harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
R_{i'j'}=\overline{R}_{i'j'}&+R(\frac{\nabla f}{|\nabla f|},e_{i'},e_{j'},\frac{\nabla f}{|\nabla f|})-\frac{1}{|\nabla f|^{2}}\sum_k(R_{i'j'}R_{k'k'}-R_{i'k'}R_{k'j'}),
J_\lambda \Bigl(\sum_{i=1}^k U_i\Bigr)=\sum_{i=1}^kJ_\lambda(U_i)+\frac{1}{2}\sum_{i=1}^k \sum_{j\neq i}\int_{\Omega}w_i^5\, U_j-\frac{1}{6}\int_{\Omega}\Bigl[\Bigl(\sum_{i=1}^k U_i\Bigr)^6-\sum_{i=1}^k U_i^6\Bigr].
\sum_{s_1\geq\dots\geq s_{r-1}\geq0}\frac{q^{s_1^2+\dots+s_{r-1}^2-s_1-\dots-s_i}}{(q)_{s_1-s_2}\dots(q)_{s_{r-1}}}=\sum_{k=0}^{i}\frac{(q^{2r+1},q^{r-i+k},q^{r+i-k+1};q^{2r+1})_\infty}{(q)_\infty}.
(y_1 s_{b_2}\cdots s_{b_2-t_2}y_2) (\overbrace{u_1 s_{b_1}\cdots s_{b_1-t_1}\tilde{u}_1}^{=u_1'}s_{b_2}\cdots s_{b_2-t_2+1}u_2')(v_1 s_{b_1}\cdots s_{b_1-t_1+1}v_2).
T r_{w}K_{H_{2}}={\frac{e^{-{\bar{s}/ 4}}}{( 4 \pi \bar{s})^{1 / 2}}}\int_{0}^{\infty}{\frac{d y \cosh y e^{-{y^{2}/ \bar{s}}}}{( \sinh^{2}y + \sin^{2}{\frac{w}{2}})}}.
2\sum_{k=1}^d \sum_{i=1}^{k-1}\sin^2(x_i/2)+\sin^2(x_k/2)+ \sin(x_i/2)\sin(x_k/2) = \sum_{k=1}^d \sum_{i=1}^d \sin^2(x_i/2) + \sin^2(x_k/2) + \sin(x_i/2)\sin(x_k/2) - 3 \omega(x) = (2d-3)\omega + \Big(\sum_{i=1}^d \sin(x_i/2)\Big)^2 \leq (3d-3) \omega.
\hat{h}^1 a^{\nu}(z^1)^{\nu}=\hat{h}^1(\hat{z}^1)^{\nu}=\hat{w}^1\circ f-\hat{w}^1\circ g= =(g_* a\circ f)(w^1\circ f)-(g_* a\circ g)(w^1\circ g)= =(w^1\circ f)\left(g_* a\circ f-g_* a\circ g\right)+(g_* a\circ g)h^1(z^1)^{\nu},
\partial_{N_{r}}K_{r}^{( 1 )}( t_{2}, t_{1}) = \sum_{p = 1 , 2}\int_{C_{p}^{( r )}}K_{r}^{( 1 )}( t_{2}, t ) d t E_{p , N_{s}}^{( 0 )}( K_{r}^{( 1 )}( t , t_{1}) )
\binom{i+\beta-\frac{\beta+1}r+\frac lr}\beta=\binom{i-2-k''(1+l)}\beta\cr=\, \begin{cases}-1 \i-2-k''(1+l)\equiv -1\pmod{\tau^4},\cr1 \i-2-k''(1+l)\equiv -2\pmod{\tau^4},\cr0 .\end{cases}
\int\varphi_{\sigma}(h)E_{\tau_1}\left ( \begin{pmatrix}I X I\end{pmatrix}\begin{pmatrix}h &h\end{pmatrix},s_1\right )E_{\tau_2}\left ( \begin{pmatrix}I X I\end{pmatrix}\begin{pmatrix}h &h\end{pmatrix},s_2\right ) \psi(X)dXdh
\lim_{n\rightarrow+ \infty}\frac{1}{n}\int (\lfloor nX \rfloor )dc =\int X dc, \ \lim_{n\rightarrow+ \infty}\frac{1}{n}\int (\lfloor nY \rfloor )dc =\int Y dc, & \lim_{n\rightarrow+ \infty}\frac{1}{n}\int (\lfloor nX \rfloor +\lfloor nY \rfloor )dc =\int (X+Y) dc.
{\cal F}_{\mathrm{F}= 0}= \frac{e^{2}}{2 \pi}\sum_{n > 0}\frac{1}{n}\sin ( \frac{2 \pi}{2 \pi}{L}n ( x - y ) ) = \frac{e^{2}}{4 \pi}\epsilon ( x - y ) - \frac{e^{2}}{2 \pi \mathrm{L}}( x - y ) .
\sum_{m=0}^{\infty}\sum_{q=1}^{m+1}(-1)^{m}\sum_{i=0}^{n}(-1)^{n+q-i-1}e^{qx}(q-1)!(q)_n{n \brack i}{m+i \brace n+q-1}\frac{y^{m}}{m!}= \sum_{r=0}^{\infty}\sum_{i=0}^{n}(-1)^{n+r-i}e^{(r+1)x}r!(r+1)_n{n \brack i}\sum_{m=r}^{\infty}{m+i \brace n+r}\frac{(-y)^{m}}{m!}.
