harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{m,n\geq 1}\dfrac{(-1)^m}{n^2+2^\ell m^2}=\begin{cases}-\dfrac{\pi^2}{24}-\dfrac{\pi\cdot\log 2}{8}, \ell=1, -\dfrac{7\pi^2}{96}-\dfrac{\pi\cdot\log 2}{32}-\dfrac{\pi\cdot\log(\sqrt{2}-1)}{8}, \ell=2, -\dfrac{31\pi^2}{384}-\dfrac{5\pi\log 2}{128}+\dfrac{\pi\cdot\log(\sqrt{2}-1)}{32}-\dfrac{\pi\cdot\log(1-2^{-1/4})}{8}, \ell=3.\end{cases}
(K^{(l)}\:V\:K^{(r)})(x,y) \;=\; \frac{1}{2^{l+r}}\left(\frac{\partial}{\partial_u}+ \frac{\partial}{\partial v}\right)^l\left(\frac{\partial}{\partial_u}- \frac{\partial}{\partial v}\right)^r\frac{\partial}{\partial u}\left. A_{uv}\right|_{u=0=v}\;.
x\cdot I(G-N[v_0]) = x \cdot (2x+1)^{2k+4}\cdot(3x^2+4x+1) &= x(3x^2+4x+1) \cdot \bigg[\sum_{i=0}^{2k+4}\binom{2k+4}{i}(2x)^{i}] &= x(3x^2+4x+1) \cdot [(2x)^{2k+1}+\dots] &= 3\cdot2^{2 k + 2}x^{2 k + 5}+\dots
n(x,y) := \phi(x)\overline{\phi(y)}+ \frac{1}{N}n^\mathrm{pair}(x,y),(\upsilon_N n)(x,y):=\upsilon_N(x-y)n(x,y) m(x,y) := \phi(x)\phi(y) + \frac{1}{N}m^\mathrm{pair}(x,y), (\upsilon_N m)(x,y):=\upsilon_N(x-y)m(x,y) \rho(x) := n(x,x), \rho^\mathrm{pair}(x) := n^\mathrm{pair}(x,x).
\nu_i + \ell - 2i_0 + 1 = \mathcal{C}_{-1}(\alpha, \nu, i, \lbrace 1, \ldots, i_0-1 \rbrace, \lbrace i_0, i_0+1, \ldots, i-1, i+1, \ldots, \ell \rbrace) \leq \mathcal{C}_{-1}(\alpha, \nu, i_0, \lbrace 1, \ldots, i_0-1 \rbrace, \lbrace i_0+1, \ldots, \ell \rbrace) = \iota_{i_0}+ \ell - 2i_0 + 1 < \nu_i + \ell - 2i_0 + 1.
Q_{2 s + 2 t}^{\mu | 2 s , 2 t}= ( - 1 )^{s t}( \sigma_{2 s + 2 t})^{\frac{3}{2}- t}( \sigma_{2 s}^{+})^{s - 2}( \sigma_{2 t}^{-})^{- 1}\det \mathcal{A}^{\mu}\mathcal{},
p_{0 \mu}= p_{\mu}- \frac{1}{2}\tilde{S}^{h_{1}h_{2}}\; \omega_{h_{1}h_{2}\mu}- \frac{1}{2}\tilde{S}^{k_{1}k_{2}}\; \omega_{k_{1}k_{2}\mu}- \frac{1}{2}\tilde{S}^{h_{1}k_{1}}\; \omega_{h_{1}k_{1}\mu},
D_1(z) = \begin{cases}\begin{pmatrix}e^{-3 \omega z^{1/3}}0 0 0 e^{-3 \omega^2 z^{1/3}}0 0 0 e^{-3z^{1/3}}\end{pmatrix}, \Im z > 0, \begin{pmatrix}e^{-3 \omega^2 z^{1/3}}0 0 0 e^{-3 \omega z^{1/3}}0 0 0 e^{-3z^{1/3}}\end{pmatrix}, \Im z < 0, \end{cases}
B_{0 i}( T = 0 ) = - \frac{2 v^{2}}{\pi \kappa}\epsilon_{0 i j}p^{j}\frac{\partial}{\partial m_{\sigma}^{2}}\left\{\frac{1}{m_{\sigma}^{2}- M^{2}}\left[ \left( M \frac{e^{- \tilde{p}M}}{( \tilde{p})^{2}}+ \frac{e^{- \tilde{p}M}}{( \tilde{p})^{3}}\right) - ( M \rightarrow m_{\sigma}) \right] \right\}
pb_{j-1}+1 =\beta_j\leqslant B_0(pq)^{\frac{j}{2}}, (q\beta_{j-1}+1)(q\beta_{j-1}+2) \leqslant (q\beta_{j-1}+2)^2=b_j^2\leqslant\widetilde{B}_0^2(pq)^{j}.
\kappa_{g_t}={}\det A_t-1 ={}\det[(1+A+e^{-2t}(1-A))^{-1}(1+A -e^{-2t}(1-A))]-1 ={}\frac{\left(1-e^{-2t}\frac{1-\lambda}{1+\lambda}\right)\left(1-e^{-2t}\frac{1+\lambda}{1-\lambda}\right)}{\left(1+e^{-2t}\frac{1-\lambda}{1+\lambda}\right)\left(1+e^{-2t}\frac{1+\lambda}{1-\lambda}\right)}-1
\partial_{\mu}(ee^{\mu}_{{f}}u^{{\{f\}}}_{{m}})+{2 \over{c (\alpha^\prime )^{{1/2}}}}\epsilon^{\mu \nu \rho}e_{\mu{f}}u^{{m\{f\}}}\partial_{\nu}\theta \Gamma_{{mn}}C^{-{1}}\partial_{\rho}\theta= 0
\liminf_{N\to\infty}\frac{2}{N^2}\log\P_{N,r_N}(U_\varepsilon^x)\geq - \lim_{N\to\infty}\frac{2}{N^2}\int_{U_\varepsilon^x}\left(\log\frac{\d\P_{N,h_N}}{\d\P_{N,r_N}}\right) \d\P_{N,h_N}=-I_r(h).
