the jacobi rosochatius problem on an ellipsoid the lax representations and billiards in this article we construct the lax representations of the geodesic flow the jacobi rosochatius problem and its perturbations by means of separable polynomial potentials on an ellipsoid we prove complete integrability in the case of a generic symmetric ellipsoid and describe analogous systems on complex projective spaces also we consider billiards within an ellipsoid under the influence of the hook and rosochatius potentials between the impacts a geometric interpretation of the integrability analogous to the classical chasles and poncelet theorems is given

nonnegative matrix factorization and i divergence alternating minimization abstract in this paper we consider the nonnegative matrix factorization nmf problem given an elementwise nonnegative matrix v r m n find for assigned k nonnegative matrices w r m k and h r k n such that v wh exact nontrivial nonnegative factorizations do not always exist hence it is interesting to pose the approximate nmf problem the criterion which is commonly employed is i divergence between nonnegative matrices the problem becomes that of finding for assigned k the factorization wh closest to v in i divergence an iterative algorithm em like for the construction of the best pair w h has been proposed in the literature in this paper we interpret the algorithm as an alternating minimization procedure a la csiszar tusnady and investigate some of its stability properties nmf is widespreading as a data analysis method in applications for which the positivity constraint is relevant there are other data analysis methods which impose some form of nonnegativity we discuss here the connections between nmf and archetypal analysis

an error bound for a discrete reduced order model of a linear multivariable system the design of feasible controllers for high dimension multivariable systems can be greatly aided by a method of model reduction in order for the design based on the order reduction to include a guarantee of stability it is sufficient to have a bound on the model error previous work has provided such a bound for continuous time systems for algorithms based on balancing in this note an linftybound is derived for model error for a method of order reduction of discrete linear multivariable systems based on balancing

hyperelliptic jacobians as billiard algebra of pencils of quadrics beyond poncelet porisms the thirty years old programme of griffiths and harris of understanding higher dimensional analogues of poncelet type problems and synthetic approach to higher genera addition theorems has been settled and completed in this paper starting with the observation of the billiard nature of some classical constructions and configurations we construct the billiard algebra that is a group structure on the set t of lines simultaneously tangent to d 1 quadrics from a given confocal family in the d dimensional euclidean space using this tool the related results of reid donagi and knorrer are further developed realized and simplified we derive a fundamental property of t any two lines from this set can be obtained from each other by at most d 1 billiard reflections at some quadrics from the confocal family we introduce two hierarchies of notions s skew lines in t and s weak poncelet trajectories s 1 0 d 2 the interrelations between billiard dynamics linear subspaces of intersections of quadrics and hyperelliptic jacobians developed in this paper enabled us to obtain higher dimensional and higher genera generalizations of several classical genus 1 results cayley s theorem weyr s theorem griffiths harris theorem and darboux theorem

billiard systems on quadric surfaces and the poncelet theorem billiard systems constrained to move on a quadric surface with confocal quadric boundaries were studied the trajectory of a billiard is described as a collection of geodesic segments joining on the boundary at the joining points the trajectory obeys the law of reflection it was found that the geodesic segments are all tangent to a common confocal quadric curve a caustic on this quadric surface if a trajectory is closed then all trajectories sharing the same caustic quadric are all closed and have the same period and length thus a generalization of poncelet s theorem on a quadric surface is achieved the elliptical billiard systems on spheres and pseudospheres were studied and results were obtained that are similar to those on a plane the results can be extended to n dimensions

balanced model reduction via the proper orthogonal decomposition a new method for performing a balanced reduction of a high order linear system is presented the technique combines the proper orthogonal decomposition and concepts from balanced realization theory the method of snapshotsisused to obtainlow rank reduced rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain theapproximationsarethenusedtoobtainabalancedreducedorder model the method is particularly effective when a small number of outputs is of interest it is demonstrated for a linearized high order system that models unsteady motion of a two dimensional airfoil computation of the exact grammians would be impractical for such a large system for this problem very accurate reducedorder models are obtained that capture the required dynamics with just three states the new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition although further development is necessary the concept also extends to nonlinear systems

approximate solution of large sparse lyapunov equations describes a simple method for efficiently estimating the dominant eigenvalues and eigenvectors of the solution to a lyapunov equation without first solving the equation explicitly the method is based on the power method and matrix vector multiplications and is particularly suitable for problems where those multiplications can be done efficiently such as where the coefficient matrices are large and sparse or low rank the same idea is directly applicable to balanced truncation order reduction of linear systems

