linearizable mappings and the low growth criterion we examine a family of discrete second order systems which are integrable through reduction to a linear system these systems were previously identified using the singularity confinement criterion here we analyse them using the more stringent criterion of nonexponential growth of the degrees of the iterates we show that the linearizable mappings are characterized by a very special degree growth the ones linearizable by reduction to projective systems exhibit zero growth i e they behave like linear systems while the remaining ones derivatives of riccati and gambier mapping lead to linear growth this feature may well serve as a detector of integrability through linearization

numeric and mesh algorithms for the coxeter spectral study of positive edge bipartite graphs and their isotropy groups we develop algorithmic techniques for the coxeter spectral analysis of the class ubigr n of connected loop free positive edge bipartite graphs d with n 2 vertices i e signed graphs in particular we present numerical and graphical algorithms allowing us a computer search in the study of such graphs d by means of their gram matrix email protected d the complex spectrum specc email protected c of the coxeter matrix cox d email protected email protected email protected d t r and the geometry of weyl orbits in the set mor d d of matrix morsifications email protected m n z of a simply laced dynkin diagram email protected a n d n e 6 e 7 e 8 associated with d and mesh root systems of type email protected our algorithms construct the coxeter gram polynomials cox d t z t and mesh geometries of root orbits of small connected loop free positive edge bipartite graphs d we apply them to the study of the following coxeter spectral analysis problem does email protected email protected hold i e the email protected email protected d arez congruent for any pair of connected positive loop free edge bipartite graphs d d inubigr nsuch thatspecc d specc d the problem if any square integer matrix email protected m n z is z congruent with its transpose a t r is also discussed we present a solution for graphs in ubigr n with email protected 6

matrices a such that as 1r ra with rk i abstract we study matrices a c n n such that a s 1 r r a where r k i n and s k are nonnegative integers with k 2 such matrices are called r s 1 k potent matrices the s 0 case corresponds to matrices such that a r a r 1 with r k i n and is studied using spectral properties of the matrix r for s 1 various characterizations of the class of r s 1 k potent matrices and relationships between these matrices and other classes of matrices are presented

on sesquilinear forms over fields with involution abstract let k be a field of characteristic 2 with an involution a matrix a is split if there is a change of variables q such that q t aq consists of two complementary diagonal blocks we classify all matrices that do not split as a consequence we obtain a new proof for the following result given a square matrix a there is a matrix s such that s t as a t and s s i

pairs of hermitian and skew hermitian quaternionic matrices canonical forms and their applications canonical forms are described for pairs of quaternionic matrices or equivalently matrix pencils where each one of the matrices is either hermitian or skew hermitian under strict equivalence and simultaneous congruence several applications are developed including structure of selfadjoint and skew adjoint matrices with respect to a regular sesquilinear quaternion valued form or with respect to a regular skew sesquilinear form another application gives canonical forms for quaternionic matrices under congruence

growth and integrability in the dynamics of mappings the growth of some numerical characteristics of the mappings under their iterations in the context of the general problem of integrability is discussed in the general case such characteristics as complexity by arnold or the number of the different images for the multiple valued mappings are growing exponentially it is shown that the integrability is closely related with thepolynomial growth the analogies with quantum integrable systems are discussed

the solution of the equation xa axt 0 and its application to the theory of orbits abstract describe how to find the general solution of the matrix equation xa ax t 0 with a c n n which allows us to determine the dimension of its solution space this result has immediate applications in the theory of congruence orbits of matrices in c n n because the set xa ax t x c n n is the tangent space at a to the congruence orbit of a hence the codimension of this orbit is precisely the dimension of the solution space of xa ax t 0 as a consequence we also determine the generic canonical structure of matrices under the action of congruence all these results can be directly extended to palindromic pencils a a t

equivalence and normal forms of bilinear forms abstract we present an alternative account of the problem of classifying and finding normal forms for arbitrary bilinear forms beginning from basic results developed by riehm our solution to this problem hinges on the classification of indecomposable forms and in how uniquely they fit together to produce all other forms we emphasize the use of split forms i e those bilinear forms such that the minimal polynomial of the asymmetry of their non degenerate part splits over ground field rather than restricting the field to be algebraically closed in order to obtain the most explicit results without resorting to the classification of hermitian symmetric and quadratic forms we merely require that the underlying field be quadratically closed

