chaos in the mixed even spin models we consider a disordered system obtained by coupling two mixed even spin models together the chaos problem is concerned with the behavior of the coupled system when the external parameters in the two models such as temperature disorder or external field are slightly different it is conjectured that the overlap between two independently sampled spin configurations from respectively the gibbs measures of the two models is essentially concentrated around a constant under the coupled gibbs measure using the extended guerra replica symmetry breaking bound together with a recent development of controlling the overlap using the ghirlanda guerra identities as well as a new family of identities we present rigorous results on chaos in temperature in addition chaos in disorder and in external field are addressed

on the overlap in the multiple spherical sk models in order to study certain questions concerning the distribution of the overlap in sherrington kirkpatrick type models such as the chaos and ultrametricity problems it seems natural to study the free energy of multiple systems with constrained overlaps one can write analogues of guerra s replica symmetry breaking bound for such systems but it is not at all obvious how to choose informative functional order parameters in these bounds we were able to make some progress for spherical pure p spin sk models where many computations can be made explicitly for pure 2 spin model we prove ultrametricity and chaos in an external field for the pure p spin model for even p 4 without an external field we describe two possible values of the overlap of two systems at different temperatures we also prove a somewhat unexpected result which shows that in the 2 spin model the support of the joint overlap distribution is not always witnessed at the level of the free energy and for example ultrametricity holds only in a weak sense

the parisi formula in this chapter we prove the parisi formula which gives the limiting value of the free energy per site for the sherrington kirkpatrick model at each temperature starting with the famous result of guerra that the parisi formula provided a bound for this free energy we make full use of poisson dirichlet cascades we then use the same techniques to prove in a strong sense that in presence of an external field the overlaps are positive

a closed form solution to the discrete time kalman filter and its applications abstract this paper presents a closed form solution to the discrete time kalman filter and its applications we first represent the kalman filter in terms of model parameters without using the riccati equation and requiring any artificial conditions such as invertibility of a system matrix and no system noises replacing the initial time with the fixed lag time to achieve the finite memory with respect to inputs and outputs and choosing the proper initial covariances on the recent finite horizon we easily obtain the minimum variance finite memory filter and then propose its iterative computation algorithm as another application a closed form solution to the difference riccati equation on the finite horizon is utilized to obtain a stabilizing gain matrix of a luenberger type filter as in ackermann s formula

eigenvalues of symmetrical matrices and graphs abstract we investigate the class c of integral domains a having the property that each totally real integral element over a is an eigenvalue of a symmetric matrix over a c is shown to contain all dedekind domains and any domain in which 1 is a sum of squares in the case a z these results imply that any totally real algebraic integer is the eigenvalue of the adjacency matrix of some regular graph thus affirming a conjecture of hoffman

an extension on a new formulation of the algebraic riccati equation problem in a recent paper 3 by utilizing the square root of a matrix approach a new formulation of the algebraic riccati equation has been presented and a closed form solution has been obtained under the hypothesis that certain commutativity conditions are fulfilled on a transformed space in this note the imposed conditions are weakened and the formulation is extended to a wider class of riccati problems

on the energy landscape of the mixed even p spin model we investigate the energy landscape of the mixed even p spin model with ising spin configurations we show that for any given energy level between zero and the maximal energy with overwhelming probability there exist exponentially many distinct spin configurations such that their energies stay near this energy level furthermore their magnetizations and overlaps are concentrated around some fixed constants in particular at the level of maximal energy we prove that the hamiltonian exhibits exponentially many orthogonal peaks this improves the results of chatterjee disorder chaos and multiple valleys in spin glasses 2009 and ding et al ann probab 43 6 3468 3493 2015 where the former established a logarithmic size of the number of the orthogonal peaks while the latter proved a polynomial size our second main result obtains disorder chaos at zero temperature and at any external field as a byproduct this implies that the fluctuation of the maximal energy is superconcentrated when the external field vanishes and obeys a gaussian limit law when the external field is present

broken replica symmetry bounds in the mean field spin glass model by using a simple interpolation argument in previous work we have proven the existence of the thermodynamic limit for mean field disordered models including the sherrington kirkpatrick model and the derrida p spin model here we extend this argument in order to compare the limiting free energy with the expression given by the parisi ansatz and including full spontaneous replica symmetry breaking our main result is that the quenched average of the free energy is bounded from below by the value given in the parisi ansatz uniformly in the size of the system moreover the difference between the two expressions is given in the form of a sum rule extending our previous work on the comparison between the true free energy and its replica symmetric sherrington kirkpatrick approximation we give also a variational bound for the infinite volume limit of the ground state energy per site

approximate and limit results for nonlinear filters with small observation noise the linear sensor and constant diffusion coefficient case recursive approximations for a class of filtering problems are presented this class is characterized by linear observation sensor constant diffusion terms and for the multidimensional problem potential like conditions on the drift for the case of small observation noise these approximations are used to demonstrate the gaussian limiting structure of the optimal nonlinear filter