\int_{\mathbb R^2}( \rho_{(1),\infty}^{\rm OCP}(\mathbf r; Q) - 1 ) | \mathbf r |^4 \, d \mathbf r ={1 \over (\pi \beta)^2}\Big ( b_0(\beta) + b_1(\beta) Q + b_2(\beta) Q^2 \Big ) Q,
\left( \frac{1}{k+1}, \frac{1}{k}\right] = \left( \frac{1}{k+1}, \frac{1}{k+1}+ \frac{1}{k(k+1)}\right] & = \bigcup_{\ell=k(k+1)}^{\infty}\left( \frac{1}{k+1}+ \frac{1}{\ell+1}, \frac{1}{k+1}+ \frac{1}{\ell}\right],
N^{(l)}(L) \; := \; \sum_{\{\alpha_a,\beta_a\}}\prod_{b=1}^l \left[\psi_{\alpha_b}^{(b)}(-L,0) \; U\Big({\cal C}(L)\Big)\;\overline{\psi}_{\beta_b}^{(b)}(+L,0) \right] \; ,
b \rightarrow b B ( r ) , \qquad B ( r ) \equiv \left[ 1 +{\frac{\varepsilon_{2}- r}{r - \varepsilon_{1}}}\left({\frac{\varepsilon_{1}}{R_{0}}}\right)^{2}\mathrm{e}^{\mp 2 a \varepsilon_{2}}\right]^{- 1}~ .
L = 2 \sqrt{-{\rm det}\Big(G_{\hat\mu \hat\nu}+ i G_{\hat\mu\rho}G_{\hat\nu\lambda}\tilde{H}^{\rho\lambda}/ \sqrt{-G_5}\Big)}+{1\over 2}\tilde{H}^{\mu\nu}\partial_5 B_{\mu\nu}-{1\over 4}\epsilon_{\mu\nu\rho\lambda\sigma}{G^{5\rho}\over G^{55}}\tilde{H}^{\mu\nu}\tilde{H}^{\lambda\sigma}.
\sum_{F \in \Pi[n]}(-1)^{|F|}\sum_{\begin{subarray}{c}S \subset \{1,\dots,|F|\}\end{subarray}}(|S|-1)! (|S^c|-1)!\ \bigg| \sum_{i \in \underset{j\in S}{\bigcup}F_j}x_i \bigg| = \sum_{A \subset \{1,\dots,n\}}\bigg| \sum_{i \in A}x_i \bigg| \sum_{\pi \in \Pi(A)}\mu(\pi) \sum_{\tilde\pi \in \Pi(A^c)}\mu(\tilde\pi) \ .
\{f_1, \ldots, f_n\}_{(AG, A^*G)}= \Pi^{\sharp}\big|_M (d_A f_1 \wedge \cdots \wedge d_A f_{n-1}) f_n = \Pi^{\sharp}( d (\beta^* f_1) \wedge \cdots \wedge d (\beta^* f_{n-1}))\big|_M f_n = \Pi (\beta^* f_1 , \ldots, \beta^* f_{n-1}, \beta^* f_n)\big|_M = (-1)^{n-1}\big( \beta^*\{f_1, \ldots, f_n\}\big) \big|_M = (-1)^{n-1}\{f_1, \ldots, f_n\}_M.
\lambda_1(J)= \min_{\begin{subarray}{c}u\in H^{1}_0(J) u\neq 0\end{subarray}}\bigg\{\frac{\int_{J}p(x)u'(x)^2 dx+ \int_J q(x) u(x)^2 dx}{\int_{J}w(x)u(x)^2 dx}\bigg\},
V^{2 - 2 h}\sum_{e n}[ \mathrm{e x p}( i \pi e / k ) - \mathrm{e x p}( - i \pi e / k ) ]^{2 h - 2}= V^{- 2 h}\left( \begin{array}{c}{h - 1}\\{2 h - 2}\\ \end{array}\right)
\psi_{1}= \frac{3}{2 \sqrt{2}}( f_{1}+ f_{2}) - \frac{1}{\sqrt{2}}f \ , \, p s i_{2}= \frac{2}{3}\bar{\phi}+ \frac{1}{4 \sqrt{2}}( f_{1}+ f_{2}) + \frac{1}{2 \sqrt{2}}f .
E(\theta_1,\ldots,\theta_n)= \sum_{k\neq j}\ln\frac{1}{|e^{\imath \theta_k}-e^{\imath \theta_j}|}+ p\sum_{j=1}^{n}\ln\frac{1}{|1-e^{\imath\theta_j}|}+q\sum_{j=1}^{n}\theta_j,\ \theta_j\in (0,2\pi),
V\tilde\pi_1(U^*)V^* e_{\sigma_2^k(m_{j''})}= V\tilde\pi_1(U^*)e_{\sigma_2^k(m_{j'})}= V e_{\sigma_1^{k}(n_{j})}= e_{\sigma_1^{k}(n_{j})}= \tilde\pi_2(U^*) e_{\sigma_2^k(m_{j''})}
m(\pi) = \lim_{a\to \infty}\frac{1}{2a}\int_{-a}^{a}Q_\pi^{ir}\varphi(\pi)Q_\pi^{-ir}dr=\lim_{a\to \infty}\frac{1}{2a}\int_{-a}^{a}Q_\pi^{ir}[(\Phi \otimes\mathrm{id}) (u^{(\pi)})]Q_\pi^{-ir}dr &=\lim_{a\to \infty}\frac{1}{2a}\int_{-a}^{a}(\Phi\circ \tau_r \otimes\mathrm{id}) (u^{(\pi)}) dr = (\Psi \otimes \mathrm{id})(u^{(\pi)}).
xg_1=\left[\begin{array}{ccccc}1_t 0 0 0 0 0 0 0 0 u 0 0 1 0 0 0 0 0 1_t 0 0 0 0 0 v\end{array}\right],yg_2=\left[\begin{array}{ccccc}1_t 0 0 0 0 0 0 0 0 u' 0 0 1 0 0 0 0 0 1_t 0 0 0 0 0 v'\end{array}\right].
