harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathrm{d}p_{Z}(t)(\mathbf{z}) = \mathrm{e}^{-\frac{|\mathbf{z}|^2}{2t}}\frac{\mathrm{d}^{2n}z}{(2\pi\,t)^n}\;,\mathbf{z}\in\mathbb{R}^{2n}\;,
\Omega_{n}= \sqrt{1 + \frac{m^{2}\alpha^{2}}{n^{2}l^{2}}}, a_{n}^{R}= ( a_{- n}^{R})^{\dag}, \tilde{a}_{n}^{R}= ( \tilde{a}_{- n}^{R})^{\dag}, \tilde{A}_{0}^{R}= ( A_{0}^{R})^{\dag}.
\begin{gathered}(Me^{-\widetilde q})(z,w) \overset{}{=}\int_{\mathbb R^k_+}\int_{\mathbb R^m_+}x^{z-I}y^{w-I}\exp(-q(x,\phi(y)) \, dx\,dy = \int_{\mathbb R^k_+}\int_0^\infty x^{z-I}t^{s-1}\exp(-q(x,t))\,dx\,dt \, (R_\phi y^{w-I})(I), \end{gathered}
\varphi(s) = (-1)^{(h-h_{0})/2}\left(\frac{\Gamma(1-s)}{\Gamma(s)}\right)^{h}\left(\frac{A}{\pi^{h}}\right)^{1-2s}\sideset{}{^{\star}}\prod_{\psi}\frac{L(2-2s, \overline{\psi})}{L(2s, \psi)}.
\mathcal{A}_T =\left\{\theta \in L^{\infty}\left(0,T; L^{1}(\mathbb{T})\right): \sup_{0\leq t<T}\mathfrak{E}(\theta(t)) + \int^{T}_{0}\left\|\Lambda^{1/2}\theta(t)\right\|^{2}_{L^{2}}dt<\infty \right\}.
\frac{d^{m_j}z^{m_j+q+k}f_j}{dz^{m_j}}(z) = CG\left(z,\frac{d^lz^{m_j+k}f_j(z)}{dz^l},\;1\leq j\leq r,\; 0\leq l\leq m_j\right).
\sigma=[\begin{matrix}-p^{*}+2(C_{1}-C_{2})+2C_{1}\gamma^{2}&2(C_{1}+C_{2})\gamma&0\\ 2(C_{1}+C_{2})\gamma&-p^{*}+2(C_{1}-C_{2})-2C_{2}\gamma^{2}&0\\ 0&0&-p^{*}+2(C_{1}-C_{2})\end{matrix}]
\bigl ( u_{n}\cdot \epsilon_{\Omega}^{U_{n - 1}G}( u_{n - 1}^{\prime}\cdot u_{n - 1}) \cdot \ldots \cdot \epsilon_{\Omega}^{U_{1}G}( u_{1}^{\prime}\cdot u_{1}) , e , \ldots , e \bigr ) .
\lim\limits_{N \rightarrow \infty}\mathbb{P}\Big(\Gamma_{1}< N^{1+\delta}\Big) = \lim\limits_{N \rightarrow \infty}1 - \mathbb{P}\Big(\Gamma_{1}\geq N^{1+\delta}\Big) = \lim\limits_{N \rightarrow \infty}1 - \Big(1 - \frac{1}{N}\Big)^{N^{1 + \delta}}= 1.
[\hat{E},\hat{x}]=-i\frac{\hbar}{m}\hat{p},\,\,\,\,\,\,[\hat{E},\hat{p}]=im\omega^{2}\hbar\hat{x},\,\,\,\,\,\,[\hat{x},\hat{p}]=i\hbar(1+\frac{1}{mc^{2}}\hat{E})\, ,
\left( \begin{array}{cc}a_1 b_1 c_1 d_1 \end{array}\right) \circ\left(\begin{array}{cc}a_2 b_2 c_2 d_2 \end{array}\right) =\left( \begin{array}{cccc}0 a_1 0 b_1 a_2 0 b_2 0 0 c_1 0 d_1 c_2 0 d_2 0 \end{array}\right)\,.
F\left({\tau - \hat \theta_{{\rm{MLE}}}^{(u)}}\right) = 1 - \underbrace{\frac{1}{{\left({M + K}\right)N}}\sum\limits_{j \in{\cal N}}{\left({\sum\limits_{m = 1}^M{u_j^{(m)}}+ \sum\limits_{i = 1}^K{u_j^{(i)}}}\right)}}_{\buildrel \Delta \over = S}
({\cal C}_{m}^{k})_{A 2}= ({\cal C}_{m}^{k})_{A 3}= ({\cal C}_{m}^{k})_{A 4}- 2 ({\cal C}_{m}^{k})_{A 5}= - ({\cal C}_{m}^{k})_{A 1}\equiv - c_{m}^{k}\; .
\tilde{Q}^n_i(\tau^n)-\tilde{Q}^n_i(\sigma^n) =y^n_i(\tau^n-\sigma^n)+ \tilde{A}^n_i(\tau^n)-\tilde{A}^n_i(\sigma^n)-(\tilde{S}^n_i(T^n_i(\tau^n))-\tilde{S}^n_i(T^n_i(\sigma^n))) &+\frac{\mu^n_i}{n}\frac{\sqrt{n}}{b_n}\rho_i(\tau^n-\sigma^n)\,,
W_{+}(z)=W_{-}(z) \begin{pmatrix}1 0 e^{\pi i \alpha}e^{-2\sqrt{ax}g(z)}1\end{pmatrix}, z\in\Sigma_1, &W_{+}(z) = W_{-}(z) \begin{pmatrix}0 1 -1 0\end{pmatrix}, z\in \Sigma_2, &W_{+}(z)=W_{-}(z)\begin{pmatrix}1 0 e^{-\pi i \alpha}e^{-2\sqrt{ax}g(z)}1\end{pmatrix}, z\in\Sigma_3, &W_{+}(z)=W_{-}(z) e^{\pi i \alpha \sigma_3}, z\in\Sigma_4.
