harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^{p-2}\, \big(u(x)-u(y)\big)}{|x-y|^{N+s\,p}}\, \big(\varphi(x)-\varphi(y)\big)\, dx\, dy=\lambda^s_{p,q}(\Omega)\, \int_\Omega |u|^{q-2}\, u\, \varphi\, dx,
\mathbf{E}[u;x_{0},\sigma R_{0}] \leq c\Big[\Psi_{q}\Big(\frac{1}{\sigma^{n+2}}\Big(\mathbf{E}[u;x_{0},R_{0}]\Big)^{\frac{1}{2n-1}}\Big)+\sigma^{2}+ \frac{1}{\sigma^{nq}}\mathbf{E}[u;x_{0},R_{0}]^{q-1}\Big]\times \times \mathbf{E}[u;x_{0},R_{0}].
\langle \nabla u_{q,r}, \nabla \nu\rangle=\frac{1+2\delta}{(1+\delta)^2}\xi^2_{\tilde K}\langle\nabla w,\nabla w\rangle+(1+u_{q,r})^{2\delta+1}\langle \nabla u_{q,r},\nabla(\xi_{\tilde K})^2\rangle,
x\vee x =x, x\vee y =y\vee x, x\vee (y\vee z) =(x\vee y)\vee z, x\vee\bot =x, x\otimes k =x, (x\otimes u)\otimes v =x\otimes (u\otimes v), \bot\otimes u =\bot, (x\vee y)\otimes u =(x\otimes u)\vee(y\otimes u);
\sum_{n=0}^{\infty}\frac{\left(a;q\right)_{n}z^{n}}{\left(q,cq;q\right)_{n}}=\frac{\left(az;q\right)_{\infty}}{\left(z;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{n^{2}}\left(a;q\right)_{n}\left(cz\right)^{n}}{\left(q,cq,az;q\right)_{n}}
\mathbb{E}(\hat{H}_n) = \int_\mathcal{X}f(x) \int_0^\infty \log u \,dF_{n,x}(u) \,dx \approx \int_\mathcal{X}f(x) \int_0^\infty \log u \,dF_x(u) \,dx & = \int_{\mathcal{X}}f(x) \int_0^\infty \log\Bigl(\frac{te^{-\Psi(k)}}{f(x)}\Bigr) e^{-t}\frac{t^{k-1}}{(k-1)!}\, dt \, dx = H.
E_f =\dfrac{X}{\hat{H}^2}E, F_f =\dfrac{X}{\hat{H}^2}F, G_f =\dfrac{X}{\hat{H}^2}G, L_f =-\dfrac{1}{\hat{H}}\left(\lambda P+ED\right), M_f =-\dfrac{1}{\hat{H}}\left(\lambda Q+FD\right), N_f =-\dfrac{1}{\hat{H}}\left(\lambda R+GD\right).
\zeta_{R,2}(s) = \frac{\splitfrac{\bigg(2^{12}x^{13}-2^{12}x^{12}+ 2^{11}x^{12}+ 2^{10}x^{11}+ 2^9x^{10}- 2^7x^8 - 2^7x^7}{+ 2^7x^6 - 2^6x^7 - 2^6x^6 - 2^5x^6 - 2^4x^5 + 2^2x^3 + 2x^2 + 1\bigg)}}{(1-x)(1-2x^2)(1-2^4x^4)(1-2^3x^3)(1-2^5x^6)}.
\frac{1}{4}\widetilde{\omega}\left(f;2\sqrt{H((e_1-x)^2;x)}\right)\cdot\widetilde{\omega}\left(g;2\sqrt{H((e_1-x)^2;x)}\right) =\frac{1}{4}\cdot (2\sqrt{H((e_1-x)^2;x)})^2 &=H((e_1-x)^2;x).
l_{\xi}(\mathfrak{s}(t_{-\lambda}))=l_{\xi}(\mathfrak{s}(t_{-\lambda})\mathfrak{h}(h))=\psi(\left(\begin{smallmatrix}I_k u\varpi^{\lambda}&I_k\end{smallmatrix}\right))l_{\xi}(\mathfrak{s}(t_{-\lambda})).
\bigl(b_{km}x_{k-m}\bigr)(t-k) =\int_{[0,1)^c+k-m}n_{km}(t-k,\sigma-k+m)\,x_{k-m}(\sigma-k+m)\,d\sigma &=\int_{[0,1)^c+k-m}n_{km}(t-k,\sigma-k+m)\,x(\sigma)\,d\sigma,t\in[0,1)^c+k.
\frac{d t_1}{ds}= -\frac{\eta - a}{2}- \frac{1}{\sqrt{\Gamma}}\Bigg\{\sigma^2 [(-a+\eta)^2 - 1]s \quad+ \frac{1}{2}[(-a+\eta)(a^2 - 1) + 2\eta]\Bigg\}, \frac{d^2 t_1}{ds^2}= \frac{\sigma^2(2a\eta - a^2 + 1)^2}{\Gamma^{\frac{3}{2}}}\geq 0.
\|u(t,x)\|_p= 2 \left \| \psi(t,x)^{-1}\frac{\partial \psi}{\partial x}(t,x) \right\|_p \le 2 \left\| \psi(t,x)^{-1}\right\|_{2p}\left\| \frac{\partial \psi}{\partial x}(t,x) \right\|_{2p}.
(\lambda_{\widetilde{\tau}, 1}, \dots, \lambda_{\widetilde{\tau}, n}, \lambda_{\widetilde{\tau}c, 1}, \dots, \lambda_{\widetilde{\tau}c, n}) \leftrightarrow a_\tau = (-\lambda_{\widetilde{\tau}c, n}, \dots, -\lambda_{\widetilde{\tau}c, 1}, \lambda_{\widetilde{\tau}, 1}, \dots, \lambda_{\widetilde{\tau}, n}).