(q-1)\left(h(q,z)-\frac12 h(q^2,z^2)\right) =(q-1)\sum_{l=1}^{\infty}\frac{k(q^{2l-1},z^{2l-1})}{2l-1}&=\sum_{l=1}^{\infty}\sum_{m=1}^{\infty}\frac{q-1}{q^{2l-1}-1}\frac{(-1)^{m-1}Q_m(q^{2l-1})}{(2l-1)q^{(2l-1)\binom{d}{2}}}z^{(2l-1)m},
\begin{aligned}\left| X_{i\bar j l}- X_{j\bar l}\frac{w_i}{w}\right|^2 = X_{i\bar jl}X_{j\bar i\bar l}- 2 \mathfrak{Re}\left\{X_{i\bar jl}X_{l\bar j}\frac{w_{\bar i}}{w}\right\}+ X_{j\bar l}X_{l\bar j}\frac{w_i w_{\bar i}}{w^2},\end{aligned}
g_{ab}^{[k](\alpha)}(z)=\epsilon_{ab}^{[k](\alpha)}(z) \prod_{i=1}^{\ell}c^{(\alpha)}_{i}\left\langle Q_{1}^{k-b}Q_{0}^{-k-1+b}v_{0},\psi_{a}^{-}(z) \prod_{s=1}^{\ell}\psi_{1}^{+}(z_{s}) \prod_{t=1}^{\ell}\psi_{0}^{-}(z_{t})v_{0}\right\rangle/\tau_{k}^{(\alpha)},
(ig^2)^m\Big({1\over 2\pi i}\Big)^m{1\over 2^m}\sum_{P_m}\int_{t_{P_m}^-<t_1<\dots <t_r< \dots<t_m< t_{P_m}^+}{\hskip-1cm}s(P_m)\prod_{r=1}^m{d z_{i_r}(t_r) - d z_{j_r}'(t_r)\over z_{i_r}(t_r) -z_{j_r}'(t_r)}R(P_m),
{\small \sigma}\left( x_{1}^{r},...,x_{\eta}^{r}\right){\small =}\frac{(-1)^{j}((r-1)(rn+p-1)-rj)...((r-1)(rn+p-1)-rj-t+1)}{\mu}\binom{rn+p-j-1}{j}_{r}{\small .}
([i_1,\ldots,i_{r-1}],S)\sim ([j_1,\ldots,j_{r-1}],S')\Leftrightarrow [i_1,\ldots,i_{r-1}] =[j_1,\ldots,j_{r-1}]{}^{k_S}H_{[i_1]}<{}^{k_{S'}}H,
r_i(x) =q_i^{\langle \mathrm{wt}x-\alpha_i, \alpha_i^{\vee}\rangle}\overline{_ir(\overline{x})}( e'_i(x) =q_i^{\langle \mathrm{wt}x+\alpha_i, \alpha_i^{\vee}\rangle}\overline{_i{e'}(\overline{x})})
\begin{array}{rcl}S_1 = \Big( \omega(\hat{x}) - \omega(x^{(k)}) \Big) + \Big( \omega(x^{(k)}) - f ( \hat{x}, q^{(k)}(\hat{x}, \hat{y}) \Big) &= \Big( \omega(\hat{x}) - \omega(x^{(k)}) \Big) + \Big(f ( x^{(k)}, q^{(k)}(x^{(k)}, y^{(k)}) ) - f(\hat{x}, q^{(k)}(\hat{x}, \hat{y}) ) \Big) \rightarrow 0.\end{array}
\partial_t c_A=\partial_t (\zeta \circ d_\Gamma)(c^{in}-\chi_+ +\chi_-)+(\zeta \circ d_\Gamma)\theta_{0}^{\prime}(\rho)(-\frac{V}{\varepsilon}-\partial_{t}^{\Gamma}h_{\varepsilon})+(\zeta \circ d_\Gamma)(\varepsilon^2\partial_t c^{in}_2+\varepsilon^3 \partial_t c^{in}_3)
C\sum_{|\beta|=m}\|D^\beta u\|^\ast_{a, Q_{2R}^+}\rho^{m+a}+C\sum_{|\alpha|=m}[f_\alpha]^\ast_{a, Q_{2R}^+}\rho^{m+a}&\quad+C\sum_{|\alpha|<m}(\|f_\alpha\|_{L_\infty(Q_{2R}^+)}+\|D^\alpha u\|_{L_\infty(Q_{2R}^+)})\rho^{m+1}.
{\mathrm d}X_{t,i}\, =\, h(X_{t, i}, X_{t, i+1}){\mathrm d}t +{\mathrm d}W_{t,i}\, ; t \ge 0 \, , i \, =\, 1, \ldots , n-1 \,,{\mathrm d}X_{t,n}\, =\, h(X_{t, n}, X_{t, 1}){\mathrm d}t +{\mathrm d}W_{t,n}\, .
G_{R}( x , x^{\prime}) = - \int{\frac{d^{4}k}{( 2 \pi )^{4}}}e^{i k_{\mu}( x^{\mu}- x^{\prime}{}^{\mu})}\Biggl [{\frac{a ( y )^{2}a ( y^{\prime})^{2}\ell^{- 1}}{{\bf k}^{2}- ( \omega + i \epsilon )^{2}}}+ \int_{0}^{\infty}d m \,{\frac{u_{m}( y ) u_{m}( y^{\prime})}{m^{2}+{\bf k}^{2}- ( \omega + i \epsilon )^{2}}}\Biggr ] ,
\partial_{\theta_1, \theta_2}F = \begin{pmatrix}-\frac{1}{4}\sin(\theta_1 - \frac{\pi}{3}) -\frac{1}{4}\sin(\theta_2 - \frac{\pi}{3}) \cos(\theta_1 - \frac{\pi}{3}) \cos(\theta_2 - \frac{\pi}{3})\end{pmatrix}.
I(x,y) = \frac{1}{\sqrt{2^{k-1}\pi^k v^{k-1}}}\cdot H_k\left( \sqrt{2 \pi v}\left( \frac{1}{2}x-y \right) \right) \cdot e^{-\pi v(\frac{1}{2}x^2 + 2 y^2)}.
