harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
[{\cal H},{\cal P}_{i}] = 0 , \qquad [{\cal P}_{i},{\cal P}_{j}] = 0 , \qquad [{\cal H},{\cal J}_{i}] = 0 , \qquad [{\cal P}_{i},{\cal J}_{k}] = - \varepsilon_{i k l}{\cal P}_{l},
\frac{1}{8}\dddot{\tau}I^2-\frac{1}{4}\ddot{\tau}+\dot{\tau}\left(K(x)+\frac{1}{2}\sqrt{a}K'(x)I(x)\right)+\frac{1}{2}\ddot{\rho}I + \rho \sqrt{a}K'(x) -\dot{\sigma}=0,
\overline{W}_{s}^{\otimes n}({p}^n)- \overline{W}_{s}^{\otimes n}({p'}^n) & =\left(1-\frac{1}{P(\mathsf{T}^n_{p,\delta})}\right)\sum_{a^n\in \mathsf{T}^n_{p,\delta}}p^n(a^n)\overline{W}_{s}^{\otimes n}(a^n)+\sum_{a^n\notin \mathsf{T}^n_{p,\delta}}p^n(a^n)\overline{W}_s^{\otimes n}(a^n)
\frac{{d^2}}{{dr^2}}{E_{p,q}(r)}= \frac{q}{{pr^2}}\left({\left({\frac{q}{p}- \frac{1}{{\left({r'}\right)^q}}- 1}\right)E_{p,q}(r) + \left({2 - \frac{q}{p}}\right)K_{p,q}(r)}\right).
W(T) =\sum_{1\leq m\leq n}\left[W_{n,m}\right]\L^{-nd}T^n &=\L^d \sum_{1\leq m\leq n}\left[V_{n,m}\right]\L^{-(n+1)(d-1)}T^n-\L^d \sum_{1\leq m\leq n}\left[V_{n,m}\right]\L^{-(n+1)(d-1)}\L^{-m}T^n &=\L^d\sum_{1\leq m\leq n}\tilde{\mu}\left(A_{n,m}\right)T^n-\L^d\sum_{1\leq m\leq n}\tilde{\mu}\left(A_{n,m}\right)\L^{-m}T^n.
\sum_{\imath=1}^{\infty}\frac{\varsigma(\imath)\Big(\sum_{\jmath=\imath}^{\infty}\varsigma(\jmath)\varrho(\jmath)\Big)^p}{\Big(\sum_{\jmath=1}^{\imath}g(\jmath)\Big)^\alpha}\leq \Big(\frac{p}{1-\alpha}\Big)^p\sum_{\imath=1}^{\infty}\varsigma(\imath)\varrho^p(\imath)\Big(\sum_{\jmath=1}^{\imath}\varsigma(\jmath)\Big)^{p-\alpha}, p>1>\alpha\geq0.
X_{1}Z_{2}Y_{3}=[\begin{matrix}c_{2}c_{3}&-s_{2}&c_{2}s_{3}\\s_{1}s_{3}+c_{1}c_{3}s_{2}&c_{1}c_{2}&c_{1}s_{2}s_{3}-c_{3}s_{1}\\c_{3}s_{1}s_{2}-c_{1}s_{3}&c_{2}s_{1}&c_{1}c_{3}+s_{1}s_{2}s_{3}\end{matrix}]
R_{x,y,z}(\textbf{a}; \textbf{c}; \textbf{b}):=\begin{cases}R^{\odot}_{x,y,z}(\textbf{a};\textbf{c};\textbf{b}) R^{\leftarrow}_{x,y,z}(\textbf{a};\textbf{c};\textbf{b}) \end{cases}
M_{\Omega,t}f(x) =\frac1{t^n}\int_{|y|<t}\Omega_0(y')f(x-y)dy+C(\Omega,n)\frac1{t^n}\int_{|y|<t}f(x-y)dy &:=M_{\Omega_0,t}f+C(\Omega,n)M_tf,
x_1 +0\cdot y_1 + (a-1)x_2 +(-b)y_2=0 & 0\cdot x_1 + y_1 + b x_2 + (a-1)y_2=0 &(a-1)x_1 +b y_1 +x_2 + 0\cdot y_2=0 &(-b)x_1 + (a-1)y_1 +0\cdot x_2 + y_2 =0.
p(r,t)=-\rho c^{2}\frac{\partial\delta}{\partial x}(r,t)=\rho c^{2}k_{x}\delta_{m}cos(k\cdot r-\omega t+\varphi_{\delta,0}+\frac{\pi}{2})=p_{m}cos(k\cdot r-\omega t+\varphi_{p,0}),
A_a(\lambda) := \frac{2 \lambda^2 + 2(a-1) \lambda + a(a-1)}{a (2\lambda+a -1)}, B_a(\lambda) :=\frac{(\lambda-1)(2\lambda+1)}{a(2\lambda+a-1)}, U_a(\lambda) :=\frac{2 \left(\lambda+ [\frac{a}{2}] \right)_{[\frac{a-1}{2}]}}{\left(\lambda + \frac{1}{2}\right)_{[\frac{a-1}{2}]}},
Z(J) = (det (-\partial^2_{\mu}))^{-1}\exp \left[ -\frac{g^2}{2}\int d^4x \left( J_i (P \frac{1}{\partial^2_{\mu}}P)_{ik}J_k +J_i ((1 - P) \frac{1}{\partial^2_0}(1 - P))_{ik}J_k \right) \right].
