A VANISHING THEOREM FOR THE HEAT CONTENT OF A RIEMANNIAN MANIFOLD

2001 ◽  
Vol 33 (6) ◽  
pp. 743-748
Author(s):  
M. VAN DEN BERG
Keyword(s):  
Riemannian Manifold ◽  
Asymptotic Behaviour ◽  
Compact Subset ◽  
Heat Content ◽  
Vanishing Theorem ◽  
Dirichlet Conditions

The paper considers the asymptotic behaviour of the heat content of a riemannian manifold M, when Dirichlet conditions are imposed on a small compact subset of M.

scholarly journals Homogenization of the spectral problem on the Riemannian manifold consisting of two domains connected by many tubes

2013 ◽  
Vol 143 (6) ◽  
pp. 1255-1289 ◽  
Author(s):  
Andrii Khrabustovskyi
Keyword(s):  
Riemannian Manifold ◽  
Small Parameter ◽  
Asymptotic Behaviour ◽  
Spectral Problem ◽  
Spectral Parameter ◽  
Beltrami Operator ◽  
Perforated Domain ◽  
Perforated Domains ◽  
Accumulation Points ◽  
Image Position

The paper deals with the asymptotic behaviour as ε → 0 of the spectrum of the Laplace–Beltrami operator Δε on the Riemannian manifold Mε (dim Mε = N ≥ 2) depending on a small parameter ε > 0. Mε consists of two perforated domains, which are connected by an array of tubes of length qε. Each perforated domain is obtained by removing from the fixed domain Ω ⊂ ℝN the system of ε-periodically distributed balls of radius dε = ō(ε). We obtain a variety of homogenized spectral problems in Ω; their type depends on some relations between ε, dε and qε. In particular, if the limitsare positive, then the homogenized spectral problem contains the spectral parameter in a nonlinear manner, and its spectrum has a sequence of accumulation points.

scholarly journals Asymptotics for Functions Associated with Heat Flow on the Sierpinski Carpet

2011 ◽  
Vol 63 (1) ◽  
pp. 153-180 ◽  
Author(s):  
B. M. Hambly
Keyword(s):  
Partition Function ◽  
Heat Flow ◽  
Periodic Function ◽  
Asymptotic Behaviour ◽  
Heat Kernel ◽  
Counting Function ◽  
Heat Content ◽  
Leading Term ◽  
Sierpiński Carpet ◽  
Sierpinski Carpets

Abstract We establish the asymptotic behaviour of the partition function, the heat content, the integrated eigenvalue counting function, and, for certain points, the on-diagonal heat kernel of generalized Sierpinski carpets. For all these functions the leading term is of the form xγΦ (log x) for a suitable exponent γ and Φ a periodic function. We also discuss similar results for the heat content of affine nested fractals.

scholarly journals On the spectrum of periodic elliptic operators

1992 ◽  
Vol 126 ◽  
pp. 159-171 ◽  
Author(s):  
Jochen Brüning ◽  
Toshikazu Sunada
Keyword(s):  
Riemannian Manifold ◽  
Band Structure ◽  
Compact Subset ◽  
Schrödinger Operator ◽  
Transformation Group ◽  
Elliptic Operators ◽  
Schrodinger Operator ◽  
Closed Intervals ◽  
Adjoint Operators

It was observed in [Su5] that the spectrum of a periodic Schrödinger operator on a Riemannian manifold has band structure if the transformation group acting on the manifold satisfies the Kadison property (see below for the definition). Here band structure means that the spectrum is a union of mutually disjoint, possibly degenerate closed intervals, such that any compact subset of R meets only finitely many. The purpose of this paper is to show, by a slightly different method, that this is also true for general periodic elliptic self-adjoint operators.

Approximation in statistical sense to B -continuous functions by positive linear operators

2010 ◽  
Vol 47 (3) ◽  
pp. 289-298 ◽  
Author(s):  
Fadime Dirik ◽  
Oktay Duman ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Statistical Approximation ◽  
Classical Version ◽  
Statistical Sense

In the present work, using the concept of A -statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued B -continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.

Sign in / Sign up

Export Citation Format

Share Document