harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
[\alpha, v_n]_{L,n}= [\alpha_m, v_m]_{E_{\rho}^m,m}=\dfrac{1}{\eta^m}\operatorname{T}_m(\lambda_{\rho}(\alpha_m) v_m^{-1}) \cdot_{\rho}v_m & = \dfrac{1}{\eta^m}\operatorname{T}_m(\lambda_{\rho}(\alpha_m) v_m^{-1}) \cdot_{\rho}(\eta^m \cdot_{\rho}v_{2m}) & = \operatorname{T}_m(\lambda_{\rho}(\alpha_m) v_m^{-1}) \cdot_{\rho}v_{2m}.
\psi(z): = \ln\left(\mathbb E\left[e^{iz X_1}\right]\right)=\lambda^+\left(\sum_{k=1}^{m^+}\sum_{j=1}^{m_k}c_{kj}\left(\frac{\eta_k}{\eta_k-iz}\right)^j-1\right) &+ iz\mu -\frac{\sigma^2}{2}z^2+\lambda^-\left(\sum_{k=1}^{n^-}\sum_{j=1}^{n_k}d_{kj}\left(\frac{\vartheta_k}{\vartheta_k+iz}\right)^j-1\right),\ \z \in \mathbb R,
xln(x)-x-\frac{2x^{3}}{3}-\frac{x(x^{2e})^{-\frac{1}{e}}((-ix^{e})^{\frac{1}{e}}\Gamma(\frac{1}{e},ix^{e})+(ix^{e})^{\frac{1}{e}}\Gamma(\frac{1}{e},-ix^{e}))}{2e}
\pi_{3, \ell/r}\circ (\chi_2 \circ \chi_1) =: (\chi_2 \circ \chi_1)_{\ell/r}\circ \pi_{1, \ell/r}. = (\chi_{2, \ell/r}\circ \chi_{1, \ell/r}) \circ \pi_{1, \ell/r}
\lim_{u\rightarrow \infty}\int_{u}^{\infty}x^{-1}\frac{\overline{\mathbf{F}}\left( x\right)}{\overline{\mathbf{F}}\left( u\right)}\mathbb{K}\left( \frac{\overline{\mathbf{F}}\left( x\right)}{\overline{\mathbf{F}}\left(u\right)}\right) dx=\gamma_{1}\int_{0}^{\infty}\mathbb{K}(s)ds=\gamma_{1}.
a-b=lim_{d\rightarrow\infty}\frac{\frac{(1/f)^{(d-1)}(b)}{(d-1)!}}{\frac{(1/f)^{(d)}(b)}{d!}}=d\frac{(1/f)^{(d-1)}(b)}{(1/f)^{(d)}(b)}.
\begin{array}{ll}u= -\tilde{u}_y \int\limits_y^{+\infty}w(t,x,\tilde{y}) \,\mathrm{d}\tilde{y}, begin{align*}8pt]\frac{u}{\tilde{u}_y}= - \int\limits_y^{+\infty}w(t,x,\tilde{y}) \,\mathrm{d}\tilde{y}.\end{array}
\begin{array}{l}{J_{\pm}^{\prime}\tilde{\psi}_{J M}({\bf x}) = \sqrt{[ J \mp M ]_{q}[ J \pm M + 1 ]_{q}}\tilde{\psi}_{J , M \pm 1}({\bf x}) ,}\\{J_{3}^{\prime}\tilde{\psi}_{J M}({\bf x}) = M \tilde{\psi}_{J M}({\bf x}) .}\\ \end{array}
P_{k}(\tau)=\big(\frac{\alpha_{0}(k+1)!}{2C_{k}}-\frac{2\alpha_{1}k(k+1)}{(2\pi)^{2}}+\tilde{C_{k}}\big)(1+\tau)^{k}+\frac{2\alpha_{1}k(k-1)}{(2\pi)^{2}}(1+\tau)^{k-2}.
\partial^2_{ji}w(t) =\sum_h(\partial_hS(\Phi(t))\partial^2_{ji}\Phi_h(t)-\partial_h \tilde S(t,\tilde\Phi(t))\partial^2_{ji}\tilde\Phi_h(t)) &+\sum_{h,k}(\partial^2_{hk}S(\Phi(t))\partial_j \Phi_k(t)\partial_i \Phi_h(t)-\partial^2_{hk}\tilde S(t,\tilde\Phi(t))\partial_j\tilde\Phi_k(t)\partial_i\tilde\Phi_h(t))=:I_1(t)+I_2(t)\,,
f(\omega)=P_r(\Delta_S)\left\{\frac{-2\pi^{(2-n)/2}}{\Gamma(\frac{n}{2})}\int\limits_{S^{n-1}}g(\sigma)\log{\frac{1}{|\omega\cdot \sigma|}}d\sigma \right\}+\frac{\Gamma(\frac{n+1}{2})}{\pi^{(n-1)/2}}\int\limits_{S^{n-1}}g(\sigma)d\sigma,
\quad{(a_1q +{n}_1(\zeta > \mathbf{e}_q), \cdots, a_{|E|}q +{n}_{|E|}(\zeta > \mathbf{e}_q))}&=(a_1 q+{n}_1(1- e^{-q\zeta}), \cdots, a_{|E|}q+{n}_{|E|}(1-e^{-q\zeta})) &=(\Phi^+_1(q,0), \cdots, \Phi^+_{|E|}(q,0)).
\frac{e}{4 m}F_{\alpha \beta}X^{\alpha \beta}= \frac{e}{4 m}F_{i j}X^{i j}= \frac{e}{4 m}\epsilon_{i j k}B^{k}X^{i j}= \frac{e}{2 m}B^{k}\left[ \frac{1}{2}\epsilon_{i j k}X^{i j}\right] = \frac{e}{2 m}B^{k}h ( \lambda_{k})
(K_{\theta_1,\Phi}\cdot K_{\theta_3,\Phi}) \cdot (K_{\theta_3,\Phi}\cdot K_{\theta_2,\Phi}) = \langle \theta_1,\theta_3\rangle \overline{\langle \theta_2,\theta_3\rangle}\cdot \K_{\theta_1,\theta_3,\Phi}\cdot K_{\theta_3,\theta_2,\Phi}.
