harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\lim_{n\rightarrow \infty}n^{2\alpha+1}\int_0^{\tau \wedge t_1 \wedge t_2}(t_1-s)^{\alpha}(t_2-s)^{\alpha}(\sigma'(X_s))^2 \Psi^{n,4}_s ds =\frac{\alpha}{2(\alpha+1)^2}\int_0^{\tau \wedge t_1 \wedge t_2}(t_1-s)^{\alpha}(t_2-s)^{\alpha}(\sigma'\sigma)^2(X_s) ds,
\rho(t,\vec{x})=\rho(t,r)=\frac{f\left( \frac{r}{a(t)}\right)}{a(t)^{2}},{\normalsize u}_{1}{\normalsize =}\frac{\overset{\cdot}{a}(t)}{a(t)}x-\frac{G(t,r)}{r}yu_{2}=\frac{G(t,r)}{r}x+\frac{\overset{\cdot}{a}(t)}{a(t)}y
\begin{array}{| c | | c | c | c | c |}\hline{{\mathrm{c u r r e n t s}}}&{{J^{+ i j}}}&{{J^{+ - i}}}&{{J^{i j k}}}&{{J^{- i j}}}\\ \hline{{\hline \mathrm{d i m e n s i o n s}}}&{{- 1}}&{{1}}&{{1}}&{{3}}\\ \hline{{}}\\ \end{array}
\int d \beta \,{\rm e}^{-\sum_k \beta_k^2 /2 \hat x - l\sum_k t_k \beta_k}{\prod_{i=1}^n \prod_{k<l}2 \sinh{\beta_k -\beta_l\over2 p_i}\over \prod_{k<l}\Bigl( 2 \sinh{\beta_k - \beta_l \over2}\Bigr)^{n-2}}
\nabla g :=\left(\frac{\partial g}{\partial x_{1}},\frac{\partial g}{\partial x_{2}},\dots,\frac{\partial g}{\partial x_{N}},\frac{\partial g}{\partial y_{12}},\dots,\frac{\partial g}{\partial y_{N-1 N}}\right),\ \ \Delta g:=\sum_{k=1}^N\frac{\partial^2 g}{\partial x_{k}^2}+\sum_{k=1}^{N-1}+\frac{\partial^2 g}{\partial y_{kk+1}^2}.
{\cal{G}}_{k_\perp}^{\pm(\eta),sc}(t,t^\prime)=-{\Gamma(-\nu) \over 4\sqrt{\pi}}e^{-{i\pi \over4}}{1 \over k_\perp}{\partial \Omega \over \partial k_\perp}S_0^\prime D_\nu \left({1+i\over \sqrt{\hbar}}S_0(t) \right) D_\nu \left(-{1+i\over \sqrt{\hbar}}S_0(t^\prime) \right).
\sum_{n \le X}\Bigl(1- \frac{\Phi(n)}n \Bigr)^2 \le \sum_{n \le X}\sum_{\substack{d,e \mid n d,e > T}}\frac1{de}= \sum_{T < d,e \le X}\frac1{de}\sum_{\substack{n \le X: [d,e]\mid n}}1 & \le X \sum_{T < d,e \le X}\frac{(d,e)}{d^2e^2}.
G_2 =\tilde F(u, u_{<0}) + \int_0^\infty (b(u,\nabla u_{<k}) - T^w_{b(u,\nabla u)}) \nabla u_k + (\tilde b(u,\nabla u_{<k}) - T^w_{\tilde b(u,\nabla u)}) \nabla \bar u_k \, dk.
+ I^{\rho \nu \mu \sigma}\left( p_{3}^{\lambda}- p_{2}^{\lambda}\right) \left( p_{2}^{\tau}- p_{1}^{\tau}\right) + I^{\lambda \sigma \nu \tau}\left( p_{1}^{\mu}- p_{3}^{\mu}\right) \left( p_{3}^{\rho}- p_{2}^{\rho}\right) + I^{\lambda \mu \nu \tau}\left( p_{1}^{\sigma}- p_{3}^{\sigma}\right) \left( p_{3}^{\rho}- p_{2}^{\rho}\right)
C_{\mathrm{WS}}\ge \frac1n\Big( \frac13\cdot|\mathcal{A}| + 1\cdot|\mathcal{B}| + 0\cdot |\mathcal{C}| + 0\cdot |\mathcal{D}| \Big) \ge\frac{|\mathcal{A}| + |\mathcal{B}|}{3n}\ = \frac{n - (|\mathcal{C}| + |\mathcal{D}|)}{3n}.
f^o(x) =\sup_{y\neq 0}\left[\frac{\langle x,y\rangle-1}{f(y)}-\frac{\langle x,y\rangle-1}{f(y)+th(y)}+\frac{\langle x,y\rangle-1}{f(y)+th(y)}\right] \leq \sup_{y\neq 0}\frac{th(y)(\langle x,y\rangle-1)}{f(y)(f(y)+th(y))}+(f+th)^*(x) \leq t\sup_{y\neq 0}\frac{\langle x,y\rangle-1}{h(y)}+(f+th)^*(x) =th(x)+(f+th)^*(x)\,.
\widetilde{\Delta}=\frac{2}{g}\sum_{i,j}\bigg\{\frac{\partial}{\partial \bar{z}_i}\bigg(g g^{i,j}\frac{\partial}{\partial z_j}\bigg)+\frac{\partial}{\partial z_j}\bigg(gg^{i,j}\frac{\partial}{\partial \bar{z}_i}\bigg)\bigg\}.
