harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\langle \rho_\gamma(u,1) P_E,\, \rho_\gamma(v,1) P_E \rangle = \langle P_E,\, \rho_\gamma[(-u,1)\cdot (v,1)] P_E \rangle &= \langle P_E,\, \rho_\gamma(v-u,[-u,v]^{1/2}) P_E \rangle &= \varphi_E(v-u,[u,v]^{-1/2}).
q_t(dy_t|y_{t-1},x_t)=\bordermatrix{0,0 e,0 1,0 0,1 e,1 1,1\cr 0 \alpha_t \gamma_t \beta_t 0 0 0\cr e 1-\alpha_t 1-\gamma_t 1-\beta_t 1-\alpha_t 1-\gamma_t 1-\beta_t\cr 1 0 0 0 \alpha_t \gamma_t \beta_t\cr},~ \alpha_t, \beta_t, \gamma_t\in[0,1].
U_{\lambda}=\left\{(x,y,t)\in \mathcal{Q}_{r_{1}}\,:\, |D^{\tau}d_{s}u(x,y,t)|\geq\lambda \right\}\subset\left(\bigcup\limits_{i\in\mathbb{N}}\mathcal{Q}_{5\rho_{{i}}}\left(z_{i}\right)\right)\bigcup \left(\bigcup\limits_{\mathcal{K}\times I\in\mathcal{A}}\mathcal{K}\times I\right)
\sum\limits_{k=n}^{m}\left( \mathcal{D}_{\eta^{k}}(\eta^{k+1},\eta^{k}; \lambda^{k}) + \beta\|r^{k+1}\|^2\right) = \mathcal{D}_{\eta^{n}}(\hat{\eta},\eta^{n}; \lambda^{n}) - \mathcal{D}_{\eta^{m}}(\hat{\eta},\eta^{m}; \lambda^{m}).
\bigg(\sqrt{\frac{\sqrt{2}}{2}}\bigg)^{4}\big(R_*^{2}-1) =\frac{1}{2}(R_*^{2}-1) <-R_*(1-R_*)-\frac{M}{K}(1+R_*) <-\bigg[R_*(1-R_*)+\frac{M}{K}(1+R_*)\bigg]^{2}.
| \phi_{( M L )}> = \prod_{( \vec{p})^{+}}\; | \phi_{M \vec{p}, L}> , \; | \phi_{M \vec{p}, L}> = ( \beta_{\vec{p}}{}^{2}| 1 > + ( - ) \sqrt{2}\; \alpha_{\vec{p}}\beta_{\vec{p}}| 2 > + \alpha_{\vec{p}}{}^{2}| 3 > ) ,
\mathfrak{X}_{k}^+(u, \lambda):=\sum_{i=0}^{\infty}g^{(i)}_{\lambda_k}(z_k) u^{i}=g_{\lambda_k}(u+z_k)=\frac{\vartheta(u+z_k+\lambda_k)}{\vartheta(u+z_k)\vartheta(\lambda_k)},
\|f|{{{W}}_{2}^{{\bf R}}(\Bbb T^d)}\| = \Big(\|f|L_2(\Bbb T^d)\|+\sum_{j=1}^{d}\Big\|\frac{\partial^{R_{j}}}{\partial x_{j}^{R_{j}}}f\,\big| L_{2}(\Bbb T^d)\Big\|^{2}\Big)^{1/2}&= \Big(\sum_{{\bf k}\in \Bbb Z^d}\big(1+\sum_{j=1}^d|k_j|^{2R_j}\big)|\hat f({\bf k})|^2\Big)^{1/2}.
R(\log N)-S(N) <\frac{0.005586}{(\log \log (10^{(10^{13})}))^2}-p_s^{-E(p_s )}&\leq T-p_s^{\frac{\log T}{\log p_s}}=T-e^{(\log p_s)\left(\frac{\log T}{\log p_s}\right)}=T-T=0.
u(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i\lambda x-\lambda^{2}t}Q(\lambda,t)d\lambda-\frac{1}{2\pi}\int_{\partial D^{+}}e^{i\lambda x-\lambda^{2}t}[2i\lambda\tilde{g_{0}}(\lambda^{2},t)+Q(-\lambda,t)]d\lambda.
\phi_1 ={\phi_0 \; (\eta_{ab}\; \zeta^a \; \zeta^b) \;\left\{\eta_{ab}\;[x^a-x_0^a] \; [x^b-x_0^b] - \eta_{ab}\; \zeta^a \; \zeta^b \right\}\over(\eta_{ab}\;[x^a-x_0^a] \; [x^b-x_0^b] - \eta_{ab}\; \zeta^a \; \zeta^b )^2 +4 (\eta_{ab}\;[x^a-x_0^a] \; \zeta^b)^2};
\liminf_{n \rightarrow \infty}\liminf_{t \rightarrow \infty}P\big( \hat{W}^{n}(t) \geq x \big) \ \ =\ \ \liminf_{n \rightarrow \infty}P\big( \hat{W}^{n}(\infty) \geq x \big) \ \ =\ \P\big( W^{\infty}(\infty) \geq x \big)
\sum_{n \geq 0}\frac{q^{{(n^2+n)}/{2}}(-1;q)_n}{(q;q)^2_n}= \frac{(-q;q)_\infty}{(q;q)_\infty}, \sum_{n \geq 0}\frac{q^{(n^2+n)/{2}}(-q;q)_n}{(q;q)^2_n}= \frac{(-q;q)_\infty}{(q;q)_\infty}\left(1+\sum_{n\geq 1}\frac{(-1)^n q^{2n-1}}{(-q;q^2)_{n}}\right) =\frac{(-q;q)_\infty}{(q;q)_\infty}\sum_{j\geq 0}q^{{(3j^2+j)}/{2}}(1-q^{2j+1}).