B_r (x_0) \, = \, \{x \in \mathbb{R}^d \, ; \, | x \, - \, x_0 | < r \}, C_r (x_0) \, = \, \{x \in \mathbb{R}^d \,; \, | x_i \, - \, x_{0,i}| < r, \; ( i = 1,2,\ldots,d) \}, P_r (z_0) \, = \, \{z=(t,x) \in Q(T) \, ; \, t_0 - r^2 < t < t_0 + r^2, | x \, - \, x_0 | < r\}, D_r (z_0) \, = \, (t_0-r^2,t_0+r^2) \times C_r (x_0).
S=\int dt \left(g_{i\bar{\jmath}}{d z_{i}\over dt}{d \bar{z}_{j}\over dt}+g_{ij}{d z_{i}\over dt}{d z_{j}\over dt}+g_{\bar{\imath}\bar{\jmath}}{d \bar{z}_{i}\over dt}{d \bar{z}_{j}\over dt}\right).
S_4 (\pi x) - \frac{1}{16}S_4 (2\pi x) = \frac{7}{8}\pi x \zeta(3) + \frac{\pi^3}{12}x^3 \log (\pi x) - \frac{\pi^3}{12}\left( \frac{11}{6}+ \log 2 \right) x^3 &{}{}+ \frac{\pi^5 x^5}{2}\sum_{n=0}^\infty \frac{| B(2n+2) | \left( 2^{2n+1}- 1 \right)}{(n+1) (2n+5)!}\pi^{2n}x^{2n}.
X^{M}(\sigma,\tau)=\hat{x}_{0}^{M}-i\alpha'\hat{p}_{0}^{M}\tau+i\sqrt{\frac{\alpha'}{2}}\sum_{n\neq 0}\left(\frac{\alpha^{M}_{n}}{n}e^{-n(\tau+i\sigma)}+ \frac{\tilde{\alpha}^{M}_{n}}{n}e^{-n(\tau-i\sigma)}\right)~,
\sum_{k=0}^{n}\binom{n+k}{2k}C_k x^{2k}(b-x^2)^{n-k}= S_n^{(b-x^2,x^2)}=bM_{n-1}^{(b+x^2,bx^2)}= b\sum_{k=0}^{n-1}\binom{n-1}{k}M_k^{(ax,bx^2)}(x^2-ax+b)^{n-1-k}= b\sum_{k=0}^{n-1}\binom{n-1}{k}M_k^{(a,b)}x^k(x^2-ax+b)^{n-1-k}.
\nu_1 = -\frac{1}{2d}\sum_{ij}A_{ii}A_{jj}\pi_i \pi_j, \nu_2 = -\frac{1}{2d^2}\sum_{ijk\ell}A_{ik}^2 A_{j\ell}^2 \pi_i \pi_j \pi_k \pi_\ell, \mbox{and}\xi_n = O(n^{-1/2}) \sum_{ij}|X_{ij}| + O(n^{-1}) \left(\sum_{ij}|X_{ij}|\right)^2.
\left( \begin{array}{c c c}{{a}}&{{{\bf O}_{+}}}&{{{\bf O}_{v}}}\\{{}}&{{}}&{{}}\\{{{\bf O}_{+}^{\dagger}}}&{{b}}&{{{\bf O}_{-}}}\\{{}}&{{}}&{{}}\\{{{\bf O}_{v}^{\dagger}}}&{{{\bf O}_{-}^{\dagger}}}&{{c}}\\ \end{array}\right)
c(k)\leq d-\sqrt{(2-\frac{1}{r})k}-(2-\frac{1}{r})^{\frac{1}{4}}k^{\frac{1}{4}}-1=\left(1-\sqrt{2-\frac{1}{r}}-o(1)\right)\sqrt{k}, c(k)\leq \frac{(\frac{1}{r}-1)k}{\sqrt{(\frac{1}{r}-1)k}+(\frac{1}{r}-1)^\frac{1}{4}k^\frac{1}{4}+1}=(1-o(1))\sqrt{(\frac{1}{r}-1)k}.
\langle A_{\mu}^{a}( x ) A_{\nu}^{b}( y ) \rangle_{R}^{\mathrm{\scriptsize ~ s i n g l e t}}: = \lim_{D \to 4}\delta^{a b}{\frac{2 N}{N^{2}- 1}}{\frac{1}{( i g_{R})^{2}}}\mu^{D - 4}\lim_{x^{\prime}\to x \atop y^{\prime}\to y}\partial_{\mu}^{x}\partial_{\nu}^{y}\, \bar{W}_{1}( x , x^{\prime}, y , y^{\prime}; D )
\pi^k\int_{\mathbf B^k_{+}}(1-\sum_{j=1}^{k}r_j)^{\alpha\cdot(\mathbf a+\mathbf 1)}r^{\mathbf c}dV=\frac{\pi^k\prod_{j=1}^{k}(c_j)!}{\big(1+\alpha\cdot(\mathbf a +\mathbf 1)\big)_{(\mathbf c\cdot\mathbf 1)+k}}.