\lim_{m,n\to\infty}\int_{\textrm{spt}\, \zeta}V| u_m|^{p-1}|\zeta| \, |T(u_m-u_n)|\,dx \le \lim_{m,n\to\infty}\left(\int_{\textrm{spt}\, \zeta}V^{q/p}|u_m|^q\,dx\right)^{(p-1)/q}\times & \times \left(\int_{\textrm{spt}\, \zeta}V^{r}|T(u_m-u_n)|^{pr}\,dx\right)^{1/(pr)}= 0.
\epsilon^{i j}\nabla_{i}^{( p , k )}{\cal A}_{( p , k ) j}= - \sum_{( p^{\prime}, k^{\prime}) \neq ( p , k )}\beta_{p p^{\prime}}\delta^{2}({\bf r}_{k}^{( p )}-{\bf r}_{k^{\prime}}^{( p^{\prime})}) .
= - \bigg [ \frac{( d - 1 ) \beta + 4 \gamma}{4 ( d - 2 )}\bigg ]{\cal R}_{G B}^{2}+ \bigg ( \frac{d - 1}{d - 2}\bigg ) ( \beta / 4 + \gamma ) C^{2}+ \bigg [ \frac{4 d \alpha + ( d + 1 ) \beta + 4 \gamma}{4 d}\bigg ] R^{2}
|J| \lesssim c_{\alpha,r}\nu \ell^2\left[\left(1-2^{\frac\alpha \ell-1-\frac2r}\right)^{-l}+\left(1-2^{\alpha-1-\frac \alpha \ell-\frac2r}\right)^{-l}\right]\sum_{q> Q}\lambda_q^\alpha\|w_q\|_\ell^\ell &\lesssim c_{\alpha,r}\nu \ell^2\left(1-2^{\frac\alpha \ell-\frac2r}\right)^{-l}\sum_{q> Q}\lambda_q^\alpha\|w_q\|_\ell^\ell
H_{k}=\Theta_*(M_{[k+1]}), (S_1^{-1})_{k,l}=\Theta_*(M^{[1]}_{[k,l+1]})\Theta_*(M_{[l+1]})^{-1}, (S_2^{-1})_{k, l}=\big(\Theta_*(M_{[l+1]})^{-1}\Theta_*(M^{[2]}_{[k,l+1]})\big)^\top,
\mathrm{ad}_{L}^{\circ}(X)(\mathsf{N}_{m}^{n}) =\pi(K_{2\rho}S^{-1}(X_{(1)})K_{2\rho}^{-1}X_{(2)})\mathsf{N}_{m}^{n}\pi(S(X_{(3)})K_{2\rho}S^{-2}(X_{(4)})K_{2\rho}^{-1}).
E(G_{r,d})=\{\{x_{i},y_{j}\}\mid 1 \leq i \leq 2, 1 \leq j \leq d \}\cup \{\{x_{1},z_{1}\}, \{x_{2},z_{2r-3}\}\}&\cup \{\{z_{i},z_{i+1}\}\mid 1 \leq i \leq 2r-4 \}.
\frac{3}{2}F^{\prime\prime}-\frac{15}{2}\frac{K'}{K}F'+6\bigg[2\bigg(\frac{K'}{K}\bigg)^2-\frac{K^{\prime\prime}}{K}\bigg]F+\bigg[\sum_i\frac{\Lambda_i}{K}\delta(z-z_i)+(\phi'_0)^2\bigg]F+\phi'_0\varphi'=0,
\tilde g_{i0}^{+,j}(s,\tau,\hat y')=\sum_{k=1}^{n-1}g_{i0}^{+,k}(s,\tau,\beta^{(i)}(s,\tau^{(i)},\hat y'))\alpha_{jy_k}^{(i)}(s,\tau,\beta^{(i)})-\alpha_{j\tau}^{(i)}(s,\tau,\beta^{(i)})=0.
\hat U_{k+1}=\begin{cases}\left( \sqrt{a_k}+ \sqrt{b_k}\right)^2 \sqrt{\frac{a_k}{b_k}}\ge \frac{\alpha_k M_{21}\delta^2}{1-\alpha_k M_{21}}a_k+b_k+ a_k \frac{1-\alpha_k M_{21}}{\alpha_k M_{21}\delta^2}+b_k \frac{\alpha_k M_{21}\delta^2}{1-\alpha_kM_{21}}\end{cases}
\sum_{n=0}^{\infty}(-1)^n q^{2n}Ai_q(q^n x)= \frac{q(1-q^2)+x}{qx(1+q)}\,_1\phi_1 (0;-q^2;q,-x)-\frac{(1-q)}{x}Ai_q(\frac{x}{q}), & \sum_{n=0}^{\infty}q^{\frac{n(n-1)}{2}}x^n A_q(q^nx) = \frac{1-q}{q^2x}A_q(\frac{x}{q}) + \frac{1-q-x}{q^2x}A_q(qx).
\frac{r!}{[(r+1)!]^2}\sum^r_{i=0}\sum^\mu_{j=0}{\mu\choose j}{\nu\choose i-j}i!(r-i)! &\quad\times \int^\infty_0(\cosh t)^{2j-2r-4}F(1+r-i,1+r+s,r+2, \tanh^2 t)D(t)dt \|\phi\|^2.
g'(x) = s\left( - \frac{2}{(2x)^{s+1}}+ \left( \frac{1}{\left(2x-\frac{1}{4}\right)^{s+1}}+ \frac{1}{\left(2x+\frac{3}{4}\right)^{s+1}}\right) \right) & = s\left(\left( \frac{1}{\left(2x-\frac{1}{4}\right)^{s+1}}- \frac{1}{(2x)^{s+1}}\right) - \left( \frac{1}{(2x)^{s+1}}- \frac{1}{\left(2x+\frac{3}{4}\right)^{s+1}}\right)\right).