dynamics of bimeromorphic maps of surfaces we classify bimeromorphic self maps f x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of compact kahler surfaces x in terms of their actions f h 1 1 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i on cohomology we observe that the growth rate of f n is invariant under bimeromorphic conjugacy and that by conjugating one can always arrange that f n f n we show that the sequence f n can be bounded grow linearly grow quadratically or grow exponentially in the first three cases we show that after conjugating f is an automorphism virtually isotopic to the identity f preserves a rational fibration or f preserves an elliptic fibration respectively in the last case we show that there is a unique up to scaling expanding eigenvector for f that is nef and that f is bimeromorphically conjugate to an automorphism if and only if 2 0 we go on in this case to construct a dynamically natural positive current representing and we study the growth rate of periodic orbits of f we conclude by illustrating our results with a particular family of examples

the moduli space of complex 5 dimensional lie algebras abstract in this paper we study the moduli space of all complex 5 dimensional lie algebras realizing it as a stratification by orbifolds which are connected only by jump deformations the orbifolds are given by the action of finite groups on very simple complex manifolds our method of determining the stratification is based on the construction of versal deformations of the lie algebras which allow us to identify natural neighborhoods of the elements in the moduli space

canonical matrices of isometric operators on indefinite inner product spaces abstract we give canonical matrices of a pair a b consisting of a nondegenerate form b and a linear operator a satisfying b ax ay b x y on a vector space over f in the following cases f is an algebraically closed field of characteristic different from 2 or a real closed field and b is symmetric or skew symmetric f is an algebraically closed field of characteristic 0 or the skew field of quaternions over a real closed field and b is hermitian or skew hermitian with respect to any nonidentity involution on f these classification problems are wild if b may be degenerate we use a method that admits to reduce the problem of classifying an arbitrary system of forms and linear mappings to the problem of classifying representations of some quiver this method was described in v v sergeichuk classification problems for systems of forms and linear mappings math ussr izv 31 3 1988 481 501

on the structure of loop free non negative edge bipartite graphs abstract we continue the study of a class of signed graphs called finite connected loop free edge bipartite graphs bigraphs for short started in simson 2013 33 and continued in g siorek et al 2016 14 and simson zaj c 2017 38 in this paper we present an algorithmic approach to the study of non negative bigraphs with n r 2 vertices of corank r 1 that is bigraphs with the symmetric gram matrix g 1 2 g g t r m n r 1 2 z positive semi definite of rank n 1 where g m n r z is the non symmetric gram matrix of one of the main results of this paper are theorem 2 1 theorem 2 17 asserting that every such a bigraph with n r 2 vertices can be obtained from a corank zero i e positive connected loop free bigraph with n 1 vertices by an r point extension procedure u 1 u r u 1 u r along the r 1 roots u 1 u r z n of the positive definite gram form q z n z v v g v t r it is also shown that the extension procedure yields a combinatorial algorithm allowing us to obtain inductively all connected loop free non negative corank r 1 bigraphs with n r 2 vertices from the corank zero bigraphs with n 1 vertices

analyzing weighted ell_1 minimization for sparse recovery with nonuniform sparse models in this paper we introduce a nonuniform sparsity model and analyze the performance of an optimized weighted l1 minimization over that sparsity model in particular we focus on a model where the entries of the unknown vector fall into two sets with entries of each set having a specific probability of being nonzero we propose a weighted l1 minimization recovery algorithm and analyze its performance using a grassmann angle approach we compute explicitly the relationship between the system parameters the weights the number of measurements the size of the two sets the probabilities of being nonzero so that when i i d random gaussian measurement matrices are used the weighted l1 minimization recovers a randomly selected signal drawn from the considered sparsity model with overwhelming probability as the problem dimension increases this allows us to compute the optimal weights we demonstrate through rigorous analysis and simulations that for the case when the support of the signal can be divided into two different subclasses with unequal sparsity fractions the weighted l1 minimization outperforms the regular l1 minimization substantially we also generalize our results to signal vectors with an arbitrary number of subclasses for sparsity

symplectic spaces and pairs of symmetric and nonsingular skew symmetric matrices under congruence abstract let f be a field of characteristic not 2 and let a b be a pair of n n matrices over f in which a is symmetric and b is skew symmetric a canonical form of a b with respect to congruence transformations s t a s s t b s was given by sergeichuk 1988 25 up to classification of symmetric and hermitian forms over finite extensions of f we obtain a simpler canonical form of a b if b is nonsingular such a pair a b defines a quadratic form on a symplectic space that is on a vector space with scalar product given by a nonsingular skew symmetric form as an application we obtain known canonical matrices of quadratic forms and hamiltonian operators on real and complex symplectic spaces

