wave chaos in acoustics and elasticity interpreting wave phenomena in terms of an underlying ray dynamics adds a new dimension to the analysis of linear wave equations forming explicit connections between spectra and wavefunctions on the one hand and the properties of a related ray dynamics on the other hand is a comparatively new research area especially in elasticity and acoustics the theory has indeed been developed primarily in a quantum context it is increasingly becoming clear however that important applications lie in the field of mechanical vibrations and acoustics we provide an overview over basic concepts in this emerging field of wave chaos this ranges from ray approximations of the green function to periodic orbit trace formulae and random matrix theory and summarizes the state of the art in applying these ideas in acoustics both experimentally and from a theoretical numerical point of view

the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

the general vector solutions of such linear coupled matrix fractional differential equations by using kronecker structures in this paper we demonstrate how kronecker products and vector operator can be combined beautifully to solve and fit the non homogenous linear matrix fractional differential equations and such coupled linear matrix fractional differential equations the way exists which transform the given coupled matrix fractional differential equations by using kronecker structures into forms for which solutions may be readily computed some important and interesting special cases of the general system of non homogeneous linear matrix fractional differential equations of fractional order are also considered which includes the non homogenous linear fractional dynamical system with delays in control finally a brief comparison between the fractional and integer order vector solutions and some examples are given to illustrate our new approaches

partially coupled stochastic gradient identification methods for non uniformly sampled systems this technical note addresses identification problems of non uniformly sampled systems for the input output representation of non uniform discrete time systems a partially coupled stochastic gradient c sg algorithm is proposed to estimate the model parameters with high computational efficiency compared with the standard stochastic gradient sg algorithm the analysis indicates that the partially c sg algorithm can give more accurate parameter estimates than the sg algorithm the parameter estimates obtained using the partially c sg algorithm converge to their true values as the data length approaches infinity

the orbifold chow ring of toric deligne mumford stacks the orbifold chow ring of a deligne mumford stack defined by abramovich graber and vistoli 2 is the algebraic version of the orbifold cohomology ring in troduced by w chen and ruan 7 8 by design this ring incorporates numerical invariants such as the orbifold euler characteristic and the orbifold hodge num bers of the underlying variety the product structure is induced by the degree zero part of the quantum product in particular it involves gromov witten invariants inspired by string theory and results in batyrev 3 and yasuda 28 one expects that in nice situations the orbifold chow ring coincides with the chow ring of a resolution of singularities fantechi and g ttsche 14 and uribe 25 verify this conjecture when the orbifold is symn 5 where 5 is a smooth projective surface with ks 0 and the resolution is hilbn the initial motivation for this project was to compare the orbifold chow ring of a simplicial toric variety with the chow ring of a cr pant resolution to achieve this goal we first develop the theory of toric deligne mumford stacks modeled on simplicial toric varieties a toric deligne mumford stack corresponds to a combinatorial object called a stacky fan as a first approximation this object is a simplicial fan with a distinguished lattice point on each ray in the fan more precisely a stacky fan s is a triple consisting of a finitely generated abelian group n a simplicial fan e in q z n with n rays and a map zn n where the image of the standard basis in zn generates the rays in e a rational simplicial fan e produces a canonical stacky fan s n e where n is the distinguished lattice and is the map defined by the minimal lattice points on the rays hence there is a natural toric deligne mumford stack associated to every simplicial toric variety a stacky fan encodes a group action on a quasi affine variety and the toric deligne mumford stack is the quotient if e corresponds to a smooth toric variety x e and s is the canonical stacky fan associated to e then we simply have x z we show that many of the basic concepts such as open and closed toric substacks line bundles and maps between toric deligne mumford stacks correspond to combinatorial notions we expect that many more results about toric varieties lift to the realm of stacks and hope that toric deligne mumford stacks will serve as a useful testing ground for general theories

toric stacks ii intrinsic characterization of toric stacks the purpose of this paper and its prequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature as well as classical toric varieties while the focus of the prequel is on how to work with toric stacks the focus of this paper is how to show a stack is toric for toric varieties a classical result says that a finite type scheme with an action of a dense open torus arises from a fan if and only if it is normal and separated in the same spirit the main result of this paper is that any artin stack with an action of a dense open torus arises from a stacky fan under reasonable hypotheses

transformations between some special matrices special matrices are very useful in signal processing and control systems this paper studies the transformations and relationships between some special matrices the conditions that a matrix is similar to a companion matrix are derived it is proved that a companion matrix is similar to a diagonal matrix or jordan matrix and the transformation matrices between them are given finally we apply the similarity transformation and the companion matrix to system identification

computation of matrix exponentials of special matrices computing matrix exponentials or transition matrices are important in signal processing and control systems this paper studies the methods of computing the transition matrices related to linear dynamic systems discusses and gives the properties of the transition matrices and explores the transition matrices of some special matrices some examples are provided to illustrate the proposed methods

gradient based iterative solutions for general linear matrix equations in this paper we present a gradient based iterative algorithm for solving general linear matrix equations by extending the jacobi iteration and by applying the hierarchical identification principle convergence analysis indicates that the iterative solutions always converge fast to the exact solutions for any initial values and small condition numbers of the associated matrices two numerical examples are provided to show that the proposed algorithm is effective

tropicalization of the moduli space of stable maps let x be an algebraic variety and let s be a tropical variety associated to x we study the tropicalization map from the moduli space of stable maps into x to the moduli space of tropical curves in s we prove that it is a continuous map and that its image is compact and polyhedral loosely speaking when we deform algebraic curves in x the associated tropical curves in s deform continuously moreover the locus of realizable tropical curves inside the space of all tropical curves is compact and polyhedral our main tools are berkovich spaces formal models balancing conditions vanishing cycles and quantifier elimination for rigid subanalytic sets