\widehat{u_j}= (2\pi)^{-\frac{d}{2}}\sum_{n}\frac{n_j}{|n|^2}u(n) e^{-i n \cdot x}= (2\pi)^{-\frac{d}{2}}\sum_{n}-i n_j a(n) e^{-in \cdot x}= \partial_{x_j}\psi,
\begin{array}{rcl}{\cal L}_{\rm g.f.}= \displaystyle-\frac{1}{2\xi_1}f_1f_1-\frac{1}{2\xi_2}f_2f_2, f_1 = \partial^{\mu}G_{1\mu}+ig_1\xi_1(\phi^{\dagger}\phi_0-\phi_0^{\dagger}\phi), f_2 = \partial^{\mu}G_{2\mu}-ig_2\xi_2(\phi^{\dagger}\phi_0-\phi_0^{\dagger}\phi).\end{array}
\sum_{k=1}^\infty|\alpha|^k\left(\Big(\frac{1}{k-1/R_0}\Big)^{2l+1}-\Big(\frac{1}{k+1/R_0}\Big)^{2l+1}\right) &\le2 \sum_{k=1}^\infty|\alpha|^k\Big(\frac{1}{k-1/2}\Big)^{2l+1}\le C(\alpha),
\mathrm{BC}(P_{y}, P_{x_0}) \ge \left(1 - \frac{a + b}{2n}+ \frac{ab}{n}- 2\frac{(a-b)^2}{n^2}\right)^{s(n-s)}= \left( 1 - \frac{(\sqrt{a}- \sqrt{b})^2}{2n}- 2\frac{(a-b)^2}{n^2}\right)^{s(n-s)}\ge \left( 1 - \frac{(\sqrt{a}- \sqrt{b})^2}{n}\right)^{s(n-s)}\ge \exp{- 2s (\sqrt{a}- \sqrt{b})^2},
J_i(f;v^1,v^2):= \dfrac{\alpha_i}{2}\iint_{\mathcal{O}_{i,d}\times (0,T)}|y-y_{i,d}|^2 dxdt + \dfrac{\mu_i}{2}\iint_{\mathcal{O}_i \times (0,T)}|v^i|^2 dxdt,
\partial_{t}( e^{-a+b+3c}\dot{\lambda}) - \partial_{r}(e^{a-b+3c}\lambda') = e^{a+b-3c}[2Q_{K}^{2}e^{\frac{8\lambda}{3}}-P^2 e^{\frac{4\lambda}{3}+ 4\nu}- Q^2 e^{\frac{4\lambda}{3}- 4\nu}].
\int_X \sum_{\lambda: d(\lambda) = n}\phi \cdot \frac{1}{|f_\lambda \circ \tau_\lambda|^2}\, d\mu = \sum_{\lambda:\,d(\lambda)=n}\int_{D_{s(\lambda)}}\phi \frac{1}{|f_\lambda\circ \tau_\lambda|^2}d \mu = \sum_{\lambda:\,d(\lambda)=n}\int_{R_\lambda}(\phi\circ \tau^n) \frac{1}{|f_\lambda|^2}d (\mu\circ(\tau_\lambda)^{-1}).
V_1 = \frac{1}{\sqrt{q^4+1}}\Biggl( \begin{matrix}0 1 0 q^2 0 0 0 0 0 \end{matrix}\, \Biggr) , V_2 = \frac{1}{\sqrt{q^4+1}}\Biggl( \begin{matrix}0 0 0 0 0 q^2 0 -1 0 \end{matrix}\, \Biggr) .
\langle F\rangle_r\cap\langle G\rangle_r =\sqrt{\langle F\rangle}\cap\sqrt{\langle G\rangle}=\sqrt{\cup_{i=0}^{\infty}F^{[i]})}\cap\sqrt{\cup_{i=0}^{\infty}G^{[i]})}=\sqrt{\cup_{i=0}^{\infty}(F^{[i]}\cap G^{[i]})}\subseteq\sqrt{\cup_{i=0}^{\infty}\sqrt{(FG)^{[i]}}}=\langle FG\rangle_r,
R_0\leq\min\{I(V_0;Y_1),I(V_0;Y_2)\}, &R_0+R_i\leq I(V_0,V_i;Y_i),i=1,2, &R_0+R_1+R_2\leq\min\{I(V_0;Y_1),I(V_0;Y_2)\}+I(V_1;Y_1|V_0)+I(V_2;Y_2|V_0)-I(V_1;V_2|V_0)
A^{2}= \left( \begin{matrix}{- \omega_{1}^{2}}&{0}&{0}&{0}\\{0}&{- \omega_{1}^{2}}&{0}&{0}&{0}\\{0}&{0}&{- \omega_{2}^{2}}&{0}\\{0}&{0}&{0}&{- \omega_{2}^{2}}\\ \end{matrix}\right) ,
\eta_\mu (\lambda) = c (\mu) \frac{\Gamma_{p, d}(\frac{d}{2}(n+1))}{\Gamma_{p, d}(\frac{1}{2}d p)}\frac{\Gamma_{p, d}(\frac{1}{2}(\lambda - \rho + k + d p))}{\Gamma_{p, d}(\frac{1}{2}(- \lambda + \rho +k))}\frac{\Gamma_{p, d}(\frac{1}{2}(- \lambda + \rho + \mu))}{\Gamma_{p, d}(\frac{1}{2}(\lambda + \rho +\mu))},
\exp\left(-\beta_t^2\left(\frac{t}{B}-2\right)\right)\leq \exp\left(-\beta_t^2\frac{t}{B}+\frac{1}{2}\right) \cr = \exp\left(-\frac{p^2t}{4B}+2p^2+\frac{1}{2}-\frac{4p^2B}{t}\right) \cr \leq \exp\left(-\frac{p^2t}{4B}+\frac{5}{2}\right) \cr \leq 13\exp\left(-\frac{p^2t}{4B}\right) \cr = 13\left(\exp\left(-\frac{pn}{2}\right)\right)^{\frac{pt}{2nB}}.
