relative entropy and the multivariable multidimensional moment problem entropy like functionals on operator algebras have been studied since the pioneering work of von neumann umegaki lindblad and lieb the best known are the von neumann entropy rho trace rhologrho and a generalization of the kullback leibler distance s rhoparsigma trace rhologrho rhologsigma referred to as quantum relative entropy and used to quantify distance between states of a quantum system the purpose of this paper is to explore and s as regularizing functionals in seeking solutions to multivariable and multidimensional moment problems it will be shown that extrema can be effectively constructed via a suitable homotopy the homotopy approach leads naturally to a further generalization and a description of all the solutions to such moment problems this is accomplished by a renormalization of a riemannian metric induced by entropy functionals as an application we discuss the inverse problem of describing power spectra which are consistent with second order statistics which has been the main motivation behind the present work

analytic interpolation with a degree constraint for matrix valued functions we consider a nehari problem for matrix valued positive real functions and characterize the class of generically minimal degree solutions analytic interpolation problems such as the one studied herein for positive real functions arise in time series modeling and system identification the degree of positive real interpolants relates to the dimension of models and to the degree of matricial power spectra of vector valued time series the main result of the paper generalizes earlier results in scalar analytic interpolation with a degree constraint where the class of generically minimal degree solutions is characterized by an arbitrary choice of spectral zeros naturally in the current matricial setting there is freedom in assigning the jordan structure of the spectral zeros of the power spectrum i e the spectral zeros as well as their respective invariant subspaces the characterization utilizes rosenbrock s theorem on assignability of dynamics via linear state feedback

many to one stable matching geometry and fairness baiou and balinski characterized the stable admissions polytope using a system of linear inequalities the structure of feasible solutions to this system of inequalities fractional stable matchings is the focus of this paper the main result associates a geometric structure with each fractional stable matching this insight appears to be interesting in its own right and can be viewed as a generalization of the lattice structure for integral stable matchings to fractional stable matchings in addition to obtaining simple proofs of many known results the geometric structure is used to prove the following two results first it is shown that assigning each agent their median choice among all stable partners results in a stable matching which can be viewed as a fair compromise second sufficient conditions are identified under which stable matchings exist in a problem with externalities in particular in the stable matching problem with couples

stable matchings optimal assignments and linear programming vande vate 1989 described the polytope whose extreme points are the stable core matchings in the marriage problem rothblum 1989 simplified and extended this result this paper explores a corresponding linear program its dual and consequences of the fact that the dual solutions have an unusually direct relation to the primal solutions this close relationship allows us to provide simple proofs both of vande vate and rothblum s results and of other important results about the core of marriage markets these proofs help explain the structure shared by the marriage problem without sidepayments and the assignment game with sidepayments the paper further explores fractional matchings which may be interpreted as lotteries over possible matches or as time sharing arrangements we show that those fractional matchings in the stable marriage polytope form a lattice with respect to a partial ordering that involves stochastic dominance thus all expected utility functions corresponding to the same ordinal preferences will agree on the relevant comparisons finally we provide linear programming proofs of slightly stronger versions of known incentive compatibility results

relative entropy and the multi variable multi dimensional moment problem author s georgiou tryphon t abstract entropy like functionals on operator algebras have been studied since the pioneering work of von neumann umegaki lindblad and lieb the most well known are the von neumann entropy trace rho log rho and a generalization of the kullback leibler distance trace rho log rho rho log sigma refered to as quantum relative entropy and used to quantify distance between states of a quantum system the purpose of this paper is to explore these as regularizing functionals in seeking solutions to multi variable and multi dimensional moment problems it will be shown that extrema can be effectively constructed via a suitable homotopy the homotopy approach leads naturally to a further generalization and a description of all the solutions to such moment problems this is accomplished by a renormalization of a riemannian metric induced by entropy functionals as an application we discuss the inverse problem of describing power spectra which are consistent with second order statistics which has been the main motivation behind the present work

characterization of stable matchings as extreme points of a polytope the purpose of this paper is to extend a modified version of a recent result of vande vate 1989 which characterizes stable matchings as the extreme points of a certain polytope our proofs are simpler and more transparent than those of vande vate

blockers and antiblockers of stable matchings an implicit linear description of the stable matching polytope is provided in terms of the blocker and antiblocker sets of constraints of the matroid kernel polytope the explicit identification of both these sets is based on a partition of the stable pairs in which each agent participates here we expose the relation of such a partition to rotations we provide a time optimal algorithm for obtaining such a partition and establish some new related results most importantly that this partition is unique

bounds for best constants in subcritical sobolev embeddings abstract we establish upper and lower estimates for the embedding constants related to the classical sobolev embeddings h 0 1 l p where r n is a bounded domain or the whole r n and p 2 2 n n 2 n 3 in dimension n 2 we also derive the sharp asymptotic behavior of the embedding constant as p