C\rho^{N-\frac{\theta}{p-1}}\ge\underset{\tau\to+0}{\mbox{{\rm ess lim}}}\int_{B(z,\rho)}u(y,\tau)\zeta(y-z)\,dy=\int_{{\bf R}^N}\zeta(y-z)\,d\mu(y)\ge\mu(B(z,\rho/2))
\sum_{k=\nu}^\mu \varphi(k)\ \le int_{\nu - 1/2}^{\mu + 1/2}\varphi(x) \, \mathrm{d}x,\mbox{and}\sum_{k=\nu}^\infty \varphi(k)\ \le int_{\nu - 1/2}^\infty \varphi(x) \, \mathrm{d}x,
V_{\rm eff}= (|\sigma + m|^{2}- D)\, |\phi|^{2}+{D\over 2\pi}\ln{\mu\over |\Lambda|}+{|\sigma|^{2}\over 4\pi}\ln{|\sigma|^{2}\over e\mu^{2}}-{|\sigma|^{2}-D\over 4\pi}\ln{|\sigma|^{2}-D \over e\mu^{2}}\, \cdotp
d_{S(T,t)}(w,w')= d_{S(T,t-j)}\left(\left(x'\right)^{t-j},x_{j+1}\cdots x_t\right)+d_{S(T,t-j)}\left(\left(y'\right)^{t-j},y_{j+1}\cdots y_t \right) &+(2^{t-j+1}-1)d_T(x,y)-2(2^{t-j}-1),
C_{\mu}^{a b}= 2 i g \, \overline{{{\psi_{A}^{a}}}}\, g^{A B}\psi_{C}^{b}\,{A_{\mu B}}^{C}+ \left( \, \overline{{{\psi_{A}^{a}}}}\, g^{A B}\, ( \partial_{\mu}\psi_{B}^{b}) + \psi_{B}^{b}\, g^{A B}\, ( \partial_{\mu}\overline{{{\psi_{A}^{a}}}}) \, \right) ,
H_{n}{}^{0n0}= \tfrac{(1 - b)}{1 + b}(y^0)^2\;,H_{n}{}^{Jn0}= - \tfrac{2 b}{1 + b}y^0 y^J\;, H_{n}{}^{nn0}= \tfrac{b A-(1+3b) y^0y^n}{1+b}\;.
y_i = \sum_{j=1}^{m}U\left[ j, k_{i-j+1}\right] , k_i = 1 + \operatorname{round}\left( \frac{\left(n-1\right) \left(x_i-x_\mathrm{min}\right)}{x_\mathrm{max}- x_\mathrm{min}}\right) ,
\begin{aligned}\dot{x}= - y + a_{20}x^2 - 2 b_{20}x y - \frac{a_{20}}{4}y^2 + x \left( a_{20}b_{20}x^2 - b_{20}^2 x y - \frac{a_{20}b_{20}}{4}y^2 \right), \dot{y}= x + b_{20}x^2 + \frac{a_{20}}{2}x y - b_{20}y^2+ y \left( a_{20}b_{20}x^2 - b_{20}^2 x y - \frac{a_{20}b_{20}}{4}y^2 \right),\end{aligned}
\phi_{B}(\textbf{x}) = I \left\lbrace \textbf{x}\in \mathcal{C}_{1}\cap \mathcal{C}_{0}, P\{Y=1\vert \textbf{X}= \textbf{x}\}> P\{Y=0 \vert \textbf{X}= \textbf{x}\}\right\rbrace + I \left\lbrace \textbf{x}\in \mathcal{C}_{1}- \mathcal{C}_{0}\right\rbrace.
\sum_{l,n}\frac{\mu_{p-1}\lambda^{k+n+l}i^kp!}{k!n!l!(p-l)!}\partial_{x^{i_1}}\ldots \partial_{x^{i_n}}C^0_{i_1'\ldots i_{2k}'j_1\ldots j_l[a_{l+1}\ldots a_p}Str \left(\partial_{a_1}\phi^{j_1}\ldots \partial_{a_l]}\phi^{j_l}\phi^{i_1}\ldots \phi^{i_n}\phi^{i_{2k}'}\phi^{i_{2k-1}'}\ldots \right)
\overline{u}\frac{\partial\overline{v}}{\partial x}+\overline{v}\frac{\partial\overline{v}}{\partial y}=-\frac{1}{\rho}\frac{\partial\overline{p}}{\partial y}+\nu(\frac{\partial^{2}\overline{v}}{\partial x^{2}}+\frac{\partial^{2}\overline{v}}{\partial y^{2}})-\frac{\partial}{\partial x}(\overline{u^{\prime}v^{\prime}})-\frac{\partial}{\partial y}(\overline{v^{\prime2}})
\omega_{x}^{( 1 )}= - 9 6 \cdot 1 6 \pi^{2}{\frac{x^{2}m^{2}d m}{( x + 2 m )^{5}}}, \quad \quad \int_{0}^{\infty}\omega_{r}^{( 1 )}= - 1 6 \pi^{2}.
D = \left \langle \begin{pmatrix}w_0 0 0 w_0 \end{pmatrix}, I + \begin{pmatrix}1 0 0 d \end{pmatrix}p\right \rangle D' = \left \langle \begin{pmatrix}w_0 0 0 w_0 \end{pmatrix}, I + \begin{pmatrix}d 0 0 1 \end{pmatrix}p \right \rangle.
U:=\left( \begin{array}{c c c c c}1 0 \cdots 0 0 1 1 \cdots 1 1 1 2 \cdots 2^{c_X-1}2^{c_X}\vdots \vdots \vdots \vdots 1 c_X \cdots (c_X)^{c_X-1}(c_X)^{c_X}\end{array}\right)\ ,
\frac{1}{(q)_{\infty}}\sum_{r=-\infty}^{\infty}(-1)^r q^{r((2k+1)r+2k-2i+1)/2}=\displaystyle\sum_{n_1 \geq \cdots \geq n_{k-1}\geq 0}\frac{q^{n_1^2 +\cdots + n_{k-1}^2 +n_i +\cdots +n_{k-1}}}{(q)_{n_1 -n_2}\cdots (q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}}.
