Datasets:
prithivMLmods
/
Latex-KIE

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\begin{align*}(n-2)q + r_{n+1} \le r_1 + \ldots + r_n + n-1\end{align*}
\begin{align*}-q + r_2 + \ldots + r_{n+1} \le r_1 + n-1\end{align*}
\begin{align*}-2q + r_1 + \ldots + r_{n+1} \le n-1\end{align*}
\begin{align*}(n-1) q \le r_1 + \ldots + r_{n+1} + n-1\end{align*}
\begin{align*}-q+ r_2 + \ldots + r_{n+1} \le r_1 + n-1.\end{align*}
\begin{align*}F(u, a) = \frac{1}{2} \int_{S_T}(u - \tilde{u})^2 z_{\delta }(t) dxdt +\frac{1}{2} \gamma \int_{\Omega}(a - a_0)^2~~ dx,\end{align*}
\begin{align*}-2q + r_1 + \ldots + r_{n+1} \le n-1\end{align*}
\begin{align*}2q \ge \lceil \frac{\sum_{i=1}^{n+1} d_i - n +1 }{2} \rceil = \lceil \frac{(n+1)q + \sum_{i=1}^{n+1} r_i - n +1}{2} \rceil\end{align*}
\begin{align*}\frac{\sum_{i=1}^4 k_i -1}{2} q + r_1 + r_2 + r_3 \ge \lceil \frac{\sum_{i=1}^4 d_i -2}{2} \rceil\end{align*}
\begin{align*} \frac{\sum_{i=1}^4 k_i + 1}{2} q + r_1 \ge \lceil \frac{\sum_{i=1}^4 d_i -2}{2} \rceil\end{align*}
\begin{align*}pq\ge \lceil \frac{\sum_{i=1}^4 d_i -2}{2} \rceil\end{align*}
\begin{align*}\frac{\sum_{i=1}^4 k_i -2}{2} q + r_1 + r_2 + r_3+r_4 \ge \lceil \frac{\sum_{i=1}^4 d_i -2}{2} \rceil\end{align*}
\begin{align*} \frac{\sum_{i=1}^4 k_i}{2} q + r_1+r_2 \ge \lceil \frac{\sum_{i=1}^4 d_i -2}{2} \rceil\end{align*}
\begin{align*} \frac{\sum_{i=1}^4 k_i+2}{2}q\ge \lceil \frac{\sum_{i=1}^4 d_i -2}{2} \rceil\end{align*}
\begin{align*}pq\ge \lceil \frac{\sum_{i=1}^4 d_i -2}{2} \rceil\end{align*}
\begin{align*}e_k(\mathbf{u}_S)=\sum_{\substack{i_1 > i_2 > \cdots > i_k \\ i_1,\dots,i_k \in S}}u_{i_1}u_{i_2}\cdots u_{i_k}\,;\end{align*}
\begin{align*} L'(v; \bar{v}) = 0 , \end{align*}
\begin{align*}h_\ell(\mathbf{u})=\sum_{1 \le i_1 \le i_2 \le \cdots \le i_\ell \le N}u_{i_1}u_{i_2}\cdots u_{i_\ell},\end{align*}
\begin{align*}h_m(\mathbf{u})-h_{m-1}(\mathbf{u})\,e_1(\mathbf{u})+h_{m-2}(\mathbf{u})\,e_2(\mathbf{u})-\cdots+(-1)^m e_m(\mathbf{u})=0.\end{align*}
\begin{align*}\Omega(\mathbf{x}, \mathbf{u})= H(x_1) H(x_2) \cdots=\prod_{j=1}^\infty\prod_{i=1}^{N}(1-x_ju_i)^{-1}=\prod_{j=1}^\infty \sum_{\ell = 0}^\infty x_j^\ell h_\ell(\mathbf{u})\end{align*}
\begin{align*}F_\gamma(\mathbf{x})=\big\langle \Omega(\mathbf{x}, \mathbf{u}) ,\gamma\big\rangle,\end{align*}