I_{Y M}^{0}= \int_{M}\mathrm{T r}F_{t}^{*}F_{t}= \frac{1}{2}\int_{M}d^{4}x \ \mathrm{T r}\left[ \left( F_{\mu \nu}+{\cal A}_{\mu \nu}\right) \left( F^{\mu \nu}+{\cal A}^{\mu \nu}\right) \right]
h^{\beta}_{\mu,N}(\lambda)=B_N^{\mu}\big(e^{i\lambda}\big)\frac{(1-e^{-i\lambda \mu})^n}{(i\lambda)^n}-A_N\bigl(e^{i\lambda}\bigr)\frac{p(\lambda)- \beta^2f(\lambda)}{(i\lambda)^np(\lambda)}-\frac{(-i\lambda)^{n}C_{\mu,N}^{\beta}(e^{i\lambda})}{(1-e^{i\lambda\mu})^np(\lambda)},
\sum_{n=0}^{\infty}\frac{1}{(n+\frac{1}{2})(n-\frac{1}{2})}x^{n}=\frac{2(xtanh^{-1}(\sqrt{x})-tanh^{-1}(\sqrt{x})-\sqrt{x})}{\sqrt{x}}=-2-2(1-x)\frac{tanh^{-1}\sqrt{x}}{\sqrt{x}}
\begin{array}{c}<0|\varphi(y,x)\,{\cal{D}}\,\varphi(y_0,-x_0)|0>_E= \begin{displaystyle}=\int_0^{\infty}\frac{\mbox{d}\theta}{2\pi}e^{-m(x+x_0)\cosh\theta}e^{im(y-y_0)\sinh\theta}\hat{T}(\theta,c)\, ,\end{displaystyle}\end{array}
R ( x , \alpha ) = - \frac{( 1 + \alpha )^{2}m^{2}[ ( 1 + \alpha )^{2}m^{2}+ \{( 1 + \alpha )^{2}- ( 1 + x ) \}( \alpha - x ) M^{2}}{\alpha M^{2}[ ( \alpha - x ) ( 1 + x ) M^{2}- ( 1 + \alpha )^{2}m^{2}] [ x ( \alpha - x ) M^{2}+ ( 1 + \alpha ) m^{2}]}.
\sup\limits_{\lambda\in \left[\tau_1, \tau_2\right]}\frac{\left|B_H(\lambda)-\lambda B_H(1)\right|}{\Bigl\{\int_0^{\lambda}\left(V_H(r; 0, \lambda)\right)^2dr+\int_{\lambda}^1 \left(V_H(r; \lambda, 1)\right)^2dr\Bigr\}^{\frac{1}{2}}}&V_H(r; r_1, r_2)=B_H(r)-B_H(r_1)-\frac{r-r_1}{r_2-r_1}\left\{B_H(r_2)-B_H(r_1)\right\},
\left|\partial_x^\alpha\partial_y^\beta \mathcal K(x,y)-\partial_x^\alpha\partial_y^\beta \mathcal K(x+u,y)\right| \leq C|u|^\delta|x-y|^{-n-|\alpha|-|\beta|-\delta}&\quad\quad\begin{cases}|\alpha|=\lfloor s\rfloor, |\beta|=\lfloor J-n\rfloor-\lfloor s\rfloor.\end{cases}
[d-\sum_{k=1}^{j+1}L_{k}]c_{j+1}=-L_{j+1}c_{j+1}+(d-\sum_{k=1}^{j}L_{k})c_{j+1}\leq-L_{j+1}c_{j+1}+(d-\sum_{k=1}^{j}L_{k})c_{j}&\leq-L_{j+1}c_{j+1}+\sum_{k=j+1}^K L_{k}c_k=\sum_{k=j+2}^K L_{k}c_k
\langle T (z) \Phi_1 (z_1) \cdots \Phi_N (z_N)\rangle = \sum^N_{j=1}\left( \frac{h_i}{(z-z_j)^2}+ \frac{1}{z-z_j}\frac{\partial}{\partial z_j}\right) \langle\Phi_1(z_1) \cdots \Phi_N (z_N) \rangle~.
h(x)=\int_{0}^{1}(1-t)^{-1+b}\, t^{-1+c}\!\left(\sum_{j=0}^{\infty}\frac{t^{j}}{(c)_{j}\, j!}\, S_{j}\!\left(x^{2};\frac{b+c-a}{2},\frac{a+c-b}{2},\frac{a+b-c}{2}\right)\!\right)\!\mbox{d}t.
\left\langle V \phi_i \phi_j, \phi_i \phi_j \right\rangle = \int_{\Omega}{V \phi_i^2 \phi_j^2 dx}&\leq \|V\|_{L^{\infty}}\max_{1 \leq i \leq n}{\|\phi_i\|^2_{L^{\infty}}}\lesssim_{V}\max_{1 \leq i \leq n}{\|\phi_i\|^2_{L^{\infty}}}
K^{-1}\leq \left\vert G^{-1}\left( x\right) \frac{y}{\left\vert y\right\vert}\right\vert \leq \frac{K^{d-1}\left( d-1\right) !{d}^{3/2}}{c_{0}}=:c_{1},x\in \mathbf{R},y\in \mathbf{R}_{0}^{d},
X=(\begin{matrix}0&1&0\\ 0&0&0\\ 0&0&0\\ \end{matrix});Y=(\begin{matrix}0&0&0\\ 0&0&1\\ 0&0&0\\ \end{matrix});Z=(\begin{matrix}0&0&1\\ 0&0&0\\ 0&0&0\\ \end{matrix}).
\sum_{i=1}^{n}\{y_{i}-\hat{f}(x_{i},z_{i})\}^{2}+\lambda\int[(\frac{\partial^{2}\hat{f}}{\partial x^{2}})^{2}+2(\frac{\partial^{2}\hat{f}}{\partial x\partial z})^{2}+(\frac{\partial^{2}\hat{f}}{\partial z^{2}})^{2}]dxdz.
S_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,k\right) =\frac{\left( r!\right)^{p}\prod_{s=2}^{L}\left( r_{s}!\right)^{p_{s}}}{k!}\sum_{i=0}^{rp+\sigma}\binom{rp+\sigma -i}{rp+\sigma -k}A_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,i\right) .
