harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
T \geq h(\tilde{X}_B^n+Z_B^n|\tilde{X}_A^{n},W_C) - h ( \tilde{X}_A^{n}+ Z_B^n|\tilde{X}_B^n,W_C) &= h(\tilde{X}_B^n+Z_B^n|W_C) - h ( \tilde{X}_A^{n}+ Z_B^n|W_C),
\left\langle N_{\nu_{\alpha_{1}}}^{( \alpha_{1})}, \cdots , N_{\nu_{\alpha_{n}}}^{( \alpha_{n})}\right\vert \left. N_{\nu_{\alpha_{1}}}^{( \alpha_{1})}, \dots , N_{\nu_{\alpha_{n}}}^{( \alpha_{n})}\right\rangle ={\cal A}^{\{{\cal N}\}}.
\sum_{k=0}^m{n+m+1\choose k}x^k (1-x)^{m-k}=\sum_{k=0}^m{n+k\choose k}x^k=(n+m+1){n+m\choose m}\sum_{k=0}^m\frac{(-1)^k}{k+n+1}{m\choose k}(1-x)^k.
\tilde\Sigma^{ij}= \sum_\rho(\tilde\gamma^{-1}\sigma)^{i}_\rho(\tilde\gamma^{-1}\sigma)^{j}_\rho=\sum_\rho\sigma^2 (\tilde\gamma^{-1})^{i}_\rho(\tilde\gamma^{-1})^{j}_\rho = \sigma^2 diag(1/(\gamma^2+B_0^2),1/(\gamma^2+B_0^2),1/\gamma^2).
S ( \lambda ) = S_{0}+ \lambda S_{1}= - 1 / g^{2}T r [ 1 / 4 [ A_{i}, A_{j}] [ A_{i}, A_{j}] - 2 / 3 i \lambda \rho \epsilon^{i j k}A_{i}A_{j}A_{k}+ \rho^{2}( 1 - \lambda ) A_{i}A_{i}]
\Im i\langle v,HD_jQ\rangle = \Im \bigl[ i\langle v,ED_jQ\rangle -i\mu\langle v,D_j(|Q|^pQ)\rangle \bigr] & = \Im\langle v,D_jDQ iEz\rangle-\Im i\mu\langle v,D_j(|Q|^p Q)\rangle,
g_{00}=-H^{-\frac{1}{2}}, g(r)=H^{-\frac{1}{2}}, g_{S}(r)=H^{\frac{1}{2}}r^{2},g_{rr}(r)=H^{\frac{1}{2}}, H=1+\frac{e^{\Phi_{0}}Q}{2r^{4}}
E^{\prime\prime}\left( 0\right) =-\int_{0}^{b}\left\langle V^{\prime\prime}+R\left( V,\dot{\gamma}\right) \dot{\gamma},V\right\rangle dt+\left.\left\langle V^{\prime}-S_{\gamma^{\prime}}V,V\right\rangle \right\vert_{0}^{b}+\sum_{i}\left\langle V^{\prime}\left( t_{i}^{-}\right) -V^{\prime}\left( t_{i}^{+}\right) ,V\left( t_{i}\right) \right\rangle ,
\frac{d\rho_i}{d t}+\sum_{j\in N(i)}\omega_{ij}(S_j-S_i)\theta_{ij}(\rho)=0, &\frac{d S_i}{dt}+\frac 12\sum_{j\in N(i)}\omega_{ij}(S_i-S_j)^2 \frac{\partial \theta_{ij}(\rho)}{\partial \rho_i}+\beta \frac{\partial I(\rho)}{\partial \rho_i}-\alpha \log(\rho_i) +\mathbb V_i+\sum_{j=1}^N\mathbb W_{ij}\rho_j=0.
|I_7| =\left|\left\langle (1-A)Bh\partial_y\tilde{h}\partial_y u, u\right\rangle_{H^{3, 0}}\right| &\lesssim\left\|(1-A)Bh)\right\|_{L^\infty}\|\partial_y\tilde{h}\partial_y u\|_{H^{3, 0}}\|u\|_{H^{3, 0}}+\|\partial_y\tilde{h}\partial_y u\|_{L^\infty}\|(1-A)Bh\|_{H^{3, 0}}\|u\|_{H^{3, 0}}.
(\partial_t-\Delta)\varphi = -f \varphi +e^{-d\rho-ft}(\partial_t-\Delta)g + e^{-ft}g (\partial_t - \Delta) (e^{-d \rho}) - 2 e^{-ft}\nabla e^{-d\rho}\cdot \nabla g \le -f \varphi +C(S,\Sigma_0,a,b,d,\eta) +C(d,\eta) \varphi - 2 e^{-ft}\nabla e^{-d\rho}\cdot \nabla g.
\psi\left(\begin{pmatrix}b_1 b_2 b_3 b_4\end{pmatrix}\right) =\begin{pmatrix}b_1^2 b_1b_2 b_2^2 2b_1b_3 b_1b_4+b_2b_3 2b_2b_4 b_3^2 b_3b_4 b_4^2\end{pmatrix}.