I = ( I^{( 1 )}| I^{( 2 )}) = ( I_{1}^{( 1 )}, I_{2}^{( 1 )}, \ldots , I_{m_{1}}^{( 1 )}\; | \; I_{1}^{( 2 )}, I_{2}^{( 2 )}, \ldots , I_{m_{2}}^{( 2 )})
\bar{L}(r, h_\beta) = P_{0}(r, h_\beta)L_0(r, h_\beta) + \left(1-P_{0}(r, h_\beta)\right)L_1(r, h_\beta) &=\underbrace{(4\pi f/c)^2 \left( r^2+h_\beta^2 \right)}_{\rm{FSPL}}\, \underbrace{\big( \eta_1 + P_{0}(r, h_\beta)(\eta_0-\eta_1) \big)}_{\rm{average\,excessive\,path\,loss}}.
\textnormal{Var}_0(\tau_i)=\sum_{j=1}^i\frac{1-\beta^{(i)}_j}{\left(\beta^{(i)}_j\right)^2},\begin{aligned}\textnormal{Var}_0(\widetilde{\tau}_i)=\sum_{j=1}^i\frac{1}{\left(\beta^{(i)}_j\right)^2}.\end{aligned}
t+\frac{1}{t}= 1-\gamma + \frac{1}{1-\gamma}= 1-\gamma + 1+\gamma +\frac{\gamma^2}{1-\gamma}& \leq 2+\frac{4.444^2}{1-4.444\cdot 0.0045}\delta_k^2 \leq 2+20.15214\delta_k^2.
X_{\alpha\circ(G,w)}=U_{(G,w)}(r_{(\alpha_1)})U_{(G,w)}(r_{(\alpha_2,\dots,\alpha_{\ell})})-U_{(G,w)}(r_{(\alpha_1+\alpha_2,\dots,\alpha_{\ell})}) =U_{(G,w)}(r_{(\alpha_1)}r_{(\alpha_2,\dots,\alpha_{\ell})}-r_{(\alpha_1+\alpha_2,\dots,\alpha_{\ell})}) = U_{(G,w)}(r_\alpha)
\varphi_{\rm diss}(t,\xi) = \chi\left(|\xi|N^{-1}\right) \chi\left((1+t)|\xi|N^{-1}\right) \varphi_{\rm hyp}^s(t,\xi) = \chi\left(|\xi|N^{-1}\right) \left(1-\chi\left((1+t)|\xi|N^{-1}\right)\right) \varphi_{\rm hyp}^\ell(\xi) = 1 - \chi\left(|\xi|N^{-1}\right)
|F_{2j,k}^{(c)}|=| \sum_{p=1, p\neq k}^{2j-1}[\bold{F}]_{j,k}F_{2j,k}^{j,p}+ [\bold{F}]_{j,2j}F_{2j,k}^{j,2j}+ [\bold{F}]_{j,j}F_{2j,k}^{j,j}|\le \| \bold{F}^{(i)}\|_2(\sum_{p=1, p\neq k}^{2j-1}|F_{2j,k}^{j,p}|^2 )^{\frac{1}{2}}\frac{j^{\frac{3}{2}}}{M^2},
P(U,V,X,Y,Z)= ( \mathring{A}_S(U,X)\nu_S^\flat(V) + \mathring{A}_S(V,X) \nu_S^\flat(U) ) \; g_S(Y,Z) & -( g_S(U,X)\nu_S^\flat(V) + g_S(V,X) \nu_S^\flat(U) ) \; \mathring{A}_S(Y,Z)
{\mathrm{G}_{\bullet}^{\bullet}}^{\ast}\mathrm{G}_{\bullet}^{\bullet}=\mathrm{I},\qquad{{G}_{+}^{-}}^{\dag}+{\mathrm{G}_{+}^{\bullet}}^{\ast}\mathrm{G}_{+}^{\bullet}+{G}_{+}^{-}=0,{\mathrm{G}_{\bullet}^{-}}^{\ast}+{\mathrm{G}_{\bullet}^{\bullet}}^{\ast}\mathrm{G}_{+}^{\bullet}=0,
h_{2}^{-1}(U) = h_{2}^{-1}((\mbox{int}(K(h_{2})))\setminus K(h_{1})) = h_{2}^{-1}(\mbox{int}(K(h_{2})))\setminus h_{2}^{-1}(K(h_{1})) = (\mbox{int}(K(h_{2})))\setminus h_{2}^{-1}(K(h_{1})) \subset (\mbox{int}(K(h_{2})))\setminus K(h_{1})=U.
Q_n^{(s)}( x ) = \frac{1}{2}\Big( \frac{4z}{1 - z}\Big)^s \int_{-\infty}^{\infty}\Big( ( 1+ z ) e^{\tau}+ 1 -z \Big)^{-s}\times \times \Big( x + (x^2 - 1)^{\frac{1}{2}}\cosh \tau \Big)^{- n - 1}d\tau .
