numerical ranges of cube roots of the identity abstract the numerical range of a bounded linear operator t on a hilbert space h is defined to be the subset w t tv v v h v 1 of the complex plane for operators on a finite dimensional hilbert space it is known that if w t is a circular disk then the center of the disk must be a multiple eigenvalue of t in particular if t has minimal polynomial z 3 1 then w t cannot be a circular disk in this paper we show that this is no longer the case when h is infinite dimensional the collection of 3 3 matrices with three fold symmetry about the origin are also classified

information geometry on hierarchy of probability distributions an exponential family or mixture family of probability distributions has a natural hierarchical structure this paper gives an orthogonal decomposition of such a system based on information geometry a typical example is the decomposition of stochastic dependency among a number of random variables in general they have a complex structure of dependencies pairwise dependency is easily represented by correlation but it is more difficult to measure effects of pure triplewise or higher order interactions dependencies among these variables stochastic dependency is decomposed quantitatively into an orthogonal sum of pairwise triplewise and further higher order dependencies this gives a new invariant decomposition of joint entropy this problem is important for extracting intrinsic interactions in firing patterns of an ensemble of neurons and for estimating its functional connections the orthogonal decomposition is given in a wide class of hierarchical structures including both exponential and mixture families as an example we decompose the dependency in a higher order markov chain into a sum of those in various lower order markov chains

quantile and quantile function estimations under density ratio model 1 introduction forestry plays a major role in the canadian economy maintaining the high quality of wood products is vital economically and socially we are designing an effective long term monitoring plan for the quality of forestry products in canada two important quality indices for a piece of lumber are the modulus of elasticity moe and the modulus of rupture mor its strength in terms of elasticity and toughness the reliability of lumber based structures may depend heavily on the lower population quantiles of these indices however it is costly time consuming and laborious to obtain these quality measurements therefore efficient estimates of the population quantiles and their functions are important the estimation of quantiles based on a single random sample is a wellresearched topic empirical quantiles have been shown to admit a bahadur representation bahadur 1966 kiefer 1967 serfling 1980 making it simple to study the joint limiting distributions of any number of sample quantiles and their smooth functions in the presence of auxiliary information the empirical likelihood el owen 1988 2001 can be utilized to improve efficiency the bahadur

non linear vibration of coupled duffing oscillators by an improved incremental harmonic balance method abstract the incremental harmonic balance method is improved by i reversing the order of increment and linearization to find the bifurcated solution branches ii using a fast fourier transform to compute the solution increments and the floquet matrix and iii finding the secondary path after a bifurcation by an improved arc length method a system of two duffing oscillators linked with a linear stiffness is taken as a numerical example the qualitative change of solution due to the change of the linking stiffness is studied in detail

continuity of quantum channel capacities we prove that a broad array of capacities of a quantum channel are continuous that is two channels that are close with respect to the diamond norm have correspondingly similar communication capabilities we first show that the classical capacity quantum capacity and private classical capacity are continuous with the variation on arguments varepsilon apart bounded by a simple function of varepsilon and the channel s output dimension our main tool is an upper bound of the variation of output entropies of many copies of two nearby channels given the same initial state the bound is linear in the number of copies our second proof is concerned with the quantum capacities in the presence of free backward or two way public classical communication these capacities are proved continuous on the interior of the set of non zero capacity channels by considering mutual simulation between similar channels

comment on some results of erdahl and the convex structure of reduced density matrices in j math phys 13 1608 1621 1972 erdahl10 1063 1 1665885 considered the convex structure of the set of n representable 2 body reduced density matrices in the case of fermions some of these results have a straightforward extension to the m body setting and to the more general quantum marginal problem we describe these extensions but cannot resolve a problem in the proof of erdahl s claim that every extreme point is exposed in finite dimensions nevertheless we can show that when 2m n every extreme point of the set of n representable m body reduced density matrices has a unique pre image in both the symmetric and anti symmetric setting moreover this extends to the quantum marginal setting for a pair of complementary m body and n m body reduced density matrices

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

dissecting the qutrit to visualize a higher dimensional object it is convenient to consider its two dimensional cross sections the set of quantum states for a three level system has eight dimensions we supplement a recent paper by goyal et al arxiv 1111 4427 by considering the set of all possible two dimensional cross sections of the qutrit each such cross section is bounded by a plane cubic curve

