complexity of matrix problems in representation theory the classification problem is called wild if it contains the problem of classifying pairs of matrices up to simultaneous similarity we show in an explicit form that the last problem contains all classification matrix problems given by quivers or posets then we prove that this problem does not contain but is contained in the problem of classifying three valent tensors hence every wild classification problem given by a quiver or poset has the same complexity moreover a solution of one of them implies a solution of each of the remaining problems the problem of classifying three valent tensors is more complicated

arveson s criterion for unitary similarity abstract this paper is an exposition of w b arveson s complete invariant for the unitary similarity of complex irreducible matrices

a multi level discontinuous galerkin method for solving the stationary navier stokes equations abstract in this paper we consider the multi level discontinuous finite element method for solving the stationary incompressible navier stokes equations on the coarsest mesh the discrete nonlinear navier stokes equations are solved by using piecewise polynomial functions which are totally discontinuous across inter element boundaries and are pointwise divergence free on each element for the velocity and are continuous functions for the pressure subsequent approximations are generated on a succession of refined grids by solving the newton linearized navier stokes equations using piecewise polynomial functions which are similar to that on the coarsest mesh finally the well posedness and the optimal error estimate for the multi level discontinuous galerkin method are provided the error analysis shows that when the mesh scales k j 1 o k j 2 h j 1 o h j 2 with j 0 1 j 1 are chosen the multi level finite element method can save a large amount of computational time compared with the one level finite element method

the intersection of the similarity and conjunctivity equivalence classes abstract let a be an n n complex matrix let sim a denote the similarity equivalence class of a let conj a denote the conjunctivity equivalence class of a and let u a denote the unitary similarity equivalence class of a define cs a sim a conj a we seek to classify the matrices that have cs a u a and show by an example that this is not true in general but we show that it is true when a is hermitian or is a scalar multiple of a hermitian for n 3 we reduce the general n n case to the case when a is non singular and not a multiple of a hermitian matrix we completely classify the 2 2 case and find that cs a u a when a is non singular or normal or both

two level method based on finite element and crank nicolson extrapolation for the time dependent navier stokes equations a fully discrete two level finite element method the two level method is presented for solving the two dimensional time dependent navier stokes problem the method requires a crank nicolson extrapolation solution u h tau 0 p h tau 0 on a spatial time coarse grid j h tau 0 and a backward euler solution u h tau p h tau on a space time fine grid j h tau the error estimates of optimal order of the discrete solution for the two level method are derived compared with the standard crank nicolson extrapolation method the one level method based on a space time fine grid j h tau the two level method is of the error estimates of the same order as the one level method in the h1 norm for velocity and the l2 norm for pressure however the two level method involves much less work than the one level method

simultaneous reduction of sets of matrices under similarity abstract let s be a set of n n matrices over a field f and a the algebra generated by s over f the problem of deciding whether the elements of s can be simultaneously reduced to block triangular form with the diagonal blocks of some specified size is considered and an account is given of various methods used to attack the problem most of the techniques use representation theory to obtain information on a the problems of simultaneous triangularization existence of common eigenvectors etc are also considered the aim of the paper is to survey the methods used to attack these problems and to give some typical results the paper does not contain many new results

a multi level stabilized finite element method for the stationary navier stokes equations abstract this paper proposes and analyzes a multi level stabilized finite element method for the two dimensional stationary navier stokes equations approximated by the lowest equal order finite element pairs the method combines the new stabilized finite element method with the multi level discretization under the assumption of the uniqueness condition the multi level stabilized finite element method consists of solving the nonlinear problem on the coarsest mesh and then performs one newton correction step on each subsequent mesh thus only solving one large linear systems the error analysis shows that the multi level stabilized finite element method provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the stationary navier stokes equations on a fine mesh for an appropriate choice of mesh widths h j h j 1 2 j 1 j moreover the numerical illustrations agree completely with the theoretical expectations therefore the multi level stabilized finite element method is more efficient than the standard one level stabilized finite element method

a conjectured property of hermitian pencils abstract if a b is any pair of hermitian matrices the power of dividing det i xa yb will be given by the number of basic singular summands in the pair contrary to conjecture this power can be greater than one even when the pair is unitarily irreducible

least squares approximation by elements from matrix orbits achieved by gradient flows on compact lie groups let s a denote the orbit of a complex or real matrix a under a certain equivalence relation such as unitary similarity unitary equivalence unitary congruences etc efficient gradient flow algorithms are constructed to determine the best approximation of a given matrix a0 by the sum of matrices in s a1 s an in the sense of finding the euclidean least squares distance

aspects of nonnormality for iterative methods abstract recently new optimal krylov subspace methods have been discovered for normal matrices in light of this novel ways to quantify nonnormality are considered in connection with various families of matrices we use as a criterion how for a given matrix these iterative methods introduced can be employed via e g inexpensive matrix factorizations the unitary orbit of the set of binormal matrices provides a natural extension of normal matrices its elements yield polynomially normal matrices of moderate degree in this context several matrix nearness problems arise

