big]\,.
\nabla_k(\nabla_iV^{(\beta,b)}\nabla_jV^{(\gamma,c)}-\nabla_iH^{(\beta,b)}\cdot \nabla_jH^{(\gamma,c)}) &=\nabla_k\nabla_iV^{(\beta,b)}\nabla_jV^{(\gamma,c)}-\nabla_k\nabla_iH^{(\beta,b)}\cdot \nabla_jH^{(\gamma,c)}&\quad+\nabla_iV^{(\beta,b)}\nabla_k\nabla_jV^{(\gamma,c)}-\nabla_iH^{(\beta,b)}\cdot \nabla_k\nabla_jH^{(\gamma,c)}.
I(p) = \frac{ck^2q^{k - 2}}{(1 - q^k)^2}, &\frac{dI}{dp}= -ck^2\left(\frac{(k - 2)q^{k-3}+ (k + 2)q^{2k - 3}}{(1 - q^k)^3}\right), &\mathrm{E}\left[\frac{d^3\ell}{dp^3}\right] = \frac{ck}{q^3}\left(\frac{k(k + 1)q^k(1 - q^k) + 2(kq^k + q^k - 1)^2}{(1 - q^k)^3}- \frac{2}{1 - q^k}\right).
\frac{n^{4d}}{m^4}\sum_{l = 2}^m\mathbb{E}_0\left[\left(D^S_{nl}\right)^4 \right] \;=\; \frac{n^{4d}}{m^4}\left\{\binom{m}{2}\mathbb{E}_0\left[\left( \bar{U}^{(12)}_h\right)^4\right] + 6 \binom{m}{3}\left(\mathrm{Var}_0\left[ \bar{U}^{(12)}_h\right]\right)^2 \right\}.
h^{(n+1)}(s) = (1-\tau)h^{(n)}(s) + \max_{a\in \mathcal{A}(s)}\left[ g(s, a) + \tau\sum_{\mathbf{H}'}\mathrm{Pr}(\mathbf{H}'|\mathbf{H})h^{(n)}(s')\right] - \Lambda^{(n+1)}(s_0).
u_{i i}= - \frac{{1-2d}}{1+ \beta d}(A + \frac{1}{2}M) + \frac{{\beta \big(-\sum\limits_j{u_j d_j}- \varphi \big)}}{1+ \beta d}+ \frac{{-\sum\limits_j{u_j d_{ji}}- D_i\varphi}}{{d_{i}}}.
\Big(\sum_{k=k_0}^\infty \Big(\sum_{r=1}^k b_r \sigma_{r \iota_0}^{-1}h_{r\iota_0}^{-\frac{1}{p_1}}\Big)^{q_2}\Big(\sum_{r=1}^{k\iota_0}h_r^{-\frac{q_2'}{p_2}}\tau_r^{-q_2'}\Big)^{-q_2}h_{k\iota_0}^{-\frac{q_2'}{p_2}}\tau_{k\iota_0}^{-q_2'}\Big)^{\frac{1}{q_2}}
G^{R}( k ) ={\frac{N^{2}( q^{2})^{\Delta - 2}\Gamma ( 3 - \Delta ) \epsilon^{2 ( \Delta - 4 )}}{8 \pi^{2}2^{2 \Delta - 5}\Gamma ( \Delta - 2 )}}\left[ \cos{\pi \Delta}- i \sin{( \pi \Delta )}\mathrm{s g n}\omega \right] .
\overrightarrow{\xi}= \frac{1}{\textsf{P}_0}\biggl((ch(lm)P_0+sh(lm)m)\overrightarrow{\bigtriangledown}+m^{-1}sh(lm) \overrightarrow{P}(\overrightarrow{P}.\overrightarrow{\bigtriangledown})\biggr)
\begin{array}[c]{c}\Delta_{b}\frac{f}{f+a}=\frac{a}{\left( f+a\right)^{2}}\Delta_{b}f-\frac{2a}{\left( f+a\right)^{3}}\left\vert \nabla_{b}f\right\vert^{2}=-2\left\langle \nabla_{b}\varphi,\nabla_{b}\frac{f}{f+a}\right\rangle-\frac{a\lambda_{1}}{f+a}\frac{f}{f+a}.\end{array}
H(u,v)=\frac{-a^{2}+b^{2}-\frac{4u^{2}}{a^{2}}+\frac{4v^{2}}{b^{2}}}{a^{2}b^{2}\sqrt{(1+\frac{4u^{2}}{a^{4}}+\frac{4v^{2}}{b^{4}})^{3}}}.
\gamma_{\textrm{FHN}}^{\textrm{NOMA}}=\frac{P_\textrm{N}{|h_{\textrm{BN}}|^2}}{{d_{\textrm{BN}}^{\alpha}\sigma_{\textrm{N}}^2}}+\frac{\beta_{\textrm{F}}\eta P_\textrm{B}|h_{\textrm{BF}}|^2 |h_{\textrm{NF}}|^2}{d_{\textrm{BF}}^{\alpha}d_{\textrm{NF}}^{\alpha}\sigma_{\textrm{N}}^2}.
\Lambda_4(\varphi)=7\left (-5{\sin{\varphi \over 9}\over \sin{\pi \over 18}}+18\sin{3\varphi \over 9}-5{\sin{5\varphi \over 9}\over \sin{5\pi \over 18}}+{\sin{7\varphi \over 9}\over \sin{7\pi \over 18}}\right ).
