harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
g_{B T Z}= - \frac{( r^{2}- r_{+}^{2}) ( r^{2}- r_{-}^{2})}{r^{2}}d t^{2}+ \frac{r^{2}}{( r^{2}- r_{+}^{2}) ( r^{2}- r_{-}^{2})}d r^{2}+ r^{2}( d \phi - \frac{r_{+}r_{-}}{r^{2}}d t )^{2}
J_e(V) = \frac{1}{c}\int_{\mathbb{R}^{n+1}_{+}}c^2 |\nabla V (x,y)|^2 + m^2 c^4 V(x,y)^2 dx dy &+ (-mc^2 + \mu) \int_{\mathbb{R}^n}V(x,0)^2 dx - \int_{\mathbb{R}^n}|V(x,0)|^p dx.
\frac{1}{|\Omega|}\int_\Omega dv(a)\;I_\pm(R) =\frac{1}{2|\Omega|}\int_\Omega dv(a)\;{\rm Tr}\;SD_a(P_\pm-D_aP_\pm D_a)^{2n+2}\chi_R^a &=\frac{1}{2|\Omega|}\int_\Omega dv(a)\;{\rm Tr}\;SD_a(P_\pm-D_aP_\pm D_a)^{2n+2}&+\frac{1}{2|\Omega|}\int_\Omega dv(a)\;{\rm Tr}\;SD_a(P_\pm-D_aP_\pm D_a)^{2n+2}(\chi_R^a-1).
P = \left( \begin{array}{cc}\begin{array}{ccc}P_{(+\,+\,+)}P_{(+\,-\,+)}P_{(+\,-\,-)}\end{array}\mbox{\Huge 0}\hspace*{-19mm}\mbox{\Huge 0}\begin{array}{ccc}\vspace*{-3.5mm}\hspace*{-0.5mm}\ddots \hspace*{-3.1mm}\ddots P_{(-\,-\,-)}\end{array}\end{array}\right)
I_{m,\epsilon}\leq I_{m}^4+I_{m,\epsilon}^5 &=\int_{[\frac{1}{m},m]^d}B_{\kappa_m,p}(f(x),g(x))x^{2\alpha}dx+\int_0^\infty \int_{[\frac{1}{m},m]^d}b_{\kappa_m,p}(x,s/\epsilon)x^{2\alpha}dx se^{-s}ds.
A_G([a,b,0,d],[0,b',c',d']) ={1\over 2i}(-(a,b')_G+(d,c')_G+(a,h_2b')_G-(h_2h_3h_4d,h_2h_3c')_G) &={1\over 2i}((b,b')_G+(d,b')_G-(d,b')_G-(d,d')_G-(h_2b,h_2b')_G - (h_2h_3h_4d,h_2b')_G+(h_2h_3h_4d,h_2h_3h_4d')_G+(h_2h_3h_4d,h_2b')_G) &= 0
\begin{bmatrix}I-VU O O I-UV\end{bmatrix}\begin{bmatrix}\frac{\phi}{\sigma_{-}}\frac{\psi}{\sigma_{+}}\end{bmatrix}= \begin{bmatrix}-P_{-}\left( z^{n}vP^{-}\left( \frac{q}{z^{n}\sigma_{+}}\right) \right) -P_{+}\left( z^{-n}uP^{+}\left( \frac{q}{\sigma_{-}}\right) \right)\end{bmatrix}.
\bar{W}(\phi)=\pm 4\frac{1}{\sqrt 2}(\frac{1}{4}{\rm sinh}2\phi-\frac{3}{2}\phi ) \hspace{0.5cm},\hspace{0.5cm}\bar{W}(\phi)=\pm\frac{\phi^2}{2}(\frac{\phi^2}{2}-1)\,.
\overset{\sim}{ch}^{[B;M,m;\epsilon]}_{j,k;\epsilon^\prime}(\tau, z_1,z_2,t)=(-1)^{j-\epsilon^\prime}\frac{\overset{\sim}{\Psi}^{[B;M,m;\epsilon]}_{j,k;\epsilon^\prime}(\tau,z_1,z_2,t)}{\widehat{R}^{(\epsilon)}_{\epsilon^\prime}(\tau,z_1,z_2,t)}, \,\,\,\,j,k \in \Omega^{[B;M;\epsilon^\prime]}.
d^{2p-3}(\lambda_1)= f^{-1}(d^{2p-3}(\sigma \bar{\xi}_1))= f^{-1}(\sigma \alpha_1)=\sigma \alpha_1, d^{2p-3}(\epsilon_1)= f^{-1}d^{2p-3}(\tilde{\tau}_1-\bar{\tau}_1)=f^{-1}(v_1)=v_1 d^{2p-3}(\mu_1)= f^{-1}(d^{2p-3}(\sigma \tilde{\tau}_1-\bar{\tau}_0\bar{\xi}_1))=f^{-1}(\sigma(v_1)-\bar{\tau}_0\alpha_1)=\sigma(v_1)
\Bigl\vert\mathbb{E}_{(m,n)\in[\vec{N}]}\bold{1}_{[cN_{1}/D]\times[cN_{2}/D]}(m,n) e(\eta(g(m,n)))\Bigr\vert\geq\frac{c^{2}}{2D^{2}}-\frac{c}{D}(\frac{1}{N_{1}}+\frac{1}{N_{2}})\geq\frac{c^{2}}{4D^{2}}.
Q=\frac{\partial L_{0}}{\partial(\partial x_{i}/\partial\sigma)}\frac{\delta x_{i}}{\delta\sigma}+\frac{\delta x_{-}}{\delta\sigma}=-\frac{dx_{i}}{d\sigma_{0}}\frac{\delta x_{i}}{\delta\sigma}+\frac{\delta x_{-}}{\delta\sigma}=L_{0}.
