matrix kadomtsev petviashvili equation matrix identities and explicit non singular solutions a new version of the backlund darboux transformation for the matrix kadomtsev petviashvili kp equation is used to construct and study explicit multi parameter solutions and wavefunctions in terms of the matrix exponents a class of the self adjoint non singular solutions of kp i is introduced using the controllability notion from the system theory a subclass of the rationally decaying self adjoint non singular solutions is studied in particular several results prove new in the scalar case also

minors of the moore penrose inverse abstract let q k n 1 k 1 1 k n denote the strictly increasing sequences of k elements from 1 n for q k n we denote by a the submatrix of a with rows indexed by columns by the submatrix obtained by deleting the rows and columns is denoted by a lsqb rsqb for nonsingular a r n n the jacobi identity relates the minors of the inverse a 1 to those of a det a 1 lsqb rsqb 1 k i 1 i k i 1 i det a lsqb rsqb det a for any q k n we generalize the jacobi identity to matrices a r m n r expressing the minors of the moore penrose inverse a in terms of the minors of the maximal nonsingular submatrices a ij of a in our notation det a lsqb rsqb 1 vol 2 a i j n det a ij a a ij for any q k m q k n 1 k r where vol 2 a denotes the sum of squares of determinants of r r submatrices of a we apply our results to questions concerning the nonnegativity of principal minors of the moore penrose inverse

analytic derivatives for estimation of linear dynamic models abstract this paper develops two algorithms algorithm 1 computes the exact gaussian log likelihood function its exact gradient vector and an asymptotic approximation of its hessian matrix for discrete time linear dynamic models in state space form algorithm 2 derived from algorithm 1 computes the exact sample information matrix of this likelihood function the computed quantities are analytic not numerical approximations and should therefore be useful for reliably quickly and accurately i checking local identifiability of parameters by checking the rank of the information matrix ii using the gradient vector and hessian matrix to compute maximum likelihood estimates of parameters with newton methods and iii computing asymptotic covariances cramer rao bounds of the parameter estimates with the hessian or the information matrix the principal contribution of the paper is algorithm 2 which extends to multivariate models the univariate results of porat and friedlander 1 by relying on the kalman filter instead of the levinson durbin filter used by porat and friedlander algorithms 1 and 2 can automatically handle any pattern of missing or linearly aggregated data although algorithm 1 is well known it is treated in detail in order to make the paper self contained

on a generalization of the szego levinson recurrence and its application in lossless inverse scattering predictor polynomials corresponding to nested toeplitz matrices are known to be connected by the szego levinson recurrence relation a generalization of that result where the relevant reduction process for toeplitz matrices of decreasing order is defined by an elementary one parameter linear transformation is addressed the descending and ascending versions of the corresponding generalized szego levinson recurrence relations are discussed in detail in particular these relations are shown to be essentially the same as the extraction formulas for canonical schur and brune sections in the dewilde dym 1984 recursive solution of the lossless inverse scattering problem some extensions of the levinson algorithm for linear prediction and of the schur cohn algorithm for polynomial stability test are presented

some schur complement identities and applications to matrix extrapolation methods the schur complement has many applications in linear algebra and in numerical analysis schur complement identities have been used for implementing some scalar and vector extrapolation methods in this paper we define some matrix sequence transformations and give two new schur complement identities that will be used to derive algorithms for implementing these transformations

a trace formula for canonical differential expressions we prove a trace formula for pairs of self adjoint operators associated to canonical differential expressions an important role is played by the associated weyl function

the principal rank characteristic sequence of a real symmetric matrix abstract given a vector u r 2 n the principal minor assignment problem asks when is there an n n matrix having its 2 n principal minors given by u this paper explores the following related problem given a sequence r 0 r 1 r n of 0s and 1s does there exist an n n real symmetric matrix that has a principal submatrix of rank k if and only if r k 1 for all 0 k n certain conditions are shown to be necessary in order for this question to have an affirmative answer several families of matrices are constructed to attain certain classes of sequences the problem is solved completely for n 6 and for 7 n 10 in the case of sequences beginning with 010