X_{(j,1)}\cdots X_{(j,k)}\cdots X_{(j,\kappa_j)}f(p) - X_{(j,1)}\cdots X'_{(j,k)}\cdots X_{(j,\kappa_j)}f(p) & = X_{(j,1)}\cdots X_{(j,k-1)}[ X_{(j,k-1)}, X_{(j,k)}- X'_{(j,k)}] X_{(j,k+1)}\cdots X_{(j,\kappa_j)}f(p) & + \cdots + [X_{(j,1)}, X_{(j,k)}- X_{(j,k)}'] X_{(j,2)}\cdots \widehat{X_{(j,k)}}\cdots X_{(j,\kappa_j)}f(p)
\rho :=\frac{c_{a4}c_{b2}}{e_{a2}e_{b4}}, \delta:=\frac{c_{a3}}{e_{a2}}-\frac{e_{b3}c_{a4}}{e_{a2}e_{b4}}\tilde{\rho}:=\frac{c_{a3}c_{b2}}{e_{a2}e_{b3}}, \tilde{\delta}:=\frac{c_{a4}}{e_{a2}}-\frac{e_{b4}c_{a3}}{e_{a2}e_{b3}}.
\begin{aligned}\Delta r_1 =\ln \Big(1+\frac{P-P_N^*}{n_1^e}\Big) =\ln \Big(1+\frac{P-P_N^*}{(\frac{N_0}{|h_1|^2}+P_1^*)e^{(K-1)r^*}}\Big) =\ln \Big(1+\frac{(P-P_N^*)|h_1|^2}{N_0 e^{Kr^*}}\Big),\end{aligned}
\xi'(x)\frac{p-2}{2}\frac{d}{dx}\log \big(h_1'^2(x)+(m-1)\cos^2h_1(x)\big)=(2-p)\tanh x \xi'(x), h'(x)(p-2)\frac{d}{dx}\big( \frac{h_1'(x)\xi'(x)-\frac{m-1}{2}\sin 2h_1(x)\xi(x)}{h_1'^2(x)+(m-1)\cos^2h_1(x)}\big) = \frac{p-2}{\cosh x}\frac{d}{dx}\big(\frac{\cosh x}{m}(\xi'(x)-(m-1)\tanh x \xi(x))\big).
\frac{\phi_{x,y-1,z}(a,b)}{\phi_{x,y-1,z-1}(a,b)}\frac{\phi_{x,y,z-1}(a,b)}{\phi_{x,y,z}(a,b)}+\frac{\phi_{x+1,y-1,z-1}(a,b)}{\phi_{x,y-1,z-1}(a,b)}\frac{\phi_{x-1,y,z}(a,b)}{\phi_{x,y,z}(a,b)}=1.
S^I(\tau)\;=\;\int^{\infty}_{-\infty}\Bigl[\;-{1\over2}\;\varepsilon\;{x^2\over \tau^2}\,\left({\partial_{x}\phi}\right)^2\;+\;{\lambda\over4}{1\over{1+{x^2\over \tau^2}\;(1-\varepsilon)}}\;\phi^4\;\Bigr]\;dx.
\sum_{k=0}^{n-1}\left(-\frac{1}{2}\right)^k P_k\left(n-1, \frac{n}{2}\right) \left(\zeta\left(k-n+2, \frac{n}{4}\right) -\zeta\left(k-n+2, \frac{n+2}{4}\right)\right) = 2^{1-n}\Gamma^2\left(\frac{n}{2}\right).
\zeta_i + \zeta_e = \frac{1}{td}\sum\limits_{\substack{uw \in K}}(uw \in R) + \frac{1}{t^2}\sum\limits_{\substack{uw \in K}}(uw \in R) &= \mathbb{E}[|R \cap (K[X]\cup K[Y])|/td] + \mathbb{E}[|R \cap \nabla(X,Y)|/t^2] &= \mathbb{E}[\xi_i(R)+\xi_e(R)] &< 1+o(1),
||P'|-|Q'|| \leqslant \left| \frac{c-a}{c}(P')^{-1}-\frac{b-a}{b}(Q')^{-1}\right| \leqslant \left|1-\frac{a}{c}\right|\frac{1}{|P'|}+\left|1-\frac{a}{b}\right|\frac{1}{|Q'|}&\leqslant \frac{1}{12}\left|1-\frac{a}{c}\right|+\frac{1}{11}\frac{2}{|b|}< \frac{5}{48}+\frac{5}{48}= \frac{5}{24},
\tilde{R}_{ij}(s)=\frac{1}{N_{T}}\sum_{r=1}^{N_{T}}R_{ij}^{(r)}(s)=\frac{1}{N_{T}}\sum_{r=1}^{N_{T}}\frac{1}{N_{S}}\sum_{t=1}^{N_{S}}X_{i}^{(r)}(t)X_{j}^{(r)}(t+s)
\int_{\hat{s}\cdot\hat{n}<0}L(\vec{r},\hat{s},t)\hat{s}\cdot\hat{n}d\Omega=\int_{\hat{s}\cdot\hat{n}>0}R_{F}(\hat{s}\cdot\hat{n})L(\vec{r},\hat{s},t)\hat{s}\cdot\hat{n}d\Omega
u(x,y)=C_n\int_{\mathbb{R}^n}\frac{y^{1-\beta}\psi(t)\,dt}{(y^2+|x-t|^2)^{n/2+(1-\beta)/2}},~ C_n=\frac{\Gamma\left(\frac n2+\frac{1-\beta}2\right)}{\pi^{n/2}\Gamma\left(\frac{1-\beta}2\right)},
\frac{1}{2}\sum_{m , n = - \infty}^{\infty}: \tilde{\alpha}_{- ( m + n )}^{( A )}\tilde{\alpha}_{n ( A )}: ( c_{m}+ \tilde{c}_{- m}) | B ; p , 0 \rangle = 0 ; A \in \{0 , . . . , 2 5 \}
\Delta_{\varepsilon ,\overline{\varepsilon},s_p,\overline{s}_p}=\frac{m-i\varepsilon \widehat{p}}{2m}\cdot\frac{m-i\overline{\varepsilon}\overline{p}}{2m}\cdot \left(\frac 12+2s_p\sigma_p^{(1/2)}\right) \left( \frac12+2\overline{s}_p\overline{\sigma}_p^{(1/2)}\right)
{\cal O}_{k}= \delta_{k , 0}+ \sum_{j = 0}^{\infty}\frac{( L_{- 1})^{j}L_{k + j}}{( k + j + 1 ) !}, \bar{{\cal O}_{k}}= \delta_{k , 0}+ \sum_{j = 0}^{\infty}\frac{( \bar{L}_{- 1})^{j}\bar{L}_{k + j}}{( k + j + 1 ) !}.