\int_{1}^{\rho_2}f(\rho,R) \, d\rho = \int_{1}^{\rho_2}\frac{1}{\rho}\, \psi \big( e^{\mu(\rho) - \mu(R)}\big) \leq \psi \big( e^{\mu(\rho_2) - \mu(R)}\big) \int_1^{\rho_2}\frac{1}{\rho}\, d\rho = \log(\rho_2) \psi \big( e^{\mu(\rho_2) - \mu(R)}\big).
D^{s,p}_0(\Omega)\ni \varphi\mapsto \langle (-\Delta_p)^s u, \varphi\rangle:=\int_{\mathbb{R}^N\times \mathbb{R}^N}\frac{J_p(u(x)-u(y))\,(\varphi(x)-\varphi(y))}{|x-y|^{N+s\,p}}\, dx\, dy
\Bigl\{\mathbf{n}\in\mathbb{Z}^3:\,\sum_{i=1}^3n_i\mathbf{v}_i\in\phi(\mathcal{R})\Bigr\}\subseteq\Bigl\{\mathbf{n}\in\mathbb{Z}^3:\,\Bigl\|\sum_{i=1}^3n_i\mathbf{v}_i\Bigr\|_2\le 2N\Bigr\}&\subseteq\Bigl\{\mathbf{n}\in\mathbb{Z}^3: \, |n_i|\le C\frac{N}{\|\mathbf{v}_i\|_2}\Bigr\}.
\zeta^a(y+2\pi r_c)=e^{-{\rm i}\beta\sigma_3}\zeta^a(y)\ ,\quad\Psi^A(y+2\pi r_c)= e^{-{\rm i}\beta\sigma_3}\Psi^A(y)\ ,\quad\xi(y+2\pi r_c)=e^{{\rm i}2\beta}\xi(y)\ .
1-\mu_R\le \sum_{t=1}^{N_T}\binom{N_T}{t}a_{0,t}+\sum_{r=1}^{N_R-1}\sum_{t=1}^{N_T}\left[\binom{N_R}{r}-\binom{N_R-1}{r-1}\right]\binom{N_T}{t}a_{r,t}= \sum_{r=0}^{N_R-1}\sum_{t=1}^{N_T}\binom{N_R-1}{r}\binom{N_T}{t}a_{r,t}.
S_{0}\equiv \int_{- \infty}^{\infty}d t \int_{- \infty}^{\infty}d x_{1}[ \bar{\psi}( x_{1}, t ) ({\gamma}^{\mu}i{\hbar}c{\partial}_{\mu}- m c^{2}){\psi}( x_{1}, t ) - e \bar{\psi}( x_{1}, t ){\gamma}^{\mu}{\psi}( x_{1}, t ) \cdot \bar{A}_{\mu}^{e x t}
\Phi_{3;x}^{\infty}(\mu)=\sqrt{\frac{\Phi_{2}\left(\left\Vert x\right\Vert +\mu\right)-\Phi_{2}\left(\mu\right)}{-\Phi_{2}\left(\frac{\mu}{c_{1}}\right)}}=\sqrt{\frac{\mu^{2}-\left(\left\Vert x\right\Vert +\mu\right)^{2}}{\left(\frac{\mu}{c_{1}}\right)^{2}}},\qquad\mu<-\left\Vert x\right\Vert .
\lim_{n\rightarrow \infty}\|\psi\varphi^n\|_*=0\,\,\,\,and\,\,\,\, \lim_{|\varphi(a)|\rightarrow 1}\big(\log\frac{2}{1-|\varphi(a)|^2}\big)\|\psi\circ\sigma_a-\psi(a)\|_{H^2}=0.
J^{\alpha\beta}(x)={1\over2}\left[\psi^{\alpha\dagger}_A(x),\psi^{\beta}_A(x)\right]=\psi^{\alpha\dagger}_A(x)\psi^{\beta}_A(x)-\delta^{\alpha\beta}{{\cal N}_{C}\over 2}=(J^{\beta\alpha}(x))^\dagger
V_{0}(m)={m^{2}\over2g^{2}}+{1\over8\pi^{2}}\left[{\Lambda^{4}\over2}\Bigl(1-{m^{2}\over\Lambda^{2}}\Bigr){\mathrm{e}}^{-{m^{2}\over\Lambda^{2}}}+{m^{4}\over2}E_{1}({m^{2}/\Lambda^{2}})\right],
\sum_{\sigma \in S_{n+1}}\prod_{i=1}^{n+1}x_{\sigma(i)}^{\log v_i}= \sum_{\sigma \in S_{n+1}}\prod_{i=1}^{n+1}x_{\sigma(i)}^{b_i}< \sum_{\sigma \in S_{n+1}}\prod_{i=1}^{n+1}x_{\sigma(i)}^{a_i}= \sum_{\sigma \in S_{n+1}}\prod_{i=1}^{n+1}x_{\sigma(i)}^{\log u_i}.
\left.D(z,t)\right|_{z=z_a(t)}=\dfrac{\left.\biggl[-\dfrac{1}{2}\,z\,\dfrac{\partial^2 A(z,t)}{\partial z^2}-z^2\,\dfrac{\partial^3 A(z,t)}{\partial z^3}-t\,\dfrac{\partial \tilde{Q}^{(1)}(z,t)}{\partial z}-t\,Q^{(2)}(z,t)-t\,z\,\dfrac{\partial Q^{(2)}(z,t)}{\partial z}\biggr]\right|_{z=z_a(t)}}{\left.\dfrac{\partial A(z,t)}{\partial z}\right|_{z=z_a(t)}}.