A=(\alpha_{i-j})=\left(\begin{array}{cccccc}\alpha_0 0 0 0 0 \cdots \alpha_1 \alpha_0 0 0 0 \cdots \alpha_2 \alpha_1 \alpha_0 0 0 \cdots \alpha_3 \alpha_2 \alpha_1 \alpha_0 0 \cdots \vdots \vdots \vdots \vdots \vdots \ddots \end{array}\right) (i,j=1,2,3,\dots),
M=\left[ \begin{array}{ccc}1 \frac{1}{3}\frac{1}{5}\frac{1}{1+n}\frac{1}{3(3+n)}\frac{1}{5(5+n)}\frac{1}{(1+2n)(1+n)}\frac{1}{3(3+2n)(3+n)}\frac{1}{5(5+2n)(5+n)}\end{array}\right]
\mathcal{G}\left( \delta_{x}- \delta_{x}^{\epsilon}, \mu_{\theta}\right) = \int_{|y-x| \leq r}h^{\delta_{x}- \delta_{x}^{\epsilon}}\mu_{\theta}(y) \, dy + \int_{|y-x| \geq r}h^{\delta_{x}- \delta_{x}^{\epsilon}}\mu_{\theta}(y) \, dy.
Ce^{\sum_{k=0}^mh_k(x)B_k}\sum_{k=0}^mh_k(x)dB_k+\dfrac{{C}}{2}e^{\sum_{k=0}^mh_k(x)B_k}\sum_{k=0}^m(h_k(x))^2dt+e^{\sum_{k=0}^mh_k(x)B_k}d{C}.
(e^{-t\mathbb{E}^{*}\mathbb{E}}f)(x) =\frac{e^{-tQ^{2}/4}}{\sqrt{4\pi t}}r^{-Q/2}\int^{\infty}_{0}e^{-\frac{(\ln r - \ln s )^{2}}{4t}}s^{-Q/2}f(sy)s^{Q-1}ds &=\frac{e^{-tQ^{2}/4}}{\sqrt{4\pi t}}|x|^{-Q/2}\int^{\infty}_{0}e^{-\frac{(\ln |x| - \ln s )^{2}}{4t}}s^{-Q/2}f(sy)s^{Q-1}ds,
[X \xleftarrow{(f_1 \sqcup f_2) \circ \overline h}(M_1 \sqcup M_2) \times_Y N \xrightarrow{k\circ \overline{g_1 \sqcup g_2}}Z] &= [X \xleftarrow{(f_1 \sqcup f_2) \circ (\widetilde h \sqcup \widetilde{\widetilde h})}\bigl (M_1 \times_Y N \bigr) \sqcup \bigl (M_2 \times_Y N \bigr) \xrightarrow{k\circ (\widetilde{g_1}\sqcup \widetilde{g_2})}Z].
y^{\prime \prime}(z)=\left[\begin{array}[c]{l}\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{\omega_{k}}{2})+\tfrac{3}{4}(\wp(z+p) +\wp(z-p))+A(\zeta(z+p)-\zeta(z-p))+B\end{array}\right] y(z)E_{\tau},
\prod_{1 < a < q_1/2 \atop (a, q_1) =1}{\xi}_a^{2d_a}~=~ \prod_{1 < b < q_2/2 \atop (b, q_2) =1}{\xi}_b^{-2e_b}\prod_{1 < c < q_3/2 \atop (c, q_3) =1}{\xi}_c^{-2f_c}~=~1.
\Delta^{(N,0)}(s)= - \sum_{n>0,\vec n^2<N}\left[ J(z_n) - 2 + 2 J(y_{n}) + \frac{J^{2}(y_{n})}{2 (1-y_{n})}- J(\tilde{z}_n) - 2 J(\tilde{y}_n) \right]\ ,
\begin{array}{l}\{n^{a}(x),n^{b}(y)\}=0 \;\;\;\;\; ,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\{{\pi}^{a}(x),n^{b}(y)\}=(\delta^{ab}-n^{a}(x)n^{b}(x))\delta(x-y) \{{\pi}^{a}(x),{\pi}^{b}(y)\}= ({\pi}^{b}(x)n^{a}(x) -{\pi}^{a}(x)n^{b}(x))\; \delta(x-y) \end{array}
\widetilde{\mathbf{H}}_k(\overline{E}):=\frac{(-1)^{k+2}}{2(k+2)!(2\pi i)^{k+2}}\int_{(\mathbb{P}^1)^{k+2}}{\rm ch}_g^0\Big({\rm tr}_2\circ \lambda\big(H_\chi(\overline{E})\big)\Big)\wedge C_{k+2}(\log\mid z_1\mid^2,\cdots,\log\mid z_{k+2}\mid^2),
\sigma_{n}=\sum_{k=1}^{n}\frac{y_{k}}{x_{k}}=\frac{a_{1}}{b_{1}}\genfrac{}{}{0pt}{}{{}}{+}\frac{a_{2}}{b_{2}}\genfrac{}{}{0pt}{}{{}}{+\cdots}\genfrac{}{}{0pt}{}{{}}{+}\dfrac{a_{2n}}{b_{2n}},
\tilde{D}^{[r,0]}_{T}(x^{2}_{M}) \rightarrow\frac{\chi_{r,0}\Lambda^{\frac{1}{2}}}{\left(2 \pi \sqrt{x^{2}_{M}}\right)^{\frac{3}{2}}}\, e^{-\gamma_{r,0}\sqrt{x^{2}_{M}}}\cos \left(\omega_{r,0}\sqrt{x^{2}_{M}}+ \varphi_{r,0}\right) \, ,
\tau_t (\tilde{\phi}(p') U(x)) \tilde{\phi}(p) = e^{i t p_0' + i x\cdot p}\tilde{\phi}(p') \tilde{\phi}(p) U(x) = e^{i t p_0' + i x\cdot p}\left(\tilde{C}(p', p) U(x) + \tilde{\phi}(p) \tilde{\phi}(p') U(x) \right)
\Gamma(2,x;0,\xi) \geq\int_{B_{1}}\cdots\int_{B_{k}}\prod_{m=1}^{k+1}\Gamma(\frac{2m}{k+1},z_{m};\frac{2(m-1)}{k+1},z_{m-1})dz_{1}\cdots dz_{k}\geq(k^{n/2}e^{-2C})^{k+1}(\vert B(1)\vert k^{-n/2})^{k}.