\|X-Y\|^2=-\frac{\lambda\tau}8\Im\left[(\overline{X}-\overline{Y})^TH(X,Y,z)\right] \leq \frac\tau8\left(\max_{1\le m\le M}|X_m+z_m||Y_m+z_m|\right)\|X-Y\|^2\le\frac{\tau}2\|X-Y\|^2,
\left( D_i\right)_{\alpha \beta}v^\beta (x)\rightarrow \frac1a\left({\cal D}_i\right)_{\alpha \beta}v^\beta (x)=\frac 1a\left\{\frac12\left( v^\alpha (x+\widehat{i})-v^\alpha (x-\widehat{i})\right) -f_{\alpha\gamma \beta}\theta^\gamma (x,x+\widehat{i})v^\beta (x)\right\}
\frac{b_{r}^{2}}{b_{s}^{2}}= \frac{( 1 + m^{2}k^{2}s^{4}) r^{4}\sin^{2}\theta_{r}}{( 1 + m^{2}k^{2}r^{4}) s^{4}\sin^{2}\theta_{s}}e^{2 i ( \varphi_{r}- \varphi_{s})}
\langle\eta,\gamma_{X_{l_{0}}(t,x)}\eta\rangle_{\mathbb{R}^{N}}=\eta^{*}\cdot JF_{l_{0}}(U_{t}^{(l_{0})},x)\cdot \gamma_{U_{t}^{(l_{0})}}\cdot JF_{l_{0}}(U_{t}^{(l_{0})},x)^{*}\eta \geq\mu_{t}^{(l_{0})}\cdot\left(\eta^{*}\cdot JF_{l_{0}}(U_{t}^{(l_{0})},x)\cdot JF_{l_{0}}(U_{t}^{(l_{0})},x)^{*}\cdot\eta\right),
u(x) =x^2(C(x)-1)+\sum_{n\geq 4}\left(\sum_{a=2}^{n-2}\sum_{t=0}^{n-2-a}\binom{n-2-a}{t}u'(n-t,a)\right)x^n &=x^2(C(x)-1)+(1-x)r\left(\frac{x}{1-x},(1-x)y\right)\mid_{y=1},
\nu_{p}(a_{n}) = \nu_{p}(a_{\rho})+\sum_{d\mid p^u}\nu_{p}(b_{\rho d})+\sum_{d\mid p^u}\sum_{\substack{e\mid m e>1}}\nu_{p}(b_{\rho de}) & = \alpha+\sum_{i=1}^u\nu_{p}(b_{\rho p^i})+0 & = \alpha+\sum_{i=1}^u1 & = \alpha+u
\sum q^{\frac{1}{2}p^{2}}= \sum_{n}q^{\frac{1 5}{2}\left( 2 n + \frac{5 m_{1}+ 3 m_{3}}{3 0}\right)^{2}}= \sum_{n}q^{3 0 \left( n + \frac{5 m_{1}+ 3 m_{3}}{6 0}\right)^{2}}= \theta_{5 m_{1}+ 3 m_{3}, 3 0}\: ( \tau ) .
|\sigma_n(x)-\sigma_n(y)|=\Bigl|\frac{\sigma(x)-\sigma(y)}{1+n^{-1/2}|x|^{\ell_\mu}}+f_n(x,y)\Bigr|\leq c\cdot (1+|x|^{\ell_{\sigma}}+|y|^{\ell_{\sigma}})\cdot |x-y|+|f_n(x,y)|.
\mathbf{v}_u\sqrt{2n-4}\leq \lambda_1 \mathbf{v}_u=\sum_{y\sim u}\mathbf{v}_y = \sum_{\substack{y\sim u y\in L}}\mathbf{v}_y + \sum_{\substack{y\sim u y\in S}}\mathbf{v}_y \leq \sum_{y\in L}\mathbf{v}_y + \sum_{\substack{y\sim u y\in S}}\mathbf{v}_y.
J_{2}^{N L S}= \left( \begin{array}{c c}{{D + \varepsilon^{2}\left\{\begin{array}{c}{{\eta^{- 1}\, D^{3}+ D^{3}\, \eta^{- 1}}}\\{{- \frac{1}{2}\left[ ( \eta^{- 1})_{x x}\, D + D \, ( \eta^{- 1})_{x x}\right]}}\\ \end{array}\right\}}}&{{\frac{1}{2}\, D \, \upsilon}}\\{{\frac{1}{2}\, \upsilon D}}&{{\frac{1}{2}( \eta \, D + D \, \eta )}}\\ \end{array}\right)
L=\frac{1}{2}(\partial_{\mu}\pi^{k})(\partial^{\mu}\pi^{k})+\frac{1}{2}(\partial_{\mu}\sigma)(\partial^{\mu}\sigma)-\frac{1}{2}(2\mu^{2})\sigma^{2}-\sqrt{\lambda}\mu\sigma^{3}-\sqrt{\lambda}\mu\pi^{k}\pi^{k}\sigma-\frac{\lambda}{2}\pi^{k}\pi^{k}\sigma^{2}-\frac{\lambda}{4}(\pi^{k}\pi^{k})^{2},
d s^{2}= \frac{d \xi^{0}d \xi^{0}}{1 + \xi^{0}\xi^{0}}- ( 1 + \xi^{0}\xi^{0}) \frac{d{\bf x}( I +{\bf x}^{\prime}{\bf x})^{- 1}d{\bf x}^{\prime}}{1 +{\bf x}{\bf x}^{\prime}}~ ,
\sum_{i=1}^{2n}D_{D_{e_i}e_i}\Omega - D_{e_i}D_{e_i}\Omega-\theta(D_{e_i}e_i)\Omega + 2\theta(e_i)D_{e_i}\Omega - (\theta(e_i))^2\Omega + e_i(\theta(e_i))\Omega=0
\mathcal{E}(\overline{D}_1)-\mathcal{E}(\overline{D}_0)=\frac{1}{n+1}\sum_{j=0}^n\int_X-\log\frac{\|\cdot\|_{\overline{D}_1}}{\|\cdot\|_{\overline{D}_0}}c_1(\overline{D}_0)^{\wedge j}\wedge c_1( \overline{D}_1)^{\wedge n-j}.
p_1p_2 \left(\frac{1-c_{1,1}^{(z)}}{\bigl|c_{1,1}^{(z)}\bigr|}\right) \!\! \left(\frac{1-c_{2,2}^{(z)}}{\bigl|c_{2,2}^{(z)}\bigr|}\right)\geqslant 1.
