harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{k=1}^{n}\frac{H_{k+r-1}}{k}=\frac{\left( H_{n}\right)^{2}+H_{n}^{\left( 2\right)}}{2}+\sum_{j=1}^{r-1}\left( \frac{H_{j}}{j}+\frac{H_{j-1}}{j+n}\right) -H_{r-1}\left( H_{n+r-1}-H_{n}\right) .
\omega(a*b*c) = \omega(b*c*\sigma_i^R(\nabla^{-1}a)) = \omega(c*\sigma_i^R(\nabla^{-1}a)*\sigma_i^R(\nabla^{-1}b)) = \omega (\nabla\sigma_{-i}^R(\sigma_i^R(\nabla^{-1}a)* \sigma_i^R(\nabla^{-1}b))*c).
Z ( t , t^{*}) = c \sum_{\{h^{e}, h^{o}\}}{\frac{\prod_{i}( h_{i}^{e}- 1 ) ! ! h_{i}^{o}! !}{\prod_{i , j}( h_{i}^{e}- h_{j}^{o})}}\chi_{\{h \}}( A ) \chi_{\{h \}}( B ) .
\int_{0}^{\infty}{\frac{d t}{t}}( \mathrm{D e t}^{\prime}\Delta_{1})^{1 / 2}J_{M}( \phi ,{\hat{g}}) \int [ d \delta \phi ]_{e^{\phi}{\hat{g}}}e^{- S_{L}[ \phi ,{\hat{g}}] - S_{\mathrm{b o u n d a r y}}[ \phi ,{\hat{e}}]}\quad ,
h_\mu(t):=\frac{t-\mu c_{2^*}^{2^*}t^{2^*-1}}{a_1 c_{2^*}|\Omega|^\frac{2^*-1}{2^*}+ a_2 c_{2^*}^q |\Omega|^\frac{2^*-q}{2^*}t^{q-1}}\mbox{for all}t\geq 0.
S_i =\begin{bmatrix}I_p 0 0 \dots (S_i)_{1,0}I_p 0 \cdots (S_i)_{2,0}(S_i)_{2,1}I_p \ddots & \ddots\end{bmatrix}, H =\begin{bmatrix}H_0 0 0 \cdots 0 H_1 0 \ddots 0 0 H_2 \ddots \vdots \vdots \ddots \ddots\end{bmatrix}, i =1,2,
\lim_{t\to 0}\int_\Omega \Phi S(t)\rho_i(t)\varphi dx =\int_\Omega \Phi S^{in}\rho^{in}_i\varphi dx,i=1, \ldots , N, &\lim_{t\to 0}\int_\Omega (\Phi E_f(t) + (1-\Phi)E_s(t))\varphi dx =\int_\Omega (\Phi E_f^{in}+ (1-\Phi)E_s^{in})\varphi dx, &\lim_{t\to 0}S(t) = S^{in}\quad\mbox{strongly in $L^1(\Omega)$,}
e^{i k \eta_{c}}c_{-}\simeq \frac{1}{2 \sqrt{2}}[ ( \frac{1}{2}- \nu ) \frac{\Gamma ( \nu )}{\pi}( \frac{k \eta_{c}}{2})^{- \nu - \frac{1}{2}}- i ( \frac{k \eta_{c}}{2})^{\nu - \frac{1}{2}}( \nu + \frac{1}{2}) \frac{1}{\nu \Gamma ( \nu )}] ,
\int_{\mathbb{R}}\frac{e^{- \frac{\pi |\xi|^2}{t^2}}}{|\xi|^{2(1-\beta)}}\, d\xi = \pi^{1/2-\beta}\Gamma\biggr(\beta - \frac{1}{2}\biggr) t^{2\beta-1}\leq \Gamma\biggr(\beta - \frac{1}{2}\biggr) t^{2\beta-1},
\mbox{}[\hat{\Phi}(x^{0},x^{1})\,,\,\hat{\Phi}(x^{0},y^{1})] \,=\,-\frac{i}{2}\left\{\frac{1}{i\pi}{\sum_{n=-\infty}^{+\infty}}^{\prime}\left(\frac{\pi}{R}\right)\frac{e^{\frac{in\pi}{R}(x^{1}-y^{1})}}{\frac{n\pi}{R}}\right\}.
\frac{p_{n+1}}{q_{n+1}}=\begin{bmatrix}a_0 0 1 1\end{bmatrix}\begin{bmatrix}a_1 0 1 1\end{bmatrix}\begin{bmatrix}a_2 0 1 1\end{bmatrix}\cdots\begin{bmatrix}a_{n+1}0 1 1\end{bmatrix}\begin{bmatrix}p_n p_{n-1}q_n q_{n-1}\end{bmatrix}
\sup_{g \in G}\big{|}D_{\pi}^{(g)}(hy)(f)\big{|}= \sup_{g \in G}\big{|}D_{\pi}^{(gh)}(y)(h \cdot f)\big{|}= \sup_{g \in G}\big{|}D_{\pi}^{(g)}(y)(h \cdot f)\big{|}= \| h \cdot f \| = \| f \|.
\begin{aligned}|\langle \mu_t-\nu,b(t,x,\cdot)\rangle\leq \gamma H\left(\nu\mid\mu_t\right)\quad t\in[0,T],x\in\mathbb{R}^d, \nu\in\mathcal{P}(\mathbb{R}^d) b(t,x,\cdot)\in L^1(\nu).\end{aligned}
\mathsf{d}(x,z) = \inf_{w \in{\rm Pic}^0(X)}\left( \mathsf{d}_{0}(v_1 , w ) + \mathsf{d}_{0}(v_3,w) \right) + \delta &\le \inf_{w \in{\rm Pic}^0(X)}\left( \mathsf{d}_{0}(v_1 , v_2 ) + \mathsf{d}_{0}(v_2 , w ) + \mathsf{d}_{0}(v_3,w) \right) + \delta &=\mathsf{d}(x,y) + \mathsf{d}(y,z),
L = \sum_{i=1}^{n-1}a_i\partial_i^2 - \sum_{i,j=1}^{n-1}a_ia_j\partial_i\partial_j = \sum_{i=1}^{n}a_i\partial_i^2 - \sum_{i,j=1}^{n}a_ia_j\partial_i\partial_j.