congruences of a square matrix and its transpose abstract it is known that any square matrix a over any field is congruent to its transpose at stas for some nonsingular s moreover s can be chosen such that s2 i that is s can be chosen to be involutory we show that a and at are congruent over any field f of characteristic not two with involution a a the involution can be the identity a t s t as for some nonsingular s moreover s can be chosen such that s s i that is s can be chosen to be coninvolutory the short and simple proof is based on sergeichuk s canonical form for congruence math ussr izvestiya 31 3 1988 481 it follows that any matrix a over f can be represented as a eb in which e is coninvolutory and b is symmetric

the moduli space of 4 dimensional nilpotent complex associative algebras abstract in this paper we study 4 dimensional nilpotent complex associative algebras this is a continuation of the study of the moduli space of 4 dimensional algebras the nonnilpotent algebras were analyzed in an earlier paper even though there are only 15 families of nilpotent 4 dimensional algebras the complexity of their behavior warranted a separate study which we give here

versal deformations of elements of classical jordan algebras every involution defining by x x the elements of one of the classical lie algebras o n c and sp 2n c over c and o p q sp 2n r u p q o 2n and sp 2p 2q over r also defines symmetric matrices y y spanning the classical jordan algebras the paper contains an explicit description of all perturbations of symmetric matrices considered up to equivalence under the action of the corresponding classical lie group such exhaustive perturbations are called versal deformations

a chaotic iterative algorithm based on linearized bregman iteration for image deblurring in this paper we propose a new chaotic iterative algorithm for image deblurring the algorithm is based on generalized inverse iteration and linearized bregman iteration for the basis pursuit problem min u r n u 1 au f numerical experiments show that the chaotic algorithm for image restoration is effective and efficient

invertibly convergent infinite products of matrices abstract the standard definition of convergence of an infinite product of scalars is extended to the infinite product p n 1 b n of k k matrices that is p is convergent according to the definition given here if and only if there is an integer n such that b n is invertible for n n and p lim n m n n b m is invertible a family of sufficient conditions for this kind of convergence is given along with examples showing that they have nontrivial applications

exact reconstruction conditions for regularized modified basis pursuit in this work we obtain sufficient conditions for exact recovery of regularized modified basis pursuit reg mod bp and discuss when the obtained conditions are weaker than those for modified compressive sensing or for basis pursuit bp the discussion is also supported by simulation comparisons reg mod bp provides a solution to the sparse recovery problem when both an erroneous estimate of the signal s support denoted by and an erroneous estimate of the signal values on are available

classification of sesquilinear forms with the first argument on a subspace or a factor space abstract let v be a vector space over a field or skew field f and let u be its subspace we study the canonical form problem for bilinear or sesquilinear forms u v f v u v f and linear mappings u v v u v u v v v u we solve it over f c and reduce it over all f to the canonical form problem for ordinary linear mappings w w and bilinear or sesquilinear forms w w f moreover we give an algorithm that realizes this reduction the algorithm uses only unitary transformations if f c which improves its numerical stability for linear mapping this algorithm can be derived from the algorithm by nazarova et al l a nazarova a v roiter v v sergeichuk v m bondarenko application of modules over a dyad for the classification of finite p groups possessing an abelian subgroup of index p and of pairs of mutually annihilating operators zap nauchn sem leningrad otdel mat inst steklov lomi 28 1972 69 92 translation in j soviet math 3 5 1975 636 654

a regularization algorithm for matrices of bilinear and sesquilinear forms abstract over a field or skew field f with an involution a a possibly the identity involution each singular square matrix a is congruent to a direct sum s as b j n 1 j n p 1 n 1 n p inwhich s is nonsingular and s s t b is nonsingular and is determined by a up to congruence and the ni by ni singular jordan blocks j n i and their multiplicities are uniquely determined by a we give a regularization algorithm that needs only elementary row operations to construct such a decomposition if f c respectively f r we exhibit a regularization algorithm that uses only unitary respectively real orthogonal transformations and a reduced form that can be achieved via a unitary congruence or congruence respectively a real orthogonal congruence the selfadjoint matrix pencil a a is decomposed by our regularization algorithm into the direct sum s a a s b b j n 1 j n 1 j n p j n p with selfadjoint summands