the noncommutative akns system projection to matrix systems countable superposition and soliton like solutions starting from the recent work on noncommutative akns systems for functions with values in the bounded operators on a banach space it is shown how their formal 1 soliton solution depending on operator parameters can be mapped to solutions of matrix akns systems the main result is rather general solution formulas for matrix akns systems the most important applications are the countable superposition of matrix solitons and explicit expressions for the soliton like solutions of the classical akns system

discrete skew selfadjoint canonical systems and the isotropic heisenberg magnet model abstract a discrete analog of a skew self adjoint canonical zakharov shabat or akns system with a pseudo exponential potential is introduced for the corresponding weyl function the direct and inverse problem are solved explicitly in terms of three parameter matrices as an application explicit solutions are obtained for the discrete integrable nonlinear equation corresponding to the isotropic heisenberg magnet model state space techniques from mathematical system theory play an important role in the proofs

equivalence groupoids and group classification of multidimensional nonlinear schrodinger equations abstract we study admissible and equivalence point transformations between generalized multidimensional nonlinear schrodinger equations and classify lie symmetries of such equations we begin with a wide superclass of schrodinger type equations which includes all the other classes considered in the paper showing that this superclass is not normalized we partition it into two disjoint normalized subclasses which are not related by point transformations further constraining the arbitrary elements of the superclass we construct a hierarchy of normalized classes of schrodinger type equations this gives us an appropriate normalized superclass for the non normalized class of multidimensional nonlinear schrodinger equations with potentials and modular nonlinearities and allows us to partition the latter class into three families of normalized subclasses after a preliminary study of lie symmetries of nonlinear schrodinger equations with potentials and modular nonlinearities for an arbitrary space dimension we exhaustively solve the group classification problem for such equations in space dimension two

tau functions grassmannians and rank one conditions in integrable systems specifically the kp hierarchy there are functions known as tau functions closely related to the schur polynomials in terms of which they are often written although they are generally viewed as the solutions to a collection of nonlinear pdes in this note they will equivalently be characterized by a quadratic difference equation sato s theorem associates tau functions to the points of a grassmann manifold to make that amazing theorem clear to non experts we will first show an analogous but easily understood example of a linear ode and its solution from a flow on the xy plane in each case the solution is created via a flow generated by a certain linear operator the question we pose is this what other operators could have been used to generate solutions in the same way although the answer is well known in the ode case the question in the nonlinear case is the main result of our new paper we will state the result and discuss its relationship to the trend of writing tau functions in terms of matrices satisfying certain rank one conditions the elucidation of a geometric interpretation of the hirota bilinear difference equation hbde is a key feature of the proof and will be briefly described

commutation methods applied to the mkdv equation an explicit construction of solutions of the modified korteweg de vries equation given a solution of the ordinary korteweg de vries equation is provided our theory is based on commutation methods i e n 1 supersymmetry underlying miura s transformation that links solutions of the two evolution equations

montgomery s method of polynomial selection for the number field sieve the number field sieve is the most efficient known algorithm for factoring large integers that are free of small prime factors for the polynomial selection stage of the algorithm montgomery proposed a method of generating polynomials which relies on the construction of small modular geometric progressions montgomery s method is analysed in this paper and the existence of suitable geometric progressions is considered

rational solutions of the master symmetries of the kdv equation in this article we characterize a certain class of rational solutions of the hierarchy of master symmetries for kdv the result is that the generic rational potentials that decay at infinity and remain rational by all the flows of the master symmetry kdv hierarchy are bispectral potentials for the schrodinger operator by bispectral potentials we mean that the corresponding schrodinger operators possess families of eigenfunctions that are also eigenfunctions of a differential operator in the spectral variable this complements certain results of airault mckean moser 4 duistermaat grunbaum 10 and magri zubelli 40 as a consequence of bispectrality the rational solutions of the master symmetries turn out to be solutions of a generalized string equation