S^0(\psi)-S^-(\psi)= \int(e^{-V^+(\psi+\psi^1)}-f_{V^+}(\psi+\psi^1))d\mu_{C_{>0}^{+(h)}}(\psi^1) &-\int(e^{-V^-(\psi+\psi^1)}-f_{V^-}(\psi+\psi^1))d\mu_{C_{>0}^{-}}(\psi^1).
\begin{aligned}V_{(1,0)}\otimes V_{(1,0)}= V_{(2,0)}\oplus V_{(0,1)}V_{(0,1)}\otimes V_{(0,1)}= V_{(0,2)}\oplus V_{(1,0)}V_{(1,0)}\otimes V_{(0,1)}= V_{(1,1)}\oplus V_{(0,0)}, \end{aligned}
\int_{1/2}^1 \frac{s^{2L-2n+1/2}}{1-s^{2L}}(1-s)^{n-1/2}s^{n-3/2}ds=\int_{1/2}^1 \frac{s^{2L-n-1}}{1-s^{2L}}(1-s)^{n-1/2}ds\ .
C_{rs}(z)T(\bar{v}+w)= (\gamma_{rs}(\bar{v}+w)-\gamma_{rs}(\bar{v}+z))T(\bar{v}+w) C_{rs}(z)\mathcal{D}T(\bar{v}+w)= \mathcal{D}^{\bar{v}}(\gamma_{rs}(\bar{v}+z))T(\bar{v}+w)+(\gamma_{rs}(\bar{v}+w)-\gamma_{rs}(\bar{v}+z))\mathcal{D}T(\bar{v}+w).
V[{\bf j}]=\sum\limits_{n=-\infty}^{+\infty}\int d^3x\left\{\Lambda |{\bf j}|\left[\ln\left[\frac{\Lambda}{2\zeta}|{\bf j}|+\sqrt{1+\left(\frac{\Lambda}{2\zeta}|{\bf j}|\right)^2}\right]+2\pi in\right]- 2\zeta\sqrt{1+\left(\frac{\Lambda}{2\zeta}|{\bf j}|\right)^2}\right\}.
\begin{aligned}\alpha' =\varphi^{-1}\frac{\partial\varphi}{\partial u}=\frac{\partial x^0}{\partial u}E_0+\frac{\partial x^1}{\partial u}e^{-\mu_1x^0}E_1+\frac{\partial x^2}{\partial u}e^{-\mu_2x^0}E_2 \end{aligned}
\Gamma_{m}\left( \overline{Z}_{n}- m \right) = \frac{1}{n}\left( \frac{T_{1}}{\gamma_{1}}- \frac{T_{n+1}}{\gamma_{n}}+ \sum_{k=2}^{n}T_{k}\left( \frac{1}{\gamma_{k}}- \frac{1}{\gamma_{k-1}}\right) + \sum_{k=1}^{n}\delta_{k}+ \sum_{k=1}^{n}\xi_{k+1}\right) .
\begin{cases}\partial_3{\cal H}=0\Leftrightarrow \partial_3 L=-p\Rightarrow\frac{d}{dt}\partial_3 L=-\dot{p}, \partial_4{\cal H}=0\Leftrightarrow \partial_4 L=-p_{\alpha}\Rightarrow{_tD_b^\alpha}\partial_4 L=-{_tD_b^\alpha}p_{\alpha}.\end{cases}
\mathcal I[u(t)] =\|\sqrt{\mathcal K_\lambda}u\|^2-\mathcal P[u] &=p\mathcal E[u]-(p-1)\|\sqrt{\mathcal K_\lambda}u\|^2 &< p\mathcal E[\varphi]-(p-1)\|\sqrt{\mathcal K_\lambda}\varphi\|^2<0,
\frac{\Omega_{n}^2}{\Omega_{n-1}\Omega_{n+1}}=\frac{\Gamma(\frac n2+\frac 12)\Gamma(\frac n2+\frac 32)}{\Gamma^2(\frac n2+1)}=\frac{n+1}{2}\frac{\Gamma^2(\frac n2+\frac 12)}{\Gamma^2(\frac n2+1)}.
ds^2 = \frac{1-c\, y}{6}( d \theta^2 + \sin^2 \theta\, d \phi^2)+ \frac{1}{w(y)q(y)}dy^2+ \frac{q(y)}{9}( d\psi - \cos \theta \, d \phi)^2 & +w(y)\left[ d\alpha + \frac{ac -2y+ c\, y^2}{6(a-y^2)}[d\psi - \cos\theta \, d\phi]\right]^2\,,
{\rm Hilb}(4NT_n,t)= \left(\sum_{k=0}^{n-1}(k+1)! N(k,n) (-t)^{k}\right)^{-1},{\rm Hilb}\big((4NT_n)^{!},t\big)=(n-1) ! L_{n-1}^{(\alpha=1)}\big({-}t^{-1}\big) t^{n-1},
\left( \begin{array}{c}{X^{0}}\\{X^{1}}\\{X^{2}}\\{X^{3}}\\ \end{array}\right) ( \tau , \sigma , \zeta ) = \left( \begin{array}{c}{\tau}\\{r ( \tau ) \sin \sigma \cos \zeta}\\{r ( \tau ) \sin \sigma \sin \zeta}\\{r ( \tau ) \cos \sigma}\\ \end{array}\right) ,
\begin{aligned}\mathbf e \approx \mathcal O(h)\begin{bmatrix}\mathbf 0 G(\mathbf S_2, S_{m+1})\end{bmatrix}+ \mathcal O(h)\begin{bmatrix}\mathbf 0 G(\mathbf S_2, S_{m+2})\end{bmatrix}+ \sum_{j=m+3}^M\mathcal O(h^5)\begin{bmatrix}\mathbf 0 G(\mathbf S_2, S_{j})\end{bmatrix}.\end{aligned}
\Gamma^{\alpha}{}_{\beta}= \frac{( - 1 )^{d ( d - 3 ) / 4}}{d !{\sqrt - \gamma}}\epsilon^{i_{1}. . i_{d}}E_{i_{1}}{}^{a_{1}}E_{i_{2}}{}^{a_{2}}\ldots E_{i_{d}}{}^{a_{d}}\Gamma_{a_{1 . . a{_d}}}~ ~ .
