harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
A^{2,2}= \Big[b_l(2+2h, 2)\Big]_{l,h = 0}^{m-2}(m-1 \times m-1) A^{2,1}= \Big[b_{l+m}(1+2h,1)\Big]_{l,h=0}^{m-2,m-1}(m-1 \times m) A^{2,0}= \Big[b_{l+2m}(2h,0)\Big]_{l,h=0}^{m-2,m}\ (m-1 \times m+1).
{\cal T}= \frac{G S}{4 \, \theta}\, | 0 \rangle \langle 0 | \, d \bar{z}\wedge d z \left( \begin{array}{r r}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right) \, , \qquad{\cal R}= 0 \, .
\omega_{\xi,\eta}((\iota \otimes m)\Delta(x)) = \lim_i \langle W^*(1 \otimes x)W(\xi \otimes \xi_i)|(\eta \otimes \xi_i) \rangle &= \lim_i \langle (1 \otimes x)(\xi \otimes \xi_i)|(\eta \otimes \xi_i) \rangle &= \omega_{\xi,\eta}(1) \lim_{i}\omega_{\xi_i}(x) = \omega_{\xi,\eta}(1) m(x).
\begin{aligned}\eta+ \overline H(f) \geq\mathrm{ess}\,\sup_{\P_0}\, \log\bigg(\sum_{e\in\mathcal U_d}\pi_\omega(0,e)\exp\big\{f(\omega,e)+ G(\omega,e)\big\}\bigg) &\geq\inf_{G\in\mathcal G_\infty}\,\mathrm{ess}\,\sup_{\P_0}\, \log\bigg(\sum_{e\in\mathcal U_d}\pi_\omega(0,e)\exp\big\{f(\omega,e)+ G(\omega,e)\big\}\bigg),\end{aligned}
\Gamma_{s}( P^{a}) ={\textstyle \frac{i R_{5}}{3 2}}\sigma^{2}\Gamma^{0 1 2 3 4}\Gamma^{a}, \Rightarrow \Gamma^{a}={\textstyle \frac{3 2 i}{R_{5}}}\mathcal{S}\Gamma_{s}( P^{a}) , \mathcal{S}= \sigma^{2}\Gamma^{0 1 2 3 4},
U_{1}( \phi_{0}, a_{1}, . . . , a_{d}) = \mu^{D}\sqrt{a_{1}\cdots a_{d}}\sum_{s = 1}^{\infty}\frac{( - 1 )^{s}}{2 s}g^{s}\phi_{0}^{2 s}\sum_{n_{1}, . . . , n_{d}= - \infty}^{+ \infty}\int \frac{d^{D - d}q}{( a_{1}n_{1}^{2}+ \cdots + a_{d}n_{d}^{2}+ c^{2}+{\bf q}^{2})^{s}}.
\Pi_{\delta}(x) = m_*^2(x) -{g^2 \over 2}\int{d^4 x' d^4 p \over (2\pi )^4}\;{d^3 k \over (2\pi )^3 E_k}\;{e^{-ip(x'-x)}\over p^2 - m_*^2 + ip_00^+}\; f_0(x',{\bf k}) \;,
{\cal G}_{a}^{(\pm)}\equiv\frac{L}{2}\eta_{ab}\epsilon^{ij}(\partial_{i}A^{(\pm)b}_{j}-\partial_{j}A^{(\pm)b}_{i}+\epsilon^{b}_{\mbox{}cd}A^{(\pm)c}_{i}A^{(\pm)d}_{j})\approx 0,
\left. - 2 \sqrt{2}\left[ 2 F_{\alpha}^{\mu \, A}C^{[ \alpha}\partial^{\nu ]}D - \frac{g^{\mu \nu}}{2}F_{\alpha_{1}\alpha_{2}}^{\, A}C^{[ \alpha_{1}}\partial^{\alpha_{2}]}D \right] + 2 \left[ 2 C_{[ \alpha}\partial^{\mu ]}D C^{[ \alpha}\partial^{\nu ]}D \right. \right.
W(e_1\odot\dots\odot e_n) = \sum_{k=0}^{n}\sum_{i_1,\dots,i_{n-k},j_{n-k+1},\dots,j_n}a_q^{\ast}(e_{i_1})\dots a_q^{\ast}(e_{i_{n-k}})a_q(I e_{j_{n-k+1}})\dots a_q(I e_{j_n}) q^{i(I_1,I_2)},
\forall \tau_p\in\left[-\eta,\varepsilon\right], \forall t\in[-\ell_{p},\ell_{p}],\Im(\gamma_{p}(t))=t, \Re(\gamma_{p}(t))=h_{p}(t):=\Psi^{-1}\left(\Psi(\tau_p)-A_It^{2\mu}\right).
\begin{aligned}\| v_i\|_{C^{(1+\alpha)/2,1+\alpha}}\leq \frac{1}{2}\|\nabla v\|_{C^0(\overline{D_T})}&+ C\big(r^{-1-2\alpha}+r^{-2}+r^{-\frac{p}{2}(1+\alpha)}\big)\big(\|v_{0}\|_{C^{1+\alpha}}+\|f\|_{L^p(D_T)}+\|g \|_{C^{\alpha/2,\alpha}}\big).\end{aligned}
|\Theta_\mathbf{F}[\varphi](p)| =|\Omega_\mathbf{F}[\varphi^+](p)- \Omega_\mathbf{F}[\varphi^-](p)| \leq \sup_{a\in A}|\varphi^+(a)-\varphi^-(a)|\cdot\mathbf{F}^*(a,p) =\sup_{a\in A}|\varphi(a)|\cdot \mathbf{F}^*(a,p)\leq \sup_{a\in A}|\varphi(a)|=\|\varphi\|.