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

numerical solution of the stationary navier stokes equations using a multilevel finite element method the numerical implementation of a multilevel finite element method for the steady state navier stokes equations is considered the multilevel method proposed here for the navier stokes equations is a multiscale method in which the full nonlinear navier stokes equations are only solved on a single coarse grid subsequent approximations are generated on a succession of refined grids by solving a linearized navier stokes problem two numerical examples are considered the first is an example for which an exact solution is known and the second is the driven cavity problem we demonstrate numerically that for an appropriate choice of grids a two or three level finite element method is significantly more efficient than the standard one level finite element method

operators on positive semidefinite inner product spaces abstract let u be a semiunitary space i e a complex vector space with scalar product given by a positive semidefinite hermitian form if a linear operator a u u is bounded i e a u c u for some c r and all u u then the subspace u 0 u u u u 0 is invariant and so a defines the linear operators a 0 u 0 u 0 and a 1 u u 0 u u 0 let a be an indecomposable bounded operator on u such that 0 u 0 u let be an eigenvalue of a 0 we prove that the algebraic multiplicity of in a 1 is not less than the geometric multiplicity of in a 0 and the geometric multiplicity of in a 1 is not less than the number of jordan blocks j t of each fixed size t t in the jordan canonical form of a 0 we give canonical forms of selfadjoint and isometric operators on u and of hermitian forms on u for an arbitrary system of semiunitary spaces and linear mappings on between them we give an algorithm that reduces their matrices to canonical form its special cases are belitskii s and littlewood s algorithms for systems of linear operators on vector spaces and unitary spaces respectively

piecewise solenoidal vector fields and the stokes problem nonconforming finite element approximations to solutions of the stokes equations are constructed optimal rates of convergence are proved for the velocity and pressure approximations for the pressure approximation c 0 piecewise polynomial functions are used the class of vector fields used to approximate the velocity field have piecewise polynomial components discontinuous across interelement boundaries on each triangle these vector fields satisfy the incompressibility condition pointwise it is shown that these piecewise solenoidal vector fields possess optimal approximation properties to smooth solenoidal vector fields on domains with curved boundaries

belitskii s canonical forms of linear dynamical systems abstract in the note all indecomposable canonical forms of linear systems with dimension less than or equals to 4 are determined based on belitskii s algorithm

belitskiĭ s canonical form for 5 5 upper triangular matrices under upper triangular similarity abstract in dimensions up to five belitski s algorithm is used to describe the set of canonical forms for upper triangular matrices under upper triangular similarity

pairs of matrices with quadratic minimal polynomials abstract let a b be n n matrices over a field f and suppose a b have quadratic minimal polynomials then the algebra generated by a b has dimension at most 2n 1 if n is odd and 2n if n is even these bounds are exact

specht s criterion for systems of linear mappings abstract w specht 1940 proved that two n n complex matrices a and b are unitarily similar if and only if trace w a a trace w b b for every word w x y in two noncommuting variables we extend his criterion and its generalizations by n a wiegmann 1961 and n jing 2015 to an arbitrary system a consisting of complex or real inner product spaces and linear mappings among them we represent such a system by the directed graph q a whose vertices are inner product spaces and arrows are linear mappings denote by q a the directed graph obtained by enlarging to q a the adjoint linear mappings we prove that a system a is transformed by isometries of its spaces to a system b if and only if the traces of all closed directed walks in q a and q b coincide

a counterexample to kippenhahn s conjecture on hermitian pencils abstract a counterexample is constructed to a conjecture of kippenhahn math nachr 6 193 288 1951 52 a pair of hermitian 8 8 matrices h k is found such that 1 h k generate m 8 c and 2 the minimal polynomial of the pencil xh yk has degree 4 recent work of h shapiro e g linear algebra appl 43 201 221 1982 has established the conjecture for n n matrices n 5