\cot \phi_{-L}\left(n,\{\omega_j\}_{j=-L}^{\lfloor{\frac{n}{\alpha}\rfloor}},E,\{V_0(j)\}_{j=-\alpha L}^{n}\right)+\tan\phi_{-L}\left(n,\{\omega_j\}_{j=-L}^{\lfloor{\frac{n-1}{\alpha}\rfloor}}, E,\{V_0(j)\}_{j=-\alpha L}^{n-1}\right)=E-\omega_{\lfloor{\frac{n}{\alpha}\rfloor}}-V_0(n).
\quad\ 4h^{ij}h_{jl}R_{\mi\}^{l\ \ \m}-4 h^{ij}h^{lm}R_{iljm}&=4h_{22}h_{22}R_{2121}+4h_{11}h_{11}R_{1212}-4h_{11}h_{22}R_{1212}-4h_{22}h_{11}R_{2121}&=4R_{1212}(h_{22}^2+h_{11}^2-2h_{11}h_{22})=4R_{1212}S^2.
\Psi \big([z_0:z_1]\big) := \frac{\imath}{|z_0|^2+|z_1|^2}\, \begin{pmatrix}\frac{1}{2}\,\left(|z_0|^2-|z_1|^2\right) z_0\cdot \overline{z}_1 \overline{z}_0 \cdot z_1 \frac{1}{2}\,\left(|z_1|^2-|z_0|^2\right)\end{pmatrix}.
\{{Q_{1 1}^{\alpha}},{Q_{1 1}^{\beta}}\}= ({\gamma}^{p})^{\alpha \beta}P_{p}+ ({\gamma}^{p_{1}p_{2}})^{\alpha \beta}Z_{p_{1}p_{2}}+ ({\gamma}^{p_{1}\dots p_{5}})^{\alpha \beta}Z_{p_{1}\dots p_{5}}\, ,
< 0 | \tilde{F}_{i j}( 0 ) | e_{\pm}^{\lambda},{\vec{p}}> = -{\frac{i}{( 2 \pi )^{3 / 2}}}\sqrt{\frac{p_{0}}{2}}\epsilon_{i j k}e_{\pm}^{k}\mathrm{l i m}_{p^{2}\rightarrow 0}{\frac{1}{1 - \Pi ( p^{2})}}
\begin{aligned}w e^\phi \leq w e^\phi |_{(p,t_0)}\leq A (u - v - \inf_{M_t}(u - v) + 1)^{\frac{3}{2}}e^{- A(u - v) + 1}|_{(p,t_0)}&\leq A e^2 e^{- A \inf_{M_t}(u - v)},\end{aligned}
z^{T}K z = z^{T}\Big(I-\frac{ll^{T}}{\Vert l \Vert^{2}}\Big)D\Big(I-\frac{ll^{T}}{\Vert l \Vert^2}\Big)z &= z^{T}\Big(I-\frac{ll^{T}}{\Vert l \Vert^{2}}\Big)^{T}D \Big(I-\frac{ll^{T}}{\Vert l \Vert^{2}}\Big)z.
\sum_{\substack{a=1 (a,q)=1}}^qf(\frac{a}{q}+\beta)^3h(-\frac{a}{q}-\beta)e(-\frac{\overline{a}\,v}{q})= \frac{1}{q^4}\mathcal{I}(\beta,q)\sum_{\substack{a=1 (a,q)=1}}^qS(q,a)^3e(-\frac{\overline{a}\,v}{q}) &+D(q,\beta),
E(A)=\sum_{p\ge0}\frac{A^p}{m(p)}=\sum_{p\geq 0}\frac{(CBC^{-1})^p}{m(p)}= \sum_{p\geq 0}\frac{CB^pC^{-1}}{m(p)}=C\sum_{p\geq 0}\frac{B^p}{m(p)}C^{-1}=CE(B)C^{-1}.
\begin{array}{rcl}P_{p}\left(V_{p}\left(t\right),F_{p}\left(t\right),\Pi\left(t\right)\right) = -\Pi\left(t_{1}\right)V_{p}\left(t_{1}\right)-\Pi\left(t_{2}\right)V_{p}\left(t_{2}\right) -C_{p}\left[V_{p}\left(t_{1}\right)+V_{p}\left(t_{2}\right),F_{p}\left(t\right),G\left(t\right)\right]\end{array}
K = \sigma_{\mu}x^{\mu}=\left[\begin{array}{cc}q(x^{0}- x^{3}) -\left[2\right]^{1/2}q^{-1/2}x^{+}-\left[2\right]^{1/2}q^{1/2}x^{-}q x^{0}+ q^{-1}x^{3}\end{array}\right] \equiv\left[\begin{array}{cc}qD B A C/q\end{array}\right]\;,
Y^n_S(v) = Y^n_S(\sum_{i=1}^{c(\boldsymbol m)}w_i) = \sum_{i=1}^{c(\boldsymbol m)}Y^n_S(w_i) = \sum_{i=1}^{c(\boldsymbol m)}\frac{1}{|G|^n}Y^n_B(\sum_{\boldsymbol \gamma \in G^n}\boldsymbol \gamma w_i) = Y^n_B(v) \, .
\frac{x^N}{(x-z_1)\cdots(x-z_m)}=x^{N-m}\left(\sum_{n=0}^\infty z_1^nx^{-n}\right)\cdots\left(\sum_{n=0}^\infty z_m^nx^{-n}\right)=x^{N-m}\cdot\sum_{n=0}^\infty a_nx^{-n}.