\mathcal{A}_{t h}( 0 , \infty ) = \mathcal{N}= \mathcal{A}_{0}( 1 , \infty ) , \, \, \, \, \, \, \, \, \, \, \mathcal{A}_{t h}( - \infty , 0 ) = \mathcal{N}^{\prime}\cap \mathcal{M = A}_{0}( 0 , 1 )
\varphi_\theta(x)=\frac{1}{|F_\theta|}\left(\int_{F_\theta}T_g\varphi(x)dg-\int_{C_\theta}T_g\varphi(x)dg\right)+\left(\frac{1}{|F_\theta|}-\frac{1}{|C_\theta|}\right)\int_{C_\theta}T_g\varphi(x)dg,
\alpha_{i,s}= \frac{1}{{w\left({\left|{{\mu_i}}\right|}\right)}}\sum\limits_{j = 0}^m{\left({{{\left({\frac{{{{\bar \mu}_i}}}{{\left|{{\mu_i}}\right|}}}\right)}^j}\mu_s^j{w_j}}\right)}\;\;\; \mbox{and}\;\;\; \beta_s = \frac{1}{k}\sum\limits_{i = 1}^k{{\alpha_{i,s}}}, \;\;\; i,s = 1, 2, \dots , k,
\left( A_p -B_j\right)^{-1}= A_p^{-1}+ A_p^{-1}B_j^{\frac{1}{2}}\left( Id - B_j^{\frac{1}{2}}A_p^{-1}B_j^{\frac{1}{2}}\right)^{-1}B_j^{\frac{1}{2}}A_p^{-1}&\preceq A_p^{-1}+ \frac{A_p^{-1}B_jA_p^{-1}}{1-{\rm Tr}\left(A_p^{-1}B_j\right)}.
\begin{aligned}\mathcal{g}_{D}(v,w) := \mathcal{W}\bigl( \hat v \hat w D^{-n}\bigr), \mathcal{G}_{D}(v,w) := \mathcal{W}\bigl( \hat v (D\hat w+\hat wD )D^{-n+1}\bigr) &\,\,=\mathcal{W}\bigl( (D\hat v+\hat vD ) \hat wD^{-n+1}\bigr),\end{aligned}
{\cal L}_{p}={\cal L}_{p}^{r e g}+ 4 p \epsilon_{\phi \rho i_{1}. . . i_{2 p - 2}}\epsilon^{\phi \rho j_{1}. . . j_{2 p - 2}}R_{( c o n ) \ \phi \rho}^{\phi \rho}R_{\ \j_{1}j_{2}}^{i_{1}i_{2}}. . . R_{\ \j_{2 p - 3}j_{2 p - 2}}^{i_{2 p - 3}i_{2 p - 2}}
[y_{(1\,l)},y_{2}]_c =[[y_{(1\,3)},y_{(4\,l)}]_c,y_2]_c &=\chi_2(g_{(4\,l)})[y_{(1\,3)},y_2]_cy_{(4\,l)}-\chi_{(4\,l)}(g_{(1\,3)})y_{(4\,l)}[y_{(1\,3)},y_2]_c &=\lambda_{122}q_{32}(1-\xi^2)\chi_2(g_{(4\,l)})\Big(y_3y_{(4\,l)}-\chi_{(4\,l)}(g_3)\chi_{(4\,l)}(g_{122})y_{(4\,l)}y_3\Big) &=\lambda_{122}(1-\xi^2)\chi_2(g_{(3\,l)})y_{(3\,l)},
Ht_+(\tau)=\tau(HE+\frac{\mu\pi}{2K}\frac{\vartheta'_4}{\vartheta_4}(\frac{\pi x}{2K}))+\frac{1}{2}\log\mid\frac{\vartheta_4(\frac{\pi(\mu\tau-x)}{2K})}{\vartheta_4(\frac{\pi(\mu\tau+x)}{2K})}\mid,
J=2\cdot \int\limits_0^\infty \int\limits_0^\infty f(\frac{u_1v_1}{xy},\frac{u_2v_2}{xy})\cdot f(\frac{u_1y}{v_1x},\frac{u_2y}{v_2x})\cdot g(xy)\cdot g(\frac{x}{y})\ \, \frac{dx}{x^3}\frac{dy}{y}
\mathcal{S}= \bigcup_{K \in \{1, 2, \ldots\}}\bigcup_{J \in \mathcal{P}(\{1,\ldots,p\})}\bigcup_{\mathbf{R}\in \{1,\ldots,|J|\wedge q\}^K}{S}_{(K,J,R)}= \bigcup_{(K,J, \mathbf{R}) \in{\mathcal{M}}}{S}_{(K,J,R)}.
\frac{N^{1+i\eta}\widetilde{w}(1+i\eta)}{\zeta(2(1+i\eta))}\zeta(1+2i\eta) =\frac{\widetilde{w}(1) N \log N}{2 \zeta(2)}\Big( \frac{1}{i \eta \log N}+ 1 + O(|\eta| \log N) \Big),
- ( \partial_{x}^{2}+ \partial_{y}^{2}) a + \sqrt{2}\, g^{4}\bar{\mu}_{D}\, \left( \tilde{q}q + \frac i 2 \, \eta + \frac{\mu_{D}}{\sqrt{2}}a \right) + 2 g^{2}a \left( \bar{q}q + \bar{\tilde{q}}\tilde{q}\right) = 0
\frac{\partial}{\partial\overline{x}^{l}}det(\frac{\partial x}{\partial\overline{x}})=det(\frac{\partial x}{\partial\overline{x}})\frac{\partial\overline{x}^{k}}{\partial x^{i}}\frac{\partial^{2}x^{i}}{\partial\overline{x}^{k}\partial\overline{x}^{l}}.