\langle \partial_t + \partial_x,\underline{L}' \rangle_e = \langle \partial_t + \partial_x,-\partial_t -{x - a \over r'}\partial_x +{y \over r'}\partial_y +{z \over r'}\partial_z \rangle_e = -1 +{a - x \over r'}={a - x - r' \over r'}.
\int \psi \left| \nabla G_{h/2}\ast u \right|^2 dx = \int \psi \left| \int \nabla G_{h/2}(z) \left(u(x) - u(x- z) \right)dz\right|^2 dx &\leq \int \psi \left( \int \left|\nabla G_{h/2}(z) \right| dz\right)\left( \int \left|\nabla G_{h/2}(z)\right| \left|u(x) - u(x- z) \right|^2 dz\right) dx.
\mathbb{E}_{\mathbf{H}, \mathbb{W}}\left \{\log_2\left(\cos^2\theta_k \right)\right \}= \log_2(e) \sum_{i=1}^{n}\binom{n}{i}(-1)^{i}\sum_{l=1}^{i\left( n_T -1 \right)}\frac{1}{l},
\forall t\in\mathbb{R},\mathcal{C}^{(N)}_t=\mathcal{C}^{(N)}_{t+\pi}\Rightarrow \min_{t\in\mathbb{R}}d_H\big(X_0^{N}(t),R_{-t}(X)\big)=\min_{t\in[0,\pi]}d_H\big(X_0^{N}(t),R_{-t}(X)\big).
\sum_{p \leq x}\frac{|g(p)|-(g(p)p^{-i\tau}}{p}= \sum_{1 \leq j \leq m}\left(\sum_{p \in E_j \atop p \leq x}\frac{|g(p)|-B_j}{p}+ B_j\sum_{p \in E_j \atop p \leq x}\frac{1-(g(p)p^{-i\tau}/B)}{p}\right) &= \sum_{1 \leq j \leq m}\left(B_j\rho_{E_j}(g;x) - \sum_{p \in E_j \atop p \leq x}\frac{|g(p)|-B_j}{p}\right).
\begin{array}{l}\displaystyle C_k(a,b)=\frac1{\sqrt\pi}2^{-a-b}\int_0^1\int_0^\infty\xi^{(-a-b)/2}e^{-\xi}G_k(\eta;a,b)(1-\eta)^{a+b-2}d\xi d\eta \qquad\hphantom{aaa}=\displaystyle\frac1{\sqrt\pi}2^{-a-b}\Gamma\left(\frac{2-a-b}2\right)\int_0^1G_k(\eta;a,b)(1-\eta)^{a+b-2}d\eta\,.\vphantom{\vrule heigh 13pt}\end{array}
2\frac54\Big(\frac{5^4}{4^4}a^4N^{4/4}\frac{75}{4^3}a^2N^{-2/4}+\frac{750}{4^4}a^3 N^{-1/4}\frac{5^3}{4^3}a^3 N^{3/4}\Big)(x-N)e^{2aN^{5/4}}&\le c_0\frac{5^2}{4^2}a^2N^{2/4}\,(x-N)\, e^{2aN^{5/4}}\le c_0\,\partial_xP_N\,e^{aN^{5/4}}\le c_0\,P_N \partial_x P_N
\begin{cases}\tilde{u}_0(x, 0)=0, \quad\,(\partial^2_y \tilde{u}_0)(x, 0)=0, \,\,\forall x\in \mathbb{R}, &(\partial^4_y \tilde u_0)(x, 0)=\big(\partial_yu^s_0(0) + (\partial_y\tilde{u}_0)(x, 0)\big) (\partial_y\partial_x\tilde{u}_0)(x, 0),\forall x\in \mathbb{R},\end{cases}
\det(\lambda E_q-A_q)= \lambda^q \exp \Bigl(\varphi(q) \log(1-\frac{q}{\lambda}) \Bigr) = \lambda^q (1-\frac{q}{\lambda})^{\varphi(q)}= \lambda^{q-\varphi(q)}(\lambda -q)^{\varphi(q)}.
\widehat{j}(0,a,b) = \frac{W_q(b)}{W_q'(b+)}\Big( \mathcal{H}^a_{q,r}(b,0) - \widehat{U}_1(b,a,b,0) - \widehat{U}_2(b,a,b) \Big)^{-1}= \mathcal{H}^{a \prime}_{q,r}(b,0)^{-1}.
2n K_1 \beta^{-1/2}|\nabla v| \psi^{-1\slash n}(\cdot,v) \leq 2 n K_1 e^{C_0^2 + 1}\psi^{-1\slash n}(\cdot,v) \leq (B - 2 K_1 -1) \sum_{p=1}^n u^{p\bar p}.
\left| w^{(k)}(x) \right| \leq (m-1) \|V^{-1}\| \left(\frac{2 k!}{h^k}\|w\|_{L^{\infty}([0,1])}+ h^{m-k}\frac{k!}{m!}\|w^{(m)}\|_{L^{\infty}([0,1])}\right) &\leq (m-1) \|V^{-1}\| \left(\frac{2 m!}{h^k}\|w\|_{L^{\infty}([0,1])}+ h^{m-k}\|w^{(m)}\|_{L^{\infty}([0,1])}\right)
S^{ND3}_{WZ}=\mu_3\int_{R^{3+1}}{\rm Tr}\left(C^{(4)}+i(2\pi\alpha^\prime)i_\Phi i_\Phi B^{(6)}+\frac12 C^{(2)}\wedge B^{(2)}-(2\pi\alpha^\prime)B^{(2)}\wedge{\tilde{\cal F}}+\dots\right)\, ,
\langle\psi_* (e_S \otimes e_T ), e_S \otimes e_T\rangle = \langle(\psi_{r+1,r+1})_* (e_{S_{r+1}}), e_{S_{r+1}}\rangle \prod_{x=1}^r \langle(\psi_{x,x})_* (e_{S_x}\otimes e_{T_x}), e_{S_x}\otimes e_{T_x}\rangle
\left(\begin{matrix}a^* -b b^* a\end{matrix}\right)=\left(\begin{matrix}\alpha_1^* \gamma_1^* \beta_1^* \delta_1^*\end{matrix}\right)\left(\begin{matrix}a_1 0 0 a_1^{-1}\end{matrix}\right)\left(\begin{matrix}1 y+y_0 0 1\end{matrix}\right).