\frac{2}{r}( f^{'}\cos 2 \theta \frac{\partial I^2}{\partial\theta}+ 2 f \sin 2 \theta \frac{\partial I^2}{\partial r}-f^{'}\sin 2 \theta \frac{\partial I^1}{\partial \theta}+ 2 f\cos 2 \theta \frac{\partial I^1}{\partial r})
\gamma = \mathrm{res}_v\left(\left(L^\mathrm{AH}(\varepsilon(v))\frac{d L^{\mathrm{AH}}(\eta(v))}{dv}- L^{\mathrm{AH}}(\varepsilon(v))d_{\mathrm{log}}(B(v)) + L^{\mathrm{AH}}(\eta(v)) d_{\mathrm{log}}(A(v))\right)\left(\frac{1}{z(v)^p-1}\right)\right).
||\mathcal{F}(\sqrt{\cdot}\left(F_{5}+F_{6}\right)(x,\cdot \lambda(x)))(\omega \lambda(x)^{2})||_{L^{2}(\rho(\omega \lambda(x)^{2}) d\omega)}^{2}=\int_{0}^{\infty}\frac{r}{\lambda(x)^{4}}\left(F_{5}+F_{6}\right)^{2}(x,r) dr
(\bar{\psi}_iM\lambda^j)(\bar{\chi}_kN\xi^l)=-\frac{1}{4}(\bar{\chi}_k\lambda^j)(\bar{\psi}_iMN\xi^l)-\frac{1}{4}(\bar{\chi}_k\gamma_a\lambda^j)(\bar{\psi}_iM\gamma^aN\xi^l)+\frac{1}{8}(\bar{\chi}_k\gamma_{ab}\lambda^j)(\bar{\psi}_iM\gamma^{ab}N\xi^l),
E \sim{\frac{I ( u_{1}, u_{2})}{\ln R_{2}- \ln R_{1}}}, \hspace{0 . 5 c m}{\frac{I ( u_{1}, u_{2})}{2 \pi}}= \left[ \int_{u_{1}}^{u_{2}}\! \! d u \sqrt{\cosh 2 u}\right]^{2}
\langle \cdot \,|\, \cdot\rangle_{\#}:=\langle \cdot\,|\, (h_{\alpha}- e_{\alpha}+ H_f^+)\,|\, \cdot \rangle, \langle \cdot\,|\, \cdot \rangle_{*}:= \langle \cdot \,|\, H_f^+ \,|\,\cdot\rangle.
d s^{2}= - \frac{\Delta}{\rho^{2}}[ d t - a \sin^{2}\theta d \phi ]^{2}+ \frac{\sin^{2}\theta}{\rho^{2}}[ ( r^{2}+ a^{2}) d \phi - a d t ]^{2}+ \frac{\rho^{2}}{\Delta}d r^{2}+ \rho^{2}d \theta^{2},
g = t^{-\sigma^-}\cdot \prod_{i=\sigma^-}^{\sigma^+-1}\big( w_i(a_1, \dots, a_d) \, t^{-1}\big) \cdot w_{\sigma^+}(a_1,\ldots,a_d) \cdot t^{\sigma^+}, g = t^{-\sigma^+}\cdot \prod_{j=\sigma^-}^{\sigma^+ -1}\big( w_{\sigma^+ + \sigma^- - j}(a_1, \dots, a_d) \, t \big) \cdot w_{\sigma^-}(a_1,\ldots,a_d) \cdot t^{\sigma^-},
\frac{2(n-4)}{r-2}FJ_{ij}= 4( J_{ij}\omega_{ij}J_{ij}+ \omega_{ij}) - \sum_{k\neq i,j}\langle J_{ij}, \omega_{ki}\rangle J_{ki}+\langle J_{ij}, \omega_{kj}\rangle J_{kj}, \frac{2(n-4)}{r-2}J_{ij}F =-4(\omega_{ij}+ J_{ij}\omega_{ij}J_{ij}) + \sum_{k\neq i,j}\langle J_{ij}, \omega_{ki}\rangle J_{ki}+\langle J_{ij}, \omega_{kj}\rangle J_{kj}.
{\cal{A}}_{lowenergy}^{relativ}\,=\,\int^\infty d^3 k I\left(k^0,\vec{k},w(p),\vec{p}\right)= \int^{\Lambda}_{0}d^2k \int^{\infty}_{-\infty}dk^0 I+\int_{\Lambda}^{\infty}d^2k \int^{\infty}_{-\infty}dk^0 I
\begin{gathered}|p^c (R_q f)(p)| \leq \|f\|_{\infty,I-c}\bigl| p^{-c}(R_q x^{c-I})(p) \bigr| = \|f\|_{\infty,I-c}\bigl| (R_q x^{c-I})(I) \bigr| \overset{\eqref{aux.MqtoRq}}{=}\Gamma(\alpha)^{-1}\|e^{-q}\|_{1,c}\|f\|_{\infty,I-c}. \end{gathered}
\widetilde{m}_2(\xi,\eta) := \widetilde{\chi}(\xi-\eta,\eta)] \big[ \xi(\xi-\eta)-|\xi||\xi-\eta| \big], \widetilde{q}_2(\xi,\eta) := \widetilde{\chi}(\xi-\eta,\eta)\frac{\eta(\xi-\eta)+|\eta||\xi-\eta|}{2}.