|v_{4,c}(t,r)| \leq C |\chi_{\geq 1}(\frac{2r}{\log^{N}(t)})| \begin{cases}\frac{1}{r^{3}t^{2}\log^{3b}(t)}, r \leq \frac{t}{2}\frac{\log(r)}{\log^{2b}(t) r^{4}|t-r|}+\frac{\log^{2b\alpha}(t)}{t^{2}r^{3}\log^{3b+1}(t)}, \frac{t}{2}\leq r \leq t-\sqrt{t}, r > t+\sqrt{t}\frac{1}{\log^{2b}(t)r^{9/2}}, t-\sqrt{t}\leq r \leq t+\sqrt{t}\end{cases}
\left( \begin{array}{c}{{a_{D}}}\\{{a}}\\ \end{array}\right) \rightarrow \left( \begin{array}{c c}{{0}}&{{- 1}}\\{{1}}&{{0}}\\ \end{array}\right) \left( \begin{array}{c}{{a_{D}}}\\{{a}}\\ \end{array}\right) = \left( \begin{array}{c}{{- a}}\\{{a_{D}}}\\ \end{array}\right) ,
Z_{\mbox{\scriptsize 1-loop}}(R,T;t) \approx t^{-{1\over 2}}\eta(t)^{-{1\over 2}}\exp \left[ -{1 \over \epsilon^2}\left({T^2 \over 2R^2 t}+{t\over 2}+ \epsilon f_1^{\mbox{\scriptsize 1-loop}}+\epsilon^2 f_2^{\mbox{\scriptsize 1-loop}}\right) \right]\;,
\int_{\rho_-}^{\rho_+}\frac{s(\xi)^{\frac1n}}{\sqrt{1-s(\xi)^{\frac2n}}}\,d\xi = \frac{\rho_+-\rho_-}{2}\int_{-1}^1 \frac{u^{\frac1n}}{\sqrt{1-u^{\frac2n}}}\,du = 0.
\lim\limits_{N \to \infty}\left\{N \sum\limits_{i = 0}^{N}{\left[ \binom{N}{i}{{p}^i}{{\left({1 - p}\right)}^{N - i}}{\ln}\left( \frac{i + 1}{pN}\right) \right]}\right\}= \frac{1}{2p}+ \frac{1}{2}.
g_{\varepsilon,ij}=(n+\varepsilon)\frac{f_{ij}}{f}-(n+\varepsilon)\frac{f_{i}f_j}{f^{2}}+\frac{f_{ki}f_{kj}}{1+|\nabla f|^{2}}+\frac{f_{kij}f_{k}}{1+|\nabla f|^{2}}-\frac{2f_kf_lf_{ki}f_{lj}}{(1+|\nabla f|^{2})^{2}}.
\mu(e^{\frac{1}{\rho(M_{\sqrt{\chi}}GM_{\sqrt{\chi}})}\langle l,\chi\rangle}-1)= \lim\limits_{y\uparrow\lambda_1}\mu(e^{y\langle \chi,l\rangle}-1)=\lim\limits_{y\uparrow\lambda_1}-\ln\det(I-yM_{\chi}VM_{\chi}) = \lim\limits_{y\uparrow\lambda_1}|-\ln\det(I-yM_{\chi}VM_{\chi})|=\infty.
\sum\limits_{i=1}^{j-1}\Big|\Big\langle{\frac{T^\ast(\tilde h_i)}{{\|h_i\|}_{F^\ast}}},{\frac{\tilde h_j}{\|h_j\|_F}}\Big\rangle\Big| = \sum\limits_{i=1}^{j-1}\Big|\Big\langle{\frac{\tilde h_i}{\|h_i\|_{F^\ast}}},{\frac{T(\tilde h_j)}{\|h_j\|_F}}\Big\rangle\Big| <{\frac{\eta_j}{2}},
\left(\frac{|1+iu|}{1+iu}\right)^k(1+u^2)^{-j-l}=(1-iu)^k(1+u^2)^{-j-l-\frac{k}2}=\sum_{m=0}^\infty b_{j,k,l,m}(-iu)^m &=\sum_{m=0}^{m_0-1}b_{j,k,l,m}(-iu)^m+O\!\left(\sum_{m=m_0}^\infty 2^{j+l+\frac{m}2}|u|^m\right) &=\sum_{m=0}^{m_0-1}b_{j,k,l,m}(-iu)^m+O_{j,l,m_0}(|u|^{m_0}).
\begin{pmatrix}5 0 0 0 0 0 \cdots 2\times5 5^{3}0 0 0 0 \cdots 9 3\times5^{3}5^{5}0 0 0 \cdots 4 22\times5^{2}4\times5^{5}5^{7}0 0 \cdots 1 4\times5^{3}8\times5^{5}5^{8}5^{9}0 \cdots \vdots \vdots \vdots \vdots \vdots \vdots \ddots\end{pmatrix}
\epsilon_{1}( \tau , \theta ) = e^{i \tau / 2}\left[ \begin{array}{c}{{\sin ( \theta / 2 + \pi / 4 )}}\\{{- \cos ( \theta / 2 + \pi / 4 )}}\\ \end{array}\right] , \epsilon_{2}( \tau , \theta ) = e^{- i \tau / 2}\left[ \begin{array}{c}{{\cos ( \theta / 2 + \pi / 4 )}}\\{{\sin ( \theta / 2 + \pi / 4 )}}\\ \end{array}\right] .