\tilde{g}_i := 1-G_i, g_i := 1-G_i^2, m_i := (A^{-1}b)_i, \tilde{m}_i := (\mathcal{A}^{-1}\beta)_i, r_i := \frac{\lambda_i^2-\tilde{\lambda}_i^2}{\lambda_i^2}, \tilde{r}_i := \frac{\lambda_i^2-\tilde{\lambda}_i^2}{\tilde{\lambda}_i^2}, \hat{r}_i := m_i - \tilde{m}_i
h_{ij}(x)=\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}h(x)=4\alpha^{2}e^{-\alpha|x-x_{0}|^{2}}\left[(x_{i}-(x_{0})_{i})(x_{j}-(x_{0})_{j})-\frac{1}{2\alpha}\delta_{ij}\right].
\begin{aligned}\langle \partial^{1/2}_t y, \partial^{1/2}_t v \rangle := \frac{T}{2}\sum_{k=1}^{\infty}k \omega \langle\boldsymbol{y}_k,\boldsymbol{v}_k\rangle_{\Omega}, \langle\sigma \partial^{1/2}_t y, \partial^{1/2}_t v\rangle := \frac{T}{2}\sum_{k=1}^{\infty}k \omega \langle\sigma \boldsymbol{y}_k,\boldsymbol{v}_k\rangle_{\Omega}.\end{aligned}
\Gamma_{l,s}^{\,\alpha,\,\alpha_{0}}(\vec p)=\Gamma_{l,s}^{\,\alpha,\,\alpha_{0}}(0)+\sum_{j=1}^{3}\,\sum_{\mu=0}^{3}\,p_{\mu,j}\int\limits_{0}^{1}d\lambda\,\left(\,\partial_{\mu,j}\,\Gamma_{l,s}^{\,\alpha,\,\alpha_{0}}\right)(\lambda\,\vec p)\quad,
C^{\mathrm{ul},k}_{3}\left( N_{\mathrm{sct}}, N_{\mathrm{scr}}\right) = \left(\frac{T_{\mathrm{ul}}}{T}\right) \min\left\lbrace c^k_{\mathrm{u2s},1}\left( N_{\mathrm{scr}}\right), c^k_{\mathrm{s2b},3}\left(N_{\mathrm{sct}}\right) \right\rbrace.
\mu_g(A)\le \frac{C}{\lambda}|z|^{\frac{n-2}{2}}\mu_g(E)^{\frac{1}{p_2}}\mu_g(A)^{\frac{1}{q_2'}}= \frac{C}{\lambda}|z|^{-\frac{1}{n+1}}\mu_g(E)^{\frac{n^2+4n-1}{2n(n+1)}}\mu_g(A)^{\frac{n+1}{2n}}R_0^\frac{n-1}{2}\,.
\left. \hat{F}_{\alpha \beta}^I\right|_{{\rm reg}}=-\frac{2\,e_\alpha^a\,\hat{\xi}_{ab}^{+}\,e_\beta^b}{\left( r^2+\rho^2\right)^2}\qquad{\rm or}\qquad\left. \hat{F}_{ab}^I\right|_{{\rm reg}}=-\frac{2\,\hat{\xi}_{ab}^{+}}{\left( r^2+\rho^2\right)^2}
\left(\begin{array}{lll}r_{1}r_{2}r_{3}r_{4}r_{5}r_{6}r_{7}r_{8}r_{9}\end{array}\right) M \left(\begin{array}{lll}r_{1}r_{2}r_{3}r_{4}r_{5}r_{6}r_{7}r_{8}r_{9}\end{array}\right)^{T}= [A]_{\mathbf{cb}_i}.
L_\varepsilon = \tfrac12 \int_\Omega\bigl[\lvert u \rvert^2+ 2 \, \varepsilon \, g( u, u_\beta')+ \varepsilon^2 \,(\lvert u' \rvert^2 + g(u, u_\beta'')\bigr] \, \mu (x)+ O (\varepsilon^3) & \equiv L_0 + \varepsilon \, L_1 + \tfrac12 \, \varepsilon^2 \, L_2+ O (\varepsilon^3) \,.
v^2=\frac{(2h\sum_{i=1}^n a_iD_i^5)}{(\sum_{i=1}^m b_iB_iC_i^2)}u^4+\frac{(20h^3\sum_{i=1}^n a_iD_i^3- \sum_{i=1}^m b_iB_i^3)}{(3\sum_{i=1}^m b_iB_iC_i^2)}u^2+\frac{(2h^5\sum_{i=1}^n a_iD_i)}{(\sum_{i=1}^m b_iB_iC_i^2)}.
\tau_V(\varphi)=\sum\limits_{\alpha=1}^n\tau(\varphi)^\alpha \frac{\partial}{\partial y_\alpha}+\sum\limits_{\alpha=1}^n\sum\limits_{i=1}^m V_i\frac{\partial \varphi^\alpha}{\partial x_i}\frac{\partial}{\partial y_\alpha}.
g(n+1)-h(n+1) >\frac{8(n^3+n^2-n-2)h(n)-128(n-1)(n+1)^2}{n(n+1)^2g(n)}begin{align*}5pt] =\frac{128(2n^2-n-2)}{n^3(n-1)(n+1)^2g(n)}>0.
= \sum_{n \geq 0}\frac{-1}{z^{n+1}}\left \langle \psi_1^{n-1}e^{\tau_2/z}T_j, \tau'^{(l)}\right \rangle^\textnormal{coh}_{0,l+1,d}= \frac{-1}{z}\left \langle \frac{e^{\tau_2/z}T_j}{z+\psi}\tau'^{(l)}\right \rangle^\textnormal{coh}_{0,l+1,d}
\frac{d^{n}}{d \hbar^n}\omega_{H, \psi}(\Psi^i(z_1) \star \Psi^j(z_2))|_{\hbar = 0}= \int_{\Sigma^{2n}}\big[ W_{\Sigma}(y_1, y_2) \cdots W_{\Sigma}(y_{2n-1}, y_{2n}) K^i_{n, \psi}(z, y_1, \ldots, y_{2n-1}) K^j_{n, \psi}(z, y_2, \ldots, y_{2n}) \big]\,\mathrm{d}y_1 \cdots \mathrm{d}y_{2n}.
