harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\psi^\dagger_1(x)\psi_1(x)=\frac{1}{8\pi}\int d\hat\epsilon \left\{\left[\psi^\dagger-1(x+\epsilon),\psi-1(x-\epsilon)\right],e^{ie\int_{x-\epsilon}^{x+\epsilon}dx_iA_i^{T}}\right\}_{|\eta|, |\xi|, |\epsilon|\propto 1/\Lambda}
x_0^{a_0-1}y_0u_1x_0^{-(a_0-1)}=x_0^{-1}y_0v_1x_0=x_0^{-1}y_0x_0v_2=x_0^{-1}y_0x_0x_1x_1^{-1}v_2 &=x_0^{-1}y_0^3x_1^{-1}v_2=x_1y_0v_3x_1^{-1}=y_0x_2v_3x_1^{-1}
a(t) := \mathrm{exp}(tx) = 1_n+\displaystyle \sum_{i < j}\frac{t^{j-i}}{(j-i)!}e_{i,j}, b(s) := \mathrm{exp}(sy) = 1_n + se_{n,1}, c(r) := \mathrm{exp}(rz) = 1_n + \sum_{j=2}^n \sum_{i=1}^{j-1}c_{j-i,j}\frac{r^{j-i}}{(j-i)!}e_{j,i}c_{j-i,j}= \prod_{k=1}^{j-i}b_{j-k}.
S_m := C_R(a + mb) &=Z \cup (Z + (a + mb)) \cup (Z + 2(a + mb)) \cup \dots \cup (Z + (p - 1)(a + mb)), 1 \leq m \leq (p - 1), S_p := C_R(a) = Z \cup (Z + a) \cup (Z + 2a) \cup \dots \cup (Z + (p - 1)a) S_{p + 1}:= C_R(b) = Z \cup (Z + b) \cup (Z + 2b) \cup \dots \cup (Z + (p - 1)b).
\begin{array}{rcl}f_{5}-D_{4}= 0 5\,D_{4}+f_{4}+D_{3}= 0 -10\,D_{4}-4\,D_{3}+f_{3}-D_{2}= 0 10\,D_{4}+6\,D_{3}+3\,D_{2}+f_{2}+D_{1}= 0 -5\,D_{4}-4\,D_{3}-3\,D_{2}-2\,D_{1}+f_{1}-D_{0}= 0, \end{array}
m_{\chi,r}^X(n)=\sum_{\rho}\sum_{m<0}\frac{a_{\rho}^X(m)}{n^{\frac{1}{4}}}\sum_{c=1}^{\infty}\frac{K^X_{\rho}(m,n,c)}{c}\cdot \mathbb{I}^X\left (\frac{4\pi \sqrt{|nm|}}{c}\right),
e^{\frac{\eta_{-}}{\varphi(x)}(\Re{z_{-}}+1)( u-v)}\frac{1}{\varphi(x)}S_{N}\Big(N\big(x+\frac{u}{N\varphi(x)}\big), N\big(x+\frac{v}{N\varphi(x)}\big)\Big) =: I_{N}+J_{N}
\mu_0(n,p,b,\beta,\ell,\psi,I_0,I_1) &= \mu \left(p,b,\beta,\frac{2I_1}{3I_0}\right)2^{-1}\| \psi^{\ell}\|_{L^1(\mathbb{R}^n}^{-(p-1)}A(n,p,\ell,\psi)^{-\frac{(p-1)^2}{p_F-p}}\left( \frac{1}{4}I_0 \right)^{\frac{1}{\frac{1}{p-1}-\frac{n}{2}}},
v(x,{\bf s}_{\rm even}) = 0, & e^{u(x,{\bf s}_{\rm even})}= \sum_{k=1}\frac1k \sum_{j_1,\dots,j_k\in \mathbb{Z}^{\rm even}_{\geq 0}, \atop j_1+\cdots+j_k=2k-2}{\rm wt}(j_1) \cdots{\rm wt}(j_k) \binom{j_1}{j_1/2}\cdots \binom{j_k}{j_k/2}s_{j_1}\cdots s_{j_k},
b^2 = (q^3 - g^4)(q^7-g^4)q^{-6}g^{-3}f^{-2}+ [\textrm{higher terms in}f] cb = (q^3 - g^4)(q^7-g^4)q^{-7}g^{-3}f^{-2}+ [\textrm{higher terms in}f]
\big\|\int^t_{t_0}e^{i(t-\tau)\triangle}\vec{F}(\tau)d\tau \big\|_{U^2_{\triangle}(h^{1};J\times \mathbb{R}^2)}\lesssim \sum^{k}_{m=1}\big\| \int_{J^m}e^{-i\tau\triangle}\vec{F}(\tau)d\tau \big\|_{L^2_x h^{1}}+ \big[\sum^{k}_{m=1}(\|\vec{F}\|_{DU^{2}_{\triangle}(h^{1};J^m\times \mathbb{R}^2)})^2 \big]^{1/2}.
Y_{h,-l}:= \lbrace \boldsymbol{v}_h \in L^2(\Omega):~\boldsymbol{v}_h|_T \circ F_T \in{\mathbb R}_l(\hat T)~ \forall T \in{\mathcal T}_h \rbrace, Y_{h,l}:= Y_{h,-l}\cap W^{1,2}(\Omega).
\sum_{|p|+q=0}^{[N/2]}|w^{q/2}\partial_\tau^p\partial_3^q\eta^r,_s|+ \sum_{|p|+q=0}^{[N/2]-1}|w^{q/2}\partial_\tau^p\partial_3^q\eta_t^r,_s|<\infty\,.