{\textstyle \frac{1}{4}}\nabla_{a}k_{( I )}^{b}\Gamma^{a}{}_{b}={\textstyle \frac{1}{4}}f_{i a}{}^{b}\Gamma_{\mathrm{A d j}}( u^{- 1})^{i}{}_{I}\gamma^{a}{}_{b}= - \Gamma_{\mathrm{A d j}}( u^{- 1})^{i}{}_{I}\Gamma_{s}( M_{i}) \, ,
H|\psi\rangle=\frac{1}{\sqrt{2}}[\begin{matrix}1&1\\ 1&-1\end{matrix}][\begin{matrix}\frac{1}{\sqrt{3}}\\ \frac{\sqrt{2}}{\sqrt{3}}\end{matrix}]=\frac{1}{\sqrt{2}}[\begin{matrix}\frac{1+\sqrt{2}}{\sqrt{3}}\\ \frac{1-\sqrt{2}}{\sqrt{3}}\end{matrix}]=\frac{1+\sqrt{2}}{\sqrt{6}}|0\rangle+\frac{1-\sqrt{2}}{\sqrt{6}}|1\rangle
G_3(k) \lesssim \frac{1}{k^{2r}}\sum_{h=\lfloor k/2\rfloor +1}^{k-1}\frac{1}{(1+k-h)^{2s}}\leq \frac{1}{k^{2r}}\sum_{h=1}^{+\infty}\frac{1}{h^{2s}}= O(\frac{1}{k^{2r}}) \k \to +\infty.
\begin{aligned}{\rm{M}}\approx{\rho_1}\exp \left({-{\lambda_2}\gamma}\right){\left({\frac{1}{{{{\bar \gamma}_0}}}\gamma +{\beta_{j,k}}}\right)^{-{\rho_2}}}\frac{1}{{\left({{\rho_1}+{\rho_2}}\right)\gamma +{\rho_3}}}\int\limits_0^\gamma{\exp \left({-{\Phi_2}z}\right)}dz\end{aligned}
\frac{G_{\mu \nu}}{\alpha^{\prime}}= \frac{1}{\alpha_{\mathrm{e f f}}^{\prime}}H^{- 1 / 2}\eta_{\mu \nu}\ , \quad G_{O S}^{2}= g H^{\frac{3 - p}{4}}\ , \quad \Theta^{0 1}= \alpha_{\mathrm{e f f}}^{\prime}\ ,
\lim_{N\rightarrow\infty}\frac{(1+2^{n_{2}-n_{1}}+\cdots+2^{n_{\tau(N)}-n_{1}})\log(1+2^{n_{2}-n_{1}}+\cdots+2^{n_{\tau(N)}-n_{1}})}{n_{1}+n_{2}2^{n_{2}-n_{1}}+\cdots+n_{\tau(N)}2^{n_{\tau(N)}-n_{1}}}=0,
P_{n+1|0}= \begin{bmatrix}P_{n|0}0 \frac{1}{2}P_{n|1}\frac{1}{2}P_{n|0}\end{bmatrix}P_{n+1|1}= \begin{bmatrix}\frac{1}{2}P_{n|1}\frac{1}{2}P_{n|0}0 P_{n|1}\end{bmatrix},
(\mathbb{A}^{-1}_R)_{11}= (d+(db^{-1}a-c)^{-1}[c,d])(\Delta')^{-1}&=(db^{-1}a-c)^{-1}((db^{-1}a-c)d+[c,d])(\Delta')^{-1}&=(db^{-1}a-c)^{-1}d\underbrace{(b^{-1}ad-c)}_{=b^{-1}\Delta'}(\Delta')^{-1}=(db^{-1}a-c)^{-1}db^{-1}=(a-bd^{-1}c)^{-1}.
f^2=1-\frac{1}{\kappa^2u^2_b}g\ \ ,\ \{\cal E}\simeq -\frac{\pi\Delta u}{u_b}(\kappa^2u^2_b)\left[1-\frac{1}{\kappa^2u^2_b}\left(L-\frac{1}{2}u^2_bb_{\rm ext}\right)^2\right]^2.
\varpi^{inst}(\zeta) = \frac{1}{2\pi R}\cdot \xi^{inst}_m \wedge \xi_e & = \frac{1}{2\pi R}\left[ 2\pi i A^{inst}+ \pi i V^{inst}\left( \frac{1}{\zeta}dc - \zeta d\overline{c}\right) \right] \wedge \xi_e
\sigma_\varepsilon: = \Big(\frac{\varepsilon^d}{a_\varepsilon^{d-2}}\Big)^{\frac{1}{2}}, \d \geqslant 3;\quad{\sigma_\varepsilon}: = \varepsilon \left| \log \frac{{a_\varepsilon}}{\varepsilon}\right|^{\frac{1}{2}}, \d = 2,
- \nabla^{\mu}\nabla_{\mu}V_{\nu}= \left[ - \nabla^{K}\nabla_{K}+ \left({\frac{\partial}{\partial r}}+{\frac{d - 1}{r}}\right) \left({\frac{\partial}{\partial r}}-{\frac{1}{r}}\right) -{\frac{1}{r^{2}}}\right] V_{\nu}
| \mathbf{v}_{1}|^{2}\equiv \rho^{2}\, \cos \phi \, , \qquad | \mathbf{v}_{2}|^{2}\equiv \rho^{2}\, \sin \phi \, , \qquad \vec{\mathbf{v}}_{1}\, \cdot \, \vec{\mathbf{v}}_{2}\equiv \rho^{2}\sin \phi \cos \phi \, \cos \theta \, .
\frac{d^{2}N}{d\eta dt}=\frac{\sqrt{3}}{2\pi}\frac{q^{2}mc}{\hbar^{2}}\frac{\chi}{\gamma\eta}F(\chi,\eta)whereF(\chi,\eta)=\frac{\eta^{2}}{\chi^{2}}yK_{\frac{2}{3}}(y)+(1-\frac{\eta}{\chi})y\int_{y}^{\infty}K_{\frac{5}{3}}(x)dx
\sum_{i=2}^{|V|}\sum\limits_{j=1}^{i-1}j= \sum_{i=2}^{|V|}\left[ \frac{1}{2}i\cdot(i-1) \right] = \frac{1}{2}\left[\sum_{i=2}^{|V|}i^2 - \sum_{i=2}^{|V|}i\right] = \frac{|V|^3-|V|}{6}\leq \frac{|V|^3}{6}.