b_{\mu}b_{\lambda}=\psi_{w_{0}}y_{\mu}\psi_{w_{0}}y_{\min}\psi_{w_{0}}y_{\lambda}\psi_{w_{0}}y_{\min}&=(-1)^{\tfrac{n(n-1)}{2}}\psi_{w_{0}}(y_{\mu}\psi_{w_{0}}y_{\lambda})\psi_{w_{0}}y_{\min}
\int_0^{+\infty}\mspace{-24mu}dx\:x\mspace{1.5mu}\sin(b \mspace{1.5mu}x)\,e^{-\beta\mspace{1.5mu}\sqrt{x^2\,+\,\gamma^2}}\!=\frac{b\mspace{1.5mu}\beta\mspace{1.5mu}\gamma^2}{\beta^2\mspace{-1.5mu}+b^2}\,\mathrm{K}_2\left(\gamma\sqrt{\beta^2\mspace{-1.5mu}+b^2}\right)\!,\quad\forall\, b\in\mathbb{R}\mspace{1.5mu}\beta,\gamma>0
\beta_k = \alpha_{2k-1}+ 2\alpha_{2k}+ \ldots + 2\alpha_{\ell-2}+ \alpha_{\ell-1}+ \alpha_\ell, k=1, \ldots, \lfloor\ell/2\rfloor-1, \beta_{\lfloor\ell/2\rfloor}= \alpha_{\ell-2}+\alpha_{\ell-1}+\alpha_\ell, \beta_{\lfloor\ell/2\rfloor}= \alpha_\ell,
\mathrm{Pf}\begin{bmatrix}\epsilon(x_{i}, x_{j}) \end{bmatrix}_{1 \leq i, j \leq m}= \sum_{k=2}^{m}(-1)^{k}\epsilon(x_{1}, x_{k}) \mathrm{Pf}\begin{bmatrix}\epsilon(x_{i}, x_{j}) \end{bmatrix}_{i, j \notin \{1, k\}}.
\|\Pr_x^{t}- \pi \|_{\ell,\pi}= \Pr_x[A] \|\Pr_x^t[\cdot \mid A ]- \pi(\cdot) \|_{\ell,\pi}+\Pr_x[A^{c}] \|\Pr_x^t[\cdot \mid A^{c}]- \pi(\cdot) \|_{\ell,\pi}.
{\cal Z}_1 \approx{1\over 1000}\left({\small \begin{array}{cccccc}0 -0.1 0 0 0 31+1.9i 0 -0.3 0 0 \ddots 0 45-2.6i 0 -0.5 0 -0.1 0 54-6.7i 0 -0.7 \ddots 0 0.15 0 63+7.7i 0 & \ddots \ddots \ddots \end{array}}\right) +{\cal O}(10^{-5})
xsinx+cosx=1-\sum_{k=1}^{\infty}\frac{(-x^{2})^{k}}{2(k+1)(2k)!}=1+\frac{1}{2}x^{2}-\frac{1}{8}x^{4}+\frac{1}{144}x^{6}-\frac{1}{5760}x^{8}+\cdot\cdot\cdot
\gamma^*(\alpha_i) (m') = \gamma^*((F_i)^*(u_{i}))(m')= \gamma u_i(F_i(\gamma^{-1}(m'))) & = \gamma \beta(u_{i_0})(F_i(\gamma^{-1}(m')))= \beta( \gamma u_{i_0})(F_i(\gamma^{-1}(m'))) & = \beta( \gamma u_{i_0}\gamma^{-1})( \gamma F_i(\gamma^{-1}(m')))= (\gamma^*F_i)^*(\gamma^*(u_i))(m').
A_{-1}^{\mbox{tr}}(s)={1\over 2\sqrt{\pi}}{\Gamma(s-{1\over 2})\over \Gamma(s+1)}\Bigr[r_{+}^{2s}-r_{-}^{2s}\Bigr]\Bigr[\zeta_{R}(2s-3)-\zeta_{R}(2s-1)\Bigr],
v_x (|\mathbf{h}|^2)_x=-\frac{\beta}{\nu}(v|\mathbf{h}|^2)_t - 2 v |\mathbf{h}_x|^2+\frac{\beta}{\nu}(v u |\mathbf{h}|^2)_x + (2 v \mathbf{h}\cdot\mathbf{h}_x)_x - \frac{2\beta}{\nu}v\mathbf{h}\cdot \mathbf{w}_x.
\epsilon ( t , z , r , \theta , \phi ) = e^{{\frac{i}{2}}\Gamma_{4}\theta}e^{{\frac{1}{2}}\Gamma_{3}\Gamma_{4}\phi}\Big ( ( 1 + \Gamma_{0}\Gamma_{1}) r^{\frac{- 1}{4}}-{\frac{( t - z )}{2 ( Z_{m})_{\mathrm{c r}}^{\frac{3}{2}}}}r^{\frac{1}{4}}( \Gamma_{0}- \Gamma_{1}) \Gamma_{2}\Big ) \kappa_{0}.
f_{}(\mathfrak{S}^{(2)}\otimes_{p_1, \mathfrak{S}}\mathfrak{M}) \subset \bigcap_{n=1}^{\infty}(\mathfrak{S}^{(2)}\otimes_{p_2, \mathfrak{S}}\mathfrak{M}+p^n\mathfrak{S}^{(2)}[E^{-1}]^{\wedge}_p\otimes_{p_2, \mathfrak{S}}\mathfrak{M})
\Vert f\Vert_{L^{2}(\mathbb{R}^{d},\rho_{s})}^{2}=\int_{\mathbb{R}^{d}}\rho_{s}(y)|f(y)|^{2}\,dy=\left( e^{\frac{s}{2}\Delta}|f|^{2}\right)(0)=\left( e^{\frac{s}{2}\Delta}|f_{\mathbb{C}}|^{2}\right) (0).
