共查询到20条相似文献,搜索用时 15 毫秒
1.
本文研究了连通分形鼓上的谱渐近,对满足“切口”条件的连通分形鼓以及一类自然连通的分形鼓,分别证明了弱Weyl-Berry猜想是成立的. 相似文献
2.
Counting function asymptotics and the weak Weyl-Berry conjecture for connected domains with fractal boundaries 总被引:1,自引:0,他引:1
In this paper, we study the spectral asymptotics for connected fractal domains and Weyl-Berry conjecture. We prove, for some
special connected fractal domains, the sharp estimate for second term of counting function asymptotics, which implies that
the weak form of the Weyl-Berry conjecture holds for the case. Finally, we also study a naturally connected fractal domain,
and we prove, in this case, the weak Weyl-Berry conjecture holds as well.
Research partially supported by the Natural Science Foundation of China-and the Royal Society of London 相似文献
3.
本文将首先在R^2和R^3中各自构造出一对混合边值条件的等谱非等距同构的基本构件。并且在R^2中利用自相似的方法构造出相应的等谱非等距同构分形鼓.在此基础上,本文讨论了这类分形鼓的波数目函数的渐近估计.得到其第二项系数的上界和下界估计。 相似文献
4.
Juan Pablo Rossetti Dorothee Schueth Martin Weilandt 《Annals of Global Analysis and Geometry》2008,34(4):351-366
We construct pairs of compact Riemannian orbifolds which are isospectral for the Laplace operator on functions such that the
maximal isotropy order of singular points in one of the orbifolds is higher than in the other. In one type of examples, isospectrality
arises from a version of the famous Sunada theorem which also implies isospectrality on p-forms; here the orbifolds are quotients of certain compact normal homogeneous spaces. In another type of examples, the orbifolds
are quotients of Euclidean and are shown to be isospectral on functions using dimension formulas for the eigenspaces developed in [12]. In the latter
type of examples the orbifolds are not isospectral on 1-forms. Along the way we also give several additional examples of isospectral
orbifolds which do not have maximal isotropy groups of different size but other interesting properties.
All three authors were partially supported by DFG Sonderforschungsbereich 647. 相似文献
5.
We construct the first non-trivial examples of compact non-isometric Alexandrov spaces which are isospectral with respect to the Laplacian and not isometric to Riemannian orbifolds. This construction generalizes independent earlier results by the authors based on Schüth’s version of the torus method. 相似文献
6.
We consider an Archimedean analogue of Tate's conjecture, and verify the conjecture in the examples of isospectral Riemann surfaces constructed by Vignéras and Sunada. We prove a simple lemma in group theory which lies at the heart of T. Sunada's theorem about isospectral manifolds. 相似文献
7.
Patrick McDonald Robert Meyers 《Proceedings of the American Mathematical Society》2003,131(11):3589-3599
Given a pair of planar isospectral, nonisometric polygons constructed as a quotient of the plane by a finite group, we construct an associated pair of planar isospectral, nonisometric weighted graphs. Using the natural heat operators on the weighted graphs, we associate to each graph a heat content. We prove that the coefficients in the small time asymptotic expansion of the heat content distinguish our isospectral pairs. As a corollary, we prove that the sequence of exit time moments for the natural Markov chains associated to each graph, averaged over starting points in the interior of the graph, provides a collection of invariants that distinguish isospectral pairs in general.
8.
Julián Fernández Bonder Juan Pablo Pinasco 《Journal of Mathematical Analysis and Applications》2005,308(2):764-774
In this paper we study the spectral counting function of the weighted p-Laplacian in fractal strings, where the weight is allowed to change sign. We obtain error estimates related to the interior Minkowski dimension of the boundary. We also find the asymptotic behavior of eigenvalues. 相似文献
9.
This paper focuses on two cases of two-dimensional wave equations with fractal boundaries. The first case is the equation with classical derivative. The formal solution is obtained. And a definition of the solution is given. Then we prove that under certain conditions, the solution is a kind of fractal function, which is continuous, differentiable nowhere in its domain. Next, for specific given initial position and 3 different initial velocities, the graphs of solutions are sketched. By computing the box dimensions of boundaries of cross-sections for solution surfaces, we evaluate the range of box dimension of the vibrating membrane. The second case is the equation with p-type derivative. The corresponding solution is shown and numerical example is given. 相似文献
10.
Huikun Jiang 《Journal of Mathematical Analysis and Applications》2009,355(1):164-169
In this paper, integrals of second kind over a rectifiable curve or a piecewise smooth surface are extended to continuous fractal curves and surfaces. Theorems for the existence of these integrals are proved. Green's, Gauss' and Stokes' theorems are developed for domains with fractal boundaries. 相似文献
11.
Müller Wolfgang Thuswaldner Jörg M. Tichy Robert F. 《Periodica Mathematica Hungarica》2001,42(1-2):51-68
In this paper we study properties of the fundamental domain F of number systems in the n-dimensional real vector space. In particular we investigate the fractal structure of its boundary F. In a first step we give upper and lower bounds for its box counting dimension. Under certain circumstances these bounds are identical and we get an exact value for the box counting dimension. Under additional assumptions we prove that the Hausdorf dimension of F is equal to its box counting dimension. Moreover, we show that the Hausdorf measure is positive and fnite. This is done by applying the theory of graphdirected self similar sets due to Falconer and Bandt. Finally, we discuss the connection to canonical number systems in number felds, and give some numerical examples. 相似文献
12.