\int_0^t DG(X_r)d\mathbf{X}_r = \int_0^t DG(X_r)f(X_r)dr + \int_0^t DG(X_r)\sigma(X_r)d\mathbf{B}^H_r & = \int_0^t DG(X_r)f(X_r)dr + \mathbf{B}^H_t
D^{m}( e^{- 2 \phi}H_{m n p}) = 1 2 m A^{m}H_{m n p}e^{- 2 \phi}+ \frac{1}{2}G_{n p q r}F^{q r}+{\frac{1}{3 6 \cdot 4 8}}\epsilon_{n p q . . . r . . .}G^{q . . .}G^{r . . .}\ ,
\begin{aligned}\Pi_1[\lambda, \xi, \dot{\lambda}, \dot{\xi}, \phi](t) = \frac{t_0^{-\varepsilon}}{R^{a-2s}}\mu_0^{n+1-4s + \sigma}(t)f(t) & + \frac{t_0^{-\varepsilon}}{R^{a-2s}}\Theta\left[\dot{\lambda},\dot{\xi}, \mu_0^{n-4s}(t)\lambda, \mu_0^{n-4s}(\xi-q), \mu_0^{n+1-4s+\sigma}\phi\right](t)\end{aligned}
H_{1}=1+\sum_{a=1}^{l_{1}}\frac{Q_{a}^{(1)}}{|x-x_{a}^{(1)}|};~~ H_{2}=1+\sum_{b=1}^{l_{2}}\frac{Q_{b}^{(2)}}{|x-x_{b}^{(2)}|},~~|x-x_{a}|=(\sum_{i=1}^{3}|x_{_{i}}-x_{a_{i}}|^{2})^{1/2}.
\left\vert \frac{(n-1)e^{\frac{t_{2}}{n}}+e^{(1-n)\frac{t_{2}}{n}}-n}{n^{3}(e^{\frac{t_{2}}{n}}-1)^2 \left( e^{\frac{t_{1}+t_{2}}{n}}-1 \right)}t_{1}^{d(r)-1}t_{2}^{\overline{d(s)}-1}\right \vert\leq \frac{t_{2}-1+e^{-t_{2}}}{t_{2}^2(t_{2}+t_{1})}t_{1}^{h(r)-1}t_{2}^{\overline{h(s)}-1}=:g(t_{1},t_{2})
{\frac{\partial^2 \psi}{\partial k^2_y}}\; +\;{\frac{\partial^2 \psi}{\partial \kappa^2_y}}\; = \; 0\qquad\qquad\left({\frac{\partial \;}{\partial p_y}}{\frac{\partial \psi}{\partial \overline{p_y}}}\; = \; 0\right)
\begin{array}{c}B^-B^+-QB^+B^-~=~Q^{-N_B}B^-B^+-Q^{-1}B^+B^-~=~Q^{N_B}Q^{N_B}B^+Q^{-N_B}~=~QB^+ Q^{N_B}B^-Q^{-N_B}~=~Q^{-1}B^- Q^{N_B}Q^{-N_B}~=~Q^{-N_B}Q^{N_B}~=~1 \end{array}
S_{i j}^{a b}( x , y ) = S^{a b}( x ) \delta_{i j}\delta ( x - y ) , \ \{\cal M}_{i j}^{a b}( x , y ) = [ S^{T a c}( x ) S^{c b}( y ) + \delta^{a b}] G^{- 1}( x - y ) \delta_{i j}
\binom{m}{\ell_i}\binom{m}{\ell_j}=\sum_{k=0}^{\ell_i}\binom{m}{\ell_j}\binom{\ell_j}{k}\binom{m-\ell_j}{\ell_i-k}=\sum_{k=0}^{\ell_i}\binom{\ell_i+\ell_j-k}{k,\ell_i-k,\ell_j-k}\binom{m}{\ell_i+\ell_j-k}.
( A , B ) = \int d^{n}x \left( \frac{\delta^{R}{A}}{\delta \phi^{A}( x )}\frac{\delta^{L}{B}}{\delta \phi_{A}^{*}( x )}- \frac{\delta^{R}{A}}{\delta \phi_{A}^{*}( x )}\frac{\delta^{L}{B}}{\delta \phi^{A}( x )}\right) .
\cos_{p,q}^q{x}+\sin_{p,q}^q{x}=1, (\sin_{p,q}{x})'=\cos_{p,q}^{\frac{q}{p}}{x}, (\cos_{p,q}{x})'=-\sin_{p,q}^{q-1}{x}\cos_{p,q}^{1-\frac{q}{p^*}}{x}, (\cos_{p,q}^{\frac{q}{p^*}}{x})'=-\frac{q}{p^*}\sin_{p,q}^{q-1}{x}, (\tan_{p,q}{x})'=\cos_{p,q}^{-1-\frac{q}{p^*}}{x}.
\frac{\#\{1\leq i\leq n:\lambda_i=0\}}{n}=\frac{\#\{1\leq i\leq 2^{k}:\lambda_i=0\}}{n}&+\sum_{l=0}^{k-1}\frac{\#\{1 \leq i \leq \epsilon_{k-l-1}2^{k-l-1}:\lambda_i=0\}}{n}.