\left( \xi ,\eta \right) \mapsto \left( \begin{array}{cc}x z y w\end{array}\right) \left( \begin{array}{cc}1 0 0 -1\end{array}\right) \left( \begin{array}{cc}w -z -y x\end{array}\right) =\left( \begin{array}{cc}wx+yz -2xz 2yw -wx-yz\end{array}\right) .
\frac{\#\mathcal{A}(x)}{W}\sum_{\mathbf{r}\in\mathcal{D}_k}\frac{y_{\mathbf{r}}^2\phi(r)}{\phi_\omega(r)^2}\sum_{A|r}\prod_{p|A}\frac{-(\omega(p)-1)}{p-1}=\frac{\#\mathcal{A}(x)}{W}\sum_{\mathbf{r}\in\mathcal{D}_k}\frac{y_{\mathbf{r}}^2}{\phi_\omega(r)}.
\langle Bf,f\rangle =\frac{1}{(2\pi)^2}\Bigl\langle\sum_{k=-\infty}^\infty q_kf_ke^{ik\theta}, \sum_{k=-\infty}^\infty f_ke^{ik\theta}\Bigr\rangle =\frac{1}{2\pi}\sum_{k=-\infty}^\infty q_k|f_k|^2\geq 0.
4sV_{\infty}\sum_{n=1}^{\infty}A_{n}sin(n\theta)=\frac{1}{2}V_{\infty}cC_{l_{\alpha}}[\alpha_{\infty}+\alpha_{geo}-\alpha_{0}-\sum_{n=1}^{\infty}\frac{nA_{n}sin(n\theta)}{sin(\theta)}](10)
\dot{\mathrm{H}}^{\alpha,p}(\mathbb{R},\mathrm{L}^p(\mathbb{R}^{n-1})) := \left\{\,u\in\mathrm{L}^{r}(\mathbb{R},\mathrm{L}^p(\mathbb{R}^{n-1}))\,\big|\, (-\partial_{x_n}^2)^{\frac{\alpha}{2}}u\in\mathrm{L}^{p}(\mathbb{R},\mathrm{L}^p(\mathbb{R}^{n-1}))=\mathrm{L}^p(\mathbb{R}^{n})\,\right\}
\frac{\varphi_{[a,b]}'(s)}{|\partial B_1|}= - \frac{\xi_{[a,b]}'(s)\zeta_{[a,b]}(s)+\xi_{[a,b]}(s)\zeta_{[a,b]}'(s)}{\xi_{[a,b]}(s)^2\zeta_{[a,b]}(s)^2}& = - s^{N-1}\left\{\left(\frac{\xi_{[a,b]}'(s)}{\xi_{[a,b]}(s)}\right)^2 -\left(\frac{\zeta_{[a,b]}'(s)}{\zeta_{[a,b]}(s)}\right)^2\right\}.
R_{j,x}^{s}=\sum_{k \in \Psi}\sum_{i \in \Omega_j}\log_2 \left(1+\frac{(|H_i^{pp}| \sqrt{T_i}+|H_{k,i}^{sp}| \sqrt{\alpha_k P_{x,i}})^2}{N_o+|H_{k,i}^{sp}|^2 (1-\alpha_k) P_{x,i}}\right)
\begin{bmatrix}A_{11}\cdots A_{1p}\vdots \ddots \vdots A_{p1}\cdots A_{pp}\end{bmatrix}\odot\begin{bmatrix}B_{11}\cdots B_{1p}\vdots \ddots \vdots B_{p1}\cdots B_{pp}\end{bmatrix}:=\begin{bmatrix}A_{11}B_{11}\cdots A_{1p}B_{1p}\vdots \ddots \vdots A_{p1}B_{p1}\cdots A_{pp}B_{pp}\end{bmatrix}.
\log \{Y_{n}^{3}f_{n}( S ) H_{n}( T ) \}= - 1 \Longrightarrow Y_{n}^{3}= e^{- 1}e^{- \frac{3 2 \pi^{2}k_{n}S}{c_{n}}}\frac{1}{H_{n}^{i}( T )}, \; i = 1 , 2 .
\exp \left[ - \frac{i}{2}\int d^{d + 1}X \hat{F}_{I}A_{I J}\hat{F}_{J}\right] = \int \left[ \prod_{X > 0}\left[ \det ~ A_{I J}( X ) \right]^{- 1}\prod_{I}{\cal D}\hat{\tilde{F}}_{I}( X ) \right]
|G^\alpha_R(x)|^2 =\left|\frac{1}{R}\int^R_0(B^{\alpha+1}_r(x)-B^{\alpha}_r(x))dr\right|^2 &\leq \int^R_0|B^{\alpha+1}_r(x)-B^{\alpha}_r(x)|^2\frac{dr}{R}&\leq \int^\infty_0|B^{\alpha+1}_r(x)-B^{\alpha}_r(x)|^2\frac{dr}{r}=:(G^\alpha(x))^2.
