An Irrotational and Incompressible Flow Around a Body in Space

We consider the problem of an irrotational and incompressible flow around a body in space. The basic existence is proved by formulating the problem into a variational problem. We also show that the solution is unique, and the maximum speed is attained on the body's boundary.     

The Problem of Eigenvalue on Noncompact Complete Riemannian Manifold

Let M be an n-dimensional noncompact complete Riemannian manifold, "Δ" is the Laplacian of M. It is a negative selfadjoint operator in L²(M). First, we give a criterion of non-existence of eigenvalue by the heat kernel. Applying the criterion yields that the Laplacian on noncompact constant curvature space form has no eigenvalue. Then, we give a geometric condition of M under which the Laplacian of M has eigenvalues. It implies that changing the metric on a compact domain of constant negative curvature space form may yield eigenvalues.     

Mixed Boundary Value Problems for Nonlinear Elliptic Systems with p‐Structure in Polyhedral Domains

The nonlinear elliptic system is investigated on a non‐smooth domain. Mixed boundary value conditions are given. The left‐hand side of the system has p‐structure (e.g., it is the p‐Laplacian and 1 < p < ∞). Global regularity results of u and |∇u|^p/2 in fractional order Sobolev spaces are proven.     

A UNIFORM GROWTH ESTIMATES OF SOLUTIONS OF ASYMPTOTICALLY ELLIPTIC OPERATORS

51.IntroductionLetusconsideranellipticoperatorLdefinedonRebywithla'jlLco(Rrt)SCandAlS(a'j(x))5AlforsomepositiveconstantsA,A.ThesolutionuofLa=0iscalledtheL-harmonicfunction.Inarecedework[61,LinstudiedtheL-harmonicfunctionsandgaveaLiouvilletypetheorem,providedtheoperatorisasymptoticallyconic.Weareconcernedherewiththe'polynomialgrowthL-harmolilcfunctions,thatis,theL-harmonicfunctionsatisfyinglulSC(1 Ixl)'.Thisproblem,ingeneral,hasnotbeenwellunderstood,exceptthecasesthatforn=2,ListheLapl…     

Coherent states attached to the spectrum of the Bochner Laplacian for the Hopf fibration

We construct coherent states attached to spectrum of the Bochner Laplacian for the Hopf fibration. This enables us to obtain a closed form for the expression of the coherent states attached to Landau levels on the Riemann sphere. We also obtain an identity for certain orthogonal polynomials.     

Does the eigenratio λ2/λN represent the synchronizability of a complex network?

In the study of complex networks,it is commonly believed that the eigenratio λ2/λ N of the Laplacian matrix of a network represents the network synchronizability,especially for symmetric networks.This paper gives two counterexamples to show that this is not true for the case where the network has a disconnected synchronized region.Consequently,a simple answer is presented to the question of when the eigenratio λ2/λ N does represent the network synchronizability.     

Scaling coupling of reflecting Brownian motions and the hot spots problem

We introduce a new type of coupling of reflecting Brownian motions in smooth planar domains, called scaling coupling. We apply this to obtain monotonicity properties of antisymmetric second Neumann eigenfunctions of convex planar domains with one line of symmetry. In particular, this gives the proof of the hot spots conjecture for some known types of domains and some new ones.

     

Perron components and algebraic connectivity for weighted graphs

The algebraic connectivity of a connected graph is the second-smallest eigenvalue of its Laplacian matrix, and a remarkable result of Fiedler gives information on the structure of the eigenvectors associated with that eigenvalue. In this paper, we introduce the notion of a perron component at a vertex in a weighted graph, and show how the structure of the eigenvectors associated with the algebraic connectivity can be understood in terms of perron components. This leads to some strengthening of Fiedler's original result, gives some insights into weighted graphs under perturbation, and allows for a discussion of weighted graphs exhibiting tree-like structure.     

Eigenvalues of the negative Laplacian for simply connected bounded domains

This paper is devoted to asymptotic formulae for functions related with the spectrum of the negative Laplacian in two and three dimensional bounded simply connected domains with impedance boundary conditions, where the impedances are assumed to be discontinuous functions. Moreover, asymptotic expressions for the difference of eigenvalues related to the impedance problems with different impedances are derived. Further results may be obtained.