=\int{\cal D}\vec\varphi\exp\Biggl\{-\int d^3x\Biggl[\frac12\left(\partial_\mu\vec\varphi-\frac{\vec\mu_\alpha}{g_m}\partial_\mu\eta\right)^2-2\zeta\sum\limits_{i=1}^{N(N-1)/2}\cos\left(g_m\vec q_i\vec\varphi\right)\Biggr]\Biggr\},
L=\frac{1}{8}\,T_{(\mu \nu )\sigma}T^{(\mu \nu )\sigma}+\frac{1}{4}T_{(\mu\nu )\sigma}T^{(\mu \sigma )\nu}+\frac{1}{2}T_{(\mu \nu )\sigma}\epsilon^{\mu \nu \alpha \beta}\partial_{\alpha}h_{\;\beta}^{\sigma}+h^{\alpha\beta}\Theta_{\alpha \beta}\ ,
L(g) \leq L(g\eta_\varepsilon)+\varepsilon L(\big(\|x\|^2+1)g+g^2\big) &=\int g\eta_\varepsilon d\mu+ \varepsilon L\big((\|x\|^2+1)g+g^2\big)\leq \int g d\mu+ \varepsilon L\big((\|x\|^2+1)g+g^2\big).
\begin{array}{ccc}\Gamma^{\hat{\mu}}=\left(\begin{array}{ll}0 \gamma^{\hat{\mu}}\tilde{\gamma}^{\hat{\mu}}0\end{array}\right)\,,~~~~ ~~~~\gamma^{\mu}=\tilde{\gamma}^{\mu}\,,~~~~\gamma^{5}=-\tilde{\gamma}^{5}=1\,.\end{array}
\left[\begin{array}{ccccc}B_k(\lambda) B_{k-1}(\lambda) B_{k-2}(\lambda) \cdots B_1(\lambda) \hline-I_n \lambda^\ell I_n & -I_n \lambda^\ell I_n & \ddots \ddots & -I_n \lambda^\ell I_n\end{array}\right]= \left[ \begin{array}{c}M_1^\ell(\lambda) \hline \phantom{\Big{(}}L_{k-1}(\lambda^\ell)\otimes I_n \phantom{\Big{(}}\end{array}\right]
\mathcal{J}_{2}(y; \mu) = \Bigl| \frac{y_1}{y_2}\Bigr|^{\frac{1}{2}\mu_2}\int_1^\infty J^{-}_{3\nu_3}\left(2\pi |y_1|^{1/2}\sqrt{u^2-1}\right) J^{-}_{3\nu_3}\left(2\pi |y_2|^{1/2}\sqrt{1-u^{-2}}\right) u^{3\mu_2}\frac{du}{u},
\|f - \tilde{f}\|_\pi^2 \leq 2^d \sum_{S \subseteq [d]}\hat{f}(S)^2 O\left(\frac{d^2}{p(1-p)n}\right) = O\left(\frac{d^22^d}{p(1-p)n}\right)\|f\|_\pi^2,
\frac{DR_n''(\rho|\rho_{\rm tx})}{R_n(\rho|\rho_{\rm tx})}+\frac{D R_n'(\rho|\rho_{\rm tx})}{\rho R_n(\rho|\rho_{\rm tx})}-\frac{Dn^2}{\rho^2}=\frac{T'_n(t|t_0)}{T_n(t|t_0)}\overset{(b)}{=}-\gamma_n^2,
\chi_{\widetilde{f}}= \;\frac{1}{12}n(n-1)(2n-1)\left(\frac{2rM}{n^2}-\sum_{k\ge 1}k^2\alpha_k\right)+\frac{1}{4}n(n-1)\left(-\frac{2M}{n}+\sum_{k\ge 1}k\alpha_k\right) = \;\frac{n-1}{6n}(r(2n-1)-3n)M-\frac{n(n-1)}{12}\sum_{k\ge 1}((2n-1)k^2-3k)\alpha_k
S_{q,r}(m) = \frac{p \left(q - 1 \right) q^v w + \left( q - 1\right)r + r}{q}& = p \left(q - 1 \right) q^{v-1}w + r & = \left(1 - \frac{1}{q}\right) \left( p\cdot q^v w \right)+ r = \left(1 - \frac{1}{q}\right) \left( m-r \right)+ r.
\phi_{*}\left(J_{1}\frac{\partial}{\partial y^{i}}\right)_{p}= -\phi_{*}\left(\frac{\partial}{\partial x^{i}}\right)_{p}= -\displaystyle\sum_{j=1}^{m}\frac{\partial a_{j}}{\partial x^{i}}(p)\frac{\partial}{\partial u^{j}}|_{\phi(p)}- \displaystyle\sum_{j=1}^{m}\frac{\partial b_{j}}{\partial x^{i}}(p)\frac{\partial}{\partial v^{j}}|_{\phi(p)}
\tilde{\rho}_\eta( \tilde{\mu}, L \mu_0) \leq \frac{2}{t^*}\int_0^{t^*}\bigl[ \tilde{\rho}_\eta( \tilde{\mu}, L( (\Pi \tilde{\mu})P_{t + t^*}^0)) +\tilde{\rho}_\eta( \tilde{\mu}, L( (\Pi \tilde{\mu})P_{t}^0)) \bigr] dt = \frac{2}{t^*}\int_0^{2t^*}\tilde{\rho}_\eta( \tilde{\mu}, L( (\Pi \tilde{\mu})P_{t}^0)) dt \,.
c_{n-1,\xi_{\boldsymbol{s}k}}^{\ast\ast}=\frac{\Gamma\left((n-1)/2\right)}{(2\pi)^{(n-1)/2}}\left[\int_{0}^{\infty}x^{(n-3)/2}\overline{\mathcal{G}}_{n}\left(\frac{1}{2}\xi_{\boldsymbol{s}k}^{2}+x\right)\mathrm{d}x\right]^{-1}
{\cal E}=\int_0^\infty du\int_{-\pi}^{\pi}d\phi\,u\left\{\left[b-b_{\rm ext}\right]^2+\left(\partial_uf\right)^2+\frac{1}{u^2}\left(\partial_\phi f\right)^2+f^2\left(j^2+\frac{1}{u^2}g^2\right)+\frac{1}{2}\kappa^2\left(1-f^2\right)^2\right\}.