iterative algorithms for solving the matrix equation axb cx td e in this paper we propose two iterative algorithms to solve the matrix equation axb cx td e the first algorithm is applied when the matrix equation is consistent in this case for any special initial matrix x 1 a solution the minimal frobenius norm solution can be obtained within finite iteration steps in the absence of roundoff errors the second algorithm is applied when the matrix equation is inconsistent in this case for any special initial matrix x 1 a least squares solution the minimal frobenius norm least squares solution can be obtained within finite iteration steps in the absence of roundoff errors some examples verify the efficiency of these algorithms

kontsevich s formula and the wdvv equations in tropical geometry abstract using gromov witten theory the numbers of complex plane rational curves of degree d through 3 d 1 general given points can be computed recursively with kontsevich s formula that follows from the so called wdvv equations in this paper we establish the same results entirely in the language of tropical geometry in particular this shows how the concepts of moduli spaces of stable curves and maps evaluation and forgetful morphisms intersection multiplicities and their invariance under deformations can be carried over to the tropical world

an iterative method for the least squares symmetric solution of the linear matrix equation axb c abstract this paper an iterative method is presented to solve the minimum frobenius norm residual problem min axb c with unknown symmetric matrix x by this iterative method for any initial symmetric matrix x0 a solution x can be obtained within finite iteration steps in the absence of roundoff errors and the solution x with least norm can be obtained by choosing a special kind of initial symmetric matrix in addition the unique optimal approximation solution x to a given matrix x in frobenius norm can be obtained by first finding the least norm solution x of the new minimum residual problem min a x b c with unknown symmetric matrix x where c c a x x t 2 b given numerical examples are show that the iterative method is quite efficient

iterative solutions of the generalized sylvester matrix equations by using the hierarchical identification principle abstract in this paper by extending the well known jacobi and gauss seidel iterations for ax b we study iterative solutions of matrix equations axb f and generalized sylvester matrix equations axb cxd f including the sylvester equation ax xb f as a special case and present a gradient based and a least squares based iterative algorithms for the solution it is proved that the iterative solution always converges to the exact solution for any initial values the basic idea is to regard the unknown matrix x to be solved as the parameters of a system to be identified and to obtain the iterative solutions by applying the hierarchical identification principle finally we test the algorithms and show their effectiveness using a numerical example

on iterative solutions of general coupled matrix equations in this paper we study coupled matrix equations which are encountered in many systems and control applications first we extend the well known jacobi and gauss seidel iterations and present a large family of iterative methods which are then applied to develop iterative solutions to coupled sylvester matrix equations the basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified and to obtain the iterative solutions by applying a hierarchical identification principle next we generalize the sylvester equations to general coupled matrix equations and propose a gradient based iterative algorithm for the solutions using a block matrix inner product the star star product we prove that the iterative algorithm always converges to the unique solutions for any initial values one advantage of the algorithms proposed is that they require less storage space in implementation than existing numerical methods finally we test the algorithms and show their effectiveness using numerical examples

bcr method for solving generalized coupled sylvester equations over centrosymmetric or anti centrosymmetric matrix abstract this paper introduces another version of biconjugate residual method bcr for solving the generalized coupled sylvester matrix equations over centrosymmetric or anti centrosymmetric matrix we prove this version of bcr algorithm can find the centrosymmetric solution group of the generalized coupled matrix equations for any initial matrix group within finite steps in the absence of round off errors furthermore a method is provided for choosing the initial matrices to obtain the least norm solution of the problem at last some numerical examples are provided to illustrate the efficiency and validity of methods we have proposed

several inequalities involving khatri rao products of positive semidefinite matrices abstract the khatri rao and tracy singh products for partitioned matrices are viewed as generalized hadamard and generalized kronecker products respectively based on the connection between the khatri rao and tracy singh products we establish new inequalities involving khatri rao products of positive semidefinite matrices the results lead to inequalities involving hadamard products as a special case

the residual based interactive stochastic gradient algorithms for controlled moving average models this paper derives a residual based interactive stochastic gradient isg parameter estimation algorithm for controlled moving average cma models and studied the performance of the residual based isg algorithm under weaker conditions on statistical properties of the noise compared with the residual based extended stochastic gradient algorithm for identifying cma models the proposed isg algorithm can give highly accurate parameter estimates by the simulation example

a pair of simultaneous linear matrix equations a1xb1 c1 a2xb2 c2 and a matrix programming problem necessary and sufficient conditions are obtained for a pair of matrix equations a1xb1 c1 a2xb2 c2 on a general field f to have a common solution along with the expression for a general common solution when certain conditions hold common solutions of minimum rank are described a matrix programming problem is solved en route

on the sylvester equation ax x b c abstract we consider the solution of the sylvester equations ax x b c for t h and a b c n n and the related linear matrix equations axb x c axb cx d e and ax x a c solvability conditions and numerical methods are considered in terms of the generalized and periodic schur and qr decompositions we emphasize the square cases where m n but the rectangular cases will be considered

iterative solutions to matrix equations of the form aixbi fi this paper is concerned with the numerical solutions to the linear matrix equations a 1xb 1 f 1 and a 2xb 2 f 2 two iterative algorithms are presented to obtain the solutions for any initial value it is proved that the iterative solutions obtained by the proposed algorithms converge to their true values finally simulation examples are given to verify the proposed convergence theorems