\gamma=\sum_{I=(\epsilon_1,i_1,\dots,\epsilon_s,i_s)\in\mathcal{I}}\omega_I(-1)^{s[\frac{|m|}{2}]}u_1^{\epsilon_1}v_1^{(p-1)i_1-\epsilon_1}\cdots u_s^{\epsilon_s}v_s^{(p-1)i_s-\epsilon_s}S_s(m)
\sum_{\mu\in \mathcal{P}_{\lambda_m}\cap \mathcal{R}_p}q^{|\mu|}=\frac{q^{|\lambda_m|}}{\prod_{i\geq 1}(1-q^{mi})^{m-1}}, \sum_{\mu\in\mathcal{P}_{\lambda_m}}t^{h_{m,0}(\mu)}q^{|\mu|}=\frac{q^{|\lambda_m|}}{\prod_{i\geq 1}(1-q^{mi})^{m-1}(1-tq^{mi})},
\hat{C}= C_{( 3 )}- \textstyle{\frac{1}{2}}\vec{A}_{( 1 )}^{T}\varepsilon \vec{A}_{( 2 )}+ \vec{A}_{( 2 )}^{T}\vec{\theta}+ \textstyle{\frac{1}{2}}A_{( 1 )}\vec{\theta}^{T}\varepsilon \vec{\theta},
Y_{l_{1}}^{m_{1}}(\Omega)Y_{l_{2}}^{m_{2}}(\Omega)=\sum_{L,M}\sqrt{\frac{(2l_{1}+1)(2l_{2}+1)}{4\pi(2L+1)}}\langle l_{1}0l_{2}0|L0\rangle\langle l_{1}m_{1}l_{2}m_{2}|LM\rangle Y_{L}^{M}(\Omega)
\lim_{t\rightarrow x^+}\int_{x-\delta}^{x}\frac{\varphi(y)dy}{t-y}-\int_{x-\delta}^{x}\frac{(x-y)\varphi(y)dy}{(t-y)^2}=\int_{x-\delta}^{x}\frac{\varphi(y)dy}{x-y}-\int_{x-\delta}^{x}\frac{\varphi(y)dy}{x-y}=0.
(\delta^{(s)})^2(c_k)+\frac{\sum_{ij}(\sum_l \Gamma^{k(s)}_{ij,l}c_l)(\delta^{(s)}c_i)^{\phi^s}(\delta^{(s)}c_j)^{\phi^s}}{(\sum_l r_lc_l)(\sum_l (\delta^{(s)}c_l)^{\phi^s})}=0,\ \k\in \{1,\ldots,n\}.
(-1)^{n}\int_{\mathcal{H}}\int_{\mathcal{H}}\|x-y\|^{2n}d\mu(x)d\mu(y) &= \sum_{l=0}^{\lfloor n/2 \rfloor}\binom{n}{2l}\binom{2l}{l}2^{n-2l}\int_{\mathcal{H}}\int_{\mathcal{H}}\langle x,y\rangle^{n-2l}\|x\|^{2l}\|y\|^{2l}d\mu(x)d\mu(y) \geq 0,
(\partial_t{\boldsymbol{u}}_h,{\boldsymbol{v}}_h) + A_G({\boldsymbol{u}}_h;\boldsymbol{\mathcal{U}}_h,{\boldsymbol{\mathcal{V}}}_h) + s_h({\boldsymbol{u}}_h;{\boldsymbol{u}}_h,{\boldsymbol{v}}_h) + t({\boldsymbol{u}}_h;{\boldsymbol{u}}_h,{\boldsymbol{v}}_h) = (\boldsymbol{f},{\boldsymbol{v}}_h)
2 \mathrm{Re}\langle u_{\alpha}\otimes R_*\,|\, H_y R_y \rangle = 2 \alpha\mathrm{Re}\langle u_{\alpha}\otimes R_*\,|\, V_y R_y \rangle &= 2 \alpha\mathrm{Re}\langle u_{\alpha}\otimes R_*\,|\, V_y \Phi_{\#}^y \rangle + 2 \alpha\mathrm{Re}\langle u_{\alpha}\otimes R_*\,|\, V_y R_y^{\#}\rangle,
H_{I}= \int d^{\nu}x \left[ \left( \mu^{2}( \phi_{0}) \, \phi_{0}-{\frac{\lambda}{3}}\phi_{0}^{3}\right) : \eta ( \vec{x}) : +{\frac{\lambda}{3 !}}\phi_{0}: \eta^{3}( \vec{x}) : +{\frac{\lambda}{4 !}}: \eta^{4}( \vec{x}) : \right] .
G^{2}=(\int_{-\infty}^{\infty}e^{-\frac{1}{2}x^{2}}dx)\cdot(\int_{-\infty}^{\infty}e^{-\frac{1}{2}y^{2}}dy)=2\pi\int_{0}^{\infty}re^{-\frac{1}{2}r^{2}}dr=2\pi\int_{0}^{\infty}e^{-w}dw=2\pi.
X_{i,j}:=\delta_{i,j+k}x_i\;,\Delta_{i,j}:=b_{i+k,j}\delta_{i,j}\;,M_{i,j}:=\delta_{i+1,j}\;, B_{i,j}:=b_i\delta_{i,j}:=-\delta_{i,j}(x_i+x_{i+1}+\cdots+x_{i+k})\;.
w(\pi_k^*(K)) =\tfrac{d}{dt}\big|_{t=0}\pi_k^*(K)(v_t) = \tfrac1{k!}\tfrac{d}{dt}\big|_{t=0}g(K,v_t^k) = \tfrac1{(k-1)!}g(K,w\cdot v^{k-1}) =\tfrac1{(k-1)!}g(w\lrcorner K,v^{k-1}),
G_{D, \epsilon}^{\delta}(X) := \inf_{\substack{P_{Y|X}: \Pr \{d(X, Y) > D \}\leq \epsilon}}H^{\delta}(Y) &= \inf_{\substack{P_{Y|X}: \Pr \{d(X, Y) > D \}\leq \epsilon}}~~\min_{\substack{{\cal W}\subset{\cal Y}: \Pr \{Y \in{\cal W}\}\geq 1-\delta}}\log |{\cal W}|,
|\psi_{n}\rangle = (A^{\dagger})^{n}|\psi_{0}\rangle, |\tilde{\psi}_{n}\rangle = A|\psi_{n}\rangle,|\tilde{\psi}_{0}\rangle=0, & H_{-}|\psi_{n}\rangle = E_{n}|\psi_{n}\rangle,\H_{+}|\tilde{\psi}_{n}\rangle = E_{n}|\tilde{\psi}_{n}\rangle.