solution estimates for linear differential equations with delay abstract in this paper we give explicit exponential estimates x t m e t t 0 where t t0 m 0 for solutions of a linear scalar delay differential equation x t k 1 m b k t x h k t f t t t 0 x t t t t 0 we consider two different cases when 0 corresponding to exponential stability and the case of in the first case together with the exponential estimate we also obtain an exponential stability test in the second case we get estimation for solution growth here both the coefficients and the delays are measurable not necessarily continuous

a structure theorem for some matrix algebras abstract let f be a field and m n f the algebra of n n matrices over f it is in general a very difficult and tedious problem to determine the structure of the subalgebra x of m n f generated by a given subset s of m n f we show that if x contains a matrix a with n distinct eigenvalues in f then x can be determined up to similarity very quickly by a graph theoretic method as a consequence we show that any such x can be generated by a pair of elements one of which can be taken to be a and the other a 0 1 matrix as an application we obtain a specht type similarity theorem

hyponormal and strongly hyponormal matrices in inner product spaces complex matrices that are structured with respect to a possibly degenerate indefinite inner product are studied based on earlier works on normal matrices the notions of hyponormal and strongly hyponormal matrices are introduced a full characterization of such matrices is given and it is shown how those matrices are related to different concepts of normal matrices in degenerate inner product spaces finally the existence of invariant semidefinite subspaces for strongly hyponormal matrices is discussed

the order of neutrality for linear operators on inner product spaces abstract let a be a symmetric linear operator defined on all of a possibly degenerate indefinite inner product space h let n be the set of all subspaces of h which are a invariant neutral in the sense of the indefinite scalar product and finite dimensional it is shown that members of n which are maximal with respect to inclusion all have the same dimension this is called the order of neutrality of a and admits immediate application to self adjoint operators on a pontrjagin space

on triangular similarity of nilpotent triangular matrices abstract let b n resp u n n n be the set of n n nonsingular resp unit nilpotent upper triangular matrices we use a novel approach to explore the b n similarity orbits in n n the belitski s canonical form of a n n under b n similarity is in q u n where q is the subpermutation such that a b n q b n using graph representations and u n similarity actions stabilizing q u n we obtain new properties of the belitski s canonical forms and present an efficient algorithm to find the belitski s canonical forms in n n as consequences we construct new belitski s canonical forms in all n n s list all belitski s canonical forms for n 7 8 and show examples of 3 nilpotent belitski s canonical forms in n n with arbitrary numbers of parameters up to o n 2

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

a criterion for unitary similarity of upper triangular matrices in general position abstract each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order let a a ij and b b ij be upper triangular n n matrices that are not similar to direct sums of square matrices of smaller sizes or are in general position and have the same main diagonal we prove that a and b are unitarily similar if and only if h a k h b k for all h c x and k 1 n where a k a ij i j 1 k and b k b ij i j 1 k are the leading principal k k submatrices of a and b and is the frobenius norm

low rank perturbations of strongly definitizable transformations and matrix polynomials abstract let a 0 be a transformation on a finite dimensional hilbert space which is self adjoint in an indefinite scalar product generated by g 0 g 0 and invertible the spectrum of a 0 is real when a 0 is g 0 strongly definitizable the problems considered here concern the number of real eigenvalues of a g self adjoint transformation a where a and g are low rank perturbations of a 0 and g 0 a notion called the order of neutrality of a with respect to g is introduced which is relevant to this problem area using linearization as well as direct methods results are obtained concerning self adjoint matrix polynomials which are low rank perturbationsof suitably defined definitizable matrix polynomials applications are made to quadratic matrix polynomials arising in the study of damped systems and gyroscopic systems

a multilevel mesh independence principle for the navier stokes equations multilevel finite element discretization methods for the navier stokes equations are considered in contrast to usual multilevel methods a superlinear scaling of the consecutive meshwidths h j 1 mathcal o h j alpha j is used on the coarsest mesh the discretized system is solved which is small and nonlinear on each subsequent mesh only one or two newton correction steps are performed so that only one or two larger linear systems are solved the scalings of the meshwidths that lead to optimal accuracy of the approximate solution in both the h 1 and l 2 norm are investigated an error analysis of independent interest is also presented for the basic finite element method using the conservation form of the nonlinear term this formulation leads to a nonlinear system whose nonlinearity is easier to resolve than the systems arising when either the convective or explicitly skew symmetric corms are used for the nonlinear term