G_tG =\frac{3(e^2-g_i^2)+6ef\varepsilon_i}{8\varepsilon_i^3}t^2-\frac{ef}{2\varepsilon_i}t -\frac{3(e^2-g_i^2)+2ef\varepsilon_i}{8\varepsilon_i}, G_t^2+G_{tt}G =\frac{3(e^2-g_i^2)+6ef\varepsilon_i}{4\varepsilon_i^3}t-\frac{ef}{2\varepsilon_i}.
s = (2k - 2\ell - 1)(|A'| - |V_0| - 2q + 1) - |B'| + 2(2k- 2\ell-1) |V_0| + 2(3k - 3\ell -1)q - (2k - 3\ell -1) = (2k - 2\ell -1) |A'| - |B'| + (2k- 2\ell-1) |V_0| + (2k - 2\ell)q + \ell.
M_1 = \left[ \Delta s - 3|\nabla n|^2 s -\frac{1}{L}(2s\partial_1\varphi + s^2 \partial_2\varphi)\right]\left(n\otimes n-\frac{1}{3}\mathbf I\right), M_2 =2n \odot (s\Delta n + 2(\nabla s\cdot \nabla)n + s|\nabla n|^2 n), M_3 = s \left[ 2\sum_k \partial_k n \otimes\partial_k n + |\nabla n|^2 \left( n\otimes n-\mathbf I\right) \right].
|K_{\alpha}(x,\vec{y}) -K_{\alpha}(x',\vec{y}) | \le \dfrac{A}{\Big (\sum\limits_{j=1}^{m}|x-y_{j}| \Big)^{mn-\alpha}}\omega \Big( \frac{|x-x'|}{|x-y_{1}|+\cdots+|x-y_{m}|}\Big)
K = A_d \oplus A_{d-1}\oplus \cdots \oplus A_1 = \sum_{\mu=1}^d A_{\otimes\mu}, A_{\otimes\mu}= I_d \otimes \cdots \otimes I_{\mu+1}\otimes A_\mu \otimes I_{\mu-1}\otimes \cdots \otimes I_1,
\begin{array}{c}{{x^{\mu}( \sigma , \tau ) = x_{R}^{\mu}( \sigma , \tau ) + x_{L}^{\mu}( \sigma , \tau )}}\\{{x_{R , L}^{\mu}( \sigma , \tau ) = \frac 1 2 x_{0}^{\mu}+ \frac 1 2 l^{2}p^{\mu}( \tau \mp \sigma ) + \frac{i l}2{\sum}\frac 1 n \alpha_{n , \pm}^{\mu}e^{- 2 i n ( \tau \mp \sigma )}}}\\ \end{array}
\begin{array}{l l l}{[ ( s x u \wedge [ r ] ) m ( s z u \wedge [ r ] ) ] j [ ( s x u \wedge [ r ] ) m ( r z d \wedge [ s ] ) ]}&{=}&{}\\{( \{s \}\wedge [ r ] ) j ( s x u \wedge r z d ) = s x u \wedge [ r ] ,}&{}&{}\\ \end{array}
I_1 \leq C \tau \sum_{k = 0}^{m-1}(t_m - t_k)^{-\frac{1}{2}}\| F_n (Y_k^n)\|_{L^2(\Omega, H_n)}\leq C L \tau \sum_{k = 0}^{m-1}(t_m - t_k)^{-\frac{1}{2}}\big( \| Y_k^n\|_{L^2(\Omega, H_n)}+ 1 \big) < \infty.
\phi_{p,q}(\lambda)= det(I_4\lambda-D_{B_1^c}) ={\lambda}^{4}+ \left( -q+4-p \right){\lambda}^{3}&+ \left( -8\,pq+2\,p+2\,q+4 \right){\lambda}^{2}&+ \left( -14\,pq+6\,p+6\,q \right) \lambda-5\,pq+2\,p+2\,q.