\left|j+\hat\theta^{\mathrm f}_{i,k-1}+\eta_{i,k}-\theta_{i,k}\right|^2 &=\left(1+\Im\{\eta_{i,k}\}\right)^2+\left(\hat\theta^{\mathrm f}_{i,k-1}+\Re\{\eta_{i,k}\}-\theta_{i,k}\right)^2.
\frac{d}{d x_3}\sigma\big(s, X (\tilde{t}, \tilde{x}_1,\tilde{x}_2,\tilde{x}_3, \tilde{m}; s)\big) =\frac{1}{u}_3\frac{d}{d s}\sigma\big(s, X (\tilde{t}, \tilde{x}_1,\tilde{x}_2,\tilde{x}_3, \tilde{m}; s)\big) .
x\sqrt{\beta -x}(16x-\beta) =\frac{5\beta - 6x}{\sqrt{\beta-x}}\left(4x^2 - \frac{\beta}{2}x +\frac{\beta^2}{48}+\frac{8}{3}\frac{x^3 (5\beta -6x)^2}{\beta-x)(16x-\beta)^2}\right)
\Lambda(i,j) =\Gamma(i)\oplus F_n(\Gamma(i))\oplus F_n(\Gamma(j)) &=\left(\bigoplus_{t=0}^{b-2}\gamma_{0,i}(t)\right)\oplus \gamma_{a,i}(n)\oplus F_n(\Gamma(i))\oplus F_n(\Gamma(j))
\displaystyle \mathop{l.i.m}\limits_{\|\bigtriangleup_{n}\| \longrightarrow 0}\sum_{\substack{j\in J\setminus J_{K}}}\Phi(\frac{B_{t_{j}}+B_{t_{j+1}}}{2},t_{j}) (B_{t_{j+1}}-B_{t_{j}})= \int_0^T \Phi_{t}\circ d B_{t}.
\lim_{t \rightarrow \infty}h_{r}( t ) \equiv \lim_{t \rightarrow \infty}{\frac{1}{t}}\int_{0}^{t}\chi ( \vec{x}( t^{\prime}) ) \d t^{\prime}= \sum_{i = 1}^{d - 1}\lambda_{r , i}= h_{\mathrm{K S}}\ .
s_3 =M \left(-2{M}^2\mu{w}^2+{M}^2\mu w+4 M\mu{w}^2+{M}^{3}-3{M}^2\mu-M\mu w-2 \mu{w}^2+{M}^2-M\mu+2 \mu \right) - a\cdot (M-1) \left[M \left(\left(M-1 \right)^2- \left(w-2 \right)^2 \right) + (1-w) \left(M^2-w-1 \right) \right]
J(\mathfrak{m}_0, \rho, w)= \int_{\mathcal X}\int_{\mathcal Z^{T}}\int_{\mathcal{R}\times \mathcal P(\mathcal X)^{T+1}}G_w(x,\phi,z,m) \rho(d\phi,dm)\nu^{\otimes T}(dz )m_0(dx),
b_{0}=\frac{1}{16\pi^{2}}\frac{1}{3}(11N_{c}-2N_{f})andb_{1}=-\frac{1}{(16\pi^{2})^{2}}(\frac{34}{3}N_{c}^{2}-\frac{1}{2}N_{f}(2\frac{N_{c}^{2}-1}{N_{c}}+\frac{20}{3}N_{c}))
W_j(z):=\underbrace{\chi(d_g(z, \xi))\mu_j^{-\frac{N-2}{2}}U\left(\frac{{\rm exp}_{\xi}^{-1}(z)}{\mu_j}\right)}_{:=\mathcal U_j(z)}+\mu_j^2 \underbrace{\chi(d_g(z, \xi))\mu_j^{-\frac{N-2}{2}}V\left(\frac{{\rm exp}_{\xi}^{-1}(z)}{\mu_j}\right)}_{:=\mathcal V_j(z)}, z\in M
g = \sum_{n}c_{n}[ 2 \theta ( y - y_{n}) - 1 ] \, , \hspace{2 c m}c_{n}= \left. - \frac{\hat{\chi}T_{n}}{2 ( \hat{d}- 2 )}\frac{1}{\sum_{m}c_{m}| y - y_{m}| + C}\right|_{y = y_{n}}\, .
\{P_{>N}\big(F(u)\big),u_{>N}\}_P &=\{F(u),u\}_P-\{F(u_{\leq N}),u_{\leq N}\}_P \quad-\{F(u)-F(u_{\leq N}),u_{\leq N}\}_P-\{P_{\leq N}\big(F(u)\big),u_{>N}\}_P =-\tfrac{p}{p+2}\nabla(\vert u\vert^{p+2}-\vert u_{\leq N}\vert^{p+2})-\{F(u)-F(u_{\leq N}),u_{\leq N}\}_P &\quad-\{P_{\leq N}\big(F(u)\big),u_{>N}\}_P &=:I+II+III.
U_{2n,N-x}= U_{2n,N+x+1}= \frac{(-1)^n}{\sqrt{2}}\tilde R_n(\lambda(x);\alpha,\beta+1,N), U_{2n+1,N-x}= -U_{2n+1,N+x+1}= -\frac{(-1)^n}{\sqrt{2}}\tilde R_n(\lambda(x);\alpha+1,\beta,N),
d(\omega) ={\inf}\{S_{\omega}(u):\, u\in W(\mathbb{R}^N) \setminus \{0 \}, I_{\omega}(u)=0\}&=\frac{1}{2}\,{\inf}\{\|u\|_{L^2}^2:u\in W(\mathbb{R}^N) \setminus \{0 \}, I_{\omega}(u)= 0 \},
F(t\, \omega)=\int_0^{t}\left(D^2F(s\, \omega) \, \omega, \omega\right)(t-s)\, ds\ge \int_0^{t}\lambda_{\rm min}(s)\, (t-s)\, ds\ge \int_0^{t/2}\lambda_{\rm min}(s)\, s\, ds=G(t/2)
G_0^{\mu\nu\alpha\beta}(x - x') = \int{dp^4\over (2\pi)^4}{{1\over2}(g^{\mu\alpha}g^{\nu\beta}+ g^{\mu\beta}g^{\nu\alpha}) -{1\over2}g^{\mu\nu}g^{\alpha\beta}+{\cal O}(p) \over p^2 - i\epsilon}e^{-ip(x - x')}~.
