generalized entropic uncertainty relations a new class of uncertainty relations is derived for pairs of observables in a finite dimensional hilbert space which do not have any common eigenvector this class contains an entropic uncertainty relation which improves a previous result of deutsch and confirms a recent conjecture by kraus some comments are made on the extension of these relations to the case where the hilbert space is infinite dimensional

fisher information prior probabilities and the state determination of spin 1 2 and spin 1 systems the fisher information matrix i2p for a p p density matrix is taken to be the 2p2 p dimensional information matrix of a 2p variate real normal distribution with an a priori zero mean vector and a covariance matrix 2p the two diagonal blocks of which are formed from the symmetric real component of and the two off diagonal blocks from the skew symmetric imaginary part of the inverse square root ni2p 1 2 where n is the number of observations serves as a unitary invariant prior distribution over the precision matrices 2p 1 and ni2p 1 2 over 2p and consequently over the p p mixed and pure density matrices the availability of these priors should facilitate the process of state determination an uncertainty principle for the estimation of is formulated the determinant of 2p proves for integral p 1 to equal 2 p 2

quantum and fisher information from the husimi and related distributions the two principal immediate influences which we seek to interrelate here upon the undertaking of this study are papers of yczkowski and slomczynski j phys a 34 6689 2001 and of petz and sudar j math phys 37 2262 1996 in the former work a metric the monge one specifically over generalized husimi distributions was employed to define a distance between two arbitrary density matrices in the petz sudar work completing a program of chentsov the quantum analog of the classically unique fisher information monotone metric of a probability simplex was extended to define an uncountable infinitude of riemannian also monotone metrics on the set of positive definite density matrices we pose here the questions of what is the specific unique fisher information metric for the classically defined husimi distributions and how does it relate to the infinitude of quantum metrics over the density matrices of petz and sudar we find a highly proximate small relative entropy relationship bet

numerical approximations to the scaled first derivatives of the solution to a two parameter singularly perturbed problem abstract a singularly perturbed problem involving two singular perturbation parameters is discretized using the classical upwinded finite difference scheme on an appropriate piecewise uniform shishkin mesh scaled discrete derivatives with scaling only used within the layers are shown to be parameter uniformly convergent to the scaled first derivatives of the continuous solution

a smoothing property of schrodinger equations in the critical case this paper deals with the critical case of the global smoothing estimates for the schrodinger equation although such estimates fail for critical orders of weights and smoothing it is shown that they are still valid if one works with operators with symbols vanishing on a certain set the geometric meaning of this set is clarified in terms of the hamiltonian flow of the laplacian the corresponding critical case of the limiting absorption principle for the resolvent is also established obtained results are extended to dispersive equations of schrodinger type to hyperbolic equations and to equations of other orders

structural resolvent estimates and derivative nonlinear schrodinger equations a refinement of a uniform resolvent estimate is given and several smoothing estimates for schrodinger equations in the critical case are induced from it the relation between this resolvent estimate and a radiation condition is discussed as an application of critical smoothing estimates we show a global existence result for derivative nonlinear schrodinger equations

applications of quantum and classical fisher information to two level complex and quaternionic and three level complex systems in the bayesian theory of statistical inference as first suggested by harold jeffreys in highly influential work one can employ the square root of the determinant of an n n fisher information matrix as a reparametrization invariant prior generally unnormalized measure over an n dimensional family riemannian manifold of probability distributions jeffreys ansatz is adopted here to the quantum context that is with regard to density matrices rather than probability distributions by computing the quantum fisher information matrices associated with helstrom and holevo for the three five and eight dimensional convex sets of two level complex two level quaternionic and three level complex systems respectively in both the two level cases the priors have been normalized to probability distributions over the 2 2 density matrices while in the much more computationally demanding three level situation no such normalization has been accomplished an argument is made for the general form in term

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

smoothing estimates for non dispersive equations this paper describes an approach to global smoothing problems for non dispersive equations based on ideas of comparison principle and canonical transformation established in authors previous paper ruzhansky and sugimoto proc lond math soc 105 393 423 2012 where dispersive equations were treated for operators a d x of order m satisfying the dispersiveness condition nabla a xi ne 0 for xi not 0 the global smoothing estimate begin aligned left left langle x right rangle s d x m 1 2 e ita d x varphi x right l 2 left mathbb r t times mathbb r n x right le c left varphi right l 2 left mathbb r n x right quad s 1 2 end aligned is well known while it is also known to fail for non dispersive operators for the case when the dispersiveness breaks we suggest the estimate in the form begin aligned left left langle x right rangle s nabla a d x 1 2 e it a d x varphi x right l 2 left mathbb r t times mathbb r n x right le c left varphi right l 2 left mathbb r n x right quad s 1 2 end aligned which is equivalent to the usual estimate in the dispersive case and is also invariant under canonical transformations for the operator a d x we show that this estimate and its variants do continue to hold for a variety of non dispersive operators a d x where nabla a xi may become zero on some set moreover other types of such estimates and the case of time dependent equations are also discussed