ellipsoidal billiards in pseudo euclidean spaces and relativistic quadrics abstract we study the geometry of confocal quadrics in pseudo euclidean spaces of arbitrary dimension d and any signature and related billiard dynamics the goal is to give a complete description of periodic billiard trajectories within ellipsoids the novelty of our approach is the introduction of a new discrete combinatorial geometric structure associated with a confocal pencil of quadrics a colouring in d colours this is used to decompose quadrics of d 1 geometric types of a pencil into new relativistic quadrics of d relativistic types a study of what we term discriminant sets of tropical lines and and their singularities provides insight into the related geometry and combinatorics this yields an analytic criterion describing all periodic billiard trajectories including light like trajectories as a case of special interest

on the partial realization problem abstract in this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra most of which were originally developed in other contexts we approach the partial realization problem from several different angles and explore the connections to such topics as factoriza tion of hankel matrices block tridiagonalization generalizations of the lanczos process for biorthogonalization the euclidean algorithm and the principal part con tinued fractions of arne magnus the pade table and the berlekamp masseyalgo rithm in this way we are able to clarify some previous results by rissanen kalman and others and place them in a broader context this leads to several results and concepts which we think are new our analysis is restricted to the scalar case but some definitions and formulations have been rigged to facilitate an extension to the matrix case

lumpable hidden markov models model reduction and reduced complexity filtering this paper is concerned with filtering of hidden markov processes hmp which possess or approximately possess the property of lumpability this property is a generalization of the property of lumpability of a markov chain which has been previously addressed by others in essence the property of lumpability means that there is a partition of the atomic states of the markov chain into aggregated sets which act in a similar manner as far as the state dynamics and observation statistics are concerned we prove necessary and sufficient conditions on the hmp for exact lumpability to hold for a particular class of hidden markov models hmm namely finite output alphabet models conditions for lumpability of all hmp representable by a specified hmm are given the corresponding optimal filter algorithms for the aggregated states are then derived the paper also describes an approach to efficient suboptimal filtering for hmp which are approximately lumpable by this we mean that the hmm generating the process may be approximated by a lumpable hmm this approach involves directly finding a lumped hmm which approximates the original hmm well in a matrix norm sense an alternative approach for model reduction based on approximating a given hmm by an exactly lumpable hmm is also derived this method is based on the alternating convex projections algorithm some simulation examples are presented which illustrate the performance of the suboptimal filtering algorithms

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

the presence of a zero in an integer linear recurrent sequence is np hard to decide we show that the problem of determining if a given integer linear recurrent sequence has a zero a problem that is known as pisot s problem is np hard with a similar argument we show that the problem of finding the minimal realization dimension of a one letter max plus rational series is np hard this last result answers a folklore question raised in the control literature on the max plus approach to discrete event systems our results are simple consequences of a construction due to stockmeyer and meyer c 2002 elsevier science inc all rights reserved

balanced truncation of linear time varying systems in this paper balanced truncation of linear time varying systems is studied in discrete and continuous time based on relatively basic calculations with time varying lyapunov equations inequalities we are able to derive both upper and lower error bounds for the truncated models these results generalize well known time invariant formulas the case of time varying state dimension is considered input output stability of all truncated balanced realizations is also proven the method is finally successfully applied to a high order model

model reduction by nice selections for linear switched systems a moment matching method for model reduction of linear switched systems lsss is presented the method can be seen as a non trivial extension of the krylov subspace methods for linear time invariant lti systems the procedure is based on the so called nice selections which represent a choice of basis in the reachability or observability space of the lss the framework can also be used for exact matching of the input output behavior of an lss with a reduced order lss for a specific switching sequence conditions for applicability of the method for model reduction are stated and finally the results are illustrated on numerical examples

the geometry and integrability of the suslov problem in this paper we discuss the integrability of a nonholonomic mechanical system a generalized klebsh tisserand case of the suslov problem using the theory of hamiltonization and the poincare hopf theorem we analyze the topology of the invariant manifolds and in particular describe their genus we contrast the results with those for hamiltonian systems

voronoi and void statistics for superhomogeneous point processes we study the voronoi and void statistics of superhomogeneous or hyperuniform point patterns in which the infinite wavelength density fluctuations vanish superhomogeneous or hyperuniform point patterns arise in one component plasmas primordial density fluctuations in the universe and jammed hard particle packings we specifically analyze a certain one dimensional model by studying size fluctuations and correlations of the associated voronoi cells we derive exact results for the complete joint statistics of the size of two voronoi cells we also provide a sum rule that the correlation matrix for the voronoi cells must obey in any space dimension in contrast to the conventional picture of superhomogeneous systems we show that infinitely large voronoi cells or voids can exist in superhomogeneous point processes in any dimension we also present two heuristic conditions to identify and classify any superhomogeneous point process in terms of the asymptotic behavior of the void size distribution