Q ( x , y ) \; = \; \frac{1}{2}\: \sum_{k = 1}^{K_{x y}}\frac{\partial{\cal{L}}( \lambda^{x y})}{\partial \lambda_{k}^{x y}}\: F_{k}^{x y}\: P ( x , y ) \; = \; \frac{1}{2}\: \sum_{k = 1}^{K_{x y}}\frac{\partial{\cal{L}}( \lambda^{x y})}{\partial \lambda_{k}^{x y}}\: P ( x , y ) \: F_{k}^{y x}\; .
\|\langle\mathbf{N}\rangle^{-\frac{r}{2}}\sum_{j=0}^{r}\frac{(i\varepsilon \pi)^{j}}{j!}(\mathrm{D}^{j}[b(z)][\xi])^{Wick}\langle\mathbf{N}\rangle^{-\frac{r}{2}}\|_{\mathcal L(\Gamma_{s}(\mathcal Z))}\leq C_{r}\langle\xi\rangle^{r}.
\rho(m_t)R_j = \epsilon([g_{\alpha}m_t g_j^{-1}]) [ \textrm{Definition}(\ref{LiftRep})] &= \epsilon([g_{\alpha}g_t^{-1}m g_t g_j^{-1}]) [m_t = g_t^{-1}m g_t] &= \epsilon([g_{\alpha}g_t^{-1}g_t g_j^{-1}]) - \epsilon([g_{\alpha}g_t^{-1}])\epsilon([g_t g_j^{-1}]) [\textrm{Skein relation}] &= R_j - \epsilon([g_t g_j^{-1}])R_t,
\widehat{U}_4(x, a,b) = \frac{W_q(x)}{W_q'(b+)}\lim_{\theta \downarrow 0}\frac \partial{\partial \theta}\widehat{U}_1^0(a,b,\theta) - \lim_{\theta \downarrow 0}\frac \partial{\partial \theta}U_1^0(a,x,\theta).
\Big|\prod_{x\in\mathcal{S}^3(\boldsymbol{\pi})}a^\varepsilon_{x,\pi_x}(\lambda)\Big| = \prod_{\substack{x\in\mathcal{S}^3(\boldsymbol{\pi}) \pi_x=x}}|\lambda+\tilde{a}^\varepsilon_{x,x}| \prod_{\substack{x\in\mathcal{S}^3(\boldsymbol{\pi}) \pi_x\neq x}}| a^\varepsilon_{x,\pi_x}| \leq (|\lambda|+ N\overline{c})^{n+p}.
\hat{\textbf{V}}_k = \left[\textbf{a}\left(\textbf{x}\right),j\textbf{a}\left(\textbf{x}\right),\beta \frac{\partial \textbf{a}\left(\textbf{x}\right)}{\partial x_1},\beta \frac{\partial \textbf{a}\left(\textbf{x}\right)}{\partial x_2}\right]\Bigg|_{{\boldsymbol{\psi}}= \hat{\boldsymbol{\psi}}_{k-1}}.
\sum_{\substack{m\in\mathcal{A}}}\Lambda_{\mathcal{C}^+(\mathbf{a};\delta)}(m)=\frac{\#\mathcal{A}}{X}\sum_{n< X}\Lambda_{\mathcal{C}^+(\mathbf{a};\delta)}(n)+O_{A,\eta}\Bigl(\frac{\#\mathcal{A}}{(\log{X})^A}\Bigr),
U_{1}=\sum_{i\in I_{c}^{N}}\int_{G_{N}}\nabla_{i}f(x)G(t,\,x_{i})dx U_{2}=\sum_{i\in I_{c}^{N}}\sum_{k:k\neq i}\int_{F_{ik}}G(t,\,x_{i})(f_{ki}(x)-f_{ik}(x))dS_{ik}(x).
V(a_1,a_2,\ldots,a_n)=\det \begin{bmatrix}1 a_1 a_1^2 \cdots a_1^{n-1}1 a_2 a_2^2 \cdots a_2^{n-1}\vdots \vdots \vdots \ddots \vdots 1 a_n a_n^2 \cdots a_n^{n-1}\end{bmatrix}=\prod_{1 \leq i < j \leq n}(a_j-a_i)
\tilde{\gamma}= -\frac{2}{3}\alpha + \frac{1}{6}= - \frac{2}{3}\beta + \frac{1}{3}, \tilde{M}_1 = \frac{1}{3}\alpha^2 + \frac{1}{6}\alpha - \frac{1}{3}, \tilde{M}_2 = \frac{1}{18}\alpha^4 + \frac{5}{54}\alpha^3 - \frac{5}{216}\alpha^2 - \frac{5}{108}\alpha + \frac{1}{2592}.
Z [ \eta , \overline{{\eta}}] : = \int [ d \psi ] [ d \overline{{\psi}}] [ d A ] e^{- S [ \psi , \overline{{\psi}}, A ] + \sum_{a}\Big \{( \overline{{\eta}}^{( a )}, \psi^{( a )}) + ( \overline{{\psi}}^{( a )}, \eta^{( a )}) \Big \}}.
\varphi(x_1,\ldots,x_k,y_1,\ldots,y_l):=x_1\in P_\sigma\wedge\bigwedge_{i=1}^kx_k=\mathfrak{s}^{u_i}(x_1)\wedge\bigwedge_{j=1}^l y_l={\mathfrak{s}}^{v_j}(x_1).
\phi b^{m_0}\sim \phi b-\sum_{s\in S_b}\sum_{i=1}^{m_0-1}\phi b^is &\sim \phi b-\sum_{s\in S_b}\sum_{i=1}^{m_0-1}(\phi bs -\sum_{j=1}^{i-1}\sum_{s'\in S_b}\phi s'b^js) &=\phi b-\sum_{s\in S_b}(m_0-1)\phi bs +\sum_{s\in S_b}\sum_{s'\in S_b}\sum_{k=1}^{m_0-2}(m_0-1-k)\phi s'b^ks=f_{m_0}.