solving the stationary liouville equation via a boundary element method energy distributions of linear wave fields are in the high frequency limit often approximated in terms of flow or transport equations in phase space common techniques for solving the flow equations in both time dependent and stationary problems are ray tracing and level set methods in the context of predicting the vibro acoustic response of complex engineering structures related methods such as statistical energy analysis or variants thereof have found widespread applications we present a new method for solving the transport equations for complex multi component structures based on a boundary element formulation of the stationary liouville equation the method is an improved version of the dynamical energy analysis technique introduced recently by the authors it interpolates between standard statistical energy analysis and full ray tracing containing both of these methods as limiting cases we demonstrate that the method can be used to efficiently deal with complex large scale problems giving good approximations of the energy distribution when compared to exact solutions of the underlying wave equation

an iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation axb c an iteration method is constructed to solve the linear matrix equation axb c over symmetric x by this iteration method the solvability of the equation axb c over symmetric x can be determined automatically when the equation axb c is consistent over symmetric x its solution can be obtained within finite iteration steps and its least norm symmetric solution can be obtained by choosing a special kind of initial iteration matrix furthermore its optimal approximation solution to a given matrix can be derived by finding the least norm symmetric solution of a new matrix equation axb c finally numerical examples are given for finding the symmetric solution and the optimal approximation symmetric solution of the matrix equation axb c

tropical geometry of genus two curves abstract we exploit three classical characterizations of smooth genus two curves to study their tropical and analytic counterparts first we provide a combinatorial rule to determine the dual graph of each algebraic curve and the metric structure on its minimal berkovich skeleton our main tool is the description of genus two curves via hyperelliptic covers of the projective line with six branch points given the valuations of these six points and their differences our algorithm provides an explicit harmonic 2 to 1 map to a metric tree on six leaves second we use tropical modifications to produce a faithful tropicalization in dimension three starting from a planar hyperelliptic embedding finally we consider the moduli space of abstract genus two tropical curves and translate the classical igusa invariants characterizing isomorphism classes of genus two algebraic curves into the tropical realm while these tropical igusa functions do not yield coordinates in the tropical moduli space we propose an alternative set of invariants that provides new length data

in out decomposition of boundary integral equations we propose a reformulation of the boundary integral equations for the helmholtz equation in a domain in terms of incoming and outgoing boundary waves we obtain transfer operator descriptions which are exact and thus incorporate features such as diffraction and evanescent coupling these effects are absent in the well known semiclassical transfer operators in the sense of bogomolny it has long been established that transfer operators are equivalent to the boundary integral approach within semiclassical approximation exact treatments have been restricted to specific boundary conditions such as dirichlet or neumann the approach we propose is independent of the boundary conditions and in fact allows one to decouple entirely the problem of propagating waves across the interior from the problem of reflecting waves at the boundary as an application we show how the decomposition may be used to calculate goos haenchen shifts of ray dynamics in billiards with variable boundary conditions and for dielectric cavities

a quantitative version of the observation that the hadamard product is a principal submatrix of the kronecker product abstract the hadamard and kronecker products of two n m matrices a b are related by a b p t 1 a b p 2 where p 1 p 2 are partial permutation matrices after establishing several properties of the p matrices this relationship is employed to demonstrate how a simplified theory of the hadamard product can be developed during this process the well known result a b a b aa bb is extended to a b a b 1 2 aa bb ab ba aa bb showing an inherent link between the hadamard product and conventional product of two matrices this leads to a sharper bound on the spectral norm of a b a b 1 2 a 2 b 2 ab 2 1 2 a b and an improvement on the weak majorization of a b 2 a b w 1 2 2 a 2 b 2 ab w 2 a 2 b for a real non singular matrix x and invertible diagonal matrices d e the spectral condition number is shown to be if scaled bounded below as follows dxe 2 x x t 2 x x t 2 1 2 x x t 1 for a 0 we have i a 2 1 2 i a 2 a a i a 2 and a 1 2 a 1 2 2 1 2 i a a 1 a a 1 when a 0 the latter inequality is compared to styan s inequality a a 1 1 2 i a a 1 when a is a correlation matrix and is shown to possess stronger properties of ordering finally the relationship a b p t 1 a b p 2 is applied to determine conditions of singularity of certain orderings of the hadamard products of matrices

the scaling conjugate gradient iterative method for two types of linear matrix equations in this paper we propose a new iteration method which is based on the conjugate gradient method for solving the linear matrix equations of the form a i x b i f i i 1 2 n and the generalized sylvester matrix equation a x b c x d e this method is compared with some existing methods in detail such as gradient based iterative gi method and least squares iterative lsi method ding et al 2010 proposed by ding et al also cyclic and simultaneous methods tang et al 2014 given by tang et al some numerical experiments demonstrate that the introduced iterative method is more efficient than the four existing methods

a representation of the general common solution to the matrix equations a1xb1 c1 and a2xb2 c2 with applications we give new necessary and sufficient conditions for the existence of a common solution to the pair of linear matrix equations a1xb1 c1 and a2xb2 c2 and derive a new representation of the general common solution to these two equations we apply this result to determine new necessary and sufficient conditions for the existence of an hermitian solution and a representation of the general hermitian solution to the matrix equation axb c

multi innovation stochastic gradient algorithm for multiple input single output systems using the auxiliary model in order to reduce computational burden and improve the convergence rate of identification algorithms an auxiliary model based multi innovation stochastic gradient am misg algorithm is derived for the multiple input single output systems by means of the auxiliary model identification idea and multi innovation identification theory the basic idea is to replace the unknown outputs of the fictitious subsystems in the information vector with the outputs of the auxiliary models and to present an auxiliary model based stochastic gradient algorithm and then to derive the am misg algorithm by expanding the scalar innovation to innovation vector and introducing the innovation length the simulation example shows that the proposed algorithms work quite well