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

periodic ellipsoidal billiard trajectories and extremal polynomials a comprehensive study of periodic trajectories of billiards within ellipsoids in d dimensional euclidean space is presented the novelty of the approach is based on a relationship established between periodic billiard trajectories and extremal polynomials on the systems of d intervals on the real line by leveraging deep but yet not widely known results of the krein levin nudelman theory of generalized chebyshev polynomials fundamental properties of billiard dynamics are proven for any d viz that the sequences of winding numbers are monotonic by employing the potential theory we prove the injectivity of the frequency map as a byproduct for d 2 a new proof of the monotonicity of the rotation number is obtained the case study of trajectories of small periods t d le t le 2d is given in particular it is proven that all d periodic trajectories are contained in a coordinate hyperplane and that for a given ellipsoid there is a unique set of caustics which generates d 1 periodic trajectories a complete catalog of billiard trajectories with small periods is provided for d 3

local density fluctuations hyperuniformity and order metrics questions concerning the properties and quantification of density fluctuations in point patterns continue to provide many theoretical challenges the purpose of this paper is to characterize certain fundamental aspects of local density fluctuations associated with general point patterns in any space dimension d our specific objectives are to study the variance in the number of points contained within a regularly shaped window ensuremath omega of arbitrary size and to further illuminate our understanding of hyperuniform systems i e point patterns that do not possess infinite wavelength fluctuations for large windows hyperuniform systems are characterized by a local variance that grows only as the surface area rather than the volume of the window we derive two formulations for the number variance i an ensemble average formulation which is valid for statistically homogeneous systems and ii a volume average formulation applicable to a single realization of a general point pattern in the large system limit the ensemble average formulation which includes both real space and fourier representations enables us to show that a homogeneous point pattern in a hyperuniform state is at a critical point of a type with appropriate scaling laws and critical exponents but one in which the direct correlation function rather than the pair correlation function is long ranged we also prove that the non negativity of the local number variance does not add a new realizability condition on the pair correlation the volume average formulation is superior for certain computational purposes including optimization studies in which it is desired to find the particular point pattern with an extremal or targeted value of the variance we prove that the simple periodic linear array yields the global minimum value of the average variance among all infinite one dimensional hyperuniform patterns we also evaluate the variance for common infinite periodic lattices as well as certain nonperiodic point patterns in one two and three dimensions for spherical windows enabling us to rank order the spatial patterns our results suggest that the local variance may serve as a useful order metric for general point patterns contrary to the conjecture that the lattices associated with the densest packing of congruent spheres have the smallest variance regardless of the space dimension we show that for d 3 the body centered cubic lattice has a smaller variance than the face centered cubic lattice finally for certain hyperuniform disordered point patterns we evaluate the direct correlation function structure factor and associated critical exponents exactly

a tutorial on the positive realization problem this paper is a tutorial on the positive realization problem that is the problem of finding a positive state space representation of a given transfer function and characterizing existence and minimality of such representation this problem goes back to the 1950s and was first related to the identifiability problem for hidden markov models then to the determination of internal structures for compartmental systems and later embedded in the more general framework of positive systems theory within this framework developing some ideas sprang in the 1960s during the 1980s the positive realization problem was reformulated in terms of a geometric condition which was recently exploited as a tool for finding the solution to the existence problem and providing partial answers to the minimality problem in this paper the reader is carried through the key ideas which have proved to be useful in order to tackle this problem in order to illustrate the main results contributions and open problems several motivating examples and a comprehensive bibliography on positive systems organized by topics are provided