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

nonisospectral integrable nonlinear equations with external potentials and their gbdt solutions auxiliary systems for matrix nonisospectral equations including coupled nls with external potential and kdv with variable coefficients were introduced explicit up to the inversion of finite matrices solutions of nonisospectral equations were constructed using the gbdt version of the backlund darboux transformation

revisiting totally positive differential systems a tutorial and new results abstract a matrix is called totally nonnegative tn totally positive tp if all its minors are nonnegative positive multiplying a vector by a tn matrix does not increase the number of sign variations in the vector in a largely forgotten paper binyamin schwarz 1970 considered matrices whose exponentials are tn or tp he also analyzed the evolution of the number of sign changes in the vector solutions of the corresponding linear system his work however considered only linear systems in a seemingly different line of research smillie 1984 smith 1991 and others analyzed the stability of nonlinear tridiagonal cooperative systems by using the number of sign variations in the derivative vector as an integer valued lyapunov function we show that these two research topics are intimately related this allows to derive important generalizations of the results by smillie 1984 and smith 1991 while simplifying the proofs these generalizations are particularly relevant in the context of control systems also the results by smillie and smith provide sufficient conditions for analyzing stability based on the number of sign changes in the vector of derivatives and the connection to the work of schwarz allows to show in what sense these results are also necessary we describe several new and interesting research directions arising from this new connection

iterated backlund darboux transform for canonical systems abstract the backlund darboux transform and its modifications are applied to the spectral theory of the canonical systems several inverse spectral problems and problems on similarity of hamiltonians are solved

can sylvester s determinantal identity equivalently muir s law of extensible minors be generalized abstract we show that sylvester s classical determinantal identity is equivalent to its generalizations given by the muhlbach gasca lopez carmona ramirez identity and the beckermann muhlbach identity we also show that the generating extension principles associated with each of these three identities muir s law of extensible minors muhlbach s extension principle and the extension principle given in camargo 2013 3 are equivalent and we use this fact to conclude that all these six statements and also jacobi s identity on the minors of the adjugate and cayley s law of complementaries are actually equivalent these findings lead naturally to enquire whether there would exist a generalization of sylvester s identity or an extension principle which could not be derived from sylvester s identity itself

harmonic maps backlund darboux transformations and line solution analogues harmonic maps from or one connected domain into and u m are treated the gbdt version of the b cklund darboux transformation is applied to the case of the harmonic maps and a new and simple algebraic procedure to construct new harmonic maps from the initial ones is given using some methods from system theory a new general formula on the gbdt transformations of the sym tafel immersions is derived a new class of the unitary harmonic maps with asymptotics along one line essentially different from the asymptotics in all other directions similar in certain ways to line solutions is obtained explicitly and studied

elimination technique from extrapolation to totally positive matrices and cagd in this survey we will show some connections between several mathematical problems such as extrapolation linear systems totally positive matrices and computer aided geometric design with elimination techniques as the common tool to deal with all of them

j unitary preserving automorphisms of rational matrix functions state space theory interpolation and factorization abstract let glm r be the multiplicative group of c m m valued rational functions of one complex variable z with determinant not vanishing identically we analyze general inner automorphisms of glm r that preserve the set of matrix functions whose values are unitary with respect to an indefinite scalar product for real or unimodular values of z the analysis is based on state space representations in particular homogeneous interpolation problems are studied for the rational matrix functions associated with such a general inner automorphism we also develop results on minimal factorizations within the group associated with j unitary preserving generalized inner automorphism

a new proof of the cross rule for the e algorithm based on schur complements abstract matrices of the form h ge 1 f are called schur complements they are closely related to elimination procedures and to determinantal identities in this paper the quotient property for schur complements is used to establish the schur complement method a useful tool for proving certain formulas as an example a new proof of the cross rule for the algorithm is presented

kac moody lie algebras and soliton equations iii stationary equations associated with a1 1 abstract the hamiltonian structure of stationary soliton equations associated with the akns eigenvalue problem is derived in two ways first it is shown to arise from the kostant kirillov symplectic structure on a coadjoint orbit in an infinite dimensional lie algebra second it is obtained as the restriction to a finite dimensional manifold of the infinite dimensional hamiltonian structure associated with a certain eigenvalue problem polynomial in the eigenvalue parameter