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

reducible correlations in dicke states we apply a simple observation to show that the generalized dicke states can be determined from their reduced subsystems in this framework it is sufficient to calculate the expression for only the diagonal elements of the reduced density matrices in terms of the state coefficients we prove that the correlation in generalized dicke states gd l n can be reduced to 2l partite level application to the quantum marginal problem is also discussed

a sharper bonferroni procedure for multiple tests of significance summary a simple procedure for multiple tests of significance based on individual p values is derived this simple procedure is sharper than holm s 1979 sequentially rejective procedure both procedures contrast the ordered p values with the same set of critical values holm s procedure rejects an hypothesis only if its p value and each of the smaller p values are less than their corresponding critical values the new procedure rejects all hypotheses with smaller or equal p values to that of any one found less than its critical value

quantum tomography under prior information we provide a detailed analysis of the question how many measurement settings or outcomes are needed in order to identify an unknown quantum state which is constrained by prior information we show that if the prior information restricts the possible states to a set of lower dimensionality then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the set of all states conversely we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the minkowski dimension of the set we apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints in particular we show that the minimal number of measurement outcomes povm elements necessary to identify all pure states in a d dimensional hilbert space is 4d 3 c d d for some c d in 1 2 and d being the number of ones appearing in the binary expansion of d 1

empirical likelihood and general estimating equations for some time so called empirical likelihoods have been used heuristically for purposes of nonparametric estimation owen showed that empirical likelihood ratio statistics for various parameters theta f of an unknown distribution f have limiting chi square distributions and may be used to obtain tests or confidence intervals in a way that is completely analogous to that used with parametric likelihoods our objective in this paper is twofold first to link estimating functions or equations and empirical likelihood second to develop methods of combining information about parameters we do this by assuming that information about f and theta is available in the form of unbiased estimating functions empirical likelihoods for parameters are developed and shown to have properties similar to those for parametric likelihood efficiency results for estimates of both theta and f are obtained the methods are illustrated on several problems and areas for future investigation are noted

empirical likelihood ratio confidence intervals for a single functional summary the empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken in this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions it is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean for a class of m estimates that includes quantiles and for differentiable statistical functionals the results are nonparametric extensions of wilks s 1938 theorem for parametric likelihood ratios the intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on student s t statistic a hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced

a semidefinite hierarchy for containment of spectrahedra a spectrahedron is the positivity region of a linear matrix pencil and thus the feasible set of a semidefinite program we propose and study a hierarchy of sufficient semidefinite conditions to certify the containment of a spectrahedron in another one this approach comes from applying a moment relaxation to a suitable polynomial optimization formulation the hierarchical criterion is stronger than a solitary semidefinite criterion discussed earlier by helton klep and mccullough as well as by the authors moreover several exactness results for the solitary criterion can be brought forward to the hierarchical approach the hierarchy also applies to the equivalent question of checking whether a map between matrix sub spaces is positive in this context the solitary criterion checks whether the map is completely positive and thus our results provide a hierarchy between positivity and complete positivity

numerical range of some doubly stochastic matrices a classification of all possible shapes is given for numerical ranges of 4 i 4 doubly stochastic matrices the tests determining the shape are also provided along with illustrating examples

multi information in the thermodynamic limit a multivariate generalization of mutual information multi information is defined in the thermodynamic limit the definition takes phase coexistence into account by taking the infimum over the translation invariant gibbs measures of an interaction potential it is shown that this infimum is attained in a pure state an explicit formula can be found for the ising square lattice where the quantity is proved to be maximized at the phase transition point by this phase coexistence is linked to high model complexity in a rigorous way

flat portions on the boundary of the davis wielandt shell of 3 by 3 matrices abstract let t be an n n matrix the davis wielandt shell of t is defined as the set w t t t x tx x t tx x c r x x 1 in this paper we discuss the existence of flat portions on the boundary of the davis wielandt shell of 3 by 3 matrices

asymptotic permutation tests in general factorial designs summary in general factorial designs where no homoscedasticity or a particular error distribution is assumed the well known wald type statistic is a simple asymptotically valid procedure however it is well known that it suffers from a poor finite sample approximation since the convergence to its 2 limit distribution is quite slow this becomes even worse with an increasing number of factor levels the aim of the paper is to improve the small sample behaviour of the wald type statistic maintaining its applicability to general settings as crossed or hierarchically nested designs by applying a modified permutation approach in particular it is shown that this approach approximates the null distribution of the wald type statistic not only under the null hypothesis but also under the alternative yielding an asymptotically valid permutation test which is even finitely exact under exchangeability finally its small sample behaviour is compared with competing procedures in an extensive simulation study

an efficient method for switching branches of period doubling bifurcations of strongly non linear autonomous oscillators with many degrees of freedom abstract popular algorithms for switching branches at a bifurcation point of strongly non linear oscillators are generally quite involved as they require the computation of the tangent of a new branch and second derivatives in this paper a simple but efficient algorithm is presented by using a perturbation incremental method for switching branches at a period doubling bifurcation of strongly non linear autonomous oscillators with many degrees of freedom to switch to a new branch at a bifurcation point a parameter is simply turned on from zero to a small positive value so as to obtain an initial solution on the emanating branch for subsequent continuation the parametric value at a period doubling bifurcation can also be determined accurately furthermore limit cycles of period 2 k k 1 can be calculated to any desired degree of accuracy