I_{n o n - p l a n a r}^{T = 0}= \frac{g}{2}\int \frac{d^{2}p}{( 2 \pi )^{2}}\frac{d^{2}k}{( 2 \pi )^{2}}\frac{e^{i \tilde{p}k}}{e^{\beta ( \frac{k^{2}}{2}- \mu )}- 1}= \frac{g}{8 \pi^{2}\theta^{2}}\frac{1}{e^{- \frac{\mu}{T}}- 1}.
m_{P_i,S}(P') =\frac{1}{d}\sum_{j=1}^d m_{P_i,S}(P_j')=\frac{1}{d}\sum_{j=1}^d m_{P_j',S}(P_i)=\frac{1}{d}m_{\phi^*(\phi(P')),S}(P_i)+O(1) &=\frac{1}{d}m_{\phi(P'),S}(\phi(P_i))+O(1)=\frac{1}{d}m_{\phi(P'),S}(Q_j)+O(1) &=\frac{1}{d}m_{Q_j,S}(\phi(P'))+O(1).
c_2(2n-1,k)=\sum_{i=k}^{2n-1}{i-1\choose k-1}c_1(2n,i)=\sum_{j=\lceil\frac{k+1}{2}\rceil}^{n}{2j-2\choose k-1}c_1(2n-1,2j-1) =\sum_{j=\lceil\frac{k+1}{2}\rceil}^{n}a^{n-j}{2j-2\choose k-1}{n+j-2\choose n-j}.
u^*\underbrace{[D,u][D,u^*]\cdots[D,u]}_{k=2n+1}\;=\; (-1)^n u^*[D,u]u^*[D,u]u^*\cdots u^*[D,u] \;=\; (-1)^n \left( u^*[D,u]\right)^k\;.
H_{W}^{+}( a m e^{i \pi / 2}) \, H_{W}^{-}( a m e^{i \pi / 2}) = \frac{4}{\pi^{2}}e^{- i \pi W}K_{W}^{2}( a m ) - \frac{4 i}{\pi}I_{W}( a m ) K_{W}( a m ) ,
\beta |f'(z^m)| +(1-\beta)|f(z^m)| + \sum_{n=1}^{\infty}|a_n|\phi_{n}(r) &\leq \beta f'_{0}(r^m)+ (1-\beta)f_{0}(r^m) +\sum_{n=1}^{\infty}M(n) \phi_{n}(r) & \leq d(0, \partial{\Omega}),
(\alpha^{d,n-i}_{max}, \alpha_i) = (\alpha_{i+1}, \alpha_i) = -1, (\alpha^{d,n-i+2}_{max}, \alpha_i) = (\alpha_{i-1}+ 2\alpha_i + 2\alpha_{i+1}, \alpha_i) = -1+4-2 = 1, \alpha^{d,n-i+2}_{max}- \alpha^{d,n-i}_{max}= \alpha_{i-1}+ 2\alpha_i + \alpha_{i+1}.
\sum_{\substack{\nu=(1^{\gamma_{1}}2^{\gamma_{2}}\ldots) \mu\preceq\nu\preceq\lambda}}(-1)^{\ell(\nu)+\ell(\lambda)}f^{\mu}_{\nu}(k+|\lambda|)\frac{(\ell(\lambda)-\ell(\nu)+|\lambda|-|\nu|)!}{(|\lambda|-|\nu|)!\prod_{i}(\alpha_{i}-\gamma_{i})!}=\frac{k!}{(k-\ell(\lambda)+\ell(\mu))!\prod_{i}(\alpha_{i}-\beta_{i})!}
Z=\begin{pmatrix}Z_1^{(1)}Z_2^{(1)}Z_3^{(1)}Z_1^{(2)}Z_2^{(2)}\end{pmatrix}=\begin{pmatrix}H+2\pi^* L H+3\pi^* L \pi^*S f_1^*(H+3\pi^* L)-E_1 E_1 \end{pmatrix}.
e\left( \bigoplus_j (R^1\pi_*L_j |_{\hat C})^\vee\right)=\prod_{j=1}^N \prod_{\substack{0\leq b<q_j+\langle q_j r \rangle+\sum_i\langle q_j(\vec l_0)_i \rangle \langle b \rangle = \langle q_j+q_j r+\sum_i q_j(\vec l_0)_i \rangle}}bz.
m<0 \; \;\;\;\psi(x)= \sum^{\infty}_{n=1}\int^{+\infty}_{-\infty}d p \; a_{np}\psi^{(+)}_{np}(x) + \sum^{\infty}_{n=0}\int^{+\infty}_{-\infty}d p \; b^{\dag}_{np}\psi^{(-)}_{np}\; ,
\frac{\partial}{\partial y}\left(y \alpha(y,t)\right)+\frac{1}{4}\frac{\partial^{2}}{\partial y^{2}}\alpha(y,t)-\frac{\partial}{\partial t}\alpha(y,t) = -\frac{1}{8}\frac{\partial^{3}}{\partial y^{3}}p_{0}(y,t)
\langle \tilde{P}(u),\tilde{Q}(v)\rangle = \langle P(\psi(u_1),...,\psi(u_N)),Q(\psi(v_1),...,\psi(v_N))\rangle &= \dot{\phi}(d(\psi(u_1),\psi(v_1))+\cdots + d(\psi(u_N),\psi(v_N))) &= \dot{\phi}\left(\tfrac{1}{2}d(u_1,v_1)+\cdots + \tfrac{1}{2}d(u_N,v_N)\right) &= \dot{\phi}\left(\tfrac{1}{2}d(u,v)\right).
\sigma_{pBf}(T) :=\{\lambda \in \mathbb{C}: \lambda I-T \thinspace \mbox{is not pseudo B-Fredholm}\}, \sigma_{pBw}(T) :=\{\lambda \in \mathbb{C}: \lambda I-T \thinspace \mbox{is not pseudo B-Weyl}\}, \thinspace \mbox{respectively.}
G_{\ast}^{N = 3}= [ G_{\ast , 8}, G_{\ast , 3}] \simeq [ 0 . 2 2 5 , 0 . 3 8 5 ] \quad \Rightarrow \quad \alpha_{\ast}^{N = 3}\simeq [ 5 . 7 , 9 . 7 ] .