\int d \mu_{C^{1}- C^{2}}[ \theta_{1}] \; \int_{\Lambda}d^{2}x \; t ( x ) \; : \cos \Big [ 2 \sqrt{\pi}\Big ( \theta_{1}^{( b )}( x ) + \theta_{2}^{( b )}( x ) \Big ) \Big ] :_{C^{1}- C^{2}, C^{2}}
n(2\ln n)^{\frac{1}{2}}\int_{I_2}D_n(\sqrt{4-y^2}/S(I) \cdot G_n(x)/2)dy \leq n(2\ln n)^{\frac{1}{2}}\int_{I_2}e^{1-(\sqrt{4-y^2}/S(I)-1)\ln n}D_n(G_n(x)/2)dy \leq n(2\ln n)^{\frac{1}{2}}\int_{I_2}e^{1-(S(I_2)/S(I)-1)\ln n}D_n(G_n(x)/2)dy = n(2\ln n)^{\frac{1}{2}}(b-b_0)e^{1-(S(I_2)/S(I)-1)\ln n}D_n(G_n(x)/2),
\begin{cases}\partial_tf_\varepsilon+\frac{1}{\varepsilon}v\cdot\nabla_{x}f_\varepsilon=\frac{1}{\varepsilon^2}\mathcal{Q}(f_\varepsilon, f_\varepsilon), f_\varepsilon|_{t=0}=f_{\varepsilon, 0},\end{cases}
\Vert f'_h - f' \Vert_1 \leq \big| f(t_1 + h) - f(t_1) \big| + \big| f(t_{M-1}+ h) - f(t_{M-1}) \big| &\enspace\enspace + \sum_{k=2}^{M-1}\Big(\big| f(t_k + h) - f(t_k) \big| + \big| f(t_{k-1}+ h) - f(t_{k-1}) \big| \Big) &\leq 2(M-1)C|h|^\mu \leq 2(L-1)C |h|^\mu,
\Gamma^{0}_{\underline\alpha\underline\beta}= \left(\begin{array}{cc}1 0 0 1\end{array}\right)\,, \quad \Gamma^{1}_{\underline\alpha\underline\beta}=\left(\begin{array}{cc}0 1 1 0\end{array}\right)\, , \quad \Gamma^2_{\underline\alpha\underline\beta}=\left(\begin{array}{cc}-1 0 0 1\end{array}\right) \,.
D_{(\gamma, \widetilde{\gamma})}^{g}((x',y'),(x'',y'')) = \sum\limits_{k=1}^{q}\int_\gamma\left|\langle \nabla g_{k}^{H,\beta}(x),dx\rangle \right | ~+~ \sum\limits_{k=1}^{q}\int_{\widetilde{\gamma}}\left|\langle \nabla g_{k}^{H,\beta}(y),dy\rangle \right | &= D_\gamma^{g}(x',x'') ~+~ D_{\widetilde{\gamma}}^{g}(y',y''),
{\frac{1}{k !}}\sum_{a_{1}, \dots , a_{k}}T r_{F}\{t_{F}^{a_{1}}t_{F}^{a_{2}}\cdots t_{F}^{a_{k}}\}t_{( i_{1})}^{( a_{1}}t_{( i_{2})}^{a_{2}}\cdots t_{( i_{k})}^{a_{k})}
\begin{array}{c c c}{{\left[ B_{a}, B_{b}\right]}}&{{=}}&{{C_{a b}^{c}B_{c}}}\\{{}}&{{}}&{{}}\\{{\left[ B_{a}, Q_{\beta}\right]}}&{{=}}&{{C_{a \beta}^{\gamma}Q_{\gamma}}}\\{{}}&{{}}&{{}}\\{{\left\{Q_{\alpha}, Q_{\beta}\right\}}}&{{=}}&{{C_{\alpha \beta}^{c}B_{c}}}\\ \end{array}
[y_{(j\,j+1)}, y_{j+2}]_c = [[y_{j},y_{j+1}]_c,y_{j+2}]_c &= y_{(j\,j+2)}-\chi_{j+1}(g_{j}) y_{j+1}[y_{j},y_{j+2}]_c+\chi_{j+2}(g_{j+1}) [y_{j},y_{j+2}]_c y_{j+1}&= y_{(j\,j+2)}+\chi_{j+2}(g_{j+1})(1+\chi_{j+1}(g_{j\,j+2}))\lambda_{jj+2}y_{j+1}&= y_{(j\,j+2)}-b_{jj+2}(0) y_{j+1}.
D(t,t) \ =\ 1 \ \ 1\leq t \leq (k+1)n \ =\ 1 \ \t=(2k+1)n +1 D\big((k+i)n+t,\, (k+i)n+t)\big)\ =\u_i(t)\ 1\leq i \leq k \ \ 1\leq t \leq n \ .
\chi\left( \rho\right) =\frac{\rho^{2}\left({4m^{2}-1}\right)}{\left({2-\rho^{2}}\right)^{2}}+\frac{7\rho^{2}-40}{4\left({4-\rho^{2}}\right)^{2}}+\frac{4m^{2}}{\left({4-\rho^{2}}\right)}.
