harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{0}^{1}\int_{0}^{v}v^{-\frac{7}{8}}u^{-\frac{3}{8}}\exp\left(-Kv^{-\frac{3}{2}}[v-u]^2\right) dudv &= \int_{0}^{1}\int_{0}^{v}v^{-\frac{7}{8}}(v-w)^{-\frac{3}{8}}\exp\left(-Kv^{-\frac{3}{2}}w^2\right) dwdv.
\left(\begin{array}{c}\psi_1 \psi_2 \end{array}\right)\longrightarrow\left(\begin{array}{c}\widetilde\psi_1 \widetilde\psi_2 \end{array}\right)=\left(\begin{array}{c}a c\end{array}\begin{array}{cc}b d\end{array}\right)\left(\begin{array}{c}\psi_1 \psi_2 \end{array}\right),
(\theta_\epsilon^\beta e^\beta + e^{\alpha + \beta}) (\theta_\iota^\beta e^\beta + e^{\alpha + \beta}) = \theta_\epsilon^\beta \theta_\iota^\beta c_\beta t_\beta + c_{\alpha + \beta}t_{\alpha + \beta}+ b_{\beta,\alpha+\beta}(\theta_\epsilon^\beta + \theta_\iota^\beta)e^\alpha
\frac{\mathrm{d}}{\mathrm{d}t}\ln x_i= \frac{1}{x_i}\dot{x}_i =\sum_{j\in N_i^+}w_{ij}(\ln x_j-\ln x_i) \Leftrightarrow \ \ \dot{x}_i = x_i\sum_{j\in N_i^+}w_{ij}(\ln x_j-\ln x_i),
S = \frac{1}{2 \kappa^{2}}\int d x^{1 0}\sqrt{- g}\left( - \frac{1}{2}\nabla_{\mu}\tau \nabla^{\mu}\bar{\tau}- \frac{i}{2}( \tau - \bar{\tau}) \nabla_{\mu}\tau \nabla^{\mu}\bar{\tau}\right) ~ ,
\begin{pmatrix}Y_{0,0}Y_{0,1}0 Y_{1,1}\end{pmatrix}\begin{pmatrix}T_0 \psi(T_0)i^* 0 T_1 \end{pmatrix}=\begin{pmatrix}T_0 \widetilde{\psi}(T_0) i^* 0 T_1 \end{pmatrix}\begin{pmatrix}Y_{0,0}Y_{0,1}0 Y_{1,1}\end{pmatrix},
S = \int d^{4}x \sqrt{| g_{\mu \nu}|}\ \frac{e^{- \sigma}}{l_{s}^{2}}\, \left[{\frac{1}{2}}R +{\frac{1}{2}}( \nabla \sigma )^{2}+{\cal L}_{m}({\cal Y}, g_{\mu \nu}, \sigma ) \right] .
\int_X |iF_A^+|^2 \,d\mu_g = \int_X \left| - \frac{1}{\sqrt{8}}\beta(|\sigma(\Phi)|) \left[ \left(\frac{2}{3}R_g + 2w_g \right)_- + \epsilon \right] \sigma(\Phi) \right|^2 \,d\mu_g &\le \frac{1}{8}\int_X \left[ \left(\frac{2}{3}R_g + 2w_g \right)_- + \epsilon \right]^2 \,d\mu_g,
a(r,s+t)|_{([r,r+s]\times[0,r]) \cup ([0,r]\times [r, r+s])}= a(r,s)|_{([r,r+s]\times[0,r])\cup ([0,r]\times [r, r+s])}a(r+s,t)|_{[r+s,r+s+t]\times[0,r]}= a(r,s+t)|_{[r+s,r+s+t]\times[0,r]},
\begin{aligned}\int\limits_{|x-e|+|x|\leq 2\tau}\frac{q(x)}{|x||x-e|}dx = \frac{1}{2}\int\limits_{\cosh\rho\leq 2\tau}\int\limits_{0}^{\pi}\int\limits_{0}^{2\pi}q(\rho,\theta,\phi)\sinh\rho\sin\phi d\theta d\phi d\rho. \end{aligned}
\big\lVert\,\cdot\,\big\rVert^\mathrm{RS}_{\det H^{\bullet}_\mathrm{bd}(X,F)}= \big\lVert\,\cdot\,\big\rVert^{L^{2}}_{\det H^{\bullet}_\mathrm{bd}(X,F)}T_\mathrm{bd}(X,g^{TX},h^{F}).
\frac{Z_{MN}}{Z_{-M,-N}}=\frac 1{2^{m(M+N)}}\frac{H_q\left(\frac{a+b+M+N}2,\frac{a+b-M-N}2+m\right)}{H_q\left(\frac{a+b-M-N}2,\frac{a+b+M+N}2+m\right)}&\quad\times\frac{H_{q^2}\left(\frac{a+c-N}2,\frac{a+c+N}2+m,\frac{b+c-M}2,\frac{b+c+M}2+m\right)}{H_{q^2}\left(\frac{a+c+N}2,\frac{a+c-N}2+m,\frac{b+c+M}2,\frac{b+c-M}2+m\right)}.
A_{4}( k_{1}, k_{2}, k_{3}, k_{4}) \rightarrow - i \int{\frac{d^{7}k}{( 2 \pi )^{7}}}{\frac{\sum_{I J \mu}A_{\J}^{I \mu}( k_{1}, k_{2}, k ) A_{\I \mu}^{J}( k_{3}, k_{4}, - k )}{k^{2}- i \epsilon}}\ ,
\sum_{t=0}^{k-1}\binom{k-1}{t}(\frac{-p}{q})^{t}U_{mk+t}^{(k)}=(-q)^{1-k}U_{m}(pU_{m+1}-qU_{m})^{k-1}&=(-q)^{1-k}U_{m}U_{m+2}^{k-1}&=(-q)^{1-k}U_{m}U_{(m+2)(k-1)}^{(k-1)},
S_{4}( \tilde{g}_{\mu \nu}( x ) ) = \int_{r_{0}}^{R}( \cdots ) d r d \varphi = \int \left\{\frac{1}{1 6 \pi G}( \tilde{R}- 2 \tilde{\lambda}) \right\}\sqrt{- \tilde{g}}\, d^{4}x .