\begin{align*}\Omega(\mathbf{x}, \mathbf{u})=\prod_{j=1}^\infty \sum_{k = 0}^\infty (x_j)^k h_k(\mathbf{u}).\end{align*}
\begin{align*}F_\Gamma&=Q_{2}+Q_{3}+Q_{12}+3Q_{13}+2Q_{14}+2Q_{23}+3Q_{24}+Q_{34}+Q_{124}+Q_{134}\\&=s_{32} + s_{311} + s_{221}\,.\end{align*}
\begin{align*} \tilde{c}(z) = c(z)\,k+r.\end{align*}
\begin{align*}\nu \cdot u_c = \begin{cases}\mu & \\0 & \end{cases}\end{align*}
\begin{align*}e_k(\mathbf{u}_{[m]}) =u_{m} e_{k-1}(\mathbf{u}_{[m-1]})+e_k(\mathbf{u}_{[m-1]}).\end{align*}
\begin{align*}\Omega^\lessdot(\mathbf{x}, \mathbf{u}) =\prod_{j=1}^\infty \bigl(c_{z_1}(x_j) c_{z_2}(x_j) \cdots c_{z_{2N}}(x_j)\bigr),\end{align*}
\begin{align*} 0 = \frac{\partial L}{\partial a}(u)(\bar{a}) = \int_{\Omega_T} (\nabla u) (\nabla \lambda) \bar{a} ~dxdt +\gamma \int_{\Omega} (a - a_0) \bar{a}~dx,~ x \in \Omega.\end{align*}
\begin{align*} a_{2i} = {a_0}^{\rho^i}, a_{2i+1} = {a_1}^{\rho^i}. \end{align*}
\begin{align*} \tau(a_1) = \tau(a_0)^{\rho^{-m}} = \rho^{m}\tau(a_0)\rho^{-m} = \tau(a_0)\rho^{-2m}, \end{align*}
\begin{align*} \tau(a_0) = \rho^{-m}\tau(a_0)\rho^{m} = \tau(a_0)\rho^{2m}, \end{align*}
\begin{align*} B = \{ a_{-2},a_{-1},a_0,a_1,a_2,s,s_2,s_2^f \}, \\ s = a_0\circ a_1, s_2 = a_0\circ a_2, s_2^f = a_{-1}\circ a_1. \end{align*}
\begin{align*} *_a\colon A\times A \to A, x*_ay = (x\circ a)y + (y\circ a)x \end{align*}
\begin{align*} B^f = \{a_3,a_2,a_1,a_0,a_{-1},s,s_2^f,s_2\}. \end{align*}
\begin{gather*} B = \{ a_{-2},a_{-1},a_0,a_1,a_2,s,s_2,s_2^f \}. \end{gather*}
\begin{align*} a_i = a_{i \bmod n}, a_ia_{i+1} = \frac{1}{64}(a_i+a_{i+1})+s, a_ia_{i+2} = \frac{1}{64}(a_i+a_{i+2}) + \begin{cases} s_2 & i\equiv_2 0, \\ s_2^f & i\equiv_2 1. \end{cases} \end{align*}
\begin{align*}C^A_\Lambda x=\lim_{\lambda\rightarrow \infty}C\lambda R(\lambda,A)x,\ D(C^A_\Lambda)=\{x\in X:\mbox{this above limit existsin}\, Y\}\end{align*}
\begin{align*} D^p(M) =\{f(\cdot)\in L^p_{loc}(R^+,X): f\in D(M) \hbox{ }for \hbox{ a.e. }t\geq 0, and \hbox{ } Mf(\cdot)\in L^p_{loc}(R^+,X)\}.\end{align*}
\begin{align*} \frac{\partial^2 \lambda}{\partial t^2} - \nabla \cdot (a \nabla \lambda) &= - (u - \tilde{u}) z_{\delta}, ~ x \in S_T, \\\lambda(\cdot, T)& = \frac{\partial \lambda}{\partial t}(\cdot, T) = 0, \\\partial _{n} \lambda& =0,~\mbox{on}~ \Omega_T\setminus S_T.\end{align*}