\sum_{p+q=i}h^p(Z,\Omega^q(\log D_\infty,\mathrm{rel}\, f^{-1}(\epsilon_0))) = \dim H^{i}(Y,f^{-1}(\epsilon_0)) = \dim \mathbb{H}^{i}(Z,\Omega^\bullet_Z(\log D_\infty, f)) = \sum_{p+q=i}\dim H^p(Z,\Omega^q(\log D_\infty, f)).
e^{x}\int_x^{+\infty}e^{-\eta}\big(vv_x^2(\eta)+v^3(\eta)\big)d\eta \geq 2e^{x}\int_{x}^{+\infty}e^{-\eta}v^2(\eta)v_x(\eta)d\eta =\dfrac{2}{3}v^3(x)-\dfrac{2e^{x}}{3}\int_{x}^{+\infty}e^{-\eta}v^3(\eta)d\eta,
C(\alpha_2, \cdots ,\alpha_{2n-1},\alpha_0) = \left\langle \left[\pi(z_{n}),\pi(z_{n+1})\right]\pi(z_{n+1})\left[\pi(z_{n}), \pi(z_{n+1})\right]^{s}u, e_{\alpha_2,\cdots ,\alpha_{2n-1},\alpha_0}\right\rangle.
|\hat{\rho}(\xi)| = |\hat{\rho}(\xi) - \hat{\rho}(0)| \leq C(\| \rho \|_{L^2}+ \| |x|^{\epsilon}\rho \|_{L_x^{1}}) | \xi - 0 |^{\epsilon}&\qquad = C(\| \rho \|_{L^2}+ \| |x|^{\epsilon}\rho(x) \|_{L_x^{1}}) | \xi |^{\epsilon}
\begin{array}{rclcrclcrclcrcl}h := E_{11}-E_{22}, e_1^+ := E_{12}, e_2^+ := E_{31}-E_{24}, e_3^+ := E_{14}+E_{32}, z := E_{33}-E_{44}, e_1^- := E_{21}, e_2^- := E_{13}+E_{42}, e_3^- := E_{41}-E_{23}.\end{array}
(u(y)-u(x))w(y) =w(x)\nabla u (x) \cdot (y-x)+ \frac{w(x)}{2}\nabla^2 u(x) \cdot(y-x,y-x) & + [\nabla u (x) \cdot (y-x)][\nabla w (x) \cdot (y-x)] + E(r) r^2.
\begin{aligned}\sigma_{+}p = \frac{q}{\sigma_{-}}+\frac{\phi}{\sigma_{-}}+\frac{z^{n}\psi}{\sigma{-}}z^{-n}\sigma_{-}p = \frac{q}{z^{n}\sigma_{+}}+ \frac{\phi}{z^{n}\sigma_{+}}+\frac{\psi}{\sigma_{+}}. \end{aligned}
(s_i \circ \beta_{i+1})(a_0 \otimes \cdots \otimes a_{i+2}) = a_0 \big(\partial_M(\psi_{i+1}\bullet z)(a_1 \otimes \cdots \otimes a_{i+1})\big) a_{i+2}\intertext{and}(d_{i+2}\circ s_{i+1})(a_0 \otimes \cdots \otimes a_{i+2}) = a_0 \big( (\partial_M (\psi_{i+1}) \bullet z)(a_1 \otimes \cdots \otimes a_{i+1})\big)a_{i+2}
P_\lambda\Big(I_t^{(O,\{e_1,y\})}\neq \emptyset\Big)+P_\lambda\Big(I_t^{(O,\emptyset)}\neq\emptyset\Big) &\leq P_\lambda\Big(I_t^{(O,e_1)}\neq \emptyset\Big)+P_\lambda\Big(I_t^{(O,y)}\neq\emptyset\Big)=2P_\lambda\Big(I_t^{(O,e_1)}\neq \emptyset\Big)
=\frac{e^{4}}{(k-k^{\prime})^{4}}(4({k^{\prime}}^{\mu}k^{\nu}-(k^{\prime}\cdot k)\eta^{\mu\nu}+k^{\prime\nu}k^{\mu})+4m^{2}\eta^{\mu\nu})(4({p^{\prime}}_{\mu}p_{\nu}-(p^{\prime}\cdot p)\eta_{\mu\nu}+p_{\nu}^{\prime}p_{\mu})+4m^{2}\eta_{\mu\nu})
h^-(\omega) = \left\{y \in \Phi_{\mathrm{b}}': \mathcal{V}_{0, y}^2 \subseteq \mathcal{C}_0^{\mathrm{E}}, y \in \Phi_{\mathrm{b}}' \right\}, &H^-\left( Y \right) = Y + h^-\left( T_Y \right).
\left\langle \tau \right\vert =\left\langle \omega \right\vert \prod_{b=1}^{\mathsf{N}_{Q}}\mathcal{\hat{B}}_{-}(\lambda_{b}),|\tau \rangle=\prod_{b=1}^{\mathsf{N}_{Q}}\mathcal{\hat{B}}_{-}(\lambda_{b})|\bar{\omega}\rangle ,
\{\mathrm{P}_{\mathrm{R}}, \mathrm{K}_{\mathrm{R}}\}= - \mathrm{H}_{\mathrm{R}}, \hspace{5 m m}\{\mathrm{H}_{\mathrm{R}}, \mathrm{K}_{\mathrm{R}}\}= - \mathrm{P}_{\mathrm{R}},
||v_{n\,m+6}^1||^2= -\frac{\overline{b_{n\,m+6}}}{c_{n+1\,m+3}}||v_{n+1\,m+3}^1||^2= \frac{\overline{b_{n\,m+6}}}{c_{n+1\,m+3}}\cdot \frac{\overline{d_{n+1\,m+3}}}{a_{nm}}||v_{nm}^1||^2.