infinite horizon optimal control of linear discrete time systems with stochastic parameters the infinite horizon optimal control problem is considered in the general case of linear discrete time systems and quadratic criteria both with stochastic parameters which are independent with respect to time a stronger stabilizability property and a weaker observability property than usual for deterministic systems are introduced it is shown that the infinite horizon problem has a solution if the system has the first property if in addition the problem has the second property the solution is unique and the control system is stable in the mean square sense a simple necessary and sufficient condition explicit in the system matrices is given for the system to have the stronger stabilizability property this condition also holds for deterministic systems to be stabilizable in the usual sense the stronger stabilizability and weaker observability properties coincide with the usual ones if the parameters are deterministic

krylov subspace methods for solving large lyapunov equations this paper considers several methods for calculating low rank approximate solutions to large scale lyapunov equations of the form ap pa bb 0 the interest in this problem stems from model reduction where the task is to approximate high dimensional models by ones of lower order the two recently developed krylov subspace methods exploited in this paper are the arnoldi method saad math comput 37 1981 pp 105 126 and the generalised minimum residual method gmres saad and schultz siam j sci statist comput 7 1986 pp 856 869 exact expressions for the approximation errors incurred are derived in both cases the numerical solution of the low dimensional linear matrix equation arising from the gmres method is discussed and an algorithm for its solution is proposed low rank solutions of discrete time lyapunov equations and continuous time algebraic riccati equations are also considered throughout this paper the authors tackle problems in which b has more than one column with the

the appell hypergeometric functions and classical separable mechanical systems a relationship between two old mathematical subjects is observed the theory of hypergeometric functions and the separability in classical mechanics separable potential perturbations of integrable billiard systems and the jacobi problem for geodesics on an ellipsoid are expressed through the appell hypergeometric functions f4 of two variables even when the number of degrees of freedom increases if an ellipsoid is symmetric the number of variables in the hypergeometric functions does not increase wider classes of separable potentials are given by the obtained new formulae automatically

intrinsic characterization of the variable separation in the hamilton jacobi equation the nonorthogonal separation of variables in the hamilton jacobi equation corresponding to a natural hamiltonian h 12gijpipj v with a metric tensor of any signature is intrinsically characterized by geometrical objects on the riemannian configuration manifold killing vectors killing tensors and killing webs comparisons with previous characterizations and some illustrative examples are given

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

tilings of space and superhomogeneous point processes we consider the construction of point processes from tilings with equal volume tiles of d dimensional euclidean space mathbb r d we show that one can generate with simple algorithms ascribing one or more points to each tile point processes which are superhomogeneous or hyperuniform i e for which the structure factor s mathbf k vanishes when the wave vector mathbf k tends to zero the exponent ensuremath gamma characterizing the leading small k behavior s k ensuremath rightarrow 0 ensuremath propto k ensuremath gamma depends in a simple manner on the nature of the correlation properties of the specific tiling and on the conservation of the mass moments of the tiles assigning one point to the center of mass of each tile gives the exponent ensuremath gamma 4 for any tiling in which the shapes and orientations of the tiles are short range correlated smaller exponents in the range 4 ensuremath dl ensuremath gamma l4 and thus always superhomogeneous for d ensuremath le 4 may be obtained in the case that the latter quantities have long range correlations assigning more than one point to each tile in an appropriate way we show that one can obtain arbitrarily higher exponents in both cases we illustrate our results with explicit constructions using known deterministic tilings as well as some simple stochastic tilings for which we can calculate s mathbf k exactly our results provide an explicit analytical construction of point processes with ensuremath gamma g4 applications to condensed matter physics and also to cosmology are briefly discussed

on the correlation structure of some random point processes on the line abstract the correlation structure of some remarkable point processes on the one dimensional real line is investigated more specifically focus is on translation invariant determinantal permanental and or renewal point processes in some cases anomalous non poissonian fluctuations for the number of points in a large window can be observed this may be read from the total correlation function of the point process we try to understand when and why this occurs and what are the anomalous behaviors to be expected from examples it is shown that determinantal fermion point processes can be super homogeneous the number variance grows slower than the number mean and even hyper uniform when variance growth saturates renewal point processes with bounded spacings variance are essentially poissonian the number variance grows like the number mean as in poisson models under certain conditions permanental boson point processes can be sub homogeneous or critical in the sense that the number variance grows faster than the number mean we give several detailed examples illustrating these properties of interest together with unexpected behaviors