\ddot{\Phi}- 3 \dot{\Phi}^{2}+ 6 \lambda^{\prime}\alpha \big \{H^{4}+ 2 H^{2}\dot{H}- 4 \dot{\Phi}H^{3}- 4 H \dot{H}\dot{\Phi}- 2 H^{2}( \ddot{\Phi}- 2 \dot{\Phi}^{2}) \big \}= 0
\begin{aligned}\mathcal{N}^k_N (s) = \max\{i: \lambda_i \leq s,\ \lambda_i \mbox{is an $ (\mathcal{E}^k,\mathcal{F}^k) $ eigenvalue}\}, \mathcal{N}_D^k(s) = \max\{i: \lambda_i \leq s,\ \lambda_i \mbox{is an $ (\mathcal{E}^k_D,\mathcal{F}^k_D)$ eigenvalue}\}.\end{aligned}
\bar F_{p+1}= \sum_{i_k, j_k}\frac{d}{dx^+}\bar A_{i_1 \cdots i_p}(-4 x^+) Q^{i_1}_{j_1}\cdots Q^{i_p}_{j_p}dx^+ \wedge dx^{j_1}\wedge \cdots \wedge d x^{j_p}.
\int D V \exp ( \frac 1 2 \int d^{4}x \varepsilon^{\mu \nu \tau \lambda}{}^{+}W_{\mu \nu}^{\alpha \beta}{}^{+}G_{\tau \lambda}^{\sigma \rho}{}\epsilon_{\alpha \beta \sigma \rho}- \frac 1 2 \int d^{4}x \varepsilon^{\mu \nu \tau \lambda}{}^{-}W_{\mu \nu}^{\alpha \beta}{}^{-}G_{\tau \lambda}^{\sigma \rho}{}\epsilon_{\alpha \beta \sigma \rho}) .
{\cal G}[ \tilde{\Phi}] (t,{\bf x};t^{\prime},{\bf x}^{\prime})=\delta (t-t^{\prime}) \, \delta^{(d)}({\bf x}-{\bf x}^{\prime})\,\left( F^{\prime}[\tilde{\Phi}(t,{\bf x})] \right)^{2}= \delta (t-t^{\prime}) \, g [\tilde{\Phi}] ({\bf x},{\bf x}^{\prime}; t)\; ,
{m_{{v_{n - 1}}\nearrow}}={m_{\searrow{v_{n - 1}}}}{m_{\uparrow{v_{n - 1}}}}{m_{\searrow{v_n}}}=\int d{{\tilde \theta}_{n - 1}}\sum\limits_{{s_{n - 1}}}{{m_{{v_{n - 1}}\nearrow}}}\psi \left({{v_{n{\rm{- 1}}}},{v_n}}\right)
\langle \alpha |\beta \rangle =\sum_{h_{1},...,h_{\mathsf{N}}=0}^{p-1}V(X_{1}^{(h_{1})},...,X_{\mathsf{N}}^{(h_{\mathsf{N}})})\prod_{a=1}^{\mathsf{N}}\alpha_{a}^{(h_{a})}\beta_{a}^{(h_{a})},
\Delta k_{A\rightarrow J}(\omega)=-\frac{4}{3}\gamma N_{A}^{0}\{\frac{A_{1}(A\rightarrow J)}{\hbar}\frac{\partial f_{ja}^{0}(\omega)}{\partial\omega}+[B_{0}(A\rightarrow J)+\frac{C_{0}(A\rightarrow J)}{k_{B}T}]f_{ja}^{0}(\omega)\}H
\begin{aligned}E(r,s) =\int_{B}\biggl[\rho f_{i}(r)u_{i}(s)-g(r)\varphi(s)-\frac{1}{T_{0}}\mathcal{H}(r)A\theta(s)\biggr]dv\ \;+\int_{\partial B}\biggl[P_{i}(r)u_{i}(s)+R_{i}(r)\mathcal{D}u_{i}(s)+\Lambda(r)\varphi(s) \;+H(r)\mathcal{D}\varphi(s)+\frac{1}{T_{0}}\left[l*q\right](r)A\theta(s)\biggr]da.\end{aligned}
\varphi_1 \circ c_iX^{q^{is}}\circ \varphi_2 = cc_i^{q^l}(gX^{q^j}+ hX^{q^{j+n}})^{q^{is+l}}+ dc_i^{q^{l+n}}(gX^{q^j}+ hX^{q^{j+n}})^{q^{is+l+n}}= (cg^{q^{is+l}}c_i^{q^l}+ dh^{q^{is+l+n}}c_i^{q^{l+n}})X^{q^{j+is+l}}& + (ch^{q^{is+l}}c_i^{q^l}+ dg^{q^{is+l+n}}c_i^{q^{l+n}}) X^{q^{j+is+l+n}}.
Z_{\mathrm{\scriptsize ~ 1 - l o o p}}( R , T ; t ) \approx t^{-{\frac{1}{2}}}\eta ( t )^{-{\frac{1}{2}}}\exp \left[ -{\frac{1}{\epsilon^{2}}}\left({\frac{T^{2}}{2 R^{2}t}}+{\frac{t}{2}}+ \epsilon f_{1}^{\mathrm{\scriptsize ~ 1 - l o o p}}+ \epsilon^{2}f_{2}^{\mathrm{\scriptsize ~ 1 - l o o p}}\right) \right] \; ,
\mathcal{F}_{D,0}(z,s) := 2^{s-1}\pi^{-\frac{s+1}{2}}|D|^{\frac{s}{2}}L_D(s) \Gamma\biggl(\frac{s+1}{2}\biggr) P_{\frac{1}{2}, 0}\biggl(z, \frac{s}{2}+ \frac{1}{4}\biggr).
