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41.
On the spectra of a Cantor measure 总被引:1,自引:0,他引:1
We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75 (1998) 185-228. A complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one-to-one correspondence with the spectral labelings of the infinite binary tree. With the help of this characterization we obtain a sufficient condition for a spectral labeling to generate a spectrum (an orthonormal basis). This result not only provides us an easy and efficient way to construct various of new spectra for the Cantor measure but also extends many previous results in the literature. In fact, most known examples of orthonormal bases of exponentials correspond to spectral labelings satisfying this sufficient condition. We also obtain two new conditions for a labeling tree to generate a spectrum when other digits (digits not necessarily in {0,1,2,3}) are used in the base 4 expansion of integers and when bad branches are allowed in the spectral labeling. These new conditions yield new examples of spectra and in particular lead to a surprizing example which shows that a maximal set of orthogonal exponentials is not necessarily an orthonormal basis. 相似文献
42.
43.
Let d be a positive integer, and let μ be a finite measure on ? d . In this paper we ask when it is possible to find a subset Λ in ? d such that the corresponding complex exponential functions e λ indexed by Λ are orthogonal and total in L 2(μ). If this happens, we say that (μ,Λ) is a spectral pair. This is a Fourier duality, and the x-variable for the L 2(μ)-functions is one side in the duality, while the points in Λ is the other. Stated this way, the framework is too wide, and we shall restrict attention to measures μ which come with an intrinsic scaling symmetry built in and specified by a finite and prescribed system of contractive affine mappings in ? d ; an affine iterated function system (IFS). This setting allows us to generate candidates for spectral pairs in such a way that the sets on both sides of the Fourier duality are generated by suitably chosen affine IFSs. For a given affine setup, we spell out the appropriate duality conditions that the two dual IFS-systems must have. Our condition is stated in terms of certain complex Hadamard matrices. Our main results give two ways of building higher dimensional spectral pairs from combinatorial algebra and spectral theory applied to lower dimensional systems. 相似文献
44.
We introduce a Fourier-based harmonic analysis for a class of discrete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space comes with a finite-to-one endomorphism which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in , this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets in of the same cardinality which generate complex Hadamard matrices.
Our harmonic analysis for these iterated function systems (IFS) is based on a Markov process on certain paths. The probabilities are determined by a weight function on . From we define a transition operator acting on functions on , and a corresponding class of continuous -harmonic functions. The properties of the functions in are analyzed, and they determine the spectral theory of . For affine IFSs we establish orthogonal bases in . These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in .
45.
We give the spectral representation for a class of selfadjoint discrete graph Laplacians Δ, with Δ depending on a chosen graph
G and a conductance function c defined on the edges of G. We show that the spectral representations for Δ fall in two model classes, (1) tree-graphs with N-adic branching laws, and (2) lattice graphs. We show that the spectral theory of the first class may be computed with the
use of rank-one perturbations of the real part of the unilateral shift, while the second is analogously built up with the
use of the bilateral shift. We further analyze the effect on spectra of the conductance function c: How the spectral representation of Δ depends on c. 相似文献
46.
V. A. Dorin 《Russian Physics Journal》1973,16(11):1485-1488
It is shown that a selenium diode biased in the reverse direction is characterized by several features special to MIS structures. The effect of ionic processes in the MIS structure on the volt-amp characteristics is studied.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 7–11, November, 1973. 相似文献
47.
48.
Dorin Ervin Dutkay Palle E. T. Jorgensen 《Journal of Fourier Analysis and Applications》2013,19(3):467-477
We show that the spectral-tile implication in the Fuglede conjecture in dimension 1 is equivalent to a Universal Tiling Conjecture and also to similar forms of the same implication for some simpler sets, such as unions of intervals with rational or integer endpoints. 相似文献
49.
Jennifer M. Kashmirian Alfred Uhlherr Alan Dorin David G. Green 《Journal of computational chemistry》2012,33(15):1364-1373
Molecular simulation models are increasingly important tools in efforts to understand the role that water plays in biochemical processes. However, existing models of water have limited capacity to deal with the characteristics of hydrogen bond networks. This article proposes a new fluctuating network (FN) algorithm as an extension of the standard molecular dynamics algorithm. The new algorithm allows for the simulation of a molecular system based on an underlying network, such as the hydrogen bond network in water. This algorithm distinguishes strong from weak network connections, applying a potential that best describes the specific connection behavior. We model liquid water with this new technique using a single‐site, isotropic, short‐range potential. We successfully reproduce liquid water's signature molecular spacing (as represented by the radial distribution function) and characterize its dynamic properties including the exponential hydrogen bond lifetime distribution, diffusion rate, and average hydrogen bonds per molecule. The FN algorithm allows exploration of the behavior of networked systems where explicit coordination limits are required. As such it could also be used to model covalent interactions, reaction dynamics, and applied to simulation of cellular networks. © 2012 Wiley Periodicals, Inc. 相似文献
50.
Let k be an algebraically closed field of characteristic 3 and i, j, t some positive integers such that 1 i < j < t, i + j t. Then there exist a finite number of nonisomorphic indecomposable maximal Cohen–Macaulay modules N over k[[x, y]] /(xt + y3) such that N / y N is a direct sum of copies of k[[x]] /(xi), k[[x]] /(xj) and we describe them completely. 相似文献