\left\langle \partial_x P_\lambda Q(\varphi_\lambda),\mathbf{u}^v\right\rangle =\left\langle \partial_x P_\lambda \tilde{\chi}Q(\varphi_\lambda),\mathbf{u}^v\right\rangle+\left\langle \chi^1\partial_x P_\lambda(1-\tilde{\chi})Q(\varphi_\lambda),\mathbf{u}^v\right\rangle
\eta'(t) =\psi^{(m+1)}_{p,q}(s+t) - \psi^{(m+1)}_{p,q}(t) &= (\ln q)^{m+2}\sum_{n=1}^{p}\left[ \frac{n^{m+1}q^{n(s+t)}}{1-q^{n}}- \frac{n^{m+1}q^{nt}}{1-q^{n}}\right] &= (\ln q)^{m+2}\sum_{n=1}^{p}\left[ \frac{n^{m+1}q^{nt}(q^{ns}-1)}{1-q^{n}}\right]\geq0. \,\,
\mathcal{L}= \partial_{\mu}\phi^{*}\, \partial^{\mu}\phi - \phi^{*}m^{2}\phi + \sum_{i = 1}^{2}\left[ \frac{1}{2}\, \partial_{\mu}\varphi_{i}\, \partial^{\mu}\varphi_{i}- \phi^{*}U_{i}( \varphi_{i}) \phi \right] ,
L_{n}^{(llm)}(\xi) = \left( \begin{array}{c}e^{p_n \Delta}~f(u_n)~ \qquad{\Delta}{\xi}e^{\eta u_n}\frac{\Delta}{\xi}e^{\eta u_n}\qquad f(u_n)~ e^{-p_n \Delta}\end{array}\right) \quad \mbox{where}\ \f(u_n)=[ 1 +{\Delta^2}e^{\eta(2u_n+i)}]^{{1 \over 2}}.\ll{Llm}
K(u) = \left( \frac{\sin{(u/2)}}{u}\right)^2 (1+\cos{(\theta+u\log{(e\log{N}\log_{2}{N})})}) +\left( \frac{\sin{(u/2)}}{u}\right)^2 (1+\cos{(\theta+u\log{(e^{3/2}\log{N}\log_{2}{N})})}) +\left( \frac{\sin{(u/2)}}{u}\right)^2 (1+\cos{(\theta+u\log{(e^{2}\log{N}\log_{2}{N})})}).
\partial_{\mu}\Psi_{a}=U_{ext}\left( x^{\mu}\right) \widetilde{U}_{int}\left( x^{\mu},x^{m}\right) \left( \partial_{\mu}+imv_{\mu \alpha}\sigma^{\alpha}+imv_{\mu c}\sigma^{c}\right) \psi_{a}^{r}\psi_{r}^{int}
e^{- \int d^{4}x \frac{\xi}{2 g^{2}}( D^{\mu}V_{\mu})^{2}}= \frac{1}{N}\int{\cal D}a \, \delta ( D^{\mu}V_{\mu}^{g}- a ) e^{- \int d^{4}x \frac{\xi}{2 g^{2}}a^{2}},
\begin{aligned}\phi(2l , 2k) =\sum_{r=l}^{k-1}\binom{r-1}{l-1}a_{k-r}+ \binom{k-1}{l -1}b_{1}=\sum_{r=l}^{k-1}\binom{r-1}{l-1}(-1) + \binom{k-1}{l -1}(-1) =-\sum_{r=l}^{k}\binom{r-1}{l-1}.\end{aligned}
X_mF_{n}(X_\delta)=q^{-\frac{n}{2}}X_{m-n}+q^{\frac{n}{2}}X_{m+n}+\sum\limits_{k=1}^{n}(\sum\limits_{l=1}^{k}q^{-\frac{k+1}{2}+l})hF_{n-k}(X_\delta).
K = \frac{1}{2}\frac{\partial^2 A}{\partial \phi^i \partial\phi^j}\dot \phi^i\dot \phi^j + i \frac{\partial^2 A}{\partial \phi^i \partial \phi^j}(\bar \psi_a^i \dot \psi^{aj}+ \psi^{ai}\dot{\bar \psi_a^j}) ,
d(v \otimes Z_1 \wedge \ldots \wedge Z_k) = \sum_{i = 1}^k (-1)^{i + 1}vZ_i \otimes Z_1 \wedge \ldots \wedge \widehat{Z_i}\wedge \ldots \wedge Z_k+ \sum_{i<j}(-1)^{i+j}v \otimes [Z_i,Z_j] \wedge \cdots \wedge \widehat{Z}_i \wedge \cdots \wedge \widehat{Z}_j \cdots \wedge Z_k,
|\omega_m-\omega_l| = c \frac{\lambda_m-\lambda_l}{\sqrt{c^2+\lambda_m}+\sqrt{c^2+\lambda_l}}\stackrel{m>l}{\geq}\frac{c\lambda_l^{1/2}}{\sqrt{c^2+\lambda_m}+\sqrt{c^2+\lambda_l}}&\gtrsim \frac{N^{\delta/3}\lambda^{1/2}}{\sqrt{N^{2\delta/3}+\lambda_m^{1/2}}+\sqrt{N^{2\delta/3}+\lambda_l^{1/2}}}>0,
\int (q^2x^2;q^2)_\infty h(x/q) \Big( \frac{1}{q}D_{q^{-1}}u(x)+\frac{1}{q}u(x) u(x/q)-\frac{q^{-n}x}{1-q}u(x/q)+ \frac{q^{1-n}[n]_q}{(1-q)}\Big)y(x) d_qx &= (x^2;q^2)_\infty h(x/q)\Big ( y(x/q)u(x/q) - D_{q^{-1}}y(x)\Big).