X_1 : y^2 = -19x^4 + 112x^3 - 142x^2 - 68x - 7 &X_2 : y^2 = -16x^4 - 82x^3 - 52x^2 + 136x - 44 &X_3 : y^2 = -x^4 - 26x^3 - 148x^2 + 274x - 111.
\frac\partial{\partial x_2}\bigg(\frac{x_2}{x_1^2}+\alpha_2(m)x_2^{-\frac{2^{m-1}}{2^{m-1}-1}}\bigg)=\frac1{x_1^2}-\frac{\alpha_2(m)2^{m-1}}{2^{m-1}-1}x_2^{-\frac{2^m-1}{2^{m-1}-1}}
2 h_{\cal O}\langle \! \langle{\cal O}|{\cal O}\rangle = 2 \langle \! \langle{\cal O}| L_{0}|{\cal O}\rangle = \langle \! \langle{\cal O}| [ L_{1}, L_{- 1}] |{\cal O}\rangle = \| L_{- 1}|{\cal O}\rangle \|^{2}\geq 0 ,
a_i\{f, \, g\}= \{a_i(f), \; g\}+ \{f, \, a_i(g)\}, b_i\{f, \, g\}= \{b_i(f), \; g\}+ \{f, \, b_i(g)\}+ a_i(f) b_i(g) - b_i(f) a_i(g)
I := \int_0^{\infty}\cdots \int_0^{\infty}k_g p_g \cdots k_2 p_2 \, e^{-(k_g p_g t_g + \cdots + k_2 p_2 t_2)}e^{-k_1 p_1 (t_2 \wedge \cdots \wedge t_g)}dt_2 \cdots dt_g.
(\alpha,\alpha)_A=\sum_{i,j\in I}{n_in_ja_{ij}}=2\sum_{i\in I}{n_i^2}-\sum_{i\neq j}{n_in_j|a_{ij}|}\leq 2\sum_{i\in I}{n_i^2}-\sum_{i\neq j}{n_in_j|b_{ij}|}=(\alpha,\alpha)_B.
\begin{array}{l}(\frac{\partial f}{\partial x})^T(x)\Pi(x) + \Pi(x)\frac{\partial f}{\partial x}(x) - \Pi(x)g(x)g^T(x)\Pi(x) begin{align*}3mm]+ (\frac{\partial h}{\partial x})^T(x)\frac{\partial h}{\partial x}(x) + \frac{\partial \Pi}{\partial x}(x)f(x)=0,\end{array}
\begin{array}{lll}\mathsf{g}_{\mathbb{D},\tilde{\mathbb{F}}(\mathsf{S})}(\tilde{F}(L)) = \tilde{F}(\mathsf{g}_{\mathbb{C},\mathsf{S}}(L))\circ\Phi_{S,S}&= \tilde{F}(\mathsf{d}_{\mathbb{C},\mathsf{S}}(R))\circ\Phi_{S,S}&= \mathsf{d}_{\mathbb{D},\tilde{\mathbb{F}}(\mathsf{S})}(\tilde{F}(R)).\end{array}
Z = \frac{{\displaystyle \int}{\cal D}{\bar{\psi}}{\cal D}\psi{\displaystyle e}^{i S ({\bar{\psi}}, \psi , A_{-}, A_{+}, A_{-}^{5}, A_{+}^{5})}}{{\displaystyle \int}{\cal D}{\bar{\psi}}{\cal D}\psi{\displaystyle e}^{i S ({\bar{\psi}}, \psi )}},
(\tilde{\kappa}_1)_{ii}=\tilde{h}_{11ii}+2\sum_{p\neq 1}\frac{\tilde{h}_{1pi}^2}{\tilde{\kappa}_1-\tilde{\kappa}_p}=h_{11ii}+2\sum_{p\neq 1}\frac{h_{1pi}^2}{\kappa_1-\tilde{\kappa}_p}.
\tilde{\hat{\Omega}}_{\underline{x}}{}^{(0)\hat{a}\hat{b}}=-\hat{\Omega}_{\underline{x}}{}^{(0)\hat{a}\hat{b}}/\hat{G}_{\underline{xx}}^{(0)}\, ,\hspace{.5cm}\tilde{\hat{\Omega}}_{\mu}{}^{(0)\hat{a}\hat{b}}=\hat{\Omega}_{\mu}{}^{(0)\hat{a}\hat{b}}-\hat{\Omega}_{\underline{x}}{}^{(0)\hat{a}\hat{b}}\hat{G}_{\underline{x}\mu}^{(0)}/\hat{G}^{(0)}_{\underline{xx}}\, ,
S = \int{\frac{d^{2}x}{2 \pi}}\left[ \sum_{i = 1}^{m}\psi_{L}^{i}\partial \psi_{L}^{i}+ \psi_{R}^{i}\bar{\partial}\psi_{R}^{i}+ \sum_{j = 1}^{n}\beta_{L}^{j}\partial \gamma_{L}^{j}+ \beta_{R}^{j}\bar{\partial}\gamma_{R}^{j}+ g \left( \psi_{L}^{i}\psi_{R}^{i}+ \beta_{L}^{j}\gamma_{R}^{j}- \gamma_{L}^{j}\beta_{R}^{j}\right)^{2}\right]
B=(\begin{matrix}w&1&w^{2}&1&w^{2}&w^{2}\\w&1&w&1&1&w\\w&w&w^{2}&w^{2}&1&0\\ 0&0&0&0&1&1\\w^{2}&1&w^{2}&w^{2}&w&w^{2}\\w^{2}&1&w^{2}&w&w^{2}&w\end{matrix}).
t ={{\cal Z}^{F_{1}}}^{3}-{\cal Z}^{F_{1}}{{\cal Z}^{F_{1}}}^{\prime \prime}-{\frac{1}{2}}\left({{\cal Z}^{F_{1}}}^{\prime}\right)^{2}+{\frac{1}{1 0}}{{\cal Z}^{F_{1}}}^{( 4 )}.