\| f\|_{\delta, \delta'}: =\sum_{n\in \mathbb{Z}^3}\sum_{|\alpha|\ge 0}\| \langle v\rangle^m \partial_v^\alpha f_n \|_{L^2( \mathbb{R}^3\times \mathbb{R}^3)}\frac{|\delta'|^{|\alpha|}}{|\alpha|!}e^{\delta |n|},
R_n^{(2)}(z) = \frac{1}{z-1}\frac{\prod_{j=0}^{n-1}\big[1- \rho_{j}^{(2)}\alpha_{j}(\delta)\big]}{\prod_{j=0}^{n-1}\big[1-\mathcal{R}e\big(\rho_{j}^{(2)}\alpha_{j}(\delta)\big)\big]}\Big[ zS_{n}(\delta; z) - \rho_{n}^{(2)}S_{n}^{\ast}(\delta; z)\Big], n \geq 0,
\dot{x}\left(t\right)=\left[\begin{array}{c}V_{p}\left(\delta\theta\right)-V_{e}+\frac{\delta ya_{e}}{V_{e}}V_{p}\left(\delta\theta\right)-\frac{\delta xa_{e}}{V_{e}}\frac{a_{p}}{V_{p}}-\frac{a_{e}}{V_{e}}\end{array}\right],
\begin{array}{lcl}X_n^{(J)}(\tau) ={\displaystyle C_n^{(J)}\sqrt{-\tau}~ J_{-\nu}\left(-\frac{2\pi}{\lambda_J}|n| \tau \right) +D_n^{(J)}\sqrt{-\tau}~ J_{\nu}\left(-\frac{2\pi}{\lambda_J}|n| \tau \right)}~,~ (n\neq 0) X_0^{(J)}(\tau) ={\displaystyle C_0^{(J)}(-\tau)^{\frac{1}{2}- \nu}+D_0^{(J)}(-\tau)^{\frac{1}{2}+ \nu}}\end{array}
P_{ij}:= \frac{1}{n-2}\left(R_{ij}- Jg_{ij}\right), W_{ijkl}:= R_{ijkl}- P_{ik}g_{jl}- P_{jl}g_{ik}+ P_{il}g_{jk}+ P_{jk}g_{il}, C_{ijk}:= \nabla_iP_{jk}- \nabla_jP_{ik}, B_{ij}:= \nabla^s C_{sij}+ W_{isjt}P^{st},
\left\vert \underset{_{\substack{(\rho )=\frac{1}{2}Y<\left\vert \gamma \right\vert \leq T}}}{\sum}\frac{x^{\rho +1}}{\rho\left( \rho +1\right)}\right\vert \leq x^{\alpha}(\log x)^{\beta}(\log\log x)^{\beta}
\sum_{n=1}^{\infty}\frac{q^n}{(zq^n;q)_{n+1}(zq^{2n+2};q^2)_{\infty}}=\sum_{n=0}^{\infty}\frac{z^nq^{2n^2+2n+1}}{(q;q^2)_{n+1}(zq;q^2)_{n+1}}, \sum_{n=0}^{\infty}q^n(-zq^{n+1};q)_{n}(-zq^{2n+2};q^2)_{\infty}=\sum_{n=0}^{\infty}\frac{z^n q^{n^2+n}}{(q;q^2)_{n+1}}.
\Phi=h_{0}h_{1}h_{2}+\frac{h_{1}^{3}}{3!}+\frac{h_{0}^{2}h_{3}}{2!}+2\frac{h_{1}^{3}}{3!}+h_{1}h_{2}h_{3}+2\frac{h_{2}h_{3}^{2}}{2}+\cdots.
\Big \| \Big ( C_{z_\gamma}(AP_\alpha + Q_\alpha)C_{-z_\gamma}- (A_{x}P_\alpha + Q_\alpha) \Big ) M_{\chi_{B(0,R)}}\Big \| &= \Big \| \Big ( C_{z_\gamma}AP_\alpha C_{-z_\gamma}- A_{x}P_\alpha) M_{\chi_{B(0,R)}}\Big \| &= \Big \| (\widetilde{C}_{z_\gamma}A \widetilde{C}_{-z_\gamma}- A_{x}) P_\alpha M_{\chi_{B(0,R)}}\Big \| & \to 0
\partial_{B}^{2}\left( f \circ \theta \right) = ( \partial_{_{w}}^{2}f ) ( x ) + \partial_{\int_{\lbrack 0 , \cdot ]^{2}}1_{r \leq \tau}[ A d_{x ( r )}B ( r ) , A d_{x ( \tau )}B ( \tau ) ] d r d \tau}f ( x ) .
R_{jkl}^{i}=\frac{\partial\Gamma_{lj}^{i}}{\partial x^{k}}-\frac{\partial\Gamma_{kj}^{i}}{\partial x^{l}}+(\Gamma_{kp}^{i}\Gamma_{lj}^{p}-\Gamma_{lp}^{i}\Gamma_{kj}^{p})
\left. C_{2}^{\left( N , M | R , S \right)}\right\vert_{N + M = R + S}= C_{2}^{S U \left( C \right)}+ \frac{1}{2 C}\left( \Delta + \left( N - R \right) C \right)^{2}+ \frac{C}{2}\left( \Delta + \left( N - R \right) C \right) .
S^{2}=\frac{n}{n-1}\tilde{S}_{Y}^{2}=\frac{n}{n-1}[\frac{1}{n}\sum_{i=1}^{n}(Y_{i}-\overline{Y})^{2}]=\frac{1}{n-1}\sum_{i=1}^{n}(Y_{i}-\overline{Y})^{2}
Z_1 = f^{-1}(X- Y) \cap Y- X Z_2 = f^{-1}(X- Y) - Y Z_3 = f^{-1}(X\cap Y) \cap Y- X Z_4 = f^{-1}(X \cap Y) - Y Z_5 = f^{-1}(X) \cap X = X \cap Z
\limsup_{t\to\infty}\frac{|X_t|}{\log^{1/2}t}= \frac{1}{\sqrt{2\alpha}}\limsup_{t\to\infty}\frac{|B_{\mathrm{e}^{2\alpha t}-1}|}{\mathrm{e}^{\alpha t}\log^{1/2}t}= \frac{1}{\sqrt{\alpha}},\;\;\;\textrm{a.s}.
\int_{D_1 \times \cdots \times D_k}e^{1, 2, \cdots, k}_k(w, z) \frac{\partial^{k-1}f_k(z)}{\partial \bar z_1 \cdots \partial \bar z_{k-1}}dV(z) = \int_{D_1 \times \cdots \times D_k}(-1)^k f_k(z) \frac{\partial^{k-1}e^{1, 2, \cdots, k}_k(w, z)}{\partial \bar z_1 \cdots \partial \bar z_{k-1}}dV(z).