{\cal L}_\rho = - \frac{1}{4}F_{\mu\nu}^a F^{a\mu\nu}- \frac{1}{4}(G_{\mu\nu}^a + f_{abc}F_{\mu\nu}^b \rho^c) (G^{a\mu\nu}+ f_{abc}F^{b\mu\nu}\rho^c) + \frac{m}{2}\epsilon^{\alpha \beta \gamma}F_{\alpha \beta}^a \phi_{\gamma}^a
\sharp ({\mathcal A}_p(H) \cap{\mathcal A}_q(H)) = \frac{X}{(pq)^n}\left( 1 + O\left(\frac{pq}{H}\right)\right) \sharp{\mathcal A}_p \sharp{\mathcal A}_q & = \frac{X}{\delta^2}+ O\left( \frac{X}{p}\right) + O\left( \frac{X}{q}\right) + O\left( pq X/H\right)= \frac{X}{\delta^2}+ R_{p,q},
a_n = \operatorname{mex}\{a_k, b_k \;|\; k < n\}b_n = \min\{b \geq a_n \;|\; b\!\neq\! b_k \;\;(b \!-\! b_k) \!-\! (a_n \!-\! a_k) \!\geq\! f(a_n) \forall k\!<\!n\}.
P^{+}= -{\frac{g^{2}}{4}}\int d x^{-}T r ( \partial_{-}^{2}\sigma ) \sigma ={\frac{g^{2}}{4}}\int d x^{-}T r ( \partial_{-}\sigma )^{2}={\frac{g^{2}}{8}}\int d x^{-}( \partial_{-}s )^{2}={\frac{g^{2}}{4 a}}
Q_{m-1}^{A}=\left(\sum_{i=m-1}^b Q_i^{A}+L^A\right)-\left(\sum_{i=m}^b Q_i^{A}+L^A\right) &\leq \left(\sum_{i=m-1}^b Q_i^{B}+L^B\right)-\left(\sum_{i=m}^b Q_i^{B}+L^B\right)= Q_{m-1}^{B}.
A_{(b)}=\frac{g^4}{\alpha'^4}\int d^{26}k~ \frac{1}{(P_1 +P_4)^2 - \frac{1}{\alpha'}}\, \frac{1}{k^2 - \frac{1}{\alpha'}}\, \frac{1}{(P_1 +P_4 -k)^2 - \frac{1}{\alpha'}}\, \frac{1}{(P_1 +P_4)^2 - \frac{1}{\alpha'}},
X_{t}^{\mathcal{P},{M_{\mathcal{P}}}}=x +\int_{0}^{t}b(X_{\tau (r)}^{\mathcal{P},{M_{\mathcal{P}}}})dr+\int_{0}^{t}\int_{B_{M_{{\mathcal{P}}}(r)}}{c}(z,X_{\tau (r)-}^{\mathcal{P},{M_{\mathcal{P}}}})N(dz,dr).
{n \choose i}_r = f_{n-i+1}\cdot \begin{cases}{r+2n\choose r+2i-1}, \mbox{$n\ge\operatorname{max}\{0,\lceil-r/2\rceil\}$,}-{-r-2i\choose -r-2n-1}, \mbox{$0\le n<\lceil-r/2\rceil$.}\end{cases}
R_{12}= R^{i_{1}i_{2}}_{j_{1}j_{2}}= \delta^{i_{1}}_{j_{1}}\delta^{i_{2}}_{j_{2}}\left( 1+(q-1) \delta^{i_{1}i_{2}}\right) + (q-q^{-1})\delta^{i_{1}}_{j_{2}}\, \delta^{i_{2}}_{j_{1}}\,\theta (i_{1}- i_{2})
\big(\boldsymbol{V}_R^{-1}\big)_{jk}=\frac{(\tilde{V}_R)_{jk}}{\det(\tilde{V}_R)}=\frac{\partial c_k}{\partial \lambda_j}\frac{1}{\prod\limits_{\stackrel{r=1}{r\neq j}}^n ( \lambda_j- \lambda_r)}.