\Delta S^{I}_l(N,z^{\beta}) = S^{I}_{p,q}\bigl( N,z^{\beta}e^{-2 li \pi}\bigr)- S^{I}_{p,q}\bigl( N,z^{\beta}e^{-2 (l-1) i \pi}\bigr) & \equiv \int_{c-i \infty \atop{{\rm Max}[N-1,-q/p] <c=\Re\,s<N}}^{c+i \infty}\!\!\!\! z^{\beta s}\; \Gamma(ps+q) \, e^{-(2l-1) i \pi s}\; ds \;\; ,
\mathbb{F}^{(l)}\left( B\right) :=\frac{1}{\left\vert \Lambda_{l}\right\vert^{1/2}}\underset{x\in \Lambda_{l}}{\sum}\left\{\chi_{x}\left( B\right) -\varrho^{(\beta ,\omega ,\vartheta ,\lambda )}\left(\chi_{x}\left( B\right) \right) \mathbf{1}_{\mathcal{U}}\right\}\ .
\delta_{[k = \frac{1}{2}]}\delta_{[h = a^2]}\bigg( 2 R'_h X^{\frac{1}{2}}\log X -4R'_h X^{\frac{1}{2}}\bigg) + \tfrac{1}{k}R_{k, h}^k X^{k}+ O\big(\frac{X^{\frac{1}{2}}\log X}{Y}\big) + O\big(\frac{X^{k}}{Y}\big) + O(X^{\frac{k-1}{2}}) + O(X^{\frac{k}{2}+ \epsilon}Y^{3k + \frac{17}{2}+ 2\epsilon}).
T_{F,G,N}^L(f_1,\dots,f_4) = \frac{1}{N^2}\sum\limits_{\substack{\mathbf{n}\in \mathbb{Z}^4 \Vert L\mathbf{n}\Vert_\infty \leqslant 10 (\begin{smallmatrix}5 1 \end{smallmatrix})L\mathbf{n}= -1,0,1}}\Big(\prod\limits_{j=1}^4 f_j(n_j)\Big)F(\mathbf{n}).
I_{22}\lesssim\sum_{i=1}^\infty \frac{\ell(Q)^{2s}}{\ell(2^iQ)^{1+2s}}\,\Big(\ell(2^iQ)^{\frac Nq+ 2s}\,\ell(Q)^{\frac N{q'}}+ i\,\ell(Q)^N\,\ell(2^iQ)^{2s}\Big) &=\frac{\ell(Q)^{2s}\ell(Q)^{N+2s}}{\ell(Q)^{1+2s}}\sum_{i=1}^\infty \,2^{i(\frac Nq-1)}(1+\frac{i}{2^i}).
\limsup_{k}{\Phi}^{k}_{\epsilon}(s)^{1/k}\leq \limsup_{k}(nk)^{1/k}{\Psi}_{\omega}^{m_{k}}(s,\epsilon)^{1/k}\leq\limsup_{k}k^{1/k}{\Psi}_{\omega}^{m_{k}}(s,\epsilon)^{1/m_{k}}=1
M\models \forall \; \overline{z}\; \exists \; \overline{b}\; \exists \; c \; \bigvee_{i=1}^{K}\;(\theta^{\prime}_{i}(c, \overline{b}) \; \& \; \xi_{0}(\overline{z}, \theta_{i}(\cdot, \overline{b}), c, \rho(\cdot, \overline{a})))
\tilde{\alpha}=\alpha \beta^{-m}=\left( \begin{array}{ccc}\omega_{k}^{-2y}\omega_{2d}^{2m}0 0 0 \omega_{k}^{y}\omega_{2d}^{-m}0 0 0 \omega_{k}^{y}\omega_{2d}^{-m}\end{array}\right)
= \Bigl( \sum_{\substack{0 \leq d_1,d_2 d_2 -d_1 =\deg g- \deg h}}q^{- d_1 \alpha_1+ d_2 \alpha_2- \frac{d_1+d_2}{2}}A_{d_2 -\deg g}+ \sum_{\substack{d_1,d_2 \leq N d_2 -d_1 =\deg g- \deg h}}q^{- d_1 \alpha_1+ d_2 \alpha_2- \frac{N-d_1+N- d_2}{2}}A_{N-d_1-\deg g}\Bigr)
|F(x) - F(y)| \leq \sum_{k=k_0}^\infty \sum_{j=1}^K |\varphi_j^k(x) - \varphi_j^k(y)| & \leq \sum_{k=k_0}^\infty KR(M)^2 \frac{r_{k_0}(x)(M-2n+1)^{k-k_0}}{M^k}& \leq KR(M)^2 \left(\frac{1}{1-(M-2n+1)/M}\right) \frac{r_{x_0}(x)}{M^{k_0}}.
W^{I}= W_{L}^{I}+ W_{R}^{I}\ , W_{L}^{I}= \int_{0}^{\pi / 2}d \sigma \partial_{\sigma}X^{I}( \sigma ) \ , W_{R}^{I}= \int_{\pi / 2}^{\pi}d \sigma \partial_{\sigma}X^{I}( \sigma ) \ ,
\mathcal{P}(\mu_0,\overline\mu_1\ldots,\overline\mu_{2n})=\Big(\varepsilon(\mu_0)\overline\mu_1,\ldots,\varepsilon(\mu_0)\overline\mu_{2n}, \frac{\mu_0 -\sin\mu_0}{2\mu_0^2}\sum_{r=1}^{2n}\overline\mu_r^2\Big)\,.
u_{0,l}(r)=\left( \begin{array}{l}\phi_{0,l,1}(r) 0\end{array}\right) ,{\rm \;}{}l\geq 1;\;\;\;u_{0}^{II}(r)=\left( \begin{array}{l}\phi_{0,1}^{ir}(r) 0\end{array}\right) ,\;{}l=0.