\sum_{i=0}^{k-1}\exp(\frac{i}{n-k}) \varphi^i(1+ \vartriangle_i) = \sum_{i=0}^{k-1}\exp(\frac{i}{n-k}) \exp(i \ln \varphi ) (1+ \vartriangle_i) = \sum_{i=0}^{k-1}\exp(ai) (1+ \vartriangle_i) &= \sum_{i=0}^{k-1}\exp(ai)+ \sum_{i=0}^{k-1}\exp(ai) \vartriangle_i = \frac{1- \exp(ak)}{1-\exp(a)}+ \frac{1+C}{2n}\sum_{i=1}^{k-1}\exp(ai) i(i-1)
\dot z(t; x_0,a, b, \gamma,{\bar r},\ell) = \left[ \begin{array}{l}\dot x_m(t) \dot e(t) \dot \phi(t)\end{array}\right] = \left[ \begin{array}{l}a x_m(t) + b{\bar r}-\ell e(t) (a+\ell) e(t) + b \phi(t) x(t) -\gamma e(t) x(t)\end{array}\right].
\displaystyle{\sum_{\lambda=-\infty}^{+\infty}\exp\left[-{{g_c^2 A_cN}\over 2}\left(\lambda+\frac{l}{N}\right)^2\right]=\sqrt{\frac{2\pi}{g_c^2A_cN}}\sum_{\lambda=-\infty}^{+\infty}\exp\left[-\frac{2\pi^2}{g_c^2 A_cN}\lambda^2-2\pi i\lambda\frac{l}{N}\right]}.
\phi \nu_y v_i = (q^{2i}-1)(q^{2(i-d-1)}-1)\phi v_{i-1}= \gamma_{i-1}(q^{2i}-1)(q^{2(i-d-1)}-1) u_{i-1}, \nu_y \phi v_i = \gamma_i \nu_y v_i = \gamma_i q(1-q^{2i-d-1}) u_{i-1}= \gamma_{i-1}(q^{2i}-1)(q^{2(i-d-1)}-1) u_{i-1}.
\sum_{i=d}^{k-1}\binom{n-i}{k-d}-(k-d-\frac{1}{2})\binom{n-d}{k-d}=\sum_{i=d}^{k-1}\left(\binom{n-i}{k-d}-\binom{n-d}{k-d}\right)+\frac{1}{2}\binom{n-d}{k-d}&=\frac{1}{2}\binom{n-d}{k-d}-\sum_{i=d+1}^{k-1}\sum_{j=d+1}^{i}\binom{n-j}{k-d-1}&\geq \frac{1}{2}\binom{n-d}{k-d}-\frac{1}{2}(k-d)(k-d-1)\binom{n-d-1}{k-d-1}>0.
\|f^{=d}\|_q \leq (1/\rho)^d \gamma' = \gamma \left( 99\cdot2 \, r \sqrt{q}\sqrt{\frac{\log(1 / \gamma)}{d}}\right)^d \leq \gamma \left( 200 r \sqrt{q}\cdot \max \left\{\sqrt{q}, \sqrt{\log(1 / \gamma)}\right\}\right)^d ,
\int_{\Omega}|\mathrm{E}|^2(\mathrm{y})d\mathrm{y}= \dfrac{1}{|1 - \alpha\lambda^{(3)}_{\mathrm{n}_{0}}|^2}\Bigg[|\mathrm{E}^{\textbf{in}}|^2(\mathrm{z})\Big(\int_\mathrm{\Omega}\mathrm{e}^{(3)}_{\mathrm{n}_{0}}(\mathrm{x})d\mathrm{x}\Big)^2 + \mathcal{O}\Big(\delta^{3}\Big)\Bigg] \mathrm{h}<1,
\sum_{k=2m}^\infty \frac{b^{\left\lceil \frac{k-2}{2}\right\rceil}}{b^{k-1}}= \sum_{k=2m}^\infty \frac{1}{b^{\left\lfloor \frac{k}{2}\right\rfloor}}= \sum_{k=m}^\infty \frac{2}{b^k}= \frac{2}{(b-1)\cdot b^{m-1}}.
\frac{d}{dt}\int_{A(k)}v^2 + \int_{A(k)}|Dv|^2 \leq \sigma p \int_{A(k)}H^2 f_\sigma^p + 5Cp \int_{A(k)}f_\sigma^p &\leq C(p) \int_{A(k)}H^2 f_\sigma^p.
Q_{8}= 8 | 1 | 26 | 1 | (8 + 48) | (1 + 26) | (1 + 26) | (1 + 8) | (1 + 8), Q_{26}= 26 | (1 + 48) | (8 + 26) | 1^{2}| (8 + 26) | (1^{2}+ 48) | 8 + 26 | 1 Q_{48}= 48 | 26 | 1 | 8 | (1 + 48) | 26 | 1
H(xy,z) = \tfrac{1}{2}(xy,z) + \tfrac{1}{2x}(x^2y,z) = \tfrac{1}{2}(xy,z) + \tfrac{x^2}{2x}(y,z) = x(\tfrac{1}{2x}(xy,z) + \tfrac{1}{2}(y,z)) = x H(y,z).