\lambda q(\eta,\eta) = \int_X \eta\wedge\eta \wedge \Omega^{n-1}\wedge \bar \Omega^{n-1}- -\frac{n-1}{n}\left(\int_X \eta \wedge \Omega^{n-1}\wedge \bar \Omega^{n}\right) \left(\int_X \eta \wedge \Omega^{n}\wedge \bar \Omega^{n-1}\right)
S_{L o c .}^{( N )}= S_{L i o u .}( \sigma ) - \int d^{2}x \det ( e ) \Biggl [{\frac{g^{2}}{2 \pi}}\phi_{2}\triangle \phi_{2}+{\frac{\pi}{2}}( N \lambda -{\frac{A}{\pi}}\sigma ) \triangle ( N \lambda -{\frac{A}{\pi}}\sigma ) - g \phi_{2}\triangle ( N \lambda -{\frac{A}{\pi}}\sigma ) \Biggr ] .
\hat{H}=-\frac{\hbar^{2}}{2M_{tot}}\sum_{\alpha=1}^{3}\frac{\partial^{2}}{\partial X_{\alpha}^{2}}+\frac{1}{2}\sum_{\alpha,\beta=1}^{3}\mu_{\alpha\beta}(P_{\alpha}-\Pi_{\alpha})(P_{\beta}-\Pi_{\beta})+U-\frac{\hbar^{2}}{2}\sum_{s=1}^{3N-6}\frac{\partial^{2}}{\partial q_{s}^{2}}+V.
\left( \begin{matrix}{{\alpha}}&{{\beta}}\\{{\gamma}}&{{\delta}}\\ \end{matrix}\right) \left( \begin{matrix}{{E_{x}}}\\{{E_{y}}}\\ \end{matrix}\right) = \left( \begin{matrix}{{\alpha E_{x}+ \beta E_{y}}}\\{{\gamma E_{x}+ \delta E_{y}}}\\ \end{matrix}\right) ,
H \left( \begin{array}{c}{{\psi_{+}}}\\{{\psi_{-}}}\\ \end{array}\right) = \left( \begin{array}{c}{{P_{-}P_{+}\qquad 0}}\\{{0 \qquad P_{+}P_{-}}}\\ \end{array}\right) \left( \begin{array}{c}{{\psi_{+}}}\\{{\psi_{-}}}\\ \end{array}\right) .
\zeta(s) = \frac{2^{s-1}}{2^{s-1}-1}\phi(s) = \frac{1}{1-2^{1-s}}\phi(s) &= \frac{1}{1-2^{1-s}}\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s},\Re(s) >0, 1-2^{1-s}\neq 0\;.
\int\limits_0^\pi Cf(u,e_n,\psi)\sin\psi d\psi = \int\limits_0^\pi C(T_uf)(0,e_n,\psi)\sin\psi d\psi = \frac{\pi}{|S^{n-1}|}\int\limits_{S^{n-1}}R(T_uf)(\omega,0)|\omega \cdot e_n|d\omega= \frac{\pi}{|S^{n-1}|}\int\limits_{S^{n-1}}Rf(\omega,\omega\cdot u)|\omega\cdot e_n|d\omega.
\langle P_h(\cdot,s),E(\cdot,\tfrac{1}{2}+it)\rangle=\frac{2\sqrt{\pi}\sigma_{2it}(h)}{\Gamma(s)(4\pi h)^{s-\frac{1}{2}}}\frac{\Gamma(s-\frac{1}{2}+it)\Gamma(s-\frac{1}{2}-it)}{h^{it}\zeta^*(1-2it)},
\left[ \frac{E ( x^{7}b , x^{6 0}) E ( x^{1 3}b , x^{6 0}) E ( x^{3 7}/ b , x^{6 0}) E ( x^{4 3}/ b , x^{6 0})}{E ( x^{7}/ b , x^{6 0}) E ( x^{1 3}/ b , x^{6 0}) E ( x^{3 7}b , x^{6 0}) E ( x^{4 3}b , x^{6 0})}\right]^{N}= b^{2 N}.
\left|\left(\frac{zf_3'(z)}{f_3(z)}\right)^2-1\right| = \left|\left(\frac{\rho_3}{1-\rho_3^2}\left(\frac{u \rho_3^2+4 \rho_3+u}{\rho_3^2+u \rho_3+1}+\frac{q \rho_3^2+4 \rho_3+q}{\rho_3^2+q \rho_3 +1}\right)\right)^2-1\right|=1.
{B}_m^pQ_m \stackrel{(b_3)}{=}Q_p{B}_m^pQ_m\stackrel{(b_2)}{=}{B}_n^p{A}_n^pQ_p{B}_m^pQ_m={B}_m^pQ_n{A}_n^pQ_p{B}_m^pQ_m= &\stackrel{(b_2)}{=}{B}_n^p{B}_m^n{A}_m^nQ_n{A}_n^pQ_p{B}_m^nQ_m={B}_n^p{B}_m^n{A}_m^pQ_p{B}_m^pQ_m= &\stackrel{(b_3)}{=}{B}_n^p{B}_m^p{A}_m^p{B}_m^pQ_m\stackrel{(b_1)}{=}{B}_n^p{B}_m^nQ_n.