irreducible multiparty correlations in quantum states without maximal rank the correlations of an n partite quantum state are classified into a series of irreducible k party ones 2 ensuremath le k ensuremath le n with the irreducible k party correlation being the correlation in the states of k parties but nonexistent in the states of k ensuremath 1 parties a measure of the degree of irreducible k party correlation is defined based on the principle of maximal entropy adopting a continuity approach we overcome the difficulties in calculating the degrees of irreducible multiparty correlations for the multipartite states without maximal rank in particular we obtain the degrees of irreducible multiparty correlations in the n qubit stabilizer states and the n qubit generalized greenberger horne zeilinger states which reveals the distribution of multiparty correlations

transformations of the explanatory variables in the logistic regression model for binary data summary some results are presented on improving the fit of the logistic regression model for binary data by transforming the vector of explanatory variables the methods are based on consideration of the distributions of these variables conditional on outcome group the transformations required are the functions of the explanatory variables which appear in the log density ratio of the conditional distributions

continuity of the von neumann entropy a general method for proving continuity of the von neumann entropy on subsets of positive trace class operators is considered this makes it possible to re derive the known conditions for continuity of the entropy in more general forms and to obtain several new conditions the method is based on a particular approximation of the von neumann entropy by an increasing sequence of concave continuous unitary invariant functions defined using decompositions into finite rank operators the existence of this approximation is a corollary of a general property of the set of quantum states as a convex topological space called the strong stability property this is considered in the first part of the paper

classification of joint numerical ranges of three hermitian matrices of size three abstract the joint numerical range w f of three hermitian 3 by 3 matrices f f 1 f 2 f 3 is a convex and compact subset in r 3 generically we find that w f is a three dimensional oval assuming dim w f 3 every one or two dimensional face of w f is a segment or a filled ellipse we prove that only ten configurations of these segments and ellipses are possible we identify a triple f for each class and illustrate w f using random matrices and dual varieties

bifurcation and chaos in geared rotor bearing system by incremental harmonic balance method abstract the periodic motions of a non linear geared rotor bearing system are investigated by the incremental harmonic balance ihb method a path following procedure using arc length continuation technique is used to trace the bifurcation diagrams the system exhibits a period doubling route and a quasiperiodic route to chaos in different regions of excitation frequency the chaotic motions are investigated by numerical integration and the lyapunov exponents are computed the periodic solutions and subharmonic solutions obtained by the ihb method compare very well with those obtained by numerical integration

a semiparametric approach to the one way layout we consider m distributions in which the first m 1 are obtained by multiplicative exponential distortions of the mth distribution which is a reference the combined data fromm samples one from each distribution are used in the semiparametric large sample problem of estimating each distortion and the reference distribution and testing the hypothesis that the distributions are identical the approach generalizes the classical normal based one way analysis of variance in the sense that it obviates the need for a completely specified parametric model an advantage is that the probability density of the reference distribution is estimated from the combined data and not only from the mth sample a power comparison with the t and f tests and with two nonparametric tests obtained by means of a simulation points to the merit of the present approach the method is applied to rain rate data from meteorological instruments

on the cone of positive semidefinite matrices abstract a survey of some general properties of the cone of positive semidefinite matrices its faces two isometric isomorphisms and linear transformations on it is given

singular points of the algebraic curves of symmetric hyperbolic forms abstract we investigate singular points of the algebraic curve defined by a hyperbolic form satisfying some symmetric properties we show that all singular points of such a curve are real nodes

quantum convex support abstract convex support the mean values of a set of random variables is central in information theory and statistics equally central in quantum information theory are mean values of a set of observables in a finite dimensional c algebra a which we call quantum convex support the convex support can be viewed as a projection of the state space of a and it is a projection of a spectrahedron spectrahedra are increasingly investigated at least since the 1990s boom in semi definite programming we recall the geometry of the positive semi definite cone and of the state space we write a convex duality for general self dual convex cones this restricts to projections of state spaces and connects them to results on spectrahedra our main result is an analysis of the face lattice of convex support by mapping this lattice to a lattice of orthogonal projections using natural isomorphisms the result encodes the face lattice of the convex support into a set of projections in a and enables the integration of convex geometry with matrix calculus or algebraic techniques

power and sample size calculations for discrete bounded outcome scores we consider power and sample size calculations for randomized trials with a bounded outcome score bos as primary response adjusted for a priori chosen covariates we define bos to be a random variable restricted to a finite interval typically a bos has a j or u shaped distribution hindering traditional parametric methods of analysis when no adjustment for covariates is needed a non parametric test could be chosen however there is still a problem with calculating the power since the common location shift alternative does not hold in general for a bos in this paper we consider a parametric approach and assume that the observed bos is a coarsened version of a true bos which has a logit normal distribution in each treatment group allowing correction for covariates a two step procedure is used to calculate the power firstly the power function is defined conditionally on the covariate values secondly the marginal power is obtained by averaging the conditional power with respect to an assumed distribution for the covariates using monte carlo integration a simulation study evaluates the performance of our method which is also applied to the ecass 1 stroke study

