25 research outputs found
The Two-Spectra Inverse Problem for Semi-Infinite Jacobi Matrices in The Limit-Circle Case
We present a technique for reconstructing a semi-infinite Jacobi operator in
the limit circle case from the spectra of two different self-adjoint
extensions. Moreover, we give necessary and sufficient conditions for two real
sequences to be the spectra of two different self-adjoint extensions of a
Jacobi operator in the limit circle case.Comment: 26 pages. Changes in the presentation of some result
Glauber dynamics for the quantum Ising model in a transverse field on a regular tree
Motivated by a recent use of Glauber dynamics for Monte-Carlo simulations of
path integral representation of quantum spin models [Krzakala, Rosso,
Semerjian, and Zamponi, Phys. Rev. B (2008)], we analyse a natural Glauber
dynamics for the quantum Ising model with a transverse field on a finite graph
. We establish strict monotonicity properties of the equilibrium
distribution and we extend (and improve) the censoring inequality of Peres and
Winkler to the quantum setting. Then we consider the case when is a regular
-ary tree and prove the same fast mixing results established in [Martinelli,
Sinclair, and Weitz, Comm. Math. Phys. (2004)] for the classical Ising model.
Our main tool is an inductive relation between conditional marginals (known as
the "cavity equation") together with sharp bounds on the operator norm of the
derivative at the stable fixed point. It is here that the main difference
between the quantum and the classical case appear, as the cavity equation is
formulated here in an infinite dimensional vector space, whereas in the
classical case marginals belong to a one-dimensional space
Bitangential interpolation in generalized Schur classes
Bitangential interpolation problems in the class of matrix valued functions
in the generalized Schur class are considered in both the open unit disc and
the open right half plane, including problems in which the solutions is not
assumed to be holomorphic at the interpolation points. Linear fractional
representations of the set of solutions to these problems are presented for
invertible and singular Hermitian Pick matrices. These representations make use
of a description of the ranges of linear fractional transformations with
suitably chosen domains that was developed in a previous paper.Comment: Second version, corrected typos, changed subsection 5.6, 47 page
Stability and convergence in discrete convex monotone dynamical systems
We study the stable behaviour of discrete dynamical systems where the map is
convex and monotone with respect to the standard positive cone. The notion of
tangential stability for fixed points and periodic points is introduced, which
is weaker than Lyapunov stability. Among others we show that the set of
tangentially stable fixed points is isomorphic to a convex inf-semilattice, and
a criterion is given for the existence of a unique tangentially stable fixed
point. We also show that periods of tangentially stable periodic points are
orders of permutations on letters, where is the dimension of the
underlying space, and a sufficient condition for global convergence to periodic
orbits is presented.Comment: 36 pages, 1 fugur
Towards Machine Wald
The past century has seen a steady increase in the need of estimating and
predicting complex systems and making (possibly critical) decisions with
limited information. Although computers have made possible the numerical
evaluation of sophisticated statistical models, these models are still designed
\emph{by humans} because there is currently no known recipe or algorithm for
dividing the design of a statistical model into a sequence of arithmetic
operations. Indeed enabling computers to \emph{think} as \emph{humans} have the
ability to do when faced with uncertainty is challenging in several major ways:
(1) Finding optimal statistical models remains to be formulated as a well posed
problem when information on the system of interest is incomplete and comes in
the form of a complex combination of sample data, partial knowledge of
constitutive relations and a limited description of the distribution of input
random variables. (2) The space of admissible scenarios along with the space of
relevant information, assumptions, and/or beliefs, tend to be infinite
dimensional, whereas calculus on a computer is necessarily discrete and finite.
With this purpose, this paper explores the foundations of a rigorous framework
for the scientific computation of optimal statistical estimators/models and
reviews their connections with Decision Theory, Machine Learning, Bayesian
Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty
Quantification and Information Based Complexity.Comment: 37 page
Time-homogeneous diffusions with a given marginal at a random time
We solve explicitly the following problem: for a given probability measure μ, we specify a generalised martingale diffusion (Xt) which, stopped at an independent exponential time T, is distributed according to μ. The process (Xt) is specified via its speed measure m. We present two heuristic arguments and three proofs. First we show how the result can be derived from the solution of [Bertoin and Le Jan, Ann. Probab. 20 (1992) 538–548.] to the Skorokhod embedding problem. Secondly, we give a proof exploiting applications of Krein's spectral theory of strings to the study of linear diffusions. Finally, we present a novel direct probabilistic proof based on a coupling argument
The exponential type of the fundamental solution of an indefinite Hamiltonian system
The fundamental solution of a Hamiltonian system whose Hamiltonian H is positive definite and locally integrable is an entire function of exponential type. Its exponential type can be computed as the integral over . We show that this formula remains true in the indefinite (Pontryagin space) situation, where the Hamiltonian is permitted to have finitely many inner singularities. As a consequence, we obtain a statement on non-cancellation of exponential growth for a class of entire matrix functions
Extending Wavelet Filters. Infinite Dimensions, the Non-Rational Case, and Indefinite-Inner Product Spaces
In this paper we are discussing various aspects of wavelet filters. While there are earlier studies of these filters as matrix valued functions in wavelets, in signal processing, and in systems, we here expand the framework. Motivated by applications, and by bringing to bear tools from reproducing kernel theory, we point out the role of non-positive definite Hermitian inner products (negative squares), for example Krein spaces, in the study of stability questions. We focus on the nonrational case, and establish new connections with the theory of generalized Schur functions and their associated reproducing kernel Pontryagin spaces, and the Cuntz relations