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

on non null distributions connected with testing reality of a complex normal distribution polynomials can be expressed in terms of the zonal polynomials on the hermitian matrices and they are used to obtain a series expansion for the density of the non null distribution of the maximal invariant corresponding to the problem of testing for reality of the covariance matrix of a complex multivariate nornal distribution 1 introduction in an article by andersson br0ns and jensen 1983 10 fundamental tests concerning the structure of covariance matrices in multivariate analysis are treated each of the 10 problems is invariant under a group of linear transformations and the distribution of the maximal invariant under the null hypothesis was found in terms of a density with respect to a lebesgue measure a series expansion for the density of the distribution of the maximal invariant under the alternative hypothesis has been obtained for some of the 10 problems by james 1964 and constantine 1963 by use of zonal polynomials and hypergeometric functions it concerns the tests for independence and the tests for identity of two sets of variates where the simultaneous covariance matrix has real or complex structure in this article one of the remaining non null distribution problems are treated by using methods similar to those of james and constantine it concerns the test that a 2m x 2m covariance matrix with complex structure has real structure this test was considered for the first time by khatri 1965 andersson and perlman 1984 study the non null distribution of the maximal invariant and we use their results as a starting point the problem is the evaluation of a certain integral over the group of nonsingular m x m matrices this integral is a function having two skew symmetric matrices as arguments the theory of group representations is used to define polynomials of a skew symmetric matrix the polynomials can be expressed in a simple way in terms of the complex zonal polynomials on the hermitian matrices finally the polynomials are used to obtain a series expansion for the integral

a local to global boundedness argument and fourier integral operators abstract we give a criterion for the global boundedness of integral operators which are known to be locally bounded as an application we discuss the global l p boundedness for a class of fourier integral operators while the local l p boundedness of fourier integral operators is known from the work of seeger sogge and stein 26 not so many results are available for the global boundedness on l p r n we give several natural sufficient conditions for them

discrete approximations of solutions and derivatives for a singularly perturbed parabolic convection diffusion equation in this paper we consider grid approximations of a boundary value problem on a segment for a singularly perturbed parabolic convection diffusion equation classical finite difference approximations on piecewise uniform meshes condensing in a neighborhood of the boundary layer are used it is necessary to find numerical approximations to both the solution and its first order derivatives with errors weakly depending on the perturbation parameter e the approximation errors in solutions and derivatives are examined in the metric which is adequate for capturing singular solutions of problems with boundary layers in this metric the errors of the solution and its derivative t u x t are determined by the absolute errors and the error in the derivative x u x t is determined by the relative error with respect to a majorant function for this derivative in the boundary layer and by the absolute error outside it in the class of meshes whose mesh size in the boundary layer does not decrease away from the boundary it is shown that there are no meshes on which the scheme converges e uniformly in the metric we establish conditions imposed on the parameters of piecewise uniform meshes under which the schemes converge in the metric almost e uniformly that is at a rate of o e vn 1 n0 1 where v 0 may be arbitrarily small n and n0 define the numbers of mesh points in x and t respectively

bayesian inference for complex and quaternionic two level quantum systems abstract jeffreys approach for generating reparameterization invariant prior distributions is applied to the three dimensional convex set of complex two level quantum systems for this purpose such systems are identified with bivariate complex normal distributions over the vectors of two dimensional complex hilbert space the trivariate prior obtained is improper or non normalizable over the convex set however its three bivariate marginals are through a limiting procedure normalizable to probability distributions and are consequently suitable for the bayesian inference of two level systems analogous results hold for the five dimensional convex set of quaternionic two level systems the complex univariate and quaternionic trivariate marginals of the improper priors are uniform distributions the bivariate marginals in the two cases are opposite in character

globally well and ill posedness for non elliptic derivative schrodinger equations with small rough data abstract we show that there exists s c 0 such that the cubic quartic non elliptic derivative schrodinger equations with small data in modulation spaces m 2 1 s r n for n 3 n 2 are globally well posed if s s c and ill posed if s s c in 2d cubic case using the gabor frame we get some time global dispersive estimates for the schrodinger semi group in anisotropic lebesgue spaces which include a time global maximal function estimate in the space l x 1 2 l x 2 t by resorting to the smooth effect estimate together with the dispersive estimates in anisotropic lebesgue spaces we show that the cubic hyperbolic derivative nls in 2d has a unique global solution if the initial data in feichtinger segal algebra or in weighted sobolev spaces are sufficiently small

information gains expected from separate and joint measurements of n identical spin 1 2 systems noninformative bayesian analyses an investigator is assumed to have n replicas of a spin 1 2 system but lacks any specific knowledge pertinent to constructing a 2 2 density matrix for it we compute within a noninformative bayesian framework the expected information gains for various measurement schemes first separate and then joint in nature in particular we obtain a result which fully accords with a certain plausibility argument of peres while the parallel result in a recent analysis of massar and popescu does not

symmetry and lattice conditional independence in a multivariate normal distribution a class of multivariate normal models with symmetry restrictions given by a finite group and conditional independence restrictions given by a finite distributive lattice is defined and studied the statistical properties of these models including maximum likelihood inference invariance and hypothesis testing are discussed