(U_{\mathbf{\tau}}^{\ast},U_{\mathbf{\nu}}^{\ast})^{T}=\left(\begin{array}[c]{cc}J_{\varphi}J_{\theta}\sin\varphi -J_{\theta}\sin\varphi J_{\varphi}\end{array}\right) (U_{\varphi}^{\ast},\frac{1}{\sin\varphi}U_{\theta}^{\ast})^{T}
\sum_{k=0}^m \int_{A'\cap D_{k}''}\widetilde{w}^2 f(\widetilde{y}, \widetilde{w}, t_i)\, d\widetilde{y}d\widetilde{w}\leq \int_{A'}\widetilde{w}^2 f(\widetilde{y}, \widetilde{w}, t_i)\d\widetilde{y}d\widetilde{w}\leq & W(X(t_i),5R(T),t_i)\leq CQ(R(T),t_i)\leq CQ,
\begin{aligned}\dim [\pi^{Sp(n)\times Sp(1)}_{d,e}]^{Sp(n-1)\times Sp(1)_\Delta}= 1 (d\ge e \ge 0), \dim\pi^{Sp(n-1)\times Sp(1)_\Delta}= 0 (\pi \not\simeq \pi^{Sp(n)\times Sp(1)}_{d,e}). \end{aligned}
\tilde{f}(s,y) =\alpha_{-1}(-\tilde{\theta}y^{2\tilde{\theta}+1})+\alpha_0y^{\tilde{\theta}+1}-\alpha_1\frac{y}{\tilde{\theta}}+\alpha_2 \frac{1}{\tilde{\theta}^{\rho}}y^{-\tilde{\theta}\rho+\tilde{\theta}+1}
\frac{\partial}{\partial t}|\nabla w|^2=(1+\frac{\alpha'}{2}R)\Delta|\nabla w|^2-2(1+\frac{\alpha'}{2}R)|\nabla^2w|^2-(r+\frac{\alpha'}{4}R^2)|\nabla w|^2+2a\langle\nabla f,\nabla w\rangle.
R_{x,C,U}^{t+1}=\prod_{D\in\mathcal{U}_{x,C}}(1- Q_{D,x,U}^t)\prod_{D\in\mathcal{S}_{x,C}}(1- Q_{D,x,S}^t)-\prod_{D\in\mathcal{U}_{x,C}}(1- Q_{D,x,*}^t)\prod_{D\in\mathcal{S}_{x,C}}(1- Q_{D,x,*}^t).
n+ 1-\operatorname{ord}_2(n!)=2^m+2-\sum_{i=1}^{\infty}\left\lfloor\frac{2^m+1}{2^i}\right\rfloor = 2^m+2-(2^{m-1}+2^{m-2}+\cdots+2+1)=2^m+2-(2^m-1)=3.
( t^{- 1})^{\prime}= t^{\mathrm{i j}}{\frac{\partial \pi^{\alpha}}{\partial p^{\mathrm{i}}}}{\frac{\partial \pi^{\beta}}{\partial p^{\mathrm{j}}}}\;{\frac{\partial}{\partial \pi^{\alpha}}}\otimes{\frac{\partial}{\partial \pi^{\beta}}}\; \in T_{\pi ( \mathrm{p})}( \Sigma ) \otimes T_{\pi ( \mathrm{p})}( \Sigma )
c_3\ : = \ \inf\left\{\int_Xe^v\omega_0^n \ | \v\in (X,c_0b_0\omega_0), \n\log\frac{(n+1)\delta c_{1}}{(2\alpha n+2\beta)c_{2}}\le\sup_Xv\le\log(c_0+b_0)^n\right\},
U_0(p^m)\big(\begin{smallmatrix}p 0 0 1\end{smallmatrix}\big) U_0(p^m) = \coprod_{i=0}^{p-1}U_0(p^m) v_i, \textrm{with}v_i = \big(\begin{smallmatrix}p 0 ip^{m}1\end{smallmatrix}\big).
\liminf_{k\to+\infty}{\mathcal{M}}_2(u_k)=\liminf_{k\to+\infty}\int_\Omega\lambda_2 Q(x)\,\chi_{\{u_k>0 \}}(x)\,dx\ge\int_\Omega\lambda_2 Q(x)\,\chi_{\{u>0 \}}(x)\,dx ={\mathcal{M}}_2(u).
y_{2}(x)=logx\times\frac{b_{0}}{(-2)_{2}}\sum_{r=3}^{\infty}\frac{(\alpha)_{r}(\beta)_{r}}{r!(r-3)!}x^{r}+b_{0}\sum_{r=0}^{3-1}\frac{(\alpha)_{r}(\beta)_{r}}{r!(1-3)_{r}}x^{r}
\frac{\sum_{\substack{u \sim v u \in E}}\mathbf{\mathrm{x}}_u}{\lambda \mathbf{\mathrm{x}}_v}\geq \frac{\lambda \mathbf{\mathrm{x}}_v - (k-1)-(2k+1)\eta}{\lambda \mathbf{\mathrm{x}}_v}\ge 1 - \dfrac{(k-1) + (2k+2) \eta}{k - \frac{1}{16k^2}}\ge \frac{4}{5k},
\begin{aligned}cap_p(X_\varepsilon) \leq (\int_0^\varepsilon x^{\frac{p-n-1}{1-p}}\cdot [Vol(\partial X,\hat{g})+O(x))]^{\frac{1}{1-p}}dx)^{1-p}&=Vol(\partial X,\hat{g})(\int_0^\varepsilon x^{\frac{p-n-1}{1-p}}(1+O(x))dx)^{1-p}&=Vol(\partial X,\hat{g})(\frac{n}{p-1})^{p-1}\varepsilon^{-n}+O(\varepsilon^{-n+1-p})\end{aligned}
\|\frac{1}{N}(\upsilon_N\ast \rho^\mathrm{pair}[n+1])\|_{L^2}\le \frac{gC}{2c}\frac{N^{3\beta}}{N}\cdot \Big(\Big\|\Theta[n-1]\Big\|_\mathrm{HS}+\mu[n-1]\Big)^{3/4}&\le \frac{gC^2}{c}\Big(\frac{N^{6\beta}}{N}\Big).