{\phi}_{(2)}^{\rm self}(x_2 ,t)= e_2 \int_0^{\rm L}dy \int_0^{\rm L}dz D(x_2,y|{\rm L})\bar{\Phi}(z,t|y,t) ({\gamma}^0 \otimes{\gamma}^0){\Phi}(z,t|y,t),
L_{f k}= \frac{e^{\frac{1}{2}\alpha \phi / M_{p}}}{\zeta + k}\left[ ( \zeta + h_{E}) f_{E}\overline{{{L_{L}}}}\, H \, e_{R}+ ( \zeta + h_{U}) f_{U}\overline{{{Q_{L}}}}\, \tilde{H}\, U_{R}+ ( \zeta + h_{D}) f_{D}\overline{{{Q_{L}}}}\, H \, D_{R}+ H . c . \right]
\tau^{-l}\frac{\Gamma\bigl(2+2l+\tau(1+\lambda_1+\lambda_2)\bigr)}{\Gamma\bigl(2+l+\tau(1+\lambda_1+\lambda_2)\bigr)}=\prod\limits_{j=0}^{l-1}\bigl(1+\lambda_1+\lambda_2+(l+j+2)/\tau\bigr).
\mu_n :=\prod_{j=1}^md^{k(n_j+nq)}\cdot(r_n)^*\left(T_{f,a_1}^k\wedge\cdots\wedge T_{f,a_m}^k\right) & =\bigwedge_{j=1}^m\left(dd^cG_{r_n(x)}\left(\sigma_j\circ a^{n,j}(x)\right)\right)^k.
\delta{\cal L}^{[ \frac{1}{2}]}= [ \kappa \ \bar{\kappa}]{\cal P}_{+}\left[ \begin{array}{c}{{- \sqrt{- g}v^{i}\tilde{\cal H}^{j}\gamma_{i j}\bar{P}}}\\{{- \sqrt{- g}v^{i}\bar{\tilde{\cal H}}^{j}\gamma_{i j}P}}\\ \end{array}\right]
\delta K_{\mu\nu}= \frac{1}{2}e^{(0)M}_{\mu}e^{(0)N}_{\nu}({\cal L}_{\delta n}g^{(0)}_{MN}+{\cal L}_{\delta Z}{\cal L}_{n^{(0)}}g^{(0)}_{MN}- 2n^{(0)L}\delta\Gamma_{LMN}).
\tau_N\left( Q(a_1^N,\dots,a_k^N)\right) = \tau\big( Q(a_1,\dots,a_k) \big) + \mathcal{O}\bigg( \frac{N^\varepsilon}{N}\max\limits_{i\neq j}| y_i^N-y_j^N |^4 + \frac{1}{\min\limits_{i\neq j}| y_i^N-y_j^N |^{\delta_P}}\bigg).
\omega_{2a}=\sum_i\frac{B^1_{1a,i}}{\mu}\omega_i=-\sum_i\frac{B^2_{2a,i}}{\mu}\omega_i,~~ \omega_{1a}=\sum_i\frac{B^1_{2a,i}}{\mu}\omega_i=\sum_i\frac{B^2_{1a,i}}{\mu}\omega_i.
\big (\sum_{j=1}^n \|\, T x_j \, |Y\|^r \Big )^{1/r}\leq C \sup_{x^* \in X^*, \|x^*|X^*\|\leq 1}\Big ( \sum_{j=1}^n |\langle x_j , x^* \rangle|^s\Big)^{1/s}
\sum_{c = 1 , 2}\Gamma_{\phi^{-}}^{b c}( p ) [ \delta^{c a}- L^{c a}( p ) G{( - 1 )}^{a + 1}] = i \frac{G}{2}\delta^{b a}{( - 1 )}^{( a + 1 )}, \ \b , a = 1 , 2 ,
K(x,x^{\prime};t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dke^{ik(x-x^{\prime})}e^{-\frac{i\hbar k^{2}t}{2m}}=(\frac{m}{2\pi i\hbar t})^{\frac{1}{2}}e^{-\frac{m(x-x^{\prime})^{2}}{2i\hbar t}}.
\langle x_1 \otimes \cdots \otimes x_m, y_1 \otimes \cdots \otimes y_n\rangle_{0}= \delta_{m,n}\prod_{i=1}^n \langle x_i, y_i\rangle, \langle \Omega, y_1 \otimes \cdots \otimes y_n\rangle_{0}= \langle x_1 \otimes \cdots \otimes x_m, \Omega\rangle_{0}=0, &\langle \Omega, \Omega\rangle_{0}=1,
\log\varrho\left(L_0^{\phi+\lambda\psi},L_1^{\phi+\lambda \psi}\right) =\lim_{n \to \infty}\sup_{x \in \Sigma_2}\sup_{k \in \mathbb{Z}}\frac{1}{n}\sum_{i=0}^{n-1}(\phi+\lambda\psi)(x_i,k+i) &\geq \liminf_{j \to \infty}\frac{1}{n_j}\sum_{i=0}^{n_j-1}(\phi+\lambda\psi)(z_i,k_j+i) &\geq \log\varrho\left(L_0^\phi,L_1^\phi\right)+\lambda N\mu([\omega]).