model reduction of multidimensional and uncertain systems model reduction methods are presented for systems represented by a linear fractional transformation on a repeated scalar uncertainty structure these methods involve a complete generalization of balanced realizations balanced gramians and balanced truncation model reduction with guaranteed error bounds based on solutions to a pair of linear matrix inequalities which generalize lyapunov equations the resulting reduction methods immediately apply to uncertainty simplification and state order reduction in the case of uncertain systems but also may be interpreted as state order reduction for multidimensional systems

non holonomic geodesic flows on lie groups and the integrable suslov problem on so 4 we study geodesic flows on lie groups with the left invariant non holonomic constraint in the case of the existence of an invariant measure we find new integrable non hamiltonian systems on so 4 and other six dimensional lie groups

balanced truncation for discrete time markov jump linear systems this technical note investigates the model reduction problem for mean square stable discrete time markov jump linear systems for this class of systems a balanced truncation algorithm is developed the reduced order model is suboptimal however the approximation error which is captured by means of the stochastic gain is bounded from above by twice the sum of singular numbers associated to the truncated states of each mode such a result allows rigorous simplification of the dynamics of each mode in an independent manner with respect to a metric which is relevant from a robust control point of view

on periodical trajectories of the billiard systems within an ellipsoid in rd and generalized cayley s condition the analytical condition for a trajectory of the billiard system within an ellipsoid of any dimension is given an interesting property is obtained every periodical trajectory with the period less than the dimension of the space is placed in a coordinate hyperplane

trajectory tracking for a particle in elliptical billiards an infinitely rigid unitary mass particle is considered moving on a planar region delimited by a rigid elliptical barrier elliptical billiards under the action of proper control forces a class of periodic trajectories involving an infinite sequence of non smooth impacts between the mass and the barrier at fixed times is found by using an lmis based procedure the jumps in the velocities at the impact times render difficult if not impossible to obtain the classical stability and attractivity properties for the dynamic system describing the tracking error behaviour hence the tracking control problem is properly stated using notions similar to the quasi stability concept in v lakshmikantham d d bainov and p s simeonov theory of impulsive differential equations 6 world scientific 1989 a controller whose state is subject to discontinuities based on the internal model principle is shown to solve the proposed tracking problem giving rise to control forces that are piecewise continuous fun

separation of variables on n dimensional riemannian manifolds i the n sphere sn and euclidean n space rn the following problem is solved what are all the different separable coordinate systems for the laplace beltrami eigenvalue equation on the n sphere sn and euclidean n space rn and how are they constructed this is achieved through a combination of differential geometric and group theoretic methods a graphical procedure for construction of these systems is developed that generalizes vilenkin s construction of polyspherical coordinates the significance of these results for exactly soluble dynamical systems on these manifolds is pointed out the results are also of importance for the analysis of the special functions appearing in the separable solutions of the laplace beltrami eigenvalue equation on these manifolds

elliptical billiards and poncelet s theorem two and three dimensional billiard systems with elliptical and ellipsoidal boundaries respectively are studied it is known that the trajectory of such a two dimensional system generates a caustic conic curve many properties of the elliptical billiard system in the language of projective geometry can be described one of these properties is that if a trajectory is closed after p bounces then all trajectories sharing the same caustic conic are also closed after p bounces in projective geometry this is known as poncelet s theorem many of the two dimensional results are extended to three dimensions in particular it is shown that all straight segments of a trajectory in a three dimensional ellipsoidal billiard system are always tangent to two caustic quadrics if a trajectory is closed after p bounces then all trajectories sharing the same two caustic quadrics are also closed after p bounces a generalized poncelet s theorem in three dimensions is thus established on the basis of numerical studies it is conjectured that this generalized poncelet s theorem is also valid for an arbitrary finite number field the implications of the results are discussed and their extension to n dimensions is outlined

