temporal homogenization of linear odes with applications to parametric super resonance and energy harvest we consider the temporal homogenization of linear odes of the form dot x ax epsilon p t x f t where p t is periodic and epsilon is small using a 2 scale expansion approach we obtain the long time approximation x t approx rm exp at left omega t int 0 t rm exp a tau f tau rm d tau right where omega solves the cell problem dot omega epsilon b omega epsilon f t with an effective matrix b and an explicitly known f t we provide necessary and sufficient conditions for the accuracy of the approximation over a mathcal o epsilon 1 time scale and show how b can be computed at a cost independent of epsilon as a direct application we investigate the possibility of using rlc circuits to harvest the energy contained in small scale oscillations of ambient electromagnetic fields such as schumann resonances although a rlc circuit parametrically coupled to the field may achieve such energy extraction via parametric resonance its resistance r needs to be smaller than a threshold kappa proportional to the fluctuations of the field thereby limiting practical applications we show that if n rlc circuits are appropriately coupled via mutual capacitances or inductances then energy extraction can be achieved when the resistance of each circuit is smaller than n kappa hence if the resistance of each circuit has a non zero fixed value energy extraction can be made possible through the coupling of a sufficiently large number n of circuits n approx 1000 for the first mode of schumann resonances and contemporary values of capacitances inductances and resistances the theory is also applied to the control of the oscillation amplitude of a damped oscillator

optimal block diagonal l_2 scaling of matrices this paper is concerned with optimal one sided and two sided scaling of a matrix with respect to the class of block diagonal matrices optimality conditions are presented a series of lower bounds for the symmetric scaling problems is introduced then for some special cases the solutions are given in a closed form finally differentiability of the best condition number as a function of a symmetric matrix is discussed

acquiescent responding in balanced multidimensional scales and exploratory factor analysis personality tests often consist of a set of dichotomous or likert items these response formats are known to be susceptible to an agreeing response bias called acquiescence the common assumption in balanced scales is that the sum of appropriately reversed responses should be reasonably free of acquiescence however inter item correlation or covariance matrices can still be affected by the presence of variance due to acquiescence to analyse these correlation matrices we propose a method that is based on an unrestricted factor analysis and can be applied to multidimensional scales this method obtains a factor solution in which acquiescence response variance is isolated in an independent factor it is therefore possible without the potentially confounding effect of acquiescence to a examine the dominant factors related to content latent variables and b estimate participants factor scores on content latent variables this method which is illustrated by two empirical data examples has proved to be useful for improving the simplicity of the factor structure

statistical inference of minimum rank factor analysis for any given number of factors minimum rank factor analysis yields optimal communalities for an observed covariance matrix in the sense that the unexplained common variance with that number of factors is minimized subject to the constraint that both the diagonal matrix of unique variances and the observed covariance matrix minus that diagonal matrix are positive semidefinite as a result it becomes possible to distinguish the explained common variance from the total common variance the percentage of explained common variance is similar in meaning to the percentage of explained observed variance in principal component analysis but typically the former is much closer to 100 than the latter so far no statistical theory of mrfa has been developed the present paper is a first start it yields closed form expressions for the asymptotic bias of the explained common variance or more precisely of the unexplained common variance under the assumption of multivariate normality also the asymptotic variance of this bias is derived and also the asymptotic covariance matrix of the unique variances that define a mrfa solution the presented asymptotic statistical inference is based on a recently developed perturbation theory of semidefinite programming a numerical example is also offered to demonstrate the accuracy of the expressions

on the unsolvability of inverse eigenvalues problems almost everywhere abstract using sard s theorem we show that the inverse eigenvalue problems are unsolvable almost everywhere if too many of the eigenvalues are equal

invariant functionals and the uniqueness of invariant norms let be a representation of a compact group g on a banach space x the question we address is whether x carries a unique invariant norm in the sense that is the unique norm on x for which is a representation we characterize the uniqueness of norm in terms of the automatic continuity of the invariant functionals in the case when x is a dual banach space and is a x x continuous representation of g on x such that g consists of x x continuous operators we illustrate the usefulness of this characterization by studying the uniqueness of the norm on the spaces lp where is a locally compact hausdorff space equipped with a positive radon measure and g acts on as a group of continuous invertible measure preserving transformations

embedding smooth diffeomorphisms in flows abstract in this paper we study the problem on embedding germs of smooth diffeomorphisms in flows in higher dimensional spaces first we prove the existence of embedding vector fields for a local diffeomorphism with its nonlinear term a resonant polynomial then using this result and the normal form theory we obtain a class of local c k diffeomorphisms for k n which admit embedding vector fields with some smoothness finally we prove that for any k n under the coefficient topology the subset of local c k diffeomorphisms having an embedding vector field with some smoothness is dense in the set of all local c k diffeomorphisms