{\rm Ind}^{(1)}(\vartheta_a,S,U) ={\rm Tr}\; SU[\vartheta_a,P_-] &={\rm Tr}\; S(P_+-P_-)(\vartheta_a P_--P_-\vartheta_a) &={\rm Tr}\; P_-S(P_+-P_-)\vartheta_a+{\rm Tr}\; SP_-\vartheta_a &={\rm Tr}\; SP_+\vartheta_a+{\rm Tr}\; SP_-\vartheta_a={\rm Tr}\; S\vartheta_a=0,
D_x^2 (\Delta^i)^{-1}(p,p) = D_x^2 \L_(p,p) - D_x^2 \L_{R_i}( p, p) + O(\epsilon^2) D_x(\Delta^i)^{-1}(p,p) = D_x \L_{R_i}( p, p) + O(\epsilon)
i \hbar \,\left\langle\frac{\partial A}{\partial t}\right\rangle_{\scriptstyle \! \! \Psi}=\left\langle [A,H]_{\rm extra}\right\rangle_{\scriptstyle \! \Psi}=-\left\langle \; [A,H]_{\rm regular}\right\rangle_{\scriptstyle \! \Psi}\; ,
\begin{aligned}\Vert \Lambda^\alpha(uv)- u\Lambda^\alpha v\Vert_{\dot{B}^{\frac{N}{2}+1-r}_{2,1}}\lesssim_{\alpha,r,r_1,N}\Vert u \Vert_{\dot{B}^{\frac{N}{2}+1+\alpha-r_1}_{2,1}}\Vert v \Vert_{\dot{B}^{\frac{N}{2}+r_1-r}_{2,1}}.\end{aligned}
p_{{\boldsymbol{\mu}},\boldsymbol{\Sigma}}({\bf{x}}) = N_{\boldsymbol{\Sigma},\nu}\Big[ 1 + \frac{1}{\nu}({\bf{x}}-{\boldsymbol{\mu}})^\top \boldsymbol{\Sigma}^{-1}({\bf{x}}-{\boldsymbol{\mu}}) \Big]_+^{-\frac{\nu + d}{2}},
\frac{d^mG_0}{dv^m}=P_{n-m+2}-\frac{\lambda v^{n-m+2}}{4\kappa^2}\sum_{l=1}^{n+3-m}\frac{(-1)^l(n+1-l)(n+2-l)^2}{(l-1)!(n+3-m-l)!}\hat{I}_{n,l-1}.
S_{DBI}\sim \int dx_4^2 \,\, d\Omega_2 \,\, e^{-\Phi_D}\sqrt{g}\left [1+ \frac{1}{2}g^{\mu\nu}g_{mn}\partial_\mu x^m \partial_\nu x^n - \frac{1}{2}g^{\mu\tau}g^{\nu\rho}F_{\mu\nu}F_{\tau\rho}\right].
\begin{array}{ccl}K_{(v:m)}= K_{(v:m)}\otimes K_{(1:m)}&= K_{(v:m)}\otimes \left(\bigoplus_{t=1}^{\frac{m-1}{2}}N_t\right) &= \bigoplus_{t=1}^{\frac{m-1}{2}}\left(K_{(v:m)}\otimes N_t\right) \end{array}
M_{mn}=\displaystyle\bigoplus\limits_{\begin{subarray}{c}1\leq i\leq l_m 1\leq j\leq l_n\end{subarray}}\mathfrak{m}_{\tilde{l}_{m-1}+i,\tilde{l}_{n-1}+j}, \textstyle \ 1\leq n<m\leq r.
\partial_y(\mathcal{L}Y^-)=-e_0Y^-,\partial_y(\mathcal{L}\textbf{}Y^+)=e_0Y^+, Y^{+}(y)=Y^-(-y),\quad\int_{\mathbb{R}}Y^\pm=0, (Y^-,\mathcal{L}Y^-)=(Y^+,\mathcal{L}Y^+)=0, (Y^-,\mathcal{L}Y^+)=(Y^+,\mathcal{L}Y^-)\not=0.
\int_{-1}^{1}\left((-\Delta)^{\frac{1}{4}}u (-\Delta)^{\frac{1}{4}}\phi + (-\Delta)^{\frac{1}{4}}v (-\Delta)^{\frac{1}{4}}\psi\right)dx = \lambda \int_{-1}^{1}\left( |u|^{p-2}|v|^{q}u\phi+|v|^{q-2}|u|^{p}v \psi\right) dx &+ \int_{-1}^{1}\left(h_1(u,v)\phi +h_2(u,v)\psi\right) e^{u^2+v^2}dx
n_{\mathbf{s}'}\geq \sum_{i=1}^{k}\;\left\lceil \frac{s'_i}{2^{i-1}}\right\rceil = \sum_{i=1}^{k}\left\lceil \frac{s_{k-i+1}}{2^{i-1}}\right\rceil = \sum_{i=1}^{k}\left\lceil \frac{s_{i}}{2^{k-i}}\right\rceil.
\begin{array}{ll}\displaystyle\int_{\Omega}|\nabla U_*|^2 =\displaystyle\int_\Omega |\nabla u_0|^2+\int_\Omega |\nabla V|^2-\displaystyle 2\int_\Omega \nabla V\nabla u_0 =\displaystyle\int_\Omega |\nabla u_0|^2+\int_\Omega |\nabla V|^2-\displaystyle 2\int_\Omega u_0^{2^*-1}V. \end{array}
m=\langle \sigma_0 \rangle =\frac{e^hg_n^{\gamma}(+)-e^{-h}g_n^{\gamma}(-)}{e^hg_n^{\gamma}(+)+e^{-h}g_n^{\gamma}(-)}=\frac{e^hx_n^{\gamma}-1}{e^hx_n^{\gamma}+1}.