U^-_M = \Big( U \setminus \{u_2\}\Big) \cup \{q_i u_2, q_{s}^{-1}u_2, q_s q_i^{-M+1}u_2, q_i^{-1}q_i^{-M+1}u_2 \}W^-_M = W \cup \{q_i q_{s}u_2, q_i^{-M+1}u_2, q_i^{-1}q_s^{-1}q_i^{-M+1}u_2 \},
E^{{\rm (surf)}}(a)=\frac{4\xi -1}{2\pi a}\sum_{l=0}^{\infty}D_{l}\int_{ma}^{\infty}\frac{zdz}{\sqrt{z^{2}-m^{2}a^{2}}}\left[ 1-\left(\beta +\frac{n}{2}\right) \frac{\left( \tilde{I}_{\nu}(z)\tilde{K}_{\nu}(z)\right)^{\prime}}{z\tilde{I}_{\nu}^{\prime}(z)\tilde{K}_{\nu}^{\prime}(z)}\right] \;
\leq\dfrac{p}{N}\beta{\P}\left\{V_i < \dfrac{pi}{2}\right\}\sum_{j=m}^{i}b[i,k-1] \binom{i+k-2}{k-1}=\O \left( e^{-\varepsilon i}i^{-(k-1)\alpha}i^{k-1}i \right) =\o \left( \dfrac{1}{i}\sum_{j=m}^{i}j^{-\alpha \left(k-1\right)}\right),
\mathcal{H}(A_{\tau,n})\le \int_X \exp\left(\lambda \left (\beta \log |DS^{n+1}_\tau(x)|-\sum_{k=0}^{n-1}f_k^\tau(x)\right)\right)d \mathcal{H}(x) \le \sum_{v_1^n\in W_\tau^n}\mathcal{H}(g_{v_1^n}(X))\sup_{x\in g_{v_1^n}(X)}\exp\left(\lambda \left (\beta \log |DS^{n+1}_\tau(x)|-\sum_{k=0}^{n-1}f_k^\tau(x)\right)\right).
\limsup_{R\rightarrow\infty}\int_{B_{R}^{c}(0)}\frac{|u(x)-u(y)|^{p-1}}{|x-y|^{n+\sigma p}}dy \leq \limsup_{R\rightarrow\infty}\left(\frac{R}{R-|x|}\right)^{n+\sigma p}\int_{B_{R}^{c}(0)}\frac{C(p)(u^{p-1}(x)+u^{p-1}(y))}{|y|^{n+\sigma p}}dy=0,
C_{k}(j)=\big[ \underbrace{B\log_{2}\left(1+\zeta_{m,k}(j)\right)}_{C_{mk}(j)}-\underbrace{B\log_{2}\left(1+\zeta_{e,k}(j)\right)}_{C_{ek}(j)}\big]^+ &k=\{1,2,3,4\},
\dot{x}(t)=[\begin{matrix}0&0&0&-d_{4}\\ 1&0&0&-d_{3}\\ 0&1&0&-d_{2}\\ 0&0&1&-d_{1}\end{matrix}]x(t)+[\begin{matrix}n_{4}\\n_{3}\\n_{2}\\n_{1}\end{matrix}]u(t)
Q_{R}= \frac{1}{\sqrt{2 T_{2}U_{2}}}( n_{2}{\bar{U}}\tau + n_{1}\tau -{\bar{U}}l_{1}+ l_{2}) ,{\bar{Q}}_{R}= \frac{1}{\sqrt{2 T_{2}U_{2}}}( n_{2}U \tau + n_{1}\tau - U l_{1}+ l_{2}) .
\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) \le F_k(\alpha').
\begin{cases}u(x,t)=0,\,\,\,(x,t)\in\Gamma\times\mathbb{R}^+, &\frac{\partial \mu}{\partial \vec{n}}=0,\,\,\,(x,t)\in\Gamma\times\mathbb{R}^+, &\frac{\partial \phi}{\partial t}=\alpha\Delta_\Gamma\phi-\frac{\partial \phi}{\partial \vec{n}}-\beta\phi-g(\phi),\,\,\,(x,t)\in\Gamma\times\mathbb{R}^+\end{cases}
\left(\langle f_2,f_3\rangle,\langle f_3,f_1\rangle,\langle f_1,f_2\rangle\right)=(\delta_1e^{i\theta_1}||f_2||||f_3||,\delta_2e^{i\theta_2}||f_3||||f_1||,\delta_3e^{i\theta_3}||f_1||||f_2||).
\Psi_{\bf D}\rightarrow \exp{( \phi_{1}{\tilde{E}}_{1}- \theta_{2}{\tilde{E}}_{2}+ \phi_{3}{\tilde{E}}_{3}+ \theta_{1}{\tilde{F}}_{1}+ \phi_{2}{\tilde{F}}_{2}+ \theta_{3}{\tilde{F}}_{3})}\cdot \Psi_{\bf D},
h(t) - \alpha(g(t)) = \sum_{\theta=1}^d (h_{p\theta+p-1}-g_{p\theta+p-1}a^{p\theta})t^{p\theta+p-1}- \sum_{\theta=1}^d r_{\theta}(t) + f_0(t) = \sum_{\theta=1}^d (h_{p\theta+p-1}-g_{p\theta+p-1}a^{p\theta})t^{p\theta+p-1}+ f_{h,g}(t),
0 =\nabla f(\bar x)+\sum\limits_{i\in I^g(\bar x)}\lambda_i\nabla g_i(\bar x)+\sum\limits_{j\in\mathcal P}\rho_j\nabla h_j(\bar x) +\sum\limits_{l\in I^{\varphi_\textup{KS}}(\bar x)}\xi_l\bigl(H_l(\bar x)\nabla G_l(\bar x)+G_l(\bar x)\nabla H_l(\bar x)\bigr).