asymptotic error rates in quantum hypothesis testing we consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies and find a closed form expression for the asymptotic exponential rate at which the error probability tends to zero this leads to the identification of the quantum generalisation of the classical chernoff distance which is the corresponding quantity in classical symmetric hypothesis testing

the holevo capacity of infinite dimensional channels and the additivity problem the holevo capacity of an arbitrarily constrained infinite dimensional quantum channel is considered and its properties are discussed the notion of optimal average state is introduced the continuity properties of the holevo capacity with respect to constraint and to the channel are explored the main result of this paper is the statement that additivity of the holevo capacity for all finite dimensional channels implies its additivity for all infinite dimensional channels with arbitrary constraints

on shor s channel extension and constrained channels several equivalent formulations of the additivity conjecture for constrained channels which formally is substantially stronger than the unconstrained additivity are given to this end a characteristic property of the optimal ensemble for such a channel is derived generalizing the maximal distance property it is shown that the additivity conjecture for constrained channels holds true for certain nontrivial classes of channels after giving an algebraic formulation for shor s channel extension its main asymptotic property is proved it is then used to show that additivity for two constrained channels can be reduced to the same problem for unconstrained channels and hence global additivity for channels with arbitrary constraints is equivalent to additivity without constraints

finding the maximizers of the information divergence from an exponential family this paper investigates maximizers of the information divergence from an exponential family e it is shown that the ri projection of a maximizer p to e is a convex combination of p and a probability measure p with disjoint support and the same value of the sufficient statistics a this observation can be used to transform the original problem of maximizing d e over the set of all probability measures into the maximization of a function dr over a convex subset of ker a the global maximizers of both problems correspond to each other furthermore finding all local maximizers of dr yields all local maximizers of d e this paper also proposes two algorithms to find the maximizers of dr and applies them to two examples where the maximizers of d e were not known before

continuity of the maximum entropy inference we study the inverse problem of inferring the state of a finite level quantum system from expected values of a fixed set of observables by maximizing a continuous ranking function we have proved earlier that the maximum entropy inference can be a discontinuous map from the convex set of expected values to the convex set of states because the image contains states of reduced support while this map restricts to a smooth parametrization of a gibbsian family of fully supported states here we prove for arbitrary ranking functions that the inference is continuous up to boundary points this follows from a continuity condition in terms of the openness of the restricted linear map from states to their expected values the openness condition shows also that ranking functions with a discontinuous inference are typical moreover it shows that the inference is continuous in the restriction to any polytope which implies that a discontinuity belongs to the quantum domain of non commutative observables and that a geodesic closure of a gibbsian family equals the set of maximum entropy states we discuss eight descriptions of the set of maximum entropy states with proofs of accuracy and an analysis of deviations

entropy distance new quantum phenomena we study a curve of gibbsian families of complex 3 3 matrices and point out new features absent in commutative finite dimensional algebras a discontinuous maximum entropy inference a discontinuous entropy distance and non exposed faces of the mean value set we analyze these problems from various aspects including convex geometry topology and information geometry this research is motivated by a theory of infomax principles where we contribute by computing first order optimality conditions of the entropy distance

an information geometric approach to a theory of pragmatic structuring 1 1 the motivation of the approach in the field of neural networks so called infomax principles like the principle of maximum information preservation by linsker 20 are formulated to derive learning rules that improve the information processing properties of neural systems see 12 these principles which are based on information theoretic measures are intended to describe the mechanism of learning in the brain there the starting point is a low dimensional and biophysiologically motivated parametrization of the neural system which need not necessarily be compatible with the given optimization principle in contrast to this we establish theoretical results about the low complexity of optimal solutions for the optimization problem of frequently used measures like the mutual information in an unconstrained and more theoretical setting in the present paper we do not comment on applications to modeling neural networks this is intended to be done in a further step where the results can be used for the characterization of good parameter sets that on the one hand are compatible with the underlying optimization and on the other hand are biologically motivated

effect of smoothing piecewise linear oscillators on their stability predictions the effects of smoothing non smooth function on estimation of dynamical stability of the periodic response determined in the frequency domain are considered in this paper for that purpose a simple single degree of freedom system with piecewise linear force displacement relationship subjected to a harmonic force excitation is analyzed stability of the periodic response obtained in the frequency domain by the incremental harmonic balance method is determined by using the floquet liapounov theorem results obtained are verified in the time domain by the method of piecing the exact solutions and runge kutta integration routine

the numerical range of 3 3 matrices abstract let a be an n n complex matrix then the numerical range of a w a is defined to be x ax x c n x x 1 in this article a series of tests is given allowing one to determine the shape of w a for 3 3 matrices reconstruction of a up to unitary similarity from w a is also examined

continuity of the maximum entropy inference convex geometry and numerical ranges approach we study the continuity of an abstract generalization of the maximum entropy inference a maximizer it is defined as a right inverse of a linear map restricted to a convex body which uniquely maximizes on each fiber of the linear map a continuous function on the convex body using convex geometry we prove amongst others the existence of discontinuities of the maximizer at limits of extremal points not being extremal points themselves and apply the result to quantum correlations further we use numerical range methods in the case of quantum inference which refers to two observables one result is a complete characterization of points of discontinuity for 3 3 matrices