\begin{align*}\Phi_{A+P,J^{A,A+P}B} =\Phi_{A+P,I}F_{A,B,P} +\Phi_{A,B}.\end{align*}
\begin{align*}F_{A+P,J^{A,A+P}B,C} =F_{A+P,I,C}F_{A,B,P} +F_{A,B,C}.\end{align*}
\begin{align*}F_{A+P,I,C}= F_{A,I,C}(I-F_{A,I,P})^{-1}.\end{align*}
\begin{align*} F_{A+P,J^{A,A+P}B,C}=F_{A,I,C}(I-F_{A,I,P})^{-1}F_{A,B,P} +F_{A,B,C}.\end{align*}
\begin{align*} \Phi_{A+P,J^{A,A+P}B}=\Phi_{A,I}(I-F_{A,I,P})^{-1}F_{A,B,P} +\Phi_{A,B}.\end{align*}
\begin{align*} \left\{ \begin{array}{ll} (\lambda-L)z=0, & \hbox{} \\ Gz=u, & \hbox{} \end{array} \right.\end{align*}
\begin{align*} G(\lambda-A_{-1})^{-1}B=I.\end{align*}
\begin{align*} \left\{ \begin{array}{ll} \frac{\partial}{\partial t}x(t,\theta)=\frac{\partial}{\partial \theta}x(t,\theta),\ x\in [-r,0] & \hbox{ } \\ x(t,0)=w(t). & \hbox{ } \end{array} \right.\end{align*}
\begin{align*} (BCS)\left\{ \begin{array}{ll} \dot{z}(t)&=Lz(t)\\ Gz(t)&=u(t) \end{array} \right.\end{align*}
\begin{align*} (\lambda-\mathbb{A})^{-1}(\lambda_0-\mathbb{A}_{-1})^{-1}\mathbb{B}=\frac{(\lambda_0-\mathbb{A}_{-1})^{-1}\mathbb{B}-(\lambda-\mathbb{A}_{-1})^{-1}\mathbb{B}}{\lambda-\lambda_0}.\end{align*}
\begin{align*} \sum_{j=0}^{n} \cos{2 l(\theta+j\pi) \over n+1}=(n+1)\cos 2s \theta\;,\end{align*}
\begin{align*}W_h^u := \{ w \in H_u^1: w|_{K \times J} \in P_1(K) \times P_1(J), \forall K \in K_h, \forall J \in J_{\tau} \}, \end{align*}
\begin{align*}Q_\Lambda^\mathbb{A}(\lambda_0-\mathbb{A}_{-1})^{-1}\mathbb{B}Gz=Q(\lambda_0-\mathbb{A}_{-1})^{-1}\mathbb{B}Gz-\mathbb{D}Gz.\end{align*}
\begin{align*} Q_\Lambda^\mathbb{A}(z-(\lambda_0-\mathbb{A}_{-1})^{-1}\mathbb{B}Gz)=Q(z-(\lambda_0-\mathbb{A}_{-1})^{-1}\mathbb{B}Gz).\end{align*}
\begin{align*} Q_\Lambda^\mathbb{A}z=Qz-\mathbb{D}Gz.\end{align*}
\begin{align*} z(t)=T_\mathbb{A}(t)x+\int_0^tT_\mathbb{A}(t-s)[Q_\Lambda^\mathbb{A}z(s)+\bar{Q}u(s)]ds+\Phi_{\mathbb{A},\mathbb{B}}(t)u.\end{align*}
\begin{align*} \mathbb{B}Gx_0=Lx_0-\mathbb{A}_{-1}x_0, \forall x_0\in D(L).\end{align*}
\begin{align*} \Phi_{\mathbb{A}+Q,J^{\mathbb{A},\mathbb{A}+Q}\mathbb{B}+\bar{Q}}=\Phi_{\mathbb{A}+Q,J^{\mathbb{A},\mathbb{A}+Q}\mathbb{B}}+\Phi_{\mathbb{A}+Q,\bar{Q}}.\end{align*}
\begin{align*} z(\cdot)=&T_{\mathbb{A}+Q}(\cdot)x+\Phi_{\mathbb{A}+Q,J^{\mathbb{A},\mathbb{A}+Q}\mathbb{B}+\bar{Q}}(\cdot)u\\=&T_{\mathbb{A}+Q}(\cdot)x+\Phi_{\mathbb{A}+Q,J^{\mathbb{A},\mathbb{A}+Q}\mathbb{B}}(\cdot)u+\Phi_{\mathbb{A}+Q,\bar{Q}}(\cdot)u\in D^p(Q^\mathbb{A}_\Lambda).\end{align*}