\mathrm{S}(1,1)=\{q^6, q^8, q^{10}, q^{12}\}.\qquad \mathrm{S}(2,1)=\{q^7, q^{11}\}. \mathrm{S}(1,2)=\{q^3, q^7\}.\qquad\qquad\qquad\quad\&\mathrm{S}(2,2)=\{q^2, q^6, q^{8}, q^{12}\}.
\frac \delta{\delta \tilde{j}_{n}\left( \mathbf{p}\right)}\equiv \frac 1{\left( 2 \pi \right)^{3}}\int d^{3}\mathbf{x \, e}^{i \mathbf{p \cdot x}}\frac \delta{\delta j_{n}\left( \mathbf{x}\right)}
J_{\lambda}N= \frac{1}{R}\left(\left(\frac{p}{2}x^{i}+\lambda \frac{\sqrt{\Delta}}{2}y^{i}\right),\left(\frac{p}{2}y^{i}+\lambda \frac{\sqrt{\Delta}}{2}x^{i}\right),\sigma_{p,q}z^{j}\right),
\bar{g}_{xy}=G_{IJ}\frac{\partial{X^{I}}}{\partial{{\phi}^{x}}}\frac{\partial{X^{J}}}{\partial{{\phi}^{y}}}=G_{xy}+G_{y0}\frac{\partial{X^{0}}}{\partial{X^{x}}}+G_{x0}\frac{\partial{X^{0}}}{\partial{X^{y}}}+G_{00}\frac{\partial{X^{0}}}{\partial{X^{x}}}\frac{\partial{X^{0}}}{\partial{X^{y}}}.
\mu_d\bigl(B_x(r_{n,u}) \cap B_y(r_{n,v})^c\bigr) \geq \mu_d\bigl(B_x(r_{n,v}) \cap B_y(r_{n,v})^c\bigr) &= V_d r_{n,v}^d \biggl\{1- I_{\frac{d+1}{2},\frac{1}{2}}\biggl(1-\frac{\|x-y\|^2}{4r_{n,v}^2}\biggr) \biggr\}\gtrsim \frac{k\|z\|}{nf(x)}.
Q\left( x\right) :=q_{\rho}\left( x,t\right) -q\left( x,t\right) & =2\left[ \int\psi_{\rho}\left( x,t;\mathrm{i}s\right) \psi\left(x,t;\mathrm{i}s\right) \mathrm{d}\rho_{t}\left( s\right) \right]^{2}+4\int\psi_{\rho}\left( x,t;\mathrm{i}s\right) \psi^{\prime}\left(x,t;\mathrm{i}s\right) \mathrm{d}\rho_{t}\left( s\right)
\mathrm{pr}^{i}_{(n+1)}(\bar{\pi}^{(n)})_{*}F^{i}_{H^{n+1}}(x) = F^{i}_{\bar{H}^{n}}(x), &\mathrm{pr}^{i}_{(n+1)}(\bar{\pi}^{(n)})_{*}(\bar{f}^{(n)})_{*}F^{i}_{H^{n+1}}(x) = F^{i}_{\bar{f}^{(n)}(\bar{H}^{n})}(x),
R_{12}(u-v)\hat l_{1}^{-1}T_{1}(u)\hat l_1\hat l_{2}^{-1}T_{2}(v)\hat l_2=\hat l_{2}^{-1}T_{2}(v)\hat l_2\hat l_{1}^{-1}T_{1}(u)\hat l_1R_{12}(u-v).\ll{trace3}
\begin{cases}-\lambda \big(\varphi_{k+1}(x)+\varphi_{k+1}(x-1)\big) + 4(\lambda+\gamma)\varphi_k(x)-\lambda\big(\varphi_{k-1}(x)+\varphi_{k-1}(x-1)\big)=0,\; k \geq 2, &(\lambda+2\gamma)\varphi_1(x) - \lambda \big(\varphi_2(x)+\varphi_2(x-1)\big) = \displaystyle \frac{\delta_0(x)}{\sqrt{m_0m_1}},\end{cases}
\lim_{\vec{p}\rightarrow 0}\, \Pi_{1}= \lim_{\vec{p}\rightarrow 0}\, \delta_{i j}\Pi_{i j}= 2 e^{2}\int{\frac{d^{2}k}{( 2 \pi )^{2}}}\,{\frac{1}{\omega_{k}}}\,{\frac{\partial}{\partial \omega_{k}}}\left({\frac{( \omega_{k}^{2}- m^{2}) n_{F}( \omega_{k})}{\omega_{k}}}\right) = 0 ,
T^2_sT_{\omega}T^2_sT_{\omega}^{-1}T^2_sT_{\omega}=T_{\omega}T^2_sT_{\omega}^{-1}T^2_sT_{\omega}T^2_s\quad\quad\mbox{if}d>1, T^2_sT_{\omega}^{-m}T^2_sT_{\omega}^m=T_{\omega}^{-m}T^2_sT_{\omega}^{m}T^2_s\quad\quad\mbox{for all}1<m<d.
( \frac{1}{4 \pi^{2}}) ( q^{\mu}q^{\nu}- g^{\mu \nu}q^{2}) \left\{\int_{0}^{1}d \alpha 2 \alpha ( 1 - \alpha ) l n \left[ \frac{\Lambda^{2}}{- \alpha ( 1 - \alpha ) q^{2}+ m^{2}}\right] - \frac{1}{3}\right\}.
T^{d}= P^\pi\sum_{j=0}^{n-1}P^j\left[\begin{array}{cc}0 0 0 (Z^d)^{j+1}\end{array}\right], X= \sum_{k=0}^{g-1}(P^{d})^{k+2}\left[\begin{array}{cc}0 0 CA^\pi 0 \end{array}\right]T^{k}T^{\pi},
\bar g_{i}= h_i(x+r_0 B y(t) ) \tilde g_i (t) = h_i(x+r_0 B y(t) ) \frac{(m_0-D_0) + (m_1-D_1) h - D_{y_1}h D_2 - D_{y_2}h D_3}{D_0 + h D_1 + D_{y_1}h D_2 + D_{y_2}h D_3}.