\sum_{\substack{m \geq 1 (m,2)=1}}\frac{\lambda_f(l_1m^2)}{m^{1+2s}}= \lambda_f(l_1)\zeta(2+4s)^{-1}L(1+2s, \operatorname{sym}^2 f) \prod_{p | 2l}\left( 1+O \left( \frac{1}{p^{1+2s}}\right) \right)^{-1}
K_{t,A}(x,y) =t\mathsf{P}_{A}^t(x,y)\frac{x-y}{x^{t}-y^{t}}\intertext{and}L_{t,A}(x,y) =t\mathsf{P}_{A'}^t(x,y)\frac{x-y}{x^{t}-y^{t}},
\oint{d}\hat{s}( \xi^{\prime \prime}){\hat{K}_{D}}( \xi , \xi^{\prime \prime}){\hat{G}_{K}}( \xi^{\prime \prime}, \xi^{\prime}) ={\hat{\delta}_{B}}( \xi - \xi^{\prime}) -{\frac{1}{\oint{d}\hat{s}( \xi^{\prime \prime \prime})}}.
(p^a-1)Q_{r,a}( X_1,\cdots,X_r) = p^a Q_{r-1,a-1}(p^{a+r-1}X_1,p^{a+r-2}X_2,\cdots,p^{a+1}X_{r-1}) \sum_{f_r\ge0}(p^aX_r)^{f_r}&-Q_{r-1,a+1}(X_1,\cdots,X_{r-1})\sum_{f_r\ge0}X_r^{f_r}.
\{x^{\mu}, ~ g_{\mu \nu}( x ) , ~ \psi_{\mu \nu . . .}( x ) \}\longrightarrow \{x^{\mu}, ~ \Omega^{2}( x ) \tilde{g}_{\mu \nu}( x ) , ~ \Omega^{- d}( x ) \tilde{\psi}_{\mu \nu . . .}( x ) \}.
W^{[1]}_{n}(x)=\frac{W_{n}(x)}{n+1}+\frac{n}{n+1}\left( x-\beta_{n}\right)W^{[1]}_{n-1}(x)-\frac{1}{n+1}\sum_{\nu=1}^{n-1}\nu \, \chi_{n-1,\nu}W^{[1]}_{\nu-1}(x)\, , n\geq 1\; , \; W^{[1]}_{0}(x)=1 \, , \; W^{[1]}_{1}(x)=x-\frac{1}{2}\left(\beta_{0}+\beta_{1}\right).
\begin{array}{l}\mathcal{G}_{\tilde{e}}^{*t}(s)\phi^{\tilde{m}}(f)=\frac{\gamma}{2}\sum\limits_{0\leq i,j\leq n}<\tilde{m},f\circ \sigma_{ij}\circ \varrho_j^{ins}-f> +\beta'\sum\limits_{i=1}^n<\tilde{m},B_i'f-f>+\beta''\sum\limits_{i=1}^n<\tilde{m},B_i''f-f> +\mathcal{G}_{\tilde{e}}^{*t,sel}(s)\phi^{\tilde{m}}(f)\end{array}
|S_{12}\rangle=\prod_{m=1}^{\infty}\exp{\left(c^{(1)}_{-m+\frac{1}{2}}b^{(2)}_{-m+\frac{1}{2}}+c^{(2)}_{-m+\frac{1}{2}}b^{(1)}_{-m+\frac{1}{2}}\right)}|0\rangle_{1}\otimes|0\rangle_{2}\;,
C^*_2 =\prod_{p>2}\left(1-\frac{1}{(p-1)^2}\right) F^*(N) =\prod_{\underset{p>2}{p\mid N}}\left(1-\frac{1}{(p-1)^2}\right)^{-1}\prod_{p\mid N}\left(1-\frac{1}{(p-1)^2(p+1)}\right) G^*(N) =\frac{N}{\varphi(N)}\prod_{\underset{p>2}{p\mid N}}\left(1-\frac{1}{(p-1)^2}\right)^{-1}\prod_{p\mid N}\left(1-\frac{1}{p^{\nu_p(N)}(p-1)}\right).
\frac{v^2}{\lambda}=\frac{(a\Lambda)^2}{8\pi^2}+\frac{1}{2}\hat{m}^2a^2\left(1+\frac{\lambda_R}{16\pi^2}\bar{c}\right)\bar{I}_2+Z_{\phi^2}\left(\frac{v_{\rm R}^2}{\lambda_{\rm R}}\right),
\max_{\alpha\!, \tilde{p}\!, p_1\!, p_2\!}\quad \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)
\sqrt{S_{k+1}}-\sqrt{S_{k}}= \frac{S_{k+1}-S_{k}}{\sqrt{S_{k+1}}+\sqrt{S_{k}}}\geq \frac{S_{k+1}-S_{k}}{2\sqrt{S_{k+1}}}\stackrel{(\ref{S_k-def})}{=}\frac{1}{2}\sqrt{1+rS_k}\geq \frac{1}{2}
K_{p}\left( u\right) \geq \int wu\d\mu - K_p^*(w) \forall u \in L^1 (\mu ), \forall w \in L^{\infty}(\mu ) K_{p}\left( u\right) = \int wu\d\mu - K_p^*(w) \iff w \in \partial K_{p}\left( u\right).
