analytic torsion of a bounded generalized cone we compute the analytic torsion of a bounded generalized cone by generalizing the computational methods of m spreafico and using the symmetry in the de rham complex as established by m lesch we evaluate our result in lower dimensions and further provide a separate computation of analytic torsion of a bounded generalized cone over s 1 since the standard cone over the sphere is simply a flat disc

sub gaussian estimates of heat kernels on infinite graphs we prove that a two sided sub gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the green kernel admits a uniform polynomial decay

cheeger muller theorem on manifolds with cusps we prove equality between the renormalized ray singer analytic torsion and the intersection r torsion on a witt manifold with cusps up to an error term determined explicitly by the betti numbers of the cross section of the cusp and the intersection r torsion of a model cone in the first step of the proof we compute explicitly the renormalized ray singer analytic torsion of a model cusp in general dimension and without the witt condition in the second step we establish a gluing formula for renormalized ray singer analytic torsion on a general class of non compact manifolds in any dimension that includes witt manifolds with cusps but also scattering manifolds with asymptotically conical ends in the final step a cheeger muller theorem on cusps follows by a combination of the previous explicit computation and the gluing formula

intersection r torsion and analytic torsion for pseudomanifolds pour une pseudovariete x n on definit un invariant i p la r torsion intersection parametrisee par la perversite p cet invariant est analogue a la torsion de reidemeister et est un invariant combinatoire

zeta determinants of sturm liouville operators with quadratic potentials at infinity abstract we consider sturm liouville operators on a half line a a 0 with potentials that are growing at most quadratically at infinity such operators arise naturally in the analysis of hyperbolic manifolds or more generally manifolds with cusps we establish existence and a formula for the associated zeta determinant in terms of the wronski determinant of a fundamental system of solutions adapted to the boundary conditions despite being the natural objects in the context of hyperbolic geometry spectral geometry of such operators has only recently been studied in the context of analytic torsion

a novel technique for inverse identification of distributed stiffness factor in structures abstract a computational inverse technique for identifying stiffness distribution in structures is proposed in this paper using structural dynamics response in the frequency domain in the present technique element stiffness factors of the finite element model of a structure are taken to be the parameters and explicitly expressed in a linear form in the system equation for forward analysis of the harmonic response of the structure this offers great convenience in applying newton s method to search for the parameters of stiffness factor inversely as the jacobian matrix can be obtained simply by solving sets of linear algebraic equation derived from the system equation examples of identifying stiffness factor distribution which is often related to damage in the elements of the structure are presented to demonstrate the present technique the advantages of the present technique for inverse parameter identification problem are 1 the number of the parameters can be very large 2 the identification process is very fast and 3 the accuracy is very high the efficiency of the proposed technique is compared with genetic algorithms

hodge cohomology of gravitational instantons we study the space of l2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type these metrics generalize the geometric structures at infinity of several different well known classes of metrics including asymptotically locally euclidean manifolds the known types of gravitational instantons and also poincare metrics on q rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in kahler manifolds the answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold the l2 signature formula implied by our result is closely related to the one proved by dai 25 and more generally by vaillant 67 and identifies dai s invariant directly in terms of intersection cohomology of differing perversities this work is also closely related to a recent paper of carron 12 and the forthcoming paper of cheeger and dai 17 we apply our results to a number of examples gravitational instantons among them arising in predictions about l2 harmonic forms in duality theories in string theory

zero patterns and unitary similarity abstract a subspace of the space l n of traceless complex n n matrices can be specified by requiring that the entries at some positions i j be zero the set i of these positions is a zero pattern and the corresponding subspace of l n is denoted by l i n a pattern i is universal if every matrix in l n is unitarily similar to some matrix in l i n the problem of describing the universal patterns is raised solved in full for n 3 and partial results obtained for n 4 two infinite families of universal patterns are constructed they give two analogues of schur s triangularization theorem

on the growth of torsion in the cohomology of arithmetic groups in this paper we consider certain families of arithmetic subgroups of mathrm so 0 p q and mathrm sl 3 mathbb r respectively we study the cohomology of such arithmetic groups with coefficients in arithmetically defined modules we show that for natural sequences of such modules the torsion in the cohomology grows exponentially

on the diagonal of an operator characterlzatlons of zero diagonal operators i e operatcors that have a diagonal whose entries consist entirely of zeros and the ncorm closure of these operators are obtalned also lncluded are new characterlzations of trace class operators self colnmutators of bcounded operatcors and others o introduction in 3 it is shown that a hermitian compact operator is a self commutator of a compact operator if and only if it has a zero diagonal motivated by this result we decided to launch an investigation into the connection between the diagonal of an operator and the operator itself we wonder to what extent an operator can be described by its diagonal it turns out that as expee ted the diagonal of an operator carries more information about the operator than its relatively small size compare to the fat matrix representation of the operator may suggest let h be a complex separable hilbert space and bf h the space of a11 bounded operators on h by zero diagonal operators we mean the operators t such that there exists an orthonormal basis bj for h satisfying tbj hx 0 for all j it is not hard to see that the set of these operators is pathwise connected in fact all its elements can be connected to the origin 0 inside itself that it is not a linear manifold 2i ed i and i d3 21 belong to it but their sum i i does not and moreover that it stays away from i indeed the distance is no less than one take a zero diagonal operator t and choose an orthonormal basis b so that tby b 0 then iit iil supllxlls t i x x t i bl bj 1 in the first section we are to characterize zero diagonal operators and hermitian