S'_{i_1, i_2,\dots,i_k}= [X'_{i_1}, S'_{i_2,\dots,i_k}] = [X_{i_1}+ Y_{i_1}, S_{i_2,\dots,i_k}+ \alpha_{i_2,\dots,i_k}] = [X_{i_1}, S_{i_2,\dots,i_k}]=S_{i_1, i_2,\dots,i_k}
\mbox [\hat{\Lambda}(k^{1}),\hat{\Lambda}(k^{\prime \, 1})]=[\hat{\Lambda}^{\dagger}(k^{1}),\hat{\Lambda}^{\dagger}(k^{\prime \,1})]=0;\,\,\,[\hat{\Lambda}(k^{1}),\hat{\Lambda}^{\dagger}(k^{\prime \,1})]=\delta(k^{1}- k^{\prime \, 1}),
\left|r_{j_k}v_k(x)\right|=\left|r_{j_k}T_k\circ \nabla u_{j_k}(x)\right|=\left|T_{r_{j_k}k}(r_{j_k}\nabla u_{j_k}(x))\right|\leq \left|T_k(r_{j_k}\nabla u_{j_k})\right|=\left|T_k(\nabla\zeta_{j_k})\right|.
\tilde{\mu}_{\scriptscriptstyle\mathcal{I}_{1}}: \mathcal{C}^{1,p}_{\ast}\longrightarrow (\mathcal{G}^{2,p})^{*}, ~~~~~~\langle \tilde{\mu}_{\scriptscriptstyle\mathcal{I}_{1}}, \gamma \rangle (q) = \frac{1}{2}\int_{\Sigma}\left(\ast F_{\mathsf{a}}\cdot \gamma - \langle \gamma, \mu_{1}\circ u\rangle \right)\omega_{\scriptscriptstyle\Sigma}
V ( s , t ) = \frac{1}{8}\{e^{-{\sqrt{2}}( 2 s +{\sqrt{3}}t )}}\ \left( 1 -{e^{2{\sqrt{2}}s}}\right)^{2}= \frac{1}{2}\e^{- \sqrt{6}t}\ \sinh^{2}( \sqrt{2}s ) \ .
\frac{\alpha_{3}( m_{\mathrm{s t r i n g}})}{\alpha_{a}( m_{\mathrm{s t r i n g}})}= | n_{a}- m_{a}R_{2}/ R_{1}| | \tilde{n}_{a}- \tilde{m}_{a}\tilde{R}_{2}/ \tilde{R}_{1}|
\left[ \Psi(x+v+Z),\Psi(y+w+Z')\right]_{\mathfrak J}&= \left[ s's(x)+s'i(v)+(s's)^{*}(Z),s's(y)+s'i(w)+(s's)^{*}(Z')\right]_{\mathfrak J}&=\Psi\left([x+v+Z,y+w+Z']_{\mathfrak d}\right).
c_{m,d}= 4^{1 - d/2}\frac{\Gamma(h_+) \Gamma(h_-)}{\Gamma(\frac{d}{2}) \Gamma(\frac{d}{2}-1)}\times \frac{\Gamma(\frac{d}{2})}{2(d-2) \pi^{d/2}}= \frac{\Gamma(h_+) \Gamma(h_-)}{(4 \pi)^{d/2}\Gamma(\frac{d}{2})}.
\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}}.
\prod_{i=1}^{k}u_i = \prod_{i=1}^{k}\left(\sum_{\chi \in \widehat{G/N}}\tau_{\chi}(u_i) \right) = \sum_{\chi_1,\ldots,\chi_{k}\in \widehat{G/N}}\tau_{\chi_1}(u_1) \cdots \tau_{\chi_{k}}(u_{k}).
(A-2I)y_{3}=(\begin{matrix}-1&0&0&0&0\\ 3&-1&0&0&0\\ 6&3&0&0&0\\ 10&6&3&0&0\\ 15&10&6&3&0\end{matrix})(\begin{matrix}0\\ 0\\ 1\\ -2\\ 0\end{matrix})=(\begin{matrix}0\\ 0\\ 0\\ 3\\ 0\end{matrix})=y_{2}.
\phi_{n,j}(y)=\frac{\rho_n}{m}(R(y,x_{n,j}^{(1)})-R(x_{n,j}^{(1)},x_{n,j}^{(1)}))+\frac{\rho_n}{m}\sum_{l\neq j}(G(y,x_{n,l}^{(1)})-G(x_{n,j}^{(1)},x_{n,l}^{(1)})),
(\mu\times\nu)(H') \gtrsim \mu(\mathbf{X}') \cdot \inf_{x\in \mathbf{X}'}|\mathcal{T}_x| \cdot \inf_{x\in \mathbf{X}', T\in\mathcal{T}'_x}\nu(T\cap B_T\cap Y_x) &\overset{\eqref{eq: eqfraction}-\eqref{form61}}{\gtrsim}r^{\eta}\cdot r^{-\sigma + 3\eta}\cdot r^{\sigma+3\eta}= r^{7\eta}.
\arctan_2\left(\tfrac{x}{y}\right)=\begin{cases}\arctan(\tfrac{x}{y}){\hbox{if}}x\geq 0, y>0 \arctan(\tfrac{x}{y})+\pi{\hbox{if}}x\geq 0, y<0 \tfrac{\pi}{2}x \geq 0, y=0\end{cases}
|\mathcal{G}(4)_{< 2^\mu}| \geq \sum_{}^{\mu}2^m\left(1-\frac{1}{\sqrt{2}}\right) -1 &=\sum_{\ell=0}^{\mu/2-2}2^{2\ell+4}\left(1-\frac{1}{\sqrt{2}}\right)-1 &=\frac{\mu}{2}-1 + 2^4\left(1-\frac{1}{\sqrt{2}}\right)\frac{4^{\mu/2-1}-1}{3}.
u \circ v = \sum_{s=0}^{\infty}{\kappa^{2s}\over 2^{2s}(2s+1)!}\omega^{i_1j_1}\ldots \omega^{i_{2s}j_{2s}}{\partial^{2s}u \over\partial x^{i_1}\ldots \partial x^{i_{2s}}}{\partial^{2s}v \over\partial x^{j_1}\ldots \partial x^{j_{2s}}}\,.