\begin{align*} \Phi_{\mathbb{A}+Q,\mathbb{D}}-\Phi_{\mathbb{A},I}F_{\mathbb{A}+Q,\bar{Q},Q}=\Phi_{\mathbb{A},\bar{Q}}.\end{align*}
\begin{align*} \left\{ \begin{array}{ll} x(t)=T_{\mathfrak{A}}(t)x+\Phi_{\mathfrak{A},\mathfrak{B}}(t)u_1, & \hbox{} \\ w(t)=T_A(t)w+\int_0^tT_A(t-s)[L_\Lambda x(s)+\bar{L}u_1(s)]ds+\Phi_{A,B}(t)u_2, & \hbox{} \end{array} \right.\end{align*}
\begin{align*} \left\{ \begin{array}{ll} \dot{z}(t)&=Lz(t)\\ Gz(t)&=Qz(t) \end{array} \right.\end{align*}
\begin{align*}W_h^{\lambda} := \{ w \in H_{\lambda}^1: w|_{K \times J} \in P_1(K) \times P_1(J), \forall K \in K_h, \forall J \in J_{\tau} \}. \end{align*}
\begin{align*} \mathbb{A}^Ix=&\mathbb{A}_{-1}x+\mathbb{B}(I-\mathbb{D})^{-1}_{Left}\mathbb{C}_\Lambda^\mathbb{A}x\\=&\mathbb{A}_{-1}\big(x-R(\lambda,\mathbb{A}_{-1})\mathbb{B}(I-\mathbb{D})^{-1}_{Left}\mathbb{C}_\Lambda^\mathbb{A}x\big)+\lambda R(\lambda,\mathbb{A}_{-1})\mathbb{B}(I-\mathbb{D})^{-1}_{Left}\mathbb{C}_\Lambda^\mathbb{A}x\in X.\end{align*}
\begin{align*} &(D-CA^{-1}B)[D^{-1}+D^{-1}C(A-BD^{-1}C)^{-1}BD^{-1}]\\ =& I+C(A-BD^{-1}C)^{-1}BD^{-1}-CA^{-1}BD^{-1}\\ & -CA^{-1}[(BD^{-1}C-A)+A](A-BD^{-1}C)^{-1}BD^{-1}\\ =& I\end{align*}
\begin{align*} &[D^{-1}+D^{-1}C(A-BD^{-1}C)^{-1}BD^{-1}](D-CA^{-1}B)\\ =& I-D^{-1}CA^{-1}B+D^{-1}C(A-BD^{-1}C)^{-1}B \\ &-D^{-1}C(A-BD^{-1}C)^{-1}[(BD^{-1}C-A)+A]A^{-1}B\\ =& I.\end{align*}
\begin{align*} F_{\mathbb{A},I,\mathbb{C}} =\left( \begin{array}{cc} 0 & F_{A,I,I} \\ F_{\mathfrak{A},I,K} & F_{A,I,M} \\ \end{array} \right),\end{align*}
\begin{align*} F_{\mathbb{A},I,\mathbb{P}}=\left( \begin{array}{cc} 0 & 0 \\ F_{\mathfrak{A},I,L} & 0 \\ \end{array} \right).\end{align*}
\begin{align*} F_{\mathbb{A},\mathbb{B},\mathbb{P}}=\left( \begin{array}{cc} 0 & 0 \\ F_{\mathfrak{A},\mathfrak{B},L} & 0 \\ \end{array} \right),\end{align*}
\begin{align*} F_{\mathbb{A},\bar{\mathbb{P}},\mathbb{P}}=0,\end{align*}
\begin{align*} F_{\mathbb{A},\mathbb{B},\mathbb{C}}=\left( \begin{array}{cc} 0 & F_{A,B,I} \\ F_{\mathfrak{A},\mathfrak{A},K} & F_{A,B,M} \\ \end{array} \right),\end{align*}
\begin{align*} F_{\mathbb{A},\bar{\mathbb{P}},\mathbb{C}}=\left( \begin{array}{cc} F_{A,\bar{L},I} & 0 \\ F_{A,\bar{L},M} & 0 \\ \end{array} \right).\end{align*}
