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\begin{gathered}h^{-2\sigma}a^{-2k + \eta}= h^{-2\sigma}(h^{\frac{\sigma}{k}}(\log(h^{-1}))^{-T})^{2k - \eta}\le e^{\frac{\sigma}{k}}\log(h^{-1})^{-\frac{T}{3}}\le C \log(h^{-1})^{-\frac{T}{3}}, h^{-2\sigma}a^{-1 - 2\delta + k + \eta}\le h^{-2\sigma}a^{-2k + \eta}\le C \log(h^{-1})^{-\frac{T}{3}}. \end{gathered} |
[F]=\beta, \ \chi(F)=n |
\widehat\alpha(I_{n+1, c}) = \frac{n+1}{c}. |
[\sigma^i(p')+\sigma^i(j(p'))+\sigma^{-i}(p')+\sigma^{-i}(j(p')) -\sigma^i(x')-\sigma^i(j(x'))-\sigma^{-i}(x')-\sigma^{-i}(j(x'))]= [p'_i+j(p'_{d-i})+p'_{d-i}+ j(p'_{i}) - x'_i-j(x'_{d-i})-x'_{d-i}-j(x'_{i})]= (1+j)([p'_i+p'_{d-i}-x'_i-x'_{d-i}]). |
F:V\rightarrow V |
\Gamma_{1}(N) |
\gamma_{1}( \alpha_{1}\beta_{2}- \alpha_{2}\beta_{1}) - \gamma_{1}\frac{\partial \gamma_{1}}{\partial \theta}= 0 |
q(u)=\exp_{A}(u-K_{p}(u)+\log_{A}p) |
\overline{\beta_{jk}\left(\left|\eta\right|\right)}=\beta_{jk}\left(-\left|\eta\right|\right). |
D_t=\left(\begin{smallmatrix}\cos t -\sin t \sin t \cos t\end{smallmatrix}\right). |
\overline{P_{1}P_{2}} |
\Delta_{R}\{\theta_{1}, \theta_{2}\}_{\mathrm{c . b .}}= \{\theta_{1}, \theta_{2}\}_{\mathrm{c . b .}}\otimes I |
{\frac{\partial}{\partial r}}\left[ ( r^{2}+ A^{2}- 2 M r ){\frac{\partial G}{\partial r}}\right] +{\frac{1}{s i n \theta}}{\frac{\partial}{\partial \theta}}\left[ s i n \theta{\frac{\partial G}{\partial \theta}}\right] = - \delta ( r - r_{o}) \delta ( c o s \theta - c o s{\theta}_{o}) \delta ( \varphi -{\varphi}_{o}) |
\lim_{n\to \infty}|\phi(\Delta_n)|=\lim_{n\to \infty}|\psi(\Delta_n)|=1 |
\pi_{3}(SL(2,\mathbb{R})) |
y_{n+1}=y_{n}(\frac{3}{2}-\frac{x}{2}y_{n}^{2})=\frac{y_{n}(3-xy_{n}^{2})}{2}. |
u\frac{\partial R(u,g)}{\partial u}= \beta (g^2)\frac{\partial R(u,g)}{\partial g^2}+\gamma (g^2)R(u,g), |
D=\frac{C\cdot A}{B}=\frac{1.63m\cdot180m}{2m}=146.7m |
(\sum_{i=1}^{3}\sqrt{a_{i}x+b_{i}}-\sum_{i=4}^{6}\sqrt{a_{i}x+b_{i}})=0 |
N_{2}(c) |
x_{j}\in\frac{[b_{i}]-\sum_{k\ne j}[a_{ik}]\cdot[x_{k}]}{[a_{ii}]} |
\rho^{\prime}\equiv\rho+\frac{\Lambda}{8\pi G} |
U_{el}(x)=\frac{1}{2}kx^{2} |