T_i f_{(\nu_1,\dots,\nu_i,\nu_{i+1},\dots,\nu_n)}=(t-1) f_{(\nu_1,\dots,\nu_i,\nu_{i+1},\dots,\nu_n)}+t f_{(\nu_1,\dots,\nu_{i+1},\nu_i,\dots,\nu_n)},\nu_i < \nu_{i+1}.
\begin{aligned}\big| \big| \nabla (e^{itA}f-f) \big| \big|_{L^2}^2\le |t|^2\sup_{|t|\le1}\big| \big| \nabla Ae^{itA}f \big| \big|_{L^2}^2\le C|t|^2 ||\nabla Af||_{L^2}^2,\end{aligned}
q(s)=\frac{-s.\Gamma(s)}{\frac{1}{x_0}.\underset{k=0}{\overset{\infty}{\sum}}\beta_{2.k+1}.\Gamma(s+2.k+1)-\frac{\Gamma(s)}{x_0}.\sum_{n=1}^{n=N}{\left(\frac{n}{x_0}\right)}^{-s}}&=s.(1-s)=\frac{1}{4}-\xi^2+Y_0^2-i.2.\xi .Y_0
z^{(l)}=(x^{(l)},s^{(l)},\lambda^{(l)},v^{(l)}), \Delta z =(\Delta x,\Delta s,\Delta \lambda,\Delta v), z^{i,(l)}=(x_{J_i}^{(l)},s^{i,(l)},\lambda^{i,(l)},v^{i,(l)}).
\zeta_{2}(k):=\left\vert D_{k}\right\vert +{\displaystyle\iint_{D_{k}}}\left\vert \nabla v\right\vert^{p}+{\displaystyle\iint_{D_{k}}}\!\left( \left\vert u\right\vert \!+\!\left\vert v\right\vert \right)^{2\theta}\rightarrow0
(1-c\mu_1)(1-c\lambda_1) = 1 \ \ \mathrm{or}\ \ (1-c\mu_1)(1-c\lambda_2) = 1 \ \ \mathrm{and}(1-c\mu_2)(1-c\lambda_1) = 1 \ \ \mathrm{or}\ \ (1-c\mu_2)(1-c\lambda_2) = 1.
H_n(s) := \pi^{-s}\Gamma(s)\Bigg(\zeta (\widetilde{\Delta}_n,s)-V_\alpha(s) \widetilde a(s)n^{2-2s}\Bigg)   = \pi^{-s}\Gamma(s) \Bigg( \zeta(\Delta,s) + \frac{s\pi^2}{3}\zeta(\Delta,s-1) + O\Bigl(n^{-4}\Bigr) \Bigg).
\frac{4n}{(1-z)^{N}(1+z)^{M}}\frac{d}{dz}B ( z ) = \big( 1-z^{2}\big) P_{n}^{(N,M)}(z) \frac{d}{dz}P_{n-1}^{(N+1,M+1)}(z){}+\big( 1-z^{2}\big) \frac{N+M+n+1}{2}\big( P_{n-1}^{( N+1,M+1)}(z) \big)^{2}{}+ [ (M-N)-z ( N+M+2 ) ] P_{n}^{(N,M)}(z) P_{n-1}^{(N+1,M+1)}(z) ,
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}}\geq \frac{r!}{\alpha^{\frac{1}{r}}}\sum_{e\in E(H)}\prod_{v\in e}(B^{\frac{1}{r}}(v,e)x^*_{v}) = \frac{r!}{\alpha^{\frac{1}{r}}}\sum_{e\in E(H)}\frac{\sum_{v\in e}(B(v,e)(x^*_{v})^{r})}{r}\geq \frac{(r-1)!}{\alpha^{\frac{1}{r}}}\|x^*\|_{r}^r.
\psi\lambda |_{S_3}= 1, \psi\delta_0 |_{S_3}= 12, \psi\delta_{2,1}|_{S_3}= -1, \lambda\delta_0 |_{S_3}= -2, \delta_0^2 |_{S_3}= -48, \delta_0\delta_{2,1}|_{S_3}=2.
\big\|F_{n,\underline{k}}(f_{\underline{k}})\big\|_{L^2}\leq C^{n+1}(1+|\widetilde{k}_{n+1}-\widetilde{k}_n|)\cdot\|f_{\underline{k}}\|_{L^\infty}\|P'_{\widetilde{k}_{n+1}}h\|_{L^2}\prod_{l=1}^n2^{\widetilde{k}_l}\|P'_{\widetilde{k}_{l}}h\|_{L^\infty}.