a direct derivation of the exact fisher information matrix of gaussian vector state space models this paper deals with a direct derivation of fisher s information matrix of vector state space models for the general case by which is meant the establishment of the matrix as a whole and not element by element the method to be used is matrix differentiation see 4 we assume the model to be gaussian and use the negative logarithm of the likelihood function as used in the definition of fisher s information in a related paper klein et al 3 establish the information matrix by assembling its elements as derived in the literature 2 5 6 and for an approximation of the hessian of the log likelihood function one can refer to 7 8

on the almost principal minors of a symmetric matrix abstract the almost principal rank characteristic sequence apr sequence of an n n symmetric matrix is introduced which is defined to be the string a 1 a 2 a n 1 where a k is either a s or n according as all some but not all or none of its almost principal minors of order k are nonzero in contrast to the other principal rank characteristic sequences in the literature the apr sequence of a matrix does not depend on principal minors the almost principal rank of a symmetric matrix b denoted by ap rank b is defined as the size of a largest nonsingular almost principal submatrix of b a complete characterization of the sequences not containing an a that can be realized as the apr sequence of a symmetric matrix over a field f is provided a necessary condition for a sequence to be the apr sequence of a symmetric matrix over a field f is presented it is shown that if b f n n is symmetric and non diagonal then rank b 1 ap rank b rank b with both bounds being sharp moreover it is shown that if b is symmetric non diagonal and singular and does not contain a zero row then rank b ap rank b

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

a generalization of sylvester s identity on determinants and some applications abstract a generalization of sylvester s identity on determinants is proved by elimination techniques as an application some inequalities for determinants of totally positive or positive definite matrices are derived

rational solutions of nonlinear evolution equations vertex operators and bispectrality abstract we prove that if q x t 2 t m and r x t 2 t m are certain rational solutions of the akns hierarchy then there are eigenfunctions x k of the akns zs operator l x q r x satisfying a differential equation in the spectral parameter k of the form b k k x where b k k is a matrix differential operator independent of x and is a nonconstant function of x we also discuss the relation between this result and a similar one for the rational solutions of the schrodinger operator with potentials in the manifold of rational solutions of the kdv hierarchy

construction of the exact fisher information matrix of gaussian time series models by means of matrix differential rules the fisher information matrix is of fundamental importance for the analysis of parameter estimation of time series models in this paper the exact information matrix of a multivariate gaussian time series model expressed in state space form is derived a computationally efficient procedure is used by applying matrix differential rules for the derivatives of a matrix function j j with respect to its vector argument an algorithm is given it is sketched for the general state space structure without specifying a parametrization it is then detailed for the vector autoregressive moving average varma model with a given parametrization where explicit recurrent relations are developed

inverse spectral problem for differential operators with rational scattering matrix functions in this paper we study the inverse scattering problem for linear canonical differential equations of a special type in the case when the scattering function is a rational matrix valued function the main result here is the form of the potential in terms of a realisation of the scattering function the difference between this publication and the previous one 3 consists in the fact that here is used the marchenko method instead of kre n s method for rational scattering functions we also prove the equivalence of the two methods

noncommutative korteweg de vries and modified korteweg de vries hierarchies via recursion methods here noncommutative hierarchies of nonlinear equations are studied they represent a generalization to the operator level of corresponding hierarchies of scalar equations which can be obtained from the operator ones via a suitable projection a key tool is the application of backlund transformations to relate different operator valued hierarchies indeed in the case when hierarchies in 1 1 dimensions are considered a backlund chart depicts links relating in particular the korteweg de vries kdv to the modified kdv mkdv hierarchy notably analogous links connect the hierarchies of operator equations the main result is the construction of an operator soliton solution depending on an infinite dimensional parameter to start with the potential kdv hierarchy is considered then backlund transformations are exploited to derive solution formulas in the case of kdv and mkdv hierarchies it is remarked that hierarchies of matrix equations of any dimension are also incorporated in the present framework