\int_0^x e^{2 \mu y}\left(G(Q(y))-Q(y)\right)\mathrm{d}y=\int_1^{Q(x)}e^{2 \mu Q^{-1}(s)}\frac{\left(G(s)-s\right)}{Q^{\prime}(Q^{-1}(s))}\mathrm{d}s.
|\cos(\nu_n x) - \cos(\lambda_nx)| =|2\sin \frac{{({\lambda_n}-{\nu_n})x}}{2}\sin \frac{{({\lambda_n}+{\nu_n})x}}{2}| &\le 2 |{\sin \frac{{({\lambda_n}-{\nu_n})x}}{2}}|\le |{{\nu_n}-{\lambda_n}}|x \le \pi |{{\nu_n}-{\lambda_n}}|,
b_h(u_h,v_h) = \int_{Q}(\partial_{t}u_hv_h + \theta h \partial_{t}u_h \partial_{t}v_h + \nabla_x u_h \cdot \nabla_x v_h + \theta h \nabla_x u_h \cdot \nabla_x \partial_{t}v_h) \,dxdt + \theta h \int_{\Sigma}n_t \nabla_x u_h \cdot \nabla_x v_h \,ds
\begin{aligned}\|f_{j}\|_{C^{\alpha}(\mathcal C^h )}\leq \|q\|_{C^{\alpha}(\mathcal C^h )}\sup_{\mathcal C^h}\vert v_{j}\vert+\|v_{j}\|_{C^{\alpha}(\mathcal C^h )}\sup_{\mathcal C^h}\vert q-1\vert. \end{aligned}
\frac{2H_{0}}{f_{2}p_{1}}=\left( 1+a^{2}\right) \left( \frac{f_{1}}{p_{1}}\right) \left( \frac{p_{2}\dot{p}_{2}}{f_{2}}\right) +2a\frac{p_{2}}{f_{2}}+\dot{p}_{1}.
\lambda_{1,2}^s(t)= \left\{\begin{aligned}\lambda_1(2st + 1 -s) : 0\leq t\leq \frac{1}{2}& \lambda_2(2t-1) : \frac{1}{2}\leq t\leq 1 \end{aligned}\right\}\lambda_i^s(t)=\lambda_i(t(1-s)).
e^{- 2 \phi}= \left( \Bigl ( e^{- 2 \hat{\phi}_{0}}+{\frac{2 \tilde{m}}{r}}\Bigr ) \Bigl ( K_{0}+{\frac{2 \hat{m}}{r}}\Bigr ) - \left[ ( A_{n})_{0}+{\frac{2 q_{n}}{r}}\right]^{2}\right)^{\frac{1}{2}}\ .
\sum_{\nu_{1}, \nu_{2}\in{\cal N}_{\varepsilon}^{+}}\frac{1}{\rho \left( \nu_{1}\right) \rho \left( \nu_{2}\right)}\left[{\Large i}\right] \rightarrow \int_{0}^{\infty}\int_{0}^{\infty}d \nu_{1}d \nu_{2}\left[{\Large i}\right] \ .
y^{2}= x^{3}+ \sum_{k = - 4}^{4}\sum_{l = 0}^{1 2 - n k}f_{k l}z^{\prime l}z^{4 + k}\ldots a^{4 - 6 k}x + \sum_{k = - 6}^{6}\sum_{l = 0}^{1 2 - n k}g_{k l}z^{\prime l}z^{6 + k}\ldots a^{6 - 6 k}
- c_0 = \left(\sum_{l=0}^k a_l z^l\right) \sum_{i=1}^n \frac{1}{z-z_i}\sum_{\substack{j=1 j\ne i}}^n \frac{2}{z_i-z_j}+ \left(\sum_{l=0}^{k-1}b_l z^l\right) \sum_{i=1}^n \frac{1}{z-z_i}+ \sum_{l=1}^{k-2}c_l z^l.
K_{a b}^{i}= \Bigl ( \frac{\partial^{2}X^{N}}{\partial \sigma^{a}\partial \sigma^{b}}+ \frac{\partial X^{P}}{\partial \sigma^{a}}\frac{\partial X^{Q}}{\partial \sigma^{b}}\Gamma_{P Q}^{N}\Bigr ) n_{N}^{i}~ .
S_n\left( \lambda \right)=\begin{cases}p_0-\lambda q_0 \mbox{for}n=0 p_0-\lambda q_0 + \sum_{k=1}^nb_k\left( \lambda \right)p_k \mbox{for}n \ge 1 \end{cases}\mbox{and}S_n^{\ast}\left( \lambda \right)=\sum_{k=1}^n \left| b_k\left( \lambda \right) \right| p_k.
\|T_kf\|_{L^r(K)}= \lim_{j\to\infty}\|T_kf_j\|_{L^r(K)}& \leq Cq^{-k}\|\Omega\|_{H^1(\mathfrak{D}^*)}\lim_{j\to\infty}\|f_j\|_{L^r(K)}\leq Cq^{-k}\|\Omega\|_{H^1(\mathfrak{D}^*)}\|f\|_{L^r(K)},
r_n^2(t) =2(n+r_n(t))\alpha_n(t)\alpha_{n-1}(t), r_n'(t) =2(n+r_n(t))(\alpha_{n-1}(t)-\alpha_n(t)), r_n(t) =\alpha_n(t)(t-\alpha_n(t))+\frac{1}{2}\alpha_n'(t), -2\sum\limits^{n-1}_{j=0}\alpha_j(t) =2tr_n(t)-2(n+r_n(t))(\alpha_n(t)+\alpha_{n-1}(t)).