{_2G_2}\left[\begin{array}{cc}\frac{1}{4}, \frac{3}{4}0, 0\end{array}\mid \frac{t}{x}\right] ={\chi_4^3\choose\varepsilon}^{-1}{_2F_1}\left(\begin{array}{cc}\chi_4, \chi_4^3 & \varepsilon\end{array}\mid\frac{x}{t}\right) &=-p\cdot{_2F_1}\left(\begin{array}{cc}\chi_4, \chi_4^3 & \varepsilon\end{array}\mid\frac{x}{t}\right).
I_0=\int_0^{\pi}\dfrac{d\theta}{(a_{2n}+\cos{2n\theta})^{2s}}=(2\rho^{2n})^{2s}\dfrac{\pi\sum_{m=0}^{2s-1}{2s-1 \choose m}{4s-m-2 \choose 2s-1}\left(\rho^{4n}-1\right)^m}{\left(\rho^{4n}-1\right)^{4s-1}}
F ( \lambda ) = x_{+}^{2 \ell - \mu + \lambda}\int_{0}^{\infty}( \rho + 1 )^{2 \ell - \mu}\rho^{\lambda - 1}d \rho = \frac{\Gamma ( \lambda ) \Gamma ( - 2 \ell + \mu - \lambda )}{\Gamma ( \mu - 2 \ell )}\, x_{+}^{2 \ell - \mu + \lambda}\, ,
\Delta_L(t) \equiv \left[\frac{\partial}{\partial x}E_{\alpha, \beta}^\gamma\left(t{\rm e}^{{\rm i}x}\right) \Big|_{x=0}\right]^2 - E_{\alpha, \beta}^\gamma(t) \cdot \frac{\partial^2}{\partial x^2}E_{\alpha, \beta}^\gamma \left(t{\rm e}^{{\rm i}x}\right) \Big|_{x=0}\geq 0,
s_{j+1,2k}^{[s]}= s_{j+1,2k}^{[s-1]}+ U_{j,k,k-1}^{[s]}s_{j+1,2k-1}^{[r+s-1]}+ U_{j,k,k}^{[s]}s_{j+1,2k+1}^{[r+s-1]}.
D_{k}=-\lambda\,\lambda_{k}-\mu(\lambda-\mu)\theta_{kl}c_{l}+\nu(\nu-\lambda)\eta_{kl}b_{l}\,, E_{k}= -\mu\,\mu_{k}-\mu(\mu-\nu)\omega_{kl}a_{l}+\lambda(\lambda-\mu)\theta_{kl}c_{l}\,, F_{k}= -\nu\,\nu_{k}-\lambda(\nu-\lambda)\eta_{kl}b_{l}+\mu(\mu-\nu)\omega_{kl}a_{l}\,.
T(m, \tau_1, \tau_2)=\sup\limits_{\lambda\in \left[\tau_1, \tau_2\right]}\frac{\left|\frac{1}{m!}\int_{\mathbb{R}}J_m(x)dF(x)(Z_m(\lambda)-\lambda Z_m(1))+c\delta_{\tau}(\lambda)\int_{\mathbb{R}}f^2(x)dx\right|}{\left\{\int_0^\lambda\left(V_{m, \tau}(r; 0, \lambda)\right)^2dr+\int_{\lambda}^1\left(V_{m, \tau}(r; \lambda, 1)\right)^2dr\right\}^{\frac{1}{2}}},
\ddot{\chi}+ \dot{\alpha}_{0}\left( 1 - \frac{\ddot{\alpha}_{0}}{\alpha_{1}^{2}}\right) \dot{\chi}- \frac{1}{3}\left( 1 + \frac{\ddot{\alpha}_{0}}{\alpha_{1}^{2}}\right) e^{- 2 \alpha_{0}}\nabla^{2}\chi = 0 \ .
V(x, k):= \frac{\pi^{\frac{1}{2}-k}}{2 \pi i}\int_{\lambda - i \infty}^{\lambda+i \infty}\frac{\Gamma(s) \Gamma\left(\frac{k}{2}-s \right)}{\Gamma\left(\frac{1-k}{2}+s \right) \zeta(1-k+2s)}\left( \frac{x}{\pi^2}\right)^{-s}{\rm d}s.
\left( \frac{C_{n,k_{1}}- (n-1)\mathbb{I}_{k_{1}=2}-\mu_{k_1}}{\sqrt{2k_{1}}}, \ldots, \frac{C_{n,k_{l}}-\mu_{k_{l}}}{\sqrt{2k_{l}}}\right) \stackrel{d}{\to}N_{l}(0, I_{l}).