\begin{align*} \bigg(I-\left( \begin{array}{cc} R(\lambda,A)L e_\lambda & R(\lambda,A_{-1})B \\ MR(\lambda,A)L e_\lambda+Ke_\lambda & MR(\lambda,A_{-1})B\\ \end{array} \right)\bigg) \left( \begin{array}{c} x \\ f \\ \end{array} \right)=0,\end{align*}
\begin{align*}C_{h}:=\{u\in L_{2}(\Omega ):u|_{K}\in P_{0}(K),\forall K\in K_h\}, \end{align*}
\begin{align*} \left\{ \begin{array}{ll} (I-R(\lambda,A)Le_\lambda)x=R(\lambda,A_{-1})Bf, & \hbox{} \\ (MR(\lambda,A)L e_\lambda+Ke_\lambda)x=(I-MR(\lambda,A_{-1})B)f. & \hbox{} \end{array} \right.\end{align*}
\begin{align*} &R(\lambda,\mathcal {A}_{L,K,M})\\=&\left( \begin{array}{cc} R(\lambda,\mathfrak{A})+e_\lambda W_1 & e_\lambda W_2 \\ W_5& W_6 \\ \end{array}\right),\end{align*}
\begin{align*} W_2=N_1R(\lambda,A)+(I-R(\lambda,A)Le_\lambda)^{-1}R(\lambda,A_{-1})BN_2MR(\lambda,A),\end{align*}
\begin{align*} W_3=&(I-MR(\lambda,A_{-1})B)^{-1}(MR(\lambda,A)Le_\lambda+Ke_\lambda)N_1R(\lambda,A)LR(\lambda,\mathfrak{A})\\&+N_2(MR(\lambda,A)LR(\lambda,\mathfrak{A})+KR(\lambda,\mathfrak{A})),\end{align*}
\begin{align*} W_4=(I-MR(\lambda,A_{-1})B)^{-1}(MR(\lambda,A)Le_\lambda+Ke_\lambda)N_1R(\lambda,A)+N_2MR(\lambda,A),\end{align*}
\begin{align*} N_1=\{I-R(\lambda,A)Le_\lambda-R(\lambda,A_{-1})B(I-MR(\lambda,A_{-1})B)^{-1}[MR(\lambda,A)Le_\lambda+Ke_\lambda]\}^{-1},\end{align*}
\begin{align*} N_2=\{I-MR(\lambda,A_{-1})B-[MR(\lambda,A)Le_\lambda+Ke_\lambda](I-R(\lambda,A)Le_\lambda)^{-1}R(\lambda,A_{-1})B\}^{-1}.\end{align*}
\begin{align*} \Phi_{\mathbb{A}+\mathbb{P},J^{\mathbb{A},\mathbb{A}+\mathbb{P}}\mathbb{B}+\bar{\mathbb{P}}}=\Phi_{\mathbb{A},I}(I-F_{\mathbb{A},I,\mathbb{P}})^{-1}F_{\mathbb{A},\mathbb{B}+\bar{\mathbb{P}},\mathbb{P}}+\Phi_{\mathbb{A},\mathbb{B}+\bar{\mathbb{P}}},\end{align*}
\begin{align*} \left\{ \begin{array}{ll} (\lambda-A_m)w_1=0, & \hbox{ } \\ Pw_1=v, & \hbox{ } \end{array} \right.\end{align*}
\begin{align*} \left\{ \begin{array}{ll} (\lambda-A_m)w_2=LR(\lambda,\mathfrak{A}_{-1})\mathfrak{B}u, & \hbox{ } \\ Pw_2=0, & \hbox{ } \end{array} \right.\end{align*}
\begin{align*}L'(v_h)(\bar{v})=0 ~\forall\bar{v} \in V_h . \end{align*}
\begin{align*} (OS)\left\{ \begin{array}{ll} \dot{z}(t)&=Lz(t)\\ Gz(t)&=Qz(t)+v(t)\\ \end{array} \right.\end{align*}