anomalous local coordination density fluctuations and void statistics in disordered hyperuniform many particle ground states we provide numerical constructions of one dimensional hyperuniform many particle distributions that exhibit unusual clustering and asymptotic local number density fluctuations growing more slowly than the volume of an observation window but faster than the surface area hyperuniformity defined by vanishing infinite wavelength local density fluctuations provides a quantitative metric of global order within a many particle configuration and signals the onset of an inverted critical point in which the direct correlation function becomes long ranged by targeting a specified form of the structure factor at small wavenumbers s k k for 0 1 using collective density variables we are able to tailor the form of asymptotic local density fluctuations while simultaneously measuring the effect of imposing weak and strong constraints on the available degrees of freedom within the system this procedure is equivalent to finding the possibly disordered classical ground state of an interacting many particle system with up to four body interactions even in one dimension the long range effective interactions induce clustering and nontrivial phase transitions in the resulting ground state configurations we provide an analytical connection between the fraction of constrained degrees of freedom within the system and the disorder order phase transition for a class of target structure factors by examining the realizability of the constrained contribution to the pair correlation function our results explicitly demonstrate that disordered hyperuniform many particle ground states and therefore also point distributions with substantial clustering can be constructed we directly relate the local coordination structure of our point patterns to the distribution of the void space external to the particles and we provide a scaling argument for the configurational entropy analogous to spin frustated system of the disordered ground states by emphasizing the intimate connection between geometrical constraints on the particle distribution and structural regularity our work has direct implications for higher dimensional systems including an understanding of the appearance of hyperuniformity and quasi long range pair correlations in maximally random strictly jammed packings of hard spheres

subspace estimation and prediction methods for hidden markov models hidden markov models hmms are probabilistic functions of finite markov chains or put in other words state space models with finite state space in this paper we examine subspace estimation methods for hmms whose output lies a finite set as well in particular we study the geometric structure arising from the nonminimality of the linear state space representation of hmms and consistency of a subspace algorithm arising from a certain factorization of the singular value decomposition of the estimated linear prediction matrix for this algorithm we show that the estimates of the transition and emission probability matrices are consistent up to a similarity transformation and that the m step linear predictor computed from the estimated system matrices is consistent i e converges to the true optimal linear m step predictor

cayley type conditions for billiards within k quadrics in bbb rd in section 4 of this paper the authors misstated theorems 3 and 4 please see pdf for corrected theorems

when is a hamiltonian system separable the hamiltonian system given by h 1 2 p2 v q with v c rn is considered a method for integrating such a system is that of separating the variables in the hamilton jacobi equation it is known that if such a separation is possible then it can take place only when the equation is expressed in terms of generalized elliptic coordinates or in a degeneration of these a criterion is proposed for deciding if separation is possible and if it is in which degeneration of elliptic coordinates it takes place

on formal power series representations for uncertain systems the concept of minimality as developed for uncertain and multidimensional systems represented by linear fractional transformations lfts is related to realization theory results for formal power series we discuss the relationship between the notions of minimality for lft and series realizations and present a method for obtaining one type of minimal realization from the opposing type an extension of an existing minimality result for formal power series to the multi input multi output mimo case is also presented

the complete whittaker theorem for two dimensional integrable systems and its application the authors prove for a particle moving in a plane under the influence of a conservative force that when the motion is constrained by a second invariant quadratic in the velocities then the potential allows separability of the hamilton jacobi equation in rectangular polar elliptical cylinder or parabolic cylinder coordinates this link shows the intimate connection between quadratic invariants and the two dimensional hamilton jacobi equation they give examples of the utility of parabolic cylinder coordinates in cases of recent study

intrinsic characterisation of orthogonal separation of one coordinate in the hamilton jacobi equation the authors extend an idea due to woodhouse 1975 to give a coordinate free characterisation of the orthogonal separation of one coordinate for the hamilton jacobi equation on a pseudo riemannian manifold in terms of an involutive family of killing tensors the coordinates can be computed from the killing tensors

nonholonomic left and right flows on lie groups nonholonomic systems on lie groups with a left invariant lagrangian and a right invariant nonintegrable distribution are studied a method for construction of the first integrals is given it is applied to three dimensional lie groups and to the four dimensional nonholonomic rigid body motion new integrable problems with an invariant measure are obtained

elliptical billiards and hyperelliptic functions the geometrical properties of the elliptical billiard system are related to poncelet s theorem this theorem states that if a polygon is inscribed in a conic and circumscribed about a second conic every point of the former conic is a vertex of a polygon with the same number of sides and the same perimeter chang and friedberg have extended this theorem to three and higher dimensions they have shown that the geometrical properties of the hyperelliptic billiard system are related to the algebraic character of a poincare map in the phase space the geometrical and algebraic properties of the system can be understood in terms of the analytical structure of the equations of motion these equations form a complete system of abelian integrals the integrability of the physical system is reflected by the topology of the riemann surfaces associated to these integrals the algebraic properties are connected with the existence of addition formulas for hyperelliptic functions the main purpose of this study is to establish such a connection and to provide an algebraic proof of poncelet s theorem in three and higher dimensions