on the resultant property of the fisher information matrix of a vector arma process a matrix is called a multiple resultant matrix associated to two matrix polynomials when it becomes singular if and only if the two matrix polynomials have at least one common eigenvalue in this paper a new multiple resultant matrix is introduced it concerns the fisher information matrix fim of a stationary vector autoregressive and moving average time series process varma the two matrix polynomials are the autoregressive and the moving average matrix polynomials of the varma process in order to show that the fim is a multiple resultant matrix two new representations of the fim are derived to construct such representations appropriate matrix differential rules are applied the newly obtained representations are expressed in terms of the multiple sylvester matrix and the tensor sylvester matrix the representation of the fim expressed by the tensor sylvester matrix is used to prove that the fim becomes singular if and only if the autoregressive and moving average matrix polynomials have at least one common eigenvalue it then follows that the fim and the tensor sylvester matrix have equivalent singularity conditions in a simple numerical example it is shown however that the fim fails to detect common eigenvalues due to some kind of numerical instability whereas the tensor sylvester matrix reveals it clearly proving the usefulness of the results derived in this paper

on lossless feedback delay networks lossless feedback delay networks fdns are commonly used as a design prototype for artificial reverberation algorithms the lossless property is dependent on the feedback matrix which connects the output of a set of delays to their inputs and the lengths of the delays both unitary and triangular feedback matrices are known to constitute lossless fdns however the most general class of lossless feedback matrices has not been identified in this contribution it is shown that the fdn is lossless for any set of delays if all irreducible components of the feedback matrix are diagonally similar to a unitary matrix the necessity of the generalized class of feedback matrices is demonstrated by examples of fdn designs proposed in literature

exact solutions of nonlinear equations and the method of operator identities abstract exact solutions and wave functions of integrable nonlinear equations are constructed by the method of operator identities

on deriving the inverse of a sum of matrices available expressions are reviewed and new ones derived for the inverse of the sum of two matrices one of them being nonsingular particular attention is given to text bf a text bf ubv 1 where textbf a is nonsingular and textbf u b and textbf v may be rectangular generalized inverses of textbf a textbf ubv are also considered several statistical applications are discussed

completion problems and scattering problems for dirac type differential equations with singularities abstract a completion problem to recover a rational matrix function which is j unitary on the line is treated a dirac type system with singularities on the semi axis is recovered explicitly from its left reflection coefficient the close relation between these two problems is discussed

a generalized nonisospectral camassa holm equation and its multipeakon solutions abstract motivated by the paper of beals sattinger and szmigielski 2000 3 we propose an extension of the camassa holm equation which also admits the multipeakon solutions the novel aspect is that our approach is mainly based on classic determinant technique furthermore the proposed equation is shown to possess a nonisospectral lax pair

the sine theorem and inequalities for volumes of simplices and determinants abstract we present two proofs of the sine theorem for a simplex and derive a well known majorization of its volume in terms of its faces we apply this to obtain some other volume and determinant majorizations

szego kac achiezer formulas in terms of realizations of the symbol abstract for rational and analytic matrix functions new formulas are obtained for the limits in the szego kac achiezer limit theorems in the rational case the new expressions are given in terms of finite matrices which come from special representations of the matrix functions these representations are known as realizations in mathematical systems theory

perturbations of orthogonal polynomials with periodic recursion coefficients the results of denisov rakhmanov szeg shohat nevai and killip simon are extended from asymptotically constant orthogonal polynomials on the real line oprl and unit circle opuc to asymptotically periodic oprl and opuc the key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations

an extended multistep shanks transformation and convergence acceleration algorithm with their convergence and stability analysis the molecule solution of an extended discrete lotka volterra equation is constructed from which a new sequence transformation is proposed a convergence acceleration algorithm for implementing this sequence transformation is found it is shown that our new sequence transformation accelerates some kinds of linearly convergent sequences and factorially convergent sequences with good numerical stability some numerical examples are also presented

other manifestations of the schur complement abstract many series and sequence transformations used in numerical analysis are defined as ratios of determinants they are implemented by recursive algorithms based on determinantal identities the aim of this paper is to study the connections of these transformations and algorithms with the schur complement of a matrix the schur s formula is extended to the vector case thus providing the same treatment for vector sequence transformations and the corresponding recursive algorithms thanks to these connections particular rules for avoiding division by zero or numerical instability are obtained for these algorithms some fixed point methods and continued fractions also fit in this framework