\Delta^{ab}_{\mu\nu}(x,y)=\int_{0}^{\infty}d{\tau}e^{- \tau M_g^2}\left[\langle x |e^{-\tau\Pi^2}|y\rangle\right]^{ac}\left(e^{-2i\tau{\cal F}[\mbox{ad}(T_3)]}\right)_{\mu\nu}^{cb}\ ,
\begin{array}{rcl}{^1\Phi}_{h\frac{n-1}{2}}=\displaystyle\sum_ia_i^h~{^1C}_i\mid1\rangle~, & \left([2]-{^1E}_h\right)a_i^h+a_{i+1}^h+a_{i-1}^h=0~, & a_0^h=a_{n+1}^h=0~.\end{array}
1 55 =5\cdot 11 210 =2\cdot 3\cdot 5\cdot 7 385 =5\cdot 7\cdot 11 21 =3\cdot 7 99 =3^{2}\cdot 11 231 =3\cdot7\cdot11 560 =2^{4}\cdot 5\cdot 7 45 =3^{2}\cdot 5 154 =2\cdot 7\cdot 11
\lambda_2(v)=\lambda^{\frac{r}{2}}_2(x)\lambda^{\frac{r}{2}}_2(s)\lambda^{\frac{1-r}{2}}_4(x)\lambda^{\frac{1-r}{2}}_4(s) & \lambda_4(v)=\lambda^{\frac{1-r}{2}}_2(x)\lambda^{\frac{1-r}{2}}_2(s)\lambda^{\frac{r}{2}}_4(x)\lambda^{\frac{r}{2}}_4(s)
{\mathcal A}_p( (X_\lambda)_{\lambda \in \Lambda_{d,n}}) := \det ( X_{(1,1)}f(p) \wedge \cdots \wedge X_{(j,1)}\cdots X_{(j,\kappa_j)}f(p) \wedge \cdots \wedge X_{(n,1)}\cdots X_{(n,\kappa_n)}f(p)).
T_w(M, \bar{u}) = \{v\in \mathcal{U}:\ \exists \{u_n\}\subset M,\ \{t_n\}\subset \mathbb{R}^+,\u_n \rightarrow \bar{u}, \t_n\rightarrow 0^+, \frac{1}{t_n}(u_n - \bar{u})\rightharpoonup v\}
\sum_{n}\frac{c_{n+1}}{h_{n+1}}\leq \sum_{n}\frac{h_{n}+ 2c_{n}+ 2r_{n}- 2}{r_{n}(h_{n}+c_{n})}\leq \sum_{n}\frac{2(h_{n}+c_{n})}{r_{n}(h_{n}+ c_{n})}= 2\sum_{n}\frac{1}{r_{n}}
\underline{\mathbf{s}}= \frac{1}{K}\sum_{k = 1}^K \mathbf{l}^{\mbox{\tiny \upshape (\itshape k\upshape)}}\leq \mathbf{s}\leq \overline{\mathbf{s}}= \frac{1}{K}\sum_{k = 1}^K \mathbf{u}^{\mbox{\tiny \upshape (\itshape k\upshape)}}.
\| \left\langle \nabla \right\rangle R(h) \|_{S'((t,+\infty),L^2)}\lesssim \sum_{n=0}^\infty \| \left\langle \nabla \right\rangle R(h) \|_{S'((t+n , t+ n+1),L^2)}\lesssim\begin{cases}e^{-(p-1) c t}, p \in (2,3), e^{-2ct}, p =2 p \ge 3.\end{cases}
\lambda^2 \beta(\lambda) = \overline{R^+(\lambda)}-\overline{R^-(\lambda)}+\eta_{11}^-(0,\lambda)\overline{R^+(\lambda)}-\eta_{11}^+(0,\lambda)\overline{R^-(\lambda)}&+ \frac{1}{4}\overline{q}^\sharp \left[ \overline \eta_{11}^+(0,\lambda) -\overline \eta_{11}^-(0,\lambda) \right]
m_{I}f_{L_{1}\ldots L_{r}}^{K_{1}\ldots K_{q}}( u ) = ( \delta_{K_{1}I}- \delta_{K_{1}N}- \delta_{L_{1}I}+ \delta_{L_{1}N}+ \ldots ) f_{L_{1}\ldots L_{r}}^{K_{1}\ldots K_{q}}( u )
\begin{aligned}m_1-M_x[c_1] =(M_{\varphi, x}[b_1]-M_x[b_1])+(M_{\varphi, x}[(\rho^{-1}-1) b_1]-M_x[(\rho^{-1}-1) b_1]) &+(M_{\varphi, x}[(B^{-1}-\mathrm{I}) b_1]-M_x[(B^{-1}-\mathrm{I}) b_1]) &+(M_{\varphi, x}[(\rho^{-1}-1)(B^{-1}-\mathrm{I}) b_1]-M_x[(\rho^{-1}-1)(B^{-1}-\mathrm{I}) b_1]).\end{aligned}
I_{\beta}^{( \pm )}( \psi ; \, k ) = \pm \frac{1}{2 \pi}\int_{0}^{a}d \eta \, \frac{\psi ( i \eta )}{- k \pm i ( \eta - \beta )}\pm \frac{1}{2 \pi i}\int_{0}^{\infty}d \xi \, \frac{\psi ( \xi \pm i a )}{\xi - k \pm i ( a - \beta )}+ \psi ( k \pm i \beta ) .
||(I-S_{\theta_i}) (\partial_t\varphi^a-u^a_1+u_2^a\partial_2\varphi^a)||_{H^{s-1}(\Gamma_T)}\le C\theta_i^{s-\alpha}||\partial_t\varphi^a-u^a_1+u_2^a\partial_2\varphi^a||_{H^{\alpha-1}(\Gamma_T)}&\le C\theta_i^{s-\alpha}(||\varphi^a||_{H^{\alpha-1}(\Gamma_T)}+||u^a||_{\alpha,\ast,T})\le C\delta\theta_i^{s-\alpha}\,,
( A_{0}(\textbf{w}+\bar{\textbf{u}})\partial_{t}{\textbf{w}},{\textbf{w}}) = \frac{1}{2}\partial_{t}(A_{0}(\textbf{w}+\bar{\textbf{u}}){\textbf{w}},{\textbf{w}}) - \frac{1}{2}(\partial_{t}A_{0}(\textbf{w}+\bar{\textbf{u}}){\textbf{w}},{\textbf{w}});
\tau_3(\alpha_{1},\alpha_2,\alpha_6) \circ \tau_3(-\alpha_{1},-\alpha_2,\alpha'_6) =\tau_4\left(\frac{\alpha'_6(1 - \alpha_{1})-\alpha_{6}(1 + \alpha_{1})}{\alpha_{2}},-\alpha_{6}-\alpha'_6\right)
\theta_b(x_1,x_2)=\begin{cases}\arctan\frac{x_2-b_2}{x_1-b_1}, x_1>b_1,\x_2\geq b_2, \frac\pi2, x_1=b_1,\x_2> b_2, \pi+\arctan\frac{x_2-b_2}{x_1-b_1}, x_1<b_1, \frac32\pi, x_1=b_1,\x_2< b_2, 2\pi+\arctan\frac{x_2-b_2}{x_1-b_1}, x_1>b_1,\x_2< b_2, \end{cases}
w^*(t)= \psi(x_0) \exp \Bigl[ -\int_{t}^{T}\frac{\lambda^*(s)}{s}ds \Bigr] - \int_t^T \exp \Bigl[ -\int_t^{\tau}\frac{\lambda^*(s)}{s}ds \Bigr] \frac{g^*(\tau)}{\tau}d\tau.