{|Y_0|}^2+\int_0^T e^{\int_0^s\eta(\tau)d\tau}\left(\eta(s){|Y_s|}^2 +{|Z_s|}^2 +\|U_s\|^2\right)ds & =M(0)+e^{\int_0^T\eta(s)ds}|\xi|^2+\int_0^T 2e^{\int_0^s\eta(\tau)d\tau}Y_sf(s,Y_s,Z_s,U_s)ds,
\begin{aligned}\|B_1(u,\tilde u)\|_X\le C\|u\|_X\|\tilde u\|_{X}, &\|B_2(u,\theta)\|_X \le C\|u\|_X\|\theta\|_Y, &\|B_3(u,\tilde\theta)\|_Y \le C\|u\|_X\|\tilde \theta\|_Y.\end{aligned}
(2\partial_{\overline z}{\mbox U}_{0}(k) -A^*_1{\mbox U}_{0}(k), 2\partial_{z}\widetilde{\mbox U}_{0}(k) -B^*_1 \widetilde{\mbox U}_{0}(k))=0\quad\mbox{in}\,\,\Omega,{\mbox U}_{0}(k)+\widetilde{\mbox U}_{0}(k)=0\mbox{on}\,\,\Gamma_0
\int_{B^2_1(0)}e_{\widehat \epsilon_i}(\widehat d_i)(x,0)\,dx=\frac{\delta_0^2}{C(3)}=\max_{z\in B^2_{\lambda_i^{-1}}}\int_{B^2_{1}(z)}e_{\widehat \epsilon_i}(\widehat d_i)(x,0)\,dx,
C^{l}_{m}(\tau)=\sqrt{\frac{\tau}{ik(k+2)}}\sum_{l^{'}=0}^{k}\sum_{m^{'}=-k+1}^{k}{\rm e}^{i\pi mm^{'}/k}\sin\left[\frac{\pi(l+1)(l^{'}+1)}{k+2}\right]C^{l^{'}}_{m^{'}}(-1/\tau)
\psi_{(m_{1},n_{1})}\otimes \psi_{(m_{2},n_{2})}=\sum_{k=|m_{1}-m_{2}|+1}^{m_{1}+m_{2}-1}\;\sum_{l=|n_{1}-n_{2}|+1}^{n_{1}+n_{2}-1}\;\left[ \psi_{(k,l)}\right],
\psi_1(N,K,s) =\frac{\sum_{i=0}^{K-1-s}\binom{K}{s+1+i}(N-1)^i}{\sum_{i=0}^{K-1-s}\binom{K-1}{s+i}(N-1)^i}&=\frac{\sum_{i=0}^{K-1-s}\frac{K}{s+1+i}\binom{K-1}{s+i}(N-1)^i}{\sum_{i=0}^{K-1-s}\binom{K-1}{s+i}(N-1)^i}& \leq \frac{\sum_{i=0}^{K-1-s}\frac{K}{s}\binom{K-1}{s+i}(N-1)^i}{\sum_{i=0}^{K-1-s}\binom{K-1}{s+i}(N-1)^i}=\frac{1}{\lambda}.
\sigma_{k}\left( D^{2}u\right) =e^{kt}2^{k-1}\left[ -\binom{n-2}{k-2}+\binom{n-2}{k-1}\right] r^{2}& +e^{kt}2^{k-1}\left[ \binom{n-2}{k-2}+\binom{n-2}{k-1}\right]e^{-t}h^{\prime\prime}& +e^{kt}2^{k-1}2\binom{n-1}{k}.
L_0(k,A) =\partial_z\left(-\mathcal{M}_k+m(k)\right)-2a\partial_z\left(\cos(z)\cdot\right)+\mathcal{O}(A^2+b^2) L_1(k,A) =\left[-\partial_z\mathcal{M}_k,z\right]+m(k)-2a\cos(z)+\mathcal{O}(A^2+b^2) L_2(k,A) =\left[\left[-\partial_z\mathcal{M}_k,z\right],z\right]+\mathcal{O}(A^2+b^2)
0 = \int_\Omega \left(\nabla u_n \nabla u_0 - \lambda_n m u_n u_0\right) - \lambda_n \int_\Omega a u_n^{p-1}u_0 - \lambda_n \int_{\partial \Omega}b u_n^{q-1}u_0 & = \lambda_n \left\{- \int_\Omega m u_n u_0 - \int_\Omega a u_n^{p-1}u_0 - \int_{\partial \Omega}b u_n^{q-1}u_0 \right\}
i = 2 \implies S(n-i+1) - S(n) = - \frac{1}{2}n^2+\frac{1}{2}n, i = 3 \implies S(n-i+1) - S(n) = -n^2+2n-1, i = 4 \implies S(n-i+1) - S(n) = - \frac{3}{2}n^2+ \frac{9}{2}n -4.
\psi_{\Gamma}(x)=\int_{[0,1)^d}f(x+t)D_{\Gamma}(t)\,dt=\sum_{j\in\mathbb{Z}^d,|j_i|\leq\Gamma_i}a_j\cos(2\pi\langle j,x\rangle)+b_j\sin(2\pi\langle j,x\rangle)
\left[P^{s,N}_{HP}(t)f\right](x_n)=\left[\mathcal{S}^N(t)f\circ \mathsf{eval}_N\right](U^*x_nU) \to \left[\mathcal{S}^N(t)f\circ \mathsf{eval}_N\right](U^*xU)=\left[P^{s,N}_{HP}(t)f\right](x).
\frac{1}{\sqrt{n}}\sum_{k>r}\frac{\tilde\lambda_k}{\tilde\lambda_1-\tilde\lambda_k}=\frac{1}{\sqrt{n}}\sum_{k>r}\frac{\lambda_k}{\tilde\lambda_1-\lambda_k}\leq \frac{1}{\sqrt{n}}\sum_{k>r}\frac{\lambda_k}{\lambda_1-\lambda_k}
-i\eta \int d\theta\, f^+\left(\theta +\frac{\pi i}3\right) z^\dagger(\theta)z\left(\theta - \frac{\pi i}{3}\right)\Psi_n = \; -i\eta P_n \int d\theta\, f^+\left(\theta +\frac{\pi i}3\right) a^\dagger(\theta)a\left(\theta - \frac{\pi i}3\right)\Psi_n.
\gamma (s,0)=\tau_{V\alpha}(\Phi_0^{\Gamma_{\tilde{L}}}(\tilde{R}_h^{e-}(g_1, g(s))))=\tau_{V\alpha}(\tilde{R}_h^{e-}(g_1,g(s)))=g_1 & \gamma (s,h)=\tau_{V\alpha}(\Phi_h^{\Gamma_{\tilde{L}}}(\tilde{R}_h^{e-}(g_1, g(s))))=\tau_{V\alpha}(\tilde{R}_h^{e+}(g_1,g(s)))=g(s)
V(T) \> = \>\exp \Biggl \{\> \int_0^T dt \> \rho^{\mu}(t) \gamma_{\mu}\> + \> \frac{1}{2}\int_0^T dt_1 \int_0^{t_1}dt_2 \>\left [ \rho_{\mu}(t_1) \gamma^{\mu}, \rho_{\nu}(t_2) \gamma^{\nu}\right ] \> + \ldots \> \Biggr \}\> .