\rm div \int \frac{d^4q}{(2\pi)^4}\frac{1}{q^2(q+p)^2\vec{p}^2(\vec{q}+\vec{p})^2}= \frac{-4\Gamma(2-\omega-\sigma)}{p^2\vec{p}^2 (4\pi)^2}\left|{\begin{array}{l}{\scriptstyle \sigma \to (1/2)^+}{\scriptstyle \omega \to (3/2)^+}\end{array}}~, \right.
\pi ( u ) = \frac{2 k - 1}{4 \pi}\int_{\tau = 0}^{\infty}d \tau \sinh \tau \int_{\theta = 0}^{2 \pi}d \theta \left[ \cosh \frac{\sigma}{2}- i \sinh \tau \cos \theta \sinh \frac{\sigma}{2}\right]^{- 2 k}
\bar{\lambda}_1 = \inf_{f \in C_0([-D/2,D/2])}\frac{\int_{-D/2}^{D/2}(f')^2}{\int_{-D/2}^{D/2}f^2}+\frac{(n-1)(n-3)K}{4}\frac{\int_{-D/2}^{D/2}\sec^2(\sqrt{K}x)f^2}{\int_{-D/2}^{D/2}f^2}-\frac{(n-1)^2K}{4},
< q | n > = \left( \frac{m \omega}{\pi \hbar}\right)^{1 / 4}\, \frac{1}{\sqrt{2^{n}\, n !}}\, e^{- \frac{m \omega}{2 \hbar}q^{2}}\, H_{n}\left( q \sqrt{\frac{m \omega}{\hbar}}\right) \ ,
\lim_{n\to\infty}\frac{1}{n}E^Q m(n) \ \ = \ \ \sum_{z\in A^c}\pi^Q(z)Q(z,A)\ = \pi^Q(A^c) - \sum_{z \in A}\pi^Q (z) Q(z,A) \ = \ (1-\pi^Q(0))- (1- \pi^Q(0))+(1-\lambda_C^{-1}) \pi^Q(0),
j\left(\frac{-1+\sqrt{-3}}{2}\right)=0, j(\sqrt{-1})=1728, j(2\sqrt{-1})=287496, &j\left(\frac{-1+\sqrt{-43}}{2}\right)=-884736000, j\left(\frac{-1+\sqrt{-67}}{2}\right)=-147197952000.
=\frac{px^2}{(1-px)(1-x)}\bigg(\frac{p^3x^3}{1-p^3x^3}- \frac{p^3x^4}{1-p^3x^4}\bigg) - \frac{p^2x^3}{(1-px)(1-px^2)}\bigg(\frac{p^4x^4}{1-p^4x^4}- \frac{p^5x^6}{1-p^5x^6}\bigg)
\delta(x-[r_0])+[r_1] \equiv 2[\phi^{-1}(r_2)] \bmod I^2 (\ref{a=b}) \Rightarrow \delta(\delta(x-[r_0])+[r_1]) \equiv [r_2]- 2[r_2] \bmod I p=2\in I \Rightarrow \delta(\delta(x-[r_0])+[r_1]) \equiv [r_2] \bmod I
\,\left(\frac{1}{4}F^2_t (u,\theta)\,-\frac{1}{4}F^2_t(\mathcal{T}_t(u),\mathcal{T}^*_t (\theta))\right) =\,(1-\,\kappa(t,\tau))(\alpha+\beta - (\alpha -\beta)) & =\,4\,(1-\,\kappa(t,\tau))\,\alpha\,\beta & =\,4\,(1-\,\kappa(t,\tau))\,\beta.
\sum_{i=1}^n b_i = \sum_{i=1}^n a_i = a_k+a_{\ell}+ \sum_{\substack{i=1 i \neq k,\ell}}^n a_i = c_k+c_{\ell}+ \sum_{\substack{i=1 i \neq k,\ell}}^n c_i = \sum_{i=1}^n c_i.