on robustness of the normal theory based asymptotic distributions of three reliability coefficient estimates this paper studies the asymptotic distributions of three reliability coefficient estimates sample coefficient alpha the reliability estimate of a composite score following a factor analysis and the estimate of the maximal reliability of a linear combination of item scores following a factor analysis results indicate that the asymptotic distribution for each of the coefficient estimates obtained based on a normal sampling distribution is still valid within a large class of nonnormal distributions therefore a formula for calculating the standard error of the sample coefficient alpha recently obtained by van zyl neudecker and nel applies to other reliability coefficients and can still be used even with skewed and kurtotic data such as are typical in the social and behavioral sciences

the asymptotic bias of minimum trace factor analysis with applications to the greatest lower bound to reliability in theory the greatest lower bound g l b to reliability is the best possible lower bound to the reliability based on single test administration yet the practical use of the g l b has been severely hindered by sampling bias problems it is well known that the g l b based on small samples even a sample of one thousand subjects is not generally enough may severely overestimate the population value and statistical treatment of the bias has been badly missing the only results obtained so far are concerned with the asymptotic variance of the g l b and of its numerator the maximum possible error variance of a test based on first order derivatives and the asumption of multivariate normality the present paper extends these results by offering explicit expressions for the second order derivatives this yields a closed form expression for the asymptotic bias of both the g l b and its numerator under the assumptions that the rank of the reduced covariance matrix is at or above the ledermann bound and that the nonnegativity constraints on the diagonal elements of the matrix of unique variances are inactive it is also shown that when the reduced rank is at its highest possible value i e the number of variables minus one the numerator of the g l b is asymptotically unbiased and the asymptotic bias of the g l b is negative the latter results are contrary to common belief but apply only to cases where the number of variables is small the asymptotic results are illustrated by numerical examples

identifiability of factor analysis some results and open problems abstract identifiability of the factor analysis model is discussed and some recent results and their relation to previous work are surveyed sufficient conditions for local identifiability are presented a counterexample shows that these conditions are not necessary in general a simple counting rule for local identifiability of the factor analysis model is given generically finally some open problems are formulated

identification of a linear system from inexact data a three variable example abstract in 1901 pearson formulated the general problem of how to fit a hyperplane in the most efficient way to a system of points in a data space this problem is still not exactly solved in all its generality as kalman and los have shown all statistical attemps to solve the problem have failed because each of them can provide only prejudicial and statistical but not objective and mathematical solutions however exact mathematical solutions do exist for special cases this paper s main principle of linear identification from inexact data provides the mathematical framework in which the problem and the deficiencies of the statistical solutions are conveniently discussed in particular those of the least squares regression and statistical common factors schemes it will be argued that the exact common factors or frisch scheme offers most promise to direct us to complete and exact solutions even though it imposes severe restrictions on the orders of the systems because of wilson s inequality throughout this paper the problem and its various solution schemes are illustrated by an empirical example consisting of three data variables describing the profitability performance of some large u s bank holding companies for this empirical example the frisch scheme provides a unique solution contrary to some earlier pessimistic conclusions

retrieving the correlation matrix from a truncated pca solution the inverse principal component problem whenr principal components are available fork variables the correlation matrix is approximated in the least squares sense by the loading matrix times its transpose the approximation is generally not perfect unlessr k in the present paper it is shown that whenr is at or above the ledermann bound r principal components are enough to perfectly reconstruct the correlation matrix albeit in a way more involved than taking the loading matrix times its transpose in certain cases just below the ledermann bound recovery of the correlation matrix is still possible when the set of all eigenvalues of the correlation matrix is available as additional information

nonuniform dichotomy spectrum and normal forms for nonautonomous differential systems abstract the aim of this paper is to study the normal forms of nonautonomous differential systems for doing so we first investigate the nonuniform dichotomy spectrum of the linear evolution operators that admit a nonuniform exponential dichotomy where the linear evolution operators are defined by nonautonomous differential equations x a t x in r n using the nonuniform dichotomy spectrum we obtain the normal forms of the nonautonomous linear differential equations finally we establish the finite jet normal forms of the nonlinear differential systems x a t x f t x in r n which is based on the nonuniform dichotomy spectrum and the normal forms of the nonautonomous linear systems

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

generic global indentification in factor analysis abstract the paper proves global indentification of the unique variances in factor analysis for almost all parameter points if the number of factors is smaller than the ledermann bound it also shows how to compute using a closed form representation the second solution in the case of six variables and three factors

bayesian estimation of sparse dynamic factor models with order independent and ex post mode identification abstract common variation in n series is extracted into k n dynamic factors we induce sparsity by using a zero point mass normal mixture prior distribution on the loadings estimation and rotational identification are independent of variable ordering sparsity helps identifying the factor space and the factors rotational identification including factor order and sign is obtained by processing the posterior output and based on factor draws rather than factor loading draws simulating data we document sampler and estimation efficiency to illustrate we estimate the model for a large panel of swiss macroeconomic and detailed price data we identify 16 factors with a clear economic interpretation