\sum_{n=0}^\infty B_n(t)\frac{x^n}{n!}:=\frac{xe^{tx}}{e^x-1}=\left(\sum_{k=0}^\infty \frac{B_k(0)}{k!}x^k\right)\left(\sum_{m=0}^\infty\frac{(tx)^m}{m!}\right),
\begin{aligned}\sum_{x \in T_{N}^{-1}(x')}T_{2}(x) = = \cos\left(2\pi u_{0}\right) + \cos\left(2\pi u_{0}+2\pi\right)+\sum_{j=1}^{\frac{N}{2}-1}\left(\cos\left(2\pi u_{0}+\frac{4\pi}{N}j\right)+\cos\left(2\pi u_{0}-\frac{4\pi}{N}j\right)\right)\end{aligned}
w^2 \, \partial_w^2 \big|_{w = z_i}\, f(w) = f(z_i) \sum_{\substack{j\neq j' (\neq i)}}^N \frac{z_i}{z_{ij}}\, \frac{z_i}{z_{ij'}}+ 2 \sum_{j(\neq i)}^N \Biggl( \, \sum_{j' (\neq i,j)}^N \frac{z_i}{z_{ij'}}\Biggr) \frac{\varphi_i(z_i)}{\varphi_j(z_j)}\, \frac{z_i}{z_{ij}}\, f(z_j) \, .
x_{k+1}= \frac{\tau_{k}}{1+\tau_{k}}z_k + \frac{1}{1+\tau_{k}}x_k \nabla h(z_{k+1}) = \nabla h(z_k) + \tau_{k}\left(\nabla h(x_{k+1}) - \nabla h(z_{k+1}) - (1/\mu)\nabla f(x_{k+1})\right),
d_i = \sum_{\substack{j\ \in\ 2\mathbb{N}_0 - 2 j\ <\i-1\ \ \}}\left(\frac{\zeta (j-i)}{j-i}-\frac{2^j E_{i-j-1}}{2^{i-j+1}-1}\right)\frac{2^{i-j+3}-4}{2^{i+1}-1}\begin{cases}\binom{i}{j+1}d_{j+1}j\neq -2 1/(i + 1) \end{cases}
{\cal L}= - \kappa \dot{B}\left( a \mp \frac{1}{\kappa}B' \right) + i\psi^\ast \dot{\psi}- A_0 (\kappa B' + \rho) - \frac{1}{2m}| (\partial_x -ia) \psi |^2 - \frac{1}{2m}B^2 \rho
A_1(n) = \begin{cases}p^{-3n/2}(-1)^n \chi_{\psi}(p^n) (1-p^{-1})(1+p-p^n) n>0, p^{3n/2}(-1)^n \chi_{\psi}(p^n) (1-p^{-1}) p^{1-n}n \leq 0. \end{cases}
C:=\sup_{0\leq t< T_{max}}\left[a_0+b_0+(|a_{3}|+|b_3|)\|u(\cdot,t)\|_{L^{1}(\overline{\Omega})}+(|a_4|+|b_4|)\|v(\cdot,t)\|_{L^{1}(\overline{\Omega})}\right]<\infty.
\mu_{Mas}(E^u_\cdot(0),E^s_\cdot(0)) =\mu^{(+,0)}_{Mas}(E^u_{(0,\cdot)}(0),E^s_{(0,\cdot)}(0))+\mu_{Mas}(E^u_{(\cdot,1)}(0),E^s_{(\cdot,1)}(0)) &-\mu^{(+,0)}_{Mas}((E^u_{(1,\cdot)}(0),E^s_{(1,\cdot)}(0))).
\left\lbrace\begin{aligned}q_{n+1}^{\epsilon,\Delta t}=q_n^{\epsilon,\Delta t}+\frac{\Delta t}{\epsilon}p_{n+1}^{\epsilon,\Delta t}p_{n+1}^{\epsilon,\Delta t}=\frac{1}{1+\frac{\Delta t}{\epsilon^2}}\Bigl(p_n^{\epsilon,\Delta t}+\frac{\Delta tf(q_n^{\epsilon,\Delta t})}{\epsilon}+\frac{\sigma(q_n^{\epsilon,\Delta t})}{\epsilon}\Delta\beta_n\Bigr).\end{aligned}\right.
\tilde{\phi}^1=\phi^0+\tau\dot{\phi}\big|_{t=0}+\frac{\tau^2}{2}\ddot{\phi}\big|_{t=0}=\phi^0+\frac{\tau^2}{2}\left(-G(\phi^0)+\lambda^0\phi^0\right), \phi^1=\tilde{\phi}^1/\|\tilde{\phi}^1\|.
{\cal S}( \hat{\Gamma}) = \int \left\{\partial_{\mu}c \; \frac{\delta \hat{\Gamma}}{\delta A_{\mu}}\; + \; B \; \frac{\delta \hat{\Gamma}}{\delta \bar{c}}\; + \; \frac{\delta \hat{\Gamma}}{\delta \underline{{{Y}}}}\; \frac{\delta \hat{\Gamma}}{\delta \underline{{{\varphi}}}}\; + \; \underline{{{q}}}\; \frac{\delta \hat{\Gamma}}{\delta \underline{{{\hat{\varphi}}}}}\right\}= 0
\psi_{K_i}\psi_{K_j}+\psi_{K_j}\psi_{K_i}=0, &\psi_{K_i}^*\psi_{K_j}+\psi_{K_j}\psi_{K_i}^*=\delta_{i,j}id,\ (\forall i,j\in \{1,2,\cdots,2bL^d\}).
\mathcal{J}_{4}(y; \mu) = \Bigl| \frac{y_1}{y_2}\Bigr|^{\frac{1}{2}\mu_2}\int_0^1 \tilde{K}_{3\nu_3}\left(2\pi |y_1|^{1/2}\sqrt{1-u^2}\right) \tilde{K}_{3\nu_3}\left(2\pi |y_2|^{1/2}\sqrt{u^{-2}-1}\right) u^{3\mu_2}\frac{du}{u},
\sum_{\alpha=1}^\infty \Upsilon_{M-1}^\alpha(t) \leq k\sum_{\alpha=1}^\infty\left[\frac{1-e^{-2\lambda_\alpha (t-t_{M-1})}}{2\lambda_\alpha}\right]+k\sum_{\alpha=1}^\infty \left[\frac{1-e^{-\lambda_\alpha (t-t_{M-1})}}{\lambda_\alpha}\right] \lesssim k^\frac{3}{2}.