\rho(n):= \frac{1}{\sqrt{2}}\sum_{w}\frac{N^{\prime\prime}}{N^{\prime}{}^{3/2}}\overline{b_{f,1/w}\big((\frac{2n}{N^{\prime}_{w}})^2N^{\prime\prime}_w/4\big)}+\sum_{w}\frac{N^{\prime\prime}}{N^{\prime}{}^{3/2}}\overline{b_{f,1/w}\big((\frac{n}{N^{\prime}_w})^2 N^{\prime\prime}\big)},
\Delta_{\mu \nu}( p ) ={\frac{1}{Z ( p^{2}) [ p^{2}+ m_{\mathrm{r e n}}^{2}( p^{2}) ]}}\left( \delta_{\mu \nu}-{\frac{p_{\mu}p_{\nu}}{p^{2}}}- m_{\mathrm{r e n}}( p^{2}) \epsilon_{\mu \nu \rho}{\frac{p^{\rho}}{p^{2}}}\right) + \xi{\frac{p_{\mu}p_{\nu}}{( p^{2})^{2}}}
\pi = w_{11}\cdot t_{7}^{2}\cdot t_{9}^{2}\cdot t_{12}^{3}| w_{14}\cdot w_{15}\cdot t_{12}\cdot t_{14}^{7}\cdot t_{16}^{4}| w_{20}\cdot t_{19}^{10}\cdot t_{22}^{5}\cdot t_{23}^{4}
\widehat{\theta}_{ab}^{\alpha}= \frac{1}{\alpha!}\frac{i^{\alpha -1}}{2^{\alpha}}\sum_{i=1}^{\alpha}\{2\theta_{ab}\prod_{k\ne i}(\theta^{\mu_{k}\nu_{k}}\partial_{\mu_k}\partial_{\nu_k})+(x_a \theta^{\mu_i}\,_{b}-x_b \theta^{\mu_i}\,_{a})\prod_{j\ne i}(\theta^{\mu_{j}\nu_{j}}\partial_{\nu_j})\prod_{k=1}^{\alpha}\partial_{\mu_k}\}.
\mu_m(X\setminus k_N) \leq \mu_m \bigg(\bigcup_{j\geq f(k_N)}A_j\bigg) \leq \psi_m \bigg(\bigcup_{j\geq f(k_N)}A_j\bigg) & \leq\sum_{j\geq f(k_N)}2^{-j}=2^{-f(k_N)+1}\leq 2^{-k_N+1}\leq 2^{-N+1}.
D^{k+1}F_N(r) = D\Big[\sum_{k \le 2j \le 2k}(-1)^j \, \alpha_{j,k}\, r^{2j-k}\, F_{N+2j}(r)\Big] &= \sum_{k \le 2j \le 2k}(-1)^j \, (2j-k) \, \alpha_{j,k}\, r^{2j-k-1}\, F_{N+2j}(r)-\!\!\sum_{k \le 2j \le 2k}(-1)^j \, \alpha_{j,k}\, r^{2j-k+1}\, F_{N+2j+2}(r). \noalign{\medskip}=: S_1+S_2.
G = \begin{bmatrix}(Z-I) A_1 (Z-I)^{-1}G_2 G_1 -(Z-I) A_1 (Z-I)^{-1}e_1 \end{bmatrix}, B = \begin{bmatrix}B_2 (Z-I) A_2^* (Z-I)^{-1}B_1 (Z-I) A_2^* (Z-I)^{-1}e_1 \end{bmatrix},
I_2 = \alpha\int^t_0\int^{t}_s e^{(t-\tau)A}A_1G(\tau-s-r)d\tau f(s)ds &= \alpha\int^t_0 e^{(t-\tau)A}A_1\Big(\int^0_{\tau}G(\tau-s-r)f(s)ds\Big)d\tau,
\exp\bigg(\sum_\alpha \partial^\alpha\otimes x_\alpha\bigg)(\cdot \otimes\blacktriangleright_\gg) (1\otimes 1) =\exp\bigg(\sum_\alpha \partial^\alpha\otimes \hat{x}_\alpha\bigg), \exp\bigg(\sum_\alpha \partial^\alpha\otimes \hat{x}_\alpha\bigg)(\cdot \otimes\triangleright) (1\otimes 1) =\exp\bigg(\sum_\alpha \partial^\alpha\otimes x_\alpha\bigg).
d) \ \prod^{p-1}_{j=1}U_{j}(N_{Z_{D}/S^{i}X}(\nu^{j}))=p^{A}m(e^{-x})^{-A}e^{\theta q(e^{-x})}\prod^{p-1}_{j=1}\left(\frac{1-e^{-x}/\nu^{j}}{1-\nu^{-j}}\right)^{-n_{j}},
\nu_{(a^-, i)}^{\hat{S}^k}= \nu_{(a_k^-, i_k)}^* - \frac{w_{(a^-, i)}^{\hat{S}^{k-1}}}{w_{(a^-, i)}^{\hat{S}^{k}}}\big(\nu_{(a_k^-, i_k)}^* - \nu_{(a^-, i)}^{\hat{S}^{k-1}}\big), (a^-, i) \in \hat{N}.
\frac{d}{d \phi^{2}}\left( v ( \phi^{2}) \, ( \exp ( 2 \sinh^{- 1}\phi^{2}) - 2 )^{\lambda / 2 - 3 / 2}\right) = \frac{1}{2}\frac{k \; ( \exp ( 2 \sinh^{- 1}\phi^{2}) - 2 )^{\lambda / 2 - 3 / 2}}{\phi^{2}( 2 + 2 V^{\prime})^{3 / 2}- 1}
D_r L_k f = L_{1+k}^* D_r f + [ (\gamma+k(\gamma-1)) D_r (\bar{\rho}^{\gamma-2}\bar{\rho}_r d) + (1+k) D_r (d_r \bar{\rho}^{\gamma-1}) ] D_r f &= L_{1+k}^* D_r f + Q_{+}D_r f
I_{n}( L , m , d , \vec{\rho}) = \frac{1}{( 2 \pi )^{d - 1}}\int{}d^{d - 1}p \frac{e^{i \vec{p}. \vec{\rho}}}{( \vec{p}^{\, 2}+ ( \frac{n \pi}{L})^{2}+ m^{2})}.