\left( \mathcal{D}^\vee \vec{F}^\vee_\lambda\right)(x) = \mathcal{D}\left({}^\vee \!\vec{F}\right)^\vee_\lambda(x) &=|x|^{-2\nu}\left( \mathcal{D}\;{}^\vee \!\vec{F}\right)\bigg( -\frac{1}{x}\bigg) &=|x|^{-2\nu}\left( \mathcal{D}^\vee\vec{F}\right)\bigg( -\frac{1}{x}\bigg),
f_{1}( s ) \equiv \int_{0}^{s}d x \frac{s i n h x}{c o s h^{5}x}\phi_{2}^{( 2 )}( x ) , \; f_{2}( s ) \equiv \int_{0}^{s}d x \frac{x}{c o s h^{2}x}\phi_{2}^{( 2 )}( x ) ,
\sum_{j=0}^{\infty}\frac{(dq)_{j}(cq^{k}/d)^{j}}{(dq^{k+1})_j(q)_j}=\frac{(cq^{k+1})_{\infty}}{(cq^{k}/d)_{\infty}}\sum_{j=0}^{\infty}\frac{(dq)_{j}q^{j^2+2jk}c^j}{(dq^{k+1})_j(cq^{k+1})_j(q)_j}.
s_{\sigma_{y}(i)-d_{i}^{-}(y)}\cdots s_{\sigma_{y}(i)-1}w_{y}=s_{\sigma_{x}(i)+d_{i}^{+}(x)-1}\cdots s_{\sigma_{x}(i)}w_{x}, a_{j}(w_{x}x)=0 (\sigma_{x}(i)\le j<\sigma_{x}(i)+d_{i}^{+}(x)), a_{j}(w_{y}y)=0 (\sigma_{y}(i)-d_{i}^{-}(y)\le j<\sigma_{y}(i))
\int_{1}^{\infty}x^{-\alpha}(x-1)^{\alpha-\beta-1}G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{q}\end{matrix}|zx)dx=\Gamma(\alpha-\beta)G_{p+1,q+1}^{m+1,n}(\begin{matrix}a_{p},\alpha\\ \beta,b_{q}\end{matrix}|z).
\mathcal{K}_{}+ A_0 \geq \frac{1}{2\lvert \Lambda \rvert}\sum_{k \in \Lambda^*}\Big( \mathcal{A}_k (b_k^{\dagger}b_k + b_{-k}^{\dagger}b_{-k})+ \widehat{g}_k (b_k^{\dagger}b_{-k}^{\dagger}+ b_{-k}b_k) \Big)
\frac{(256-3125a^{4})}{1155}\frac{d^{4}\phi}{da^{4}}-\frac{6250a^{3}}{231}\frac{d^{3}\phi}{da^{3}}-\frac{4875a^{2}}{77}\frac{d^{2}\phi}{da^{2}}-\frac{2125a}{77}\frac{d\phi}{da}+\phi=0
|\hat{V}(\xi)| = \Big| \int_0^1 f^\prime(t){\rm d}t - f(1)\Big| &\leq |\hat{V}(\xi + \frac{\rm i}{\zeta}\xi)| + \frac{1}{\zeta}\sup_{0\leq t\leq 1}|\nabla \hat{V}(\xi + \frac{{\rm i}t}{\zeta}\xi)\cdot\xi|.
a_n := \lim_{d \to \infty}\frac{A_n( d )}{d^n}, b_n^1 := \varlimsup_{d \to \infty}\frac{\mu_n( d )}{d^n}begin{align*}1ex] b_n^2 := \varliminf_{d \to \infty}\frac{\mu_n( d )}{d^n}, c_n := \binom{n}{\lfloor n / 2 \rfloor}2^{-n}.
\exp{{i\over\hbar}S_{\rm IF}[x_c]}\approx e^{{i\over\hbar}\left[\int^{\cal C}ds_1 F(s_1) f(x_c(s_1)) +{1\over 2}\int^{\cal C}ds_1ds_2 f(x_c(s_1)) D(s_1,s_2) f(x_c(s_2))\right]}
H_{+}( x ) = \left( \begin{matrix}{{h_{G}^{( 1 )}( x )}}&{{0}}\\{{0}}&{{h_{G}^{( 1 )}( x )}}\\ \end{matrix}\right) \quad \mathrm{a n d}\quad H_{-}( x ) = \left( \begin{matrix}{{h_{G + 1}^{( 1 )}( x )}}&{{0}}\\{{0}}&{{h_{G - 1}^{( 1 )}( x )}}\\ \end{matrix}\right)
\begin{array}{ll}\displaystyle\int_{S}| U_*|^{2^*}=\displaystyle\int_S V^{2^*}-\displaystyle 2^*\int_S V^{2^*-1}u_0+O(\int_S v^{2^*-1-\delta}u_0^\delta) =\displaystyle\int_S V^{2^*}-\displaystyle 2^*\int_S V^{2^*-1}u_0+O(\lambda^{-\frac{(1+\delta)(N-2)}{2}}), \end{array}
\psi_{0,1}( x, y ) \, v_1 = \big(\tfrac{\partial}{\partial y}\psi( x, y )\big)( v_1 ) , \psi_{2,0}( x, y )( v_1, v_2 ) = \big( \tfrac{\partial^2}{\partial x^2}\psi( x, y ) \big)( v_1, v_2 ) ,
\beta_{i j}^{( k )}\equiv \partial_{x}^{- 1}\left({\cal L}^{k}( \Phi_{j}) \Psi_{i}\right) ; \beta_{i j}^{* ( k )}\equiv \partial_{x}^{- 1}\left( \Phi_{j}{\cal L}^{* k}( \Psi_{i}) \right)
a(t){\Vert u(t)\Vert}^2 ={\Vert x \Vert}^2 + \int_0^t\frac{d}{ds}\big( a(s){\Vert u(s)\Vert}^2\big)\, ds ={\Vert x \Vert}^2 + \int_0^t qb{\Vert u(s)\Vert}^2\, ds + 2\int_0^ta(u,u^{\prime})\, ds \le{\Vert x \Vert}^2 + M({\Vert u \Vert}_X^2 +{\Vert u \Vert}_X{\Vert u^{\prime}\Vert}_X) < \infty,
\Sigma=[\begin{matrix}\langle x\cdot x\rangle&\langle x\cdot x^{\prime}\rangle\\ \langle x\cdot x^{\prime}\rangle&\langle x^{\prime}\cdot x^{\prime}\rangle\end{matrix}]=[\begin{matrix}\sigma_{11}&\sigma_{12}\\ \sigma_{21}&\sigma_{22}\end{matrix}]
Q_{p - 1}={\frac{1}{4}}\sqrt{- d e t ( \eta + F )}e^{-{\frac{p - 3}{2}}\phi}E^{a_{1}}\wedge . . . \wedge E^{a_{p - 1}}\epsilon_{a_{1}. . . a_{p - 1}a b}( \eta + F )^{- 1 a b}.
