Narrow Results

We give a lower bound for the first Dirichlet eigenvalue for a compact manifold with positive Ricci curvature in terms of the lower bound of the Ricci curvature and the interior radius. The result sharpens earlier estimates.

UHF flows are the flows obtained as inductive limits of flows on full matrix algebras. We will revisit universal UHF flows and give an explicit construction of such flows on a UHF algebra M_k^∞ for any k and also present a characterization of such flows. Those flows are UHF flows whose cocycle perturbations are almost conjugate to themselves.

For a bounded domain Ω with a piecewise smooth boundary in a complete Riemannian manifold M, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of L²(Ω) in place of the Rayleigh–Ritz formula, we obtain inequalities for eigenvalues of the Laplacian. In particular, for lower order eigenvalues, our results extend the results of Chen and Cheng [D. Chen and Q.-M. Cheng, Extrinsic estimates for eigenvalues of the Laplace operator, J. Math. Soc. Japan 60 (2008) 325–339].

In this paper, we consider the boundary value problem

We show that for any two $n\times n$ matrices $X$ and $Y$ we have the inequality $s_j^2(I+XY)\le {\lambda }_j((I+X^{\ast }X)(I+Y^{\ast }Y)),$ where $s_j(T)$ and ${\lambda }_j(T)$ denote the decreasingly ordered singular values and eigenvalues of $T$. As an application, we show that for $2n\times 2n$ real positive definite matrices the symplectic eigenvalues $d_j,$ under some special conditions, satisfy the inequality $d_j(A+B)\ge d_j(A)+d_1(B)$.

A method is proposed for finding the normal mode eigenvalues in shallow water waveguide. We transform the problem determining eigenvalues in complex wave number plane into solving a "true" one dimensional Hamilton system. Simulations are performed, the result agrees very well with that calculated by the normal mode program KrakanC.

In this article, we have determined the remainder term for Hardy–Sobolev inequality in H¹(Ω) for Ω a bounded smooth domain and studied the existence, non existence and blow up of first eigen value and eigen function for the corresponding Hardy–Sobolev operator with Neumann boundary condition.

In this paper, we prove the long-time existence of the CR Yamabe flow on the compact strictly pseudoconvex CR manifold with positive CR invariant. We also prove the convergence of the CR Yamabe flow on the sphere by proving that: the contact form which is pointwise conformal to the standard contact form on the sphere converges exponentially to a contact form of constant pseudo-Hermitian sectional curvature. We also show that the eigenvalues of some geometric operators are non-decreasing under the unnormalized CR Yamabe flow provided that the pseudo-Hermitian scalar curvature satisfies certain conditions.

The purpose of this paper is two-fold. Firstly, we state and prove a Berezin–Li–Yau-type estimate for the sums of eigenvalues of , the fractional Laplacian operators restricted to a bounded domain Ω ⊂ ℝ^d for d ≥ 2 and α ∈ (0, 2]. Secondly, we provide an improvement to this estimate by using a pure analytical approach.

In this paper we will use variational methods and limiting approaches to give a complete solution to the maximization problems of the Dirichlet/Neumann eigenvalues of the one-dimensional p-Laplacian with integrable potentials of fixed L¹-norm.

A key basis for seeking solutions of the Camassa–Holm equation is to understand the associated spectral problem

This study aimed to determine the best constants in the Wirtinger inequality

This paper gives a kind of sharp simultaneous approximation error estimation of Birkhoff interpolation $\text{for all }1\le p,q\le \infty ,0\le s\le r-1$,

This study gives a kind of sharp Wirtinger inequalities (Pizone inequalities)

Given an arbitrary strictly increasing infinite sequence of positive integers, let S_n = {x₁,…, x_n} for any integer n ≥ 1. Let q ≥ 1 be a given integer and f an arithmetical function. Let be the eigenvalues of the matrix (f(x_i, x_j)) having f evaluated at the greatest common divisor (x_i, x_j) of x_i and x_j as its i, j-entry. We obtain a lower bound depending only on x₁ and n for if (f * μ)(d) < 0 whenever d|x for any x ∈ S_n, where g * μ is the Dirichlet convolution of f and μ. Consequently we show that for any sequence , if f is a multiplicative function satisfying that f(p^k) → ∞ as p^k → ∞, f(2) > 1 and f(p^m) ≥ f(2)f(p^m−1) for any prime p and any integer m ≥ 1 and (f *μ)(d) > 0 whenever d|x for any , then approaches infinity when n goes to infinity.

The initial discovery and implementation by the author of a particular kind of nonstandard finite difference (NSFD) scheme called a “Mickens finite difference” (MFD) for approximating the radial derivatives of the Laplacian in cylindrical coordinates is reviewed. The development of a similar scheme for the spherical coordinates case is also recounted. Examples of application of the schemes to several related (singular and nonsingular, linear and nonlinear) boundary value problems are given. Examples of applying Buckmire's MFD scheme to the bifurcatory, nonlinear eigenvalue problems of Bratu and Gel'fand are also presented. The results support the utility and versatility of MFD schemes for boundary value problems with singularities or bifurcations.

We present some recent results on variable exponent Lebesgue spaces, which include: A variant of the definition of the norm in the variable exponent Lebesgue space; The Amemiya norm equals the Orlicz norm in the variable exponent Lebesgue space; An exact inequality involving the Luxemburg norm and the conjugate-Orlicz norm in the variable exponent Lebesgue space. We also present some results and open problems on the solutions of the p(x)–Laplacian equations.

Let X be an infinite-dimensional Banach space with unit closed ball B(X) and unit sphere S(X). It is well known that there is a retraction of B(X) onto S(X), that is, a continuous mapping R: B(X) → S(X) with Rx = x for all x ∈ S(X), and moreover such a retraction can be chosen among Lipschitz mappings. Many authors have looked for the smallest possible Lipschitz constant for special classes of Banach spaces. A similar and related problem can be considered by replacing the Lipschitz constant by the γ-Lipschitz constant when γ is the Hausdorff measure of noncompactness.

Aim of this talk is to present recent contributions related to the above problem, and give some applications to fixed point and eigenvalue theory of nonlinear operators.