J. B. Kennedy 《Archiv der Mathematik》2018,110(3):261-271
We introduce an analogue of Payne’s nodal line conjecture, which asserts that the nodal (zero) set of any eigenfunction associated with the second eigenvalue of the Dirichlet Laplacian on a bounded planar domain should reach the boundary of the domain. The assertion here is that any eigenfunction associated with the first nontrivial eigenvalue of the Neumann Laplacian on a domain \(\Omega \) with rotational symmetry of order two (i.e. \(x\in \Omega \) iff \(-x\in \Omega \)) “should normally” be rotationally antisymmetric. We give both positive and negative results which highlight the heuristic similarity of this assertion to the nodal line conjecture, while demonstrating that the extra structure of the problem makes it easier to obtain stronger statements: it is true for all simply connected planar domains, while there is a counterexample domain homeomorphic to a disk with two holes. 相似文献
13.
Overlap coincidence in a self-affine tiling in Rd is equivalent to pure point dynamical spectrum of the tiling dynamical system. We interpret the overlap coincidence in the setting of substitution Delone set in Rd and find an efficient algorithm to check the pure point dynamical spectrum. This algorithm is easy to implement into a computer program. We give the program and apply it to several examples. In the course of the proof of the algorithm, we show a variant of the conjecture of Urbański (Solomyak (2006) [40]) on the Hausdorff dimension of the boundaries of fractal tiles. 相似文献
14.
A conjecture of Kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra. In this paper we prove these conjectures for indivisible dimension vectors. Dedicated to Idun Reiten on the occasion of her sixtieth birthdayMathematics Subject Classification (1991) 16G20, 17B67 相似文献
15.
Although the “hot spots” conjecture was proved to be false on some classical domains, the problem still generates a lot of interests on identifying the domains that the conjecture hold. The question can also be asked on fractal sets that admit Laplacians. It is known that the conjecture holds on the Sierpinski gasket and its variants. In this note, we show surprisingly that the “hot spots” conjecture fails on the hexagasket, a typical nested fractal set. The technique we use is the spectral decimation method of eigenvalues of Laplacian on fractals. 相似文献
16.
Julian Edward 《偏微分方程通讯》2013,38(7-8):1249-1270
The Neumann operator maps the boundary value of a harmonic function tc its normal derivative. The inverse spectral properties of the Neumann operator associated to smooth, planar, Jordan curves are studied. The Riemann mapping theorem is used tc parametrize the set of planar Jordan curves by positive functions on the unit circle. By studying the zeta function associated to the spectrum, it is shown that isospectral sets of these functions are pre-compact in the topology of the L2-Sobolev space of order 5/2 - [euro]. Spectral criteria are given for the limiting curves of an isospectral set to be Jordan. A spectrally determined lower bound on the area of the interior of the curve is given. 相似文献
17.
We give a criterion for the Kac conjecture asserting that the free term of the polynomial counting the absolutely indecomposable representations of a quiver over a finite field of given dimension coincides with the corresponding root multiplicity of the associated Kac–Moody algebra. Our criterion suits very well for computer tests. 相似文献
18.
Ruth Gornet 《Journal of Geometric Analysis》2000,10(2):281-298
The purpose of this paper is to present the first continuous families of Riemannian manifolds that are isospectral on functions
but not on 1-forms, and, simultaneously, the first continuous families of Riemannian manifolds with the same marked length
spectrum but not the same 1-form spectrum. Examples of isospectral manifolds that are not isospectral on forms are sparse,
as most examples of isospectral manifolds can be explained by Sunada’s method or its generalizations, hence are strongly isospectral.
The examples here are three-step Riemannian nilmanifolds, arising from a general method for constructing isospectral Riemannian
nilmanifolds previously presented by the author. Gordon and Wilson constructed the first examples of nontrivial isospectral
deformations, continuous families of Riemannian nilmanifolds. Isospectral manifolds constructed using the Gordon-Wilson method,
a generalized Sunada method, are strongly isospectral and must have the same marked length spectrum. Conversely, Ouyang and
Pesce independently showed that all isospectral deformations of two-step nilmanifolds must arise from the Gordon-Wilson method,
and Eberlein showed that all pairs of two-step nilmanifolds with the same marked length spectrum must come from the Gordon-Wilson
method.
To the memory of Hubert Pesce, a valued friend and colleague. 相似文献
19.
Two groups are said to be isospectral if they share the same set of element orders. For every finite simple linear group L
of dimension n over an arbitrary field of characteristic 2, we prove that any finite group G isospectral to L is isomorphic
to an automorphic extension of L. An explicit formula is derived for the number of isomorphism classes of finite groups that
are isospectral to L. This account is a continuation of the second author's previous paper where a similar result was established
for finite simple linear groups L in a sufficiently large dimension (n > 26), and so here we confine ourselves to groups of
dimension at most 26.
Supported by RFBR (project Nos. 08-01-00322 and 06-01-39001), by SB RAS (Integration Project No. 2006.1.2), and by the Council
for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-344.2008.1) and Young Doctors and Candidates
of Science (grants MD-2848.2007.1 and MK-377.2008.1).
Translated from Algebra i Logika, Vol. 47, No. 5, pp. 558–570, September–October, 2008. 相似文献
20.
Spectral Asymtotic Behavior for a Class of Schrodinger Operators on 1-dimensional Fractal Domains 下载免费PDF全文
In this paper, we study the spectral asymptotic behavior for a class of Schrödinger operators on 1-dimensional fractal domains. We have obtained, if the potential function is locally constant, the exact second term of the spectral asymptotics. In general, we give a sharp estimate for the second term of the spectral asymptotics. 相似文献