p_{V_{\geq t}}\left(RV_{\geq t}\ \middle|\G_{<t}, V_{<t}\right) =\frac{\mathbb{P}\left( G_{<t}\ \middle|\V \right)p_V(RV)}{\mathbb{P}\left( G_{<t}\ \middle|\V_{<t}\right)p_{V_{<t}}(V_{<t})}
\mathbf{y}_{1}= \mathbf{D}^{q-m}\mathbf{x}_{1}\oplus \mathbf{D}^{q-n}\mathbf{x}_{2}; \ \mathbf{y}_{2}= \mathbf{D}^{q-m}\mathbf{x}_{2}\oplus \mathbf{D}^{q-n}\mathbf{x}_{1},
\mu\Big(E_f(Q_{\rho_{m+1}},c_0 \lambda_0 M^{m+1})\Big) \leq \alpha \left[ \mu\Big(E_f(Q_{\rho_m}, c_0 \lambda_0 M^m)\Big) +\nu\Big(E_g(Q_{\rho_m}, c_0 \lambda_0 M^m)\Big) + \hat\nu\Big(E_{\hat g}(Q_{\rho_m}, c_0 \lambda_0 M^m)\Big)^{\hat p}\right]\forall m=0,1,\dots
R_{a L b L}= W_{a L b L}R_{a L b\underline{L}}= W_{a\underline{L}b\underline{L}}- \kappa^2 g_{ab}R_{a L L \underline{L}}= W_{a L L \underline{L}}R_{L \underline{L}L \underline{L}}= W_{L \underline{L}L \underline{L}}- \kappa^2.
f^{(\tau_0,\omega_0)}(t) = -\frac{(4\pi\omega_0^2{\rm Im}(\tau_0))^2}{24\omega_0^2 (t+4\pi\omega_0^2{\rm Im}(\tau_0))^2}E_2\left(\frac{\bar{\tau_0}t +4\pi\omega_0^2{\rm Im}(\tau_0)\tau_0}{t+4\pi\omega_0^2{\rm Im}(\tau_0)}\right) + \frac{1}{2(t+4\pi\omega_0^2{\rm Im}(\tau_0))}.
(i)xy^{2}+\frac{2\omega}{K_{0}}y=4\mathfrak{S}_{0}& (ii)\frac{z}{y}-\omega\lbrack J_{3}\frac{1}{xy}+J_{4}y\ ]=J_{2}& (iii)x(z^{2}-2h)+\frac{\omega^{2}}{x}-\frac{\omega}{2K_{0}}y=J_{1}
B := \{t \in \rho A : |z_0^* (f_k (t)) - z_0^* (f_k (t_0)) | < 1, \, k=1,\ldots, n\}, &D := \{z^* \in \Gamma_{Z}: |z^* (f_k (t_0)) - z_0^* (f_k (t_0)) | < 1, \, k=1,\ldots, n\}.
\|V_{n}\|^{\frac{10}{3}}= \|V_{n}\|^{2}\|V_{n}\|^{\frac{4}{3}}\le \frac{2}{3}\|V_{n}\|^{3}+ \frac{1}{3}\|V_{n}\|^{4}\le \frac{1}{3}\|V_{n}\|^{2}+ \frac{2}{3}\|V_{n}\|^{4}.
\mbox{dim}(\mbox{Im}\,(U^*U-I)) = \overline{\mbox{dim}(\mbox{Im}\,(U^*U-I))}= \mbox{dim}(\mbox{Im}\,W^*QW)\leq \mbox{dim}(\mbox{Im}\,Q) &= \mbox{dim}((\mbox{Im}\,U)^{\perp}) = e(f_n).
\left|-\int_{0}^{T}\int_{\Omega}\varepsilon\,\displaystyle\sum_{j=1}^{d}B(u^{\varepsilon})\eta^{\prime}(u^{\varepsilon})\frac{\partial u^{\varepsilon}}{\partial x_{j}}\,\frac{\partial \phi}{\partial x_{j}}\,dx\,dt\right| \leq \varepsilon \|B\|_{L^{\infty}(\mathbb{R})}\|\eta^{'}\|_{L^{\infty}(I)}\|\nabla u^{\varepsilon}\|_{\left(L^{2}(\Omega_{T})\right)^{d}}.
F_1 (a,b_1,b_2,c,x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n}(b_1)_m (b_2)_n}{(c)_{m+n}}\frac{x^m y^n}{m!n!}&= F\left[ \begin{array}{c}\{a,\{1,1\}\}, \{b_1,\{1,0\}\}, \{b_2,\{0,1\}\}\{c,\{1,1\}\}\end{array}\;\middle|\;\{x,y\}\right]
\gamma \left\{\Vert\varphi_{t}\Vert^2_{L^2_\gamma(\mathbb R; H^2(\mathbb T))}+\Vert\varphi_{x}\Vert^2_{L^2_\gamma(\mathbb R;H^2(\mathbb T))}\right\}\le \frac{C}{\gamma}\Vert g\Vert^2_{L^2_\gamma(\mathbb R; H^2(\mathbb T))}\,.
I m I_{1_{\beta}}\approx \frac{- g^{2}}{8}\int \frac{d^{3}l}{( 2 \pi )^{2}E_{1}E_{2}}2 \delta ( E_{1}- E_{2}) \Bigg ( \frac{2}{e^{\beta E_{1}}- 1}+ \frac{2}{( e^{\beta E_{1}}- 1 )^{2}}\Bigg ) .
\frac{1}{2}\frac{d}{dt}\int_{\Omega}\Big(\mu|\nabla u|^2+(\mu+\lambda)|u|^2-\Big(P+\frac{1}{2}|H|^2\Big) u+ H\cdot \nabla u\cdot H\Big)x+C|\nabla^2 u|^2_2 \leq C\Big(|\nabla u|^2_2+\Big(1+\frac{1}{\sigma}\Big)|\nabla H|^2_2+|\nabla \rho|^2_2+1\Big)(|\nabla u|^2_2+|\nabla \theta|^2_2+|D(u)|_\infty+1).