\Bigg|\int_{t_0-\delta}^{t_0}h(t)\chi'_{\delta}(t)\,dt+h(t_0)\Bigg| =\Bigg|\int_{t_0-\delta}^{t_0}(h(t)-h(t_0))\chi'_{\delta}(t)\,dt\Bigg| &\leq\int_{t_0-\delta}^{t_0}|h(t)-h(t_0)||\chi'_{\delta}(t)|\,dt &\leq2\sigma,
\prod_{a}d \widetilde{A}_{a}^{M i}({\bf x}) = \left( \prod_{a}d A_{a}^{M i}({\bf x}) \right) \det \left\{{\frac{\partial \widetilde{A}_{b}^{M i}({\bf x})}{\partial A_{c}^{M i}({\bf x})}}\right\}.
\overline{\lim}_{t\to\infty}\frac{h(t)}{t}=\overline{\lim}_{t\to\infty}\frac{\tilde{h}(t-T)+l^*}{t}\leq\overline{\lim}_{t\to\infty}\frac{\xi(t)}{t}&=\overline{\lim}_{t\to\infty}\frac{(1-\epsilon)^{-2}\int^t_0\mu(v_\epsilon)_x(s,0)ds+L+\tilde{h}(\bar{T})}{t}&=(1-\epsilon)^{-2}\lim_{t\to\infty}\frac{\int^t_0\mu(v_\epsilon)_x(s,0)ds}{t}.
V_{m}\bar{\Phi}_{A}^{( n )}= \left( - \right)^{\varepsilon_{A}}\varepsilon_{m n p}\Phi_{A}^{* ( p )}, V_{m}\tilde{\Phi}_{A}= \left( - \right)^{\varepsilon_{A}+ 1}\bar{\Phi}_{A}^{( m )},
\Psi(v_i)v_i^* = \Psi(v_i)v_i^*p = \Psi(v_i)v_i^* (v'_1{v'_1}^*+\cdots+v'_n{v'_n}^*) &= \Psi(v_i)(v_i^*v'_1){v'_1}^*+\cdots+\Psi(v_i)(v_i^*v'_n){v'_n}^* &= \Psi(v_iv_i^*v'_1){v'_1}^*+\cdots+\Psi(v_iv_i^*v'_n){v'_n}^*,
X^{2}= b \left( \begin{array}{c c c c c}{\ddots}&{\ddots}&{\ddots}&{}&{}\\{\ddots}&{0}&{e^{i \omega t}}&{\ddots}&{}\\{\ddots}&{e^{- i \omega t}}&{0}&{e^{i \omega t}}&{\ddots}\\{\ddots}&{\ddots}&{e^{- i \omega t}}&{0}&{\ddots}\\{}&{}&{\ddots}&{\ddots}&{\ddots}\\ \end{array}\right)
\vartheta_2= X_{2,4}+X_{2,3}X_{2,4}+X_{2,2}X_{2,3}X_{2,4}+X_{2,1}X_{2,2}X_{2,3}X_{2,4}, \vartheta_5= \vartheta_{7-2}=X_{3,2}+X_{2,2}X_{3,2}+X_{1,2}X_{2,2}X_{3,2}.
J_{l}\left( \rho\right) :=\int_{\mathbb{R}}\frac{\varphi_{0}\left(\tau\right) \left( \frac{1}{2}-i\tau\right)^{l-1}}{\left( \tau-\rho\right)^{l+1}}d\tau =\sum_{j=0}^{l-1}C_{l-1}^{j}\left( \frac{1}{2}\right)^{j}\left(-i\right)^{l-1-j}\int_{\mathbb{R}}\frac{\varphi_{0}\left( \tau\right)\tau^{l-1-j}}{\left( \tau-\rho\right)^{l+1}}d\tau.
\forall x,y\in\Lambda, 0 \leq \big<\varphi_x;\varphi_y\big>^{h}_{A,g,\nu}\leq \big<\varphi_x;\varphi_y\big>^{0}_{A,g,\nu}= \big<\varphi_x\varphi_y\big>^{0}_{A,g,\nu}.
V(x)=\sup_{\theta \in\mathcal{A}}K(\theta) = \sup_{\theta \in\mathcal{A}}E_x\left[\int_{0}^{\infty}p\left(\theta_t\right) q(t,X_t)e^{-rt}\,dt-\int_{0}^{\infty}k_{t}e^{-rt}\,d\theta_{t}\right]
\begin{cases}x_0(t)=\begin{cases}x(t)\,, \qquad\qquad \hbox{when $0\le t\le T_k$}\,, x(T_k)\,, \qquad\qquad \hbox{when $T_k \le t\le T_{k+1}$}\,, \end{cases}& x_{n+1}(t)=F(t, x_n)\,, \qquad \hbox{for all $0\le t\le T_{k+1}$}\,, \end{cases}
M_{\varPsi}(2n,j,0;l-1)=(-1)^j\frac{\binom{2l}{l}\binom{2n}{n}\binom{2j}{j}\binom{2(n+l-j)}{n+l-j}\binom{2n-j}{n}}{\binom{n+l}{n}\binom{2n+l-j}{n}}
H^{(9)}_{2A,r}(\tau) :=(-1)^{r+1}H^{(9)}_{1A,r}(\tau), H^{(9)}_{2C,r}(\tau) :=(-1)^{r+1}H^{(9)}_{2B,r}(\tau), H^{(9)}_{6A,r}(\tau) :=(-1)^{r+1}H^{(9)}_{3A,r}(\tau).
\S_1 = \, \left\{(0, A_2 ,A_3 , \ldots, A_t ) \mid A_i \in \mathbf{M_1}\right\}, \mbox{and}\S_2 = \, \left\{(A_1, 0,A_3 , \ldots, A_t ) \mid A_i \in \mathbf{M_2}\right\}.