kronecker operational matrices for fractional calculus and some applications the problems of systems identification analysis and optimal control have been recently studied using orthogonal functions the specific orthogonal functions used up to now are the walsh the block pulse the laguerre the legendre haar and many other functions in the present paper several operational matrices for integration and differentiation are studied we introduce the kronecker convolution product and expanded to the riemann liouville fractional integral of matrices for some applications it is often not necessary to compute exact solutions approximate solutions are sufficient because sometimes computational efforts rapidly increase with the size of matrix functions our method is extended to find the exact and approximate solutions of the general system matrix convolution differential equations the way exists which transform the coupled matrix differential equations into forms for which solutions may be readily computed finally several systems are solved by the new and other approaches and illustrative examples are also considered

tropicalizing the space of admissible covers we study the relationship between tropical and classical hurwitz moduli spaces following recent work of abramovich caporaso and payne we outline a tropicalization for the moduli space of generalized hurwitz covers of an arbitrary genus curve our approach is to appeal to the geometry of admissible covers which compactify the hurwitz scheme we study the relationship between a combinatorial moduli space of tropical admissible covers and the skeleton of the berkovich analytification of the classical space of admissible covers we use techniques from non archimedean geometry to show that the tropical and classical tautological maps are compatible via tropicalization and that the degree of the classical branch map can be recovered from the tropical side as a consequence we obtain a proof at the level of moduli spaces of the equality of classical and tropical hurwitz numbers

joint independent subspace analysis uniqueness and identifiability this paper deals with the identifiability of joint independent subspace analysis jisa jisa is a recently proposed framework that subsumes independent vector analysis iva and independent subspace analysis isa each underlying mixture can be regarded as a dataset therefore jisa can be used for data fusion in this paper we assume that each dataset is an overdetermined mixture of several multivariate gaussian processes each of which has independent and identically distributed samples this setup is not identifiable when each mixture is considered individually given these assumptions jisa can be restated as coupled block diagonalization cbd of its correlation matrices hence jisa identifiability is tantamount to cbd uniqueness in this work we provide necessary and sufficient conditions for uniqueness and identifiability of jisa and cbd our analysis is based on characterizing all the cases in which the fisher information matrix is singular we prove that non identifiability may occur only due to pairs of underlying random processes with the same dimension our results provide further evidence that irreducibility has a central role in the uniqueness analysis of block based decompositions our contribution extends previous results on the uniqueness and identifiability of isa iva coupled matrix and tensor decompositions we provide examples to illustrate our results

auxiliary model based least squares identification methods for hammerstein output error systems abstract the difficulty in identification of a hammerstein a linear dynamical block following a memoryless nonlinear block nonlinear output error model is that the information vector in the identification model contains unknown variables the noise free true outputs of the system in this paper an auxiliary model based least squares identification algorithm is developed the basic idea is to replace the unknown variables by the output of an auxiliary model convergence analysis of the algorithm indicates that the parameter estimation error consistently converges to zero under a generalized persistent excitation condition the simulation results show the effectiveness of the proposed algorithms

the mgpbicg method for solving the generalized coupled sylvester conjugate matrix equations in this paper we extend the generalized product type bi conjugate gradient gpbicg method for solving the generalized sylvester conjugate matrix equations a 1 x b 1 c 1 y d 1 s 1 a 2 x b 2 c 2 y d 2 s 2 by the real representation of the complex matrix and the properties of kronecker product and vectorization operator some numerical experiments demonstrate that the introduced iteration approach is efficient

density and correlation functions of vortex and saddle points in open billiard systems we present microwave measurements for the density and spatial correlation of current critical points in an open billiard system and compare them with new and previous predictions of the random wave model rwm in particular due to an improvement of the experimental setup we determine experimentally the spatial correlation of saddle points of the current field an asymptotic expression for the vortex saddle and saddle saddle correlation functions based on the rwm is derived with experiment and theory agreeing well we also derive an expression for the density of saddle points in the presence of a straight boundary with general mixed boundary conditions in the rwm and compare with experimental measurements of the vortex and saddle density in the vicinity of a straight wall satisfying dirichlet conditions

hierarchical least squares identification methods for multivariable systems for multivariable discrete time systems described by transfer matrices we develop a hierarchical least squares iterative hlsi algorithm and a hierarchical least squares hls algorithm based on a hierarchical identification principle we show that the parameter estimation error given by the hlsi algorithm converges to zero for the deterministic cases and that the parameter estimates by the hls algorithm consistently converge to the true parameters for the stochastic cases the algorithms proposed have significant computational advantage over existing identification algorithms finally we test the proposed algorithms on an example and show their effectiveness

auxiliary model based rels and mi els algorithm for hammerstein oema systems this paper considers the identification problem for hammerstein output error moving average oema systems an auxiliary model based recursive extended least squares rels algorithm and an auxiliary model based multi innovation extended least squares mi els algorithm are presented using the multi innovation identification theory the basic idea is to express the system output as a linear combination of the parameters by using the key term separation principle and auxiliary model method the proposed algorithms can give highly accurate parameter estimates the simulation results show the effectiveness of the proposed algorithms