\Psi = \begin{bmatrix}\psi(x_1) \psi(x_2) \cdots \psi(x_n) \psi(x_1)u_{1,1}\psi(x_2)u_{1,2}\cdots \psi(x_n)u_{1,n}\vdots \vdots \ddots \vdots \psi(x_1)u_{m,1}\psi(x_2)u_{m,2}\cdots \psi(x_n)u_{m,n}\end{bmatrix},
\frac{L'}{L}(s,\chi) = -\sum_{n\leq xy}\frac{\Lambda_{x,y}(n)\chi(n)}{n^s}+ \frac{1}{\log{y}}\sum_{\rho_{\chi}}\frac{x^{\rho_{\chi}-s}-(xy)^{\rho_{\chi}-s}}{\left(\rho_{\chi}-s\right)^{2}}+ \frac{1}{\log{y}}\sum_{n=0}^{\infty}\frac{x^{-2n-\mathfrak{a}-s}-(xy)^{-2n-\mathfrak{a}-s}}{\left(2n+\mathfrak{a}+s\right)^{2}}
s_1 =\frac{t_1}{2}, s_2 =\frac{4t_1+7t_1^3-8t_2+2t_1^2t_2+(-8-10t_1^2-t_1^4-4t_1t_2+t_1^3t_2)\sqrt{-3}}{2(4t_1+t_1^3-4t_2+t_1^2t_2-(4+3t_1^2)\sqrt{-3})}, s_3 =-\frac{(t_1^2-4)t_1t_3(2+t_1\sqrt{-3})}{2(-4t_1-7t_1^3+8t_2-2t_1^2t_2+(8+10t_1^2+t_1^4+4t_1t_2-t_1^3t_2)\sqrt{-3})}.
r!\sum_{\{x_{v_{1}},x_{v_{2}},\ldots,x_{v_{r}}\}\in E(H)}x_{v_{1}}x_{v_{2}}\cdots x_{v_{r}}= \frac{r!}{\alpha^{\frac{1}{r}}}\sum_{e\in E(H)}\prod_{v\in e}(B^{\frac{1}{r}}(v,e)x_{v}) \leq \frac{r!}{\alpha^{\frac{1}{r}}}\sum_{e\in E(H)}\frac{\sum_{v\in e}(B(v,e)x_{v}^{r})}{r}= \frac{(r-1)!}{\alpha^{\frac{1}{r}}}\|x\|_{r}^r.
\eta(\tau_n^q\tau_i, \tau_j) =\eta(\tau_i, \eta(\tau_n^q, \tau_j))=\eta(\tau_i, \eta(\tau_n^{q-1}, \eta(\tau_n, \tau_j)))=\eta(\tau_i, \eta(\tau_n^{q-1}, \tau_n^{q-1+D(j)}\tau_{j+1})) &=\eta(\tau_i, \eta(1, \tau_n^{D(j)}\tau_{j+1}))=\eta(\tau_i, \tau_n^{D(j)}\tau_{j+1}).
2 G^{i j i^{\prime}j^{\prime}}= ( h^{i i^{\prime}}h^{j j^{\prime}}+ h^{i j^{\prime}}h^{j i^{\prime}}- \frac{2}{D}h^{i j}h^{i^{\prime}j^{\prime}}) + C h^{i j}h^{i^{\prime}j^{\prime}},
\left[ E^{2}-P_{z}^{2}-\frac{m^{2}}{4}-4\omega \left( \frac{E}{m}+(N+\left|m\right| +\frac{1}{2})\right) \right] +i\omega P_{z}b_{-}+i\omega \left(N+\left| m\right| +\frac{1}{2}\right) P_{z}b_{+}=0
{C_{\beta,n_1,1}}=(2\pi)^{n_1+1}M_{n_1}(\beta,\beta,\beta/2) = (2\pi)^{n_1+1}\frac{\Gamma(\beta (n_1+3)/2+1)\Gamma(\beta (n_1+2)/2+1)}{\Gamma(3\beta/2 +1)\Gamma(\beta+1)\Gamma(\beta (n_1+1)/2+1)(\Gamma(\beta/2+1))^{n_1-1}}.