stability and accuracy of the iterative differential quadrature method in this paper the stability and accuracy of an iterative method based on differential quadrature rules will be discussed the method has already been proposed by the author in a previous work where its good performance has been shown nevertheless discussion about stability and accuracy remained open an answer to this question will be provided in this paper where the conditional stability of the method will be pointed out in addition to an examination of the possible errors which arise under certain conditions the discussion will be preceded by an overview of the method and its foundations i e the differential quadrature rules and followed by a numerical case which shows how the method behaves when applied to reduce continuous systems to two degree of freedom systems in the non linear range in particular here the case of oscillators coupled in non linear terms will be treated the numerical results used to draw poincare maps will be compared with those obtained by using the runge kutta method with a high precision goal copyright 2003 john wiley sons ltd

on flat portions on the boundary of the numerical range abstract the paper is devoted to matrices with flat portions on the boundary of their numerical range a constructive criterion for such portions to exist is obtained in case of tridiagonal matrices and a particular case of continuant matrices is considered as an application the cases of arbitrary 3 3 and 4 4 matrices are treated it is shown in particular that the sharp bound for the number of flat portions on the boundary of the numerical range for 4 4 matrices is four three if the matrices are assumed unitarily irreducible

strict convexity of the joint c numerical range abstract let h h 1 h 2 h m be an m tuple of n n hermitian matrices and c c 1 c 2 c n r n the joint c numerical range w c h of h is defined for x x 1 x 2 x m r m the form associated with h is defined as f h t x det ti n x 1 h 1 x 2 h 2 x m h m we prove that the convex hull of w c h is strictly convex if the roots of f h t x 0 have constant multiplicities for all x on the unit sphere in r m the result is illustrated with several examples

joint numerical ranges quantum maps and joint numerical shadows abstract we associate with a k tuple of hermitian n n matrices a probability measure on r k supported on their joint numerical range the joint numerical shadow of these matrices when k 2 we recover the numerical range and the numerical shadow of the complex matrix corresponding to a pair of hermitian matrices we apply this material to the theory of quantum information thus we show that quantum maps on the set of quantum states defined by kraus operators satisfying the identity resolution assumption shrink joint numerical ranges

the use of bootstrap methods for estimating sample size and analysing health related quality of life outcomes health related quality of life hrqol measures are increasingly used in trials as primary outcome measures investigators are now asking statisticians for advice on how to plan and analyse studies using such outcomes hrqol outcomes like the sf 36 are usual measured on an ordinal scale although most investigators assume that there exists an underlying continuous latent variable and that the actual measured outcomes the ordered categories reflect contiguous intervals along this continuum the ordinal scaling of hrqol measures means they tend to generate data that have discrete bounded and skewed distributions thus standard methods of analysis that assume normality and constant variance may not be appropriate for this reason conventional statistical advice would suggest non parametric methods be used to analyse hrqol data the bootstrap is one such computer intensive non parametric method for estimating sample sizes and analysing data we describe three methods of estimating sample sizes for two group cross sectional comparisons of hrqol outcomes we then compared the power of the three methods for a two group cross sectional study design using bootstrap simulation the results showed that under the location shift alternative hypothesis conventional methods of sample size estimation performed well particularly whitehead s method whitehead s method is recommended if the hrqol outcome has a limited number of discrete values 7 and or the expected proportion of cases at either of the bounds is high if a pilot data set is readily available then bootstrap simulation will provide a more accurate and reliable estimate than conventional methods finally we used the bootstrap for hypothesis testing and the estimation of standard errors and confidence intervals for parameters in an example data set we then compared and contrasted the bootstrap with standard methods of analysing hrqol outcomes in the data set studied with the sf 36 outcome the use of the bootstrap for estimating sample sizes and analysing hrqol data produces results similar to conventional statistical methods these results suggest that bootstrap methods are not more appropriate for analysing hrqol outcome data than standard methods

quantum codes give counterexamples to the unique preimage conjecture of the n representability problem it is well known that the ground state energy of many particle hamiltonians involving only 2 body interactions can be obtained using constrained optimizations over density matrices which arise from reducing an n particle state while determining which 2 particle density matrices are n representable is a computationally hard problem all known extreme n representable 2 particle reduced density matrices arise from a unique n particle preimage satisfying a conjecture established in 1972 we present explicit counterexamples to this conjecture through giving hamiltonians with 2 body interactions which have degenerate ground states that cannot be distinguished by any 2 body operator we relate the existence of such counterexamples to quantum error correction codes and topologically ordered spin systems