ds_{D}^2 = H^{-\frac{d-2}{D-2}}\left[ - f dt^2 + \sum_{i=1}^p (dx^i)^2 + H R_T^2\left( f^{-1}A dR^2 + \frac{A}{K^{d-2}}dv^2+ K R^2 d\Omega_{d-2}^2 \right) \right]
\beta=\frac{16\sigma^2\eta^2}{\alpha}\int_0^\infty{\rm d}\xi\;\xi^2\phi_\sigma^2\left(\xi\right)\,,\quad\phi_\sigma\left(\xi\right)={\rm e}^{-2\sigma\xi}\int_0^\infty{\rm d}s\;{\rm e}^{-2\xi s}\,{\cal V}_\sigma\left(\eta,\zeta,s\right)
\int d^3 \zeta - \sqrt{-g (Z(\zeta ) )}=\int - h^{-1}\sqrt{1 - hv^2}\Big[ 1 + \frac{1}{2}h^{2/3}\big[ 2 h^{-1/3}\bar{\theta}\Gamma^{\tilde{1}}\partial_1 \theta + 2 h^{-1/3}\bar{\theta}\Gamma^{\tilde{2}}\partial_2 \theta
2\pi \cdot \widehat{T_a f}(\xi) = \int_{|\xi-\eta|\leq \frac{M}{2}}\hat{P}_{>M}(\xi)\hat{a}(\xi-\eta)\hat{P}_{>M}(\eta)\hat{f}(\eta)\,d\eta + \int_{|\xi-\eta|>\frac{M}{2}}\hat{P}_{>M}(\xi)\tilde{\chi}(\xi-\eta,\xi+\eta)\hat{a}(\xi-\eta)\hat{P}_{>M}(\eta)\hat{f}(\eta)\,d\eta
\alpha_{0}=\operatorname{Id}_V, \alpha_{1}(v) =\begin{cases}f v=c; e v=d; d v=e; c v=f; v v\in \{a,b\},\end{cases}\alpha_{2}(v) =\begin{cases}e v=d; d v=e; v v\neq d,e,\end{cases}\alpha_{3}=\alpha_{1}\alpha_{2}.
\frac{-2q}{1+q}\frac{(q^2;q)_\infty}{(-q;q)_\infty}{}_2\phi_2 \left(\genfrac{}{}{0pt}{}{-q,q}{-q^2,q^2};q,q^2\right) = 2\frac{(q;q)_\infty}{(-q;q)_\infty}\sum_{n\geq 1}\frac{(-1)^{n}q^{n(n+1)/2}}{(1+q^{n})(q;q)_{n}}.
F(x)-F(x+1)= \sum_{k=0}^{\infty}\frac{1}{(k+x+\frac a2)^2-\frac{a^2-4b}{4}}-\sum_{k=0}^{\infty}\frac{1}{(k+x+1+\frac a2)^2-\frac{a^2-4b}{4}}= \frac{1}{(x+\frac a2)^2-\frac{a^2-4b}{4}}=\frac{1}{x^2+a x+b}.
w_{1i}= (1 +\varphi_u d)u_{1i}+ \varphi d_{1i}+ (\varphi_{1i}+\varphi_{1u}u_i+\varphi_{iu}u_1+ \varphi_{uu}u_1 u_i) d + (\varphi_1 + \varphi_{u}u_1 ) d_i + (\varphi_i +\varphi_{u}u_i )d_1, i =1, \cdots, n; w_{1t}= (1 +\varphi_u d)u_{1t}+ (\varphi_{1u}u_t+ \varphi_{uu}u_1 u_t)d +\varphi_{u}u_t d_1.
\sum_{k=1}^nC_f^n\int\limits_{[t,T]}^{\sim}\int\limits_{[s_1,T]}^{\sim}\cdots \int\limits_{[s_{k-1},T]}^{\sim}v_{n-k}(s_k)ds_k\cdots ds_1 \leq \left(\dfrac{C_1}{\sqrt{M}}\right)^{n}e^{C_f\sqrt{M}(T-t)}\sum_{k=1}^n{\dfrac{(C_f\sqrt{M}(T-t))^k}{k!C_1^k}}\leq \left(\dfrac{C_1}{\sqrt{M}}\right)^{n}e^{C_f\sqrt{M}(T-t)\left(1+\frac{1}{C_1}\right)}
\mathfrak{\varphi}_{i,k}=\{((0,y_{0}),(1,y_{1}),\ldots,(n+1,y_{n+1})): & y_{0}=i+k, \y_{n+1}=n+1-i+k, y_{i+1}-y_{i}\in \{1,-1\}, \ 0\leq i\leq n\}.
F(z+\zeta_0) = \frac1{2\pi\mathrm{i}}\int_{\Gamma}F(\zeta)\bigg( \frac{1}{\zeta-\zeta_0-z}- \frac{1}{\zeta-\zeta_0+z}\bigg)\,\mathrm{d}\zeta = \int_\Gamma K_z(\zeta,\zeta_0)F(\zeta)\,\mathrm{d}\zeta.
I_2+\overline{\Psi}^{-T}\overline{K}^{-1}\overline{L}=I_2 + \sum_{j\in\mathcal{I}^-}L(A_j)\overline{\Psi}^T_jK^{-1}= (K + \sum_{j\in\mathcal{I}^-}L(A_j)\overline{\Psi}_j )K^{-1}
\mathcal{M}_{\sigma,\alpha,\gamma}f(z) := \sup_{\substack{I\subset\mathbb{R}z\in Q_I}}\frac{1}{|Q_I|_{\sigma ,\alpha}^{1-\frac{\gamma}{2+\alpha}}}\int_{Q_I}|f(w)|\sigma(w)dV_{\alpha}(w).