poincare type theorems for non autonomous systems in this paper we establish analytic equivalence theorems of poincare and poincare dulac type for analytic non autonomous differential systems based on the dichotomy spectrum of their linear part as applications of the theorem normal forms linearize for two illustrative examples

complex factor analysis and extensions many subspace based array signal processing algorithms assume that the noise is spatially white in this case the noise covariance matrix is a multiple of the identity and the eigenvectors of the data covariance matrix are not affected by it if the noise covariance is an unknown arbitrary diagonal e g for an uncalibrated array the eigenvalue decomposition leads to incorrect subspace estimates and it has to be replaced by a more general factor analysis decomposition fa which then reveals all relevant information we consider this data model and several extensions where the noise covariance matrix has a more general structure such as banded sparse block diagonal and cases where we have multiple data covariance matrices that share the same noise covariance matrix starting from a nonlinear weighted least squares formulation we propose new estimation algorithms for both classical fa and its extensions the optimization is based on gauss newton gradient descent generally this leads to an iteration involving the inversion of a very large matrix using the structure of the problem we show how this can be reduced to the inversion of a matrix with dimension equal to the number of unknown noise covariance parameters this results in new algorithms that have faster numerical convergence and lower complexity compared to several maximum likelihood based algorithms that could be considered state of the art the new algorithms scale well to large dimensions and can replace eigenvalue decompositions in many applications even if the noise can be assumed to be white

weighted minimum trace factor analysis in the last decade several authors discussed the so called minimum trace factor analysis mtfa which provides the greatest lower bound g l b to reliability however the mtfa fails to be scale free in this paper we propose to solve the scale problem by maximization of the g l b as the function of weights closely related to the primal problem of the g l b maximization is the dual problem we investigate the primal and dual problems utilizing convex analysis techniques the asymptotic distribution of the maximal g l b is obtained provided the population covariance matrix satisfies sone uniqueness and regularity assumptions finally we outline computational algorithms and consider numerical examples

the prejudices of least squares principal components and common factors schemes we prove mathematically that the least squares regression scheme is of little use for identification from inexact data even in the low noise case its results depend solely on the prejudice concerning which subsets of variables are chosen as regressands and which as regressors from a given set of data variables in practice least squares regression results are always biased and depend on the relative noise levels while even the signs of the estimates are completely determined by the arbitrary choice of the regressands if the number of underlying relationships is misspecified the principal components or statistical common factor scheme fares a similar fate since the choice of how many principal components or common factors to retain is essentially prejudiced and not determined by the data both schemes produce artificial and unexplained correlations among the residuals for the principal components scheme this occurs because it usually violates wilson s inequality a fortiori we prove exactly why in practice factor indeterminancy occurs the exact ideal multiple common factor scheme of frisch and thurstone has not yet been solved except for small numbers six and less of data variables

test theory and psychometrika the past twenty five years on the occasion ofpsychometrika s fiftieth anniversary the past twenty five years developments in mental test theory are reviewed with special emphasis on the topics receiving attention in the pages of this journal analogous reviews forpsychometrika s first quarter century were given by gulliksen and guilford in 1961 much of the recent progress in test theory and in other branches of psychometrics as well has been made by treating the problems in this field as being essentially ones of statistical inference it is concluded that a research in test theory is in a healthy state and b psychometrika is an important source of information about that research

uniqueness of translation invariant norms let a be a banach function space and let m be a family of multipliers on a we provide conditions on m so that the original topology of a is the only complete norm topology on a making all of the maps from m continuous as a corollary we show that for a compact abelian group g and a circle group t for a lp t 1 p the lp norm is the only one that makes all translations continuous while for a c g a l g or a l1 g there are other norms with that property for noncompact groups the situation is different on the space l1 r the l1 norm is the only one that makes a single nontrivial translation continuous

alpha dimension free and model based internal consistency reliability as pointed out by sijtsma in press coefficient alpha is inappropriate as a single summary of the internal consistency of a composite score better estimators of internal consistency are available in addition to those mentioned by sijtsma an old dimension free coefficient and structural equation model based coefficients are proposed to improve the routine reporting of psychometric internal consistency the various ways to measure internal consistency are also shown to be appropriate to binary and polytomous items

koenigs embedding flow problem with global c1 smoothness which c1 orientation preserving diffeomorphisms on the compact interval 0 1 that of course fixing the end points can be embedded into a c1 flow on 0 1 is a difficult problem in the iteration theory such maps were proved to be rare in some references in this paper this problem is reduced to the relations between solutions of an ordinary differential equation and solutions of a functional equation by discussing those relations we describe a relatively wide class of orientation preserving diffeomorphisms on 0 1 which can achieve such a c1 flow our results are demonstrated by practical calculation for polygons

analytic normalization of analytic integrable systems and the embedding flows abstract this paper provides the normal forms of analytic integrable differential systems and diffeomorphisms via analytic normalizations furthermore we consider the existence of embedding flows of an analytic integrable diffeomorphism