hyperuniformity quasi long range correlations and void space constraints in maximally random jammed particle packings i polydisperse spheres hyperuniform many particle distributions possess a local number variance that grows more slowly than the volume of an observation window implying that the local density is effectively homogeneous beyond a few characteristic length scales previous work on maximally random strictly jammed sphere packings in three dimensions has shown that these systems are hyperuniform and possess unusual quasi long range pair correlations decaying as r 4 resulting in anomalous logarithmic growth in the number variance however recent work on maximally random jammed sphere packings with a size distribution has suggested that such quasi long range correlations and hyperuniformity are not universal among jammed hard particle systems in this paper we show that such systems are indeed hyperuniform with signature quasi long range correlations by characterizing the more general local volume fraction fluctuations we argue that the regularity of the void space induced by the constraints of saturation and strict jamming overcomes the local inhomogeneity of the disk centers to induce hyperuniformity in the medium with a linear small wave number nonanalytic behavior in the spectral density resulting in quasi long range spatial correlations scaling with r d 1 in d euclidean space dimensions a numerical and analytical analysis of the pore size distribution for a binary maximally random jammed system in addition to a local characterization of the n particle loops governing the void space surrounding the inclusions is presented in support of our argument this paper is the first part of a series of two papers considering the relationships among hyperuniformity jamming and regularity of the void space in hard particle packings

stability results for discrete time linear systems with markovian jumping parameters stability results for discrete time linear systems subject to random jumps in the parameters are obtained first necessary and sufficient conditions for mean square stability mss including the case in which the system is driven by an independent wide sense stationary random sequence are derived it is shown that mss is equivalent to the spectral radius of an augmented matrix being less than one or to the existence of a solution of a certain lyapunov equation in addition it is shown that the lyapunov equation can be derived in four equivalent forms and as a by product each one gives risc to an easier to check sufficient condition these results give inter alia a unified and rather complete picture for mss of markovian jump linear systems next we derive sufficient conditions for almost sure stability of the noiseless ease finally we conclude by presenting an illustrative application to adaptive filtering where a mild condition for almost sure convergence is provided

error bounds for balanced model reduction of linear time varying systems error bounds are developed for balanced truncation of linear time varying systems leading to an extension of the twice the sum of the tail formulas well known in the time invariant case the approach relies on an operator theoretic framework for analysis of linear time varying systems this provides a multivariable notion of frequency for such systems which are thus characterized by rational functions of many complex variables allowing the problem to be formulated in the linear fractional framework using a time varying version of standard necessary conditions for reduced order modeling based on convex operator inequalities we show that these error bounds for balanced truncation are related to the closest possible reduced order modeling error in a sense which parallels the time invariant case

balanced truncation for a class of stochastic jump linear systems and model reduction for hidden markov models this paper develops a generalization of the balanced truncation algorithm applicable to a class of discrete time stochastic jump linear systems the approximation error which is captured by means of the stochastic l 2 gain is bounded from above by twice the sum of singular numbers associated to the truncated states similar to the case of linear time invariant systems a two step model reduction algorithm for hidden markov models is also developed the first step relies on the aforementioned balanced truncation algorithm due to a topological equivalence established between hidden markov models and a subclass of stochastic jump linear systems in a second step the positivity constraints which reflect the hidden markov model structure are enforced by solving a low dimensional optimization problem

an improved error estimate for reduced order models of discrete time systems the authors derive under weaker conditions a discrete time counterpart of k glover s 1984 error estimate for reduced order models following the same lines a simple proof of glover s result for continuous time systems can be given

on the balanced truncation and coprime factors reduction of markovian jump linear systems abstract the paper deals with the balanced truncation and coprime factors reduction of markovian jump linear mjl systems which can have mode varying state input and output dimensions we develop machinery for balancing mean square stable mjl system realizations using generalized gramians and strict lyapunov inequalities and provide an improved a priori upper bound on the error induced in the balanced truncation process we also generalize the coprime factors reduction method and in doing so extend the applicability of the balanced truncation technique to the class of mean square stabilizable and detectable mjl systems we provide tools to establish mean square stabilizability and detectability of the considered mjl systems in addition a notion of right coprime factorization of mjl systems and methods to construct such factorizations are given the error measure in the coprime factors reduction approach while still norm based does not directly capture the mismatch between the nominal system and the reduced order model as is the case in the balanced truncation approach where mean square stable models are considered instead the error measure is given in terms of the distance between the coprime factors realizations and thus has an interpretation in terms of robust feedback stability the paper concludes with an illustrative example which demonstrates how to apply the coprime factors model reduction approach