\nu = \sin^{- 1}( \sqrt{\frac{( b - d ) ( 1 / 2 - a )}{( a - d ) ( 1 / 2 - b )}}) q = \sqrt{\frac{( b - c ) ( a - d )}{( a - c ) ( b - d )}}\; \; \rho = \sqrt{( a - c ) ( b - d )}
= \frac{( a - 3 b )^{2}}{a ( a - 2 b )}\times \frac{a - 2 b}{( 3 a \times 2 b + a )}- \frac{1}{a}= \frac{( a - 3 b )^{2}}{a ( 9 b^{2}- a^{2})}- \frac{1}{a}
\max_{O \in \O}\|f_{O}\|_2^2 \lesssim_{\varepsilon}\Big(\prod_{i=1}^{m}r_{i}^{-1/2}D_{i}^{\delta}\Big)\Big(\prod_{i=1}^{\ell}r_{i}^{-1/2}\Big)\Big( \prod_{i = \ell+1}^{m+1}D_{i}^{-(n-i)}\Big) R^{O(\varepsilon_{\circ})}\|f\|_{L^{\infty}(B^{n-1})}^2
\tilde{\Lambda}^{(m)}_{\alpha, 1, 1}( \beta) = \sum_{k=E+1}^{E(\lambda)}\partial_\nu\phi^{(\alpha)}_k\Big\vert_{\partial B_R}\frac{1}{(\lambda^{(\alpha)}_k - \beta)^{m + 1}}&\times\Big(\int_{\partial B_R}\partial_\nu\phi^{(\alpha)}_k f{\rm d}s(y) + \int_{\partial B_R}\partial_\nu(\Delta\phi^{(\alpha)}_k)g{\rm d}s(y)\Big)
\begin{aligned}S_{\nu,\boldsymbol\mu}= O\Biggl(\!\binom{N}{\nu}\!\left(\frac{M}{eN}\right)^{2M}\frac{f_{k_1}(z_i)^Nf_{k_2}(z_o)^N}{z_i^M\,z_o^M}&\times\,\binom{M}{\boldsymbol\mu}\nu^{2\mu_1+\mu_{1,2}}(N-\nu)^{\mu_{1,2}+2\mu_2}\!\Biggr),\end{aligned}
\delta_{(\mathcal{X}_s,\mathcal{D}_s)}(\epsilon\mathcal{H}_s+f_s^*\mathcal{O}(1))=\lim_{l\to\infty}\delta_{lm_\epsilon,(\mathcal{X}_s,\mathcal{D}_s)}(\epsilon\mathcal{H}_s+f_s^*\mathcal{O}(1))\ge u-\delta_0
1+t^2 + t^3 + \cdots=1+\frac{t^{2}}{1-t}=\frac{1-t+t^{2}}{1-t}=\frac{1+t^{3}}{(1+t)(1-t)}=\frac{1-t^{6}}{\left( 1-t^{2}\right) \left(1-t^{3}\right)}\,.
\left|z+\dfrac{a_{n-1}}{a_n}-(t_1-t_2)\right|\leq \left(t_2+\left|\dfrac{a_{n-1}}{a_n}\right|\right)(\sin\alpha-\cos\alpha) &+\dfrac{2t_2|a_{k+1}|+2|a_k|}{|a_n|t_1^{n-k-1}}\cos\alpha+2\dfrac{\sin\alpha}{|a_n|}\sum\limits_{\nu=0}^{n-2}\dfrac{|a_{\nu}|}{t_1^{n-\nu-1}}.
b_{4s+3}= b_{2s+1}+b_{2s+2}= b_{2s+1}-\frac{1}{k}b_{s+1}\geq b_{2s+1}-\frac{1}{4}|b_{s+1}| > b_{2s+1}- \frac{1}{2}b_{2s+1}= \frac{1}{2}b_{2s+1}.
\eta(x,\pi,{\rm d}\theta) :=\left(\frac{c(x,\pi)}{r(x,\pi)}1_{\{r(x,\pi)>0\}}+1_{\{r(x,\pi)=0\}}\right)\delta_{0}({\rm d}y) &+\frac{\theta 1_{\{\theta\in (0,+\infty)\}}\Pi^{NL}(x,\pi,{\rm d}\theta)}{r(x,\pi)}1_{\{r(x,\pi)>0\}}.
\mathrm{e}^{\phi^{i}( x )}= \xi_{1}^{( i )}( x ) \bar{\xi}_{1}^{( i )}( x ) ,{}~ \psi_{+}^{i}( x ) = \frac{\xi_{1}^{( i )}( x ) \bar{\xi}_{2}^{( i )}( x )}{\xi_{1}^{( i )}( x ) \bar{\xi}_{1}^{( i )}( x )},{}~ \psi_{-}^{i}( x ) = \frac{\xi_{2}^{( i )}( x ) \bar{\xi}_{1}^{( i )}( x )}{\xi_{1}^{( i )}( x ) \bar{\xi}_{1}^{( i )}( x )}.
\Delta M(\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{b})=\mathbf{b\cdot n}_{12}\left\vert \mathbf{n}_{12}\cdot \mathbf{v}_{12}^{(+)}\right\vert\mathbf{v}_{12}^{(+)}\cdot \mathbf{b}-\left( \mathbf{b\cdot n}_{12}\right)^{2}\left( \mathbf{n}_{12}\cdot \mathbf{v}_{12}^{(+)}\right)^{2},
\log (\widetilde W n+ A) = \log (\widetilde W n'+ A) + \log \frac{\widetilde W (n' + n - n') + A}{\widetilde W n'+ A}&= \log (\widetilde W n'+ A) + \log \bigg(1 + O\bigg( \frac{1}{M(\log \log N)^{1/(2R)}}\bigg) \bigg) &= \log (\widetilde W n'+ A) + O\bigg( \frac{1}{M(\log \log N)^{1/(2R)}}\bigg)
\limsup_{l\to\infty}\limsup_{k\to\infty}\|\nabla(\zeta_l\tilde u_k)\|_{L^p}= \limsup_{l\to\infty}\limsup_{k\to\infty}\|\nabla\zeta_l\otimes\tilde u_k + \zeta_l\nabla\tilde u_k\|_{L^p}&\leq \limsup_{k\to\infty}\|\nabla\tilde u_k\|_{L^p}< \infty\,.