the stable allocation or ordinal transportation problem the stable allocation problem generalizes the 0 1 stable matching problems one to one one to many and many to many to the allocation of real valued hours or quantities a strongly polynomial algorithm proves the existence of stable allocations the set of stable allocations is shown to be a distributive lattice in general but in the nondegenerate case it is a complete linear order indeed in the generic case when a problem is strongly nondegenerate there exists a single stable allocation a simple algorithm finds row optimal and column optimal stable allocations given any stable allocation when a problem is nondegenerate it finds all stable allocations

bose fluids and positive solutions to weakly coupled systems with critical growth in dimension two abstract we prove using variational methods the existence in dimension two of positive vector ground states solutions for the bose einstein type systems u 1 u 1 u e u 2 1 v e u v 1 in v 2 v 2 v e v 2 1 u e u v 1 in u v h 0 1 where is a bounded smooth domain 1 2 1 the first eigenvalue of h 0 1 1 2 0 and is either positive small or large or negative small the nonlinear interaction between two bose fluids is assumed to be of critical exponential type in the sense of j moser for small solutions the system is asymptotically equivalent to the corresponding one in higher dimensions with power like nonlinearities

of stable marriages and graphs and strategy and polytopes this expository paper develops the principal known results and some new ones on the stable matchings of marriage games in the language of directed graphs this both unifies and simplifies the presentation and renders it more symmetric in addition it yields a new algorithm and a new proof for the existence of stable matchings new proofs for many known facts and some new results notably concerning players strategies and the properties of the stable matching polytope

equivalent moser type inequalities in r2 and the zero mass case abstract we first investigate concentration and vanishing phenomena concerning moser type inequalities in the whole plane which involve complete and reduced sobolev norms in particular we show that the critical ruf inequality is equivalent to an improved version of the subcritical adachi tanaka inequality which we prove to be attained then we consider smooth compactly supported functions with respect to the dirichlet norm 2 and we prove an optimal lorentz zygmund type inequality with explicit extremals and from which can be derived classical inequalities in h 1 r 2 such as the adachi tanaka inequality and a version of ruf s inequality

a new approach to spectral estimation a tunable high resolution spectral estimator traditional maximum entropy spectral estimation determines a power spectrum from covariance estimates here we present a new approach to spectral estimation which is based on the use of filter banks as a means of obtaining spectral interpolation data such data replaces standard covariance estimates a computational procedure for obtaining suitable pole zero arma models from such data is presented the choice of the zeros ma part of the model is completely arbitrary by suitable choices of filter bank poles and spectral zeros the estimator can be tuned to exhibit high resolution in targeted regions of the spectrum

information matrix and d optimal design with gaussian inputs for wiener model identification we present a closed form expression for the fischer s information matrix associated with the identification of wiener models in the derivation we assume that the input signal is gaussian the analysis allows the linear sub system in the wiener model to have a generic rational transfer function of arbitrary order it also allows the static nonlinearity of the wiener model to be a polynomial of arbitrary degree in addition we show how this analysis can be used to design tractable algorithms for d optimal input design the idea is further extended to design optimal inputs consisting of a sequence of gaussian signals with different mean values and variances by combining gaussian inputs with different means we can tune the amplitude distribution of the input to achieve the best identification accuracy in d optimal sense the analytical results are also illustrated with some numerical simulations

signal estimation via selective harmonic amplification music redux the technique known as multiple signal classification music is a semi empirical way to obtain pseudo spectra that highlight the spectral energy distribution of a time series it is based on a certain canonical decomposition of a toeplitz matrix formed out of an estimated autocorrelation sequence the purpose of this paper is to present an analogous canonical decomposition of the state covariance matrix of a stable linear filter driven by a given time series accordingly the paper concludes with a modification of music the new method starts with filtering the time series and then estimating the covariance of the state of the filter this step in essence improves the signal to noise ratio snr by amplifying the contribution to the actual value of the state covariance of a selected harmonic interval where spectral lines are expected to reside then the method capitalizes on the canonical decomposition of the filter state covariance to retrieve information on the location of possible spectral lines the framework requires uniformly spaced samples of the process

ground states for a system of schrodinger equations with critical exponent abstract we study the following system of nonlinear schrodinger equations u u u p 1 u v x r n v v v 2 2 v u x r n where n 3 2 2 n n 2 1 p 2 1 and are positive parameters satisfying 0 we show that there is some critical value 0 0 1 such that this system has a positive ground state solution if 0 0 in the case 0 there exists 0 such that this system has no ground state solutions if while this system has a positive ground state solution if in particular if p 2 1 the system has no nontrivial solutions some further properties of the ground state solutions are also studied this seems to be the first result for such a critical schrodinger system