monotone metrics on matrix spaces abstract the study of monotone inner products under stochastic mappings on the space of matrices was initiated by morozova and chentsov motivated by information geometry they did not show a monotone metric but proposed several candidates the main result of the present paper is to provide an abundance of monotone metrics by means of operator monotone functions and to characterize them it turns out that there is a correspondence between monotone metrics and operator means in the sense of kubo and ando it follows that all proposals of morozova and chentsov are indeed monotone metrics

circularity of the numerical range abstract an equivalent condition on a 3 square complex or a 4 square real upper triangular matrix is found for its numerical range to be a circular disk centered at the origin sufficient conditions for the circularity of the numerical range of certain sparse matrices are also given in terms of graphs

network information and connected correlations entropy and information provide natural measures of correlation among elements in a network we construct here the information theoretic analog of connected correlation functions irreducible n point correlation is measured by a decrease in entropy for the joint distribution of n variables relative to the maximum entropy allowed by all the observed n 1 variable distributions we calculate the connected information terms for several examples and show that it also enables the decomposition of the information that is carried by a population of elements about an outside source

poincare maps of duffing type oscillators and their reduction to circle maps ii methods and numerical results bifurcation diagrams and plots of lyapunov exponents in the r plane for duffing type oscillators exhibit a regular pattern of repeating self similar tongues with complex internal structure we demonstrate here how this behaviour is easily understood qualitatively and quantitatively from a poincare map of the system in action angle variables in the limit of large driving force or equivalently small driving frequency this map approaches the one dimensional form as derived in paper i this second paper describes our approach to calculating the various constants and functions introduced in paper i it gives numerical applications of the theory and tests its range of validity by comparison with results from the numerical integration of duffing type equations finally we show how to extend the range in the parameter space where the map is applicable

a bootstrap semiparametric homogeneity test for the distributions of multigroup proportional data with applications to analysis of quality of life outcomes in clinical trials this article is concerned about the test for the difference in the distributions of multigroup proportional data which is motivated by the problem of comparing the distributions of quality of life qol outcomes among different treatment groups in clinical trials the proportional data such as qol outcomes assessed by answers to questions on a questionnaire are bounded in a closed interval such as 0 1 with continuous observations in 0 1 and in addition excess observations taking the boundary values 0 and or 1 common statistical procedures used in practice such as t and rank based tests may not be very powerful since they ignore the specific feature of the proportional data in this article we propose a three component mixture model for the proportional data and a density ratio model for the distributions of continuous observations in 0 1 a semiparametric test statistic for the homogeneity of distributions of multigroup proportional data is derived based on the empirical likelihood ratio principle and shown to be asymptotically distributed as a chi squared random variable under null hypothesis a nonparametric bootstrap procedure is proposed to further improve the performance of the semiparametric test simulation studies are performed to evaluate the empirical type i error and power of the proposed test procedure and compare it with likelihood ratio tests lrts under parametric distribution assumptions rank based kruskal wallis test and wald type test the proposed test procedure is also applied to the analysis of qol outcomes from a clinical trial on colorectal cancer that motivated our study

complementation in the face lattice of a proper cone abstract barker proved that the lattice of faces f k of a finite dimensional proper cone k is always complemented the proof however contained a gap in this paper we offer a correct proof of the result at the same time new characterizations of lattices of finite length which are section complemented or relatively complemented are found it is also proved that the lattice of exposed faces of a proper cone k is complemented finally we show that the class of proper cones with a relatively complemented face lattice is strictly contained in the class of proper cones with a dual atomistic face lattice for a few natural related questions counterexamples are given it turns out that one of our examples also answers in the negative an open problem posed recently by the second author

non linear dynamics of a two dimensional airfoil by incremental harmonic balance method periodic oscillations and bifurcations of a two dimensional airfoil in plunge and pitching motions with cubic pitching stiffness in incompressible flow is investigated using the incremental harmonic balance ihb method the bifurcations are obtained with the parametric continuation technique and the stability of the periodic motions is investigated using the floquet theory the autonomous non linear system of the airfoil undergoes initial hopf bifurcation leading to limit cycle oscillation as the airspeed parameter is increased further increase in the airspeed causes symmetry breaking saddle node and period doubling bifurcations leading to chaos the frequency of the limit cycle oscillation is also determined in the ihb method

quantum sphere packing bounds with polynomial prefactors we study lower bounds on the optimal error probability in classical coding over classical quantum channels at rates below the capacity commonly termed quantum sphere packing bounds winter and dalai have derived such bounds for classical quantum channels however the exponents in their bounds only coincide when the channel is classical in this paper we show that these two exponents admit a variational representation and are related by the golden thompson inequality reaffirming that dalai s expression is stronger in general classical quantum channels second we establish a finite blocklength sphere packing bound for classical quantum channels which significantly improves dalai s prefactor from the order of subexponential to polynomial furthermore the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of o log n n indicating our sphere packing bound is almost exact in the high rate regime finally for a special class of symmetric classical quantum channels we can completely characterize its optimal error probability without the constant composition code assumption the main technical contributions are two converse hoeffding bounds for quantum hypothesis testing and the saddle point properties of error exponent functions

optimization of nonlinear structural resonance using the incremental harmonic balance method abstract we present an optimization procedure for tailoring the nonlinear structural resonant response with time harmonic loads a nonlinear finite element method is used for modeling beam structures with a geometric nonlinearity and the incremental harmonic balance method is applied for accurate nonlinear vibration analysis an optimization procedure based on a gradient based algorithm is developed and we use the adjoint method for efficient computation of design sensitivities we consider several examples in which we find optimized beam width distributions that minimize or maximize fundamental or super harmonic resonant responses