I(i,j) \le \left( \int_0^1(\log f_i^{(k)}(x))^2 \d \mu\right)^{1/2}\left( \int_0^1 (\log f_j^{(k)}(x))^2\d \mu\right)^{1/2}= \left( \int_0^1(\log f_k(x))^2 \d \mu\right)^{1/2}\left( \int_0^1 (\log f_k (x))^2\d \mu\right)^{1/2}&\le C.
{B_{(k+1) n}\over B_{k n}}={1+ \rho_* [r_{k n-1}, r_{k n -2}, \ldots,r_0] \over 1+ \rho_* [r_{(k+1) n-1}, r_{(k+1) n -2}, \ldots,r_0]}{1+{p_{(k+1)n}\over q_{(k+1) n}}\over 1 +{p_{k n}\over q_{k n}}}\le (1+C \theta^{2 k n}).
R_{ij}(c_{r_1}^{(1)}\cdots c_{r_i}^{(i)}\cdots c_{r_j}^{(j)}\cdots c_{r_m}^{(m)}) = c_{r_1}^{(1)}\cdots c_{r_i+1}^{(i)}\cdots c_{r_j-1}^{(j)}\cdots c_{r_m}^{(m)}.
{{\left\langle T\left[ \left({{\psi}_{{{i}_{1}}}}{{{\bar{\psi}}}_{{{j}_{1}}}}\right)\ldots \left({{\psi}_{{{i}_{n}}}}{{{\bar{\psi}}}_{{{j}_{n}}}}\right) \right] \right\rangle}_{0}}=\det \left({{\Delta}_{{{i}_{\alpha}}{{j}_{\beta}}}}\right) \ ,
\hat{H}\psi_{0}=\hbar\omega(a^{\dagger}a+\frac{1}{2})\psi_{0}=\hbar\omega a^{\dagger}a\psi_{0}+\frac{\hbar\omega}{2}\psi_{0}=0+\frac{\hbar\omega}{2}\psi_{0}=E_{0}\psi_{0}.
\partial_{\gamma}v(y)=\left( \frac{d}{dt_1}\right)^{\gamma_1}\left( \frac{d}{dt_2}\right)^{\gamma_2}\ldots \left( \frac{d}{dt_n}\right)^{\gamma_n}\bigl( v(y\exp(t_1E_1)\exp (t_2E_2)\ldots &\ldots \exp (t_nE_n)\bigr)\bigr\vert_{t_1=t_2=\ldots=t_n=0}.
\Vert x_k-y_{f(k^*)}\Vert^2 \le\Vert x_{f(k^*)}-y_{f(k^*)}\Vert^2+2M^2(\theta_{f(k^*)}-\theta_k)\sum_{j=f(k^*)}^{k-1}\lambda_j+4M^2\sum_{j=f(k^*)}^{k-1}\lambda_j^2 \le\Vert x_{f(k^*)}-y_{f(k^*)}\Vert^2+2M^2(\theta_{f(k^*)}-\theta_{f(k^*+1)})\sum_{j=f(k^*)}^{f(k^*+1)}\lambda_j+4M^2\sum_{j=f(k^*)}^{f(k^*+1)}\lambda_j^2.
(T(a \chi^{-1})-H_{\alpha}(b\chi))c_+^{-1}= Pbc \chi^{-1}c_+^{-1}-Pb\chi QJ_{\alpha}c_+^{-1}=Pbc_+\chi c_- \chi^{-1}c_+^{-1}-Pb\chi(c_+^{-1})_{\alpha}\chi^{-1}=0,
F_{1}(r,0) = -r\int_{0}^{2\pi}\log\left(2\sin^{2}\left(\frac{x-y}{2}\right)\right)(\sin(x-y))dy F_{2}(r,0) = -r\int_{0}^{2\pi}\log\left((1+r^2)+(r^2-1)\cos(x+y)\right)(\sin(x-y))dy
\sigma \big( P_0(b_1,\cdots,b_d) \big) = \sigma_{\textup{ess}}\big( P_0(b_1,\cdots,b_d) \big) = \bigg\lbrace 2 \sum_{j = 1}^d b_j (q_j - 1) : (q_1,\ldots,q_d) \in \mathbb{N}^d \bigg\rbrace.
e^{- S_{\mathrm{d u a l}}( A ) \;}= \int D B_{\mu}D B_{\mu}^{\dagger}D \eta_{\mu}e^{- \left( S_{\mathrm{t o p}}( B , ( \partial + i \eta ) B ) + \int d^{3}x A_{\mu}\epsilon^{\mu \nu \rho}\partial_{\nu}\eta_{\rho}\right) \;}\; .
\Pi_{W_n}= \sum_{t=1}^T \big(\bar{\tau}_n^t\bar{p}_{W_n}^t +s_{n}^t[w_n^t-\bar{p}_{W_n}^t]^{+}- b_{n}^t[\bar{p}_{W_n}^t-w_n^t]^{+}\big).