generic families of matrix pencils and their bifurcation diagrams abstract v i arnold russian math surveys 26 2 1971 29 constructed smooth generic families of matrices with respect to similarity transformations depending smoothly on the entries of matrices and got bifurcation diagrams of such families with a small number of parameters we extend these results to pencils of matrices

on the congruence transformation of a pencil of real symmetric matrices to a pencil with identical characteristic polynomial abstract simultaneous nonorthogonal congruence transformations for pairs a b of 2 2 real symmetric matrices which leave the characteristic polynomial of the pencil x 1 a x 2 b invariant are studied

on asymptotically autonomous retarded functional differential equations the aim of the present paper is to investigate the asymptotic behavior of the solutions of the asymptotically autonomous retarded functional differential equation t lxt m t xt we obtain a convergence result without assuming that m t is in lp 1 p moreover we give bounds on the rate of convergence as well

a generalization of the motzkin taussky theorem abstract in this paper we generalize the motzkin taussky theorem to matrices with polynomial entries

the constrained bilinear form and the c numerical range abstract let v be an n dimentional unitary space with inner product and s the set x v x x 1 for any a hom v v and q c with q 1 we define w a q ax y x y s x y q if q 1 then w a q is just the classical numerical range ax x x s the convexity of which is well known another generalization of the numerical range is the c numerical range which is defined to be the set w c a tr cu au u unitary where c hom v v in this note we prove that w a q is always convex and that wc a is convex for all a if rank c 1 or n 2

unitarily achievable zero patterns and traces of words in a and a abstract our primary objective is to identify a natural and substantial problem about unitary similarity on arbitrary complex matrices which 0 patterns may be achieved for any given n by n complex matrix via some unitary similarity of it to this end certain restrictions on achievable 0 patterns are mentioned both positional and more important on the maximum number of achievable 0 s prior results fitting this general question are mentioned as well as the first unresolved pattern for 3 by 3 matrices in the process a recent question is answered a closely related additional objective is to mention the best known bound for the maximum length of words necessary for the application of specht s theorem about which pairs of complex matrices are unitarily similar which seems not widely known to matrix theorists in the process we mention the number of words necessary for small size matrices

on the reduction of pairs of hermitian or symmetric matrices to diagonal form by congruence abstract let a 1 a 2 be given n by n hermitian or symmetric matrices and consider the simultaneous transformations a i sa i s if a i is hermitian or a i sa i s t if a i is symmetric we give necessary and sufficient conditions for the existence of a unitary s which reduces both a 1 and a 2 to diagonal form in this way when at least one of a 1 or a 2 is nonsingular we give necessary and sufficient conditions for a reduction of this sort with a nonsingular s these results are a generalization of the classical theorem from mechanics that a positive definite matrix and a hermitian matrix can always be diagonalized simultaneously by a nonsingular congruence

symmetric matrices with respect to sesquilinear forms abstract simple forms are obtained for matrices that are symmetric with respect to degenerate sesquilinear forms on finite dimensional complex linear spaces of column vectors symmetric matrices and the sesquilinear forms are then representable as block diagonals having simple forms as the diagonal blocks the notion of indecomposability for symmetric matrices is studied an example shows that in contrast with the nondegenerate sesquilinear forms an indecomposable symmetric matrix with respect to a degenerate sesquilinear form may have arbitrarily many jordan blocks all indecomposable symmetric matrices are characterized in two situations when the sesquilinear form has only one degree of degeneracy and when the form is semidefinite

canonical matrices for linear matrix problems abstract we consider a large class of matrix problems which includes the problem of classifying arbitrary systems of linear mappings for every matrix problem from this class we construct belitskii s algorithm for reducing a matrix to a canonical form which is the generalization of the jordan normal form and study the set c mn of indecomposable canonical m n matrices considering c mn as a subset in the affine space of m n matrices we prove that either c mn consists of a finite number of points and straight lines for every m n or c mn contains a 2 dimensional plane for a certain m n

the hartman wintner theorem for functional differential equations abstract in this paper we present a generalization of the hartman wintner theorem about the asymptotic behavior of the solutions of linear ordinary differential equations to functional differential equations

a canonical form for nonderogatory matrices under unitary similarity abstract a square matrix is nonderogatory if its jordan blocks have distinct eigenvalues we give canonical forms for nonderogatory complex matrices up to unitary similarity and pairs of complex matrices up to similarity in which one matrix has distinct eigenvalues the types of these canonical forms are given by undirected and respectively directed graphs with no undirected cycles