_{2}\phi_{2}\left(\sqrt{b},-\sqrt{b};\sqrt{c}q^{1/4},-\sqrt{c}q^{1/4};\sqrt{q},-\frac{c}{b}\right)=\frac{\left(cq^{1/2}/b;q\right)_{\infty}}{\left(cq^{1/2};q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{\binom{n}{2}}}{\left(q;q\right)_{n}}\left(-\frac{c}{b}\right)^{n}A_{q}\left(-cq^{n-1}\right)
(xy)^{\omega}x = ((xy)')^{n+1}(xy)^{n+1}x = ((xy)')^{n+1}x (yx)^{n+1}= ((xy)')^{n+1}x (yx)^n (yx)' &= ((xy)')^{n+1}(xy)^n x (yx)' = (xy)'x(yx)'\,.
dY_t=\left(\lambda(\varepsilon- \theta^2) - (a +\lambda \theta) \phi_t + \theta\frac{\theta+\phi_t}{1+\frac{1}{\lambda}\phi_t}\phi_t \right)(m(t) - X_t) dt+Q_t\,dW_t+R_{t}\,d\widetilde N_t
\#\{(x, y)\in \mathbb{F}_q^2: y^2=x^3+ax+b\}=q-q\cdot \phi(-ab)\cdot{_{2}}G_2\left[\begin{array}{cc}\frac{1}{4}, \frac{3}{4}\frac{1}{6}, \frac{5}{6}\end{array}|\frac{-27b^2}{4a^3}\right]_q,
\frac{i}{\hbar}A ( Q , Q^{\prime}) = \frac{1}{\hbar^{2}}\int_{0}^{t}d s \int_{0}^{s}d s^{\prime}r ( s ) [ - i \mu ( s - s^{\prime}) R ( s^{\prime}) - \nu ( s - s^{\prime}) r ( s^{\prime}) ]
\langle\tilde{G},G\rangle =G_0\tilde{G}_1^\ast \frac{ic^2 y}{(2\pi)^2}\left ( J_0(y,-\lambda y)-\lambda y \sum_{n\geq 1}\dfrac{J_n(y,-\lambda y)}{n}\right ) &= G_0\tilde{G}_1^\ast \frac{ic}{2\pi}\partial_\lambda\Lambda(\lambda).
N_{\mu \nu \rho}\left( 0 , x , y \right) = - \frac{i g^{3}f^{2}}{5 1 2 \pi^{8}}\, x^{6}y^{6}\, J_{\nu \nu^{\prime}}( x^{\prime}) \, J_{\rho \rho^{\prime}}\left({y^{\prime}}\right) \, \frac{\mathrm{T r}\, \gamma_{5}\gamma_{\mu}\gamma_{\rho^{\prime}}\gamma_{\nu^{\prime}}\, \left( / \hspace{- 2 m m}x^{\prime}- / \hspace{- 2 m m}y^{\prime}\right)}{\left( x^{\prime}- y^{\prime}\right)^{4}}\, .
K ( \Lambda_{1}, \Lambda_{2}; t ) ) = \frac{1}{( 2 \pi i t )^{( m + n ) / 2}}\frac{1}{m !}\frac{1}{n !}d e t ( e^{i ( \lambda_{1 , i}- \lambda_{2 , j})^{2}/ 2 t}) d e t ( e^{- i ({\bar{\lambda}}_{1 , \alpha}-{\bar{\lambda}}_{2 , \beta})^{2}/ 2 t}) ,
\begin{array}{l}\left[ :X^{-}\,P^{+}X^{-}:\,-ik\partial_zX^{-}+2X^{-}S\right] (z)\,\times\frac{T^{\prime}(w)}{P^{+}(w)}\rightarrow \,\quad \,\frac{-2iT^{\prime}(w)}{\left( z-w\right)}\left(:X^{-}\frac 1{P^{+}}:\,+S\right) (w) \quad \quad \quad +\frac{k-2}{(z-w)^2}\frac{T^{\prime}(w)}{\left(P^{+}(w)\right)^2}+\cdots\end{array}
\begin{aligned}\int_0^T<[S\mathcal{L}_{\mathcal{I}_l}V(x_j,\cdot)u_\epsilon](t),v(t)>dt \lesssim_{\alpha,p,\widetilde{q},q}\sum_{\mu}Z_\mu T^{1/\theta}\|\mathcal{L}_{\mathcal{I}_l}u_\epsilon\|_X\|v\|_{L_t^{\theta_p'}(L^{p',2}_D)},\end{aligned}
\bar D_{3,a,3}= \frac{|X_1-X_2|}{|z|}\big| \big[ D_\xi Q(X_\tau,z+t_0(X_\tau),D_t \tilde h(X_\tau,z+t_0(X_\tau))) &\qquad- D_\xi Q(X_\tau,z+t_0(X_\tau),D_t \tilde h(X_\tau,t_0(X_\tau))) \big] D^2_{X,t}\tilde h(X_\tau,z+t_0(X_\tau))\big|
\begin{bmatrix}1 t_1 \cdots t_1^{n-2}t_1^{n-1}\vdots \vdots \vdots \vdots 1 t_{n-1}\cdots t_{n-1}^{n-2}t_{n-1}^{n-1}0 0 \cdots 0 \lambda\end{bmatrix}.
k_{mn\bar{p}\bar{q}}= e_{mn}(1-\delta_{pq}) N_{pm}\bar{N}_{nq}+ e_{pq}(1-\delta_{mn}) N_{qm}\bar{N}_{np}+\delta_{pq}\delta_{mn}N_{pm}\bar{N}_{nq}\hphantom{k_{mn\bar{p}\bar{q}}}{}=\frac{1}{2}( N_{qn}\bar{N}_{mp}+ N_{qm}\bar{N}_{np}).