\frac{d^{l+m}}{dx^{l+m}}(x^{2}-1)^{l}=\sum_{i=0}^{l+m}(\begin{matrix}l+m\\i\end{matrix})\frac{d^{l+m-i}}{dx^{l+m-i}}(x+1)^{l}\frac{d^{i}}{dx^{i}}(x-1)^{l}
\begin{cases}\Bar{\Phi}^{(r)}\colon S_\mathbb{K}^{r+1}(\mathfrak{g}[1])\xrightarrow{\Phi_{r}}\mathfrak{X}_{-r}(E)[1] &\Bar{\Psi}^{(r)}\colon S_\mathbb{K}^{r+1}(\mathfrak{g}[1])\xrightarrow{\Psi_{r}}\mathfrak{X}_{-r}(E)[1] \end{cases},\;
f_{n}^{(k)}= \sum_{j=0}^{\lfloor n/(k+1)\rfloor}(-1)^j\,\frac{(n-jk)+j+\delta_{n,0}}{2(n-jk)+\delta_{n,0}}\, \binom{n-jk}{j}\, 2^{n-j(k+1)}\,.
\{\theta_{1}, \theta_{2}\}_{\mathrm{c . b .}}= ( \beta_{1}I + \beta_{2}P_{1 2}) + \{\theta_{1}[ \theta_{2}, r ] \}+ \beta_{3}( \theta_{1}\theta_{1}+ \theta_{2}\theta_{2}) \pm \sqrt{\alpha}( \theta_{1}P \theta_{2}+ \theta_{2}P \theta_{1})
\begin{bmatrix}J^{}_{0}(x^{(0)}) 0 \cdots 0 begin{align*}2ex]\dfrac{\partial J^{}_{1}}{\partial x^{(0)}}(x^{(0)})\,x^{(1)}J^{}_{1}(x^{(0)}) \cdots 0 \vdots \vdots \ddots \vdots \dfrac{\partial J^{}_{\ell}(x^{(0)})}{\partial x^{(0)}}(x^{(0)})\,x^{(\ell)}0 \cdots J^{}_{\ell}(x^{(0)})\end{bmatrix}.
ds^2_{(n+2)}=-\left( (n-1)^2\frac{\alpha^2x^2}{r_0^2}-2\frac{\Delta M}{M_0}\right) dt^2+\frac{\alpha^2dx^2}{\frac{(n-1)^2\alpha^2x^2}{r_0^2}-2\frac{\Delta M}{M_0}}+r_0^2d\Omega_{(n)}^2\ ,
R \leq \frac{1}{n\log n}\left[ (1+\kappa+b) \, n\log n + \frac{1}{2}n \log \left( 1 + 1 / n^{2(1+\kappa+b)}\right) - 0.599n \right] = 1 + \kappa + b + \log \left( 1 + 1 / n^{2(1+\kappa+b)}\right) / \log n - 0.599 / \log n \;,\,
g(k) =\begin{cases}z^{-\frac{k - L_N - 1}{2}}g(L_N + 1) z^{\frac{k - L_N - 2}{2}}h(L_N + 1) \end{cases}h(k) =\begin{cases}z^{\frac{k - L_N - 1}{2}}h(L_N + 1) z^{-\frac{k - L_N - 2}{2}}g(L_N + 1) \end{cases}
\pi_\lambda(k)(\varphi\, K_{k\cdot z_0})(z) = j_\lambda(k^{-1},z)\varphi(k^{-1}\cdot z) K_{k\cdot z_0}(k^{-1}\cdot z) = \varphi(k^{-1}\cdot z) \pi_\lambda(k) (K_{k\cdot z_0})(z) = \overline{j_\lambda(k^{-1},k\cdot z_0)}\varphi(z)K_{z_0}(z),
\Bigl [ \varepsilon ({\bf r}, \omega ) \, \omega^{2}\delta_{j m}- \mathrm{r o t}_{j l}\mathrm{r o t}_{l m}\Bigr ] D_{m k}( \varepsilon ,{\bf r},{\bf r}^{\prime}, \omega ) = - \omega^{2}\delta ({\bf r}-{\bf r}^{\prime}) \delta_{j k}
\sum_{k=D}^{N}\frac{k!}{(k-D)!}=D!\sum_{k=D}^{N}(\begin{matrix}k\\D\end{matrix})=D!\sum_{k=0}^{N-D}(\begin{matrix}k+D\\D\end{matrix})=D!(\begin{matrix}N+1\\D\end{matrix})=\frac{(N+1)!}{(N+1-D)!}
\frac{1}{3}\frac{y\cdot r}{x}\cdot\frac{y^{2}}{x^{2}}+\frac{1}{7}\frac{y\cdot r}{x}\cdot\frac{y^{2}}{x^{2}}\cdot\frac{y^{2}}{x^{2}}\cdot\frac{y^{2}}{x^{2}}+\cdot\cdot\cdot
C_{p \varrho}- C_{\varrho p}\&=\ \tfrac{1}{648}A_p + \tfrac{1}{324}\varrho A_{p\varrho}+\tfrac{1}{60}f_{ppp}+\tfrac{1}{10}h_y+\tfrac{1}{72}B_{pp}+\tfrac{2}{3}B_\varrho\ =\ 0,
\frac{1}{\ell_i}\sum_{n\leq \ell_i}\mathbf{1}_{\underline{A}\times A}(\underline{\varphi}(R^n (\Delta(0))),\varphi(R^n (\Delta(0))))=\frac{1}{\ell_i}\sum_{n\leq \ell_i}\mathbf{1}_{\underline{C}}(R^n(\Delta(0)))\mathbf{1}_{C}(R^n(\Delta(0))).
- \frac 1t \log u_2(t) \xrightarrow{t\to\infty}\ 1 - p^k - \hat\alpha k + p^k\Big(1 - \frac1{p^k\lambda}+ \frac1{p^k\lambda}\log\frac1{p^k\lambda}\Big) begin{align*}1em] =\ 1 - \hat\alpha k - \frac1\lambda\Big(1 - \log\frac1\lambda - k\log\frac1p\Big) = A_{\lambda}.