\aligned \quad||\sigma(t,\cdot,x)-\sigma(t,\cdot,y)||^2+2\langle x-y,b(t,\cdot,x)-b(t,\cdot,y)\rangle &\quad+\sum_{i=1}^2\int_U|f_i(t,\cdot,x,u)-f_i(t,\cdot,y,u)|^2\nu(du) &\quad+2\int_U\langle x-y,f_2(t,\cdot,x,u)-f_2(t,\cdot,y,u)\rangle\nu(du) &\le g(t,\cdot)\eta_R(|x-y|^2),\t\ge 0, |x|\vee|y|\le R, a.s.\endaligned
\int_{|\mu|\le |\eta|^{3/2}/2}e^{3i\mu^2/4}z\left(\eta+\frac{\mu}{\sqrt{\eta}}\right)w\left(\eta-\frac{\mu}{\sqrt{\eta}}\right)=|\eta|^{3/2}\int_{|\nu|<1/2}e^{3i\eta^3\nu^2/4}z(\eta(1+\nu))w(\eta(1-\nu))d\nu
\int \langle\phi,g\rangle |\eta_0| \,d|T| \, =\, \int \langle\phi, g\rangle \,d|T_0| =\, \lim_{j\to+\infty}\int \langle\phi,\frac{\eta^j_1}{|\eta_0^j|}\rangle\,d|T_0^j| \,=\, \lim_{j\to\infty}\int \langle\phi ,\eta_1^j\rangle \,d|T^j| \,=\, \int \langle\phi,\eta_1 \rangle\,d|T|
\displaystyle\frac{1}{k}(e_{\boldsymbol\sigma}^l,\bar{\boldsymbol \sigma}_h) + (B_h e_{\boldsymbol\sigma}^l,\bar{\boldsymbol \sigma}_h)+2(u^{l-1}_h e_u^l,\nabla\cdot\bar{\boldsymbol \sigma}_h)=-(\vert e_u^{l-1}\vert^2,\nabla\cdot\bar{\boldsymbol \sigma}_h), \ \ \forall \bar{\boldsymbol\sigma}_h\in{\boldsymbol \Sigma}_h.
(i+1-dz_1)(j+1-dz_2)\Tilde{B}(\frac{i}{d},x,\frac{j}{d},y)+(i+1-dz_1)(dz_2-j)\Tilde{B}(\frac{i}{d},x,\frac{j+1}{d},y) &+(dz_1-i)(j+1-dz_2)\Tilde{B}(\frac{i+1}{d},x,\frac{j+1}{d},y)+(dz_1-i)(dz_2-j)\Tilde{B}(\frac{i+1}{d},x,\frac{j+1}{d},y) &=\Tilde{B}(z_1,x,z_2,y)+o(1)
\left( \sigma.(g_{1}^{m_{1}}g_{2}^{m_{2}}\dots g_{r}^{m_{r}})\right)(b) = \sigma.(g_{1}^{m_{1}}g_{2}^{m_{2}}\dots g_{r}^{m_{r}})(\tau a) = (\sigma_{1}g_{1})\dots (\sigma_{q}g_{r})(\tau a) = g_{1}(\sigma_{1}^{-1}(\tau a))\dotsb g_{r}(\sigma_{q}^{-1}(\tau a)).
F_{\omega_0,\mu}(\psi)\,=\, J_{\omega_0}(\psi) \,-\,\frac{1}{V}\int_M\,\psi\,\omega^n_0\,-\,\frac{1}{\mu}\log\left(\frac{1}{V}\,\int_M \,\frac{1}{|S|_{H_0}^{2\beta}}\,e^{h_0-\mu\psi}\,\omega^n_0\right),
S_{B F}[ e , \omega , \phi ] = \int d^{2}x \, \left[ - \phi^{0}\left( d \omega - \frac{\Lambda}{2}e^{a}\wedge e^{b}\epsilon_{a b}\right) + \Lambda \delta_{a b}\phi^{a}\left( d e^{b}+ \omega_{\, \, c}^{b}\wedge e^{c}\right) \right] .
\sum_{n=0}^{\infty}ne^{-n\beta\hbar\omega}=-\frac{\partial}{\partial(\beta\hbar\omega)}(\sum_{n=0}^{\infty}e^{-n\beta\hbar\omega})=\frac{e^{-\beta\hbar\omega}}{(1-e^{-\beta\hbar\omega})^{2}}.
\displaystyle{e_{1}(U_{n+k})(0) \le \rho_{n,k}(T) \sum_{l=0}^k\int_0^T \| \mathcal{\mathbf{B^{\ast}_{n+l}}}(U_{n+l}) \|_{G_{n+l}}^2 dt \,,}\forall \k=1, \ldots, p \,,
f_{D_{4}}= \frac{4 \pi}{\sqrt{y^{2}+ x_{0}^{2}}}\, K_{0}\left( \frac{\sqrt{( y^{2}+ x_{0}^{2}+ x_{1}^{2}) ( y^{2}+ x_{0}^{2}+ x_{2}^{2}) ( y^{2}+ x_{0}^{2}+ x_{3}^{2})}}{y^{2}+ x_{0}^{2}}\right) e^{- i \, \frac{x_{0}x_{1}x_{2}x_{3}}{y ( y^{2}+ x_{0}^{2})}}\, .
g_{c r}\ge \frac{1}{1 - \frac{L \pi}{2 \Lambda^{2}}+ \frac{M^{2}}{3 \Lambda^{2}}( \frac{\pi}{2 L}- 1 / \Lambda^{2}) - \frac{M}{\Lambda^{2}}\mathrm{l n}2}\approx \frac{1}{1 - \frac{\pi | e | | f_{1}|^{1 / 2}}{2 \Lambda^{2}}- \frac{| e |}{\Lambda^{2}}( \frac{f_{2}^{2}}{| f_{1}|})^{1 / 2}\mathrm{l n}2}.