u_{-1}= -5 u_{1}+ v_{-1}+ 2 w_{1}\, a^{- \gamma}+ (\mbox{$\frac{9}{2}$}u_{1}^2 - \mbox{$\frac{1}{2}$}v_{-1}^2 - 3 u_{1}v_{-1}+ w_{2}) a^{-2 \gamma}
s \ge\begin{cases}\frac{n+1}{4},\, \, 0 < \alpha \le 1, \frac{n+1}4 +\frac{1-\alpha}{2q},\, \,1 < \alpha \le n, \frac{n+1}4+ \frac{n+1-2\alpha}{2q},\, \, n < \alpha \le n+1.\end{cases}
\begin{array}{l}LG_{2X}=X^{-1/3}_{R}X^{1/3}_{L}LG_{2}X^{-1/3}_{L}X^{1/3}_{R}G_{2X}= X^{-1/3}_{R}X^{1/3}_{L}G_{2}X^{-1/3}_{L}X^{1/3}_{R}. \end{array}
\kappa ( h ) = \frac{2 \Gamma ( h / 2 )}{\pi \Gamma ( 1 - h / 2 )}\left( \frac{\sqrt{\pi}\Gamma \left( \frac{1}{2 - h}\right)}{2 \Gamma \left( \frac{h}{4 - 2 h}\right)}\right)^{2 - h}\, .
v_A^*(S_{a_i}S_{a_i}^*\otimes p_{\bar{0}}) v_A = v_1^* (S_{a_i}S_{a_i}^*\otimes p_{\bar{0}})v_1 & = (P_C \otimes s_{1_0,1})^* (S_{a_i}S_{a_i}^*\otimes p_{\bar{0}}) (P_C \otimes s_{1_0,1}) = S_{a_i}S_{a_i}^*\otimes s_{1,1_0}p_{\bar{0}}s_{1_0,1}.
D_{111}\ll \frac{t_3^{1-\sigma_1}}{t_3}\int_1^{at_3}\frac{1}{v^{\sigma_2}(at_3+v)^{\sigma_3}}dv \ll \begin{cases}t_3^{-\sigma_1-\sigma_3}(\sigma_2>1) t_3^{-\sigma_1-\sigma_3}(\sigma_2=1) t_3^{1-\sigma_1-\sigma_2-\sigma_3}(\sigma_2<1) \end{cases}
I_{s}={\frac{1}{4 \epsilon^{2}}}{\frac{\Gamma ( 1 + 2 \epsilon ) \Gamma ( 1 - \epsilon )^{3}}{( 1 - 3 \epsilon ) ( 1 - 2 \epsilon ) \Gamma ( 1 - 3 \epsilon )}}={\frac{1}{2}}\left[{\frac{1}{2 \epsilon^{2}}}+{\frac{1}{2 \epsilon}}+{\frac{1}{\epsilon}}( 2 - C ) \right]
C_0=\max\limits_{1\leq j\leq k}(|x_j|+|u_j|),\ \varepsilon_1=\min\limits_{0\leq i<j\leq k}\big||u_i|-|u_j|\big|,\ \varepsilon_0=\varepsilon_1S(I)^2/(2+4\varepsilon_1).
\hat{G}_{a}^{\prime}= \frac{1}{\sqrt{n}}\hat{G}_{a}, \hspace{0 . 3 c m}\hat{G}_{a b}^{\prime}= \frac{1}{\sqrt{n}}\hat{G}_{a b}, \hspace{0 . 3 c m}\hat{G}_{a 5}^{\prime}= \frac{1}{\sqrt{n}}\hat{G}_{a 5}.
\nu \left( \left \{\omega \in \Omega :\int_{\Omega}gd\nu \leq g\left(\omega \right) \leq \int_{\Omega}gd\bar{\nu}\right \}\right) =\nu \left(\left \{\omega \in \Omega :t^{\star}\leq g\left( \omega \right) \leq-t_{\star}\right \}\right) =1.
c_1\:H_{\Lambda_k\cup\Lambda_{k+1}}\leq\sum_{i : \{i+1,...,i+P\}\subset\Lambda_k\cup\Lambda_{k+1}}\tau^i\left(\sum_{j=1}^{P-1}h_{j,j+1}\right)\leq c_2\:H_{\Lambda_k\cup\Lambda_{k+1}}.
\frac{1}{m!}\Big|\sum_{\sigma \in{\rm Sym}(I)}\prod_{j\in I}\cos(2\pi y_{\sigma^{-1}(j)}\xi_j)\Big| \le\sum_{\substack{S\subseteq I \emptyset\not=S\not=I}}a_S(\xi)e^{-\frac{cl|S|}{m}}+\sum_{\substack{S\subseteq I \emptyset\not=S\not=I}}a_S(\xi)e^{-\frac{cl(m-|S|)}{m}}&+e^{-\sum_{i\in I}\sin^2(\pi\xi_i)}+ e^{-\sum_{i\in I}\cos^2(\pi\xi_i)},
\delta_{c_3}\circ \delta_{c_4}= p \delta_{c_0}+ r \delta_{c_1}+ s \delta_{c_2}+ x \delta_{c_3}+ x \delta_{c_4}, &\delta_{c_4}\circ \delta_{c_3}= p \delta_{c_0}+ s \delta_{c_1}+ r \delta_{c_2}+ x \delta_{c_3}+ x \delta_{c_4},
\phi'_\omega(t)= t\|\omega\|_X^2-t^{2_\alpha^*-1}\displaystyle\int_{\mathbb{R}^N}b(x)|\omega(x,0)|^{2_\alpha^*}dx &-\lambda t\left(\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2\ln|\omega(x,0)|dx+\ln t\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2dx\right)
\partial^2_t\big[u(x',t)\big] =-\sum_{j=1}^{n-1}\partial_t\Big[R^{(j)}_t\ast\big(\partial_jf\big)\Big](x') begin{align*}0pt] =\sum_{i=1}^{n-1}\sum_{j=1}^{n-1}\Big[R^{(ij)}_t\ast\big(\partial_i\partial_jf\big)\Big](x')\,\,\mbox{for every}\,\,(x',t)\in{\mathbb{R}}^n_{+},
\partial_{t}^{\alpha}u =[a^{ij}u_{x^{i}x^{j}}+b^{i}u_{x^{i}}+cu+f(u)] \qquad+\partial_{t}^{\beta}\int_{0}^{t}[\sigma^{ijk}u_{x^{i}x^{j}}+\mu^{ik}u_{x^{i}}+\nu^{k}u+g^{k}(u)]dw_{s}^{k}
d\mathbb{K}_3^{Ch}(\lambda) =\sqrt{-1}[\mathbf{T}^1\wedge\overline{\varphi^1}-\overline{\mathbf{T}^1}\wedge\varphi^1+\mathbf{T}^2\wedge\overline{\varphi^2}~~~~~-\overline{\mathbf{T}^2}\wedge\varphi^2 +\lambda^2(\Psi_2^1\wedge\overline{\varphi^3}-\overline{\Psi_2^1}\wedge\varphi^3)],
\int_{- \epsilon}^{\infty}d l \: \mathrm{e}^{- l \zeta}\int_{- \epsilon}^{\infty}d l^{\prime}\mathrm{e}^{- l^{\prime}\zeta}l l^{\prime}{\frac{l^{\prime}- l}{l + l^{\prime}}}\{3 \delta^{\prime \prime}( l ) -{\frac{3}{4}}t \delta ( l ) \}= 0 .