P^{KB}_\mu(T,f) = P^{B}_\mu(T,f),~\underline{CP}^K_\mu(T,f)= \underline{CP}_\mu(T,f), \overline{CP}^K_\mu(T,f) \leq\overline{CP}_\mu(T,f),~P^{KP}_\mu(T,f)=P^{P}_\mu(T,f).
\rm \Pi^{af}_{\mu\nu}(p) = \frac1{3!}\int\!\!\int \frac{d^4q~d^4k}{(2\pi)^8}\,V^{abcd}_{\mu\lambda\sigma\rho}\,G^{dg}_{\rho\alpha}(q) \,G^{ch}_{\sigma\beta}(p-k-q) \,G^{be}_{\lambda\xi}(k) \,V^{ghef}_{\alpha\beta\xi\nu}~,
p_{l_1,l_2}= \frac{1}{h^\theta(T)}\frac{l_1(l_1+\theta-1)l_2(l_2+\theta-1)}{(l_1+l_2+\theta-1)(l_1+l_2+\theta-2)}q_{l_1}^\theta(t)q_{l_2}^\theta(T-t), h^\theta(T) = g_\theta(0;T) = \sum_{l=1}^\infty e^{-l(l+\theta-1)T/2}(2l+\theta - 1)l(l+\theta-1).
\begin{array}{r c l}{{h ( z )}}&{{=}}&{{\frac{z}{( 1 - 2 z^{2})^{1 / 2}}}}\\{{f_{n}( z )}}&{{=}}&{{\frac{z}{( 1 + 2 z^{2^{n}})^{1 / 2^{n}}}.}}\\ \end{array}
\partial_y \log \widetilde Z_{n,\alpha}(y;r) = \frac{n}{4\pi i}\int^\infty_{r}x \left(\mathrm{Tr}\left(\widetilde{Y}^{-1}_{+}(x)\widetilde{Y}^{\prime}_{+}(x)\sigma_3 \right) - \mathrm{Tr}\left(\widetilde{Y}^{-1}_{-}(x)\widetilde{Y}^{\prime}_{-}(x) \sigma_3 \right)\right) dx.
\sum{\gamma_i^2\theta_i^2}= \frac{-k}{2}\sqrt{9k^2 + 12}- 2rp \mp \frac{\sqrt{3}}{2}i(k^2 - 4rp) \pm \frac{\dim(\mathcal{C})}{d}&= \frac{-k}{2}\sqrt{9k^2 + 12}- 2rp \mp \frac{\sqrt{3}}{2}i(k^2 - 4rp) \pm \sqrt{9k^2 + 12}.
\bar{d}\le \big( \sqrt{8 \cdot 4^{d_1^2+1}}+1\big)^{2n^2}-\big(\sqrt{8 \cdot 4^{d_1^2+1}}-1\big)^{2n^2}\le \big(\sqrt{32}\cdot 2^{8^{(2n^3)^{149n^8}}}+1 \big)^{2n^2}- \big(\sqrt{32}\cdot 2^{8^{(2n^3)^{149n^8}}}-1 \big)^{2n^2}\le \big(2\sqrt{32}\cdot 2^{8^{(2n^3)^{149n^8}}}\big)^{2n^2}= 2^{10n^2}4^{n^2 8^{(2n^3)^{149n^8}}}
\chi_m^{(1)}(t)=-\frac{(u_m-v_m)^2}{[(z_m-u_m)(z_m-v_m)Q_m^2(t)]^{3/2}}\,,\quad\phi_m^{(1)}(t)=\frac{\vartheta_mQ_m^2(t)\sqrt{(z_m-u_m)(z_m-v_m)]}}{(u_m-v_m)}
x^n = \frac 1{x^{\frac pq}},\;\;\;\;\;\;\;\;\;\;\;\;\frac{dx^n}{dx}=\left(-\frac pq \right) \left(\frac{x^{\frac pq -1}}{x^{\frac{2p}q}}\right) = \left(-\frac pq\right)\left( x^{-\frac pq -1}\right).
\mathcal{L}_{\mathrm{b u l k}}= i \bar{\psi}_{i}\gamma^{\mu}\partial_{\mu}\psi^{i}+ \bar{\psi}_{2}\partial_{5}\psi^{1}- \bar{\psi}_{1}\partial_{5}\psi^{2}- \omega ( \bar{\psi}_{1}\psi^{1}+ \bar{\psi}_{2}\psi^{2})
\theta_g(x^i, 1; \|\Sigma^i\|_g) - \theta(x^i; \|\Sigma^i\|_g) \leq \delta; \theta(x^0; \|\Sigma^0\|_g) = \theta(x^1; \|\Sigma^1\|_g) = \theta(x^2; \|\Sigma^2\|_g).
P(\mbox{\boldmath $p$}) = l\left( |f_0|^2 \left[|g_{p0}|^2 -(-\eta)^3\frac{H^2}{2\sqrt{(-p\eta)^2}}\right] + \int_{3/2}^{\infty}\,dm \ |f_m|^2 \left[|g_{pm}|^2-(-\eta)^3\frac{H^2}{2\omega}\right] \right)\ .