\phi_{c}^{( 2 )}( \tau ) = 2 \tan^{- 1}\frac{1}{\alpha}\left( 1 + \sqrt{1 - \alpha^{2}}{\frac{e^{- \sqrt{1 - \alpha^{2}}\omega_{0}( \tau - \tau_{0})}+ f^{( 2 )}( \alpha )}{e^{- \sqrt{1 - \alpha^{2}}\omega_{0}( \tau - \tau_{0})}- f^{( 2 )}( \alpha )}}\right)
\begin{aligned}p_N(a) = P_N(a) - P_N\left(a + \frac 1N\right) &\sim K(a) N^{-\frac \alpha 2}e^{-N^\alpha J(a)}- K\left(a + \frac 1N\right) N^{-\frac \alpha 2}e^{-N^\alpha J\left(a + \frac 1 N\right)}\,.\end{aligned}
\mu(X^{\underline m}Y^{\underline k-2-\underline m})=\rho\circ s(\mu)(X^{\underline m}Y^{\underline k-2-\underline m})=C_n\cdot \rho\circ\varphi_n((x^{2n})^\vee)(X^{\underline m}Y^{\underline k-2-\underline m})=C_n\frac{(-1)^{m_{\rm id}+m_c}}{n+M+1}\binom{n+M}{m_{\rm id}}^{-1}
f_{{\bf{k}},{\bf{q}}}({\bf{x}}) = e^{i{\bf{q . x}}}( \frac{1}{2 \mathrm{~}N \mathrm{~}\epsilon_{{\bf{q}}}}) \Lambda_{{\bf{k}}}( -{\bf{q}}) \omega_{{\bf{k}}}({\bf{q}}) + i \mathrm{~}U_{-{\bf{q}}}({\bf{x}}) \Lambda_{{\bf{k}}}( -{\bf{q}})
\prod_{j=0}^k\cos^2\left(\pi x\b_j\right)d\,x \leq \prod_{j=0}^k \exp\left(-\pi^2\left\|xb_j\|\right\|^2 \right) =\prod_{j=0}^k\exp\left(-\pi^2 \left(xb_j\right)^2 \right) =\exp\left(-\pi^2 x^2\sum_{j=0}^k b_j^2 \right) =\exp\left(-\pi^2 x^2 S_{n,B-k}\right).
(1-q^{N})(-q)_{N-1}\cdot\left.G(z, q, N)\right|_{z=q}&= \frac{q^N}{1-q^N}(-1)_N - \sum_{m=N}^{\infty}\frac{2q^m}{1-q^{2m}}+ \sum_{m=1}^{\infty}\frac{q^{m}}{1-q^{m}}\frac{(-1)_m}{(-q^N)_m}-\sum_{m=1}^{N}\frac{q^m}{1-q^m}.
d_K(\xi(s_j;s_0,x)) = d_K(\xi(s_j;s_{j-1},\xi(s_{j-1};s_0,x))) \leq \Phi_{\beta+L,0}(d_K(\xi(s_{j-1};s_0,x)),\epsilon,s_j-s_{j-1}) \leq \Phi_{\beta+L,0}(\Phi_{\beta+L,0}(d_K(x),\epsilon,s_{j-1}-s_0),\epsilon,s_j-s_{j-1}) = \Phi_{\beta+L,0}(d_K(x),\epsilon,s_j-s_0),
{\cal W}_{a}= \tilde{I}_{a a^{\prime}}\int \! V_{b}^{\mu}z_{b b^{\prime}}\hat{\epsilon}_{b^{\prime}c^{\prime}a^{\prime}}( \theta_{W}+ \theta_{V}) \tilde{I}_{c^{\prime}c}z_{c d}\frac{\delta}{\delta V_{d}^{\mu}}
={1 \over \Gamma(2z+2k-2l)}\int_{0}^{\infty}dy~{y^{2z+2k-l-1}\over 1-e^{-y}}(-1)^{p-l}\left({d \over dy}\right)^{p-l}\left[{\exp\left(-{d-1 \over 2}y\right) \over e^{\gamma y}-1}\right]~~~.
u^N_{t}(x) - u_{t}(x) = e^{t\Delta}(u^N_{0}- u_{0})(x) + \int_0^t \nabla \cdot e^{(t-s)\Delta}\left(u_{s}F(K\ast u_s) - u^{N}_{s}F(K\ast u^{N}_s)\right)(x) ~ ds &+ E_{t}(x) - M^N_{t}(x),
\left(1-c+(a^{p^m}-a)^p\left(1-\dfrac{1-c}{(1-c)^{p^m}}\right)\right)\beta+\left((a-a^{p^m})\left(1-\dfrac{(1-c)}{(1-c)^{p^m}}\right)\right)^{p^{m-1}}\beta^{p^{m-1}}=0,
\frac{(x-1)! \cdot 2^{x+1}}{q}\leq \frac{1}{2}, \frac{3x}{q^{n-2k+1}}\leq \frac{3x}{q^3}\leq \frac{3x}{2^{3(x+2)}}\leq \frac{3}{256}, x^2 \cdot 2^{x}\cdot q^{-\delta}\leq \frac{x^2 \cdot 2^x}{\left(2^{x+2}\right)^2}\leq \frac{x^2}{2^{x+4}}\leq \frac{1}{8}, \frac{q}{\binom{x}{2}}\geq \frac{(x-1)! \cdot 2^{x+3}}{x(x-1)}\geq 16,
I ( \nu ) = \frac{1}{( \nu^{2}- \frac{1}{4})}\{8 [ W ( 1 ) + ( W^{\prime}( \alpha ) )_{\alpha = 1}] + \int_{1}^{\infty}W_{1}( \alpha ) [ \alpha^{\frac{\nu}{2}}+ \alpha^{\frac{- \nu}{2}}] d \alpha \},
\sum_{m_1=0}^{N-1}\sum_{\scriptsize\begin{array}{c}m_2=0 m_2\neq m_1\end{array}}^{N-1}\frac{1}{(m_1-m_2)^2}\rightarrow 2N\frac{\pi^2}{6}-\sum_{k=1}^{\infty}\frac{1}{k}.
K_{D}({\bf r^{\prime \prime}},{\bf r^{\prime}}; t^{'^{\prime}}, t^{'}) = \left\langle{\bf r^{\prime \prime}}\left| \hat{T}\exp \left[ - \frac{i}{\hbar}\int_{t^{\prime}}^{t^{\prime \prime}}\hat{H}d t \right] \right|{\bf r^{\prime}}\right\rangle \; ,
\Delta (t_i)=t_i\otimes t_i, \varepsilon(t_i)=1, S(t_i)=t_i^{-1}, & \Delta(\theta_i)=t_i\otimes \theta_i+\theta_i\otimes 1, \varepsilon(\theta_i)=0, S(\theta_i)=-\theta_i t_i^{-1}.