\mathbb{P}_{q,p}\left\{0\longleftrightarrow x\ ,\ \left\vert \mathbf{C}_{\{0,x\}}\right\vert <\infty\right\}=\frac{\Phi_{q,p}\left( \hat{x}\right)}{\sqrt{\left( 2\pi\left\Vert x\right\Vert \right)^{d-1}}}e^{-\tau_{q,p}\left( x\right)}\left( 1+o\left( 1\right) \right) \ ,
(\overline{\partial}_{E,p}+ \overline{\partial}^t_{E,p})_{\min}(\phi_i\omega)=\phi_i(\overline{\partial}_{E,p}+ \overline{\partial}^t_{E,p})\omega+(\overline{\partial}_{\min}\phi_i)\wedge \omega-{\rm Int}(\overline{\partial}_{\min}\phi_i)\omega.
\int_{|y-x|=r}|y|^{-\kappa}dS(y) \leq 2 \int_0^\pi \left(\frac{4 r^2 \theta^2}{\pi^2}\right)^{-\kappa/2}r d\theta \leq 2^{1-\kappa}\pi^{\kappa}r^{1-\kappa}\int_0^\pi \theta^{-\kappa}d \theta \leq C(\kappa) |x|^{1-\kappa}.
\mathbf{y}=\mathbf{H}_k^H\mathbf{P}_{\mathcal{K}}\mathbf{s}_{\mathcal{K}}+\sum_{g=1}^{G}\mathbf{H}_k^H\mathbf{P}_{\mathcal{K}_g}\mathbf{s}_{\mathcal{K}_g}+\sum_{j=1}^{K}\mathbf{H}_k^H\mathbf{P}_{j}\mathbf{s}_{j}+\mathbf{n}_k.
(\nabla_X\phi)Y =\eta(Y)A_NX-\langle A_NX,Y\rangle\xi \nabla_X\xi =\phi A_NX; \nabla^\perp_XN=Ch(X,\xi) h(X,\phi Y) =-\langle \phi A_NX,Y\rangle N-\eta(Y)Ch(X,\xi)+Ch(X,Y)
\varrho(\mathsf{A})=\lim_{n \to\infty}\max_{A_1,\ldots,A_n \in \mathsf{A}}\left\|A_n\cdots A_1\right\|^{\frac{1}{n}}= \inf_{n \geq1}\max_{A_1,\ldots,A_n \in \mathsf{A}}\left\|A_n\cdots A_1\right\|^{\frac{1}{n}}
\sum_{k=1}^\infty \mathbb P(H_k) = \sum_{k=1}^{K_0-1}\mathbb P(H_k) + \sum_{k = K_0}^\infty \mathbb P(H_k) &\leq \sum_{k=1}^{K_0-1}\mathbb P(H_k) + \sum_{k = K_0}^\infty \frac{P(k/2)}{2^{k/2}}&< \infty.
\exp \Bigl \{\frac{i}{\hbar}\int_{0}^{t}\theta_{n}( t^{\prime}) d t^{\prime}\bigr \}= \exp \Bigl ( - i ( n + \frac{1}{2}) \varphi ( t ) \Bigr ) = \Bigl ( \frac{k_{1}+ k_{2}- k_{3}\tilde{\varphi}( t ) + 2 i \sqrt{\kappa}\tilde{\varphi}( t )}{k_{1}+ k_{2}- k_{3}\tilde{\varphi}( t ) - 2 i \sqrt{\kappa}\tilde{\varphi}( t )}\Bigr )^{- ( n + \frac{1}{2}) / 2}.
R < \min\left\{\begin{aligned}3C - F_1,C + 2C_0 + F_2, &2C + 4C_0 + F_3 -F_4, & F_5, \frac{1}{2}\left[C + 2C_0 + F_2+F_6\right], & C + 2C_0 + \frac{1}{2}\left[F_3+F_6 - F_4\right], & C + 2C_0 + \frac{1}{3}\left[2F_6 - F_1\right] .\end{aligned}\right\}
S [ \{\nu^{j}\}, \{\mu^{j}\}] = \sum_{i \neq k = 1}^{N}\frac{1}{2}\left[ \frac{\mu_{i}^{3}\mu_{k}^{3}+ \mu_{i}^{4}\mu_{k}^{4}}{g}+ g ( \epsilon_{i}^{3}\epsilon_{k}^{3}+ \epsilon_{i}^{4}\epsilon_{k}^{4}) \right] \ln \left| \frac{r_{i}- r_{k}}{a}\right|
i_{abc}(t)=T^{-1}i_{\alpha\beta\gamma}(t)=[\begin{matrix}1&0&1\\ -\frac{1}{2}&\frac{\sqrt{3}}{2}&1\\ -\frac{1}{2}&-\frac{\sqrt{3}}{2}&1\end{matrix}][\begin{matrix}i_{\alpha}(t)\\i_{\beta}(t)\\i_{\gamma}(t)\end{matrix}].
W^{(\alpha)}(\Omega) := \sum_{n \in \mathbb{Z}}\frac{\left|\left\langle u_{D,\alpha \alpha}^{\infty}, \varphi^{(n)}_{\Omega,\alpha}\right\rangle_{\mathbb{S}}\right|^{2}}{\left|\lambda^{(n)}_{\Omega,\alpha}\right|},
\Sigma^{}_6(\mu) \ll \sum_{j=1}^3 \left( \frac{T^{3+\varepsilon}L^{41j/4}}{M^{1/2}}+ T^{5/2 + \varepsilon}L^{29j/4}\right) \ll T^{\frac{9}{4}+ \frac{\eta}{2}+ 31\lambda +\varepsilon}+ T^{\frac{5}{2}+ 22\lambda+\varepsilon}.
