Results 1  10
of
243
The phase transition in inhomogeneous random graphs, preprint available from http://www.arxiv.org/abs/math.PR/0504589
"... Abstract. The ‘classical ’ random graph models, in particular G(n, p), are ‘homogeneous’, in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, powerlaw degree distributions. ..."
Abstract

Cited by 101 (30 self)
 Add to MetaCart
Abstract. The ‘classical ’ random graph models, in particular G(n, p), are ‘homogeneous’, in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, powerlaw degree distributions. Thus there has been a lot of recent interest in defining and studying ‘inhomogeneous ’ random graph models. One of the most studied properties of these new models is their ‘robustness’, or, equivalently, the ‘phase transition ’ as an edge density parameter is varied. For G(n, p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogenous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this already, and others can be approximated by models with
Quantum Dynamics and Decompositions of Singular Continuous Spectra
 J. Funct. Anal
, 1995
"... . We study relations between quantum dynamics and spectral properties, concentrating on spectral decompositions which arise from decomposing measures with respect to dimensional Hausdorff measures. 1. Introduction Let H be a separable Hilbert space, H : H ! H a self adjoint operator, and / 2 H (wi ..."
Abstract

Cited by 80 (11 self)
 Add to MetaCart
. We study relations between quantum dynamics and spectral properties, concentrating on spectral decompositions which arise from decomposing measures with respect to dimensional Hausdorff measures. 1. Introduction Let H be a separable Hilbert space, H : H ! H a self adjoint operator, and / 2 H (with k/k = 1). The spectral measure ¯/ of / (and H ) is uniquely defined by [24]: h/ ; f(H)/i = Z oe(H) f(x) d¯/ (x) ; (1:1) for any measurable (Borel) function f . The time evolution of the state / , in the Schrodinger picture of quantum mechanics, is given by /(t) = e \GammaiHt / : (1:2) The relations between various properties of the spectral measure ¯/ (with an emphasis on "fractal" properties) and the nature of the time evolution have been the subject of several recent papers [7,13,1518,20,22,33,36,39]. Our purpose in this paper is twofold: First, we use a theory, due to Rogers and Taylor [28,29], of decomposing singular continuous measures with respect to Hausdorff measures to i...
The Divisor of Selberg's Zeta Function for Kleinian Groups
 DUKE MATH. J
, 2000
"... We compute the divisor of Selberg's zeta function for convex cocompact, torsionfree discrete groups ; acting on a real hyperbolic space of dimension n +1. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on X = ;nH n+1 together with the Euler characteristic of X ..."
Abstract

Cited by 67 (8 self)
 Add to MetaCart
We compute the divisor of Selberg's zeta function for convex cocompact, torsionfree discrete groups ; acting on a real hyperbolic space of dimension n +1. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on X = ;nH n+1 together with the Euler characteristic of X compactified to a manifold with boundary.Ifn is even, the singularities of the zeta funciton associated to the Euler characteristic of X are identified using work of Bunke and Olbrich.
Compressive Sensing and Structured Random Matrices
 RADON SERIES COMP. APPL. MATH XX, 1–95 © DE GRUYTER 20YY
"... These notes give a mathematical introduction to compressive sensing focusing on recovery using ℓ1minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to ..."
Abstract

Cited by 59 (13 self)
 Add to MetaCart
These notes give a mathematical introduction to compressive sensing focusing on recovery using ℓ1minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to providing conditions that ensure exact or approximate recovery of sparse vectors using ℓ1minimization.
On the conditioning of random subdictionaries
 Appl. Comput. Harmonic Anal
"... Abstract. An important problem in the theory of sparse approximation is to identify wellconditioned subsets of vectors from a general dictionary. In most cases, current results do not apply unless the number of vectors is smaller than the square root of the ambient dimension, so these bounds are too ..."
Abstract