rank preserving multiplicative maps on b x abstract let x h be a banach space hilbert space and let b x b h be the algebra of all bounded linear operators on x h in this paper we get some characterizations of rank preserving multiplicative maps on b x as applications we show that every multiplicative local approximate automorphism of b x with the set of all rank 1 idempotents contained in its range is in fact an automorphism we describe the structure of corank preserving multiplicative maps on b h we also get a characterization of a isomorphism or a conjugate isomorphism on b h by showing that there exists a unitary or conjugate linear unitary operator u b h k such that t utu for all t b h if and only if is multiplicative with the range containing all rank 1 projections and for any a b b h a b 0 a b 0

flat portions on the boundary of the davis wielandt shell of 3 by 3 matrices abstract let t be an n n matrix the davis wielandt shell of t is defined as the set w t t t x tx x t tx x c r x x 1 in this paper we discuss the existence of flat portions on the boundary of the davis wielandt shell of 3 by 3 matrices

on tridiagonalization of matrices abstract we consider the question is every n n complex matrix unitarily similar to a tridiagonal one it is shown that the answer is negative if n 6 and is affirmative if n 3 additionally some positive partial answers and related results are given for example 1 every pair of hermitian projections is simultaneously unitarily similar to a pair of tridiagonal matrices 2 if a a has a rank one then a is unitarily similar to a tridiagonal matrix

numerical ranges of companion matrices we show that an n by n companion matrix a can have at most n line segments on the boundary w a of its numerical range w a and it has exactly n line segments on w a if and only if for n odd a is unitary and for n even a is unitarily equivalent to the direct sum a1 a2 of two n 2 by n 2 companion matrices a1

canonical forms for symmetric skew symmetric real matrix pairs under strict equivalence and congruence a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the reals one of which is skew symmetric and the other is either symmetric or skew symmetric these reductions are by strict equivalence and by congruence over the reals or over the complex numbers and essentially complete proofs are presented the proofs are based on canonical forms attributed to jordan and kronecker some closely related results which can be derived from the canonical forms of pairs of symmetric skew symmetric real forms are also included they concern simultaneously neutral subspaces hamiltonian and skew hamiltonian matrices and canonical structures of real matrices which are selfadjoint or skew adjoint in a regular skew symmetric indefinite inner product and real matrices which are skew adjoint in a regular symmetric indefinite inner product the paper is largely expository and continues the comprehensive account of the reduction of pairs of matrices started in p lancaster l rodman canonical forms for hermitian matrix pairs under strict equivalence and congruence siam rev in press

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

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

gaussian upper bounds for the heat kernels of some second order operators on riemannian manifolds abstract we describe a method of obtaining gaussian upper bounds on heat kernels which unifies and improves recent results for hypoelliptic operators in divergence form and for laplace beltrami operators on complete riemannian manifolds

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

band diagonal operators abstract we show that there are operators on a five dimensional hilbert space which are not tridiagonal and that there are compact operators and normal operators on separable infinite dimensional spaces which are not band diagonal

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

analytic torsion for group actions the reidemeister torsion is a classical topological invariant for nonsimply connected manifolds 14 let m be a closed oriented smooth manifold with the fundamental group m nontrivial let p x m o n be an orthogonal representation of nx m if the twisted real cohomology h m ep vanishes then one can define the reidemeister torsion p gr which is a homeomorphism invariant of m the original interest of p was that it is not a homotopy invariant and so can distinguish spaces which are homotopy equivalent but are not homeomorphic ray and singer asked whether as for many other topological quantities one can compute p by analytic methods 17 given a riemannian metric g on m they defined an analytic torsion t e r as a certain combination of the eigenvalues of the laplacian acting on twisted differential forms they showed that under the above acyclicity condition t is independent of the metric g and they conjectured that the analytic expression t equals the combinatorial expression this conjecture was proven to be true independently by cheeger 4 and m ller 15 one can look at the above situation in the following way the group n m acts freely on the universal cover m and so one has an invariant for free group actions a natural question is whether the reidemeister torsion can be extended to an invariant for more general group actions for a finite group acting not necessarily freely on a closed oriented pl manifold x a reidemeister torsion was defined algebraically by rothenberg 19 one can then ask whether there is a corresponding analytic torsion when x is smooth and whether the analytic torsion equals the combinatorial torsion in 11 we define the analytic torsion tp for a finite group action and show that if the relevant cohomology groups vanish then it is independent of the g invariant metric used in its definition this was shown previously in unpublished work by cheeger 5 the analysis involved to