\mathbf{0}=\begin{pmatrix}a_{j-N+1,f_{l-N+1}}\hdots a_{j-N+1,f_{l}}\vdots \vdots a_{j,f_{l-N+1}}\hdots a_{j,f_{l}}\end{pmatrix}\begin{pmatrix}s_{l-N+1}\vdots s_{l}\end{pmatrix}
\displaystyle{\lim_{k\to +\infty}\int_{Q_i}f(x, \xi^{(i)}_k,\overline{v}^{(i)}_k,\nabla \xi^{(i)}_k)dx=\int_{\partial^\ast E\cap Q_i}K_p(x,0,c,d,e_N)d{\cal H}^{N-1}+\int_{Q_i}f(x,u(x),v(x),0)dx,}
G_{\alpha_1\ldots\alpha_n}(p_1,\ldots,p_n)\;\prod_{i=1}^{n}\; J_{\alpha_i}^i(p_i) = \hat{G}_{\alpha_1\ldots\alpha_n}(p_1,\ldots,p_n)\;\prod_{i=1}^{n}\;\hat{J}_{\alpha_i}^i(p_i) \ \ \ \ \ .
\mathbb{E}h(X)-\mathbb{E}h(\pi) =\mathbb{E}f(X)-\mathbb{E}\sum_{j=0}^\infty P_{X,j}f(j) &=\sum_{j=0}^\infty f(j)\left[\mathbb{P}(X=j)-\sum_{i=0}^\infty\mathbb{P}(X=i)\mathbb{P}(Z_1=j|Z_0=i)\right] &=\sum_{j=0}^\infty\Delta f(j)\left[\mathbb{P}(X>j)-\sum_{k=j+1}^\infty\sum_{i=0}^\infty\mathbb{P}(X=i)\mathbb{P}(Z_1=k|Z_0=i)\right],
\frac{da}{dr}=\frac{2PQ}{R_{+}R_{-}^{2}}\;ab-\frac{r}{2R_{-}^{2}}a, &\frac{da}{dr}=\frac{rR_{+}}{Q^{2}}ab-\frac{r}{2R_{-}^{2}}a, &\frac{db}{dr}=\frac{PQ}{R_{-}^{2}R_{+}}(1-b^{2})-\frac{PQ}{R_{+}^{3}}(1-a^{2})-\frac{r}{2R_{+}^{2}}b, &\frac{db}{dr}=\frac{rR_{+}}{2Q^{2}}(1-b^{2})-\frac{R_{-}^{2}r}{2R_{+}Q^{2}}(1-a^{2})-\frac{r}{2R_{+}^{2}}b.
\Delta^{(m+n)}_{l}= \left| c \bigl( \xi^{(m+n)}_{l};q_0\bigr) \right|^2 \left( \sum_{0\leq\nu <m+n}\left| p_{\nu}(\xi^{(m+n)}_{l};q_0) \right|^2 \right)^{-1}= \Delta_{\texttt{b};l}^{(m+n-1,1)}= \left( 2(m+n)+ u_{q_0}\bigl( \xi^{(m+n)}_l \bigr)+ u_{q_1}\bigl( \xi^{(m+n)}_l \bigr) \right)^{-1},
(\mathcal{B}_{m_{\tilde{k}}}^{d}F)(Z) = \frac{-\tilde{k}}{2i\pi}\int_{\gamma_{\tilde{k}}}F(u) \exp( (\frac{Z}{u})^{\tilde{k}}) \frac{Z^{\tilde{k}}}{u^{\tilde{k}+1}}du
\psi_{k,k-n+1}^{\ast}(Q^{\prime})=\frac{L_{k,k-n+1}^{\ast}(q^{\prime})}{q^{\prime}}\leq \min_{1\leq u\leq k-n+1}\frac{L_{k,R_{u}}^{\ast}(q^{\prime})}{q^{\prime}}\leq \Phi_{k,n}(\psi_{n,1}^{\ast}(Q))
a_{k b}^{\prime}={\cal K}_{k n}^{- 1}\Bigl ( G^{1 1}({H^{n}}_{b}+ e^{\rho}\alpha_{n}\gamma_{b}) - G^{n 1}({H^{b}}_{J}b^{J}+{\frac{1}{2}}e^{\rho}\gamma^{b}b^{I}d_{I J}b^{J}) \Bigr ) ,
\{\{...\{QA_{0}(x)\xi_{0},\underbrace{S\},S\},...\},S\}}_{n}=\left\{\begin{array}{lr}Q(2i)^{n}\Phi_{n}(p,x,\vec{\xi})\xi_{5}\xi_{0}n-odd & Q(2i)^{n}\Phi_{n}(p,x,\vec{\xi})\xi_{0}n-even\end{array}\right. +{\cal O}(Q^{2}),
s_{2}[f;m^{2}]=-{1\over 16\pi m^{2}}\,\int\! d^{2}x\,\phi^{2}(x) -{1\over 96\pi m^{4}}\,\int\! d^{2}x\, \phi(x)\partial^{2}\phi(x) +{\cal O}(\partial^{4}).
\limsup_{n\to\infty}\left|\Delta_n^{-\frac 12}\langle f_1,f_2\rangle \sum_{i=1}^n \int_{(i-1)\Delta_n}^{i\Delta_n}\langle (\mathcal S(i\Delta_n-s)-I) f_1,f_2\rangle ds\right|=\infty.
H_\eta(u_1(t)|u_2(t)) :=\int_\Omega \bigg[f_1(t)\ln\Big(\frac{f_1(t)+\eta}{f_2(t)}\Big)-(f_1(t)-f_2(t))\bigg]\,\mathrm{d}x begin{align*}1ex] \quad+ \cfrac{b}{c}\int_\Omega\bigg[g_1(t)\ln\Big(\frac{g_1(t)+\eta}{g_2(t)}\Big)-(g_1(t)-g_2(t))\bigg]\,\mathrm{d}x, t\in[0,T]\,.