\Gamma^*\rho_0(\overrightarrow{\eta})= b\bigl[(m+\eta_1-1)\gamma(\eta_1-1)-(m+\eta_1)\gamma(\eta_1)\bigr]\times\prod_{i=2}^N\gamma(\eta_i) & +b\bigl[(m+\eta_N)\gamma(\eta_N)-(m+\eta_N-1)\gamma(\eta_{N}-1)\bigr]\times\prod_{i=1}^{N-1}\gamma(\eta_i).
I_2\leq \sum_{q\geq0}\int_0^t c_q\lambda_q^{\alpha-\beta}e^{-\mu\lambda_q^\alpha (t-\tau)}\|B(\tau)\|^2_{H^{\frac52-\alpha+\beta}}\, d\tau \lesssim \int_0^t (t-\tau)^{-1+\frac{\beta}{\alpha}}\|B(\tau)\|^2_{H^{\frac52-\alpha+\beta}}\, d\tau.
\tilde\tau_1 = \frac{(2\phi_2^2-\phi_1^2) - (2\phi_2^1-\phi_1^1)}{2\omega_2-\omega_1}\mbox{and}\tilde\tau_2 = \frac{(\phi_2^2-2\phi_1^2) - (\phi_2^1-2\phi_1^1)}{\omega_2-2\omega_1}.
\begin{aligned}{k_{00}}_{|t=0}= -2 \lambda_{\Sigma}^2 \cdot{F_0}_{|t=0}+ 2 \lambda_{\Sigma}^3 \cdot_{g^{\Sigma}}W,{k_{0 i}}_{|t=0}= \lambda_{\Sigma}^2 \cdot \left[-F_{i}+ \frac{1}{2}g^{jk}(2 \partial_{j}g_{ki}- \partial_{i}g_{jk}) +\partial_i(\lambda_{\Sigma})\right]_{|t=0}.\end{aligned}
\mathrm{D e t}^{g}( d ) = \mathrm{D e t}^{r}( d ) \cdot e^{{\frac{i}{8 \pi^{2}}}\int d^{4}x \int d^{4}y B^{\mu}( x ){\frac{\partial}{\partial x^{\mu}}}G ( x , y ) \left[ f e^{2}\tilde{F}^{\mu \nu}( y ) F_{\mu \nu}( y ) +{\frac{1}{3}}f^{3}\tilde{G}^{\mu \nu}( y ) G_{\mu \nu}( y ) \right]}.
H_k = \frac{-1}{4n^2}\left(\frac{a^2}{4x_k}+ \frac{a\sqrt{n}G_{k}}{\sqrt{\beta x_k}}+ \frac{aG^{(2)}_k}{\beta x_{k}}\right) A_{k,k+1}= -4 (1-r_k) p_k \exp(H_k), A_{k,k-1}= -4 (1-r_k) q_k \exp(H_k).
H_k^{(s,t)}:=\left|\begin{array}{cccc}f_{s}^{(t)}f_{s+1}^{(t)}\cdots f_{s+k-1}^{(t)}f_{s+1}^{(t)}f_{s+2}^{(t)}\cdots f_{s+k}^{(t)}\vdots \vdots \vdots \vdots f_{s+k-1}^{(t)}f_{s+k}^{(t)}\cdots f_{s+2k-2}^{(t)}\end{array}\right|,s,t=0,1,\dots,
\lambda_{A_{\mathfrak{p}}}\left( \operatorname{Tor}_{2i}^{A_{\mathfrak{p}}}(M_{\mathfrak{p}}, N_{\mathfrak{p}}) \right) = P_1(i) \quad\mbox{for all}i \gg 0; \lambda_{A_{\mathfrak{p}}}\left( \operatorname{Tor}_{2i+1}^{A_{\mathfrak{p}}}(M_{\mathfrak{p}}, N_{\mathfrak{p}}) \right) = P_2(i) \quad\mbox{for all}i \gg 0.
\hat{\varphi}^{( 1 )}( x ) = \mathrm{i}\int d^{6}y \Big < \hat{\varphi}( x ) J ( y ) \hat{\varphi}( y ) \Big > = \frac{\kappa_{o r b}M}{2}\int \frac{d^{5}k}{( 2 \pi )^{5}}\frac{\mathrm{e}^{\mathrm{i}k \cdot x}}{k^{2}}.
\mathrm{mult}\, (\alpha) = \sum_{\substack{\mathcal D: \, , \ \mathrm{wt}(\mathcal D)=\alpha{\phi(\mathcal D):}}}c(\mathcal D) +\sum_{\substack{\mathcal D: \, \mathrm{wt}(\mathcal D)=\alpha/2}}\left \lfloor \tfrac{1-c(\mathcal D)}2 \right \rfloor .
\overline{H}(\rho, t) \approx r^{-n}e^{-\frac{\rho^2}{4t}-\lambda_1(M)t}=\max\left(1, \frac{\rho}{t}\right)^n e^{-\frac{\rho^2}{4t}-\lambda_1(M)t}\approx \left(\frac{\rho+t}{t}\right)^ne^{-\frac{\rho^2}{4t}-\lambda_1(M)t}\lesssim (1+\rho^n)e^{-\frac{\rho^2}{4t}-\frac{(n-1)^2a^2}{4}t},
\frac{1 + |\nabla_X \theta|^2}{h}\partial_z^2 \tilde u = \tilde{f}- \mu \nabla_X\cdot(h \nabla_X \tilde u ) + \mu \nabla_X\cdot (\nabla_X \theta \partial_z \tilde u )+ \mu \partial_z(\nabla_X \theta \cdot \nabla_X \tilde u ) - \frac{(\partial_z|\nabla_X \theta|^2)}{h}\partial_z \tilde u ,
Z[A_{+},\eta,{\bar \eta}] = \int \! D \bar{\psi}D \psi \exp\{i\int \! d^2x \,[\bar{\psi}i \gamma^{\mu}\partial_{\mu}\psi + \psi^{\dagger}_-A_{+}\psi_- + \eta \psi^{\dagger}_{+}\psi_{-}+{\ \bar \eta}\psi^{\dagger}_{-}\psi_{+}] \}\quad.