an algebraic method to construct the darboux matrix we present an effective procedure to construct the 1 soliton darboux matrix our approach based on the zakharov shabat mikhailov s dressing method is especially useful in the case of non canonical normalization and for non isospectral linear problems the construction is divided into two steps first we represent a given linear problem as a system of some algebraic constraints on two matrices in this context we introduce and discuss invariants of the darboux matrix second we derive the darboux matrix demanding that it preserves the algebraic constraints in particular we consider in details the restrictions imposed by various reduction groups on the form of the darboux matrix

stability of the procedure of explicit recovery of skew selfadjoint dirac systems from rational weyl matrix functions abstract procedures to recover explicitly discrete and continuous skew selfadjoint dirac systems on semi axis from rational weyl matrix functions are considered their stability is shown some new facts on asymptotics of pseudo exponential potentials i e of explicit solutions of inverse problems are proved as well the gbdt version of backlund darboux transformation methods from system theory and results on algebraic riccati equations are used for this purpose

rational solutions of the kp hierarchy and the dynamics of their poles ii construction of the degenerate polynomial solutions a general approach to constructing the polynomial solutions satisfying various reductions of the kadomtsev petviashvili kp hierarchy is described within this approach the reductions of the kp hierarchy are equivalent to certain differential equations imposed on the function of the hierarchy in particular the l reduction and the k constraint as well as their generalized counterparts are considered a general construction of the rational solutions to these reductions is found and the particular solutions are explicitly derived for some typical examples including the kdv and gardner equations the boussinesq and classical boussinesq systems the nls and yajima oikawa equations it is shown that the degenerate rational solutions of the kp hierarchy are related to stationary manifolds of the calogero moser cm hierarchy of dynamical systems the scattering dynamics of interacting particles in the cm systems may become complicated due to an anomalously slow fractional power rate of the particle motion along the stationary manifolds

the approximation of a totally positive band matrix by a strictly banded totally positive one every nonsingular totally positive m banded matrix is shown to be the product of m totally positive one banded matrices and therefore the limit of strictly m banded totally positive matrices this result is then extended to bi infinite m banded totally positive matrices with linearly independent rows and columns in the process such matrices are shown to possess at least one diagonal whose principal sections are all nonzero as a consequence such matrices are seen to be approximable by strictly m banded totally positive ones

inequalities for two simplices we establish in this paper an inequality for two simplices which combine altitudes and edge lengths of one simplex with distances from an interior point to its facets of the other simplex and give some applications thereof

likelihood ratio testing in linear state space models an application to dynamic stochastic general equilibrium models abstract this paper considers the problem of hypothesis testing in linear gaussian state space models we consider two hypotheses of interest a simple null and a hypothesis of explicit parameter restrictions we derive the asymptotic distributions of the corresponding likelihood ratio test statistics and compute the bartlett adjustments the results are non trivial because the unrestricted state space model is not even locally identified we apply our analysis to test the validity of the dynamic stochastic general equilibrium dsge models a monte carlo exercise illustrates our findings and confirms the importance of bartlett corrections at sample sizes typically encountered in macroeconomics

hyperdeterminantal relations among symmetric principal minors abstract the principal minors of a symmetric n n matrix form a vector of length 2 n we characterize these vectors in terms of algebraic equations derived from the 2 2 2 hyperdeterminant

an algorithm for the exact fisher information matrix of vector armax time series abstract in this paper an algorithm is developed for the exact fisher information matrix of a gaussian vector armax or varmax process the algorithm proposed in this paper is composed by chandrasekhar recursion equations at a vector matrix level and some of these recursions consist of derivatives based on appropriate differential rules applied to a state space model for a vector process the chosen representation is such that the recursions extracted from the state space model are given in terms of expectations of derivatives of innovations and not the process and observation disturbances the algorithm will be illustrated by an example on that example a comparison is made with results from e4 a toolbox for matlab and with the asymptotic information matrix

the time dependent schrodinger equation of dimension k 1 explicit and rational solutions via gbdt and multinodes a version of the binary darboux transformation is constructed for a non stationary schrodinger equation of dimension k 1 where k 1 is the number of space variables this is an iterated generalized backlund darboux transformation version new families of non singular and rational potentials and solutions are obtained some results are also new for the case that k 1 a certain generalization of a colligation introduced by m s livsic and a generalization of the s node introduced by l a sakhnovich have been successfully used in our construction