the greatest lower bound to the reliability of a test and the hypothesis of unidimensionality to assess the reliability of congeneric tests specifically designed reliability measures have been proposed this paper emphasizes that such measures rely on a unidimensionality hypothesis which can neither be confirmed nor rejected when there are only three test parts and will invariably be rejected when there are more than three test parts jackson and agunwamba s 1977 greatest lower bound to reliability is proposed instead although this bound has a reputation for overestimating the population value when the sample size is small this is no reason to prefer the unidimensionality based reliability firstly the sampling bias problem of the glb does not play a role when the number of test parts is small as is often the case with congeneric measures secondly glb and unidimensionality based reliability are often equal when there are three test parts and when there are more test parts their numerical values are still very similar to the extent that the bias problem of the greatest lower bound does play a role unidimensionality based reliability is equally affected although unidimensionality and reliability are often thought of as unrelated this paper shows that from at least two perspectives they act as antagonistic concepts a measure based on the same framework that led to the greatest lower bound is discussed for assessing how close is a set of variables to unidimensionality it is the percentage of common variance that can be explained by a single factor an empirical example is given to demonstrate the main points of the paper

moment inequalities for sums of random matrices and their applications in optimization in this paper we consider various moment inequalities for sums of random matrices which are well studied in the functional analysis and probability theory literature and demonstrate how they can be used to obtain the best known performance guarantees for several problems in optimization first we show that the validity of a recent conjecture of nemirovski is actually a direct consequence of the so called non commutative khintchine s inequality in functional analysis using this result we show that an sdp based algorithm of nemirovski which is developed for solving a class of quadratic optimization problems with orthogonality constraints has a logarithmic approximation guarantee this improves upon the polynomial approximation guarantee established earlier by nemirovski furthermore we obtain improved safe tractable approximations of a certain class of chance constrained linear matrix inequalities secondly we consider a recent result of delage and ye on the so called data driven distributionally robust stochastic programming problem one of the assumptions in the delage ye result is that the underlying probability distribution has bounded support however using a suitable moment inequality we show that the result in fact holds for a much larger class of probability distributions given the close connection between the behavior of sums of random matrices and the theoretical properties of various optimization problems we expect that the moment inequalities discussed in this paper will find further applications in optimization

uniqueness of norm on lp g and c g when g is a compact group let g be a compact group if the trivial representation of g is not weakly contained in the left regular representation of g on l 2 g and x is either l p g for 1oppn or c g then we show that every complete norm jj on x that makes translations from x jj into itself continuous is equivalent to jjjj p or jjjj n respectively if 1oppn and every left invariant linear functional on l p g is a constant multiple of the haar integral then we show that every complete norm jj on l p g that makes translations from l p g jj into itself continuous and that makes the map t lt from g into l l p g jj bounded is equivalent to jjjj p

three mode factor analysis by means of candecomp parafac a three mode covariance matrix contains covariances of n observations e g subject scores on j variables for k different occasions or conditions we model such an jk jk covariance matrix as the sum of a common covariance matrix having candecomp parafac form and a diagonal matrix of unique variances the candecomp parafac form is a generalization of the two mode case under the assumption of parallel factors we estimate the unique variances by minimum rank factor analysis the factors can be chosen oblique or orthogonal our approach yields a model that is easy to estimate and easy to interpret moreover the unique variances the factor covariance matrix and the communalities are guaranteed to be proper a percentage of explained common variance can be obtained for each variable condition combination and the estimated model is rotationally unique under mild conditions we apply our model to several datasets in the literature and demonstrate our estimation procedure in a simulation study

discontinuous translation invariant functionals let g be an infinite a compact locally compact group we shall study the existence of many discontinuous translation invariant linear functionals on a variety of translation invariant frechet spaces of radon measures on g these spaces include the convolution measure algebra m g the lebesgue spaces lv g where i p oo and certain combinations thereof among other things it will be shown that c g has many discontinuous translation invariant functionals provided that g is amenable this solves a problem of g h meisters let g be a locally compact group in the present paper we shall consider various translation invariant spaces of functions and measures on g and prove the existence of discontinuous invariant functionals on such spaces among other things it will be shown that c g admits many discontinuous left invariant functionals provided that g is an infinite a compact locally compact amenable group our hypotheses are a little weaker than amenability this resolves the problem of g h meisters 11 as to whether there exists a discontinuous invariant functional on c qu where h denotes the circle group as far as the compact groups are concerned the problems for c g and for l g are equivalent see remark iv stated below as some basic references on this subject we refer to e granirer s papers 3 4 on invariant means the nice survey article 14 by meisters and g s woodward s paper 22 on discontinuous invariant functionals on a variety of invariant spaces let g be an arbitrary locally compact lc group all of the lebesgue spaces lp g 1 s p x are taken with respect to a fixed left haar measure ag on g if g is compact we normalize ag so that xg g 1 let c g and cc g denote the banach space of all continuous bounded complex valued functions on g and the subspace of all members of c g with compact supports respectively we denote by co g the uniform closure of cc g in c g thus the dual of co g may be identified with the convolution measure algebra m g of g cf hewitt and ross 7 a net fa in cc g converges to the zero function if and only if i iaf 0 uniformly on g and ii the f s are eventually supported by a fixed compact set depending on the net this topology for cjg is different from the uniform topology and will be used throughout the paper unless otherwise mentioned the continuous dual of cc g will be denoted c g an element of c g is often called received by the editors february 8 1983 and in revised form april 19 1983 1980 mathematics subject classification primary 43a15 secondary 43a05