complex orthogonal and symplectic matrices depending on parameters versal deformations of elements of complex orthogonal and symplectic lie algebras are studied for a general element m of o n c or sp 2n c a normal form ma is found which unlike the jordan normal form mj depends holomorphically on m and on the similarity transformation ma gmg 1 from the corresponding group orthogonal and symplectic cases are treated simultaneously in order to underline their close relation bundles of matrices of low codimension are listed and bifurcation diagrams of two parameter families of orthogonal matrices are shown finally versal deformations of all elements of o 6 c are explicitly shown

change of the congruence canonical form of 2 by 2 and 3 by 3 matrices under perturbations and bundles of matrices under congruence we construct the hasse diagrams g2 and g3 for the closure ordering on the sets of congruence classes of 2 2 and 3 3 complex matrices in other words we construct two directed graphs whose vert

miniversal deformations of matrices under congruence and reducing transformations arnold 1971 1 constructed a miniversal deformation of a square complex matrix under similarity that is a simple normal form to which not only a given square matrix a but all matrices b close to it can be reduced by similarity transformations that smoothly depend on the entries of b we give miniversal deformations of matrices of sesquilinear forms that is of square complex matrices under congruence and construct an analytic reducing transformation to a miniversal deformation analogous results for matrices under congruence were obtained by dmytryshyn futorny and sergeichuk 2012 11

algebraic entropy and the space of initial values for discrete dynamical systems a method to calculate the algebraic entropy of a mapping which can be lifted to an isomorphism of a suitable rational surface the space of initial values is presented it is shown that the degree of the nth iterate of such a mapping is given by its action on the picard group of the space of initial values it is also shown by construction that the degree of the nth iterate of every painleve equation in sakai s list is o n2 and therefore its algebraic entropy is zero

congruence and conjunctivity of matrices to their adjoints abstract every square matrix over a field f is involutorily congruent over f to its transpose and hence each such matrix is the product of a symmetric matrix and an involutory matrix over f in the usual complex case every matrix which is conjunctive with its adjoint conjugate transpose is involutorily conjunctive with its adjoint and hence is the product of a hermitian matrix and an involutory matrix furthermore every such matrix is conjunctive with a real matrix these three conditions on a matrix 1 being conjunctive with its adjoint 2 being involutorily conjunctive with its adjoint and 3 being conjunctive with a real matrix are studied in the more general context of a field f with involution and it is shown in general that 3 implies 2 that 2 implies 3 if char f 2 a 2 2 counterexample exists for each f with char f 2 and that 1 does not in general imply 2 a 2 2 counterexample in the complexification of the rational field is presented the problem of deciding which matrices satisfy 2 is equivalent even in this general context to the problem of deciding which pairs of self adjoint hermitian matrices are involutorily conjunctive for the general 2 2 case the three conditions are characterized in terms of norms

coxeter spectral classification of almost tp critical one peak posets using symbolic and numeric computations following an idea of spectral graph theory we give several characterizations of finite one peak posets i with almost p critical quadratic tits form q i zi z by applying linear algebra methods combinatorial algorithms an old idea of s a ovsienko and recent results by v m bondarenko and m v stepochkina our study is mainly motivated by an important application of the quadratic tits form of a poset in constructing linear algebra invariants that measure a geometric complexity of nazarova roiter matrix problems over a field k in particular our study is inspired by a well known result of ju a drozd explained in introduction one of our main results asserts that given a finite poset i 1 n t such that is its unique maximal element and q i is almost p critical we have i the form q i is non negative and the subgroup kerq i v zn 1 q i v 0 of zi zn 1 is generated by a vector h h1 hn h with h1 0 hn 0 ii 5 i 9 and the coxeter polynomial coxi t z t coincides with the coxeter polynomial f i t of a uniquely determined euclidean diagram i d 4 e 6 e 7 e 8 iii the z bilinear tits form b i zn 1 zn 1 z of i is z equivalent to the gram z bilinear form b i zn 1 zn 1 z of the euclidean diagram i iv there is a z invertible matrix c such that c itr ctr c i c and c2 e where c i mn 1 z is the tits matrix of i and v i is one of the posets l1 l132 listed in the paper in particular we get in an alternative way the list of p critical posets t in the form t i presented by v m bondarenko and m v stepochkina 6 and we determine their coxeter euclidean types there are 115 one peak posets i l1 l115 with q i p critical and 17 exceptional posets i l116 l132 with q i almost p critical but not p critical our lists are obtained by using computer algebra tools mainly symbolic computation in maple and numeric computation in c