a two level method with backtracking for the navier stokes equations we consider a two level method for resolving the nonlinearity in finite element approximations of the equilibrium navier stokes equations the method yields l2 and h1 optimal velocity approximations and an l2 optimal pressure approximation the two level method involves solving one small nonlinear coarse mesh system one oseen problem hence linear with positive definite symmetric part on the fine mesh and one linear correction problem on the coarse mesh the algorithm we study produces an approximate solution with the optimal asymptotic in h accuracy for any fixed reynolds number we do not consider the behavior of the error for fixed h as re rightarrow infty i e for flows in transition to turbulence

matrices with some extremal properties abstract for a complex matrix a there are many inequalities related to its eigenvalues diagonal elements and three types of singular values we investigate the conditions on the matrices for which those inequalities become equalities it is shown that in many cases the conditions are both necessary and sufficient for the matrices to be unitarily similar to matrices of lower orders moreover we get alternative proofs of various characterization theorems of different classes of matrices

a nonconforming finite element method for the stationary navier stokes equations approximations to solutions of the inhomogeneous boundary value problem for the navier stokes equations are constructed via a nonstandard finite element method the velocity field is approximated using piecewise solenoidal functions that are totally discontinuous across interelement boundaries but which are pointwise divergence free on each element the pressure is approximated by c0 functions optimal rates of convergence results are obtained requiring only local quasi uniformity assumptions on the meshes

asymptotically exponential solutions in nonlinear integral and differential equations abstract in this paper we investigate the growth decay rate of solutions of an abstract integral equation which frequently arises in quasilinear differential equations applying a variation of constants formula these results are applicable to some abstract equations which appear in the theory of age dependent population models and also to some quasilinear delay differential equations with bounded and unbounded delays examples are given to illustrate the sharpness of the results

on simultaneous diagonalization of one hermitian and one symmetric form abstract it is remarked that if a b m n c a a t and b b t b positive definite there exists a nonsingular matrix u such that 1 t bu i and 2 u t au is a diagonal matrix with nonnegative entries some related actions of the real orthogonal group and equations involving the unitary group are studied

asymptotic behavior in volterra difference systems with unbounded delay we show that under certain conditions there is a one to one correspondence between weighted bounded solutions of a linear volterra difference equation with unbounded delay and its perturbation this correspondence is elevated to asymptotic equivalence under suitable conditions

canonical forms for hamiltonian and symplectic matrices and pencils abstract we study canonical forms for hamiltonian and symplectic matrices or pencils under equivalence transformations which keep the class invariant in contrast to other canonical forms our forms are as close as possible to a triangular structure in the same class we give necessary and sufficient conditions for the existence of hamiltonian and symplectic triangular jordan kronecker and schur forms the presented results generalize results of lin and ho w w lin t c ho on schur type decompositions for hamiltonian and symplectic pencils technical report institute of applied mathematics national tsing hua university taiwan 1990 and simplify the proofs presented there

belitskii s canonical forms of upper triangular nilpotent matrices under upper triangular similarity abstract all indecomposable canonical forms are determined for upper triangular nilpotent matrices of size less than or equal to 7 under upper triangular similarity via belitskii s algorithm furthermore we show that there exists an indecomposable canonical form of upper triangular nilpotent n n matrix which admits at least n 2 2 parameters for n 8

selfadjoint operators in s spaces abstract we study s spaces and operators therein an s space is a hilbert space s with an additional inner product given by u where u is a unitary operator in s we investigate spectral properties of selfadjoint operators in s spaces we show that their spectrum is symmetric with respect to the real axis as a main result we prove that for each selfadjoint operator a in an s space we find an inner product which turns s into a krein space and a into a selfadjoint operator therein as a consequence we get a new simple condition for the existence of invariant subspaces of selfadjoint operators in krein spaces which provides a different insight into this well know and in general unsolved problem

conjugacy classes in sylow p subgroups of gl n q taking s to consist of lower unitriangular matrices higman notes that for i 3 ic g j

asymptotic integration of nonautonomous delay differential systems we prove existence theorems and asymptotic formulas for the solutions of a class of delay differential equations with time state dependent lag