skeletons and tropicalizations abstract let k be a complete algebraically closed non archimedean field with ring of integers k and let x be a k variety we associate to the data of a strictly semistable k model x of x plus a suitable horizontal divisor h a skeleton s x h in the analytification of x this generalizes berkovich s original construction by admitting unbounded faces in the directions of the components of h it also generalizes constructions by tyomkin and baker payne rabinoff from curves to higher dimensions every such skeleton has an integral polyhedral structure we show that the valuation of a non zero rational function is piecewise linear on s x h for such functions we define slopes along codimension one faces and prove a slope formula expressing a balancing condition on the skeleton moreover we obtain a multiplicity formula for skeletons and tropicalizations in the spirit of a well known result by sturmfels tevelev we show a faithful tropicalization result saying roughly that every skeleton can be seen in a suitable tropicalization we also prove a general result about existence and uniqueness of a continuous section to the tropicalization map on the locus of tropical multiplicity one

a modified cg algorithm for solving generalized coupled sylvester tensor equations abstract in this paper a modified conjugate gradient mcg algorithm is proposed for solving generalized coupled sylverster tensor equations if the tensor equations are consistent we show the solution can be obtained within finite steps in the absence of roundoff errors for any initial value at last some numerical examples are provided to illustrate the efficiency and validity of the proposed algorithm

an iterative algorithm for the reflexive solutions of the generalized coupled sylvester matrix equations and its optimal approximation abstract the generalized coupled sylvester matrix equations ay zb cy zd e f with unknown matrices y z are encountered in many systems and control applications also these matrix equations have several applications relating to the problem of computing stable eigendecompositions of matrix pencils in this work we construct an iterative algorithm to solve the generalized coupled sylvester matrix equations over reflexive matrices y z and when the matrix equations are consistent for any initial matrix pair y 0 z 0 a reflexive solution pair can be obtained within finite iteration steps in the absence of roundoff errors and the least frobenius norm reflexive solution pair can be obtained by choosing a special kind of initial matrix pair also we obtain the optimal approximation reflexive solution pair to a given matrix pair y z in the reflexive solution pair set of the generalized coupled sylvester matrix equations ay zb cy zd e f moreover several numerical examples are given to show the efficiency of the presented iterative algorithm

the semiclassical theory of discontinuous systems and ray splitting billiards we analyze the semiclassical limit of spectral theory on manifolds whose metrics have jump like discontinuities such systems are quite different from manifolds with smooth riemannian metrics because the semiclassical limit does not relate to a classical flow but rather to branching ray splitting billiard dynamics in order to describe this system we introduce a dynamical system on the space of functions on phase space to identify the quantum dynamics in the semiclassical limit we compute the principal symbols of the fourier integral operators associated to reflected and refracted geodesic rays and identify the relation between classical and quantum dynamics in particular we prove a quantum ergodicity theorem for discontinuous systems in order to do this we introduce a new notion of ergodicity for the ray splitting dynamics

reconstruction of continuous time systems from their non uniformly sampled discrete time systems a continuous time system cannot be recovered solely from its uniformly sampled discrete time model through the zero order hold discretization or step invariant transformation but our studies indicate that it can be recovered uniquely from its non uniformly sampled discrete time model in this paper we discuss some related issues of non uniformly sampled systems including model derivation controllability and observability computation of single rate models with different sampling periods reconstruction of continuous time systems and parameter identification of non uniformly sampled discrete time systems a numerical example is also given for illustration

block matrix generalizations of schur s basic theorems on hadamard products abstract in a classic 1911 paper i schur gave several useful bounds for the spectral norm and eigenvalues of the hadamard entrywise product of two matrices motivated by applications to the theory of monotone and convex matrix functions we are led to consider hadamard products in which both factors are conformally partitioned block matrices and the entries of one factor are constant within each block such products are special cases of a block kronecker tensor product and it is in this context that we present generalizations of schur s results

iterative algorithms for the minimum norm solution and the least squares solution of the linear matrix equations abstract in this paper two iterative algorithms are proposed to solve the linear matrix equations a 1 xb 1 c 1 x t d 1 m 1 a 2 xb 2 c 2 x t d 2 m 2 when the matrix equations are consistent by the first algorithm a solution x can be obtained within finite iterative steps in the absence of roundoff error for any initial value furthermore the minimum norm solution can be got by choosing a special kind of initial matrix additionally the unique optimal approximation solution to a given matrix x 0 can be derived by finding the minimum norm solution of a new matrix equations a 1 x b 1 c 1 x t d 1 m 1 a 2 x b 2 c 2 x t d 2 m 2 when the matrix equations are inconsistent we present the second algorithm to find the least squares solution with the minimum norm finally two numerical examples are tested by matlab the results show that these iterative algorithms are efficient

non axisymmetric bending of thin annular plates due to circumferentially distributed moments abstract the non linear deformation of a thin annular plate subjected to circumferentially distributed bending moments is studied a von karman plate model is adopted to formulate the equations of motion free free boundary conditions have been applied at both inner and outer edges the matrix formulation of the differential quadrature method is used to discretise and solve the governing equations linear analysis predicts that the annular disk deforms axisymmetrically into a spherical dome however the proposed non linear analysis shows that a symmetry breaking bifurcation may occur after which the linear solution becomes unstable and the plate transitions into a non axisymmetric cylindrical deformation this is the case when at least one of two parameters reaches a critical value these parameters are the non dimensionalised ratio between applied moment and bending stiffness and the ratio between inner and outer radius furthermore it is noted that free free boundary conditions and circumferentially distributed bending moments do not break the circular symmetry of the annular disk hence the principal axes of curvature in the deformed configuration do not have a preferred orientation therefore the present work describes a shell possessing infinite identical equilibria having different yet no favoured direction and hence links to previous researches on neutrally stable structures