\phi_{1}= \frac{m}{\sqrt \lambda}\left\{- \frac{3}{4}m^{2}t^{2}{\eta}_{0}- \frac{1}{4}{\eta}_{k{_0}}+ a_{1}e^{i \omega_{1}t}\frac{~ \mathrm{{s h}}~ z}{\mathrm{{c h}}^{2}~ z}+ \sum_{k = 0}^{\infty}a_{k}{\widetilde C}_{k}( t ) \eta_{k}( x ) \right\}
\sigma_z^2 = \frac{\theta_1}{\theta_1 + \theta_2}\frac{P_1}{h_d^2 + h_c^2}, P_{p1}= \frac{\theta_2}{\theta_1 + \theta_2}P_1, P_{d2}= (P_2 - (h_d^2 + h_c^2)\sigma_z^2)^+.
g^{(1)}(r_{1},t_{1};r_{2},t_{2})=\frac{\langle E^{*}(r_{1},t_{1})E(r_{2},t_{2})\rangle}{[\langle|E(r_{1},t_{1})|^{2}\rangle\langle|E(r_{2},t_{2})|^{2}\rangle]^{\frac{1}{2}}},
g^{i_{0}}\rightarrow g^{i_{0}}\; , \; g^{i_{1}}\rightarrow\lambda g^{i_{1}}\; , \; \ldots , g^{i_{n}}\rightarrow\lambda^n g^{i_{n}}\quad \quad (g^{i_{p}}\rightarrow \lambda^pg^{i_{p}}, \; p=0,\ldots,n)
\beta_{\pm j}= \frac{\mp\mu_\mp B_\mp}{\mu_{+}B_++\mu_{-}B_-}\hat h_j(0) &+\frac{1}{\mu_{+}B_++\mu_{-}B_-}\{- \mu_{+}(B_+-A)\alpha_{+j}-\mu_{-}(B_--A)\alpha_{-j}+i\xi_j\mu_+\beta_{+N}-i\xi_j\mu_-\beta_{-N}\}.
\frac{d I_{\epsilon}}{d R}( R ) = - \frac{4 \pi}{R_{0}}\left( \frac{R^{2}- R_{0}^{2}}{R R_{0}}\right) \left( \frac{R^{2}+ R_{0}^{2}}{R^{2}}\right) \int_{\epsilon}^{\infty}\frac{d \tau_{2}}{\tau_{2}^{1 / 2}}e^{- \pi \tau_{2}\left( \frac{R^{2}- R_{0}^{2}}{R R_{0}}\right)^{2}}
i \frac{d}{dt}\left( u_0 \bar{v}_0 + \bar{u}_0 v_0 \right) = \epsilon \left[\bar{u}_0 (u_1 + u_{-1}) - u_0 (\bar{u}_1 + \bar{u}_{-1}) + \bar{v}_0 (v_1 + v_{-1}) - v_0 (\bar{v}_1 + \bar{v}_{-1}) \right].
( P , Z ) \equiv \frac{1}{2}t r ( P Z ) = \frac{1}{2}t r ( \bar{Z}\bar{P}) = \frac{1}{2}( p^{1}z^{3}+ p^{3}z^{1}- q^{2}( p^{2}z^{4}+ p^{4}z^{2}) ) ,
\small \nabla^{(m, n)}\eta(z, y)[h_1]^m [h_2]^n = \tfrac{(p-1)!}{(p-m)!}\cdot \begin{cases}\sum_{i=1}^d (z^{(i)})^{p-m}(h_1^{(i)})^m h_2^{(i)}, \textnormal{if}n = 1, \sum_{i=1}^d (z^{(i)})^{p-m}y^{(i)}(h_1^{(i)})^m, \textnormal{if}n = 0, 0, \textnormal{otherwise}. \end{cases}
{\cal D}_{l}^{+ ( 1 )}( P ){\cal A}= \frac{( - 1 )^{l}}{( l + 1 )}\frac{\Delta ( \lambda ( 1 ) )}{\Delta ( \lambda ( 2 ) )}: Q_{l + 1}^{( + )}[ j_{1}( P ) + \bar{j}_{1}( P ) ] :_{-}\exp ( U )
\vartheta_{\mathbf{d}}:\mathbb{C}[x_{\alpha}:\,\alpha\in[n]]\longrightarrow\mathbb{C}(\mathbf{t}),\quad\vartheta_{\mathbf{d}}(x_{\alpha}):=\mathbf{d}(\mathbf{t})_{\alpha},\,\alpha\in [n]
\sum_{n=-\infty}^{\infty}q^{n(n+k+\ell)}\left(B_{n+k}(z)B_{n+\ell}(z)+B_{n+k}\!\left(-z^{-1}\right)B_{n+\ell}\!\left(-z^{-1}\right)\right) \hskip136pt=2q^{-k^{2}}\left(q^{2};q^{2}\right)_{\!\infty}^{\!2}\theta_{q^{2}}\left(-qz^{2}\right)\delta_{k,\ell}
d(A,B)=\sum_{i=1}^{n}(\sum_{j=1}^{n}\frac{|p_{ij}|}{max_{k}|p_{ik}|}-1)+\sum_{j=1}^{n}(\sum_{i=1}^{n}\frac{|p_{ij}|}{max_{k}|p_{kj}|}-1),P=A^{-1}B
\boldsymbol{\eta}_{u,t}^n:= \boldsymbol{u}(t_n)- \boldsymbol{w}_t^n \widetilde{\boldsymbol{\eta}}_{u,t}^n := \boldsymbol{u}(t_n)- \widetilde{\boldsymbol{w}}_t^n, \eta_{p,t}^n:=p(t_n)-r_t^n.
g (\mu L)^\varepsilon L^{-3+\varepsilon}c_1(\varepsilon) =\left\{2^{-9+2\varepsilon}\pi^{-4+\varepsilon}\Gamma^2(1-\varepsilon/2)g (\mu L)^\varepsilon L^{-3+\varepsilon}\right\}&\times\left\{\zeta^2(\alpha)+(1-\cos\pi\alpha)B(1-\alpha,2\alpha-1)\zeta(2\alpha-1)\right\}.
[T_{a,b}G - G T_{a,b}]_{x,y}= \frac{1}{\sqrt 2}[ \Delta_{a,b}G - \Delta_{b,a}G - G \Delta_{a,b}+ G \Delta_{b,a}]_{x,y}= \frac{1}{\sqrt{2}}[ \delta_{a,x}G_{b,y}- \delta_{b,x}G_{a,y}- G_{x,a}\delta_{b,y}+ G_{x,b}\delta_{a,y}] \, .