\lefteqn{(-1)^n\left(\sum_{j=0}^k(\lambda-1)_{k-j}\binom{k}{j}\left(x^{j}f(x)\right)^{(j)}\right)^{(n)}}&=(-1)^n\sum_{m=0}^k\binom{k}{m}\frac{\Gamma(n+k+\lambda)}{\Gamma(n+m+\lambda)}x^{m}f^{(n+m)}(x)=T^{\lambda}_{n,k}(f)(x).
\bigl\{\lambda_{2}\bigl(\psi(\alpha),\psi(\beta),1,\psi(\gamma_1)\bigr), \rho_{2}\bigl(\psi(\alpha),\psi(\beta),1,\psi(\gamma_1)\bigr)\bigr\}\quad=\{\varepsilon(XYW_1YX),\varepsilon(YXW_1XY)\}.
\deg D^{\bar{A}}_2=\int_{[\mathrm{G}(3,V)]}\sigma_1^2\cdot (\sigma_2^2-\sigma_1\sigma_3)\cdot (16\sigma_1^3-12\sigma_1\sigma_2+12\sigma_3)=\int_{[\mathrm{G}(3,V)]}36 \sigma_1^2\sigma_2^3\sigma_3=72.
\left\vert\int_\Omega 2R\xi_E\xi_{\tilde K}u^2dV_g\right\vert=\left\vert\int_\Omega \tilde R u^2dV_g\right\vert\leq \epsilon \|\nabla u\|^2_{L^2(\Omega\cap \tilde K)}+C'_{\epsilon}\|u\|^2_{L^2(\Omega\cap\tilde K)}.
m_{j}(q_{j}):=L_{nj}\left( \begin{smallmatrix}1 0 0 q_{j}\end{smallmatrix}\right)_{n}\cdots L_{2j}\left( \begin{smallmatrix}1 0 0 q_{j}\end{smallmatrix}\right)_{2}L_{1j}\left( \begin{smallmatrix}1 0 0 q_{j}\end{smallmatrix}\right)_{1}\;.
|v^+_\infty(r) \,v^+_0(\rho)\,\mathbf{1}_{(1,+\infty)}(r)\,\mathbf{1}_{(1,+\infty)}(\rho)|\; \lesssim\; e^{-r}\,e^\rho \, (\rho/r)^B\qquad\textrm{if}\,0<\rho<r |v^+_0(r)\, v^+_\infty(\rho)\,\mathbf{1}_{(1,+\infty)}(r)\,\mathbf{1}_{(1,+\infty)}(\rho)|\; \lesssim\; e^r\,e^{-\rho}\, (r/\rho)^B \qquad\textrm{if}\,0<r<\rho\,,
e_\alpha v = \sum \limits_{\mu \in \mathrm E \setminus \mathrm E_\varrho}c_\mu e_\alpha e_{-\alpha}e_{-\beta}v_\mu = \sum \limits_{\mu \in \mathrm E \setminus \mathrm E_\varrho}c_\mu \langle \alpha^\vee, \mu \rangle e_{-\beta}v_\mu.
H\tbinom{n_1,\ldots,n_r}{x_1,\ldots,x_r}:=\sum_{0<d_1<\cdots<d_r}e^{d_1x_1}\left(\frac{q^{d_1}}{1-q^{d_1}}\right)^{n_1}\cdots e^{d_rx_r}\left(\frac{q^{d_r}}{1-q^{d_r}}\right)^{n_r},
\int_{0}^{v}v^{-\frac{7}{8}}(v-w)^{-\frac{3}{8}}\exp\left(-Kv^{-\frac{3}{2}}w^2\right) dw &= \int_{0}^{v/2}v^{-\frac{7}{8}}(v-w)^{-\frac{3}{8}}\exp\left(-Kv^{-\frac{3}{2}}w^2\right) dw & + \int_{v/2}^{v}v^{-\frac{7}{8}}(v-w)^{-\frac{3}{8}}\exp\left(-Kv^{-\frac{3}{2}}w^2\right) dw &=: H_1(v)+ H_2(v).
H_0 =1, H_1 =\frac1{q^2(q-\delta)}\Big(q+1+\delta(\delta+1)-q^{1/2}(\delta+1)(A+B)+\delta AB\Big), H_2 =\frac1{q^5(q-\delta)}\Big(q(\delta+1)+\delta^2(q+1)-q^{1/2}\delta(\delta+1)(A+B)+\delta q AB\Big), H_3 =-q^{-7}\delta.