\left\{N_t\Big|_{\left[0,\frac{\tau(t)}{\widetilde{d}(t)}\right]\times(x,\infty]\times[0,\infty]}=0\right\}\ \mbox{and}\ \left\{N_t\Big|_{\left[0,\frac{\tau(t)}{\widetilde{d}(t)}\right)\times(x,\infty]\times[0,\infty]}=0\right\}\,.
\partial_t\bar{\eta}(x,t) = \partial_t\bar{\eta}(\pi(x,t),t)+\partial_t\pi(x,t)\cdot\nabla\bar{\eta}(\pi(x,t),t) = \partial^\circ\eta(\pi(x,t),t)+\partial_t\pi(x,t)\cdot\nabla_\Gamma\eta(\pi(x,t),t) = \overline{\partial^\circ\eta}(x,t)+\partial_t\pi(x,t)\cdot\overline{\nabla_\Gamma\eta}(x,t).
= B_n A_{n+1}- A_n B_{n+1}= B_n B_{n+1}(h(A_n/B_n) - A_n/B_n) = B_n B_{n+1}( p(A_n/B_n) - q(A_n/B_n) A_n/B_n) q(A_n/B_n)^{-1}= C' B_n B_{n+1}(A_n - \gamma_1 B_n)^{c_1}\cdots (A_n - \gamma_u B_n)^{c_u}B_n^{-c_1-\cdots -c_u}B_{n+1}^{-1}B_n^{{\rm deg}(p)}
\pi_i = -C\sum_{j=1}^i \frac{1}{q_j}\bigg(\prod_{k=j+1}^i \frac{p_{k-1}}{q_k}\bigg) + \bigg(\prod_{j=1}^i \frac{p_{j-1}}{q_j}\bigg) \pi_0 &= -C \,\frac{1}{q_i}\bigg[1+\sum_{j=1}^{i-1}\bigg(\prod_{k=j}^{i-1}\frac{p_k}{q_k}\bigg)\bigg] + \frac{q_0}{q_i}\bigg(\prod_{j=0}^{i-1}\frac{p_j}{q_j}\bigg) \pi_0,i=1,2,3,\ldots,n.
\tilde{\Gamma}_{5}( p , p^{\prime}) \to \gamma_{5}\left( 1 - \frac{\alpha}{2 \pi}\right) - \gamma_{5}\frac{\alpha}{2 \pi}\, P^{2}\int_{4 m^{2}}^{\infty}\frac{d M^{2}}{M^{2}}\, \frac{a ( M^{2})}{M^{2}+ P^{2}},
\begin{array}{c}{{E_{1 , 2}^{2}= p_{\bot}^{2}+ m^{2}+ \frac{g^{2}}4 \left( \tau_{1}+ \tau_{2}\right) \pm g \sqrt{\tau_{1}p_{1}^{2}+ \tau_{2}p^{2}+ \frac{\left( H_{z}^{3}\right)^{2}}4}}}\\{{E_{3}^{2}= p_{\bot}^{2}+ m^{2}, \ \ \left( p_{\bot}^{2}= p_{1}^{2}+ p_{2}^{2}\right) .}}\\ \end{array}
k_1(\delta_tV^{\frac{1}{2}}, V^{1}) + k_1 (AV^{1}, V^{1}) + k_1 \sum\limits_{j=1}^{m}\left(I^{(\alpha_j)}B_jV^{1/2}, V^{1}\right) - \kappa k_1 (V^1,V^{1}) = k_1\left(g^{\frac{1}{2}},V^{1}\right),
\tilde U_{k+1}= a_k(1+\tfrac{1}{m\alpha}) + b_k(1+m\alpha) &= \tilde \rho^2 \tilde U_k + \bigl(\alpha^2 ( c^2+2\delta^2 G^2)(1+\tfrac{1}{m\alpha}) +2\alpha^2 G^2(1+m\alpha) \bigr)
\alpha_{n,2}\left( x\right) =n\left( n-1\right) \frac{8n\left(n-1\right) x^{2}+\left( 2n+1\right) \left( 2n-1\right) \left(n-2\right) \left( n-3\right)}{8\left( 2n+1\right) \left( 2n-1\right)^{2}\left( 2n-3\right)}x^{n-4}.
F(t;x,y,1,u,1,v) =\sum_{m=0}^{\infty}\frac{rv(1-yr)(1-r)^m}{(x-xu+u(1-yr)(1-r)^m)(1-ytuv)}\quad\times\prod_{i=0}^{m}\frac{x(1-(1-yr)(1-r)^i)(x-xu+u(1-yr)(1-r)^i)}{(x-u(x-1)(1-yr)(1-r)^i)(x-xu+u(1-rv)(1-yr)(1-r)^i)},
\dot z= \Gamma \frac{\partial R}{\partial x}(x{''}) (x-x_e)= \sum_{\ell=1}^s \rho_{\ell}(t) \Gamma E^{\ell}z = \sum_{\ell=1}^s \rho_{\ell}(t) \Gamma_{i_\ell}e_{j_\ell}^T z,
\int\limits_0^\infty{{x^{- \frac{s}{2}- 1}}\left[{\bar R\left( x \right) + \bar R\left({- x}\right)}\right]}dx = 2{\left({- 1}\right)^{1 - \frac{s}{4}}}\frac{{\Gamma \left({1 - \frac{1}{2}s}\right)}}{{\zeta \left( s \right)}}\cos \frac{{\pi s}}{4}.