W_{\wedge}= \left\{\begin{array}{c}L_{-1}\quad L_{0}\quad L_{1}J_{-2}\quad J_{-1}\quad J_{0}\quad J_{1}\quad J_{2}\Lambda_{-3}\quad \Lambda_{-2}\quad \Lambda_{-1}\quad \Lambda_{0}\quad \Lambda_{1}\quad \Lambda_{2}\quad \Lambda_{3}\ldots\ldots \ldots \ldots\ldots\ldots \ldots \ldots \ldots \ldots \ldots \end{array}\right\}
\omega_j := \omega_j(c) = c \sqrt{c^2+\lambda_j}\; = \; c^2 \; + \frac{\lambda_j}{1+\sqrt{1+\lambda_j/c^2}}&= \; c^2 \; + \frac{\lambda_j}{2}-\frac{\lambda_j^2}{2c^2}\frac{1}{(1+\sqrt{1+\lambda_j/c^2})^2},
M^{2}\left( \begin{matrix}{\phi^{\prime}}\\{{\overline{{\phi}}}^{* \prime}}\\ \end{matrix}\right) = \left( \begin{matrix}{m^{2}}&{m^{2}x}\\{m^{2}x}&{m^{2}}\\ \end{matrix}\right) \left( \begin{matrix}{\phi^{\prime}}\\{{\overline{{\phi}}}^{* \prime}}\\ \end{matrix}\right) \ .
\mbox{Med-bias}_{\theta_0}(\widehat{\theta}_n) ~=~ \left(\frac{1}{2}- \max\left\{\mathbb{P}(\dot{\mathbb{M}}_n(\theta_0) \ge 0),\,\mathbb{P}(\dot{\mathbb{M}}_n(\theta_0) \le 0)\right\}\right)_+.
C =\left(\begin{array}{@{\hskip 2pt}c@{\hskip 2pt}c@{\hskip 2pt}c@{\hskip 2pt}c@{\hskip 2pt}}1 3 2 3 3 1 3 2 2 3 1 3 3 2 3 1 \end{array}\right)\,,\,C' =\left(\begin{array}{@{\hskip 2pt}c@{\hskip 2pt}c@{\hskip 2pt}c@{\hskip 2pt}c@{\hskip 2pt}}1 3 3 2 3 1 3 3 3 3 1 3 2 3 3 1 \end{array}\right).
{\overline A}_{\mu}(x,\beta) = -\frac{i}{g}e^{i\theta_{a}T^{a}}\left[{\overline\sigma}_{\mu\nu}\frac{\rho^{2}(x-x_{0})_{\mu}}{(x-x_{0})^{2}[(x-x_{0})^{2}+\rho^{2}]}\right]e^{-i\theta_{a}T^{a}}\, ,
b^{(k)}= i(b^{(k_1)})s^{(k)}, \textit{where}s^{(k)}\textit{is a small piece (possibly empty)}, &U^{(k)}= A^{(k_1)}b^{(k)}B^{(k)}, \textit{where}A^{(k_1)}\textit{is a prefix of}U^{(k)},\B^{(k)}\textit{is a suffix of}U^{(k)}, &k = k_1, \ldots, k_2.
\kappa_\chi(z_{\alpha(1)}, \ldots, z_{\alpha(n)}) = \sum_{\substack{\pi \in BNC(\chi_{\widehat{\alpha}}) \pi \vee \sigma = 1_{\chi_{\widehat{\alpha}}}}}\kappa_\pi\left(z^{\epsilon(1)}_{\alpha(1)}, z^{\epsilon(2)}_{\alpha(1)}, \ldots, z^{\epsilon(2n-1)}_{\alpha(n)}, z^{\epsilon(2n)}_{\alpha(n)}\right)
M(0,x,y) =1152-576x^3+(1024-1152x-896x^2+1152x^3-128x^4)y^2 &=576(2-x^2)+128(8-x)(1-x)^2(1+x)y^2 &\leq576(2-x^2)+128(8-x)(1-x)^2(1+x) &=1024-896x^2+576x^3-128x^4\;\leq1024,(x,y)\in(0,1)\times(0,1)
\forall \gamma^1,x^X\big(\gamma>_\mathbb{R}0\land \exists y^X,z^X(z\in Ay\land x=_Xy+_X\gamma z) &\qquad\qquad\qquad\qquad\qquad\rightarrow \gamma^{-1}(x-_XJ^A_{\gamma}x)\in A(J^A_{\gamma}x)\big),
\int_{\mathbb{R}^{n}}e^{2\pi ix\cdot\xi}(Ff)(\xi)d\xi=lim_{\epsilon\rightarrow0}\int_{\mathbb{R}^{n}}e^{-\pi\epsilon^{2}|\xi|^{2}+2\pi ix\cdot\xi}(Ff)(\xi)d\xi.
\frac{det(-\frac{d^{2}}{dx^{2}}+A)}{det(-\frac{d^{2}}{dx^{2}})}=\prod_{n=1}^{+\infty}\frac{\frac{n^{2}\pi^{2}}{L^{2}}+A}{\frac{n^{2}\pi^{2}}{L^{2}}}=\prod_{n=1}^{+\infty}(1+\frac{L^{2}A}{n^{2}\pi^{2}}).
h_{T,S}=\frac{[Ext_U^1(j_{*}\hat{T},\mathbb{G}_m)][H^1(K,\hat{T})]}{[\mathrm{cok}(\Theta^1_S)][\mathbb{III}^1(T)][\mathbb{III}^2(T)]\prod_{v\in S}[H^1(K_v,T)]\prod_{v\notin S}[H^0(\hat{\mathbb{Z}},H^1(I_v,\hat{T}))]}.