\zeta_1+\frac{\gamma_{11}^1}{4d_1}+\frac{d_2^2}{d_1\gamma_{22}^1}\left(\zeta_2+\frac{\gamma_{22}^2}{4d_2}+\frac{\gamma_{22}^1}{d_2}-cz_2\right)^2-cz_1 =0, \frac{2d_2}{\gamma_{22}^1}\left(\zeta_2+\frac{\gamma_{22}^2}{4d_2}+\frac{\gamma_{22}^1}{d_2}-cz_2\right) =\frac{x_1}{x_2}.
V_{K_{n}}(t) =t^{2}V_{K_{n-1}}(t)+t(t^{\frac{1}{2}}-t^{-\frac{1}{2}})V_{J}(t) =t^{2}\bigl(t^{2}V_{K_{n-2}}(t)+t(t^{\frac{1}{2}}-t^{-\frac{1}{2}})V_{J}(t)\bigr)+t(t^{\frac{1}{2}}-t^{-\frac{1}{2}})V_{J}(t) \cdots =t^{2n}V_{K_{0}}(t)+t(t^{\frac{1}{2}}-t^{-\frac{1}{2}})V_{J}(t)(1+t^{2}+\dots+t^{2(n-1)}).
\begin{cases}-\Delta u(x)=u_1(x), x\in\Omega, (-\Delta)^{s_1}u_1(x)=v^p(x), x\in\Omega, -\Delta v(x)=v_1(x), x\in\Omega, (-\Delta)^{t_1}v_1(x)=u^q(x), x\in\Omega, u(x)=v(x)=u_1(x)=v_1(x)=0, x\in \mathbb{R}^n\backslash \Omega,\end{cases}
l_R(-cd)l_R(ad) =l_R(-cd)l_R(a)+l_R(-cd)l_R(d)= =l_R(-cd)l_R(a)+l_R(c)l_R(d)=l_R(a)l_R(-cd)+l_R(a)l_R(b) =l_R(a)l_R(-bcd)=l_R(a)l_R(-a)=0.
\int_{0}^\infty |J_{\nu(k)}(r)|^2r^{1-\tau}\,\mathrm{d}r=2^{1-\tau}\frac{\Gamma(\tau-1)\Gamma(k+\frac{n-\tau}{2})}{\Gamma(\frac{\tau}{2})^2\Gamma(k+\frac{n+\tau}{2})}
PSp(2n,q) =Sp(2n,q)/\langle-I\rangle, PGSp(2n,q) =GSp(2n,q)/Z, P\Sigma Sp(2n,q) =\Sigma Sp(2n,q)/\langle-I\rangle, P\Gamma Sp(2n,q) =\Gamma Sp(2n,q)/Z
|b|^{2}\frac{\overline{a}b}{a\overline{b}}\,\alpha^{2}=|e|^{2}\frac{\overline{d}e}{d\overline{e}}\,\beta^{2}\ \Leftrightarrow\ \left(\frac{\overline{a}b}{|a|}\right)^{2}\alpha^{2}=\left( \frac{\overline{d}e}{|d|}\right)^{2}\beta^{2}.
- \frac{c^{- 2}}{4}r^{2 ( 1 - \alpha )}\left[ \frac{d^{2}}{d r^{2}}+ ( 3 - 2 \alpha ) \frac{1}{r}\frac{d}{d r}- \frac{1}{r^{2}}( m^{2}+ 2 ( \alpha - 1 ) m ) \right] f_{m}= \lambda^{2}f_{m}
\partial_t u - \sum\nolimits_{i,j=1}^na_{ij}(x,t,u)\partial^2_{x_i x_j}u = b(x,t,u,\nabla_x u) \mbox{in}\ \Omega\times (0,T], \sum\nolimits_{i,j=1}^na_{ij}(x,t,u)\nu_j \partial_{x_i}u = \psi_1(x,t,u) \mbox{on}\ \partial\Omega\times (0,T], u|_{t=0}=\psi_0(x),
H_1(z):= \begin{cases}\frac{H(z)}{c_{1,p}z^{p}}\,\, c_{1,p}\neq0 \frac{H(z)}{c_{2,p}z^{p}\log (z)}\,\,\c_{2,p/2}\neq 0 \frac{H(z)}{z^{p}(c_{1,p}+c_{2,p}\log (z) )}\,\,\c_{1,p}, c_{2,p/2}\neq 0 \end{cases}
u^{(i_1)}= em_v\left(u^{(i_1)}\right)f,\ \widetilde{u}^{(i_1)}= \widetilde{e}m_v\left(u^{(i_1)}\right)\widetilde{f},\textit{and}&\widetilde{u}^{(i_1)}= \widetilde{e}\left(e^{-1}\cdot u^{(i_1)}\cdot f^{-1}\right)\widetilde{f},
\sum\limits_{k = 1}^n{\int\limits_0^\infty{{x^{s - 1}}{e^{- kx}}dx}=}\Gamma \left( s \right)\left({\zeta \left( s \right) + \frac{{{{\left({n + \frac{1}{2}}\right)}^{1 - s}}}}{{1 - s}}+{\rm O}\left({{n^{- s - 1}}}\right)}\right).