on the discrete spectrum of the nonstationary schrodinger equation and multipole lumps of the kadomtsev petviashvili i equation the discrete spectrum of the nonstationary schrodinger equation and localized solutions of the kadomtsev petviashvili i kpi equation are studied via the inverse scattering transform it is shown that there exist infinitely many real and rationally decaying potentials which correspond to a discrete spectrum whose related eigenfunctions have multiple poles in the spectral parameter an index or winding number is asssociated with each of these solutions the resulting localized solutions of kpi behave as collection of individual humps with nonuniform dynamics

non hermitian matrix schrodinger equation backlund darboux transformation weyl functions and script pscript t symmetry the non hermitian matrix schrodinger equation obtained by the backlund darboux transformation bdt is treated the potentials fundamental solutions and weyl functions are constructed explicitly a symmetric reduction of the bdt is introduced and this case is studied in greater detail including potentials fundamental solutions bound states the reality of the discrete spectrum and spontaneous break of the symmetry the sign indefinite scalar product and examples

computation of the exact information matrix of gaussian time series with stationary random components the paper presents an algorithm for efficient recursive computation of the fisher information matrix of gaussian time series whose random components are stationary and whose means and covariances are functions of a parameter vector the algorithm is first developed in a general framework and then specialized to the case of autoregressive moving average processes with possible additive white noise the asymptotic behavior of the algorithm is explored and a termination criterion is derived finally the algorithm is used to demonstrate the behavior of the exact cramer rao bound for unbiased estimates for some arma processes as a function of the number of data points it is shown that for processes with zeros near the unit circle and short data records the exact cramer rao bound differs dramatically from its common approximation based on asymptotic theory

an effective method to compute n fold darboux matrix and n soliton surfaces fairly simple formulas for an n fold darboux matrix and an n soliton addition to an arbitrary soliton surface embedded in e3 are proven they depend directly on the background wave function and are evidently symmetric with respect to n added solitons as an example an explicit expression for the interaction of n solitons on a single vortex filament is presented moreover there is given a simplified version of the n soliton formulas of neugebauer and meinel

solitons and almost intertwining matrices we define the algebraic variety of almost intertwining matrices to be the set of triples x y z of n n matrices for which xz yx t for a rank one matrix t a surprisingly simple formula is given for tau functions of the kp hierarchy in terms of such triples the tau functions produced in this way include the soliton and vanishing rational solutions the induced dynamics of the eigenvalues of the matrix x are considered leading in special cases to the ruijsenaars schneider particle system

cauchy binet for pseudo determinants abstract the pseudo determinant det a of a square matrix a is defined as the product of the nonzero eigenvalues of a it is a basis independent number which is up to a sign the first nonzero entry of the characteristic polynomial of a we prove det f t g p det f p det g p for any two n m matrices f g the sum to the right runs over all k k minors of a where k is determined by f and g if f g is the incidence matrix of a graph this directly implies the kirchhoff tree theorem as l f t g is then the laplacian and det 2 f p 0 1 is equal to 1 if p is a rooted spanning tree a consequence is the following pythagorean theorem for any self adjoint matrix a of rank k one has det 2 a p det 2 a p where det a p runs over k k minors of a more generally we prove the polynomial identity det 1 x f t g p x p det f p det g p for classical determinants det which holds for any two n m matrices f g and where the sum on the right is taken over all minors p understanding the sum to be 1 if p 0 it implies the pythagorean identity det 1 f t f p det 2 f p which holds for any n m matrix f and sums again over all minors f p if applied to the incidence matrix f of a finite simple graph it produces the chebotarev shamis forest theorem telling that det 1 l is the number of rooted spanning forests in the graph with laplacian l

a study of solitonic combinations based on the theory of commuting non self adjoint operators abstract an intimate connection between two mathematical concepts the theory of solitons on the one hand and the theory of commuting non self adjoint operators on the other hand is established the link is provided by the marchenko method for solving nonlinear partial differential equations in general and the korteweg de vries equation in particular the solution of this equation is cast in a special form in terms of solitonic combinations such that the role played by a pair of commuting non self adjoint operators is evident and lends itself to generalization solitonic combinations are closely related to collective motions of spatiotemporal systems whose theory is reviewed here and extended to the physical space of four dimensions one time and three space a novel interpretation of the physical concept of quasiparticle is presented