\rho_n(a)=\sum_{\lambda\in \Lambda^{n_+}}\sum_{\mu \in \Lambda^{n_-}}T_\lambda\rho_0(t_\lambda^*t_\lambda a_{\lambda,\mu}t_\mu^* t_\mu)T_\mu^*=\sum_{\lambda\in \Lambda^{n_+}}\sum_{\mu \in \Lambda^{n_-}}T_\lambda\rho_0(a_{\lambda,\mu})T_\mu^*.
-\frac{1}{e^{2\varepsilon}-1}\mathrm{Re}(F_N(e^{i\theta + \varepsilon}))=\|g\|^2\geq\sum_{k = 0}^\infty |g(L_N+2k+1)|^2=\frac{1}{1-e^{-2\varepsilon}}|g(L_N+1)|^2
\begin{array}{l}(x + a)\cdot (y + b) =\big(x\cdot_A y + l_{B}(a)y + r_{B}(b)x\big)+ \big (a\cdot_B b + l_{A}(x)b + r_{A}(y)a\big), (\alpha\oplus\beta)(x + a)=\alpha(x) + \beta(a).\end{array}
(A\rho)_\mu = \mathcal{F}\big\{\bar{d}_\ell(s, t) \rho(s, t) \big\}(p, q) (A^* y)_\nu = \sum_{l=1}^L\sum_{p=0}^{M-1}\sum_{q=0}^{N-1}d_\ell(s, t) e^{2\pi i ps/M}e^{2\pi i qt/N}y_\ell(p, q) &= MN~\sum_{l=1}^L d_\ell(s, t)~\mathcal{F}^{-1}\big\{y_\ell(p, q) \big\}(s, t),
A = (\{\!\{a_{1,1}\}\!\},\{\!\{a_{2,1},a_{2,2}\}\!\},\ldots,\{\!\{a_{n,1},a_{n,2},\ldots,a_{n,n}\}\!\}), a_{i,j}\leq a_{i,j+1}(\forall 1 \leq j < i \leq n ),
\partial_\alpha D(p||p_{\alpha,\mu}) = - \frac{1}{\alpha}+ \mathbb{E}_p[(x-\mu)] - \mathbb{E}_p[ e^{-\alpha(x-\mu)}(x-\mu)], \partial_\mu D(p||p_{\alpha,\mu}) = -\alpha + \alpha \mathbb{E}_p[e^{-\alpha(x-\mu)}].
\zeta_1(s,x\b a)= a^{-s}\cdot \zeta_H\left(\frac{x}{a}\right),\Gamma_1(x\b a)= \frac{1}{\sqrt{2\pi}}\cdot \Gamma\left(\frac{x}{a}\right) \cdot a^{\frac{x}{a}-\frac{1}{2}},
F^{\hat \nabla}= \left(\begin{array}{ccc}0 f_{13}\vartheta -f_{12}\vartheta -f_{13}\vartheta 0 f_{23}\vartheta^{2}f_{12}\vartheta -f_{23}\vartheta 0 \end{array}\right) + \left( \begin{array}{ccc}1 0 0 0 0 \vartheta 0 -\vartheta 0 \end{array}\right) \, .
\hat W_{n}^T(k_1,\ldots,k_n)=\tilde c_n \int_{R^n}\hat W_{n,m_1,\ldots,m_n}^T(k_1,\ldots,k_n)\prod_{l=1}^n\rho_{\alpha,m_0}(m_l^2)dm_1^2\cdots dm_n^2
\hat{f}_\theta(k) = e^{-d \theta/2}\left( \tilde{Q}_{\sigma,a}(e^{-\theta}k ) e^{-\frac{\pi}{\sigma}\frac{\delta}{4}|k|^2}\right) \left( e^{-\frac{\pi}{\sigma}e^{-2\theta}|k|^2}e^{\frac{\pi}{\sigma}\frac{\delta}{2}|k|^2}\right) \left( e^{- e^{-\theta}k \cdot u}e^{- \frac{\pi}{\sigma}\frac{\delta}{4}|k|^2}\right) .
I_{4}^{1}\leq \frac{C}{(\frac{\lambda_{i}}{\lambda_{j}}+\lambda_{i}\lambda_{j}G^{\frac{2}{2-n}}_{a_{j}}(a_{i}))^{\frac{n+2}{2}}}\underset{[\vert \frac{x}{\lambda_{j}}\vert \leq \tilde E d(a_{i},a_{j})]}{\int}(\frac{1}{1+r^{2}})^{\frac{n-2}{2}}
\varphi(1,\phi) = \frac{\phi}{\ln 2}\int^{\infty}_{0}\frac{1}{1+x\phi}\left(1-\gamma_1 \frac{\left[\gamma(1,x) \right]^{K}}{K}\right) dx = \frac{1}{\ln 2}\left( \sum^{K}_{l=1}{K \choose l}(-1)^l e^{\frac{l}{\phi}}\mathbf{E_i}\left(-\frac{l}{\phi}\right) \right) .
\|z_{i,t}-\bar{z}_{t}\| \le \tau \lambda^{t-2}\sum_{j=1}^n\|z_{j,1}\|+\tau\sum_{s=1}^{t-2}\lambda^{t-s-2}\sum_{j=1}^n\|\epsilon^z_{j,s}\|+\|\epsilon^z_{i,t-1}\|+\frac{1}{n}\sum_{j=1}^n\|\epsilon^z_{j,t-1}\|,
{\cal L}_{M a x w e l l}^{S U S Y}= \frac{1}{4 e^{2}}\left[ \int \! d^{2}\theta ~ W^{2}\! \left( y , \theta \right) + \int \! d^{2}\bar{\theta}~ \bar{W}^{2}( y^{\dagger}, \bar{\theta}\, ) \right]
\rho_1(a)=\min\left\{|\lambda-\mu|\mid \lambda\neq\mu,\lambda,\mu\in\mathrm{supp}(a)\right\}\rho_2(a)=\min\left\{|\lambda-\mu|-1\mid|\lambda-\mu|>1,\lambda,\mu\in\mathrm{supp}(a)\right\}\rho(a)=\frac{1}{2}\min(\rho_1(a),\rho_2(a),1)>0.
