通过Lidskii定理,可以通过特征值求 Trace,而 AB,BA 的非零特征值 \lambda , 在其中一个代数重数有限 (我不清楚这一说法是否普遍,我是指 \mathrm{dim}\ker(AB-\lambda)<\infty )的情况下,两个算子关于此特征值的重数相等(这甚至与拓扑无关),因此计入 Trace 的部分相同。由此可以认为(并且他在 P67 用过)...
sin \theta &{}\cos \theta \end{pmatrix}\begin{pmatrix} \chi \\ \phi _R \end{pmatrix}\quad \nonumber \\{} & {} \quad \text {where},\ \theta =\tan ^{-1}\Bigg [\frac{2\sqrt{2}\mu v}{\mu _{\phi }^2-\mu _{\chi }^2+(\lambda _4-\lambda _5)v^2}\Bigg ]....
wherehandgare suitable measure-preserving maps, in whichis a local minimizer of the Dirichlet energy . The proof of this fact requires a careful calculation of the second variation of, which quantity turns out to be non-negative in general and zero only when. ...
Explain step by step how when given T(L, \lambda):= \int_0^\infty \frac{T_0 \cdot \sin(\lambda y)}{e^y} dy you get T(L, \lambda):= \frac{T_0 \cdot \lambda}{\lambda^2 + 1}. Derive the forumla: \sum_{n=0}^{N} ar^n = a \cdot \frac{1-r^{N+1{1-r} ...
Analogously to the case of Virasoro, where we can find a basis of the Verma module \mathcal {V}_{\varvec{\theta }} labeled by partitions, in the W_N case we can find a basis labeled by N-1-tuples of partitions \lambda ^{(j)}=(\lambda _1^{(j)},\dots ,\lambda _k^{(j...
\begin{aligned}&dX^\mu \wedge \star dX^\nu g_{\mu \nu } + \cos \theta ~ \omega _{\mu \nu } dX^\mu \wedge dX^\nu \nonumber \\&= 4i \Big ( \cos ^2 \frac{\theta }{2}~\partial _{{\bar{z}}} X^i \partial _z X^{{\bar{j}}} + \sin ^2 \frac{\theta }{...
wheref\equiv (f^1, \ldots ,f^k):{\mathbb {R}}^n\rightarrow {\mathbb {R}}^kare smooth functions such that the Jacobian matrixJ=\partial f/\partial xis of rankkon all{\mathbb {R}}^n; or, more generally, thatfis well defined andJis of rankkon an open subset\mathcal{D}_f\...
\begin{aligned} u_{\sigma }(\theta ):=\left\{ \begin{array}{lr} (\cos \theta ,\sin \theta );\quad 0\le \theta<\frac{3\pi }{2} -\frac{\sigma }{2} ,\\ \left( \cos \theta ,2\sin \left( \frac{3\pi }{2}-\frac{\sigma }{2}\right) -\sin \theta \right) ;\qua...