\begin{align*} \left\{ \begin{array}{ll} \dot{z}(t)=\mathbb{A}_{-1}z(t)+\mathbb{B}u(t), & \hbox{ } \\ y(t)=\mathbb{C}_\Lambda^\mathbb{A}z(t)+\mathbb{D}u(t) & \hbox{ } \end{array} \right.\end{align*}
\begin{align*} F_{A^I,B^I,C_\Lambda^A}&=\left( \begin{array}{cc} 0 & I \\ \end{array} \right)F_{A^{I_{X\times U}},\tilde{B}^{I_{X\times U}},\tilde{C}^{I_{X\times U}}}\left( \begin{array}{c} I \\ 0 \\ \end{array} \right)\\ &=F_{A,B,C}(I-F_{A,B,P})^{-1}.\end{align*}
\begin{align*} \left\{ \begin{array}{ll} \dot{z}(t)&=Lz(t)\\ Gz(t)&=Qz(t)+v(t)\\ y(t)&=Kz(t) \end{array} \right.\end{align*}
\begin{align*} Kz=K_\Lambda^Az+\bar{K}Gz,\ z\in Z.\end{align*}
\begin{align*} Kz=K_\Lambda^Az+\bar{K}Q_\Lambda^Az,\ z\in D(A^I).\end{align*}
\begin{align*} w(t)=&T_A(t)w+\int_0^tT_A(t-s)[L_\Lambda x(s)+\bar{L}u_1(s)]ds+\Phi_{A,B}(t)u_2\\=&T_A(t)w+\int_0^tT_A(t-s)[L_\Lambda w_s+\bar{L}w(s)]ds+\Phi_{A,B}(t)v.\end{align*}
\begin{align*} 1=K_ie_\lambda(I-R(\lambda,A)Le_\lambda)^{-1}e^{-\int_0^\cdot(\lambda+\mu(s))ds},\ i=1,2.\end{align*}
\begin{align*} (I-R(\lambda,A)Le_\lambda)f=e^{-\int_0^\cdot(\lambda+\mu(s))ds}.\end{align*}
\begin{align*} (\lambda-A)g=-\alpha(\cdot)e^{-\lambda r}f(\cdot)\end{align*}
\begin{align*}&- \int_{\Omega_{FEM_T}} \frac{\partial \bar{\lambda}}{\partial t} \frac{\partial u_h}{\partial t}~ dxdt - \int_{\partial \Omega_{FEM}} \partial_n u_h \bar{\lambda} ~ dxdt+ \int_{\Omega_{FEM_T}} ( a_h \nabla u_h) (\nabla \bar{\lambda}) ~ dxdt = 0,\end{align*}
\begin{align*} f(a)+e^{-\int_0^a(\lambda+\mu(s))ds}\int_0^a\alpha(s)e^{-\lambda r}e^{\lambda s+\int_0^s\mu(\sigma)d\sigma}f(s)ds=e^{-\int_0^a(\lambda+\mu(s))ds}, \ a\geq 0,\end{align*}
\begin{align*} e^{\int_0^a(\lambda+\mu(s))ds}f(a)+\int_0^a\alpha(s)e^{-\lambda r}e^{\lambda s+\int_0^s\mu(\sigma)d\sigma}f(s)ds=1, \ a\geq 0.\end{align*}
\begin{align*} m(a)+\int_0^a\alpha(s)e^{-\lambda r}m(s)ds=1, \ a\geq 0,\end{align*}
\begin{align*} \left\{ \begin{array}{ll} m'(a)+\alpha(a)e^{-\lambda r}m(a)=0, & \hbox{ } \\ m(0)=1. & \hbox{ } \end{array} \right.\end{align*}
\begin{align*} 1=&K_1e_\lambda(I-R(\lambda,A)Le_\lambda)^{-1}e^{-\int_0^\cdot(\lambda+\mu(s))ds}\\=&K_1e_\lambda f\\=&\int_0^{+\infty}\int_{-r}^0\beta_1(\sigma,a)e^{\lambda \sigma}e^{-\int_0^a(\lambda+\mu(s)+e^{-\lambda r}\alpha(s))ds}d\sigma da,\end{align*}