\begin{array}{l}{{\Phi_{3}\times \Phi_{\overline{{3}}}= \Phi_{1}+ \Phi_{8}\ , \ \Phi_{6}\times \Phi_{\overline{{6}}}= \Phi_{1}+ \Phi_{8}\ , \ \Phi_{1 0}\times \Phi_{\overline{{{1 0}}}}= \Phi_{1}}}\\{{\Phi_{8}\times \Phi_{8}= \Phi_{1}+ 2 \Phi_{8}+ \Phi_{1 0}+ \Phi_{\overline{{{1 0}}}}\ , \ \Phi_{1 5}\times \Phi_{\overline{{{1 5}}}}= \Phi_{1}+ \Phi_{8}}}\\ \end{array}
\int g'(\bar{x}) \cdot h(\bar{x}) d\mu_{\bar{n}}(\bar{x}) = \int \left( \int g \left( \bar{x}\oplus \bar{y}\right) d\mu_{\bar{m}}(\bar{y}) \right) \cdot h(\bar{x}) d\mu_{\bar{n}}(\bar{x}) = \int \left( \int g(\bar{x}\oplus \bar{y}) \cdot h(\bar{x}) d\mu_{\bar{n}}(\bar{x}) \right) d\mu_{\bar{n}}(\bar{y}) = 0
\overline{{U}}_{W}( T ) = \frac{\Delta n^{2}}{\pi a}\left( - \frac{1}{9 6}+ \frac{\zeta ( 3 )}{4 \pi^{2}}t^{3}- \frac{1}{3 0}t^{4}+ \frac{8}{5 6 7}t^{6}- \frac{8}{1 1 2 5}t^{8}+{\cal O}( t^{1 0}) \right){.}
A_{i}^{\prime}( \vec{x}, t ) = e^{- i t \tilde{A}_{0}}\left[ A_{i}^{\prime}( \vec{x}, 0 ) + \int_{0}^{t}d t^{\prime}e^{i t^{\prime}\tilde{A}_{0}}( \partial_{i}\tilde{A}_{0}) e^{- i t^{\prime}\tilde{A}_{0}}\right] e^{i t \tilde{A}_{0}}
\begin{aligned}\|\lambda^{1/2}|u_{\varepsilon}| + |D_v u_{\varepsilon}| + |(-\Delta_x)^{1/6}u_{\varepsilon}|\| &\le N \||\vec f_{\varepsilon}| + \lambda^{-1/2}|g_{\varepsilon}|\| \le N \||\vec f| + \lambda^{-1/2}|g|\|.\end{aligned}
\int_{t_0}^s d\tau\int_{t_0}^\tau d\xi \, |E(X(\xi),\xi)-E(Y(\xi),\xi)|\leq 2C_2 P^{\frac43}Q^{\frac13}\frac{|s-t_0|^2}{2}\leq &C_2 P^{\frac43}Q^{\frac13}\Delta |s-t_0|\leq P^\gamma \frac{|s-t_0|}{4}.
K_p = - \sum_{i=1}^{p-1}\dfrac{1}{i}\binom{p-1}{i-1}b^{i-1}\bigg( \sum_{j=1}^{p-i-1}(-1)^j \binom{p-i}{j}x^{p-i-j}a^{j-1}\bigg) & - \sum_{i=1}^{p-1}\dfrac{(-1)^i}{i}\binom{p-1}{i-1}b^{p-i-1}a^{i-1}.
\begin{cases}-\Delta\eta ' =(k^2) ' \eta +k^2 \eta ' \,\,\,\,\Omega begin{align*}0.3cm] \frac{\partial \eta'}{\partial n}=\left(-\frac{\partial^2 \eta}{\partial n^2}\right) V\cdot n+\nabla \eta\cdot\nabla_\Gamma(V\cdot n)\,\,\,\,\partial\Omega.\end{cases}
P_{EBPP}= \begin{cases}0 \eta \geq \frac{\sum_{i=1}^{M_R}l_i}{\lambda N_0}P_{max}\eta \leq \sum_{i=1}^{M_R}\frac{{l_i}}{\lambda\left(N_0 + P_{max}l_i\right)}\rho\left(\sum_{i=1}^{M_R}\frac{l_i}{N_0 + P l_i}- \lambda \eta =0 \right) \end{cases}
V_n(A;z,z') =(-i)^n\int_{z'}^z d\zeta_1\int_{z'}^{\zeta_1}d\zeta_2\dots \int_{z'}^{\zeta_{n-1}}d\zeta_n\, A(\zeta_1) A(\zeta_2) \dots A(\zeta_n) &=(-i)^n\int_{z'}^z d\zeta_n\int_{\zeta_n}^z d\zeta_{n-1}\dots \int_{\zeta_2}^z d\zeta_1\,A(\zeta_1) A(\zeta_2) \dots A(\zeta_n).
0 \longrightarrow R(-2n-1)^{\binom{n+1}{2}}\xrightarrow{\; F_n^t \;}R(-2n)^{\binom{n+2}{2}}& \xrightarrow{\; G_n \;}R(-n)^{\binom{n+2}{2}}\xrightarrow{\; F_n \;}R(-n+1)^{\binom{n+1}{2}}\longrightarrow 0
u_{i}^{p-1}(y)-\sum^{3}_{j=1}|\mathcal{K}_{j}|\leq\Theta=u_{i}^{p-1}(y)+\sum^{3}_{j=1}\mathcal{K}_{j}\leq u_{i}^{p-1}(y)+\sum^{3}_{j=2}|\mathcal{K}_{j}|.
g (x)= x^{-k_2}\sum_{n=-\infty}^{+\infty}q^n\xi\frac{\big(q^{n + \lambda +1}\xi / x, q^{n- l_1 + 3/2}\xi /t_1, q^{n-l_2 +3/2}\xi /t_2 ; q\big)_{\infty}}{\big(q^{n+1}\xi / x, q^{n + \lambda -h_1 +1/2}\xi /t_1, q^{n + \lambda -h_2 +1/2}\xi /t_2 ; q\big)_{\infty}},