|\nabla\zeta_{j}(x)|\lesssim\begin{cases}\frac{1}{|x|}|\xi'(j|x|)| \ |x|\in [\frac{1}{2j},\frac{1}{j}], 0 \ |x|\in [0,\frac{1}{2j}]\cup [\frac{1}{j},j]\cup[2j,\infty], \frac{1}{j}\ |x|\in [j,2j].\end{cases}
\begin{cases}\displaystyle (z_i^*)^{p-1}\le \frac{m_0+\gamma}{\alpha+\beta}|v_i|,\,\,\,\, i=1,2\ldots,m_0, \displaystyle (z_i^*)^{p-1}= \frac{m_0+\gamma}{\alpha+\beta}|v_i| ,\,\,\,\, i=m_0+1,\ldots,n. \end{cases}
\widetilde{M}= \{db;\bigg{(}\frac{\alpha_1}{\lambda_1(\alpha_1)}, \beta_1 \bigg{)}, \dots, \bigg{(}\frac{\alpha_1}{\lambda_{r_1}(\alpha_1)}, \beta_1 \bigg{)}, \dots, \bigg{(}\frac{\alpha_k}{\lambda_k(\alpha_k)}, \beta_k \bigg{)}, \dots, \bigg{(}\frac{\alpha_k}{\lambda_{r_k}(\alpha_k)}, \beta_k \bigg{)}\}
(4) ADB(\hat{\beta}_n^{S}) = -(p-2) \delta E\left[\chi^{-2}_{p+2}(\Delta^2)\right] & ADQR(\hat{\beta}_n^{S}) = \sigma^2\left[\mbox{tr}C^{-1}\right]\left[1-(p-2)\left\{2E\left[\chi^{-2}_{p+2}(\Delta^2)\right]-(p-2)E\left[\chi^{-4}_{p+2}(\Delta^2)\right]\right\}\right] & + \sigma^2(p^2-4) \Delta^2 E\left[\chi^{-4}_{p+4}(\Delta^2)\right]
T_{F,G,N}^L(1,\dots,1) \ll_{c,C,\varepsilon}\sigma_G + \frac{\log X}{\sigma_G X}+ X^{O(1)}\max\limits_{\substack{\mathbf{k}\in \mathbb{Z}^m 0<\Vert \mathbf{k}\Vert_\infty \leqslant X}}\Big( \prod\limits_{j=m+1}^d \min (1, N^{-1}\Vert \mathbf{k}\cdot \mathbf{v_j}\Vert^{-1}_{\mathbb{R}/\mathbb{Z}})\Big),
\begin{array}{rcl}U_c/m^4 = \displaystyle +D_m^2(k_1)D(k_1+p)[9D_m(k_2)D_m(k_1+k_2) & \displaystyle +D(k_2)D(k_1+k_2)](1+K_{12}) & \displaystyle +4D^2(k_1)D_m(k_1+p)D_m(k_2)D(k_1+k_2) & \displaystyle +4(N-1)D_m^2(k_1)D(k_1+p)D(k_2)D(k_1+k_2), \end{array}
E(F_N;G_N)\ll N^2+N^\epsilon\displaystyle\sum_{1\leq s\leq [N^\epsilon]+1}\sum_{n,n_1\in I_s}\sum_{\substack{1\leq n<m\leq N 1\leq n_1<m_1\leq N f(n)+f(m)=f(n_1)+f(m_1) g(n)+g(m)=g(n_1)+g(m_1)}}1.
\begin{tabular}[c]{l}$i)f\left( t,u\right) \equiv f\left( u\right) ,\;B\left(t,u\right) \equiv B\left( u\right) $ ($f$ and $B$ are independent of$t$)\medskip $ii)\quad(\widetilde{H}_{1}),\;$(\ref{4.1}) with $a>0,$ and (\ref{4.4}) aresatisfied\medskip $iii)\quad\beta_{0}=a-L_{1}-\frac{1}{2}L>0$\end{tabular}
f(x) = \sum_{i=1}^k \frac{f^{(m_i)}(0^+)}{m_i!}x^{m_i}+o(x^{m_k})(x \in (0,R)), & f(x) = \sum_{i=1}^k (-1)^{m_i}\frac{f^{(m_i)}(0^-)}{m_i!}x^{m_i}+o(x^{m_k}) (x \in (-R, 0)).
f_{(3,1,0,2)}=(1-q t) (1-q^{2}t^{2}) x_1 x_2 x_4\left({\rm Tr}[ k_3 k_2 k_3 s ] f_{(2,0,0,1)}+{\rm Tr}[ a_3^\dag k_3 k_2 a_3 k_3 s] f_{(0,0,2,1)}\right. &+\left.{\rm Tr}[ k_3 k_2 a_2 k_3 a_2^\dag s] f_{(2,0,1,0)}+{\rm Tr}[ a_3^\dag a_2 k_3 k_2 a_3 k_3 a_2^\dag s] f_{(1,0,2,0)}\right).
\widehat{H}=\displaystyle\frac{2\pi}{L}\left( \widehat{L_{0}}+\widehat{\overline{L}_{0}}-\displaystyle\frac{c}{12}\widehat{I}+\lambda \displaystyle\frac{L^{2-h}}{\left( 2\pi \right)^{1-h}}\widehat{B}\right) \: ,
{\frac{d W [ \bar{\phi}( \beta ) ]}{d \beta}}={\frac{\partial W [ \bar{\phi}, \beta ]}{\partial \beta}}+ \left. \int_{M^{2}}{\frac{\delta W [ \phi , \beta ]}{\delta \phi}}\right|_{\phi = \bar{\phi}( \beta )}{\frac{\partial \bar{\phi}( \beta )}{\partial \beta}}.
q_{c}( a , b , r ) = \frac{1}{8 \pi}\sum_{m = - \infty}^{+ \infty}\int_{- \infty}^{+ \infty}{d k \sum_{\gamma , \lambda}}\beta_{\lambda m}^{2}f_{\lambda m}^{( q )}( \gamma a , \gamma r ) , \quad q_{c}= \varepsilon , p_{i},
\begin{array}{l}{{L G_{2 ( 1 , Z )}= Z_{R}^{1 / 3}Z_{L}^{2 / 3}L G_{2}Z_{L}^{- 2 / 3}Z_{R}^{- 1 / 3},}}\\{{G_{2 ( 1 , Z )}= Z_{R}^{1 / 3}Z_{L}^{2 / 3}G_{2}Z_{L}^{- 2 / 3}Z_{R}^{- 1 / 3}.}}\\ \end{array}
\delta\left(w_1,w_2\right) = -\sum_{k,l = 0}^{+ \infty}\left(-1\right)^{l}\frac{B_{2k+2}}{\left(2k+2\right)!}\left(w_{1}^{2k+l+1}w_{2}^{-l-1}+w_{1}^{l}w_{2}^{2k-l}\right).
h_k \left( f_\sigma (x) \, dx_{\sigma (1)}\wedge \ldots \wedge dx_{\sigma (k)}\right) = & \int_0^1 t^{k - 1}f_\sigma (t x) \, dt \cdot \sum_{j = 1}^k (- 1)^{j + 1}x_{\sigma (j)}dx_{\sigma (1)}\wedge \ldots \wedge dx_{\sigma (j - 1)}\wedge dx_{\sigma (j + 1)}\wedge \ldots \wedge dx_{\sigma (k)}.