\sum_{n=1}^{N}\left[\begin{array}{c}N n\end{array}\right] \frac{(q)_n\left(\frac{b}{a}\right)_n(a)_{N-n}a^n}{(b)_n(1-q^n)(a)_{N}}=\sum_{n=1}^{N}\left(\frac{aq^{n-1}}{1-aq^{n-1}}-\frac{bq^{n-1}}{1-bq^{n-1}}\right) &=\sum_{m=1}^{\infty}(a^m-b^m)\sum_{n=0}^{N-1}(q^n)^m &=\sum_{m=1}^{\infty}\frac{(a^m-b^m)(1-q^{mN})}{1-q^m}.
z_m = \frac{1}{2\pi}\cdot \left[ Re(c -c \ln c)+ S \right] , z_e = c_2 \tilde{\theta}_e = \theta_e + \frac{2\pi \cdot R \sigma}{S_1 - \ln|c|}, \tilde{\theta}_m = -\theta_m - \frac{S_2 + \arg c}{S_1 - \ln|c|}\cdot R \sigma
\mathcal{Z}\left(\hat{\mu}\right)=\bigcup_{n=1}^{\infty}\mathcal{Z}\left( M_{\mathcal{D}_{n}}(\rho^{n}\xi)\right)= \bigcup_{n=1}^{\infty}\frac{\rho^{-n}a_{n}}{N_{n}},\a_{n}\in\mathbb{Z}\backslash N_{n}\mathbb{Z}.
4\rho^3 cos\theta -\left[3s+lcos(2\theta - \omega)\right]\rho^2+2m\rho cos(\theta-\alpha)-2d=0 4\rho^3 sin\theta -\left[3t+hsin(2\theta - \varphi)\right]\rho^2+2k\rho sin(\theta+\mu)-2f=0
\mathcal{H}^{d-1}(\Gamma_n) = \frac{1}{2h}\int_{U_{h-t}(\Gamma_\infty)}\det(dT_n)\, dy + \frac{1}{2h}\int_{U_h(\Gamma_n) \setminus U_{h-t}(\Gamma_\infty)}\det(dT_n) \, dy
\sum_{[b]}Y_{[b]}\sum_{(a)}(\bar{\psi}_L)_r^{a \tilde{a}}\tilde{\phi}^{(a)}_{\tilde{a}}\phi^{(1)r}(\psi_R)_a^{[b][1]}+ \sum_{[b]}Y'_{[b]}\sum_{(a)}(\bar{\psi}_L)_r^{a \tilde{a}}\tilde{\phi}_{\tilde{a}}^{(a)}\phi^{(2)r}(\psi_R)_a^{[b][2]},
N_6(f,H)=-f+S(H,f)-2S(H_2,2f)-\frac{3}{2}S(H_3,3f)+3S(H_6,6f)=\begin{cases}-2a-1 \hbox{if $a\equiv 0\pmod 6$,}-3 \hbox{if $a\equiv 2,3\pmod 6$,}2a+1 \hbox{if $a\equiv 5\pmod 6$.}\end{cases}
\|\hat x +\hat y \|_1 \leq \|\hat x\|_1 +\|\hat y\|_1 \leq K_1\|x\|_\infty +K_1 \|y\|_\infty \leq K_1 \|x+y\|_\infty +2K_1 \|y\|_\infty & \leq K_1 \|x+y\|_\infty +2K_1K_2 \|x+y\|_\infty =(K_1+2K_1K_2)\|x+y\|_\infty.
\lim_{n \rightarrow \infty}\left\|\frac{1}{\lambda_{j,n}^{(d-2)/2}}V_j \left(\frac{x - x_{j,n}}{\lambda_{j,n}}, t_j\right)\cdot \frac{1}{\lambda_{j',n}^{(d-2)/2}}V_{j'}\left(\frac{x - x_{j',n}}{\lambda_{j',n}}, t_{j'}\right)\right\|_{L^{2^\ast/2}}= 0.
I_{N}:= \left(\frac{2a\phi^{\prime}(a)}{2a\phi^{\prime}(a)+2}N+\frac{2a\phi^{\prime}(a)-1}{2a\phi^{\prime}(a)+2}\ ,\ \frac{2b\phi^{\prime}(b)}{2b\phi^{\prime}(b)+2}N+\frac{2b\phi^{\prime}(b)-1}{2b\phi^{\prime}(b)+2}\right)\cap (0,N).