analytic integrable systems analytic normalization and embedding flows abstract in this paper we mainly study the existence of analytic normalization and the normal form of finite dimensional complete analytic integrable dynamical systems more details we will prove that any complete analytic integrable diffeomorphism f x b x f x in c n 0 with b having eigenvalues not modulus 1 and f x o x 2 is locally analytically conjugate to its normal form meanwhile we also prove that any complete analytic integrable differential system x a x f x in c n 0 with a having nonzero eigenvalues and f x o x 2 is locally analytically conjugate to its normal form furthermore we will prove that any complete analytic integrable diffeomorphism defined on an analytic manifold can be embedded in a complete analytic integrable flow we note that parts of our results are the improvement of moser s one in j moser the analytic invariants of an area preserving mapping near a hyperbolic fixed point comm pure appl math 9 1956 673 692 and of poincare s one in h poincare sur l integration des equations differentielles du premier order et du premier degre ii rend circ mat palermo 11 1897 193 239 these results also improve the ones in xiang zhang analytic normalization of analytic integrable systems and the embedding flows j differential equations 244 2008 1080 1092 in the sense that the linear part of the systems can be nonhyperbolic and the one in n t zung convergence versus integrability in poincare dulac normal form math res lett 9 2002 217 228 in the way that our paper presents the concrete expression of the normal form in a restricted case

an upper bound on the number of linear relations identified from noisy data by the frisch scheme abstract the frisch scheme is one of many approaches to the identification of linear relations from noisy data solution of the associated matrix optimization problem leads to the identification of a maximal number of such relations to date no general solution procedure is known here an upper bound on the maximal number of relations that can be identified is presented the upper bound is derived using the theory of schur complements in conjunction with a result for the case where only one relation can be identified the possibility of developing a completely general solution procedure using these ideas is also briefly discussed

pre scaling optimization for space shift keying based on semidefinite relaxation the performance of space shift keying ssk is known to be dominated by the minimum euclidean distance med in the received ssk constellation in this paper we propose a method of enhancing the med in the received ssk constellation and improving both the attainable performance and the power efficiency by means of symbol scaling at the transmitter to this aim we formulate a pair of optimization problems one for maximizing the med subject to a specific transmit power constraint and one for minimizing the transmit power subject to a med threshold as these problems are np hard we re formulate their optimization using semidefinite relaxations which results in convex problem formulations that can be efficiently solved using standard approaches moreover we design pre scaling techniques for imperfect channel state information at the transmitter where the existing approaches are inapplicable our results show that the proposed schemes substantially improve the power efficiency of ssk systems with respect to state of the art techniques by offering an improved performance for specific transmit power requirements or equivalently a transmit power reduction for a given med threshold

statistical inference of semidefinite programming in this paper we consider covariance structural models with which we associate semidefinite programming problems we discuss statistical properties of estimates of the respective optimal value and optimal solutions when the true covariance matrix is estimated by its sample counterpart the analysis is based on perturbation theory of semidefinite programming as an example we consider asymptotics of the so called minimum trace factor analysis we also discuss the minimum rank matrix completion problem and its sdp counterparts

shift invariant signal norms for fault detection and control two desirable properties of a signal norm to be used for the purpose of optimal control or fault detection are highlighted these properties are shift invariance and the permitting of persistent signals a class of norms called the window norms is analyzed among norms considered for control purposes they appear to be the only ones save the l norm that satisfy both properties the signal spaces defined by the window norms however include some interesting signals that are not in the l space the window norms have been found suitable for application in fault detection and are here also considered for optimal control it is shown that they can be taken as a support for the concept of l1 control but may also suggest a new class of optimal controllers