\pi(w^l)\mu_b(w_{l+1}^{\ell_2}|w^l,u^{\ell_1}) = \pi(w^{\ell_2}){\mu_b(w_{l+1}^{\ell_2}|w^l,u^{\ell_1})\over \pi(w_{l+1}^{\ell_2}|w^l)}.
e^{i W}= \int d g_{\mu \nu}~ d \bar{\Gamma}_{~ \mu \nu}^{\sigma}e^{i \left( S_{c l a s}+ S_{g f}\right)}\left( d e t \omega_{\mu \nu}\right)^{\frac{1}{2}}\left( d e t \omega_{\mu \alpha}\triangle_{~ \nu}^{\alpha}\right) \left( d e t \zeta_{\mu \nu}\right)^{\frac{3}{2}}\left( d e t \varsigma_{\mu \nu}\right)^{\frac{3}{2}}
\frac{1}{Z_n}\det \left[x_j^{\frac{\kappa+i-1}{2}}I_{\kappa+i-1}(2\alpha\sqrt{x_j})\right]_{i,j=1}^n\det\left[x_j^{\frac{\nu-\kappa+i-1}{2}}K_{\nu-\kappa+i-1}(2\beta\sqrt{x_j})\right]_{i,j=1}^n,
\mathbf{S}_L (t) (u_0,u_1) \doteq (u(\cdot, t), \partial_t u (\cdot, t)), \mathbf{S}_L (t) \begin{pmatrix}u_0 u_1 \end{pmatrix}\doteq \begin{pmatrix}u(\cdot, t) \partial_t u(\cdot, t) \end{pmatrix}.
\delta_{\alpha{\mathbf i}}{\cal R}^\mu = 16i \left\{\frac{1}{192}\left[ \frac{3}{2}J^\mu_{\alpha{\mathbf i}}+ \sigma_\alpha^{\mu\nu\beta}J_{\nu\beta{\mathbf i}}\right] - \frac{3}{4}(8+a) \partial^\mu D_{\alpha{\mathbf i}}T| + \frac{1}{2}(a-24) \sigma_\alpha^{\mu\nu\beta}\partial_\nu D_{\beta{\mathbf i}}T| \right\}.
f_n(z) = \frac{z(1+\kappa(q(z))^{2^{n}})}{1 - \kappa(q(z))^{2^{n}}}- z= \frac{2z \, \kappa(q(z))^{2^{n}}}{1 - \kappa(q(z))^{2^{n}}},{\rm Re}\, z > 0,
\min ~\sum_{i\in\mathcal{N}}\sigma_n^2\hat{r}_0(1+\hat{r}_0)^{i-1}/(P\zeta_i), & \sum_{k=0}^{i-1}\gamma_{i,k}\bigg(1-\exp(-\delta_{i,k}\zeta_i)\bigg)\le t, ~\zeta_i\ge 0, ~i\in\mathcal{N}.
\sum_{\overline{\beta}'}\sum_{J}\dfrac{\prod_{a=1}^r\mathfrak{r}_{n+k+a}}{r!}\int_{Q_{\overline{\beta}'}}(-1)^{s-r}\prod_{a=1}^r(\tilde{\psi}_{n+k+a}+\tilde{\psi}_{a}')^{j_a},
\pi_2^{*}H^2(\tilde Z, \mathbb Z)=\Omega_2^{\perp_{H^2(X, \mathbb Z)}}=\Omega_2^{\perp_{\Lambda_{\mathrm{K3}}}}, \widehat{\pi_2}^{*}H^2(\tilde Y, \mathbb Z)=\Omega_2^{\perp_{H^2(\tilde Z, \mathbb Z)}}=\Omega_2^{\perp_{\Lambda_{\mathrm{K3}}}}.
\bigg\langle \sum_{j}(H_{\infty}W(n_{j}))\prod_{i\not=j}W(n_{i}) +{1\over N^{2}}\sum_{j}\sum_{i\not=j}n_{i}n_{j}W(n_{i}+n_{j})\prod_{k\not=i,j}W(n_{k}) \bigg\rangle= 0
\begin{aligned}\tilde e=\tilde e[\Lambda,\phi,\alpha]:= ~{}\kappa \partial_t\alpha \left[ (\partial_t\Lambda)^2+(\partial_x\Lambda)^2+4\sinh^2(\Lambda)\Big((\partial_t \phi)^2+(\partial_x\phi)^2\Big) \right] &~{}-2\kappa \partial_x\alpha\left(\partial_x\Lambda \partial_t\Lambda +4\partial_x \phi \partial_t \phi \sinh^2(\Lambda) \right),\end{aligned}
X^\pi(z) = V_\pi(z) e^{iq}z^{\alpha_0}:= \sum_{n\in{\mathbb Z}}z^{n+{\alpha_0}}X^\pi_{-n}\quad\mbox{and}X^{*\pi}(z) = V_\pi^*(z) z^{-\alpha_0}e^{-iq}:= \sum_{n\in{\mathbb Z}}z^{-n-{\alpha_0}}X^{*\pi}_{{n}}\,.