\frac{d^pH_{\lambda}(z)}{dz^p}=(-1)^p p!\frac{i}{2\pi}\oint\limits_{{\Gamma}_1}\oint\limits_{{\Gamma}_2}dk_1\;dk_2 \;\frac{k_1^{\Lambda}F(k_1) k_2^{\Lambda}G(k_2)}{(z-k_1-k_2)^p}
\int_{{\tau_{k - 1}}}^{{y_i^{(k)}}}{g\left({{\tau^{(k)}}}\right)\,d{\tau^{(k)}}}= \sum\limits_{j = 0}^{{m_k}}{_k{p}_{OB,i,j}^{(1)}\,g\left({\hat z_{{m_k},i,j}^{(k),\alpha_i^{(k),*}}}\right)}+ E_{{m_k}}^{\left({\alpha_i^{(k),*}}\right)}\left({y_i^{(k)},\xi_i^{(k)}}\right) ,
\begin{cases}w(s) \sim \phi_1(s) = s^{-\frac{1}{\alpha}}w(s) \sim \phi_2(s) = \begin{cases}s^{-\frac{1}{\alpha}+\frac{N-2}{2}}N \ge 3 s^{-\frac{1}{\alpha}}\log s N = 2 .\end{cases}\end{cases}
\det\left( \zeta\mathbf{1}-\Omega\right) =\det\left( \zeta^{2}\mathbf{1}-\alpha^{-1}\eta\right) , \det\left( \zeta\mathbf{1}-B\right) =\zeta^{N}\det\left( \zeta\mathbf{1}-\alpha^{-1}R\right) ,
(d_{e}+d_{e^{\prime}})(TLL^{\prime}-\partial^{\mu}TV_{\mu}L^{\prime}-\partial^{\mu}TLV_{\mu}^{\prime}+\partial^{\mu}\partial^{\nu}TV_{\mu}V_{\nu})=0 TL^{P}L^{P\prime}\equiv(d_{e}+d_{e^{\prime}})(TLL^{\prime}-\partial^{\mu}TV_{\mu}L^{\prime}-\partial^{\mu}TLV_{\mu}^{\prime}+\partial^{\mu}\partial^{\nu}TV_{\mu}V_{\nu})
f(p^{-1}\|X_{i_1}-X_{i_2}\|_{2}^2)=f(\widetilde{X}_{i_1,i_2}+\tau_1)=\sum^{l}_{s=0}\frac{f^{(s)}(\tau_1)\widetilde{X}_{i_1,i_2}^s}{s!}+c_{l+1,\tau_1}(\widetilde{X}_{i_1,i_2})\widetilde{X}_{i_1,i_2}^{l+1}.
q_{13}'(t,k,l,\xi,\eta)=-\mathbf{1}_{(k,l)=(\pm 1,0)}\frac{\varepsilon_0\widehat{\partial_v\varphi_b}(\xi-\eta)}{2},|q_{13}''|\lesssim\varepsilon_0^2e^{-(C_0/5)(|k-l|+|\xi-\eta|)}.
\begin{aligned}\mu(B) \geq K(n,p) m(A)^{1 - \frac{p}{q}}\sum_{j \;: \;x_j \in A}\tilde{\nu}_j^{\frac{p}{q}}+ K(n,p) \sum_{i \;:\; x_i \not\in A}|Gx_i|^{1-\frac{p}{q}}\tilde{\nu}_i^{\frac{p}{q}}\geq &\geq K(n,p) (m(A)^{1-\frac{p}{q}}\alpha^{\frac{p}{q}}+ m(G)^{1 - \frac{p}{q}}(1-\alpha)^{\frac{p}{q}}),\end{aligned}
\begin{aligned}\theta^2_{[i_o]}\ge \sigma^2\big[\log(\tfrac{en}{i_o})-(i_o-1)\log(\tfrac{i_o}{i_o-1})\big], \theta^2_{[i_o]}+\theta^2_{[i_o+1]}\ge \sigma^2\big[2\log(\tfrac{en}{i_o})-(i_o-2)\log(\tfrac{i_o}{i_o-2})\big], &\ldots, \textstyle{\sum_{i=1}^{i_o}}\theta_{[i]}^2 \ge\sigma^2 i_o \log(\tfrac{en}{i_o}).\end{aligned}
(d^k((\phi) P^{1,k-1}))_b \overset{(\ast)}{=}- d^{k-1}(((\phi)P^{1,k-1})_b) =- d^{k-1}(\phi_b \cdot P) & =- (d^{k-1}\phi_b ) \cdot P \overset{(\ast\ast)}{=}(d^k \phi)_b \cdot P = ((d^k \phi)P^{k,1})_b
\Delta_n(x)=\frac{(-1)^{n}}{\pi x}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{I_1(2x\sin t)}{\sin t}\cos(2n t)dt=\sum_{m\geq n}C_{2m}^{m-n}\frac{\left(\frac{1}{2}x\right)^{2m}}{m!(m+1)!}.
c_{\gamma \mu}= \big\langle g_\gamma^{(k+1)}, \sum_\beta c_{\beta \mu}\,G_\beta^{(k+1)}\big\rangle= \big\langle g_\gamma^{(k+1)}, G_{1^\ell}^{(k+1)}G_\mu^{(k+1)}\big\rangle = \big\langle (G_{1^\ell}^{(k+1)})^\perp g_\gamma^{(k+1)}, G_\mu^{(k+1)}\big\rangle\,.