N(\varphi_a, r,\lambda_a,A_a)+1 \leq (1+\delta p)\big(\tfrac k2+\delta p+1\big)+\frac{c_{\delta,r_\delta}}{\mu^{1-2\delta}}=1+\frac k2+\delta\Big(2p+p\frac k2+\delta p^2\Big)+\frac{c_{\delta,r_\delta}}{\mu^{1-2\delta}}<1+\frac k2+\frac12\delta+\frac{c_{\delta,r_\delta}}{\mu^{1-2\delta}}.
\begin{cases}\partial^{k}_{x}U=0, \Gamma_{1}=\{x=0 \},\,\forall\, 0\leq k\leq m, \partial^{k}_{y}U=0, \Gamma_{3}=\{y=0 \},\,\forall\, 0\leq k\leq m,\end{cases}
F{}{}\circ K{}{}^{1/\varepsilon}= \left( (F{}{}\circ K{}{}^{1/\varepsilon})^\varepsilon \right)^{1/\varepsilon}= \left( \left(K{}{}^{1/\varepsilon}\circ (A_c + r^{1/\varepsilon}) \right)^\varepsilon \right)^{1/\varepsilon}= K{}{}^{1/\varepsilon}\circ (A_c + r^{1/\varepsilon}).
F_{n}(x)=(x-\frac{d}{dx})^{n}e^{-\frac{x^{2}}{2}}=(-1)^{n}e^{\frac{x^{2}}{2}}\frac{d^{n}}{dx^{n}}(e^{-x^{2}})=(2^{n}x^{n}+\cdot\cdot\cdot)e^{-\frac{x^{2}}{2}}.
|\Delta_{l,1}^a \cap \Delta_{l,2}^{Q_1,R_i}| = |\Delta_{l,1}^b \cap \Delta_{l,2}^{P_1,R_i}| = |\Delta_{l,1}^c \cap \Delta_{l,2}^{P_1,Q_i}| = = (q^4-q)(q^4-q^2)(q^2-q)+ (q^4-q)(q^4-q^2)q(q^2-q-2).
\begin{array}{l}{{\hat{\partial}_{i}\hat{\partial}_{j}= q^{- 1}\hat{\cal R}_{j i}^{l k}\hat{\partial}_{k}\hat{\partial}_{l}~ ,}}\\{{\hat{\partial}_{i}\partial_{j}= q \hat{\cal R}_{j i}^{l k}\partial_{k}\hat{\partial}_{l}~ .}}\\ \end{array}
\begin{array}{c}<O_{k}O_{l}P>_{c}= ( \frac{1}{\rho^{2}}\frac{\partial \rho}{\partial t_{l}}+ \frac{1}{\rho^{2}}\frac{\partial \rho}{\partial \mu}\frac{\partial \mu}{\partial t_{l}})<O_{k}P> - \frac{1}{\rho}(<O_{k}O_{l}P> + \frac{\partial \mu}{\partial t_{l}}<O_{k}PP>). \end{array}
( \tilde{S}\tilde{T}^{p}\tilde{S})_{\alpha \beta}= - e^{- \frac{i \pi}{4}p}\sqrt{\frac{i}{8 K p}}\sum_{n = 0}^{p - 1}\sum_{\mu_{1 , 2}= \pm 1}\mu_{1}\mu_{2}\exp \left[ - \frac{i \pi}{2 K p}( \mu_{1}\alpha + \mu_{2}\beta + 2 K n )^{2}\right]
T_{a}= \left( \begin{array}{c c c}{V_{a}^{( \mathrm{c )}}}&{0}&{0}\\{0}&{( \omega V^{( \mathrm{c )}})_{a}}&{0}\\{0}&{0}&{( \omega^{2}V^{( \mathrm{c )}})_{a}}\\ \end{array}\right)
F_u(\sigma_k) =\left\lbrace \frac{f}{D}u_k^K \;\middle|\; f\in \mathbb{C}_{m'-1}[z]\right\rbrace , E_u(\sigma_k) = \left\lbrace \frac{zg}{D}u_k^K \;\middle|\; g\in \mathbb{C}_{m'-2}[z]\right\rbrace ,
M_{r}^{(1)}=\begin{cases}\emptyset, r=1,2, \prod_{k=1}^{\frac{r-1}{2}}(C_{2k}, C_{2k-1}, C_{2k-2}, \ldots, C_{1}), r \equiv 1 \pmod2, r>1, \prod_{k=1}^{\frac{r-2}{2}}(C_{2k},C_{2k-1},C_{2k-2},\ldots,C_{1}) , r\equiv 0\pmod2, r>2.\end{cases}
\int_{B(R)}|E f(x)|^{p_0}\lesssim \int_{B(R)}|\mathbf{B}_{\alpha}[Ef](x)|^{p_0}&+ \alpha^{-p_0}\Big( \sum_{\tau}\int_{B(R)}|Ef_{\tau}(x)|^{p_0}+ \sum_{L = L_{\parallel e_1}L_{\parallel e_2}}\int_{B(R)}|\sum_{\tau: \bar\tau \subset L}Ef_{\tau}(x)|^{p_0}\Big).
\begin{aligned}F(\{x_{i,r}\}_{1\leq i\leq 2}^{1\leq r\leq k_{i}})=0 x_{1,1}=v^{6}x_{1,2}=v^{3}x_{2,1}, x_{2,1}=v^{2}x_{2,2}=v^{4}x_{2,3}=v^{6}x_{2,4}=v^{3}x_{1,1}.\end{aligned}
\left[ B_{a}^{i j}\left( x \right) , \pi_{k l}^{b}\left( y \right) \right]_{x^{0}= y^{0}}^{*}= \frac 1 2 \delta_{a}^{b}\delta_{k}^{\left[ i \right.}\delta_{l}^{\left. j \right]}\delta^{3}\left({\bf x}-{\bf y}\right) .