pseudo canonical systems with rational weyl functions explicit formulas and applications abstract the paper extends earlier results of the authors for canonical systems with spectral functions of which the absolutely continuous part has a rational derivative to a class of differential systems with skew selfadjoint potentials the corresponding direct and inverse spectral problems are solved explicitly using state space methods from mathematical system theory applications to nonlinear integrable partial differential equations are included

kac moody lie algebras and soliton equations ii lax equations associated with a1 1 the soliton equations associated with sl 2 eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment they are shown to be generated by a single family of commuting hamiltonians on a subalgebra of the loop algebra of sl 2 the conserved densities and fluxes of the usual anks hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems the hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment

determinantal identities gauss schur cauchy sylvester kronecker jacobi binet laplace muir and cayley abstract we give a common concise derivation of some important determinantal identities attributed to the mathematicians in the title we also give a formal treatment of determinantal identities of the minors of a matrix

principal minors and rhombus tilings the algebraic relations between the principal minors of a generic n n matrix are somewhat mysterious see e g lin and sturmfels 2009 j algebra 322 4121 31 we show however that by adding in certain almost principal minors the ideal of relations is generated by translations of a single relation the so called hexahedron relation which is a composition of six cluster mutations we give in particular a laurent polynomial parameterization of the space of n n matrices whose parameters consist of certain principal and almost principal minors the parameters naturally live on vertices and faces of the tiles in a rhombus tiling of a convex gon a matrix is associated to an equivalence class of tilings all related to each other by yang baxter like transformations by specializing the initial data we can similarly parameterize the space of hermitian symmetric matrices over or the quaternions moreover by further specialization we can parametrize the space of positive definite matrices over these rings this article is part of a special issue of journal of physics a mathematical and theoretical devoted to cluster algebras mathematical physics

on the general structure of nonlinear evolution equations and their backlund transformations connected with the matrix non stationary schrodinger spectral problem the general form of nonlinear evolution equations in 1 2 dimensions integrable by the matrix non stationary schrodinger spectral problem is found the infinite dimensional group of backlund transformations for these equations is constructed and the nonlinear superposition principle is obtained

on minimality in the partial realization problem abstract a compression algorithm is constructed which allows one to reduce an arbitrary realization of a finite sequence of matrices m 1 m r to a minimal partial realization of m 1 m r furthermore a state space criterion of minimality of a partial realization and a formula for the minimal state space dimension are obtained

new proofs of two representations and minor of generalized inverse a 2 t s abstract in this paper we prove the limit representation for the generalized inverse a t s 2 of a given complex matrix a in a simpler way then using this representation two known results the generalized inverse a t s 2 as an analogs of convex combination with the minor of adja and the representation of the minor of the generalized inverse a t s 2 were derived anew as a special case of the minor a determinantal representation of the generalized inverse a t s 2 is obtained

the generalized sine theorem and inequalities for simplices abstract for an n dimensional simplex in en we establish a sine theorem and some inequalities for k order vertex angles which are defined in this paper to prove these results we give a formula about the inverse matrix of the metric matrix of an n parallelotope

the bitangential inverse spectral problem for canonical systems the theory of the direct and bitangential inverse input impedance problem is used to solve the direct and bitangential inverse spectral problem the analysis of the direct spectral problem uses and extends a number of results that appear in the literature special attention is paid to the class of canonical integral systems with matrizants that are strongly regular j inner matrix valued functions in the sense introduced in 7 the bitangential inverse spectral problem is solved in this class in our considerations the data for this inverse problem is a given nondecreasing p p matrix valued function on r and a normalized monotonic continuous chain of pairs b3t b4t 0 t d of entire inner p p matrix valued functions each such chain defines a class of canonical integral systems in which we find a solution of the inverse problem for the given spectral function a detailed comparison of our investigations of inverse problems with those of sakhnovich is presented