\begin{array}{l}\nabla f(x^*)+\left[\displaystyle \sum_{i=1}^p[y^I]^*_j\nabla^2 c_j(x^*)-A^TA-{\cal J}c_{\gamma}(x^*)^T{\cal J}c_{\gamma}(x^*)-{\cal J}c_{\beta}(x^*)^T{\rm Diag}([v_b]_{\beta}){\cal J}c_{\beta}(x^*)\right]\eta^*_1=0. \end{array}
R_{Q,k}\leq \log_2 e \left( \sum_{m = K-1}^{n_T-2}\sum_{l=0}^{n_T-m-2}\frac{(n_T-2)!}{m!~ l!~ (n_T -m-2 - l) !}(-1)^{l}\frac{1}{m+l}+ \frac{1}{n_T-1}\sum_{l=1}^{n}\frac{1}{l}\right),
R_2=\left(\begin{array}{lll}x=0: 0 a_1 x=1: 0 a_2 x=\infty: a_3 a_4\end{array}\right)= \left(\begin{array}{llc}x=0: 0 1-c x=1: 0 c-a-b \xi=\infty: a b\end{array}\right)= R_{abc},
q ( \hat{A}_{4}= 0 ) = \oint K^{\alpha}\, d S_{\alpha}= - \left. \int \frac{\, d^{3}\vec{x}}{1 6 \pi^{2}}\epsilon^{i j k}\, \mathrm{t r \,}\left( \hat{A}_{i}\, \hat{F}_{j k}+ \frac{2 i}3 \hat{A}_{i}\hat{A}_{j}\hat{A}_{k}\right) \right|_{x_{4}= - \infty}^{x_{4}= + \infty}.
\tilde{\rho}_{B^{2}}( s ) = \frac{\beta}{2 \pi}\rho_{B^{2}}( s ) = \frac{\beta}{2 \pi}\sqrt{\frac{b^{2}- s}{s - c^{2}}}\rho_{B}( \eta ) = \frac{\beta \eta}{\pi}\sqrt{\frac{b^{2}- \eta^{2}}{\eta^{2}- c^{2}}}
V ( r ) = r \sum_{i = 1}^{1 6}( N_{i}^{B}+ 1 / 2 ) + r \sum_{i = 1}^{8}( N_{i}^{F}- 1 / 2 ) + \frac{r}{2}\sum_{i = 1}^{1 6}( N_{i}^{f}- 1 / 2 )
C=\frac{1}{\sqrt{2}}[\begin{matrix}2&0&0&0\\ -1&1&1&1\end{matrix}],D=\frac{1}{\sqrt{2}}[\begin{matrix}2&0&0&0\\ 1&1&1&-1\end{matrix}],L=[\begin{matrix}2&0&0&0\\ 1&1&-1&1\end{matrix}]
\mathcal{L}=-\frac 12\left[ \overline{\Psi}(x)\left( \Gamma_\mu\overrightarrow{D_\mu^{(-)}}+m\right) \Psi (x)-\overline{\Psi}(x)\left( \Gamma_\mu \overleftarrow{D_\mu^{(+)}}-m\right) \Psi(x)\right] -\frac 14 \mathcal{F}_{\mu \nu}
L=-m\sqrt{-\dot{x}^2\left (1-\frac{2s}{m}\frac{\dot{\varphi}}{(\dot{x},n)}-\frac{\varrho^2}{m^2}\frac{\dot{\varphi}^2}{(\dot{x},n)^2}\right )}+\varrho\frac{(\dot{x},\partial_\varphi n)}{(\dot{x},n)}\dot{\varphi}\ .
( Z , Z ) \equiv \frac{1}{2}t r ( \bar{Z}Z ) = z^{1}z^{3}- q^{2}z^{2}z^{4}\qquad ( \bar{Z}= \left( \begin{array}{c c}{z^{3}}&{- q^{2}z^{4}}\\{- q^{2}z^{2}}&{z^{1}}\\ \end{array}\right) ) .
\tilde{G}_{p,p^{\prime}}(s,c):=\frac{2}{1+\frac{1}{\pi p^{\prime}}}\,\eta_{p}(2s)+\frac{2\sqrt{\pi}\,c^{\frac{1}{2}-s}}{\Gamma(s)}\cdot\frac{\Gamma\left(s-\frac{1}{2}\right)}{1+\frac{1}{\pi p}}\,\zeta_{p^{\prime}}(2s-1).
H_{+}\phi(r,\theta)= \left(\begin{array}{cc}m ie^{- i\theta}\left(\partial_{r}-\frac{i}{r}\partial_{\theta}-\frac{eB}{2}r\right) ie^{i\theta}\left(\partial_{r}+\frac{i}{r}\partial_{\theta}+\frac{eB}{2}r\right) -m\end{array}\right)\phi(r,\theta)=E\phi(r,\theta).
\int_0^1 e^{i (b_k-b_j) \l t}\sum_{l=1}^n q_{jl}(t)\rho_{lk}(t,\l)dt = \sum_{l=1}^n o\left(\int_0^1 \left|e^{i (b_k-b_j) \l t}q_{jl}(t) \right|dt\right) = o(\l^{-1}),
\phi(e^{-\varepsilon}x_0 + \sqrt{1-e^{-2\varepsilon}}z)\le \phi(y_0) + \langle \nabla \phi(y_0), e^{-\varepsilon}x_0 + \sqrt{1-e^{-2\varepsilon}}z -y_0 \rangle = \phi(y_0) + \langle x_0-y_0, e^{-\varepsilon}x_0 + \sqrt{1-e^{-2\varepsilon}}z -y_0 \rangle
[D_uG_{m-n_0}(hx,z)-D_uG_{m-n_0}(hx,z')] \Dh(x) \qquad\qquad=[D_uG_{m-n_0}(k(\ell x),z)-D_uG_{m-n_0}(k(\ell x),z')] \Dk(\ell x) \D\ell(x).