boundary conditions for integrable discrete chains a method of searching for boundary conditions consistent with the integrability property is suggested for discrete chains generating functions of conserved quantities of the corresponding finite dimensional reductions are given in terms of the l a pairs

compressed sensing with prior information strategies geometry and bounds we address the problem of compressed sensing cs with prior information reconstruct a target cs signal with the aid of a similar signal that is known beforehand our prior information we integrate the additional knowledge of the similar signal into cs via ell 1 ell 1 and ell 1 ell 2 minimization we then establish bounds on the number of measurements required by these problems to successfully reconstruct the original signal our bounds and geometrical interpretations reveal that if the prior information has good enough quality ell 1 ell 1 minimization improves the performance of cs dramatically in contrast ell 1 ell 2 minimization has a performance very similar to classical cs and brings no significant benefits in addition we use the insight provided by our bounds to design practical schemes to improve prior information all our findings are illustrated with experimental results

exploiting prior information in block sparse signals we study the problem of recovering a block sparse signal from under sampled observations the non zero values of such signals appear in few blocks and their recovery is often accomplished using an ell 1 2 optimization problem in applications such as dna micro arrays some extra information about the distribution of non zero blocks is available i e the number of non zero blocks in certain subsets of the blocks is known a typical way to consider the extra information in recovery procedures is to solve a weighted ell 1 2 problem in this paper we consider a block sparse model which is accompanied with a partitioning of the blocks besides the overall block sparsity level of the signal we assume to know the block sparsity of each subset in the partition our goal in this work is to minimize the number of required linear measurements for perfect recovery of the signal by tuning the weights of a weighted ell 1 2 problem for this goal we apply tools from conic integral geometry and derive closed form expressions for the optimal weights we show through precise analysis and simulations that the weighted ell 1 2 problem with optimal weights significantly outperforms the regular ell 1 2 problem we further show that the optimal weights are robust against the inaccuracies of prior information

classification of linear mappings between indefinite inner product spaces abstract let a u v be a linear mapping between vector spaces u and v over a field or skew field f with symmetric or skew symmetric or hermitian forms b u u f and c v v f we classify the triples a b c if f is r or c or the skew field of quaternions h we also classify the triples a b c up to classification of symmetric forms and hermitian forms if the characteristic of f is not 2

skew symmetric matrix polynomials and their smith forms we characterize the smith form of skew symmetric matrix polynomials over an arbitrary field f showing that all elementary divisors occur with even multiplicity restricting the class of equivalence transformations to unimodular congruences a smith like skew symmetric canonical form for skew symmetric matrix polynomials is also obtained these results are used to analyze the eigenvalue and elementary divisor structure of matrices expressible as products of two skew symmetric matrices as well as the existence of structured linearizations for skew symmetric matrix polynomials by contrast with other classes of structured matrix polynomials e g alternating or palindromic polynomials every regular skew symmetric matrix polynomial is shown to have a structured strong linearization while there are singular skew symmetric polynomials of even degree for which a structured linearization is impossible for each odd degree we develop a skew symmetric companion form that uniformly provides a structured linearization for every regular and singular skew symmetric polynomial of that degree finally the results are applied to the construction of minimal symmetric factorizations of skew symmetric rational matrices

a gram classification of non negative corank two loop free edge bipartite graphs we continue the coxeter spectral study of finite connected loop free edge bipartite graphs with m 2 3 vertices a class of signed graphs started in simson 2013 49 by means of the non symmetric gram matrix g mm 2 z of its symmetric gram matrix g 12 g g tr mm 2 12z the gram quadratic form q zm 2 z and the coxeter spectrum specc c i e the complex spectrum of the coxeter matrix cox g g tr gl m 2 z in the present paper we study non negative edge bipartite graphs of corank two in the sense that the symmetric gram matrix g mm 2 z of is positive semi definite of rank m 1 one of our aims is to get a complete classification of all connected corank two loop free edge bipartite graphs with m 2 3 vertices up to the weak gram z congruence z where z means that g btr g b for some b mm 2 z such that det b 1 by one vertex extensions of the simply laced euclidean diagrams a m m 1 d m m 4 e 6 e 7 e 8 we construct a family of connected loop free corank two diagrams a m 2 d m 2 e 6 2 e 7 2 e 8 2 called simply extended euclidean diagrams such that they classify all connected corank two loop free edge bipartite graphs with m 2 3 vertices up to the weak gram z congruence z a structure of connected corank two loop free edge bipartite graphs is described it is shown that every such contains a connected positive edge bipartite subgraph that is z congruent with a simply laced dynkin diagram dyn called the dynkin type of such that is a two point extension u w of along two roots u w of the positive definite gram form q zm z this yields a combinatorial algorithm u w u w allowing us to construct all connected corank two loop free edge bipartite graphs with m 2 3 vertices and d dyn from the triples u w where is positive of the dynkin type d and u w are roots of the positive definite gram form q zm z