\beta=\frac{E}{RT_{f}}\frac{T_{f}-T_{o}}{T_{f}} |
\Delta_p^\pm(H(\omega))\leq\sum_{\ell=1}^L\Delta_p^\pm(H(\omega_\ell)),\delta_p^\pm(H(\omega))\geq\sum_{\ell=1}^L\delta_p^\pm(H(\omega_\ell)). |
{}\displaystyle\sum_{n=2}^{\infty}\Big|\Big(\frac{n-1}{n}\Big)^{\frac{\beta}{2}}\frac{A_n}{\sqrt{\lambda_{n}}}-\Big(\frac{n}{n-1}\Big)^{\frac{\beta}{2}}\frac{A_{n-1}}{\sqrt{\lambda_{n}}}\Big|^{2}\geq \displaystyle\sum_{n=1}^{\infty}(\widetilde{w_{n}}(\lambda,\beta)-w_{n}(\lambda,\beta))|A_{n}|^{2}\geq 0. |
0\le\lambda<1-\epsilon |
( a = 1 , b = 9 , c = 8 , d = 5 ) = 1 9 8 5 |
H^{\prime}=\sum_{n=1}^{N}\beta_{n}H_{n}. |
h''(r)+ \left(\frac{3}{r}-\frac{r}{2}\right)h'(r)=h(r) |
\Delta{K}^{s,\infty}=\log\Big((2^{\ell_1}-1)+\sqrt{(1-2^{\ell_1})^2+2^{\ell_0+2}}\Big)-\mu_0-1 |
\sigma(a_1+ a_2)\setminus \{0\}= \{\lambda: \lambda^2 \in \sigma(a_1a_2)\}\setminus\{0\}= \{\lambda: \lambda^2 \in \sigma(a_2a_1)\}\setminus\{0\}. |
\langle f_i \ |\ \bar{x}\rangle = \eta_i \iff \sum_{j\in \mathbb{N}}\lambda_j^i \bar{x}_j = \eta_i . |
\mathcal{C}:= \int_{D}W(x) \, dx = \int_{D}\left(I + \eta \nabla M^{-k}- k^{2}\, \eta \, N^{-k}\right)^{-1}\left( \mathcal{P}(\cdot,z)\right)(x) \, dx. |
H(x)\triangleq[\begin{matrix}h_{1}(x)\\h_{2}(x)\\h_{3}(x)\\ \vdots\\h_{n}(x)\end{matrix}]\triangleq[\begin{matrix}h(x)\\L_{f}h(x)\\L_{f}^{2}h(x)\\ \vdots\\L_{f}^{n-1}h(x)\end{matrix}] |
2\notin Y |
(R,P) |
\mu_{\Lambda|\Phi} |
f(x)=\frac{1}{2}x^{T}Ax-x^{T}b,x\in R^{n}. |
\langle \zeta^{\prime}| \zeta^{\prime \prime}\rangle = \delta ( \zeta^{\prime}- \zeta^{\prime \prime}) |
\langle X\rangle=\frac{1}{n}\sum_{j=1}^{n}X_{j} |
\neg\neg A\rightarrow A. |
\sim[(1-i)^2+(1+i)^2]\ \biggl\langle e^{i/2\H}(x) e^{-i/2\H}(y)\biggl\rangle |
\lambda_{\alpha}(X)=\frac{\mathbb{E}[X^{\alpha}]}{\Gamma(\alpha+1)},\alpha>0. |
v^{\prime}=\frac{3}{2}v |
\Sigma = \Sigma_{gf}+ \Sigma_{ext} |
\frac{1}{2}(L_Z g)(X,Y) + \frac{1}{\lambda^2}\{Ric^N (\tilde{X}, \tilde{Y}) + f_3 h(\tilde{X},\tilde{Y}) \}= 0, |
|F_1|\le \underset{j=0}{\overset{3}{\sum}}\|\langle x\rangle^{j/4}\partial_x^{3-j}u_1\|_{\infty}\int w^2 \phi_N\,dx |
S^{-1}M^{*}S=N^{*}. |
\|h\| = q^{\deg h}\mbox{and}\|0\|=0. |