using stacks to impose tangency conditions on curves we define a deligne mumford stack xp r which depends on a scheme x an effective carrier divisor d c x and a positive integer r then we show that the abramovich vistoli mod uli stack of stable maps into xq r provides compactifications of the locally closed substacks of mgtn x 3 corresponding to relative stable maps 1 introduction this paper has two main purposes the first is to introduce the rth root construction which is a way of creating stack structure along a divisor given a scheme x an effective cartier divisor d c x and a positive integer r which is invertible on x it produces a deligne mumford stack xo r when r is not invertible the construction still makes sense and produces an artin stack which has finite diagonal over x since the output is in general an artin stack we give the definition and basic properties when x is an artin stack this simplifies the exposition since we need to apply the construction multiple times this construction was known to several mathematicians before we became aware of it but very little appeared in the literature aside from a brief mention in agv 3 5 3 secondly we compare twisted stable maps into xp r with ordinary stable maps into x the theory of twisted stable maps was developed by chen and ruan cr in the symplectic category and by abramovich and vistoli av in the algebraic category the latter showed that there is a deligne mumford stack xg n x 3 parametrizing morphisms x from twisted rc marked genus g nodal curves into a deligne mumford stack x such that 3 in the case where x is a scheme this is just the space of kontsevich stable maps mg n x 3 the theory of relative stable maps was developed by several symplectic ge ometers and the algebraic definition was worked out partially by andreas gath mann ga and in general by jun li li let x be a smooth complex projective veriety d c x a smooth divisor and 3 g b x such that d 3 0 here b x denotes the group of 1 cycles modulo algebraic equivalence choose a partition d 3 q qn where each qi is a nonnegative integer then a stable map c x from a smooth curve c having marked points x xn such that f c 3 and d y qixi ls a relative stable map the space of such mor phisms was compactified by ga and li and they used this to define relative gromov witten invariants

combinatorics of compactified universal jacobians abstract we show that the combinatorial structure of the compactified universal jacobians over m g in degrees g 1 and g is governed by orientations on stable graphs in particular for a stable curve we exhibit graded stratifications of the compactified jacobians in terms of totally cyclic respectively rooted orientations on its dual graph we prove functoriality under edge contraction of the posets of totally cyclic and rooted orientations on stable graphs

block kronecker products and the vecb operator this paper is concerned with two generalizations of the kronecker product and two related generalizations of the vec operator it is demonstrated that they pairwise match two different kinds of matrix partition viz the balanced and unbalanced ones relevant properties are supplied and proved a related concept the so called tilde transform of a balanced block matrix is also studied the results are illustrated with various statistical applications of the five concepts studied

toric stacks i the theory of stacky fans the purpose of this paper and its sequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature as well as classical toric varieties in this paper we define a toric stack as the stack quotient of a toric variety by a subgroup of its torus we also define a generically stacky version any toric stack arises from a combinatorial gadget called a stacky fan we develop a dictionary between the combinatorics of stacky fans and the geometry of toric stacks stressing stacky phenomena such as canonical stacks and good moduli space morphisms we also show that smooth toric stacks carry a moduli interpretation extending the usual moduli interpretations of p n and a 1 g m indeed smooth toric stacks precisely solve moduli problems specified by generalized effective cartier divisors with given linear relations and given intersection relations smooth toric stacks therefore form a natural closure to the class of moduli problems introduced for smooth toric varieties and smooth toric dm stacks in papers by cox and perroni respectively we include a plethora of examples to illustrate the general theory we hope that this theory of toric stacks can serve as a companion to an introduction to stacks in much the same way that toric varieties can serve as a companion to an introduction to schemes

gradient based and least squares based iterative algorithms for matrix equations axb cxtd f abstract this paper develops a gradient based and a least squares based iterative algorithms for solving matrix equation axb cx t d f the basic idea is to decompose the matrix equation system under consideration into two subsystems by applying the hierarchical identification principle and to derive the iterative algorithms by extending the iterative methods for solving ax b and axb f the analysis shows that when the matrix equation has a unique solution under the sense of least squares the iterative solution converges to the exact solution for any initial values a numerical example verifies the proposed theorems

vector least squares solutions for coupled singular matrix equations the weighted least squares solutions of coupled singular matrix equations are too difficult to obtain by applying matrices decomposition in this paper a family of algorithms are applied to solve these problems based on the kronecker structures subsequently we construct a computationally efficient solutions of coupled restricted singular matrix equations furthermore the need to compute the weighted drazin and weighted moore penrose inverses and the use of tian s work and lev ari s results are due to appearance in the solutions of these problems the several special cases of these problems are also considered which includes the well known coupled sylvester matrix equations finally we recover the iterative methods to the weighted case in order to obtain the minimum d norm g vector least squares solutions for the coupled sylvester matrix equations and the results lead to the least squares solutions and invertible solutions as a special case

matrix results on the khatri rao and tracy singh products abstract we establish a connection between the khatri rao and tracy singh products introduced by khatri and rao c g khatri c r rao sankhya 30 1968 167 180 and tracy and singh d s tracy r p singh statistica neerlandica 26 1972 143 157 respectively and present further results including matrix equalities and inequalities involving the two products also we give two statistical applications

the accelerated gradient based iterative algorithm for solving a class of generalized sylvester transpose matrix equation in this paper we present an accelerated gradient based algorithm by minimizing certain criterion quadratic function for solving the generalized sylvester transpose matrix equation a x b c x t d f the idea is from ding and chen 2005 niu et al 2011 wang et al 2012 in which some efficient algorithms were developed for solving the sylvester matrix equation and the lyapunov matrix equation on the basis of the information generated in the previous half step we further introduce a relaxation factor to obtain the solution of the generalized sylvester transpose matrix equation we show that the iterative solution converges to the exact solution for any initial value provided that some appropriate assumptions finally some numerical examples are given to illustrate that the introduced iterative algorithm is efficient