\bigg(\frac{Q}{\phi(Q)}\frac{1}{2^j X}\sum_{\substack{X < m < 2^j X \gcd(m,Q)=1}}|h(m)|^2 \bigg)^{1/2}\ll \bigg( \frac{1}{\log T}\frac{Q}{\phi(Q)}\prod_{\substack{p \leq T p\nmid Q}}\left(1 + \frac{|h(p)|}{p}\right)\bigg)^{1/2}.
\begin{aligned}f_1 \lambda_1 = \lambda_1 \sigma_{k-1}(\lambda|1) = \sigma_k(\lambda) - \sigma_{k}(\lambda|1) &\geq \frac{\mu_2 (k-1)}{\mu_1^2 (n-1)(n-2+k)(n-k+1)}\lambda_1 \sum_i f_i &\geq \frac{\mu_2^2 (k-1)}{\mu_1^2 (n-1)(n-2+k)(n-k+1)}\lambda_1 \sum_i f_i,\end{aligned}
t=-(\frac{E_{0}}{B_{0}}+\frac{A_{0}D_{0}}{B_{0}^{2}}+\frac{C_{0}A_{0}^{2}}{B_{0}^{2}})ln(1-\frac{X}{X_{max}})-\frac{C_{0}}{2B_{0}}X^{2}-(\frac{D_{0}}{B_{0}}+\frac{C_{0}A_{0}}{B_{0}^{2}})X
\langle \mathrm{V o l}^{2}\rangle Z ( \Lambda ,{\cal M}) = \langle \mathrm{V o l}^{2}\rangle_{g r}Z_{g r}( \Lambda ,{\cal M}) + \overline{{\langle \mathrm{V o l}^{2}\rangle_{g r}Z_{g r}( \Lambda ,{\cal M})}}.
\frac{1}{2}\left [ \left( \frac{\partial}{\partial t}- \frac{\partial}{\partial x}\right)^\beta +\left( \frac{\partial}{\partial t}+ \frac{\partial}{\partial x}\right)^\beta \right] q(x,t) =[\delta(x-t)+\delta(x+t)]\frac{t^{-\beta}}{\Gamma(1-\beta)}.
f(a_i^{(n)}, b_i^{(n)}) = f(\bar a_i, \bar b_i) + \frac{1}{n^r}\sum_{i = 1}^k \left(\frac{\partial f}{\partial a_i}(\bar a_i, \bar b_i) \tilde a_i^{(n)}+ \frac{\partial f}{\partial b_i}(\bar a_i, \bar b_i) \tilde b_i^{(n)}\right) + \sum{}^*.
d_{m-3,k}^{m,l}= \left(\left(\theta+m-1\right)\left(\theta+m-2\right)\left(\theta+m-3\right)\right)^{k}\binom{l}{l-k}& \cdot\left(\left(\alpha + \theta \right)_{m-1\uparrow}+\left(\theta+m-1\right)\left(\alpha+\theta\right)_{m-2\uparrow}+\left(\theta+m-1\right)\left(\theta+m-2\right)\left(\alpha+\theta\right)_{m-3\uparrow}\right)^{l-k}.
\frac{\partial^{4}W}{\partial x_{1}^{4}}+2\frac{\partial^{4}W}{\partial x_{1}^{2}\partial x_{2}^{2}}+\frac{\partial^{4}W}{\partial x_{2}^{4}}=\frac{2\rho h\omega^{2}}{D}W=:\lambda^{4}W
\xi_{n,1}(s) = \xi_{n,1}^{()}(s)\cdot\xi_{n,1}^{()}(s) \xi_{n,1}^{()}(s)= \frac{\xi\left(\frac{n+3}{2}-2s \right)}{\xi\left(2s+\frac{n-1}{2}\right)}\frac{\prod_{j=1}^{\frac{n-3}{2}}\xi(n+3-4s-2j)}{\prod_{j=1}^{\frac{n-1}{2}}\xi(4s+n-1-2j)}, & \xi_{n,1}^{()}(s)=\xi (4-4s).
\left(\overline{\partial_{0,\nu'}M_{0}(\mathrm{m}_{0})+M_{1}(\mathrm{m}_{0})+A_{\nu'}}\right)^{-1}(f)=\left(\overline{\partial_{0,\nu}M_{0}(\mathrm{m}_{0})+M_{1}(\mathrm{m}_{0})+A_{\nu}}\right)^{-1}(f).
\begin{array}{rcl}\sum_{c\in C}V_{c}+\sum_{p\in P}V_{p}-B_{M}^{\top}\eta_{M}= 0 \left(B_{M}\mathbb{E}^{\mathbb{P}}\left[\Pi\right]-b_{M}\right)^{\top}\eta_{M}= 0 B_{M}\mathbb{E}^{\mathbb{P}}\left[\Pi\right]-b_{M}\leq 0 \eta_{M}\geq 0,\end{array}
C{\bf e}\ = \ \begin{bmatrix}A JBJ B JAJ\end{bmatrix}\begin{bmatrix}{\bf e}{\bf e}\end{bmatrix}= \begin{bmatrix}(A+JB){\bf e}(B+JA){\bf e}\end{bmatrix}= \begin{bmatrix}\lambda_{1}{\bf e}\lambda_{1}{\bf e}\end{bmatrix},
F_T(x)-1-xF_T(x) =xC(x)(F_T(x)-1)+\frac{1}{1-x}\left(\frac{(x^4+x^5C^4(x)+x^5C^3(x))C^2(x)}{1-x}+x^3C^2(x)\right) &+x^3C^3(x)(F_T(x)-1)+\frac{x^5C^3(x)}{(1-x)^2}.