A(x,y;\xi,\zeta)=\int_0^1 \int_0^1|y-\zeta+s(\zeta-\xi)+t (x-y)|^{p-2}D^2h\left(\frac{y-\zeta+s(\zeta-\xi)+t (x-y)}{|y-\zeta+s(\zeta-\xi)+t (x-y)|}\right)dt \, ds
\begin{array}{ll}C^{l}_{p}(\tau)_{m^{\prime},m^{\prime \prime}}= \frac{1}{M}\sum_{k^{m}_{l+1}}e^{\tau(\epsilon^l(k^{m}_{l+1})-\mu)+ik^{m}_{l+1}(m^{\prime}-m^{\prime \prime})}+\frac{q^{p}}{M}\sum_{k^{m}_{l}}e^{(\tau-\beta) (\epsilon^l(k^{m}_{l})-\mu)+ik^m_l (m^{\prime}-m^{\prime \prime})}(1 + (q^{-1}-1)\theta (L-m^\prime))\end{array}
[a_l,a_m,a_k,a_{\infty}]= \exp(4 \pi i (\eta_{m}-\eta_{l})_1(\eta_k)_2)\left( \frac{\theta[\eta_{S_l \circ U_\eta}](\Omega)\theta[\eta_{T_m \circ U_\eta}](\Omega)}{\theta[\eta_{S_m \circ U_\eta}](\Omega)\theta[\eta_{T_l \circ U_\eta}](\Omega)}\right)^2.
P= \begin{pmatrix}p_0 p_1 \dots p_{m-1}0 p_0 \dots p_{m-2}\vdots 0 \ddots \vdots 0 0 0 p_{0}\end{pmatrix}Q = \begin{pmatrix}\overline{p_{m}}\overline{p_{m-1}}\dots \overline{p_1}0 \overline{p_m}\dots \overline{p_2}\vdots 0 \ddots \vdots 0 0 0 \overline{p_m}\end{pmatrix}
\beta^{-3}\nabla_q\beta\cdot \left( \frac{3n+2}{6}- \int_0^\infty Tr[\gamma e^{-2y \gamma}]e^{-y \gamma}dy\right)\gamma^{-1}\nabla_q\beta = \frac{n+2}{6}\beta^{-3}\gamma^{-1}\|\nabla_q\beta\|^2.
\sum^n_{j=0}(-1)^j\mathrm{dim\,}H^j_{b,m}(X,E)=\Bigr(\sum\limits^{p}_{s=1}e^{\frac{2\pi(s-1)}{p}mi}\Bigr)\lim_{t\rightarrow0^+}\int_X\sum^n_{\ell=0}t^{-\ell}\Bigr(\mathrm{Tr\,}\widetilde\alpha^+_\ell(x)-\mathrm{Tr\,}\widetilde\alpha^-_\ell(x)\Bigr)dv_X(x).
u\left( x,k_{0}\right) \mid_{\Gamma_{tr}=}g_{0}\left(x_{1},x_{2},x_{0}\right) ,\partial_{n}u\left( x,k_{0}\right) \mid_{\Gamma_{tr}=}g_{1}\left( x_{1},x_{2},x_{0}\right) ,\forall x_{0}\in \left[ 0,1\right] .
{\vec{P}}= ({\overline{q}}+ 3 A^{I}p^{I}w ,{\overline{p}}_{2},{\overline{p}}_{3}\vert \vert q + ( 3 A^{I}p^{I}+ \alpha_{2}) w , p_{2}, p_{3}\vert p^{I}+ \alpha_{2}A^{I}) ,
\rho_{m_{0}\rightarrow 0 + 0 + 0}^{5}\! = \frac{1}{6 4 \pi^{4}}\int_{0}\! \! \int_{0}\! \! \int_{0}\! \! d \mu_{1}d \mu_{2}d \mu_{3}\frac{\mu_{1}\mu_{2}\mu_{3}\theta ( \lambda_{E})}{\lambda_{E}( \mu_{1}^{2}, \mu_{2}^{2}, \mu_{3}^{2})}\left( 1 \! - \frac{\mu_{1}+ \mu_{2}+ \mu_{3}}{m_{0}}\right) .
\left\langle Q^{m-n-1}v_0,\psi^{-}(z) \prod_{i=1}^{m}\psi^{+}(w_i) \prod_{i=1}^{n}\psi^{-}(y_i)v_0\right\rangle{}= \frac{\prod\limits_{i=1}^{m}(z-y_{i}) \prod\limits_{1\le i<j\le m}(w_i-w_j) \prod\limits_{1\le i<j\le n}(y_i-y_j)}{\prod\limits_{i=1}^{n}(z-w_i) \prod\limits_{i=1}^{m}\prod\limits_{j=1}^{n}(w_i-y_j)}.
G^{\phi ss}_{I_1, I_2 I_3}= -\frac{9 N^3}{64 \pi^5}\, \frac{\alpha_1(\alpha_1-1)(2\alpha_1-3)\Sigma(\Sigma+1)(\Sigma+3)}{k_2 (2k_2+1) k_3 (2k_3+1)}\, a(k_1,k_2,k_3) \, \langle T^{I_1}C^{I_2}C^{I_3}\rangle. \vbox{\vskip8mm}
\delta_k(t) = \,\prod_{\alpha\in R_+}\big\vert e^{i\langle\alpha,t\rangle/2}- e^{-i\langle\alpha,t\rangle/2}\big\vert^{2k_\alpha}=\, const\cdot\prod_{j=1}^q \sin^{2p-2q+1}\!t_j\, \cos^{2|l|+1}\! t_j \cdot\prod_{1\leq i<j\leq q}(\cos(2t_i) - \cos(2t_j))^2
M_{p, 1}(\delta, \delta^{1/2^{N+1}}) & \leq C^{\sum_{j = 0}^{N + 1}(\frac{2}{p - 2})^j}\delta^{-\frac{1}{2^{N + 1}}\frac{1}{p}\sum_{j = 0}^{N}(\frac{2}{p - 2})^{j}}(\log\frac{1}{\delta})^{\frac{1}{2}\sum_{j = 0}^{N}(\frac{2}{p - 2})^{j}}\prod_{j = 0}^{N}D_{p}(\delta^{1 - \frac{1}{2^{j + 1}}})^{\frac{p - 4}{p - 2}(\frac{2}{p - 2})^{N - j}}.