\dfrac{m_{n}(X\cap B(0,\varepsilon))}{m_{n}(B(0,\varepsilon))}\geq(1+g_{\alpha}^{2})^{\frac{-n}{2}}\dfrac{m_{n}( B(0,\delta))}{m_{n}(B(0,\varepsilon))}&=(1+g_{\alpha}^{2})^{\frac{-n}{2}}\left(\dfrac{(R-r)\tan \alpha}{R \sin 2\alpha}\right)^{n}&=(1+g_{\alpha}^{2})^{\frac{-n}{2}}\left(\dfrac{R-r}{2R}\right)^{n}\cos^{-2n}\alpha,
[s, x_1] [x_2, x_3] + [s, x_2][x_1, x_3] = - [s, x_1] [x_3, x_2] - [s, x_2][x_3, x_1] = - [s x_3 , x_1, x_2] + [s , x_1, x_2] x_3 + s [x_3, x_1, x_2] .
\begin{aligned}\nu_c^{\pm}(Y_1,Y_2,\dots,Y_m) =\sum\limits_{j=2}^{m-1}\nu_c^{\pm}(Y_1,Y_j,Y_{j+1})=\sum\limits_{j=1}^{m-2}\nu_c^{\pm}(Y_j,Y_{j+1},Y_m)\end{aligned}
||\partial_{t}v_{5}||_{L^{2}(r dr)}+ ||\left(\partial_{r}+\frac{1}{r}\right)v_{5}||_{L^{2}(r dr)}\leq \int_{t}^{\infty}||N_{2}(f_{v_{5}})(x,r)||_{L^{2}(r dr)}dx &\leq C \frac{\log^{3}(t)}{t^{7/4}}
\lambda\|\nabla v\|_{L^2(\Omega)}^2 \leq \int_\Omega \nabla v^\top A_1\nabla v = -\int_\Omega\nabla u_2^\top(A_1-A_2)\nabla v &\leq \|A_1 - A_2\|_{L^\infty(\Omega)}\|\nabla u_2\|_{L^2(\Omega)}\|\nabla v\|_{L^2(\Omega)}.
f(t,m,x) \ = \ \int_0^{m-x}m \frac{2}{\sqrt{T-t}}\varphi\Big(\frac{z}{\sqrt{T-t}}\Big) dz + \int_{m-x}^\infty (z+x) \frac{2}{\sqrt{T-t}}\varphi\Big(\frac{z}{\sqrt{T-t}}\Big) dz.
(I_0^\delta(a))^2 \leq \frac{1}{4a}\int\limits_{0}^{2\pi}\int\limits_{\delta \sqrt{a}}^{\infty}\frac{d}{dr}\left[-\frac{1}{2}e^{-r^2}\right] dr d\phi =\frac{1}{4a}\int\limits_{0}^{2\pi}\frac{1}{2}e^{-\delta^2a}d\phi = \frac{\pi}{4a}(e^{-\frac{\delta^2a}{2}})^2.
\pi_{\mu \nu}= - \frac{1}{4}T_{\mu \alpha}T_{\nu}^{\; \alpha}+ \frac{1}{1 2}T T_{\mu \nu}+ \frac{1}{8}g_{\mu \nu}T_{\; \beta}^{\alpha}T_{\alpha}^{\; \beta}- \frac{1}{2 4}g_{\mu \nu}T^{2}
\left. \frac{d^{n}}{d\rho^{n}}W_{1}^{-1}(\cos(\theta+\varphi)|\rho,q)W_{1}^{-1}(\cos(\theta-\varphi)|\rho,q)\right\vert_{\rho=0}& =\frac{n!}{(q)_{n}}\sum_{m=0}^{n}\genfrac{[}{]}{0pt}{}{n}{m}_{q}h_{m}(\cos(\theta+\varphi)|q)h_{n-m}(\cos(\theta-\varphi)|q).
s^{\gamma(l)}_\alpha = -\sum_{m=0}^{l-1}s^{\beta(m)}_\alpha \theta^{\gamma(l-m+1)}(\hat{X}_\beta) - \sum_{m=0}^{l}r^{j(m)}_\alpha \theta^{\gamma(l-m+2)}(\hat{Z}_j), r^{i(l)}_\alpha = -\sum_{m=0}^{l-1}s^{\beta(m)}_\alpha \eta^{i(l-m+1)}(\hat{X}_\beta) - \sum_{m=0}^{l-1}r^{j(m)}_\alpha \eta^{i(l-m+2)}(\hat{Z}_j)
m_1(s) =\frac{1}{\sqrt{t_{j}-t_{j-1}}}, m_i(s) = \sqrt{\frac{2}{t_{j}-t_{j-1}}}\cos \left( \frac{(i-1) \pi s}{t_{j}-t_{j-1}}\right), i \geq 2, s \in [t_{j-1},t_{j}].
{A}=\left(\begin{array}{ccc}S^{N_h^0\times N_h^0}0 S^{N_h^0 \times N_h^\mathcal{B}}0 S^{N_h^0\times N_h^0}S^{N_h^0 \times N_h^\mathcal{B}}(S^{N_h^0 \times N_h^\mathcal{B}})^T (-S^{N_h^0 \times N_h^\mathcal{B}})^T S^{N_h^\mathcal{B}\times N_h^\mathcal{B}}-S^{N_h^B\times N_h^\mathcal{B}}\end{array}\right),
\sup_{f \in \mathcal{F}_{d,\theta}}\biggl| \mathbb{E}_f(\hat{H}_n) - H - \sum_{l=1}^{\lceil \beta/2 \rceil -1}\frac{\Gamma(k+2l/d) \Gamma(n)}{\Gamma(k) \Gamma(n+2l/d)}\lambda_l\biggr| = O \biggl( \max \biggl\{\frac{k^{\frac{\alpha}{\alpha+d}- \epsilon}}{n^{\frac{\alpha}{\alpha+d}- \epsilon}}\, , \, \frac{k^{\frac{\beta}{d}}}{n^{\frac{\beta}{d}}}\biggr\}\biggr)
S_{F P}= \int d^{4}x \biggl [ -{\frac{1}{4}}F_{\mu \nu}F^{\mu \nu}+{\bar{\psi}}( i \gamma^{\mu}D_{\mu}- m ) \psi -{\frac{1}{2 \xi}}( \partial_{\mu}A^{\mu})^{2}+ i{\bar{C}}\partial_{\mu}\partial^{\mu}C \biggr ] \, ,