quantile lower bounds to reliability based on locally optimal splits extending the theory of lower bounds to reliability based on splits given by guttman in psychometrika 53 63 70 1945 this paper introduces quantile lower bound coefficients 4 q that refer to cumulative proportions of potential locally optimal split half coefficients that are below a particular point q in the distribution of split halves based on different partitions of variables into two sets interesting quantile values are q 0 05 0 50 0 95 1 00 with 4 0 05 4 0 50 4 0 95 4 1 0 only the global optimum 4 1 0 guttman s maximal 4 has previously been considered to be interesting but in small samples it substantially overestimates population reliability the three coefficients 4 0 05 4 0 50 and 4 0 95 provide new lower bounds to reliability the smallest 4 0 05 provides the most protection against capitalizing on chance associations and thus overestimation 4 0 50 is the median of these coefficients while 4 0 95 tends to overestimate reliability but also exhibits less bias than previous estimators computational theory algorithm and publicly available code based in r are provided to compute these coefficients simulation studies evaluate the performance of these coefficients and compare them to coefficient alpha and the greatest lower bound under several population reliability structures

the embedding flows of c hyperbolic diffeomorphisms abstract in weigu li j llibre xiang zhang extension of floquet s theory to nonlinear periodic differential systems and embedding diffeomorphisms in differential flows amer j math 124 2002 107 127 we proved that for a germ of c hyperbolic diffeomorphisms f x a x f x in r n 0 if a has a real logarithm with its eigenvalues weakly nonresonant then f x can be embedded in a c autonomous differential system its proof was very complicated which involved the existence of embedding periodic vector field of f x and the extension of the floquet s theory to nonlinear c periodic differential systems in this paper we shall provide a simple and direct proof to this last result next we shall show that the weakly nonresonant condition in the last result on the real logarithm of a is necessary for some c diffeomorphisms f x a x f x to have c embedding flows finally we shall prove that a germ of c hyperbolic diffeomorphisms f x a x f x with f x o x 2 in r 2 0 has a c embedding flow if and only if either a has no negative eigenvalues or a has two equal negative eigenvalues and it can be diagonalizable

computational aspects of the greatest lower bound to the reliability and constrained minimum trace factor analysis in the last decade several algorithms for computing the greatest lower bound to reliability or the constrained minimum trace communality solution in factor analysis have been developed in this paper convergence properties of these methods are examined instead of using lagrange multipliers a new theorem is applied that gives a sufficient condition for a symmetric matrix to be gramian whereas computational pitfalls for two methods suggested by woodhouse and jackson can be constructed it is shown that a slightly modified version of one method suggested by bentler and woodward can safely be applied to any set of data a uniqueness proof for the solution desired is offered

formal first integrals for periodic systems in the present paper we give an elementary proof for the result of li et al 2003 6 about nonexistence of formal first integrals for periodic systems in a neighborhood of a constant solution moreover we present a criterion about partial existence of formal first integrals for the periodic system by using the floquet s theory

a new approach to center conditions for simple analytic monodromic singularities abstract in this paper we present an alternative algorithm for computing poincare lyapunov constants of simple monodromic singularities of planar analytic vector fields based on the concept of inverse integrating factor simple monodromic singular points are those for which after performing the first generalized polar blow up there appear no singular points in other words the associated poincare return map is analytic an improvement of the method determines a priori the minimum number of poincare lyapunov constants which must cancel to ensure that the monodromic singularity is in fact a center when the explicit laurent series of an inverse integrating factor is known in generalized polar coordinates several examples show the usefulness of the method

convergence of estimates of unique variances in factor analysis based on the inverse sample covariance matrix if the ratio m p tends to zero where m is the number of factors m and p the number of observable variables then the inverse diagonal element of the inverted observable covariance matrix pjj 1 tends to the corresponding unique variance jj for almost all of these guttman 1956 if the smallest singular value of the loadings matrix from common factor analysis tends to infinity as p increases then m p tends to zero the same condition is necessary and sufficient for pjj 1 to tend to jj for all of these several related conditions are discussed psycinfo database record c 2009 apa all rights reserved journal abstract

minimum rank and minimum trace of covariance matrices this paper considers some mathematical aspects of minimum trace factor analysis mtfa the uniqueness of an optimal point of mtfa is proved and necessary and sufficient conditions for a point x to be optimal are established finally some results about the connection between mtfa and the classical minimum rank factor analysis will be presented

rank reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis one of the intriguing questions of factor analysis is the extent to which one can reduce the rank of a symmetric matrix by only changing its diagonal entries we show in this paper that the set of matrices which can be reduced to rankr has positive lebesgue measure if and only ifr is greater or equal to the ledermann bound in other words the ledermann bound is shown to bealmost surely the greatest lower bound to a reduced rank of the sample covariance matrix afterwards an asymptotic sampling theory of so called minimum trace factor analysis mtfa is proposed the theory is based on continuous and differential properties of functions involved in the mtfa convex analysis techniques are utilized to obtain conditions for differentiability of these functions

multiple hopf bifurcation in r3 and inverse jacobi multipliers abstract in this paper we study the maximum number of limit cycles that can bifurcate from a singular point of saddle focus type of an analytic autonomous differential system in r 3 under any analytic perturbation that keeps the location and nature of the singularity we only consider those foci on center manifolds having associated two nonzero purely imaginary and one nonzero real eigenvalues our approach is different from the classical one in the sense that we do not use any center manifold reduction to compute poincare lyapunov constants instead we study the multiple hopf bifurcation first doing a lyapunov schmidt reduction to the associated poincare map obtaining in this way an analytic reduced displacement map next we prove that the order of this displacement map coincides with the vanishing multiplicity denoted m of any locally smooth and non flat inverse jacobi multiplier finally the cyclicity of the focus is given in terms of m