singularity complexity and quasi integrability of rational mappings we investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models by exploiting their nature of birational transformations of projective spaces we give an algorithmic analysis of the structure of invariants of such mappings we discuss some characteristic conditions for their quasi integrability and in particular its links with their singularities in the 2 plane finally we describe some of their propertiesqua dynamical systems making contact with arnol d s notion of complexity and exemplify remarkable behaviours

similarity of some dual operators in characteristic two abstract let k be a field of characteristic two with involution x x let v be a finite dimensional hermitian space over k and v v a linear operator whose dual is 1 we prove that 1 1 where is an isometry and 2 1 if e is a given subspace of v then can also be chosen to stabilize e but the equality 1 1 is only true modulo a combination of certain bracket operators as a corollary we solve the following congruence problem given a square matrix a over k there is a non singular matrix s satisfying a t s t as and s s 1

stratification of moduli spaces of lie algebras similar matrices and bilinear forms abstract in this paper the authors apply a stratification of moduli spaces of complex lie algebras to analyzing the moduli spaces of n n matrices under scalar similarity and bilinear forms under the cogredient action for similar matrices we give a complete description of a stratification of the space by some very simple projective orbifolds of the form p n g where g is a subgroup of the symmetric group n 1 acting on p n by permuting the projective coordinates for bilinear forms we give a similar stratification up to dimension 4

topological classification of sesquilinear forms reduction to the nonsingular case abstract two sesquilinear forms c m c m c and c n c n c are called topologically equivalent if there exists a homeomorphism c m c n i e a continuous bijection whose inverse is also a continuous bijection such that x y x y for all x y c m r a horn and v v sergeichuk in 2006 constructed a regularizing decomposition of a square complex matrix a that is a direct sum s a s r j n 1 j n p in which s and r are nonsingular and each j n i is the n i by n i singular jordan block in this paper we prove that and are topologically equivalent if and only if the regularizing decompositions of their matrices coincide up to permutation of the singular summands j n i and replacement of r c r r by a nonsingular matrix r c r r such that r and r are the matrices of topologically equivalent forms c r c r c analogous results for bilinear forms over c and over r are also obtained

construction of miniversal deformations of lie algebras abstract we consider deformations of finite or infinite dimensional lie algebras over a field of characteristic 0 there is substantial confusion in the literature if one tries to describe all the non equivalent deformations of a given lie algebra it is known that there is in general no universal deformation of a lie algebra l with a commutative algebra base a with the property that for any other deformation of l with base b there exists a unique homomorphism f a b that induces an equivalent deformation thus one is led to seek a miniversal deformation for a miniversal deformation such a homomorphism exists but is unique only at the first level if we consider deformations with base spec a where a is a local algebra then under some minor restrictions there exists a miniversal element in this paper we give a construction of a miniversal deformation

opuc cmv matrices and perturbations of measures supported on the unit circle abstract let us consider a hermitian linear functional defined on the linear space of laurent polynomials with complex coefficients in the literature canonical spectral transformations of this functional are studied the aim of this research is focused on perturbations of hermitian linear functionals associated with a positive borel measure supported on the unit circle some algebraic properties of the perturbed measure are pointed out in a constructive way we discuss the corresponding sequences of orthogonal polynomials as well as the connection between the associated verblunsky coefficients then the structure of the matrices of the perturbed linear functionals which is the main tool for the comparison of their corresponding cmv matrices is deeply analyzed from the comparison between different cmv matrices other families of perturbed verblunsky coefficients will be considered we introduce a new matrix named fundamental matrix that is a tridiagonal symmetric unitary matrix containing basic information about the family of orthogonal polynomials however we show that it is connected to another family of orthogonal polynomials through the takagi decomposition