-\theta\left(\frac{\partial r_{\ell}}{\partial x_j}\right) = \theta\left(\frac{\partial r_{\ell}'}{\partial x_j}\right)-\theta\left(\frac{\partial r_{\ell-1}'}{\partial x_j}\right)-\theta\left(\frac{\partial r_{\ell+1}'}{\partial x_j}\right)
\|\partial_{\Lambda}\partial^pC_{2m}^{(p,q)}\|\leq{1\over\Lambda}P\left(ln{\Lambda\over\Lambda_R}\right){1\over\Lambda^{2m-6}}\left({\Lambda\over M}\right)^{2-\epsilon}\left({M\over\Lambda_R}\right)^{\epsilon'},\qquad q>1, M>\Lambda_R
X_{ab}={- e^4 \over 16 m^2 (pk) (pk')}{\rm Tr}\left[ \gamma^{\beta}(\gamma^{\mu}(p+k)_{\mu}+m) \gamma^{\alpha}( \gamma^{\nu}p_{\nu}+m)\gamma_{\beta}(\gamma^{\rho}(p-k')_{\rho}+m) \gamma_{\alpha}( \gamma^{\lambda}{p'}_{\lambda}+m) \right]
Q^{N_{p}/ 2}= Q^{N_{n - h}/ 2}Q^{N_{h}/ 2}= \sum_{\{\epsilon_{i}= \pm 1 \}}\cos \left[ \frac{\pi}{2}e_{0}\sum_{i}\epsilon_{i}\right] \sum_{\{\epsilon_{j}= \pm 1 \}}\exp \left[ 4 i \lambda \sum_{j}\epsilon_{j}\right]
\operatorname*{l i m}_{x \rightarrow 0}x \frac{d \hat{G}( x )}{d x}= \operatorname*{l i m}_{x \rightarrow 0}\frac{1}{\pi^{2}N}\frac{\hat{G}^{3}( x ) \, x^{3}}{m^{2}\left( x^{2}+ m^{2}\right)}\rightarrow 0 ,
S = \left( \begin{array}{c c c c c c}{{0}}&{{1}}&{{0}}&{{0}}&{{0}}&{{\rightarrow}}\\{{0}}&{{0}}&{{1}}&{{0}}&{{0}}&{{}}\\{{0}}&{{0}}&{{0}}&{{1}}&{{0}}&{{}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{1}}&{{}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{0}}&{{}}\\{{\downarrow}}&{{}}&{{}}&{{}}&{{}}&{{\ddots}}\\ \end{array}\right)
g(W(X,Y)Z,U) =g(R(X,Y)Z,U)+g(Y,U)L(X,Z)-g(X,U)L(Y,Z)+ &+\stackrel{*}{L}(Y,U)g(X,Z)-\stackrel{*}{L}(X,U)g(Y,Z) g(\stackrel{*}{W}(X,Y)U,Z) =g(\stackrel{*}{R}(X,Y)U,Z)+g(Y,Z)\stackrel{*}{L}(X,U)-g(X,Z)\stackrel{*}{L}(Y,U)+ &+L(Y,Z)g(X,U)-L(X,Z)g(Y,U)
\begin{aligned}H(\rho_{T_0}| M_{\Omega_{\rho_{T_0}}}) \ge H(\rho_{T_0+L/2}| M_{\Omega_{\rho_{T_0+L/2}}})\ge\int_{T_0+L/2}^{T_0+L}I(\rho_s | M_{\Omega_{\rho_s}})ds & \ge \frac{L}{2}\min_{s\in[T_0+L/2,T_0+L]}I(\rho_s | M_{\Omega_{\rho_s}}).\end{aligned}
\langle N_{1,2}\psi, N_{2,2}\psi\rangle_{L^2(\Gamma_T)}=- \lambda \int_{\Gamma_T}\left(s\xi +\frac{\tau}{2}\right) \left(\frac{\tau}{2}-s\alpha\right) \partial_\nu^A \eta^0 (\partial_t \log \gamma) \psi^2 \,\d S\,\d t
\dot{V}=x_{1}\dot{x}_{1}+x_{2}\dot{x}_{2}=x_{1}x_{2}-x_{1}x_{2}+\epsilon\frac{x_{2}^{4}}{3}-\epsilon x_{2}^{2}=\epsilon\frac{x_{2}^{4}}{3}-\epsilon x_{2}^{2}.
F_G (q)\left({\zeta_q ( - 1)}\right)^{|E(G)|}=\sum\limits_{H:H \subseteq G}( - 1)^{|E(G)| -|E(H)|}\frac{P_H (q)}{q^{|V(H)|}}\left({\zeta_q (1)}\right)^{|E(H)|},
O_i(P^{Cont})=\left\langle \prod_{j\in I_i}\tau_{d_j}^{(a_j,b_j)}\sigma_1^{k_1(i)}\sigma_2^{k_2(i)}\sigma_{12}\right\rangle^{\mathbf{s}^{\Gamma_{0,k_1(i),k_2(i),1,I_i},o}}.
{\rm Var}\,\bar{S}_{N_{\beta,\omega}}= \varphi''(0) \left(\frac{1}{\beta}- \frac{1}{\beta+\omega}\right) + \varphi'(0))^2\left(\frac{1}{\beta^2}- \frac{1}{(\beta+\omega)^2}\right) - \left(\frac{1}{\psi(\beta)}\right)^2 + \left(\frac{1}{\psi(\beta+\omega)}\right)^2 ,
\begin{cases}u_\epsilon (x )=\widetilde{u}_0 (x )+\epsilon \widetilde{u}_1 (x, \frac{x}{\epsilon}), m_\epsilon (x )=\widetilde{m}_0 (x ) (\widetilde{m}_1 (x,\frac{x}{\epsilon})+\epsilon m_2 (x,\frac{x}{\epsilon}) )\end{cases}
\begin{array}{c l l}{( 3 , 6 ) \otimes ( 2 , \overline{{4}})}&{=}&{( 2 \oplus 4 , 4 \oplus \overline{{2 0}}) ,}\\{{}}&{{}}&{{}}\\{( 1 , 1 0 ) \otimes ( 2 , \overline{{4}})}&{=}&{( 2 , 4 \oplus 3 6 ) ,}\\ \end{array}