\langle \operatorname{c o s}^{2}\theta \rangle ={\frac{\int_{0}^{\pi}\operatorname{c o s}^{2}\theta \operatorname{s i n}^{d - 1}\theta \, d \theta}{\int_{0}^{\pi}\operatorname{s i n}^{d - 1}\theta \, d \theta}}={\frac{1}{d + 1}},
\lim_{\xi\rightarrow\overline{c_{0}}^{+}}\frac{p_{\mu_{1}\boxtimes\mu_{2}}(\xi)}{\left|\xi-\overline{c_{0}}\right|^{1/3}}=\frac{-2\left|a_{1}\right|}{\sqrt[3]{\left|c_{3}\right|}}\cos\left(\frac{\theta}{3}-\frac{5\pi}{6}\right),
P\{M(s) \le\beta\ |\V(0) = c,\N(t) = 2k\}= \sum_{j=0}^{2k}P\{M(s) \le\beta,\N(s) = j\ |\V(0)=c,\N(t) = 2k\}& =\sum_{j=0}^{2k}P_j^+\{M(s) \le\beta \}\,\frac{P\{N(s) = j\}\,P\{N(t-s)=2k-j\}}{P\{N(t) = 2k\}}&=\frac{1}{t^{2k}}\sum_{j=0}^{2k}P_j^+\{M(s) \le\beta \}\binom{2k}{j}s^j(t-s)^{2k-j},
\widehat{R}_{\mu \nu}^{a b}= R_{\mu \nu}^{a b}+ i \theta^{\rho \tau}R_{\mu \nu \rho \tau}^{a b}+ \theta^{\rho \tau}\theta^{\kappa \sigma}R_{\mu \nu \rho \tau \kappa \sigma}^{a b}+ O ( \theta^{3})
R^a_{\mu\nu}=\partial_\mu A^a_\nu - \partial_\nu A_\mu^a+g\epsilon^{abc}A^b_\mu A^c_\nu\ , \hskip 0.5cm\bar R^a_{\mu\nu}=\partial_\mu\bar A_\nu- \partial_\nu\bar A_\mu+g\epsilon^{abc}\bar A^b_\mu\bar A^c_\nu \ .
\| u^{*}(\varphi(f\otimes a) \oplus \overbrace{\bigoplus_{k=1}^m f(z_k)\sigma (a) \oplus \cdots \oplus \bigoplus_{k=1}^m f(z_k)\sigma (a)}^m) u & - \psi(f\otimes a)\oplus \overbrace{\bigoplus_{k=1}^m f(z_k)\sigma (a) \oplus \cdots \oplus\bigoplus_{k=1}^m f(z_k)\sigma (a)}^m\| < \varepsilon
\frac{1}{N^2}\int_{0}^{1/2}\sum_{k,m \in \mathbb{Z}\atop k \neq 0 \neq m}\widehat \chi_{\left[-s,s\right]}(k) \widehat \chi_{\left[-s,s\right]}(m) \left| \sum_{n=1}^{N}{e^{2 \pi i k x_n}}\right|^2 \left| \sum_{n=1}^{N}{e^{2 \pi i m x_n}}\right|^2 ds,
\begin{aligned}\Phi_3(z,s,a) =\sum_{m_1,m_2,m_3=0}^\infty\frac{z^{m_1+m_2+m_3}}{(m_1+m_2+m_3+a)^{s}}&=\sum_{m_1=1}^\infty\sum_{m_2,m_3=0}^\infty\frac{z^{m_1+m_2+m_3}}{(m_1+m_2+m_3+a)^{s}}+\Phi_2(z,s,a).\end{aligned}
+{\frac{1 2}{y^{2}}}\sqrt{{\frac{1}{4}}- x y}\left( \int_{\frac 1 2}^{\frac 1 2 + \sqrt{\sigma}}- \int_{\frac 1 2}^{\frac 1 2 - \sqrt{\sigma}}\right) u ( u - 1 ) \psi ( u ) d u .
\forall V \in H^{-1}_{\rm per, r}, \mathcal{J}_{b}(V):= \frac{1}{2}\int_{\Gamma^*}|b(q) - \varepsilon_{q}^V|^2\,dq = \int_0^{1/2}|b(q) - \varepsilon_{q}^V|^2\,dq.
0 ~ = ~ \frac{( m - 1 ) ( m + 2 )}{m}y q ( \beta , \omega ) ~ - ~ q ( \gamma , \omega ) \left[ 2 z ~ - ~ ( 1 + q ( \alpha , \omega ) ) q ( \beta , \omega ) \left[ z ~ + ~ \frac{( m - 1 ) ( m + 2 )}{2 m}y \right] \right] \, .
{\cal B}_d(N) \le \frac{4 L V(x_0, x^*)}{\nu N (N+1) (1 - P_{T_1})}+ \frac{4 M^2}{L N (N+1)}\sum_{k=1}^N \frac{k^2}{T_k+3}&\le \frac{6 L V(x_0, x^*)}{\nu N (N+1)}+ \frac{4 M^2}{L N (N+1)}\sum_{k=1}^N \frac{k^2}{T_k+3},
\det\begin{pmatrix}a t^{6k + 3}c^p t^{6k + 3 + (5k + 2)p}b^{p^2}b t^{3k + 1}a^p t^{3k + 1 + (6k + 3)p}c^{p^2}c t^{5k + 2}b^p t^{5k + 2 + (3k + 1)p}a^{p^2}\end{pmatrix}= 0
A = \frac{V_{0}^{2}+ g^{2}X_{0}^{2}Y_{0}^{2}}{Y_{0}}, \qquad \varphi_{0}= \operatorname{a r c c o s}\sqrt{\frac{X_{0}^{2}Y_{0}^{2}}{V_{0}^{2}+ g^{2}X_{0}^{2}Y_{0}^{2}}}
\mathcal{J}_{5}(y;\mu) = \Bigl|\frac{y_1}{y_2}\Bigr|^{\frac{1}{2}\mu_2}\int_0^\infty \tilde{K}_{3\nu_3}\left(2\pi |y_1|^{1/2}\sqrt{1+u^2}\right) \tilde{K}_{3\nu_3}\left(2\pi |y_2|^{1/2}\sqrt{1+u^{-2}}\right) u^{3\mu_2}\frac{du}{u}.
\Phi_{\mathbf{m}}^{(d)}(\mathbf{z}) = \sum_{\mathbf{n}\subset \mathbf{m}}\frac{1}{|\mathbf{m}|-|\mathbf{n}|+1}\binom{\mathbf{m}}{\mathbf{n}}^{(d)}B_{\mathbf{n}}^{(d)}(\mathbf{z}).