3 3 matrices with a flat portion on the boundary of the numerical range the set of unitarily irreducible complex 3 3 matrices whose numerical range has a flat portion is explicitly described it is proved that the set is connected and real analytic properties of the set are studied connectedness of the sets of unitarily irreducible 3 3 matrices having two other possible shapes of their numerical ranges is proved as well

closures of exponential families the variation distance closure of an exponential family with a convex set of canonical parameters is described assuming no regularity conditions the tools are the concepts of convex core of a measure and extension of an exponential family introduced previously by the authors and a new concept of accessible faces of a convex set two other closures related to the information divergence are also characterized

a generalized partially linear mean covariance regression model for longitudinal proportional data with applications to the analysis of quality of life data from cancer clinical trials motivated by the analysis of quality of life data from a clinical trial on early breast cancer we propose in this paper a generalized partially linear mean covariance regression model for longitudinal proportional data which are bounded in a closed interval cholesky decomposition of the covariance matrix for within subject responses and generalized estimation equations are used to estimate unknown parameters and the nonlinear function in the model simulation studies are performed to evaluate the performance of the proposed estimation procedures our new model is also applied to analyze the data from the cancer clinical trial that motivated this research in comparison with available models in the literature the proposed model does not require specific parametric assumptions on the density function of the longitudinal responses and the probability function of the boundary values and can capture dynamic changes of time or other interested variables on both mean and covariance of the correlated proportional responses copyright 2017 john wiley sons ltd

the parts determine the whole in a generic pure quantum state we show that almost every pure state of multiparty quantum systems each of whose local hilbert space has the same dimension is completely determined by the state s reduced density matrices of a fraction of the parties this fraction is less than about two thirds of the parties for states of large numbers of parties in other words once the reduced states of this fraction of the parties have been specified there is no further freedom in the state

tight uniform continuity bounds for quantum entropies conditional entropy relative entropy distance and energy constraints we present a bouquet of continuity bounds for quantum entropies falling broadly into two classes first a tight analysis of the alicki fannes continuity bounds for the conditional von neumann entropy reaching almost the best possible form that depends only on the system dimension and the trace distance of the states almost the same proof can be used to derive similar continuity bounds for the relative entropy distance from a convex set of states or positive operators as applications we give new proofs with tighter bounds of the asymptotic continuity of the relative entropy of entanglement er and its regularization e r infty as well as of the entanglement of formation ef using a novel quantum coupling of density operators which may be of independent interest we extend the latter to an asymptotic continuity bound for the regularized entanglement of formation aka entanglement cost e c e f infty second we derive analogous continuity bounds for the von neumann entropy and conditional entropy in infinite dimensional systems under an energy constraint most importantly systems of multiple quantum harmonic oscillators while without an energy bound the entropy is discontinuous it is well known to be continuous on states of bounded energy however a quantitative statement to that effect seems not to have been known here under some regularity assumptions on the hamiltonian we find that quite intuitively the gibbs entropy at the given energy roughly takes the role of the hilbert space dimension in the finite dimensional fannes inequality

simulating non linear coupled oscillators by an iterative differential quadrature method abstract an iterative method based on differential quadrature rules is proposed as a new unified frame of resolution for non linear two degree of freedom systems dynamical systems with duffing type non linearity have been considered differential quadrature rules have been applied with a careful distribution of sampling points to reduce the governing equation of motion to two second order non linear non autonomous ordinary differential equations and to solve the time domain problem the time domain of the problem is discretized by means of time intervals with the same distribution of sampling points used to discretize the space domain which can be seen as a single interval it will be shown that accurate solutions depend not only on the choice of the distribution of sampling points but also on the length of the time interval one refers to in the computations the numerical results utilized to draw poincare maps are successfully compared with those obtained using the runge kutta method

generic pure quantum states as steady states of quasi local dissipative dynamics we investigate whether a generic pure state on a multipartite quantum system can be the unique asymptotic steady state of locality constrained purely dissipative markovian dynamics in the tripartite setting we show that the problem is equivalent to characterizing the solution space of a set of linear equations and establish that the set of pure states obeying the above property has either measure zero or measure one solely depending on the subsystems dimension a complete analytical characterization is given when the central subsystem is a qubit in the n partite case we provide conditions on the subsystems size and the nature of the locality constraint under which random pure states cannot be quasi locally stabilized generically also allowing for the possibility to approximately stabilize entangled pure states that cannot be exact steady states in settings where stabilizability is generic our results offer insights into the extent to which random pure states may arise as unique ground states of frustration free parent hamiltonians we further argue that to a high probability pure quantum states sampled from a t design enjoy the same stabilizability properties of haar random ones as long as suitable dimension constraints are obeyed and t is sufficiently large lastly we demonstrate a connection between the tasks of quasi local state stabilization and unique state reconstruction from local tomographic information and provide a constructive procedure for determining a generic n partite pure state based only on knowledge of the support of any two of the reduced density matrices of about half the parties improving over existing results