lifting tropical curves in space and linear systems on graphs abstract tropicalization is a procedure for associating a polyhedral complex in euclidean space to a subvariety of an algebraic torus we study the question of which graphs arise from tropicalizing algebraic curves by using baker s specialization of linear systems from curves to graphs we are able to give a necessary condition for a balanced weighted graph to be the tropicalization of a curve our condition reproduces a generalization of speyer s well spacedness condition and also gives new conditions in addition it suggests a new combinatorial structure on tropicalizations of algebraic curves

gradient based iterative algorithms for solving a class of matrix equations in this note we apply a hierarchical identification principle to study solving the sylvester and lyapunov matrix equations in our approach we regard the unknown matrix to be solved as system parameters to be identified and present a gradient iterative algorithm for solving the equations by minimizing certain criterion functions we prove that the iterative solution consistently converges to the true solution for any initial value and illustrate that the rate of convergence of the iterative solution can be enhanced by choosing the convergence factor or step size appropriately furthermore the iterative method is extended to solve general linear matrix equations the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

developments in random matrix theory in this introduction to the journal of physics a special issue on random matrix theory we give a review of the main historical developments in random matrix theory a short summary of the papers that appear in this special issue is also given

iterative least squares solutions of coupled sylvester matrix equations abstract in this paper we present a general family of iterative methods to solve linear equations which includes the well known jacobi and gauss seidel iterations as its special cases the methods are extended to solve coupled sylvester matrix equations in our approach we regard the unknown matrices to be solved as the system parameters to be identified and propose a least squares iterative algorithm by applying a hierarchical identification principle and by introducing the block matrix inner product the star product for short we prove that the iterative solution consistently converges to the exact solution for any initial value the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

counting function for a sphere of anisotropic quartz we calculate the leading weyl term of the counting function for a monocrystalline quartz sphere in contrast to other studies of counting functions the anisotropy of quartz is a crucial element in our investigation hence we do not obtain a simple analytical form but we carry out a numerical evaluation to this end we employ the radon transform representation of the green s function we compare our result to a previously measured unique data set of several tens of thousands of resonances

tropical geometry and correspondence theorems via toric stacks in this paper we generalize correspondence theorems of mikhalkin and nishinou siebert providing a correspondence between algebraic and parameterized tropical curves we also give a description of a canonical tropicalization procedure for algebraic curves motivated by berkovich s construction of skeletons of analytic curves under certain assumptions we construct a one to one correspondence between algebraic curves satisfying toric constraints and certain combinatorially defined objects called stacky tropical reductions that can be enumerated in terms of tropical curves satisfying linear constraints similarly we construct a one to one correspondence between elliptic curves with fixed j invariant satisfying toric constraints and stacky tropical reductions that can be enumerated in terms of tropical elliptic curves with fixed tropical j invariant satisfying linear constraints our theorems generalize previously published correspondence theorems in tropical geometry and our proofs are algebra geometric in particular the theorems hold in large positive characteristic

universal spectral statistics in quantum graphs we prove that the spectrum of an individual chaotic quantum graph shows universal spectral correlations as predicted by random matrix theory the stability of these correlations with regard to nonuniversal corrections is analyzed in terms of the linear operator governing the classical dynamics on the graph

wall crossing and recursion formulae for tropical jucys covers hurwitz numbers enumerate branched genus covers of the riemann sphere with fixed ramification data or equivalently certain factorisations of permutations double hurwitz numbers are an important class of hurwitz numbers obtained by considering ramification data with a specific structure they exhibit many fascinating properties such as a beautiful piecewise polynomial structure which has been well studied in the last 15 years in particular using methods from tropical geometry it was possible to derive wall crossing formulae for double hurwitz numbers in arbitrary genus further double hurwitz numbers satisfy an explicit recursive formula in recent years several related enumerations have appeared in the literature in this work we focus on two of those invariants so called monotone and strictly monotone double hurwitz numbers monotone double hurwitz numbers originate from random matrix theory as they appear as the coefficients in the asymptotic expansion of the famous harish chandra itzykson zuber integral strictly monotone double hurwitz numbers are known to be equivalent to an enumeration of grothendieck dessins d enfants these new invariants share many structural properties with double hurwitz numbers such as piecewise polynomiality in this work we enlarge upon this study and derive new explicit wall crossing and recursive formulae for monotone and strictly monotone double hurwitz numbers the key ingredient is a new interpretation of monotone and strictly monotone double hurwitz numbers in terms of tropical covers which was recently derived by the authors an interesting observation is the fact that monotone and strictly monotone double hurwitz numbers satisfy wall crossing formulae which are almost identical to the classical double hurwitz numbers

boundary element dynamical energy analysis a versatile method for solving two or three dimensional wave problems in the high frequency limit dynamical energy analysis was recently introduced as a new method for determining the distribution of mechanical and acoustic wave energy in complex built up structures the technique interpolates between standard statistical energy analysis and full ray tracing containing both of these methods as limiting cases as such the applicability of the method is wide ranging and additionally includes the numerical modelling of problems in optics and more generally of linear wave problems in electromagnetics in this work we consider a new approach to the method with enhanced versatility enabling three dimensional problems to be handled in a straightforward manner the main challenge is the high dimensionality of the problem we determine the wave energy density both as a function of the spatial coordinate and momentum or direction space the momentum variables are expressed in separable polar coordinates facilitating the use of products of univariate basis expansions however this is not the case for the spatial argument and so we propose to make use of automated mesh generating routines to both localise the approximation allowing quadrature costs to be kept moderate and give versatility in the code for different geometric configurations

