Curvature formulas of holomorphic curves on C∗C^*-algebras and Cowen-Douglas Operators

For $\Omega\subseteq \mathbb{C}$ a connected open set, and ${\mathcal U}$ a unital $C^*$-algebra, let ${\mathcal I} ({\mathcal U})$ and ${\mathcal P}({\mathcal U})$ denote the sets of all idempotents and projections in ${\mathcal U}$ respectively. ${\mathcal P}({\mathcal U})$ is called as the Grassmann manifold of $\mathcal U$ and ${\mathcal I} ({\mathcal U})$ is called as the extended Grassmann manifold. If $P:\Omega \rightarrow {\mathcal P}({\mathcal U})$ is a real-analytic ${\mathcal U}$-valued map which satisfies $\overline{\partial} PP=0$, then $P$ is called a holomorphic curve on ${\mathcal P}({\mathcal U})$. In this note, we will define the formulaes of curvature and it's covariant derivatives for holomorphic curves on $C^*$-algebras. It can be regarded as the generalization of curvature and it's covariant derivatives of the classical holomorphic curves. By using the curvature formulae, we give the unitarily and similarity classifications for the holomorphic curves and extended holomorphic curves on $C^*$-algebras respectively. And we also give a description of the trace of the covariant derivatives of curvature for any Hermitian holomorphic vector bundles. As applications, we also discuss the relationship between holomorphic curves, extended holomorphic curves, similarity of holomorphic Hermitian vector bundles and similarity of Cowen-Douglas operators.Comment: 28pages; Revised and enlarged versio

Singularities of mean curvature flow and isoperimetric inequalities in H^3

In this article, by following the method in \cite{PT}, combining Willmore energy with isoperimetric inequalities, we construct two examples of singularities under mean curvature flow in $\mathbb{H}^3$. More precisely, there exists a torus, which must develop a singularity under MCF before the volume it encloses decreases to zero. There also exists a topological sphere in the shape of a dumbbell, which must develop a singularity in the flow before its area shrinks to zero. Simultaneously, by using the flow, we proved an isoperimetric inequality for some domains in $\mathbb{H}^3$.Comment: 12 page

Preconditioning rectangular spectral collocation

Rectangular spectral collocation (RSC) methods have recently been proposed to solve linear and nonlinear differential equations with general boundary conditions and/or other constraints. The involved linear systems in RSC become extremely ill-conditioned as the number of collocation points increases. By introducing suitable Birkhoff-type interpolation problems, we present pseudospectral integration preconditioning matrices for the ill-conditioned linear systems in RSC. The condition numbers of the preconditioned linear systems are independent of the number of collocation points. Numerical examples are given.Comment: 12 pages, 2 figure

On well-conditioned spectral collocation and spectral methods by the integral reformulation

Well-conditioned spectral collocation and spectral methods have recently been proposed to solve differential equations. In this paper, we revisit the well-conditioned spectral collocation methods proposed in [T.~A. Driscoll, {\it J. Comput. Phys.}, 229 (2010), pp.~5980-5998] and [L.-L. Wang, M.~D. Samson, and X.~Zhao, {\it SIAM J. Sci. Comput.}, 36 (2014), pp.~A907--A929], and the ultraspherical spectral method proposed in [S.~Olver and A.~Townsend, {\it SIAM Rev.}, 55 (2013), pp.~462--489] for an $m$th-order ordinary differential equation from the viewpoint of the integral reformulation. Moreover, we propose a Chebyshev spectral method for the integral reformulation. The well-conditioning of these methods is obvious by noting that the resulting linear operator is a compact perturbation of the identity. The adaptive QR approach for the ultraspherical spectral method still applies to the almost-banded infinite-dimensional system arising in the Chebyshev spectral method for the integral reformulation. Numerical examples are given to confirm the well-conditioning of the Chebyshev spectral method.Comment: 17 pages, 8 figure

First eigenvalue for p-Laplacian with mixed boundary conditions on manifolds

In this paper, we mainly study eigenvalue problems of p-Laplacian on domains with an interior hole. Firstly we prove Faber-Krahn-type inequalities, and Cheng-type eigenvalue comparison theorems on manifolds. Secondly, we prove a comparison theorem for eigenvalues with inner Dirichlet and outer Neumann boundary in minimal submanifolds in Euclidean space. Lastly we give a sharp estimate of the eigenvalue (with outer Dirichlet and inner Neumann boundaries) in terms of the Dircihlet eigenvalue, and also we give an upper bound of the eigenvalue with inner Dirichlet and outer Neumann problems by the diameter of the hole inside.Comment: Some issues in Theorem 1.

Two spectral methods for 2D quasi-periodic scattering problems

We consider the 2D quasi-periodic scattering problem in optics, which has been modelled by a boundary value problem governed by Helmholtz equation with transparent boundary conditions. A spectral collocation method and a tensor product spectral method are proposed to numerically solve the problem on rectangles. The discretization parameters can be adaptively chosen so that the numerical solution approximates the exact solution to a high accuracy. Our methods also apply to solve general partial differential equations in two space dimensions, one of which is periodic. Numerical examples are presented to illustrate the accuracy and efficiency of our methods.Comment: 15 pages, 2 figure

Preconditioning fractional spectral collocation

Fractional spectral collocation (FSC) method based on fractional Lagrange interpolation has recently been proposed to solve fractional differential equations. Numerical experiments show that the linear systems in FSC become extremely ill-conditioned as the number of collocation points increases. By introducing suitable fractional Birkhoff interpolation problems, we present fractional integration preconditioning matrices for the ill-conditioned linear systems in FSC. The condition numbers of the resulting linear systems are independent of the number of collocation points. Numerical examples are given.Comment: 9 pages, 2 figure

Any admissible harmonic Ritz value set is possible for prescribed GMRES residual norms

We show that any admissible harmonic Ritz value set is possible for prescribed GMRES residual norms, which is a complement for the results in [Duintjer Tebbens and Meurant, {\it SIAM J. Matrix Anal. Appl.}, 33 (2012), no. 3, pp. 958--978].Comment: 5 page

On the mean value of a kind of Zeta functions

Let $d_{\alpha, \beta}(n)=\sum\limits_{\substack{n=kl \alpha l<k\leq\beta l}}1$ be the number of ways of factoring n into two almost equal integers. For rational numbers $0<\alpha <\beta$, we consider the following Zeta function $\zeta_{\alpha,\beta}(s)=\sum\limits_{n=1}^{\infty}\frac{d_{\alpha, \beta}(n)}{n^{s}}$ for $\Re s>1.$ It has an analytic continuation to $\Re s>1/3.$ We get an asymptotic formula for the mean square of $\zeta_{\alpha,\beta}(s)$ in the strip $1/2<\Re s<1$. As an application, we improve an result on the distribution of primitive Pythagorean triangles

Quantum invariants of links and new quantum field models

We propose a gauge model of quantum electrodynamics (QED) and its nonabelian generalization from which we derive knot invariants such as the Jones polynomial. Our approach is inspired by the work of Witten who derived knot invariants from quantum field theory based on the Chern-Simon Lagrangian. From our approach we can derive new knot and link invariants which extend the Jones polynomial and give a complete classification of knots and links. From these new knot invariants we have that knots can be completely classified by the power index $m$ of $TrR^{-m}$ where $R$ denotes the $R$-matrix for braiding and is the monodromy of the Knizhnik-Zamolodchikov equation. A classification table of knots can then be formed where prime knots are classified by prime integer $m$ and nonprime knots are classified by nonprime integer $m$.Comment: 17 pages, 7 figure