\;\;\;\;\;\;- \frac{1}{M^2}[ C^2 (\frac{6 f_{1}(s)}{cosh^{2}s}+ \frac{\phi^{(2)}_{2}(s)}{cosh^{6}s}) + 4 K^b_a K^a_b (\frac{f_{2}(s)}{cosh^{2}s}+ f_{3}(s) \phi^{(2)}_{2}(s)) ],
\int_{\Omega}u^{\gamma-1}(t)\nabla u (t)\cdot\nabla w(t) =-\frac{\lambda}{d_3\gamma}\int_{\Omega}w(t)u^{\gamma}(t)+\frac{k}{d_3\gamma}\int_{\Omega}u^{\gamma+1}(t) +\frac{l}{d_3\gamma}\int_{\Omega}u^{\gamma}(t)v(t).
c_{j}={\frac{j_{j + 1 / 2}( i S_{n}\bar{\Omega}_{n}) h_{j + 1 / 2}^{( 1 )}( i S_{n}\bar{\Omega}_{n}) - j_{j - 1 / 2}( i S_{n}\bar{\Omega}_{n}) h_{j - 1 / 2}^{( 1 )}( i S_{n}\bar{\Omega}_{n})}{j_{j + 1 / 2}^{2}( i S_{n}\bar{\Omega}_{n}) - j_{j - 1 / 2}^{2}( i S_{n}\bar{\Omega}_{n})}},
\Big|\Psi^{\mathrm{var}}_l(x)- \Psi^{\mathrm{var}}_l(0) - 1 + \sqrt{2\pi}\phi\big(\frac{wx}{r}\big)\big[1+ \frac{\zeta}{\kappa -\zeta}\big( \frac{wx}{r}\big)^2\big] \Big|\leq \frac{4}{\kappa -\zeta}\big( r+ 4r^{-1}\big) \phi(r) =\delta \ ,
{}_{\mathrm{N S}}^{( 0 ) T}\langle \mathrm{D}p_{\psi}, \eta | = \langle \widetilde M | \langle L | \left( \widehat C{\widehat{\gamma}}^{6}\dots{\widehat{\gamma}}^{5 + s}\frac{1 - \mathrm{i}\eta \widehat \gamma}{1 - \mathrm{i}\eta}\right)_{L M}
\bigg(\sum_{j=0}^{k}\dbinom{k}{j}(1-q)^{k-j}q^j \dbinom{k}{j}\Big(\frac{1}{2}\Big)^k \bigg)^{1/p_k}\leq \Big(\frac{1-q}{2}\Big)^{k/p_k}+ \Big(\frac{q}{2}\Big)^{k/p_k}
E_{0}^{(1)}=\lambda(\frac{\alpha}{\pi})^{\frac{1}{2}}\frac{\partial^{2}}{\partial\alpha^{2}}(\frac{\pi}{\alpha})^{\frac{1}{2}}=\lambda\frac{3}{4}\frac{1}{\alpha^{2}}=\frac{3}{4}\frac{\hbar^{2}\lambda}{m^{2}\omega^{2}}.
L^{A}= \frac{1}{2 R}\sum_{n \geq 0}\left[ \omega_{n + \frac{1}{2}}( a_{n}\dot{b_{n}}- \dot{a_{n}}b_{n}) - \omega_{n + \frac{1}{2}}^{2}( a_{n}^{2}+ b_{n}^{2}) \right] .
=\alpha L^2\sum_{L^2\leqslant p\leqslant\exp(\log^2L)}\frac{\omega(p)}{p\log p}+O\bigg(\alpha^2L^2\sum_{L^2\leqslant p\leqslant\exp(\log^2L)}\frac{\omega(p)}p\bigg) &\leqslant\alpha a_{\omega}\cdot\frac{L^2}{2\log L}+O_{\omega}\left(\alpha\frac{L^2}{\log^2L}+\alpha^2L^2\log\log L\right),
|\theta^j_{\alpha \beta}(x', t; y')| = t |\partial_j K^L_{\alpha \beta}(x'-y', t)| \lesssim \frac{t}{|(x'-y', t)|^n}, |\nabla_{y'}\theta^j_{\alpha \beta}(x', t; y')| \le t |\nabla^2 K^L_{\alpha \beta}(x'-y', t)| \lesssim \frac{t}{|(x'-y', t)|^{n+1}}.
q(z,s+h)-q(z,s)={}\biggl(\int_{\mathbb R}{p}(t-s-h,x_1-y)\sigma(s+h,y)\, dy &{}-\int_{\mathbb R}{p}(t-s-h, x_2-y)\sigma(s+h,y)\,dy\biggr) &{}- \biggl(\int_{\mathbb R}{p}(t-s, x_1-y)\sigma(s,y)\,dy &{}-\int_{\mathbb R}{p}(t-s, x_2-y)\sigma(s,y)\,dy\biggr) :=E_1-E_2.
{\overline{{{V_{0}+ V_{1}+ V_{2}}}}}= ( -{\frac{1}{2}}({\frac{1}{6}}~{\frac{1}{3}}) ( -{\frac{1}{3}}{\frac{5}{6}})^{2}\vert \vert{\frac{e_{1}}{2}}~ ({\frac{5}{6}})^{2}\vert ({\frac{1}{3}})^{5}~ 0_{r}^{5}~ ({\frac{2}{3}})_{r}) \,
\varphi^{[B;m]}_{add}(\tau,u,v,t):=\frac{1}{2}e^{2\pi i(m+\frac{1}{2})t}\sum\limits^{2m}_{j=0}R^{[B]}_{j+\frac{1}{2},m+\frac{1}{2}}(\tau,v)(\Theta^{-}_{j+\frac{1}{2},m+\frac{1}{2}}+\Theta^{-}_{-(j+\frac{1}{2}),m+\frac{1}{2}})(\tau,2u)
\widetilde{\psi}(x, t)=-\int_{\Omega(t)}\frac{\widetilde{\rho}(y, t)}{|x-y|}dy=-\int_{\mathbb{R}^3}\frac{\widetilde{\rho}(y, t)}{|x-y|}dy\in C^1(\mathbb{R}^3)\cap W^{1, \infty}(\mathbb{R}^3).