iterative method to solve the generalized coupled sylvester transpose linear matrix equations over reflexive or anti reflexive matrix abstract the iterative method of generalized coupled sylvester transpose linear matrix equations a x b c y t d s 1 e x t f g y h s 2 over reflexive or anti reflexive matrix pair x y is presented on the condition that the coupled matrix equations are consistent we show that the solution pair x y proposed by the iterative method can be obtained within finite iterative steps in the absence of roundoff error for any initial value given a reflexive or anti reflexive matrix moreover the optimal approximation reflexive or anti reflexive matrix solution pair to an arbitrary given reflexive or anti reflexive matrix pair can be derived by searching the least frobenius norm solution pair of the new generalized coupled sylvester transpose linear matrix equations finally some numerical examples are given which illustrate that the introduced iterative algorithm is quite efficient

the modified conjugate gradient methods for solving a class of generalized coupled sylvester transpose matrix equations in this paper we consider the iteration solutions of generalized coupled sylvester transpose matrix equations a1xb1 c1ytd1 f1 a2yb2 c2xtd2 f2 when the coupled matrix equations are consistent we propose a modified conjugate gradient method to solve the equations and prove that a solution x y can be obtained within finite iterative steps in the absence of roundoff error for any initial value furthermore we show that the minimum norm solution can be got by choosing a special kind of initial matrices when the coupled matrix equations are inconsistent we present another modified conjugate gradient method to find the least squares solution with the minimum norm finally some numerical examples are given to show the behavior of the considered algorithms

the matrix equations ax c xb d for the pair of matrix equations ax c xb d this paper gives common solutions of minimum possible rank and also other feasible specified ranks

parametric correlations of the energy levels of ray splitting billiards parameter dependent statistical properties of the spectra of ray splitting billiards are studied experimentally and theoretically the autocorrelation functions c x and c omega x of level velocities as well as the generalized conductance c 0 are calculated for two different classically chaotic ray splitting billiards experimentally a modified sinai ray splitting billiard is studied consisting of a thin microwave rectangular cavity with two quarter circle shaped teflon inserts the length of the cavity serves as the experimentally adjustable parameter for the theoretical estimates of the parametric correlations we compute the quantum spectrum of a scaling triangular ray splitting billiard our experimental and numerical results are compared with each other and with the predictions of random matrix theory

enumeration of rational curves with cross ratio constraints abstract in this paper we prove the algebraic tropical correspondence for stable maps of rational curves with marked points to toric varieties such that the marked points are mapped to given orbits in the big torus and in the boundary divisor the map has prescribed tangency to the boundary divisor and certain quadruples of marked points have prescribed cross ratios in particular our results generalize the results of nishinou siebert 14 the proof is very short involves only the standard theory of schemes and works in arbitrary characteristic including the mixed characteristic case

some new connections between matrix products for partitioned and non partitioned matrices this paper is concerned with a long list of matrix products for partitioned and non partitioned matrices including kronecker khatri rao of first kind hadamard tracy singh khatri rao block kronecker block hadamard and box products sums our contribution is to gather the most useful connections of the matrix products from various sources and present them in one place several new attractive connections are also added to this collection we give as an example two applications in order to show that these connections play a central role in many applications these applications are a block diagonal least squares problem and a generalization of matrix inequalities involving khatri rao products of positive definite matrices

a riemann roch theorem in tropical geometry recently baker and norine have proven a riemann roch theorem for finite graphs we extend their results to metric graphs and thus establish a riemann roch theorem for divisors on abstract tropical curves

schur complements and matrix inequalities in the lowner ordering abstract the purpose of this paper is to present a matrix inequality on the kronecker product that unifies the proofs of many existing matrix inequalities in the lowner partial ordering on the sum ordinary product and hadamard schur product schur complements serve as the basic tool

an iterative method for symmetric solutions and optimal approximation solution of the system of matrix equations a1xb1 c1 a2xb2 c2 abstract the symmetric solutions of the system of matrix equations a1xb1 c1 a2xb2 c2 are too difficult to be obtained by applying matrices decomposition in this paper an iterative method is applied to solve this problem with it the solvability of this system of matrix equations can be determined automatically when this system of matrix equations is consistent its solution can be obtained within finite iterative steps and its least norm solution can be obtained by choosing a special kind of initial iterative matrix furthermore its optimal approximation solution to a given matrix can be derived by finding the least norm symmetric solution of a new system of matrix equations a 1 x b 1 c 1 a 2 x b 2 c 2 finally numerical examples are given

on the existence of a common solution x to the matrix equations aixbj cij i j γ in this paper conditions are derived for the existence of a common solution x to the matrix equations aixbj cij i j where the matrices ai bj cij and x have suitable dimensions and the i j s are index pairs in some set the purpose of this paper is to present for certain specific sets of index pairs verifiable necessary and sufficient solvability conditions that are stated directly in terms of the matrices and that do not use kronecker products

least squares solution with the minimum norm for the matrix equation axb gxh c d based on the projection theorem in hilbert space by making use of the generalized singular value decomposition and the canonical correlation decomposition an analytical expression of the least squares solution for the matrix equation axb gxh c d with the minimum norm is derived an algorithm for finding the minimum norm solution is described and some numerical results have been given