quantum fisher bures information of two level systems and a three level extension braunstein and caves have recently demonstrated that the bures metric on the mixed quantum states is equivalent up to a proportionality factor of four to the statistical distinguishability or quantum fisher information metric the volume element of these metrics can then adapting a fundamental bayesian principle of jeffreys to the quantum context serve as a reparametrization invariant prior measure over the quantum states the implications of this line of reasoning for the two level systems in general and an embedding of them into a certain set of three level systems are investigated

numerical approximation of solution derivatives in the case of singularly perturbed time dependent reaction diffusion problems numerical approximations to the solution of a linear singularly perturbed parabolic problem are generated using a classical finite difference operator on a piecewise uniform shishkin mesh first order convergence of these numerical approximations in an appropriately weighted c 1 norm is established numerical results are given to illustrate the theoretical error bounds

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

symmetric matrix derivatives with applications abstract dwyer 1967 provided extensive formulas for matrix derivatives many of which are for derivatives with respect to symmetric matrices the results of his article are only for symmetric matrices whose j i element is considered to differ from the i j element even though their scalar values are equal a simple result is given that extends the use of dwyer s formulas to symmetric matrices in which the i j element is considered functionally equal to the j i element application of the results are illustrated by deriving the multivariate normal information matrix

reformulation for arbitrary mixed states of jones bayes estimation of pure states abstract jones has cast the problem of estimating the pure state of a d dimensional quantum system into a bayesian framework the normalized uniform ray measure over such states is employed as the prior distribution the data consist of observed eigenvectors k k 1 n from an n trial analyzer that is a collection of n bases of the hilbert space c d the desired posterior inferred distribution is then simply proportional to the likelihood of k 1n k 2 here jones approach is extended to the more realistic experimental case of mixed input states as the unnormalized prior over the d d density matrices the recently developed reparameterization and unitarily invariant measure 2d 1 is utilized the likelihood is then taken to be k 1n k k reducing to that of jones when corresponds to a pure state the case of a pure state however the associated prior and posterior probabilities are then zero some analytical results for the case d 2 are presented

generalized likelihood ratios for testing the properness of quaternion gaussian vectors in a recent paper the second order statistical analysis of quaternion random vectors has shown that there exist two different kinds of quaternion widely linear processing which are associated with the two main types of quaternion properness in this paper we consider the problem of determining from a finite number of independent vector observations whether a quaternion gaussian vector is proper or not specifically we derive three generalized likelihood ratio tests glrts for testing the two main kinds of quaternion properness and show that the glrts reduce to the estimation of three previously proposed quaternion improperness measures interestingly the three glrt statistics improperness measures can be interpreted as an estimate of the entropy loss due to the quaternion improperness additionally we analyze the case in which the orthogonal basis for the representation of the quaternion vector is unknown which results in the problem of estimating the principal bbc properness direction i e the pure unit quaternion minimizing the bbc improperness measure although this estimation problem is not convex we propose a technique based on successive convex approximations which can be solved in closed form finally some simulation examples illustrate the performance and practical application of the proposed tests

graphical gaussian models with edge and vertex symmetries summary we introduce new types of graphical gaussian models by placing symmetry restrictions on the concentration or correlation matrix the models can be represented by coloured graphs where parameters that are associated with edges or vertices of the same colour are restricted to being identical we study the properties of such models and derive the necessary algorithms for calculating maximum likelihood estimates we identify conditions for restrictions on the concentration and correlation matrices being equivalent this is for example the case when symmetries are generated by permutation of variable labels for such models a particularly simple maximization of the likelihood function is available

symmetry models and hypotheses for structured data layouts on peut utiliser des proprietes de symetrie pour deriver des tests valides d hypotheses nulles et pour construire des modeles statistiques appropries pour les donnees on developpe des idees generales et on les illustre dans le contexte d une classification a deux criteres

geometry of canonical correlation on the state space of a quantum system a riemannian metric is defined on the state space of a finite quantum system by the canonical correlation or kubo mori bogoliubov scalar product this metric is infinitesimally induced by the nonsymmetric relative entropy functional or the von neumann entropy of density matrices hence its geometry expresses maximal uncertainty it is proven that the metric is monotone under stochastic mappings however an example shows that it is not the only such riemannian metric this fact is remarkable because in the probabilistic case the markovian monotonicity property characterizes the fisher information metric the essential difference appears in the curvatures of a classical state space and a quantum one a conjecture is made that the scalar curvature is monotone with respect to the more mixed statistical partial order of density matrices furthermore an information inequality resembling the cramer rao inequality of classical statistics is established the inequality provides a lower bound for the c

information theoretic asymptotics of bayes methods in the absence of knowledge of the true density function bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior the authors examine the relative entropy distance d sub n between the true density and the bayesian density and show that the asymptotic distance is d 2 log n c where d is the dimension of the parameter vector therefore the relative entropy rate d sub n n converges to zero at rate log n n the constant c which the authors explicitly identify depends only on the prior density function and the fisher information matrix evaluated at the true parameter value consequences are given for density estimation universal data compression composite hypothesis testing and stock market portfolio selection