on closed form solutions for the differential matrix riccati equation problem a new formulation of the differential matrix riccati equation is presented and a closed analytical solution is obtained under the hypothesis that certain commutativity conditions are fulfilled on a transformed space the formulation generalizes the results of 1 on algebraic equations to differential matrix riccati equations to illustrate the usefulness of the method a closed analytical solution of the differential matrix riccati equation is obtained in r 2 times 2

characteristic polynomials of symmetric matrices over the univariate polynomial ring abstract viewing a bivariate polynomial f r x t as a family of univariate polynomials in t parametrized by real numbers x we call f real rooted if this family consists of monic polynomials with only real roots if f is the characteristic polynomial of a symmetric matrix with entries in r x it is obviously real rooted in this article the converse is established namely that every real rooted bivariate polynomial is the characteristic polynomial of a symmetric matrix over the univariate real polynomial ring as a byproduct we present a purely algebraic proof of the helton vinnikov theorem which solved the 60 year old lax conjecture on the existence of definite determinantal representation of ternary hyperbolic forms

minimal polynomials of integral symmetric matrices abstract we investigate what the possible minimal polynomials are for integral symmetric matrices we show that the obvious necessary conditions are sufficient for polynomials of degree at most 4 we show that necessary conditions are sufficient for minimal polynomials of self adjoint operators on positive definite unimodular lattices we also give a relatively elementary proof of the result of estes that any total real algebraic integer is the eigenvalue of an integral symmetric matrix this question was asked by alan hoffman who also showed that this result implies that any totally real algebraic integer is the eigenvalue of the adjacency matrix of some graph

free energy of the spherical mean field model we compute at any temperature the free energy of the multi p spin spherical model when only terms for p even are considered

aging of spherical spin glasses sompolinski and zippelius 1981 propose the study of dynamical systems whose invariant measures are the gibbs measures for hard to analyze statistical physics models of interest in the course of doing so physicists often report of an aging phenomenon for example aging is expected to happen for the sherrington kirkpatrick model a disordered mean field model with a very complex phase transition in equilibrium at low temperature we shall study the langevin dynamics for a simplified spherical version of this model the induced rotational symmetry of the spherical model reduces the dynamics in question to an n dimensional coupled system of ornstein uhlenbeck processes whose random drift parameters are the eigenvalues of certain random matrices we obtain the limiting dynamics for n approaching infinity and by analyzing its long time behavior explain what is aging mathematically speaking what causes this phenomenon and what is its relationship with the phase transition of the corresponding equilibrium invariant measures

disorder chaos in the sherrington kirkpatrick model with external field we consider a spin system obtained by coupling two distinct sherrington kirkpatrick sk models with the same temperature and external field whose hamiltonians are correlated the disorder chaos conjecture for the sk model states that the overlap under the corresponding gibbs measure is essentially concentrated at a single value in the absence of external field this statement was first confirmed by chatterjee disorder chaos and multiple valleys in spin glasses 2009 preprint in the present paper using guerra s replica symmetry breaking bound we prove that the sk model is also chaotic in the presence of the external field and the position of the overlap is determined by an equation related to guerra s bound and the parisi measure

an approach to chaos in some mixed p spin models we consider the problems of chaos in disorder and temperature for coupled copies of the mixed p spin models under certain assumptions on the parameters of the models first we prove a weak form of chaos by showing that the overlap is concentrated around its gibbs average depending on the disorder and then obtain several results toward strong chaos by providing control of the overlap between two systems in terms of their parisi measures

a new formulation of the algebraic riccati equation problem a new formulation of the algebraic riccati equation is presented and a closed form solution is obtained under the hypothesis that the coefficient matrices fulfill certain commutativity conditions on a transformed space the formulation extends the results of 1 to a wider class of riccati problems

eigenvalues of symmetric matrices over integral domains abstract given an integral domain a we consider the set of all integral elements over a that can occur as an eigenvalue of a symmetric matrix over a we give a sufficient criterion for being such an element in the case where a is the ring of integers of an algebraic number field this sufficient criterion is also necessary and we show that the size of matrices grows linearly in the degree of the element the latter result settles a questions raised by bass estes and guralnick

the geometry of the gibbs measure of pure spherical spin glasses we analyze the statics for pure p spin spherical spin glass models with p ge 3 at low enough temperature with f n beta denoting the free energy we compute the logarithmic second order term of nf n beta and prove that for an appropriate centering c n beta nf n beta c n beta is a tight sequence we further establish the absence of temperature chaos those results follow from the following geometric picture we prove for the gibbs measure of interest by itself asymptotically the measure splits into infinitesimal spherical bands centered at deep minima playing the role of pure states for the pure models the latter makes precise the picture of many valleys separated by high mountains and significant parts of the tap analysis from the physics literature