\begin{aligned}[b] |S_1^*|-|S_2^*|-\frac{2E_{S_1^*}-E_{S_1^*,S_3^*}+E_{S_2^*,V_1}-E_{S_1^*,V_1}}{\tau-1}&>|S_1^*|-1-|S_2^*|-\frac{2E_{S_1^*}-E_{S_1^*,S_3^*}-d_{i,S_1^*}-d_{i,S_3^*}+E_{S_2^*,V_1}-E_{S_1^*,V_1}+d_{i,V_1}}{\tau-1},\end{aligned} |
\delta T_{\pm\pm}= - (2\partial_\pm \xi^\pm + \xi^\pm\partial_\pm) T_{\pm\pm}+{c\over 24\pi}\partial^3_\pm \xi^\pm\ . |
\frac{2}{\pi\sqrt{3}(1+\frac{t^{2}}{3})^{2}} |
\lambda_{k,n}(v,\sigma) = \frac{\int_a^b \frac{|\phi_{k,n}'|^2 v}{\sigma}+ \frac{k^2 |\phi_{k,n}|^2 \sigma}{v}\,dt}{\int_a^b |\phi_{k,n}|^2 v \sigma \,dt} |
M \geq \tau_p v_p + \tau_{p'}v_{p'} |
= \frac{4 4}{5}\div \frac{8 8}{1 0} |
\cdot A E = 7 - \sqrt{1 3} |
X_\beta:=\sum_{e\in H}\widehat{Z_\beta}(e) \chi_e Y_\beta:=\sum_{\substack{S\subset H |S|\ge 2}}\widehat{Z_\beta}(S) \chi_S |
52-50-9=-7lbs |
\frac{d g^{2}}{d t}= - \epsilon g^{2}- \frac{8 g^{4}}{( 4 \pi )^{2}} |
\frac{8!\times3^{7}\times24!^{2}}{24^{7}}\approx7.40\times10^{45}. |
( \hat{\nu}_{\epsilon}, a ) = \sum_{i , k}\hat{\nu}_{\epsilon \, i}( \delta_{i j}- \frac{1}{2}{\cal C}_{i j}) \, a_{j}= \sum_{i \in \Gamma_{\epsilon}^{( 0 )}}\hat{\nu}_{\epsilon \, i}\, a_{i}- c o s \frac{\pi}{h}\sum_{i \in \Gamma_{\epsilon + 1}^{( 0 )}}\hat{\nu}_{\epsilon + 1 \, i}\, a_{i} |
A(\tau,s)(c'(s))=0, A\wedge A = 0. |
P_{n+1}(x)=A_{n}(x)P_{n}^{\prime}(x)+B_{n}(x)P_{n}(x),n\geq0, |
A(n)\simeq\mathbb{R}^{n} |
\lambda_{Z}=\sum_{j=1}^{\infty}\left\Vert \tilde{k}_{z}\right\Vert_{H^{2}}^{-2}\delta_{z_{j}}. |
{\bf B}_5=\begin{pmatrix}32 0 41472 0 1620000 0 3872 0 355552 0 3872 0 80352 0 2024352 0 80352 0 737792 0 355552 0 737792 0 4220000 \end{pmatrix}. |
g_{55} |
\dot{h}v + h \dot{v}= \dot{\lambda}v + \lambda \dot{v}. |
[u, v] = D_{u}v - D_{v}u + (Du)^{*}v - T^{D}(u, v) |
\frac{-1}{\pi\sqrt{u^{2}-t^{2}}}\frac{d}{du} |
T_{zz}^{\Psi}=T_{zz},\hspace{1em}T_{z\bar{z}}^{\Psi}={\cal O}_{0}\Psi\hspace{1em}T_{\bar{z}\bar{z}}^{\Psi}=T_{\bar{z}\bar{z}},\hspace{1em}T_{\bar{z}z}^{\Psi}=\overline{\cal O}_{0}\Psi . |
(6)_{t}^{4}(2)_{t^{2}}^{2} |
\sqrt{a^{6}}= \sqrt{3^{- 2}}= \frac{1}{3} |
(-\Delta_p)^s u=\lambda\,\frac{|u|^{p-2}\,u}{d_\Omega^{s\,p}},\mbox{in}\Omega, |