non self adjoint jacobi matrices with a rank one imaginary part we develop direct and inverse spectral analysis for finite and semi infinite non self adjoint jacobi matrices with a rank one imaginary part it is shown that given a set of n not necessarily distinct nonreal numbers in the open upper lower half plane uniquely determines an n n jacobi matrix with a rank one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity algorithms of reconstruction for such finite jacobi matrices are presented a new model complementing the well known livsic triangular model for bounded linear operators with a rank one imaginary part is obtained it turns out that the model operator is a non self adjoint jacobi matrix we show that any bounded prime non self adjoint linear operator with a rank one imaginary part acting on some finite dimensional respectively separable infinite dimensional hilbert space is unitarily equivalent to a finite respectively semi infinite non self adjoint jacobi matrix this obtained theorem strengthens a classical result of stone established for self adjoint operators with simple spectrum we establish the non self adjoint analogs of the hochstadt and gesztesy simon uniqueness theorems for finite jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non self adjoint tri diagonal matrices to the case of mixed eigenvalues real and nonreal a unique jacobi matrix unitarily equivalent to the operator of integration ff x 2i xlf t dt in the hilbert space l2 0 l is found as well as spectral properties of its perturbations and connections with the well known bernoulli numbers we also give the analytic characterization of the weyl functions of dissipative jacobi matrices with a rank one imaginary part

rank constrained optimization and its applications abstract this paper investigates an iterative approach to solve the general rank constrained optimization problems rcops defined to optimize a convex objective function subject to a set of convex constraints and rank constraints on unknown rectangular matrices in addition rank minimization problems rmps are introduced and equivalently transformed into rcops by introducing a quadratic matrix equality constraint the rank function is discontinuous and nonconvex thus the general rcops are classified as np hard in most of the cases an iterative rank minimization irm method with convex formulation at each iteration is proposed to gradually approach the constrained rank the proposed irm method aims at solving rcops with rank inequalities constrained by upper or lower bounds as well as rank equality constraints proof of the convergence to a local minimizer with at least a sublinear convergence rate is provided four representative applications of rcops and rmps including system identification output feedback stabilization and structured h 2 controller design problems are presented with comparative simulation results to verify the feasibility and improved performance of the proposed irm method

applications of kazhdan s property t to automatic continuity let g be a compact group which has the strong kazhdan s property t and let be an isometric representation of g on l p for some measure space and 1 p we characterize the continuity of all the linear operators from l p which commute with translations and the property of p being the unique norm up to equivalence on l p with respect to which all translations are continuous in terms of the finite dimensionality of the subspace of all invariant func

asymptotic theory of overparameterized structural models abstract a theory of overparameterized structural models is presented in such a model some redundant parameters are involved the parameter vector is not identified and the information matrix is not nonsingular the minimum discrepancy function mdf test statistic is shown to have an asymptotic chi squared distribution almost everywhere for a wide class of discrepancy functions asymptotic distribution properties of the mdf estimators are investigated the factor analysis model is discussed as an example

on similarity of quasinilpotent operators bounded linear operators on separable banach spaces algebraically similar to the classical volterra operator v acting on c 0 1 are characterized from this characterization it follows that v does not determine the topology of c 0 1 which answers a question raised by armando villena a sufficient condition for an injective bounded linear operator on a banach space to determine its topology is obtained from this condition it follows for instance that the volterra operator acting on the hardy space hp of the unit disk determines the topology of hp for any p 1

rank regularized estimation of approximate factor models abstract it is known that the common factors in a large panel of data can be consistently estimated by the method of principal components and principal components can be constructed by iterative least squares regressions replacing least squares with ridge regressions turns out to have the effect of removing the contribution of factors associated with small singular values from the common component the method has been used in the machine learning literature to recover low rank matrices we study the procedure from the perspective of estimating an approximate factor model under the rank constraint the common component is estimated by the space spanned by factors whose singular values exceed a threshold the desire for minimum rank and parsimony lead to a data dependent penalty for selecting the number of factors the new criterion is more conservative than the existing deterministic penalties and is appropriate when the nominal number of factors is inflated by the presence of weak factors or large measurement noise we provide asymptotic results that can be used to test economic hypotheses

a numerical approach to the approximate and the exact minimum rank of a covariance matrix a concept of approximate minimum rank for a covariance matrix is defined which contains the exact minimum rank as a special case a computational procedure to evaluate the approximate minimum rank is offered the procedure yields those proper communalities for which the unexplained common variance ignored in low rank factor analysis is minimized the procedure also permits a numerical determination of the exact minimum rank of a covariance matrix within limits of computational accuracy a set of 180 covariance matrices with known or bounded minimum rank was analyzed the procedure was successful throughout in recovering the desired rank