equality of higher numerical ranges of matrices and a conjecture of kippenhahn on hermitian pencils let mn be the algebra of all n n complex matrices for 1 k n the kth numerical range of a mn is defined by wk a 1 k jk 1xj axj x1 xk is an orthonormal set in n it is known that tr a n wn a wn 1 a w1 a we study the condition on a under which wm a wk a for some given 1 m k n it turns out that this study is closely related to a conjecture of kippenhahn on hermitian pencils a new class of counterexamples to the conjecture is constructed based on the theory of the numerical range

upper triangular similarity of upper triangular matrices abstract we consider the following equivalence relation in the set of all complex upper triangular n n matrices a and b are called u similar if there exists an invertible upper triangular matrix s such that a s 1 bs if a b are u similar then they must have the same diagonal and the same jordan form it is known that for n 6 there are infinitely many mutually non u similar nilpotent upper triangular matrices with the same jordan form we introduce an appropriate generalization of the jordan block called an irreducible matrix and we prove that each upper triangular matrix is u similar to a generalized direct sum of irreducible blocks where the location and the order of the blocks is fixed and each block is determined uniquely up to u similarity

on the gau wu number for some classes of matrices abstract for a given n n matrix a let k a stand for the maximal number of orthonormal vectors x j such that the scalar products a x j x j lie on the boundary of the numerical range w a this number was recently introduced by gau and wu and we therefore call it the gau wu number of the matrix a we compute k a for two classes of n n matrices a a simple and explicit expression for k a for tridiagonal toeplitz matrices a is derived furthermore we prove that k a 2 for every pure almost normal matrix a note that for every matrix a we have k a 2 and for normal matrices a we have k a n so our results show that pure almost normal matrices are in fact as far from normal as possible with respect to the gau wu number finally matrices with maximal gau wu number k a n are considered

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

unitary and euclidean representations of a quiver abstract a unitary euclidean representation of a quiver is given by assigning to each vertex a unitary euclidean vector space and to each arrow a linear mapping of the corresponding vector spaces we recall an algorithm for reducing the matrices of a unitary representation to canonical form give a certain description of the representations of canonical form and reduce the problem of classifying euclidean representations to the problem of classifying unitary representations we also describe the set of dimensions of all indecomposable unitary euclidean representations of a quiver and establish the number of parameters in an indecomposable unitary representation of a given dimension

estimate of the number of one parameter families of modules over a tame algebra abstract the problem of classifying modules over a finite dimensional algebra a reduces to a linear matrix problem whose indecomposable matrices can be parametrized by certain normal forms called canonical matrices in linear algebra appl 317 2000 53 if the algebra a is tame then the set of isomorphism classes of indecomposable a modules of dimension at most d is given by a finite number f d a of discrete modules and one parameter families of modules accordingly there are for every fixed size only a finite number of discrete and one parameter families of canonical matrices we prove that in the tame case the number of one parameter families of canonical matrices of size m n and a given partition into blocks is bounded by 4 mn based on this estimate we prove that f d a d r r 4 d 2 1 2 r 2 where r is the number of nonisomorphic indecomposable projective a modules and 1 r are their dimensions this estimate improves significantly the double exponential estimate from c r acad sci paris 322 serie i 1996 211

geometry of the numerical range of matrices abstract some techniques for the study of the algebraic curve c a which generates the numerical range w a of an n n matrix a as its convex hull are developed these enable one to give an explicit point equation of c a and a formula for the curvature of c a at a boundary point of w a applied to the case of a nonnegative matrix a a simple relation is found between the curvature of the function a p 1 a a t p being the perron root at 1 2 and the curvature of w a at the perron root of 1 2 a a t a connection with 2 dimensional pencils of hermitian matrices is mentioned and a conjecture formulated

b orbits in solutions to the equation x2 0 in triangular matrices abstract consider in k the set of solutions to the equation x2 0 in n n strictly upper triangular matrices over field k in this paper we construct the bijection from the set of adjoint orbits in in k onto the set of involutions in symmetric group sn for a finite field k we compute the number of points in the orbits

a novel two grid method for semilinear elliptic equations a new finite element discretization technique based on two coarse and fine subspaces is presented for a semilinear elliptic boundary value problem the solution of a nonlinear system on the fine space is reduced to the solution of two small one linear and one nonlinear systems on the coarse space and a linear system on the fine space it is shown both theoretically and numerically that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation as a result the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation

an asymptotic theory for nonlinear functional differential equations abstract the hartman wintner theorem on asymptotic integration is established for a certain class of functional differential equations with nonlinear perturbations by looking them as abstract ordinary differential equations results for delay differential equations are included in this study consequences and applications are also shown