V_{\mathcal{D}_{d}}^{\mu[\nu_{N}]}(x) = \begin{cases}-\log w(x) - K(\alpha_{N}) (x \in [-\alpha_{N}, \alpha_{N}]), \tilde{\upsilon}_{N}(x,0) (x \in \mathbf{R}\setminus [-\alpha_{N}, \alpha_{N}]).\end{cases}
\times\int\left(\prod\limits_{i}^{}{\cal D}x^{(i)}(\xi){\cal D}h_{\mu\nu}^i\right)\delta\left(\sum\limits_{i}^{}\Sigma_{\mu\nu}^i\right)\exp\Biggl\{-\int d^4x\Biggl[\frac{1}{24\eta^2}\left(H_{\mu\nu\lambda}^i\right)^2+\frac{Ng_m^2}{8}\left(h_{\mu\nu}^i\right)^2-i\pi h_{\mu\nu}^i\Sigma_{\mu\nu}^{i{\,}(\alpha)}\Biggr]\Biggr\}.
\begin{gathered}\int_0^\infty u^{2\alpha}\left[C_1(u)+C_2(u)\right] J_\nu (u\rho; q^2) \,d_q u- \int_0^\infty u^{2\alpha}\left[C_1(u)+C_2(u)\right]\dfrac{w(u)}{1+w(u)}J_\nu (u\rho; q^2) \,d_q u= f_1 (\rho), \rho\in A_{q,a}, \end{gathered}
f \{\otimes}_r \g (t_1, \ldots, t_{p+q-2r})= \sum_{1 \le z_1, \ldots, z_r \le n}f(t_1, \ldots, t_{p-r}, z_{1}, \ldots, z_r) \cdot g(t_{p-r+1}, \ldots, t_{p+q-r}, z_{1}, \ldots, z_r).
\begin{aligned}i. \qquad\overline{\omega}'=-\,\omega' ii. \qquad\omega\omega'=-\,\omega'\omega \mbox{where}\omega=\frac{x}{|x|}iii. \qquad\big(\omega'\big)^2=-\frac{1}{|x|^2}\Big(|x|'-|x'|\Big)^2 iv. \qquad\omega\omega''+\omega''\omega=2|\omega'|^2 \end{aligned}
m_{G}^{2}\; = \; - i e^{2}\, \int_{\mathcal B}\frac{d^{2}p}{( 2 \pi )^{2}}\, \frac{\partial}{\partial p_{\mu}}\mathrm{t r}\left[ \gamma_{\mu}\frac{a}{i \not \! \! C ( a p ) + a m}\right] \; ,
Z(s)= \underset{n=1}{\overset{N}{\sum}}n^{-s}+\underset{X\rightarrow \infty}{\lim}\frac{X^{1-s}-\left(x_0\right)^{1-s}}{1-s}+ &\underset{K,N\rightarrow \infty}{\lim}\left[\frac{\left(x_0\right)^{-s-1}}{\Gamma (s)}.\underset{k=0}{\overset{k=K}{\sum}}\alpha_{2.k+1}.\Gamma (s+2.k+1).\left(x_0\right)^{-2.k}\right]
\mathcal{P}(\theta_1,\theta_2,\theta_3)=\sum_{k=1}^3\frac{i\theta_k}{2}\Big(\partial_k\otimes \partial_{\sigma_c(k)}\otimes\partial_{\sigma_c\circ\sigma_c(k)}-\partial_k\otimes \partial_{\sigma_c\circ\sigma_c(k)}\otimes\partial_{\sigma_c(k)}\Big),
(\begin{matrix}-&1&1&0&1&0&0\\ 0&-&1&1&0&1&0\\ 0&0&-&1&1&0&1\\ 1&0&0&-&1&1&0\\ 0&1&0&0&-&1&1\\ 1&0&1&0&0&-&1\\ 1&1&0&1&0&0&-\end{matrix})
{\bf E}\Bigl[\Bigl(\int\limits_0^1 f(t)\,M_\mu(dt)\Bigr)^n\Bigr] = n!\int\limits_{0<t_1<\cdots<t_n<1}\prod\limits_{i=1}^n f(t_i)\prod\limits_{k<p}^n |t_p-t_k|^{-\mu\,d(p-k)}\,dt.
\frac{c_{l_{3}}^{n}}{c_{l_{1}}^{n}c_{l_{2}}^{n}}\rightarrow b_{l_{1}}^{n}b_{l_{2}}^{n}b_{l_{3}}^{n}=(n!\sqrt{n+1})^{3}{\displaystyle\prod\limits_{k=1}^{3}}\frac{1}{\sqrt{(n+l_{k}+1)!(n-l_{k})!}}\ .
\lim_{m\rightarrow\infty}\int_0^1 \Big(\big\|\rho_m u_m\big\|_{L^{\frac{6\gamma}{\gamma+6}}(\Omega)}^2+\big\|\rho_m |u_m|^2\big\|_{L^{\frac{3\gamma}{\gamma+3}}(\Omega)}^2\Big)=0.