\sum_{n=0}^\infty \frac{q^{n^2}X^n}{\prod_{i=0}^{n-1}(q^n-q^i)}= \left(\sum_{n=0}^\infty \frac{q^{n^2-n}X^n}{\prod_{i=0}^{n-1}(q^n-q^i)}\right) \cdot \left(\sum_{n=0}^\infty X^n\right).
\left(\begin{array}{c}F'(t,z) G'(t,z)\end{array}\right)=\left(\begin{array}{cc}1 0 0 e^{i\xi_0 t}\end{array}\right)\left(\begin{array}{cc}e^{ic_3 t}0 0 e^{-ic_3 t}\end{array}\right)\left(\begin{array}{c}F_0'(z) G_0'(z)\end{array}\right)\,,
P\left(\frac{n-A_n^*}{f_2(n)}\leq x+\delta\right) -P\left(\frac{n-A_n^*}{f_2(n)}\leq x-\delta\right)\geq\mathrm{e}^{-(H(\ell_2(x+\delta))+\eta)b_c^{(n)}}-\mathrm{e}^{-(H(\ell_2(x-\delta))-\eta)b_c^{(n)}}&=\mathrm{e}^{-(H(\ell_2(x+\delta))+\eta)b_c^{(n)}}\left(1-\mathrm{e}^{-(H(\ell_2(x-\delta))-H(\ell_2(x+\delta))-2\eta)b_c^{(n)}}\right).
\lim_{x\to0^+}\int_0^\infty\frac{xt^{x-1}}{e^t+1}dt = \lim_{x\to0^+}\int_0^\infty\frac{t^xe^t}{(e^t+1)^2}dt =\int_1^\infty\frac{du}{(u+1)^2}=\frac{1}{2}.
s_{\mathrm{e f f}}[ \alpha , \varphi ] = \int \! d^{2}x \left({\frac{1}{2}}\, \partial_{\alpha}\varphi \partial_{\alpha}\varphi +{\frac{\alpha - v}{2}}\, \varphi^{2}-{\frac{\alpha}{2 N g^{2}}}\right) + s [ \alpha ] .
\vert \partial_{p}^{\omega}\, G_{l , s}^{\, \alpha}( \vec{\alpha}, \vec{\lambda}, \vec{p}) \vert \ \leq \e^{\, - m^{2}\sum_{j = 1}^{s}\alpha_{j}}\, P ( \vert \vec{p}\vert ) \, Q ( \sqrt \alpha_{1}, \ldots , \sqrt \alpha_{s}) \quad ,
{h^{\mu}}_{\nu}= D i a g \left[ H_{0}( r ) , H_{1}( r ) , - \frac{1}{d - 2}\left( H_{0}( r ) + H_{1}( r ) \right) , \ldots , - \frac{1}{d - 2}\left( H_{0}( r ) + H_{1}( r ) \right) \right] ,
X^s(e_1\wedge \cdots \wedge e_n) = s! \sum_{\substack{I \subseteq \{1,\ldots,n\}|I|=2s}}\pm P_I \Big(\bigwedge_{\substack{1 \leq r \leq k r \notin I}}e_r\Big) \in \bigwedge^{n - 2s}F_1
\phi_{\mathrm{\bf ~ \tiny ~ k}}\left( r \right) = - \frac{\Omega_{n}G_{n + 4}\, \mu \, \psi_{\mathrm{\bf \tiny ~ k}}^{\star}\left( 0 \right)}{2}\frac{1}{r}e^{- \frac{\left| \mathrm{\tiny \bf ~ k}\right|}{R}}.
X_1X_2+X_2X_1 = -uQ^{-1}v = - \begin{pmatrix}0 0 0 1 \end{pmatrix}\begin{pmatrix}0 X_1 X_2 -1 X_1 0 -1 0 X_2 -1 0 0 -1 0 0 0\end{pmatrix}^{-1}\begin{pmatrix}0 0 0 1 \end{pmatrix},
(L_{1}(u_{1})-L_{1}(u_{2}))(t,r) &= \left(\frac{\sin(2u_{1}(t,r))-\sin(2u_{2}(t,r))}{2r^{2}}\right)\left(\cos(2Q_{1}(\frac{r}{\lambda(t)}))\left(\cos(2v_{corr}(t,r))-1\right)-\sin(2Q_{1}(\frac{r}{\lambda(t)}))\sin(2v_{corr}(t,r))\right)
\sigma_{n}\left( i\right) =r\left( g^{i-1}\left( 0\right) \right)+\frac{1}{2}\left( n-i\right) & =\left( 4d^{2}\right) 2d\left( d+1\right) \frac{\left( 4d^{2}\right)^{i}-1}{4d^{2}-1}+\left( 3d+2\right) d+\frac{1}{2}\left( n-i\right) ,
\frac{f_X(t)-\nu g_X(\nu t)}{u_X(\nu t)-v_X(t)}= \psi_X(t;\nu) \left( \frac{f_X(t)-g_X(t)}{u_X(t)-v_X(t)}+ \frac{g_X(t)-\nu g_X(\nu t)}{u_X(t)-v_X(t)}\right).
\prod_{i = 1}^{k}\bigl ( 1 + \eta_{i}n ( x_{i}) \bigr ) - \prod_{i = 1}^{k}\eta_{i}n ( x_{i}) ={\frac{\prod_{i = 1}^{k}n ( x_{i})}{\tilde{n}( \sum_{i = 1}^{k}x_{i})}}\, .