first integrals and normal forms for germs of analytic vector fields abstract for a germ of analytic vector fields the existence of first integrals resonance and the convergence of normalization transforming the vector field to a normal form are closely related in this paper we first provide a link between the number of first integrals and the resonant relations for a quasi periodic vector field which generalizes one of the poincare s classical results h poincare sur l integration des equations differentielles du premier order et du premier degre i and ii rend circ mat palermo 5 1891 161 191 11 1897 193 239 on autonomous systems and theorem 5 of weigu li j llibre xiang zhang local first integrals of differential systems and diffeomorphism z angew math phys 54 2003 235 255 on periodic systems then in the space of analytic autonomous systems in c 2 n with exactly n resonances and n functionally independent first integrals our results are related to the convergence and generic divergence of the normalizations lastly for a planar hamiltonian system it is well known that the system has an isochronous center if and only if it can be linearizable in a neighborhood of the center using the euler lagrange equation we provide a new approach to its proof

commentary on coefficient alpha a cautionary tale the general use of coefficient alpha to assess reliability should be discouraged on a number of grounds the assumptions underlying coefficient alpha are unlikely to hold in practice and violation of these assumptions can result in nontrivial negative or positive bias structural equation modeling was discussed as an informative process both to assess the assumptions underlying coefficient alpha and to estimate reliability

the rank of reduced dispersion matrices psychometricians working in factor analysis and econometricians working in regression with measurement error in all variables are both interested in the rank of dispersion matrices under variation of the diagonal elements psychometricians concentrate on cases in which low rank can be attained preferably rank one the spearman case econometricians cocentrate on cases in which the rank cannot be reduced below the number of variables minus one the frisch case in this paper we give an extensive historial discussion of both fields we prove the two key results in a more satisfactory and uniform way we point out various small errors and misunderstandings and we present a methodological comparison of factor analysis and regression on the basis of our results

the strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications for a locally optimal solution to the nonlinear semidefinite programming problem under robinson s constraint qualification the following conditions are proved to be equivalent the strong second order sufficient condition and constraint nondegeneracy the nonsingularity of clarke s jacobian of the karush kuhn tucker system the strong regularity of the karush kuhn tucker point and others

linear models based on noisy data and the frisch scheme we address the problem of identifying linear relations among variables based on noisy measurements this is a central question in the search for structure in large data sets often a key assumption is that measurement errors in each variable are independent this basic formulation has its roots in the work of charles spearman in 1904 and of ragnar frisch in the 1930s various topics such as errors in variables factor analysis and instrumental variables all refer to alternative viewpoints on this problem and on ways to account for the anticipated way that noise enters the data in the present paper we begin by describing certain fundamental contributions by the founders of the field and provide alternative modern proofs to certain key results we then go on to consider a modern viewpoint and novel numerical techniques to the problem the central theme is expressed by the frisch kalman dictum which calls for identifying a noise contribution that allows a maximal number of simultaneous linear relations a

brief identification of static errors in variables models the rank reducibility problem the problem of identifying linear static relations from noisy data is investigated under the assumptions of the frisch scheme the work presents an attempt to simplify the search for solutions of the related rank reducibility problem the proposed approach is based on a relaxation of the original problem by means of algebraic manipulations of the non linear equations which characterize the solution set it is shown that this method can be advantageously used when particular conditions regarding the number of variables and the number of linear relations are satisfied

identification and estimation of dynamic errors in variables models abstract we consider here the factor analysis approach to the analysis of linear dynamic errors in variables models identification of the most general models and the question of nonuniqueness are discussed we show how these models are generalizations of the total least squares model and how the grassmann manifold provides a natural setting for their analysis under certain assumptions we analyze the problem of maximum likelihood estimation and indicate how this involves maximization of a functional on a grassmann manifold

modeling individualized coefficient alpha to measure quality of test score data individualized coefficient alpha is defined it is item and subject specific and is used to measure the quality of test score data with heterogenicity among the subjects and items a regression model is developed based on 3 sets of generalized estimating equations the first set of generalized estimating equation models the expectation of the responses the second set models the response s variance and the third set is proposed to estimate the individualized coefficient alpha defined and used to measure individualized internal consistency of the responses we also use different techniques to extend our method to handle missing data asymptotic property of the estimators is discussed based on which inference on the coefficient alpha is derived performance of our method is evaluated through simulation study and real data analysis the real data application is from a health literacy study in hunan province of china

extension of floquet s theory to nonlinear periodic differential systems and embedding diffeomorphisms in differential flows this paper has two parts in the first one we generalize the floquet theory to nonlinear periodic differential systems in the second part we apply the first one to obtain a sufficient condition in order that a germ of diffeomorphism can be embedded in a differential flow