elliptical billiard systems and the full poncelet s theorem in n dimensions in this work is presented a generalization of poncelet s theorem to n dimensions which is refered to as the full poncelet s theorem the theorem states that if the reflections of a trajectory by a sequence of confocal quadrics lead to a closed skew polygon then there exists an n 1 parameter family of polygons having the same property a physical realization and a projective geometrical proof of this theorem are given if all the reflecting quadrics coincide the above theorem reduces to the n dimensional poncelet s theorem presented by chang and friedberg the geometrical proof is a finite construction based on a preliminary theorem which extends hart s lemma the full poncelet s theorem may thus be extended to projective geometries over most fields including discrete ones

reachability and observability reduction for linear switched systems with constrained switching we present an algorithm for reducing the number of continuous states of a discrete time linear switched system such that the reduced system has the same input output behavior as the original system for a subset of switching sequences the procedure can be interpreted as reachability and observability reduction for a linear switched system with constrained switching the proposed method is expected to be useful for abstraction based control synthesis methods for hybrid systems

statistical properties of determinantal point processes in high dimensional euclidean spaces the goal of this paper is to quantitatively describe some statistical properties of higher dimensional determinantal point processes with a primary focus on the nearest neighbor distribution functions toward this end we express these functions as determinants of n n matrices and then extrapolate to n this formulation allows for a quick and accurate numerical evaluation of these quantities for point processes in euclidean spaces of dimension d we also implement an algorithm due to hough et al for generating configurations of determinantal point processes in arbitrary euclidean spaces and we utilize this algorithm in conjunction with the aforementioned numerical results to characterize the statistical properties of what we call the fermi sphere point process for d 1 4 this homogeneous isotropic determinantal point process discussed also in a companion paper s torquato a scardicchio and c e zachary j stat mech theory exp 2008 p11019 i s the high dimensional generalization of the distribution of eigenvalues on the unit circle of a random matrix from the circular unitary ensemble in addition to the nearest neighbor probability distribution we are able to calculate voronoi cells and nearest neighbor extrema statistics for the fermi sphere point process and we discuss these properties as the dimension d is varied the results in this paper accompany and complement analytical properties of higher dimensional determinantal point processes developed in a prior paper

on integrable potential perturbations of the jacobi problem for the geodesics on the ellipsoid all integrable mechanical systems describing motion of a particle on an ellipsoid surface in three dimensional space are described in the class of the loran polynomial potentials two countable families of the basic solutions are obtained explicit formulae are given the limit when the considered system goes into the billiard system within an ellipse is analysed and the results are compared with those obtained previously in relation to the billiard system

on the polynomial integrability of kirchoff s equations abstract we prove that kirchoff s equations either are completely integrable or have at most four functionally independent polynomial first integrals

generalizations of classical integrable nonholonomic rigid body systems nonholonomic systems with an invariant measure the suslov chaplygin and veselov veselova problem are considered new families of integrable potential perturbations of these non hamiltonian systems are constructed we also obtain a similar result for a classical hamiltonian system of the motion of a rigid body fixed at a point

on integrable potential perturbations of the billiard system within an ellipsoid generalizing kozlov s method of integrable potential perturbations for systems with three degrees of freedom new integrable billiard systems within ellipsoid in r3 are constructed two countable families of the basic solutions in the class of the laurent polynomials are obtained explicit formulas are given

conditions of cayley s type for ellipsoidal billiard the billiard system within an ellipsoid in three dimensional space is considered we describe periodical trajectories of that mechanical system deriving analytical condition of cayley s type for generalized poncelet s theorem using methods of algebrogeometric integration

separability and lax pairs for henon heiles system the hamiltonian system corresponding to the generalized henon heiles hamiltonian h 1 2 px2 py2 1 2ax2 1 2by2 x2y ey3 is known to be integrable in the following three cases a b e 1 3 e 2 b 16a e 16 3 in the first two the system has been integrated by making use of genus one and genus two theta functions we show that in the third case the system can also be integrated by making use of elliptic functions finally using the fairbanks theorem we find lax pairs for each of the three integrable systems under consideration