\int_{-\infty}^\infty z^i\,\phi_s(z)\,[\Phi_s(z)]^{j+\alpha-1}\mathrm{d}z= \int_0^\infty z^i\,\phi_s(z)\,[\Phi_s(z)]^{j+\alpha-1}\mathrm{d}z &+ \int_0^\infty (-1)^i\,z^i\,\phi_s(z)[1-\Phi_s(z)]^{j+\alpha-1}\mathrm{d}z.
\displaystyle \sum_{n=-\infty}^{\infty}f \left( \frac{2 \pi n}{L}i \right)=\frac{L}{2 \pi i}\int_{-i \infty}^{i \infty}dp\, \frac{1}{2}\left[f(p)+f(-p) \right] + \frac{L}{2 \pi i}\int_{-i \infty + \epsilon}^{i \infty+ \epsilon}dp \, \frac{f(p)+f(-p)}{\exp{(Lp)}-1}.
\begin{aligned}U(T) a_1 U^{\dagger}(T)=\cos \theta\,a_1-\sin \theta\,a_2-\alpha \cos \theta+\beta \sin \theta, U(T) a_2 U^{\dagger}(T)=\sin \theta\,a_1+\cos \theta\,a_2-\alpha \sin \theta-\beta \cos \theta,\end{aligned}
=\frac{Lu^{2}+Lc^{2}(1-u^{2}/c^{2})}{2uc^{2}\sqrt{...}}=\frac{Lu^{2}+L(c^{2}-u^{2})}{2uc^{2}\sqrt{...}}=\frac{Lc^{2}}{2uc^{2}\sqrt{...}}=\frac{L}{2u\sqrt{...}}\ne\frac{L/2}{u}
\begin{aligned}\frac{d}{dt}|_{t=0}\int 1_{A^{(t)}}(y)G(x,y)dy =\frac{d}{dt}|_{t=0}\int_{A^{(t)}}G(x,y)dy=\frac{d}{dt}|_{t=0}\int_{A}G(x,\Psi(y,t))J\Psi(y,t)dy &=\int_{A}\mathrm{div}_{y}(G(x,y)V(y))dy=\int_{\partial A}G(x,y)\langle V(y),N(y)\rangle dy.\end{aligned}
J(x) = \int_{0}^{\infty}dA(t)\int_{0}^{x+t}dF_{n}(u) G(u) - \int_{0}^{\infty}dA(t)\int_{0}^{t}dF_{n}(u) G(u) = \int_{z=x}^{\infty}\int_{u=0}^{z}G(u)f_n(u)\lambda e^{-\lambda(z-x)}du dz - \int_{z=x}^{\infty}\int_{u=0}^{z-x}G(u)f_n(u)\lambda e^{-\lambda(z-x)}du dz =: J_1(x) + J_{2}(x).
R_q^{-1}(A\times B) = \{s\in S:s\otimes q\in\widehat{A\times B}\}&= \{s\in S:A\times B\in s\otimes q\}&=\{s\in S:\{u:\{t:(u,t)\in A\times B\}\in q\}\in s\}&= \{s\in S:\{t:(s,t)\in A\times B\}\in q\}&= \{s\in S:s\in A, B\in q\}=A\in\mathfrak{m}
\frac{\delta^{2}\mathcal{A}}{\delta x_{\mu}( s ) \delta x_{\mu}( s^{\prime})}\propto \frac{1}{| s - s^{\prime}|^{4}}( 1 - D ) ( 1 - \dot{\phi}^{2}) + \frac{1}{| s - s^{\prime}|^{2}}\left( \frac{1 - D}{1 2}\dot{\phi}\stackrel{{\, . . .}}{\phi}+ \frac{3 - D}{4}\ddot{\phi}^{2}\right) .
F^{(3)}_{j_1 j_2 j_3}:=\begin{cases}\dfrac{-\mathrm{i}\,c_1\, j_1 j_2 j_3-c_2\,j_1 j_2+c_3}{\mathrm{i}(j_1^3+j_2^3+j_3^3)}\mbox{if}\,\,(j_1, j_2, j_3)\in\mathcal{A}_3, 0 \qquad\,\,\,\,\mbox{otherwise}.\end{cases}
(u, \psi_{0,0,h})_{L^2}=h^{-\frac{n}{4}}\int \psi(h^{-\frac{1}{2}}x)b(x)e^{i\phi(x)/h}dx &=h^{\frac{n}{4}}\int \psi(x)b(h^{\frac{1}{2}}x)e^{i\phi(h^{\frac{1}{2}}x)/h}dx=\mathcal{O}(h^{\infty}).
R_{1313}=\frac{{{e}^{q_3+3 q_2+2 q_1}}}{\Delta_1},R_{1213}=-\frac{{{e}^{2 q_3+2 q_2+2 q_1}}}{\Delta_1},R_{1223}=\frac{{{e}^{2 q_3+3 q_2+q_1}}}{\Delta_1},R_{1323}=-\frac{{{e}^{q_3+4 q_2+q_1}}}{\Delta_1},
R_\Omega:=\tilde{\tilde R}_\Omega +\tilde{\tilde K}_\Omega\tilde{\tilde R}_\Omega+\tilde{\tilde K}_\Omega\underline{R}\tilde{\tilde K}_\Omega\quad\mbox{and}K_\Omega:=\tilde{\tilde K}_\Omega{\mathcal{P}}_{{\rm{\sf N}}(D_{{}^\Vert})}\tilde{\tilde K}_\Omega,
\langle 0 p_{1}p_{2}p_{3}p_{4}\rangle ={\frac{\xi_{1 2 3}^{2}\xi_{1 2 4}^{2}}{( 1 2 )^{2}}}{\cal F}^{p_{1}- 2 \vert p_{2}- 2 \vert p_{3}- 2 \vert p_{4}- 2}( x , u ) + O ( \theta^{5}\bar{\theta}) .