\begin{aligned}A_{j;i}=\partial_{i}A_{j}-\left\{_{ji}^{k}\right\}A_{k}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, A_{\,\,;i}^{j}=\partial_{i}A^{j}+\left\{_{ki}^{j}\right\}A^{k}\end{aligned}
||\sqrt{\omega}\lambda(x) \mathcal{F}(\sqrt{\cdot}\left(F_{5}+F_{6}\right)(x,\cdot \lambda(x)))(\omega \lambda(x)^{2})||_{L^{2}(\rho(\omega \lambda(x)^{2}) d\omega)}^{2}= \frac{1}{\lambda(x)^{2}}\int_{0}^{\infty}(L(\left(F_{5}+F_{6}\right)(x,\cdot \lambda(x))))^{2}(R) R dR
\begin{aligned}\Phi_k(x)=T_k \circ T_{k-1}\circ \dots \circ T_1(x)=T_k\circ\Phi_{k-1}(x),\ \k \in \mathbb{N}, \Psi_k(x)=T_1 \circ T_2 \circ \dots \circ T_k(x)=\Psi_{k-1}\circ T_k(x),\ \k \in \mathbb{N}.\end{aligned}
\|\tilde\rho Q\nabla_y w\|_k^k = \int_{Q(a_1,b_1)}|\tilde\rho Q\nabla_y w|^k\, ds\,dy \ \leq c_2^k\int_{Q(a_1,b_1)}\tilde\rho^k w^{k-1}W_1\, ds\, dy \leq c_2^k\|\tilde\rho w\|_\infty^{k-1}\int_{a_1}^{b_1}\zeta_{W_1}(s,x)\, ds.
\sum_{j=2}^3\phi_j^af_{1t}^j=-\sum_{j=2}^3[\phi_{jt}^af_1^j+\phi_{j1}^af_t^j+\sum_{k=2}^3(\phi_{jk}^af_t^k)f_1^j]-\phi_{1t}^a\quad \quad a=1,2,3
\frac{\ddot{a}}{a}+ 2 \frac{\dot{a}^2}{a^2}+ D \frac{\dot{a}}{a}\frac{\dot{b}}{b}+ \frac{2}{a^2}=\frac{8 \pi \overline{G}}{D+2}\left[\rho + (D-1) p_a - D p_b\right]
C = \sum_{l=1}^L C_l \leq \sum_{l=1}^L \chi_{l}+ \left(\sum_{l=1}^L K_l\right) \log_2 \mathrm{tr}\left[ \left(\mathbf{W}^{\mathrm{RF}}\right)^\mathrm{H}\left(\frac{\sum_{l=1}^L \sum_{k_l=1}^{K_l}\mathbf{R}_{k_l}}{\sum_{l=1}^L K_l}\right) \mathbf{W}^{\mathrm{RF}}\right] .
b_{11}(0) + \left (2p_1 -q_1^2 \frac{a(0)}{b_{11}(0)}\right )\tau \leq b_{11}(\tau)\leq b_{11}(0) +\frac{2n_1}{n_1+1}p_1 \tau, \tau \leq 0, & b_{12}(0) + 2p_2\tau \leq b_{12}(\tau) \leq b_{12}(0) + \left (2p_2 -q_2^2 \frac{a(0)}{b_{12}(0)}\right ) \tau,\tau \leq 0.
||\alpha_k(x)||_{L^2}^2 =\begin{pmatrix}\int_{S^n}\alpha_k^2(x)~d\sigma(x)\end{pmatrix}&=\sum_{i=1}^{h_k}\phi_{i,k}(y)^2\int_{S^n}\phi_{i,k}(x)^2~d\sigma(x) =\sum_{i=1}^{h_k}\phi_{i,k}(y)^2 =h_k
\begin{aligned}Z_t - a^{kl}Z_{kl}+ b^k Z_k = \zeta^2 (W_t - a^{kl}W_{kl}+ b^k W_k) + 2 \zeta \zeta_t W + \mu & \geq \zeta^2 \big(\theta^2 |\nabla^2 v|^2 - C|\nabla v|^3 - C |\nabla v|^2 - C\big) - C \zeta |W| + \mu.\end{aligned}
\rho=e\int(Z_{i}f_{i}-f_{e})d^{3}p,j=e\int(Z_{i}f_{i}v_{i}-f_{e}v_{e})d^{3}p,v_{\alpha}=\frac{\frac{p}{m_{\alpha}}}{(1+\frac{p^{2}}{(m_{\alpha}c)^{2}})^{1/2}}
Q_{1}\left( x,y,z\right) =x^{2}+2cxy+2y^{2}+4\left( c^{2}-1\right)xz+6cyz+z^{2}\left( 4c^{2}+5\right) Q_{2}\left( x,y,z\right) =y^{2}-xz+2cyz+2z^{2}.
Y_n =\int_0^T \left[\varphi(t,\gamma_n(t,X_n(t^-)))-\varphi(t,\gamma_n(t,X(t^-)))\right]dt Z_n =\int_0^T\int_A \varphi(t,a)[\delta_{\gamma_n(t,X(t^-))}-\widehat{\gamma}(t,X(t^-))](da)dt.
0=\frac{\partial}{\partial y}(y\beta(y,t))+\frac{1}{4}\frac{\partial^{2}}{\partial y^{2}}\beta(y,t)+\biggl[\frac{1}{2}y\frac{\partial}{\partial y}+\frac{1}{2}(y^{2}+1)\frac{\partial^{2}}{\partial y^{2}}+\frac{1}{4}y\frac{\partial^{3}}{\partial y^{3}}+\frac{1}{64}\frac{\partial^{4}}{\partial y^{4}}-\frac{1}{2}\biggr]p_{0}(y,t)