X(t) = x + \int_{\Gamma\times[0,t]}b_{\mathfrak{m}}\bigl(t_{0}+s,X(s),\gamma\bigr)\rho(d\gamma,ds) &+ \int_{0}^{t}\sigma_{\mathfrak{m}}\bigl(t_{0}+s,X(s)\bigr)dW(s),t\in [0,T-t_{0}],
f_{n}( t-t_{0n},x-x_{n}):= a(t,x)-\chi\lambda v(t,x;t_{0},x_n,u_{0n}(\cdot+x_n)) \cr -(b(t,x)-\chi\mu)(u(t,x;t_{0},x_n,u_{0n}(\cdot+x_n))+u(t,x;t_{0},x_n,\tilde u_{0n})).
\Omega^{\mu\nu}(\tau)=S^{\mu\nu}-\frac{1}{2}\epsilon^A_\alpha(p)\eta_{AB}\left(\frac{\partial\epsilon^B_\beta(p)}{\partial p_\mu}p^\nu-\frac{\partial\epsilon^B_\beta(p)}{\partial p_\nu}p^\mu\right)S^{\alpha\beta}.
\pi(u) \leq\sum_{i=1}^n |\lambda_i|\left\|y_i\right\|=\sum_{i=1}^n|\lambda_i|\left(\sup_{y^*\in B_{F^*}}\left|y^*(y_i)\right|\right) &\leq\sum_{i=1}^n|\lambda_i|\max_{1\leq i\leq n}\left(\sup_{y^*\in B_{F^*}}\left|y^*(y_i)\right|\right)=\left\|(\lambda_1,\ldots,\lambda_n)\right\|_{\ell^n_1}\left\|(y_1,\ldots,y_n)\right\|_{\ell^{n,w}_\infty(F)},
\left\langle{^0t},{(l^{\alpha \beta})^{\rho}}_{\mu}x^{\mu}\partial_{\rho}wx^{\alpha_1}x^{\alpha_2}\right\rangle={(l^{\alpha \beta})^{\alpha_1}}_{\sigma}c^{\sigma \alpha_2}+{(l^{\alpha \beta})^{\alpha_2}}_{\sigma}c^{\sigma \alpha_1}.
\frac{p_{r}^{2}}{g_{r r}}- \frac{{g^{\prime}}_{t t}}{- \cal D}\left( p_{\phi}+ \frac{g_{t \phi}+ \Omega_{0}g_{\phi \phi}}{{g^{\prime}}_{t t}}E \right)^{2}= - \left( \frac{E^{2}}{{g^{\prime}}_{t t}}+ V \right) ,
e^{-i{t}^{a}{\theta}_{a}}\partial_{\mu}e^{i{t}^{b}{\theta}_{b}}=i{t}^{a}\left \{\bar C\partial_{\mu}S_a-\bar S_a\partial_{\mu}C+({f}_{abc}-i{d}_{abc})\bar S^{b}\partial_{\mu}{S}^{c}\right \},
\begin{cases}u_t - \alpha^2u_{txx}+ c_0u_x + 3uu_x = \alpha^2(2u_xu_{xx}+uu_{xxx}) - \tilde{\rho}\tilde{\rho_x}-\tilde{\rho_x}, t > 0, x\in \mathbb{R}, \tilde{\rho_t}+(u\tilde{\rho})_x+u_x = 0 t > 0, x\in \mathbb{R}, \tilde{\rho}(0,x) = \tilde{\rho_0}, u(0,x) = u_0.\end{cases}
-\int_{\Omega_z^1}\partial_x\left(A_x\partial_xw^\prime_1 \right)\mathrm{d}z+\frac{A_z\left(x,z_\iota\right)}{\varepsilon\left(z_\iota\right)}\partial_z\xi^\prime_2\left(x,z_\iota\right) =\int_{\Omega_z^1}\partial_x\left(A_x\partial_x\overline{w}\right)\mathrm{d}z,
G(x, y) \asymp \frac{1}{|x-y|^{d-\alpha}}\left( \frac{x_d \wedge y_d}{|x-y|}\wedge 1 \right)^p\left(\frac{x_d \vee y_d}{|x-y|}\wedge 1\right)^{p-{a_p}_+}\log\left(2+{\bf 1}_{a_p\le 0}\frac{|x-y|}{(x_d \vee y_d)\wedge |x-y|}\right)^{\beta_4 +{\bf 1}_{a_p= 0}}.
Y_i E_\mu = t^{-\frac12\ell(v_\mu^{-1})}\tau^\vee_{u_\mu}1= t^{-\frac12\ell(v_\mu^{-1})}\tau^\vee_{u_\mu}Y_{u^{-1}_\mu(i)}1= t^{-\frac12\ell(v_\mu^{-1})}\tau^\vee_{u_\mu}q^{-\mu_i}Y_{v_\mu(i)}) 1 &= t^{-\frac12\ell(v_\mu^{-1})}\tau^\vee_{u_\mu}q^{-\mu_i}t^{-(v_\mu(i)-1)+\frac12(n-1)}1= q^{-\mu_i}t^{-(v_\mu(i)-1)+\frac12(n-1)}E_\mu.