G_{i_{1}}\wedge\cdot\cdot\cdot\wedge G_{i_{t}}\mapsto\sum_{j=1}^{t}(-1)^{j+1}g_{i_{j}}G_{i_{1}}\wedge\cdot\cdot\cdot\wedge\hat{G}_{i_{j}}\wedge\cdot\cdot\cdot\wedge G_{i_{t}},
\|(w_n)_+\|_{H^1(\Omega)}^2=\|(w_n)_+\|_{L^2(\Omega)}^2+\|\nabla (w_n)_+\|_{L^2(\Omega)}^2\le\|w_n\|_{L^2(\Omega)}^2+\|\nabla w_n\|_{L^2(\Omega)}^2=\|w_n\|_{H^1(\Omega)}^2.
v_{j}(x):=j^{-s}w_{j}(R_{j}^{-1}x),w_{j}(x):=\begin{cases}j^{s}+j^{t}\phi(x),\quad\mathrm{in}\;B_{6}, (1-\psi(x))(j^{s}+j^{t}),\quad\mathrm{in}\; B_{6}^{c},\end{cases}
\left( \begin{matrix}{{P^{I}}}\\{{i Q_{I}}}\\ \end{matrix}\right) \rightarrow \Gamma \left( \begin{matrix}{{P^{I}}}\\{{i Q_{I}}}\\ \end{matrix}\right) = \left( \begin{matrix}{{U}}&{{Z}}\\{{W}}&{{V}}\\ \end{matrix}\right) \left( \begin{matrix}{{P^{I}}}\\{{i Q_{I}}}\\ \end{matrix}\right) ,
\hskip 0.3 cm=\big(\frac{1}{f_2^v}\big)^2\sum_{j=1}^{m_1}G_{f_1f_2}(\nabla_{e_j^h}grad(f_1^h),e_j^h)+\hskip -0.4 cm\sum_{j=m_1+1}^{m_1+m_2}\frac{1}{\|u'\|^2}G_{f_1f_2}(\nabla_{u'_j}grad(f_1^h),u'_j)
L^1(J)+L^2(J)-L^1(I)-L^2(I) \leq (K^{*}_{11}|K_{11}|+K^{*}_{25}|K_{15}|-K^{*}_{15})|\alpha_5|-\sum\limits_{i=2,3,4}K^{*}_{1i}|\alpha_i|+M\Delta''(\boldsymbol{\alpha}^{*},\beta_1).
\begin{array}{ll}Z_{23}= \frac{1}{s}(\sum_{l=1}^{g(s)}e^{2\pi i (l-1)/ns}+e^{-2\pi i/n}\sum_{l=g(s)+1}^{s}e^{2\pi i (l-1)/ns}) \times(\sum_{p=0}^{p_{max}}e^{-4 \pi i p/n}\sum_{u(p) > j \geq l(p)}e^{4\pi i(j-1)/M}),\end{array}
\gamma_{k}= \left( \begin{matrix}{0}&{- i \sigma_{k}}\\{i \sigma_{k}}&{0}\\ \end{matrix}\right) \qquad \gamma_{4}= \left( \begin{matrix}{0}&{I}\\{I}&{0}\\ \end{matrix}\right) \qquad \gamma_{5}= \left( \begin{matrix}{I}&{0}\\{0}&{I}\\ \end{matrix}\right)
u\,=\,\sqrt{\rho}\,u\,+\,\left(\frac{1}{\sqrt{\rho}}\,-\,1\right)\,\sqrt{\rho}\,u\quad\in\,L^\infty_T\bigl(L^2\bigr)+L^2_T\bigl(L^{3/2}\bigr)\,\hookrightarrow\,L^2_T\bigl(L^{3/2}_{loc}\bigr)\,,
\mathbf{H}_{\mathrm{b2u},k}\mathbf{G}= \left( \mathbf{H}_{\mathrm{b2u},k}\mathbf{R}\right) \left(\mathcal{H}_{\mathrm{b2s}}\mathbf{R}\right)^{\dagger}= \mathbf{0}_{N_{\mathrm{ue}}\times \left(N_{\mathrm{bs}}- KN_{\mathrm{ue}}\right)}\left(\mathcal{H}_{\mathrm{b2s}}\mathbf{R}\right)^{\dagger}= \mathbf{0}_{N_{\mathrm{ue}}\times KN_{\mathrm{scr}}}.
\{J_{1}\equiv J^{1 2}, J_{3}\equiv J^{3 4}, J_{5}\equiv J^{5 6}, J_{7}\equiv J^{7 8}, D_{1}\equiv D^{1 2 9}, D_{3}\equiv D^{3 4 9}, D_{5}\equiv D^{5 6 9}, D_{7}\equiv D^{7 8 9}\}
\left( \begin{array}{l}\theta^{1}\theta^{2}\end{array}\right) \rightarrow \left( \begin{array}{ll}\cos \frac{\alpha}{2}-\sin \frac{\alpha}{2}\sin \frac{\alpha}{2}\cos \frac{\alpha}{2}\end{array}\right) \left( \begin{array}{l}\theta^{1}\theta^{2}\end{array}\right) .
a=\frac{e^{2x_{3}+x_{2}}-e^{2x_{3}+x_{1}}+e^{2x_{2}+x_{1}}-e^{2x_{2}+x_{3}}+e^{2x_{1}+x_{3}}-e^{2x_{1}+x_{2}}}{y_{3}e^{x_{2}}-y_{3}e^{x_{1}}+y_{2}e^{x_{3}}+y_{1}e^{x_{3}}-y_{1}e^{x_{2}}}
\int_{2R^N(t)\leq |\mu-x|}\frac{g(|x|)}{|\mu-x|^2}\,dx\leq &\sum_{\substack{i\in \mathbb{Z}^3\setminus \{0\}|\mu-i|\geq 1}}\frac{4}{|\mu-i|^2}\int_{|i-x|\leq 1}g(|x|)\,dx\leq 4C_1\sum_{\substack{i\in \mathbb{Z}^3\setminus \{0\}|\mu-i|\geq 1}}\frac{1}{|\mu-i|^2|i|^{2+\epsilon}}.