(B^t B) A =- J_0 B^{-1}J_0 B A=- J_0 B^{-1}J_0 C B=- J_0 B^{-1}C J_0 B =- J_0 A B^{-1}J_0 B=- A J_0 B^{-1}J_0 B = A (B^t B).
\begin{aligned}A_n =\frac{4}{\sinh(2n\pi M)}\int_0^{1/2}\int_{0}^{\infty}\left[\frac{1}{x^{1/2+\theta}}-\frac{1}{x^{1/2-\theta}}\right]\frac{\sin\left(M\log x\right)}{\left(e^{x}+1\right)}\sin(2n\pi\theta) dx d\theta \end{aligned}
R_1\leq\frac{1}{2}\log\frac{{C^{2}_{y_{12}}}-{|\tilde{C}_{y_{12}}|^{2}}}{{C^{2}_{s_1}}-{|\tilde{C}_{s_1}|^{2}}}=L_1, R_2\leq\frac{1}{2}\log\frac{{C^{2}_{y_{11}}}-{|\tilde{C}_{y_{11}}|^{2}}}{{C^{2}_{s_1}}-{|\tilde{C}_{s_1}|^{2}}}=L_2, R_1+R_2\leq\frac{1}{2}\log\frac{{C^{2}_{y_1}}-{|\tilde{C}_{y_1}|^{2}}}{{C^{2}_{s_1}}-{|\tilde{C}_{s_1}|^{2}}}=L_4,
\nu_j = \frac{h}{\mu}\sum_{i=1}^{j+1}\left(1 + \cdots + \rho^{i-1}\right) =\begin{cases}\displaystyle \frac{h}{\mu}\, \left[\frac{\rho^{j+2}-1}{(\rho-1)^2}- \frac{j+2}{\rho-1}\right] \rho \neq 1 \displaystyle \frac{h}{\mu}\, \frac{(j+1) \, (j+2)}{2}\rho = 1.\end{cases}
\begin{dcases}\Delta(B\Delta \eta) + (1-\nu)\left(2B_{xy}\eta_{xy}- B_{xx}\eta_{yy}- B_{yy}\eta_{xx}\right) - f = 0 \Gamma_c \Delta \eta -(1-\nu)\eta_{\tau\tau}= 0 \partial\Gamma_c (B\Delta \eta)_n + (1-\nu)(2 B_\tau \eta_{n\tau}- B_n \eta_{n\tau}+ B\eta_{n\tau\tau}) = 0 \partial\Gamma_c\end{dcases},
G_t(x,y)=\sum_{\alpha=1}^\infty \phi_\alpha(t)\varphi_\alpha(x)\varphi_\alpha(y):\phi_\alpha^I(t)=e^{-\lambda_\alpha t},\phi_\alpha^{II}(t)=\frac{\sin(\sqrt{\lambda_\alpha}t)}{\sqrt{\lambda_\alpha}}.
H=-t\sum_{\langle i,j\rangle}(\hat{b}_{i}^{\dagger}\hat{b}_{j}+\hat{b}_{j}^{\dagger}\hat{b}_{i})+\frac{U}{2}\sum_{i}\hat{n}_{i}(\hat{n}_{i}-1)-\mu\sum_{i}\hat{n}_{i}
\mathrm{R}\left(\frac{c_0}{\Phi_0(\nu;x)}-U(x)\frac{c_0}{\Phi_0(\nu;x+1)}-V(x)\right) = \max_{c, \Phi(\nu;x)}\mathrm{R}\left(\frac{c}{\Phi(\nu;x)}-U(x)\frac{c}{\Phi(\nu;x+1)}-V(x)\right).
\tilde{\mathbf{E}}_{\beta,s}^{\pm,(k)}\coloneqq \begin{cases}\frac{(\tilde{E}^{\pm}_{\beta,s})^{k}}{[k]_{v_{\beta}}!}\ \beta=[i,j] \frac{(\tilde{E}^{\pm}_{\beta,s})^{k}}{([2]_{v}!)^{k}[k]_{v_{\beta}}!}\ \beta=[i,n,j] \end{cases}.
\int_{0}^{1}\mathrm{d}\rho_{1}\ldots \int_{0}^{1}\mathrm{d}\rho_{N}\delta ( \sum \rho_{\mu}- 1 ) \prod_{\nu = 1}^{N}\rho_{\nu}^{n_{\nu}}= \frac{\prod_{\nu = 1}^{N}n_{\nu}!}{\left( \sum_{\nu = 1}^{N}n_{\nu}+ N - 1 \right) !}\: ,
(J_{x_{0}}^{k}f)(z)=f(x_{0})+(Df(x_{0}))\cdot z+\frac{1}{2}(D^{2}f(x_{0}))\cdot z^{\otimes2}+\cdot\cdot\cdot+\frac{D^{k}f(x_{0})}{k!}\cdot z^{\otimes k}
\left. \frac{\delta \Omega^{a}}{\delta C^{\alpha b}}\right|_{C = \pi ={\cal P}= \lambda = 0}= T_{\alpha}\delta_{b}^{a}, \; \left. \frac{\delta \Omega^{a}}{\delta \pi^{\alpha}}\right|_{C = \pi = \lambda = 0}= \varepsilon^{a b}{\cal P}_{\alpha b},
\|u-(u)^{\Lambda\eta^2}_{B}\|_{2,B,\Lambda\eta^2}\lesssim M |B|^{-1}\int_{r/2}^{r}F(u,s)\nu(ds) &\lesssim M C_P^{B,\Lambda}|B|^\frac{2-d}{d}\int_{r/2}^{r}\int_B a\nabla u\cdot\nabla u 1_{B_s}dx\,\gamma(ds) &= M C_P^{B,\Lambda}|B|^\frac{2-d}{d}\int_B \eta^2 a\nabla u\cdot\nabla u dx.
{\mathcal D}^{-1}={\mathcal D}_0^{-1}-{\mathcal D}_0^{-1}{\mathcal V}{\mathcal D}_0^{-1}+{\mathcal D}_0^{-1}{\mathcal V}{\mathcal D}_0^{-1}{\mathcal V}{\mathcal D}_0^{-1}-...
S(x,y)=2\sum_{j=0}^{(N-1)/2-1}\frac{1}{h_j}\left(\hat{q}_{2j}(x)\hat{\tau}_{2j+1}(y)-\hat{q}_{2j+1}(x)\hat{\tau}_{2j}(y)\right)+\frac{q_{N-1}(x)}{\mu_N}.