H_{\mathrm{s p i n \frac{1}{2}}}^{( 1 )}[ N ] = - \frac{i}{2 \kappa^{2}}\int d^{3}x N ( x ) \left( \epsilon^{i j k}\epsilon^{a b c}4 \frac{\{A_{a}^{i}( x ) , V ( x , \delta ) \}\{A_{b}^{j}( x ) , V ( x , \delta ) \}}{\sqrt{q}( x )}\right) [ \pi^{T}\tau_{k}{\cal D}_{c}\xi - c . c . ]
\left( \begin{array}{c}{\psi_{+}( z , t )}{\psi_{-}( z , t )}\end{array}\right) = \mathcal{F}\left( \begin{array}{c}{\psi_{+}( z - \Delta t , t - \Delta t )}{\psi_{-}( z + \Delta t , t - \Delta t )}\end{array}\right) ,
\Phi^*\circ T_S(f)(z) = \Phi^*\left( \int_{\mathbb{R}}f(t)e^{i2\pi zt}\d t\right) = \left(\int_{\mathbb{R}}f(t)e^{i2\pi \Phi(z)t}\d t\right)\cdot \Phi'(z) = \int\limits_{\mathbb{R}}f(t)\dfrac{z^{i2\pi t}}{z}\d t,
\frac{5 a - 2 1 + 5 b}{1 2 m n - 3}= \frac{5 a + 5 b - 2 1}{1 2 \times 1 - 3}= \frac{5 \times ( a + b ) ) - 2 1}{1 2 - 3}= \frac{5 \times 0 - 2 1}{9}= \frac{0 . 2 1}{9}= \frac{- 2 1}{9}= \frac{- 7}{3}= - \frac{7}{3}
{}\gamma F_k(\alpha^{(1)}) + (1-\gamma) F_k(\alpha^{(2)})\le\alpha'\!\log_2\!\Big(\!1\!+\!\frac{\tilde{p}'|H_{\tilde{i}k}|^2}{\sigma^2_n}\!\Big) \!+ \! (\!1\!-\!\alpha'\!)\sum_{i=1}^{2}\log_2\Big(\!1\!+\!\frac{p_{i}'}{\sigma^2_n}\!\Big)
= 1 + (d-1)(1-t)^{d-1}t + (1-t)^{d-1}- 1 - dt^{r}(1-t)^{d-1}&= (1-t)^{d-1}[1 + (d-1)t - dt^r] &= (1-t)^{d-1}[(1-t^r) + t(d-1)(1-t^{r-1})] &= (1-t)^{d-1}[(1-t)(1+\cdots+t^{r-1}) + t(d-1)(1-t)(1+\cdots+t^{r-2})] &= (1-t)^d[1 + dt + dt^2 + \cdots + dt^{r-1}].
\sigma(Y_t) - \sigma(Y_s) = \sigma'(Y_s)Y_s'(B^H_{s,t}) + R^{\sigma(Y)}_{s,t}= \sigma'(Y_s)Y_{0,s}'(B^H_{s,t}) + \sigma'(Y_s)\sigma(x_0)(B^H_{s,t}) + R^{\sigma(Y)}_{s,t}.
\widehat{\gamma}_{1,\mathbb{K}}=\frac{\sum\limits_{i=1}^{k}a_{n}^{\left(i\right)}g_{\mathbb{K}}\left( \dfrac{\overline{\mathbf{F}}_{n}\left(X_{n-i+1:n}\right)}{\overline{\mathbf{F}}_{n}\left( X_{n-k:n}\right)}\right) \log \dfrac{X_{n-i+1:n}}{X_{n-k:n}}}{\sum\limits_{i=1}^{k}a_{n}^{\left( i\right)}}.
P_0\colon J_2(\mathcal{O})\longrightarrow\mathcal{O}^3 &(z,z_x,z_u,z_{xx},z_{xu},z_{uu})\mapsto \left(\begin{array}{c}z_{xx}+p_3z_{x}+q_3z_{u}+r_3z z_{xu}+p_2z_{x}+q_2z_{u}+r_2z z_{uu}+p_1z_{x}+q_1z_{u}+r_1z \end{array}\right).
\frac{( n - 1 ) \cdots 3}{n \cdots 4}\left( 4 ( x^{2})^{\frac{n - 2}{2}}x^{( \alpha_{1}}x^{\alpha_{2}}\eta^{\alpha_{3}\alpha_{4}}\cdots \eta^{\alpha_{n - 1}\alpha_{n})}- ( x^{2})^{\frac{n}{2}}\eta^{( \alpha_{1}\alpha_{2}}\cdots \eta^{\alpha_{n - 1}\alpha_{n})}\right){^0 t},
\mathcal{K}^{[\infty]}\colon x_m=0,y_n=0,\Delta x_m\, \delta\Delta^2 u_{m,n}-\Delta^2 x_m\, \delta\Delta u_{m,n}=(\Delta x_m)^3\, \delta y_n, u_{m+k,n}=u_{m,n+k}=1, k \geq 0,
\begin{aligned}4 \pi - \frac{(1+k)^{2}}{(1-k)^{2}}(\eta(r_{0}))^{2}\int_{\Sigma}|\nabla u|^2 \ge \frac{(1+k)^{2}}{2k}\frac{(1-a\eta(r_{0}))}{\eta(r_{0})}\left\{4 \pi \frac{(1-k)^{2}}{(1+k)^{2}}- \eta(r_{0}) \int_{\Sigma}H |\nabla u|+ (\eta(r_{0}))^{2}\int_{\Sigma}|\nabla u|^2\right\},\end{aligned}
\omega_1^{-1}(h)=\lambda_2 \nu^{-1}, \omega_2^{-1}(h)=\lambda_2, \omega_3^{-1}(h)=\lambda_2\nu, \omega_4^{-1}(h)=\lambda_1\lambda_2\nu^{-1}, \omega_5^{-1}(h)=\lambda_1\lambda_2
P_{\lambda}(x, s) = (x/s)^{\lambda}\frac{(x/s;q)_{\infty}}{\big(q^{-\lambda}x/s ;q\big)_{\infty}}, y(x)=x^{\mu '}\frac{(q/(\beta_1 x), q/(\beta_2 x) ;q)_{\infty}}{(q/(\alpha_1 x), q/(\alpha_2 x) ;q)_{\infty}}
{\cal L}'= \bar{\psi}^{(i)}{\rm i}\partial \!\!\!/ \, \psi^{(i)}+\frac{g^2}{2}\left(\psi_R^{(i)\dagger}\psi_L^{(i)\dagger}+ \psi_L^{(i)}\psi_R^{(i)}\right)^2 \ .