spectral estimation via selective harmonic amplification the state covariance of a linear filter is characterized by a certain algebraic commutativity property with the state matrix of the filter and also imposes a generalized interpolation constraint on the power spectrum of the input process this algebraic property and the relationship between state covariance and the power spectrum of the input allow the use of matrix pencils and analytic interpolation theory for spectral analysis several algorithms for spectral estimation are developed with resolution higher than state of the art

the stable admissions polytope the stable admissions polytope the convex hull of the stable assignments of the university admissions problem is described by a set of linear inequalities it depends on a new characterization of stability and arguments that exploit and extend a graphical approach that has been fruitful in the analysis of the stable marriage problem

the structure of state covariances and its relation to the power spectrum of the input we study the relationship between power spectra of stationary stochastic inputs to a linear filter and the corresponding state covariances and identify the structure of positive semidefinite matrices that qualify as state covariances of the filter this structure is best revealed by a rank condition pertaining to the solvability of a linear equation involving the state covariance and the system matrices we then characterize all input power spectra consistent with any specific state covariance the parametrization of input spectra is achieved through a relation to solutions of an analytic interpolation problem which is analogous but not equivalent to a matricial nehari problem

a maximum entropy enhancement for a family of high resolution spectral estimators structured covariances occurring in spectral analysis filtering and identification need to be estimated from a finite observation record the corresponding sample covariance usually fails to possess the required structure this is the case for instance in the byrnes georgiou lindquist three like tunable high resolution spectral estimators there the output covariance of a linear filter is needed to initialize the spectral estimation technique the sample covariance estimate however is usually not compatible with the filter in this paper we present a new systematic way to overcome this difficulty the new estimate is obtained by solving an ancillary problem with an entropic type criterion extensive scalar and multivariate simulation shows that this new approach consistently leads to a significant improvement of the spectral estimators performances

erratum the stable allocation or ordinal transportation problem an incorrect version of this paper was printed in mathematics of operations research vol 27 no 3 august 2002 pp 485 503 the paper is reprinted correctly below the stable allocation problem generalizes the 0 1 stable matching problems one to one one to many and many to many to the allocation of real valued hours or quantities a strongly polynomial algorithm proves the existence of stable allocations the set of stable allocations is shown to be a distributive lattice in general but in the nondegenerate case it is a complete linear order indeed in the generic case when a problem is strongly nondegenerate there exists a single stable allocation a simple algorithm finds row optimal and column optimal stable allocations given any stable allocation when a problem is nondegenerate it finds all stable allocations

solution of the general moment problem via a one parameter imbedding this paper presents a computational theory for the general scalar moment problem the formalism is sufficiently general to encompass problems in sensor arrays with arbitrary geometry and dynamics and in nonuniform multidimensional sampling given a finite set of moments the theory provides a test for the existence of a positive measure which is consistent with such data at the same time the theory also provides a characterization of all such consistent positive measures it should be noted that classical results e g in the theory of the trigonometric moment problem hamburger stieljes nevanlinna pick interpolation etc are not applicable to the general setting sought herein where there is no natural shift operator in the space spanned by the integration kernels the centerpiece of the theory is a differential equation which depends on the given finite set of moments and on an arbitrary positive function spl psi which plays the role of a free parameter the differential equation has an exponentially attractive point of equilibrium if and only if there exists a consistent positive measure for each spl psi the fixed point determines a corresponding measure suitable choice of spl psi allows recovering any measure which is consistent with the data the fixed point of the differential equation corresponds to an extremum of an entropy like functional and the differential equation is constructed via an appropriate homotopy that follows changes in the lagrange multipliers from a convenient starting value to a value for the multipliers that corresponds to the given moments

time and spectral domain relative entropy a new approach to multivariate spectral estimation the concept of spectral relative entropy rate is introduced for jointly stationary gaussian processes using classical information theoretic results we establish a remarkable connection between time and spectral domain relative entropy rates this naturally leads to a new spectral estimation technique where a multivariate version of the itakura saito distance is employed it may be viewed as an extension of the approach called three introduced by byrnes georgiou and lindquist in 2000 which in turn followed in the footsteps of the burg jaynes maximum entropy method spectral estimation is here recast in the form of a constrained spectrum approximation problem where the distance is equal to the processes relative entropy rate the corresponding solution entails a complexity upper bound which improves on the one so far available in the multichannel framework indeed it is equal to the one featured by three in the scalar case the solution is computed via a globally convergent matricial newton type algorithm simulations suggest the effectiveness of the new technique in tackling multivariate spectral estimation tasks especially in the case of short data records