transformation to versal deformations of matrices abstract in the paper versal deformations of matrices are considered the versal deformation is a matrix family inducing an arbitrary multi parameter deformation of a given matrix by an appropriate smooth change of parameters and basis given a deformation of a matrix it is suggested to find transformation functions the change of parameters and the change of basis dependent on parameters in the form of taylor series the general method of construction of recurrent procedures for calculation of coefficients in the taylor expansions is developed and used for the cases of real and complex matrices elements of classical lie and jordan algebras and infinitesimally reversible matrices several examples are given and studied in detail applications of the suggested approach to problems of stability singularity and perturbation theories are discussed

structure and a coxeter dynkin type classification of corank two non negative posets inspired by famous results of drozd 1974 13 nazarova ro ter 1975 42 and bondarenko stepochkina 2005 8 on the importance of finite posets i with the non negative tits quadratic form q i zi z in the representation theory we study the finite non negative posets i of corank two that is such posets i that the symmetric gram matrix gi 12 ci citr mi q of i is positive semi definite of corank two where ci mi z is the incidence matrix of i a structure of such posets is described in the paper it is shown that every such a poset i can be obtained in a natural way from a corank zero poset j and a pair of roots of j following main ideas of the coxeter spectral analysis of posets we associate with any connected corank two poset i the incidence quadratic form qi zi z the coxeter matrix coxi ci ci tr the coxeter polynomial coxi t det t e coxi z t its complex coxeter spectrum specci the coxeter number ci n the reduced coxeter number c i n the incidence defect homomorphism i zi kerqi zi and a simply laced dynkin diagram dyni in particular we study the question when the coxeter spectrum specci of i together with dyni determines the incidence matrix ci of i hence the poset i uniquely up to a z congruence by applying linear algebra techniques and a combinatorial computer aided analysis of posets we prove that the question has an affirmative answer for connected corank two posets i with at most 15 elements a complete list of such posets is presented and their coxeter type invariants are computed it follows that i 6 there is only one such a poset i of cardinality 6 and 8 such posets of cardinality 7 up to the poset duality i iop the existence of a i mesh translation quiver ri kerqi i on the set ri kerqi is discussed where ri v zi qi v 1 is the set of roots of qi

matrices that are self congruent only via matrices of determinant one abstract ocovic and szechtman d ocovic f szechtman characterization of bilinear spaces with unimodular isometry group proc amer math soc 133 2005 2853 2863 considered a vector space v endowed with a bilinear form they proved that all isometries of v over a field f of characteristic not 2 have determinant 1 if and only if v has no orthogonal summands of odd dimension the case of characteristic 2 was also considered their proof is based on riehm s classification of bilinear forms coakley et al e s coakley f m dopico c r johnson matrices with orthogonal groups admitting only determinant one linear algebra appl 428 2008 796 813 gave another proof of this criterion over r and c using thompson s canonical pairs of symmetric and skew symmetric matrices for congruence let m be the matrix of the bilinear form on v we give another proof of this criterion over f using our canonical matrices for congruence and obtain necessary and sufficient conditions involving canonical forms of m for congruence of m t m for equivalence and of m t m if m is nonsingular for similarity

versal deformations of elements of real classical lie algebras versal deformations of normal forms of elements of real classical lie algebras o p q sp 2n r o 2n sp 2p 2q and u p q are computed examples involving the lie algebras o p q p q or 6 are considered

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

canonical matrices of bilinear and sesquilinear forms abstract canonical matrices are given for i bilinear forms over an algebraically closed or real closed field ii sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution and iii sesquilinear forms over a field f of characteristic different from 2 with involution possibly the identity up to classification of hermitian forms over finite extensions of f the canonical matrices are based on any given set of canonical matrices for similarity over f a method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings is used to construct the canonical matrices this method has its origins in representation theory and was devised in v v sergeichuk classification problems for systems of forms and linear mappings math ussr izv 31 1988 481 501