\left(\begin{array}{c}\hat{a}_i({\bf k}) \hat{a}_i^\dag(-{\bf k})\end{array}\right)=\sum_{j=1}^2 \left( \begin{array}{cc}\alpha_{ij}^{k\ast}(t) -\beta_{ji}^k(t) -\beta_{ji}^{k\ast}(t) \alpha_{ij}^k(t)\end{array}\right)\left(\begin{array}{c}\hat{a}_j({\bf k},t) \hat{a}_j^\dag(-{\bf k},t)\end{array}\right)\;.
P(Z_0(\Delta\ell_2)=m|W^k_{\ell_1\Delta,t}(j)=1,Z_0(\Delta \ell_1)=n) &\cdot P(B_{\ell_1\Delta}(\ell_2\Delta)=i|Z_0(\Delta\ell_2)=m,Z_0(\Delta \ell_1)=n,W^k_{\ell_1\Delta,t}(j)=1) &= \frac{P(B_{\ell_1\Delta}(\ell_2\Delta)=i,Z_0(\Delta\ell_2)=m,Z_0(\Delta \ell_1)=n,W^k_{\ell_1\Delta,t}(j)=1)}{P(W^k_{\ell_1\Delta,t}(j)=1,Z_0(\Delta \ell_1)=n)}
\sum_{i=1}^M\mathbb{E}\left[y_i^*y_i|\right.\left.\hat{\mathbf{H}}\right] = \sum_{j=1}^{M}a_j^2\left(\sum_{l=1}^M\lvert\hat{\phi}^{\left(l,j\right)}\rvert^2+M\sigma_{e_i}^2\lVert\mathbf{p}_j\rVert^2\right)+a_c^2\left(\sum_{i=1}^M\lvert\hat{\phi}^{\left(i,c\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_c\rVert^2\right)+M\sigma_n^2.
{\hat H}\psi_s=\frac{1}{r^2}\frac{d}{dr}r^2{\rho}^{\prime}\left(\begin{array}{c}\begin{array}{cc}-1-E^2_c E^2_c E^2_c 1-E^2_c \end{array}\end{array}\right)\left(\begin{array}{c}1 1 \end{array}\right)=\frac{1}{r^2}\frac{d}{dr}r^2{\rho}^{\prime}\left(\begin{array}{c}-1 1 \end{array}\right)
e^{-W(J,\xi )}=\int{\cal D}\phi_i{\cal D}\eta{\cal D}\overline{\eta}e^{-S-\frac 1{2\xi}\int d^dxF^2+\int d^dx\overline{\eta}(x)\frac{\delta F}{\delta \phi_i(x)}\Delta_i\eta (x)+\int d^dxJ_i(x)\phi_i(x)},
\tilde{R}( \tilde{g}) = e^{U}\tilde{R}( \tilde{\gamma}) + ( d - 2 ) e^{U}D^{2}U -{\frac{1}{4}}( d^{2}- 5 d + 8 ) e^{U}( D U )^{2}-{\frac{1}{4}}e^{3 U}V_{i j}V^{i j}\ ,
V_\phi[D^\beta f](x,\xi) =\sum_{\alpha\leq\beta}C'_{\alpha,\beta}\xi^{\beta-\alpha}\iint V_\phi f(x-y,\xi-\eta) V_{D^\beta\phi}\phi(y,\eta) e^{-i\langle x-y,\eta \rangle}d y \, d \eta,
\int_{\O^\#}\int_{B_1(0)}D^2_xu^\#(x,y,\epsilon) W_\delta(x,y)\phi(h) \, dxdy dh =\!\int_{\Omega^\#}\!\!\int_{B_1(0)}\!\!\frac{(u^\#(x+\varepsilon h,y)-u^\#(x,y))(W_\delta(x+\varepsilon h,y)-W_\delta(x,y))}{\epsilon^2}\phi(h)\, dxdy dh\,.
\ddot{\vec{\phi}}= - [ ( \vec{A_{1}^{2}}+ \vec{A_{2}^{2}}) \vec{\phi}- ( \vec{A_{1}}\cdot \vec{\phi}\vec{A_{1}}+ \vec{A_{2}}\cdot \vec{\phi}\vec{A_{2}}] - \frac{\kappa}{2}\vec{\phi}( \vec{\phi}^{2}- 1 ) .
\eta_{\underline{ab}}\, V^{\underline{b}}_m \, \eta^{i\ell_1}\, \epsilon^{m\ell_2 \dots \ell_d}\, \epsilon_{\ell_1 \dots \ell_d}\, +\, 2 \, (d!) \, a_1 \, \eta_{\underline{ab}}\, \Pi^{\underline{b}}_j \, h^{ij}= 0
\langle x_j, x_k \rangle =\frac{\|x_j+x_k\|^2-\|x_j-x_k\|^2+i\|x_j+ix_k\|^2-i\|x_j-ix_k\|^2}{4}&=\frac{(\|x_j\|^2+\|x_k\|^2)-(\|x_j\|^2+\|-x_k\|^2)+i(\|x_j\|^2+\|ix_k\|^2)-i(\|x_j\|^2+\|-ix_k\|^2)}{4}=0.