\Big [ \begin{array}{c}{v}\\{\emptyset}\\ \end{array}\Big ] , \qquad \Big [ \begin{array}{c}{\emptyset}\\{v}\\ \end{array}\Big ] , \qquad \Big [ \begin{array}{c}{v}\\{v + F}\\ \end{array}\Big ] \sim \Big [ \begin{array}{c}{v}\\{v}\\ \end{array}\Big ] ,
\mathrm M_{t,n}^{(j)}\, :=\, \frac{1}{\, n \,}\sum_{i=1}^{n}\delta_{(X_{t,i}^{(u)}, \, \ldots , \,X_{t,i+j-1}^{(u)})}\, ; j \, =\, 2, \ldots , k \,
\|W_tf\|_{\ell^2(\mathbb{N})}=\|c_n(e^{(\cdot-s^{+})t}F)\|_{\ell^2(\mathbb{N})}=\|e^{(\cdot-s^{+})t}F\|_{L^2(X,d\mu)}&\le \|F\|_{L^2(X,d\mu)}=\|f\|_{\ell^2(\mathbb{N})}.
I_\lambda(\tilde w_1, \tilde w_2)+ o(1) = I_\lambda(w_{k_1},w_{k_2}) &= \frac{1}{2}\|w_k\|^2-\frac{\lambda}{p+q}\int_{-1}^1f(x)|w_{k_1}|^p|w_{k_2}|^q dx- \int_{-1}^1 G (w_{k_1}, w_{k_2}) dx .
(\phi \psi)(X) = (\phi \otimes \psi) \Delta(X), 1(X) = \varepsilon(X), \Delta(\phi)(X \otimes Y) = \phi(X Y), \varepsilon(\phi) = \phi(1), S(\phi)(X) = \phi(S(X)), \phi^*(X) = \overline{\phi(S(X)^*)}.
f=2\zeta(3)\tau_2^{3/2}+{2\pi^{2}\over 3}\tau_2^{-1/2}+2\tau_2^{3/2}\sum_{m,n\neq 0}\int_0^{\infty}dy \exp\left(-{\pi^2 m^2\over y}+2\pi i m n\tau_1 - yn^2 \tau_2^2\right),
+\left(\frac{1}{2}\bar{C}^\dagger_\gamma U^{\dagger\gamma}_{\alpha\beta}C^\alpha C^\beta+\frac{1}{2}C^{\dagger\beta}C^{\dagger\alpha}U^\gamma_{\alpha\beta}\bar{C}_\gamma+C^{\dagger\alpha}\bar{U}^\gamma_{\alpha\beta}\bar{C}_\gamma C^\beta+C^{\dagger\beta}\bar{C}^\dagger_\gamma\bar{U}^{\dagger\gamma}_{\alpha\beta}C^\alpha\right)(-1)^{\varepsilon_\beta}.
\hat{\theta}_{BAYES}=\frac{n+1}{n}[1-\frac{\frac{(\frac{X_{(1)}}{1-k})}{(\frac{X_{(n)}}{1+k})}-1}{{[\frac{(\frac{X_{(1)}}{1-k})}{(\frac{X_{(n)}}{1+k})}]}^{n+1}-1}]\frac{X_{(n)}}{1+k}
E_q(g) = E^1_q(g) + \delta_2 E^2_q(g) + \delta_3 E^3_q(g), &E^1_q(g)=\sum_{i}(B_i (\Delta_q g_2), \partial_i \Delta_q c), &E^2_q(g)=\sum_{i,j}(A_{ij}(\Delta_q g_2 )+2\Delta_q c\delta_{ij}, \partial_j \Delta_qb_i + \partial_i \Delta_qb_j), &E^3_q(g)=(\Delta_q b_,\nabla_x \Delta_q a).
Z(a_{1},b_{1},a_{2},b_{2},a_{3})+Z(a_{2},b_{1},a_{3},b_{2},a_{1})+Z(a_{3},b_{1},a_{1},b_{2},a_{2}) = Z(a_{1},b_{2},a_{2},b_{1},a_{3})+Z(a_{2},b_{2},a_{3},b_{1},a_{1})+Z(a_{3},b_{2},a_{1},b_{1},a_{2}).
\lvert\Xi^{*}\rvert (\epsilon c_{\min})^{d}=\sum_{x\in\Xi^{*}}(\epsilon c_{\min})^{d}\leq \sum_{x\in\Xi^{*}}\lvert S_{x}(\Delta)\rvert^{d}\leq \lvert B(z,2\epsilon)\rvert^{d}\leq (4\epsilon)^{d};
\int \Pi_{\mu \nu \lambda \sigma}A_{\mu}A_{\nu}A_{\lambda}A_{\sigma}= \frac{1}{2 8 8 0 m^{4}}\int d^{4}x \, [ 1 4 F_{\mu \nu}F_{\nu \alpha}F_{\alpha \beta}F_{\beta \mu}- 5 ( F_{\mu \nu}F_{\mu \nu})^{2}] + \cdots .
C_{v}(T)=\frac{4D(E_{F})}{k_{B}T^{2}}\int_{-\infty}^{\infty}\frac{e^{\beta\epsilon}}{(e^{\beta\epsilon}+1)^{2}}\epsilon^{2}d\epsilon=\frac{\pi^{2}}{3}D(E_{F})k_{B}^{2}T
( b \eta_{k,i,t}+ \eta_{k-1,i,t}b) c = \, (-1)^{i-1}\Psi_{k,i-1,t}c \, + \, \Psi_{k,i-1,t}\sum_{j=0}^{i-2}\, (-1)^j \, s_{k-1,i-2}b_{k,j}c \, + + (-1)^i \Psi_{k,i,t}c \, + \Psi_{k,i,t}\sum_{j=0}^{i-1}\, (-1)^j \, s_{k-1,i-1}b_{k,j}c ,
H_{j,6}= \real \Big[ i^j \sum_{|\alpha|=j-4}(-1)^{\alpha_1+\alpha_2}\int u^{(\alpha_1)}u^{(\alpha_2)}u \overline{u^{(\alpha_3)}u^{(\alpha_4)}u^{(\alpha_5)}}+ u^{(\alpha_1)}u^{(\alpha_2)}\bar u \frac{d^{\alpha_3}}{dx^{\alpha_3}}\Big( u \overline{u^{(\alpha_4)}u^{(\alpha_5)}}\Big) dx\Big] .