error analysis of reweighted l 1 greedy algorithm for noisy reconstruction sparse solutions for an underdetermined system of linear equations x u can be found more accurately by l 1 minimization type algorithms such as the reweighted l 1 minimization and l 1 greedy algorithms than with analytical methods in particular in the presence of noisy data recently a generalized l 1 greedy algorithm was introduced and applied to signal and image recovery numerical experiments have demonstrated the convergence of the new algorithm and the superiority of the algorithm over the reweighted l 1 minimization and l 1 greedy algorithms although the convergence has not yet been proven theoretically in this paper we provide an error bound for the reweighted l 1 greedy algorithm a type of the generalized l 1 greedy algorithm in the noisy case and show its improvement over the reweighted l 1 minimization

one peak posets with positive quadratic tits form their mesh translation quivers of roots and programming in maple and python by applying linear algebra and computer algebra tools we study finite posets with positive quadratic tits form our study is motivated by applications of matrix representations of posets in representation theory where a matrix representation of a partially ordered set t p1 pn with a partial order means a block matrix m m1 m2 mn over a field k of size d d1 dn determined up to all elementary row transformations elementary column transformations within each of the substrips m1 m2 mn and additions of linear combinations of columns of mi to columns of mj if pi pj drozd 10 proves that t has only a finite number of direct sum indecomposable representations if and only if its quadratic tits form q x1 xn x x12 xn2 x 2 pi pjxixj x x1 xn is weakly positive i e q a1 an a 0 for all non zero vectors a1 an a with integral non negative coefficients in this case there exists an indecomposable representation m of size d d1 dn if and only if d1 dn d is a root of q i e q d1 dn d 1 bondarenko and stepochkina 8 give a list of posets t with positive tits form consisting of four infinite series and 108 posets up to duality in the paper we construct this list in an alternative way by applying computational algorithms implemented in maple and python moreover given any poset t of the list we show that a the coxeter polynomial coxi t of the poset i t obtained from t by adding a unique maximal element called a peak is a coxeter polynomial f i t of a simply laced dynkin diagram i an dn e6 e7 e8 and b the set of orbits of the set rq of integral roots of q admits a mesh translation quiver structure rq of a cylinder shape where is the coxeter transformation of i t in the sense of drozd 10

miniversal deformations of matrices of bilinear forms arnold v i arnold on matrices depending on parameters russian math surveys 26 2 1971 29 43 constructed miniversal deformations of square complex matrices under similarity that is a simp

canonical forms for complex matrix congruence and congruence abstract canonical forms for congruence and congruence of square complex matrices were given by horn and sergeichuk in linear algebra appl 389 2004 347 353 based on sergeichuk s paper math ussr izvestiya 31 3 1988 481 501 which employed the theory of representations of quivers with involution we use standard methods of matrix analysis to prove directly that these forms are canonical our proof provides explicit algorithms to compute all the blocks and parameters in the canonical forms we use these forms to derive canonical pairs for simultaneous congruence of pairs of complex symmetric and skew symmetric matrices as well as canonical forms for simultaneous congruence of pairs of complex hermitian matrices

congruence of multilinear forms abstract let f u u k g v v k be two n linear forms with n 2 on finite dimensional vector spaces u and v over a field k we say that f and g are symmetrically equivalent if there exist linear bijections 1 n u v such that f u 1 u n g i 1 u 1 i n u n for all u 1 u n u and each reordering i 1 i n of 1 n the forms are said to be congruent if 1 n let f and g be symmetrically equivalent we prove that i if k c then f and g are congruent ii if k r f f 1 f s 0 g g 1 g r 0 and all summands f i and g j are nonzero and direct sum indecomposable then s r and after a suitable reindexing f i is congruent to g i

searching for integrable lattice maps using factorization we analyse the factorization process for lattice maps searching for integrable cases the maps were assumed to be at most quadratic in the dependent variables and we required minimal factorization one linear factor after two steps of iteration the results were then classified using the algebraic entropy some new models with polynomial growth strongly associated with integrability were found one of them is a nonsymmetric generalization of the homogeneous quadratic maps associated with kdv modified and schwarzian for this new model we have also verified the consistency around a cube

tridiagonal canonical matrices of bilinear or sesquilinear forms and of pairs of symmetric skew symmetric or hermitian forms abstract tridiagonal canonical forms of square matrices under congruence or congruence pairs of symmetric or skew symmetric matrices under congruence and pairs of hermitian matrices under congruence are given over an algebraically closed field of characteristic different from 2

a square matrix is congruent to its transpose for any matrix x let xdenote its transpose we show that if a is an n by n matrix over afi eldk t hena and aare congruent over k i e pap afor some p gln k