N = O O^{*}= c^{2}t^{2}+ \frac{\hbar^{2}}{P_{x}^{2}}+ \frac{\hbar^{2}}{P_{y}^{2}}+ \frac{\hbar^{2}}{P_{z}^{2}}- x^{2}- y^{2}- z^{2}- \frac{c^{2}\hbar^{2}}{E^{2}}~ ,
\frac{d}{dt}\mathfrak{M}_{\mathbf{X}}(t,\mathbf{z})= \mathfrak{H}_{\mathbf{X}}(\mathfrak{M}_{\mathbf{X}}(t, \mathbf{z})),\mathfrak{M}_{\mathbf{X}}(0,\mathbf{z})=\mathbf{z}.
\overline{R}^k_{ijl}:= \delta^{(c)}_i \pi \cdot (\overline{\Gamma}^{k(c)}_{jl})^{p^c}-\delta^{(c)}_j \pi \cdot (\overline{\Gamma}^{k(c)}_{il})^{p^c}-\overline{\Gamma}_{i\star j,l}^{k(2c)}+\overline{\Gamma}_{j\star i,l}^{k(2c)}\in k.
{\cal D}( s , \nu ,{\frac{\theta}{2 \pi}}) ={\frac{\sqrt{\pi}}{\Gamma ( s )}}\left[{\frac{\Gamma ( s -{\frac{1}{2}})}{\nu^{2 s - 1}}}+ 4 \sum_{m = 1}^{+ \infty}\cos ( m \theta ) \left({\frac{m \pi}{\nu}}\right)^{s -{\frac{1}{2}}}K_{s -{\frac{1}{2}}}( 2 m \pi \nu ) \right] \; ,
Z(x,y,t) :=\left\langle\nabla \log u (y,t),\gamma'\left(\tfrac{d}{2}\right)\right\rangle -\left\langle\nabla \log u (x,t),\gamma'\left(-\tfrac{d}{2}\right)\right\rangle -2\psi\left(\frac{d(x,y)}{2},t\right),
\lim_{n\to\infty}\frac{1}{b_c^{(n)}}\log P\left(\frac{n-A_n^*}{f_2(n)}\leq\varepsilon\right) =\lim_{n\to\infty}\frac{1}{b_c^{(n)}}\log P(B_4^{(n)}\cap B_5^{(n)}) &=\lim_{n\to\infty}\frac{1}{b_c^{(n)}}\log P(B_5^{(n)})=-H(\ell_2\varepsilon),
\left. \begin{array}{l l l}{W}&{=}&{\lambda \wp ( z - a ) ,}\\{E}&{=}&{8 | \lambda |^{2}\frac{| \wp ( z - a ) | | \wp^{2}( z - a ) - \wp^{2}( L / 2 ) |}{[ 1 + | \lambda \wp ( z - a ) |^{2}]^{2}}.}\\ \end{array}\right.
\rho_{t}(\varphi) \, =\, \pi_{0}(\varphi) + \int^{t}_{0}\rho_{s,2}( \varphi b){\mathrm d}X_{s}+ \int^{t}_{0}\rho_{s,3}( \widetilde{\mathcal A}_{s}\varphi){\mathrm d}s\, , 0 \le t \le T \, ,
\frac{z-w}{qz-q^{-1}w}\ell^-_{1j}(z)\ell^-_{jj}(w)+\frac{(q-q^{-1})z}{qz-q^{-1}w}\ell^-_{jj}(z)\ell^-_{1j}(w)= \ell^-_{jj}(w)\ell^-_{1j}(z).
\int_{X_j}|u_{\epsilon,j}|^2_{h}e^{v_\epsilon(\Psi)-\delta \max\{\psi+T,2\log|F|\}}c(-v_\epsilon(\Psi))\le\int_{X_j}v''_{\epsilon}(\Psi)|fF|^2_h e^{-u(-v_{\epsilon}(\Psi))}<+\infty.
\left( \frac{2 i \, \rho}{z - \bar{z}}\right)^{2 \sigma}\, \int \, d \mu ( G ) \, ( ( a^{2}+ b^{2}+ c^{2}+ d^{2}+ 2 ) - \nu^{2}( c^{2}+ d^{2}) )^{- 2 \sigma}.
R^\psi_{[a,b]}=P(T,\psi([0,b]))-P(T,\psi([0,a))), R^\psi_{(a,b]}=P(T,\psi([0,b]))-P(T,\psi([0,a])), R^\psi_{(a,b)}=P(T,\psi([0,b)))-P(T,\psi([0,a])), R^\psi_{[a,b)}=P(T,\psi([0,b)))-P(T,\psi([0,a))).
d \mathbf{P}_{N, \beta}(X_{N}) = \frac{1}{\widetilde{K}_{N, \beta}}\exp \left( - N^{2}\beta{\rm F}_{N}(X_{N}, \mu_{V}) + N\log \omega_{N}\right) \Pi_{i=1}^{N}\rho_{N}(x_{i}) d X_{N}.
\Psi_m(x,t) := \sum_{k=1}^m d^k_m(t) \Phi_k(x) &=\frac{1}{\sqrt{N}}\sum_{k=1}^{m}\begin{pmatrix}d_{m,1}^k(t) 0 &\ddots & d_{m,j}^k(t) & \ddots 0 d_{m,N}^k(t)\end{pmatrix}\begin{pmatrix}\phi_{k}(x) \vdots \phi_{k}(x) \vdots \phi_{k}(x)\end{pmatrix},