\begin{array}{c}{{\varphi ( n ) = q^{n - 1}( S_{0}( n , q^{2}) + 2 \nu S_{0}( n , - q^{2})}}\\{{= q^{1 - n}\{[ n ]_{q^{2}}^{( a )}+ 2 \nu ( - )^{n - 1}[ n ]_{- q^{2}}^{( a )}\}}}\\{{= [ n ]_{q}^{( b )}+ 2 \nu ( i )^{1 - n}[ n ]_{i q}^{( b )}.}}\\ \end{array}
P[\hat{K}_U \neq K_U \hat{K}_W \neq K_W] \longrightarrow 0 I(K_{\overline{U}}; V_B,V_C| U = W ) \longrightarrow 0 I(U ; V_A,V_C) \longrightarrow 0 I(W ; V_A,V_B) \longrightarrow 0 I(U,W; V_A) \longrightarrow 0 I(W,K_{\overline{U}}; V_B) \longrightarrow 0 I(U,K_{\overline{W}}; V_C) \longrightarrow 0.
\partial_{x}\rho_{n}= - \rho_{n}^{2}\partial_{x}\left( \frac{1}{\rho_{n}}\right), ~~ \partial_{x}^{2}\rho_{n}= 2\rho_{n}^{3}\partial_{x}\left( \frac{1}{\rho_{n}}\right)^{2}- \rho_{n}^{2}\partial_{x}^{2}\left( \frac{1}{\rho_{n}}\right).
x_{m,j}+ x_{m,j+1}+ x_{1,n+1-j}+ x_{1,n-j}=x_{m,j}+ x_{m,j+1}+ (\tfrac{1}{2}S - x_{1,j}) + (\tfrac{1}{2}S - x_{1,j+1}) = a_j +S -a_j =S.
\omega = \frac{1}{\gamma^2}\,\int_\tau^{\tau +2\pi}\,\, \frac{1}{\psi^2(z)}\,\delta\psi'(z)\wedge\,\delta\psi(z)\,dz +\frac{1}{\gamma^2}\,\delta\lambda \wedge\, \frac{\delta\psi(\tau)}{\psi(\tau)}.
(\delta g^{(1)}, \delta g^{(2)}) = \int \! \sqrt{g}\; \delta g^{(1)}_{\mu\nu}\left( g^{\mu\alpha}g^{\nu\beta}+ g^{\mu\beta}g^{\nu\alpha}+ C g^{\mu\nu}g^{\alpha\beta}\right) \delta g^{(2)}_{\alpha\beta},
\tilde{D}(r) = r^{-1}\left(\int_{t-r^2}^{t}\left(\int_{B^+(x,r)}|\Pi(y,s)-\overline{\Pi}_{B^+(x,r)}(s)|^{\kappa^*}d y\right)^{\frac{\lambda}{\kappa^*}}ds\right)^{\frac{1}{\lambda}}, \tilde{D}_1(r) = r^{-1}\left(\int_{t-r^2}^{t}\left(\int_{B^+(x,r)}|\nabla \Pi(y,s)|^{\kappa}d y\right)^{\frac{\lambda}{\kappa}}ds\right)^{\frac{1}{\lambda}},
\varepsilon_{n}\stackrel{\mathrm{def}}{=}\sup_{H\in \mathcal{H}_{0}}\;\sup_{x\in E_{n}}\left| h_{n}^{-1}\int_{\mathbf{R}}f(z)\,H\left(\frac{x-z}{h_{n}}\right) \,dz-f(x)\int_{\mathbf{R}}H(z)\,dz\right|\rightarrow 0,
\tilde{h}(P(\alpha_{0}, \ldots, \alpha_{d});t) =\sum_{i = 0}^{d}\sum_{j = 0}^{\alpha_i - 1}t^{\sum_{0 \le k \le d, k \ne i}(\frac{j \alpha_k}{\alpha_i}- \lfloor \frac{j \alpha_k}{\alpha_i}\rfloor ) + \sum_{k = i + 1}^d \phi( \frac{j \alpha_k}{\alpha_i})},
\mathcal{A}_{c}=\{\phi\in[0,\pi] \mid 2\cos\phi\in[-2,2]\cap\mathcal{A}\}\quad\mbox{and}\mathcal{A}_{d}=\{m>\lfloor\log|\alpha|\rfloor \mid \alpha^{-1}q^{m}+\alpha q^{-m}\in\mathcal{A}\}.
\int k\psi^{k-1}(1+ |\nabla f|^{2})^{\frac{k-2}{2}}\nabla f\nabla\psi -\int\frac{n}{f}\psi^{k}(1+ |\nabla f|^{2})^{\frac{k-2}{2}}&\qquad+(k-1)\int\psi^{k}(1+ |\nabla f|^{2})^{\frac{k-2}{2}}\frac{f_{i}f_{j}}{1+|\nabla f|^{2}}f_{ij}=0.
\zeta_{0}( \nu ) ={\cal A}_{c}{\frac{( \ell \mu )^{2 \nu}z_{-}^{2 \nu - ( D - 1 )}\Gamma ( \nu - \eta )}{( 4 \pi )^{\eta}\Gamma ( \nu )}}\hat{\zeta}( 2 \nu - ( D - 1 ) ) ,
n_k = \left\lfloor\frac{n_{k-1}+1}{2}\right\rfloor = \left\lfloor\frac{2^{m-k+1}+c_{k-1}+ 1}{2}\right\rfloor = 2^{m-k}+ \left\lfloor\frac{c_{k-1}+ 1}{2}\right\rfloor = 2^{m-k}+ c_k,
b^{-\frac{1}{2}}\partial_\xi \phi_{i,int, \beta}\left( \frac{y_0}{\sqrt{b}}\right) \phi_{i, out, \beta}(y_0) - \phi_{i,int, \beta}\left( \frac{y_0}{\sqrt{b}}\right) \partial_y \phi_{i,out, \beta}(y_0)=0