on a conjecture of kippenhahn about the characteristic polynomial of a pencil generated by two hermitian matrices i abstract let a be an n n matrix write a h ik where i 2 1 and h and k are hermitian let f x y z det zi xh yk we first show that a pair of matrices over an algebraically closed field which satisfy quadratic polynomials can be put into block upper triangular form with diagonal blocks of size 1 1 or 2 2 via a simultaneous similarity this is used to prove that if f x y z g x y z n 2 where g has degree 2 then for some unitary matrix u the matrix u au is the direct sum of n 2 copies of a 2 2 matrix a 1 where a 1 is determined up to unitary similarity by the polynomial g x y z we use the connection between f x y z and the numerical range of a to investigate the case where f x y z has the form z ax y r g x y z s where g x y z is irreducible of degree 2

a survey of canonical forms and invariants for unitary similarity abstract matrices a and b are said to be unitarily similar if u au b for some unitary matrix u this expository paper surveys results on canonical forms and invariants for unitary similarity the first half gives a detailed description of methods developed by several authors brenner littlewood mitchell mcrae radjavi serge chuk and benedetti and cragnolini using inductively defined reduction procedures to transform matrices to canonical form the matrix is partitioned and successive unitary similarities applied to reduce the submatrices to some nice form at each stage one refines the partition and restricts the set of permissible unitary similarities to those that preserve the already reduced blocks the process ends in a finite number of steps producing both the canonical form and the subgroup of the unitary group that preserves that form depending on the initial step various canonical forms may be defined the method can also be used to define canonical forms relative to certain subgroups of the unitary group and canonical forms for finite sets of matrices under simultaneous unitary similarity the remainder of the paper surveys results on unitary invariants and other topics related to unitary similarity such as the specht pearcy trace invariants the numerical range and unitary reducibility

a complete unitary similarity invariant for unicellular matrices abstract we present necessary and sufficient conditions for an n n complex matrix b to be unitarily similar to a fixed unicellular i e indecomposable by similarity n n complex matrix a

a multilevel decoupling method for the navier stokes darcy model abstract this paper considers a multilevel decoupling method for the coupled navier stokes darcy model describing a free flowing fluid over a porous medium the method utilizes a sequence of meshes on which a low dimensional fully coupled nonlinear problem is solved only on a very coarse initial mesh on subsequent finer meshes the approximate solution in each flow region is obtained by solving a linear decoupled problem and performing a correction step the correction step in each domain is achieved by solving a linear system that differs from the original decoupled system only in the right hand side we prove optimal error estimates and demonstrate that for a sequence of meshes with spacing h j h j 1 2 the decoupling method is computationally efficient and achieves the same order of approximation as the fully coupled method

hermitian pencils with a cubic minimal polynomial let a be an n n complex matrix and write a h ik where h and k are hermitian matrices we show that if the minimal polynomial of the pencil xh yk has degree 3 then there is a unitary matrix u such that u 1au is block diagonal with blocks of size 3 3 or smaller this is a special case of a conjecture made by kippenhahn in 1951

simultaneous similarity of matrices abstract in this paper we solve completely and explicitly the long standing problem of classifying pairs of n n complex matrices a b under the simultaneous similarity tat 1 tbt 1 roughly speaking the classification decomposes to a finite number of steps in each step we consider an open algebraic set m 0n 2 r mn mn mn the set of n n complex valued matrices here r and are two positive integers then we construct a finite number of rational functions o1 os in the entries of a and b whose values are constant on all pairs similar in m n 2 r to a b the values of the functions oi a b i 1 s determine a finite number at most n 2 r of similarity classes in m n 2 r let sn be the subspace of complex symmetric matrices in mn for a b sn sn we consider the similarity class tatt tbtt where t ranges over all complex orthogonal matrices then the characteristic polynomial i a xb determines a finite number of similarity classes for almost all pairs a b sn sn

nonconforming mixed finite element approximation to the stationary navier stokes equations on anisotropic meshes abstract the main aim of this paper is to study the error estimates of a rectangular nonconforming finite element for the stationary navier stokes equations under anisotropic meshes that is the nonconforming rectangular element is taken as approximation space for the velocity and the piecewise constant element for the pressure the convergence analysis is presented and the optimal error estimates both in a broken h 1 norm for the velocity and in an l 2 norm for the pressure are derived on anisotropic meshes