Cited by 48 (5 self)
 Add to MetaCart
Abstract. An important problem in the theory of sparse approximation is to identify wellconditioned subsets of vectors from a general dictionary. In most cases, current results do not apply unless the number of vectors is smaller than the square root of the ambient dimension, so these bounds are too weak for many applications. This paper shatters the squareroot bottleneck by focusing on random subdictionaries instead of arbitrary subdictionaries. It provides explicit bounds on the extreme singular values of random subdictionaries that hold with overwhelming probability. The results are phrased in terms of the coherence and spectral norm of the dictionary, which capture information about its global geometry. The proofs rely on standard tools from the area of Banach space probability. As an application, the paper shows that the conditioning of a subdictionary is the major obstacle to the uniqueness of sparse representations and the success of ℓ1 minimization techniques for signal recovery. Indeed, if a fixed subdictionary is well conditioned and its cardinality is slightly smaller than the ambient dimension, then a random signal formed from this subdictionary almost surely has no other representation that is equally sparse. Moreover, with overwhelming probability, the maximally sparse representation can be identified via ℓ1 minimization. Note that the results in this paper are not directly comparable with recent work on subdictionaries of random dictionaries. 1.
R.: Derivation of the GrossPitaevskii Equation for rotating Bose gases
 Comm. Math. Phys
, 2006
"... We prove that the GrossPitaevskii equation correctly describes the ground state energy and corresponding oneparticle density matrix of rotating, dilute, trapped Bose gases with repulsive twobody interactions. We also show that there is 100 % BoseEinstein condensation. While a proof that the GP e ..."
Abstract

Cited by 35 (4 self)
 Add to MetaCart
We prove that the GrossPitaevskii equation correctly describes the ground state energy and corresponding oneparticle density matrix of rotating, dilute, trapped Bose gases with repulsive twobody interactions. We also show that there is 100 % BoseEinstein condensation. While a proof that the GP equation correctly describes nonrotating or slowly rotating gases was known for some time, the rapidly rotating case was unclear because the Bose (i.e., symmetric) ground state is not the lowest eigenstate of the Hamiltonian in this case. We have been able to overcome this difficulty with the aid of coherent states. Our proof also conceptually simplifies the previous proof for the slowly rotating case. In the case of axially symmetric traps, our results show that the appearance of quantized vortices causes spontaneous symmetry breaking in the ground state. 1
Vishik: Determinants of elliptic pseudo–differential operators
, 1994
"... Abstract. Determinants of invertible pseudodifferential operators (PDOs) close to positive selfadjoint ones are defined through the zetafunction regularization. We define a multiplicative anomaly as the ratio det(AB)/(det(A)det(B)) considered as a function on pairs of elliptic PDOs. We obtained a ..."
Abstract

Cited by 34 (1 self)
 Add to MetaCart
Abstract. Determinants of invertible pseudodifferential operators (PDOs) close to positive selfadjoint ones are defined through the zetafunction regularization. We define a multiplicative anomaly as the ratio det(AB)/(det(A)det(B)) considered as a function on pairs of elliptic PDOs. We obtained an explicit formula for the multiplicative anomaly in terms of symbols of operators. For a certain natural class of PDOs on odddimensional manifolds generalizing the class of elliptic differential operators, the multiplicative anomaly is identically 1. For elliptic PDOs from this class a holomorphic determinant and a determinant for zero orders PDOs are introduced. Using various algebraic, analytic, and topological tools we study local and global properties of the multiplicative anomaly and of the determinant Lie group closely related with it. The Lie algebra for the determinant Lie group has a description in terms of symbols only. Our main discovery is that there is a quadratic nonlinearity hidden in the definition of determinants of PDOs through zetafunctions. The natural explanation of this nonlinearity follows from complexanalytic properties of a new trace functional TR on PDOs of noninteger orders. Using TR we easily reproduce known facts about noncommutative residues of PDOs and obtain several new results. In particular, we describe a structure of derivatives of zetafunctions at zero as of functions on logarithms of elliptic PDOs. We propose several definitions extending zetaregularized determinants to general elliptic PDOs. For elliptic PDOs of nonzero complex orders we introduce a canonical determinant in its natural domain of definition. Contents