\mathrm{t r ~ l n}( - \nabla^{2}+ 4 b^{2}) = - ( Y / \sqrt{4 \pi}) ( \partial / \partial \beta ) ( \Gamma ( \beta - 1 / 2 ) / \Gamma ( \beta ) ) \sum_{n = 1}^{\infty}( ( n \pi / L )^{2}+ ( 4 b^{2}) )^{- \beta + 1 / 2}
\begin{aligned}U_\eta \to U L^2(0,T;L^{2}(\mathbb T^d_{\ell}))-w, \sqrt{R_\eta}U_\eta \to \sqrt{R}U C([0,T] ; L^2(\mathbb T^d_{\ell})-w), R_\eta^{\frac14}U_\eta \to R^{\frac14}U L^4(0,T;L^4(\mathbb T^d_{\ell}))-w, R_\eta U_\eta \to RU L^2(0,T;L^2(\mathbb T^d_{\ell})) .\end{aligned}
\phi_t =d_2\Delta \phi-\chi_2\nabla \cdot(\phi\nabla w)-\chi_2\nabla \cdot(v^{**}\nabla(w-w^{**})) &+ \psi\Big(b_0(t,x)-b_1(t,x)u-b_2(t,x)(v+v^{**})\Big)-b_1(t,x)v^{**}\psi.
\hat P_n(x) =\frac{1}{2^n}Q\begin{bmatrix}U_n(x)- U_{n-1}(x) 0 0 U_n(x)+ U_{n-1}(x)\end{bmatrix}Q^\top, Q := \frac{1}{\sqrt 2}\begin{bmatrix}1 1 1 - 1\end{bmatrix}.
\sum_{n=0}^{\infty}\frac{z^{2n}q^{n^2+n}}{(zq;q^2)_{n+1}}=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}b^4_\nu(m,n)z^mq^n=\sum_{n=0}^{\infty}(-zq;q^2)_n(zq)^n.
\mathcal A_{p\mu\rho}(\delta p,\delta\mu,\delta\rho,\delta \b u)=\begin{pmatrix}\mathcal L_{p\mu\rho}\;(\delta p,\delta\mu,\delta\rho,\delta\b u) \mathcal B \;(\delta \b u)|_{\partial\Omega}\end{pmatrix},
\frac{\textrm d}{\textrm dt}\left[V(x(t))+Q_k(t-t_k,x_{k})\right]+2\alpha V(x(t)) + 2\alpha Q_k(t-t_k,x_{k}) &=\nabla V(x(t))^T f(x(t),x(t_k)) + \nabla_3 F(t,x(t_k),x(t))^T f(x(t),x(t_k)) & + \nabla_1 F(t,x(t_k),x(t)) +2\alpha V(x(t)) + 2\alpha F(t,x(t_k),x(t)) &\le 0
\int_0^1 P_{4n,-}^p \leq \sum_{i=1}^{n}\Big[\cos\Big(\frac{2i-\frac{3}{2}}{4n+\frac{1}{2}}\pi\Big)-\cos\Big(\frac{2i}{4n+\frac{1}{2}}\pi\Big)\Big]y_{2i-1,4n}^p = 2\sum_{i=1}^{n}\sin\Big(\frac{2i-\frac{3}{4}}{4n+\frac{1}{2}}\pi\Big)\sin\Big(\frac{3}{4(4n+\frac{1}{2})}\pi\Big)y_{2i-1,4n}^p.
\widehat{F}(z):=\left(\sum_{j=1}^{\lceil \mu/2\rceil}F_j(x_{j,1},\ldots, x_{j,n_j},x_n)\right)-\sum_{j=\lceil \mu/2\rceil+1}^{\mu}F_j(x_{j,1},\ldots, x_{j,n_j},x_n).
H=\int\! d^2\!{\bf r}\,\left\{\sum_{p=1}^M\frac{\hbar^2}{2m_p}\mbox{\boldmath $\Pi$}_p^\dagger({\bf r},t) \cdot\mbox{\boldmath $\Pi$}_p({\bf r},t)+U(\Psi^\dagger,\Psi)\right\},
\lim_{m\rightarrow+\infty}\frac{x_{2}^{\left(t_{0}+m\right)}}{\varpi\circ W\left(z^{\left(t_{0}+m\right)}\right)}=\begin{cases}\qquad0 \mbox{if}\gamma_{2}=0,\delta_{1}\neq0, \frac{\delta_2x_{2}^{(t_{0})}}{x_{1}^{\left(t_{0}\right)}+x_{2}^{\left(t_{0}\right)}}\mbox{if}\gamma_{2}=\delta_{1}=0, \qquad\delta_2 \mbox{if}\gamma_{2}\neq0,\delta_{1}=0,\end{cases}