\mathbf{x}^{\star}=\underset{\mathbf{x}=(x_{k})_{k=1}^{K}}{\arg\min}\;\mathbb{E}\left[\sum_{n=1}^{N}\left(y_{n}-\mathbf{g}_{n}^{T}\mathbf{x}\right)^{2}\right] =\underset{\mathbf{x}}{\arg\min}\;\frac{1}{2}\mathbf{x}^{T}\mathbf{G}\mathbf{x}-\mathbf{b}^{T}\mathbf{x},
\sigma_{\mathrm{t o t}\, i}^{( 1 )}= \frac{( 2 \pi )^{4}\hbar^{2}}{v_{i}}\frac{1}{\pi}\mathrm{I m}\, R_{i i}^{( 1 )}\exp \left[ - \frac{1}{\hbar}\int_{- \infty}^{\infty}d t \Gamma^{( 1 )}( t ) \right] \; ,
\left(\mathrm{curl}\ \boldsymbol{u}, \mathrm{curl}_w\ \boldsymbol{v}_h\right)_{\mathcal{T}_h}+\left(\boldsymbol{u}, \boldsymbol{v}^0_h \right)_{\mathcal{T}_h}-\left(\boldsymbol{f}, \boldsymbol{v}^0_h \right)_{\mathcal{T}_h}-E(\boldsymbol{u}, \boldsymbol{v}_h)=0,
\prod_{l=1}^d2^{(a_l+b_l)/2}\left(\int_{\mathcal{I}_{\boldsymbol{b}}}\prod_{l=1}^d2^{b_l}\|\psi_{k_l}\|_{\infty}\|\psi_{k_l}^{'}\|_{\infty}d\boldsymbol{x}+\int_{\mathcal{I}_{\boldsymbol{a}}}\prod_{l=1}^d2^{a_l}\|\psi_{k_l}\|_{\infty}\|\psi_{k_l}^{'}\|_{\infty}d\boldsymbol{x}\right),
p_D(t,x,y)\ge \int_{B_1}\ldots\int_{B_{k-1}}p_D(t/k,x,y_1)\ldots p_D(t/k, y_{k-1},y) dy_{k-1}\ldots dy_1\ge \left( \frac{c_4\,t}{Tk}\right)^k\Pi^{k-1}_{i=1}|B_i| &\ge \left(\frac{c_5\,t}{Tk}\right)^{k}\ge c_6 \left(\frac{c_7\,t}{Tr}\right)^{k}\ge c_6 \exp\left(-c_8r\log\frac{Tr}{c_7t}\right) \ge \exp\left(-c_9r\left(1+\log\frac{Tr}{t}\right)\right).
\left\|\sum\limits_{Q\in \partial G_N}(f-m_Q)\cdot 1_Q\right\|_q= \left\{\sum\|f-m_Q\|_{L_q(Q)}^q \right\}^{1/q}:= \left\{\sum\limits_{Q\in \partial G_N}E_k(f;Q;L_q)^q\right\}^{1/q}\leqslant
\widetilde{\bf y}_{R_p}[f_k]=\widetilde{\bf s}_{{\mathrm{dir}}_{p}}[f_k]+\widetilde{\bf s}_{\mathrm{ind}_p}[f_k] +\widetilde{\bf c}_{\mathrm{dir}}[f_k]+\widetilde{\bf c}_{\mathrm{ind}}[f_k]+\widetilde{\bf n}_R[f_k],
\begin{array}{l}{{\delta_{1}^{( 1 )}h_{\alpha \beta}= g ( \partial_{\pi \kappa}\partial_{\kappa \nu}\xi_{\alpha \pi}h_{\nu \beta}+ \partial_{\kappa \nu}\partial_{\nu \beta}\xi_{\kappa \pi}h_{\alpha \pi}- s \partial_{\alpha \pi}\partial_{\kappa \nu}\xi_{\pi \kappa}h_{\nu \beta}) + ( \alpha \Leftrightarrow \beta )}}\\ \end{array}
\left( \begin{matrix}{t^{{\vec{\Lambda}}_{1}}= p^{{\vec{\Lambda}}_{1}}+ \mathrm{i}q_{{\vec{\Lambda}}_{1}}}\\{{\bar{t}}_{{\vec{\Lambda}}_{1}}= p^{{\vec{\Lambda}}_{1}}- \mathrm{i}q_{{\vec{\Lambda}}_{1}}}\\ \end{matrix}\right)
\rho_0(|x|) =\frac{\bar{\rho}(|\zeta_0^{-1}(x)|)}{[\chi_0(|\zeta_0^{-1}(x)|)]^2 (\chi_0(|\zeta_0^{-1}(x)|) +\chi_0'(|\zeta_0^{-1}(x)|) |\zeta_0^{-1}(x)|)}S_0(|x|) =\ln d( | \zeta_0^{-1}(y) | ),
\mathbf{S}[F_{N}](t,x) =\frac{1}{N^{2}}\underset{l,r}{\sum}H(x-x_{l}(t))m_{l}(t)m_{r}(t)S(x_{l}(t)-x_{r}(t)) =\frac{1}{N^{2}}\stackrel[r=1]{N}{\sum}\stackrel[l=1]{(i-1)^{\ast}(j)}{\sum}m_{l}(t)m_{r}(t)S(x_{l}(t)-x_{r}(t)),
A(\hat x,\hat y)=\frac{1}{(2\pi)^{2n}}\int\bigr(\int^{\infty}_0e^{i(\psi(t,\hat x,\hat\eta)-<\hat y,\hat\eta>)}a(t,\hat x,\hat\eta)(1-\chi(\hat\eta))\tau(\hat x,\hat\eta)dt\bigr)d\hat\eta.