|m_H-m_{\mu_A\boxplus\mu_B}| =|m_{H}m_{\mu_A\boxplus\mu_B}|\Big|\frac{1}{m_H(z)}-\frac{1}{m_{\mu_A\boxplus\mu_B}(z)}\Big| &\leq C|m_{H}m_{\mu_A\boxplus\mu_B}|\,\Lambda\, \leq C|m_{H}-m_{\mu_A\boxplus\mu_B}| \,\Lambda\,+C|m_{\mu_A\boxplus\mu_B}|^2 \,\Lambda\, .
\rho ( l_{0}, l , d ) = \lim_{\tau \to 0}{\frac{\displaystyle{\left({\frac{\partial}{\partial \tau}}\right)^{n}\left[ N_{l_{0}, l}( d ) G_{l}/ l \right]}}{\displaystyle{\left({\frac{\partial}{\partial \tau}}\right)^{n}\left[ G_{l_{0}}/ l_{0}\right]}}}\ .
\sum_{i=2}^{\infty}[z^i]\left(-\frac{z^3}{(1+z)^3}+\frac{z^2}{(1+z)^2}\eta\right)\left(-\frac{(1+z)^2}{z^2}\eta\right)u^{i-2}\alpha_i = \sum_{i=2}^{\infty}[z^i]\left(\frac{z}{1+z}\eta-\eta^2 \right)u^{i-2}\alpha_i.
J_\theta(s,W,\phi)=\int W\left(\sigma \begin{pmatrix}I_m z I_m 1 \end{pmatrix}\!\!\begin{pmatrix}g g 1 \end{pmatrix}\!\!\begin{pmatrix}I_m I_m x 1 \end{pmatrix}\!\!\right)\phi(x)\theta(Tr(-z))|g|^{s-1}dxdzdg
S^*_{n}(p^d) \equiv \frac{(1+kp)^{\phi(p^d)}-1}{(1+kp)-1}\equiv \frac{\phi(p^d)kp+\binom{\phi(p^d)}{2}(kp)^2+\dotsb +(kp)^{\phi(p^d)}}{kp}\equiv \phi(p^d)\pmod{p^d}.
S_{\mathrm{g a u g e ~ f i e l d}}= \int d x^{0}\left\{-{\frac{i n}{2 r}}\overline{{{\pi}}}_{a}\pi_{a}\sum_{i}( \phi_{i}\widetilde{D}_{0}\overline{{{\phi}}}_{i}- \overline{{{\phi}}}_{i}\widetilde{D}_{0}\phi_{i}) \right\}
f^*(x)= k\,x^{\frac{rp^l-1}{2}}+(2-k)\,\binom{r}{r-1}\,x^{\frac{(r-1)p^l}{2}}+k\,\binom{r}{e-2}\,x^{\frac{(r-2)p^l-1}{2}}\cr +(2-k)\,\binom{r}{r-3}\,x^{\frac{(r-3) p^l}{2}}+\cdots+(2-k)\,\binom{r}{2}\,x^{\frac{2p^l}{2}}+k\,\binom{r}{1}\,x^{\frac{p^l-1}{2}}
\frac{d^2u}{dz^2}+\bigg[\frac{1-\alpha-\alpha'}{z-a}+\frac{1-\beta-\beta'}{z-b}+ \frac{1-\gamma-\gamma'}{z-c}\bigg]\frac{du}{dz}& + \bigg[\frac{\alpha\alpha'(a-b)(a-c)}{z-a}+\frac{\beta\beta'(b-a)(b-c)}{z-b}+\frac{\gamma\gamma'(c-a)(c-b)}{z-c}\bigg]\frac{u}{(z-a)(z-b)(z-c)}=0,
\begin{aligned}(d/dt)|_{t=0}G(x,\Psi(y,t))(d/dt)J\Psi(y,t) &\qquad\stackrel{\eqref{Btwo3}\wedge\eqref{Btwo5}}{=}\langle \nabla_{y}G(x,y),X(y)\rangle\mathrm{div}(X(y))+G(x,y)\mathrm{div}(X(y)\mathrm{div}(X(y))) &\qquad=\mathrm{div}_{y}(G(x,y)X(y)\mathrm{div}(X(y))).\end{aligned}
B^{(\theta)}f = \sum_{I\in \mathcal{D}_{\leq n}}\frac{\langle f, h_I\rangle}{\|h_I\|_2^2}b_I^{(\theta)}, f\in W_n, A^{(\theta)}f = \sum_{I\in \mathcal{D}_{\leq n}}\frac{\langle f, b_I^{(\theta)}\rangle}{\|b_I^{(\theta)}\|_2^2}h_I, f\in W_N
\begin{aligned}u^{-\kappa}\sigma(u) =\frac{(2e^\gamma)^{-\kappa}}{\Gamma(1+\kappa)}\mbox{for}u\in(0,2], \frac{d}{du}\bigl(u^{-\kappa}\sigma(u)\bigr) =-\kappa u^{-\kappa-1}\sigma(u-2) \mbox{for}u>2.\end{aligned}
Z_{\phi_1}(T, v, k) \geq (n-s-1)c^{n+k}+1+sc^{s+t+k+1}\left(c^{(n-s-1)+t}\prod_{i=1}^t (|T_i|-1)+1\right) &= (n-s-1)c^{n+k}+1+sc^{n+2t+k}\prod_{i=1}^t (|T_i|-1)+sc^{s+t+k+1}