P_{(p,M)}\left[\sum_{i=1}^{n}\xi_{i}^{(j)}\geq \frac{\rho_j (j-1)n}{j}+(1-\frac{1}{j})M_j\right] \geq P_{(p,M)}\left[\sum_{i=1}^{n}\xi_{i}^{(j)}\geq \frac{\rho_j (j-1)n}{j}+j\right] = P_{(p,M)}\left[n\rho_j - \sum_{i=1}^{n}\xi_{i}^{(j)}\leq \frac{n\rho_j}{j}-j\right].
\begin{array}{r c l}{{Y}}&{{=}}&{{\frac{1}{2}( x + y ) = \frac{1}{2}( z_{1}^{2 n - 4}+ z_{2}^{2 n - 4})}}\\{{X}}&{{=}}&{{\frac{1}{2}z ( x - y ) = \frac{1}{2}z_{1}z_{2}( z_{1}^{2 n - 4}- z_{2}^{2 n - 4})}}\\{{Z}}&{{=}}&{{z^{2}= z_{1}^{2}z_{2}^{2}}}\\ \end{array}
\left( \begin{matrix}{0}&{1}\\{1}&{0}\\ \end{matrix}\right) \left( \begin{matrix}{1}\\{\epsilon}\\ \end{matrix}\right) = \epsilon \left( \begin{matrix}{1}\\{\epsilon}\\ \end{matrix}\right) , \quad \left( \begin{matrix}{1}&{0}\\{0}&{- 1}\\ \end{matrix}\right) \left( \begin{matrix}{1}\\{\epsilon}\\ \end{matrix}\right) = \left( \begin{matrix}{1}\\{- \epsilon}\\ \end{matrix}\right) .
\Big( \sum_{n=1}^{N}\Vert x_{n}\Vert^{q}\Big)^{\frac{1}{q}}\leq C \bigg( \int_{\mathbb{T}^{N}}\Big\Vert \sum_{n=1}^{N}x_{n}z_{n}\Big\Vert^{q}dz \bigg)^{\frac{1}{q}}\,,
\!\!\!f(k,x)\!=e^{ikx}\left[\!I_n\!+\!\frac{Q(x)}{ik}\!+\!o\left(\!\frac{1}{k}\!\right)\!\right]\!,\; Q(x)\!:=\frac{\int_x^\infty V(t)dt}{2},\; |k|\to\infty\; \;\overline{\mathbb{C}}^+.
{V}^{\mathrm{g a u g e}}( r ) = - \, \frac{g_{\mathrm{Y M}}^{2}}{2}~ \frac{1}{7 \, \Omega_{8}\, r^{7}}\left( \delta^{A B}\, \delta^{C D}- \delta^{A C}\, \delta^{D B}\right) ~ ~ ,
Pr(Y_{i}=y|X_{i})=(\begin{matrix}n_{i}\\y\end{matrix})p_{i}^{y}(1-p_{i})^{n_{i}-y}=(\begin{matrix}n_{i}\\y\end{matrix})(\frac{1}{1+e^{-\beta\cdot X_{i}}})^{y}(1-\frac{1}{1+e^{-\beta\cdot X_{i}}})^{n_{i}-y}.
T\in(0,T_R)T_R:= \min\left\{1,\left(\frac{1}{\tilde C[R^3 + 2|\phi |_{\gamma (W^{\alpha ,p})}(1+|v_0|_{W^{1,p}})]}\right)^{\frac{1}{\min\left(1,\frac{p-2}{2p}\right)}}\right\},
| B_{m n}^{( + )}| \leq ( n + m )^{- k}\sqrt{{\frac{n}{m}}}s u p \{|{\frac{d^{k + 1}\varphi_{t}}{d y^{k + 1}}}| \, : \, 0 \leq y \leq 2 \pi , 0 \leq t \leq 1 \}.
\widehat{P_{\le N}f}(\xi):= \widehat{f_{\le N}}(\xi):= \varphi(\tfrac{\xi}{N}) \widehat{f}(\xi) ,\widehat{P_{> N}f}(\xi):= \widehat{f_{> N}}(\xi):= \bigl(1-\varphi(\tfrac{\xi}{N})\bigr) \widehat{f}(\xi) , &\widehat{P_{N}f}(\xi):= \widehat{f_{N}}(\xi):= \bigl(\varphi(\tfrac{\xi}{N})-\varphi(\tfrac{2\xi}{N})\bigr)\widehat{f}(\xi).
{\cal W}_{\Gamma{_0}}={\frac{1}{N}}\langle 0 | \mathrm{T r}\left[{\cal T}{\cal P}\mathrm{e x p}\left( i g \oint_{\Gamma{_0}}d x^{\mu}\A_{\mu}^{a}( x ) T^{a}\right) \right] | 0 \rangle ,
\begin{array}{ccccc}SO(2n-1) \subset SO(2n) R_M={\bf 2n}Sp(2n) \subset SU(2n) R_M={\bf n (2n-1)}G_2 \subset SO(7) R_M={\bf 7}{\rm \or\}{\bf 8}F_4 \subset E_6{\bf 27}\end{array}
g(x)= g\bigg(m\bigg(1-\frac{x-ma}{b-ma}\bigg)a+\bigg(\frac{x-ma}{b-ma}\bigg)b\bigg) \leq m \bigg(1-\frac{x-ma}{b-ma}\bigg)^s g(a)+ \bigg(\frac{x-ma}{b-ma}\bigg)g(b).
\begin{aligned}(g^{p}\circ\tilde f^{p})^{*}=(g^{p}\circ (D_{N}^{-1}) \circ (f^{m-p})^{*}\circ D_{M})^{*}=D_{M}\circ f^{m-p}\circ (D_{N}^{-1})\circ (g^{p})^{*}&=D_{M}\circ f^{m-p}\circ (D_{N}^{-1})\circ (g^{p})^{*}\circ D_{M}\circ D_{M}^{-1}=D_{M}\circ f^{m-p}\circ \tilde g^{m-p}\circ D_{M}^{-1}.\end{aligned}