geometry of the numerical range of matrices abstract some techniques for the study of the algebraic curve c a which generates the numerical range w a of an n n matrix a as its convex hull are developed these enable one to give an explicit point equation of c a and a formula for the curvature of c a at a boundary point of w a applied to the case of a nonnegative matrix a a simple relation is found between the curvature of the function a p 1 a a t p being the perron root at 1 2 and the curvature of w a at the perron root of 1 2 a a t a connection with 2 dimensional pencils of hermitian matrices is mentioned and a conjecture formulated

on a conjecture of kippenhahn about the characteristic polynomial of a pencil generated by two hermitian matrices i abstract let a be an n n matrix write a h ik where i 2 1 and h and k are hermitian let f x y z det zi xh yk we first show that a pair of matrices over an algebraically closed field which satisfy quadratic polynomials can be put into block upper triangular form with diagonal blocks of size 1 1 or 2 2 via a simultaneous similarity this is used to prove that if f x y z g x y z n 2 where g has degree 2 then for some unitary matrix u the matrix u au is the direct sum of n 2 copies of a 2 2 matrix a 1 where a 1 is determined up to unitary similarity by the polynomial g x y z we use the connection between f x y z and the numerical range of a to investigate the case where f x y z has the form z ax y r g x y z s where g x y z is irreducible of degree 2

sets of matrices with given joint numerical range abstract the following question is addressed to what extent the n tuple of m m hermitian matrices is determined by its joint numerical range the cases m 2 n arbitrary and m n 3 are considered in detail

the capacity of hybrid quantum memory the general stable quantum memory unit is a hybrid consisting of a classical digit with a quantum digit qudit assigned to each classical state the shape of the memory is the vector of sizes of these qudits which may differ we determine when n copies of a quantum memory spl ascr embed in n 1 spl ogr 1 copies of another quantum memory spl bscr this relationship captures the notion that spl bscr is as at least as useful as spl ascr for all purposes in the bulk limit we show that the embeddings exist if and only if for all p spl ges 1 the p norm of the shape of spl ascr does not exceed the p norm of the shape of spl bscr the log of the p norm of the shape of spl ascr can be interpreted as the maximum of s spl rho h spl rho p quantum entropy plus discounted classical entropy taken over all mixed states spl rho on spl ascr we also establish a noiseless coding theorem that justifies these entropies the noiseless coding theorem and the bulk embedding theorem together say that either spl ascr blindly bulk encodes into spl bscr with perfect fidelity or a admits a state that does not visibly bulk encode into spl bscr with high fidelity in conclusion the utility of a hybrid quantum memory is determined by its simultaneous capacity for classical and quantum entropy which is not a finite list of numbers but rather a convex region in the classical quantum entropy plane

additivity of the classical capacity of entanglement breaking quantum channels we show that for the tensor product of an entanglement breaking quantum channel with an arbitrary quantum channel both the minimum entropy of an output of the channel and the holevo schumacher westmoreland capacity are additive in addition for the tensor product of two arbitrary quantum channels we give a bound involving entanglement of formation for the amount of subadditivity for minimum entropy output or superadditivity for classical capacity that can occur

an asymptotic error bound for testing multiple quantum hypotheses we consider the problem of detecting the true quantum state among r possible ones based of measurements performed on n copies of a finite dimensional quantum system a special case is the problem of discriminating between r probability measures on a finite sample space using n i i d observations in this classical setting it is known that the averaged error probability decreases exponentially with exponent given by the worst case binary chernoff bound between any possible pair of the r probability measures define analogously the multiple quantum chernoff bound considering all possible pairs of states recently it has been shown that this asymptotic error bound is attainable in the case of r pure states and that it is unimprovable in general here we extend the attainability result to a larger class of r tuples of states which are possibly mixed but pairwise linearly independent we also construct a quantum detector which universally attains the multiple quantum chernoff bound up to a factor 1 3

a goodness of fit test for logistic regression models based on case control data summary we test the logistic regression assumption under a case control sampling plan after reparameterisation the assumed logistic regression model is equivalent to a two sample semiparametric model in which the log ratio of two density functions is linear in data by identifying this model with a biased sampling model we propose a kolmogorov smirnovtype statistic to test the validity of the logistic link function moreover we point out that this test statistic can also be used in mixture sampling we present a bootstrap procedure along with some results on simulation and on analysis of two real datasets