lim_{i}m(x_{i})\le lim_{i}\frac{f^{\prime}(x_{i})}{g^{\prime}(x_{i})}\le lim_{i}M(x_{i}) |
dX^{(k)}(t)=\mu^{(k)}dt+\Sigma^{(k)\frac{1}{2}}dW^{(k)}(t) |
b_{3}=3\frac{y_{2}-y_{1}}{(x_{2}-x_{1})^{2}} |
H_{4}= T r \Phi_{2}^{2}= \sum_{i \neq j}f_{i j}f_{j i}. |
\eta_{C}=\frac{IsentropicCompressorWork}{ActualCompressorWork}=\frac{W_{s}}{W_{a}}\cong\frac{h_{2s}-h_{1}}{h_{2a}-h_{1}} |
S^{(1)>}_c (x^0 - i \beta - y^0, \vec{p}) = - e^{- \beta \mu}S^{(1)<}_c (x^0 - y^0, \vec{p}) . |
2d_{0,0}(n,0)=d_{0,0}(0,i)+d_{0,0}(n,-i),ni\ne 0. |
{P^{IJ}}_{MN} |
\frac{a(+)}{a(-)}=\frac{b(+)}{b(-)}=\frac{c(+)}{c(-)}, |
w\ =\v + \sum_a \beta_a(b)(1,a)\ +\c \ =\ \sum_s (\beta_a(b)+\nu)(1,a)\ +\c\,, |
U(N)=\{u{\in}SU(N+1):uY^{(N+1)}u^{-1}=Y^{(N+1)}\}. |
S=S_{g,b,n} |
Q_{i}\left( z \right)=\Gamma_{i}^{+}\left( A_{i}-zE_{i}\right)^{-1}\Gamma_{i}\in N_{\kappa_{i}}\left( H \right),\thinspace i=1,\thinspace \thinspace \mathellipsis ,\thinspace r. |
u(1+i)^{e_{0}}{p_{1}}^{e_{1}}\cdot\cdot\cdot{p_{k}}^{e_{k}}, |
0=\lim_{n\to \infty}\left(\mu(s_n)-\phi(s_n)(\pi)\right)\geq \lambda \pi>0. |
\nu=r_{p}v_{p}\int_{t_{p}}^{t}\frac{1}{r^{2}}dt |
D^{\mu}D_{\mu}\Phi + \frac{\lambda}{2}\Phi \; ( | \Phi |^{2}- \frac{M^{2}}{\lambda}) = 0 , |
2K(\int_{0}^{\infty}\frac{dt}{\pi t^{2}+t+A/K^{2}})^{-1} |
f(X)=X^{3}-aX+b |
e_{a_j}\left[t+1\right]= e_{a_j}\left[t\right]+2\mu\Re{\left\{\hat{\phi}^{\left(j,j\right)}\right\}}-4\mu\left(\sum_{l=1}^M\lvert\hat{\phi}^{\left(l,j\right)}\rvert^2+M\sigma_{e,i}^2\lVert\mathbf{p}_j\rVert^2\right.\left.+\sum_{q=1}^{M-1}\sum_{r=q+1}^Mf_{q,r}^{\left(j\right)}\right)a_j\left[t\right]. |
1/2=0.5 |
E F = \frac{1}{2}B D |
\widetilde{g}(\cdot,\cdot) = \langle\cdot,\cdot\rangle_T\, . |
\frac{d h_{p p^{\prime}}( l )}{d l}= \! \! - \sum_{q}\! \left( \frac{d h_{p q}( l )}{d l}\frac{1}{E_{p}( l ) - E_{q}( l )}h_{q p^{\prime}}( l ) + h_{p q}( l ) \frac{1}{E_{p^{\prime}}( l ) - E_{q}( l )}\frac{d h_{q p^{\prime}}( l )}{d l}\right) \, , |
\sigma \rightarrow \sqrt{\lambda}( \sigma - \mu^{\varepsilon}\sqrt{\lambda}Z_{2}\phi_{c}\phi ) |
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