normal matrices subordinate to a tree and flat portions of the field of values abstract matrices subordinate to trees are considered an efficient normality characterization for any such matrix is given and several consequences not valid for general normal matrices of it are established in addition the existence and enumeration of flat portions on the boundary of the field of values of matrices subordinate to a tree is characterized

tridiagonal normal forms for orthogonal similarity classes of symmetric matrices abstract let f be an algebraically closed field of characteristic different from 2 define the orthogonal group o n f as the group of n by n matrices x over f such that xx i n where x is the transpose of x and i n the identity matrix we show that every n by n symmetric matrix over f is orthogonally similar to a tridiagonal symmetric matrix if further the characteristic is 0 we construct the tridiagonal normal form for the o n f similarity classes of symmetric matrices we point out that in this case the known normal forms as presented in the well known book by gantmacher are not tridiagonal

the operator factorization problems we survey various results concerning operator factorization problems more precisely we consider the following setting let h be a complex hilbert space and let g h be the algebra of all bounded linear operators on h for a given subset of 3 h we are interested in the characterization of operators in q h which are expressible as a product of finitely many operators in v and for each such operator the minimal number of factors in a factorization the classes of operators we consider include normal operators involutions partial isometries together with their various subclasses and other miscellaneous classes of operators most of the known results are for operators on finite dimensional spaces or finite matrices the paper concludes with some applications due to hochwald concerning the uniqueness of the adjoint operation on operators

a combined genetic algorithm and nonlinear least squares method for material characterization using elastic waves abstract the inverse problem of material characterization is formulated as a parameter identification problem in which a set of parameters corresponding to the material property can be found by minimizing error functions formulated using the measured displacement response and the one computed by a forward solver based on projected candidates of parameters a hybrid numerical method is employed as the forward solver to calculate the dynamic displacement response on the surface of the composite plate for given material property a combined method is used as the inverse operator to determine the material property of composite plate in this method genetic algorithm is first used to select a set of better solutions close to the optima then the nonlinear least squares method is applied using these better solutions as the initial guesses finally the identification results can be determined from the solutions of nonlinear least squares method by comparing their corresponding error function values actual material characterizations of composites demonstrate the higher efficiency of the present method

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

analytic torsion versus reidemeister torsion on hyperbolic 3 manifolds with cusps for a non compact hyperbolic 3 manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of mathrm sl 2 mathbb c to the corresponding reidemeister torsion while the analytic torsion is a spectral invariant of the manifold the reidemeister torsion is of combinatorial nature our proof rests on an expression of the analytic torsion in terms of special values of ruelle zeta functions as well as on recent work of pere menal ferrer and joan porti

once more on algebras generated by two projections we consider von neumann algebras generated by two arbitrary orthoprojections on a hilbert space a canonical decomposition is obtained for elements a of these algebras in terms of the operator angle between the ranges of the above mentioned projections this decomposition leads to explicit descriptions and formulas for kernels ranges spectra and essential spectra generalized inverses and other objects related to a

multiplicative preservers and induced operators abstract let v be an n dimensional hilbert space suppose h is a subgroup of the symmetric group of degree m and h c is a character of degree 1 on h consider the symmetrizer on the tensor space m v s v 1 v m 1 h h v 1 1 v 1 m defined by h and the subspace v m h of m v spanned by s m v is called the symmetry class of tensors over v associated with h and the elements in v m h of the form s v 1 v m are called decomposable tensors and are denoted by v 1 v m for any linear operator t acting on v there is an unique induced operator k t or just k t for notational simplicity acting on v m h satisfying k t v 1 v m tv 1 tv m we characterize multiplicative maps such that f t f t for all operators t acting on v where f are various scalar or set valued functions including the spectral radius decomposable numerical radius spectral norm spectrum decomposable numerical range of t or k t

differential topology of numerical range abstract the numerical range of an n n matrix also known as its field of values is reformulated as the image of a smooth quadratic mapping from the n 1 dimensional complex projective space to the complex plane this paper investigates the numerical range from the perspective of differential topology morse theory more specifically the boundary of the range is interpreted as a rank 1 critical value curve and its sharp points are interpreted as rank 0 critical values more importantly the map is shown to have additional critical value curves in the interior of the numerical range these additional curves are shown to have such singularity phenomena as cusps and swallow tails to be the caustic envelopes of families of lines and to exhibit the so called normal bifurcation when an eigenvalue becomes unitarily decoupled

on the numerical ranges of some tridiagonal matrices abstract for certain tridiagonal matrices of small size we give a complete description of flat portions on the boundary of their numerical range we also discuss the conditions for these numerical ranges to be elliptical

boundary generating curves of the c numerical range abstract let a be an n n matrix and c c 1 c 2 c n be a real n tuple the c numerical range of a is the set w c a j 1 n c j x j ax j x 1 x 2 x n is an orthonormal basis of c n we obtain parametric representations of the boundary generating curve of the c numerical range of a matrix applying this result we generalize the result of anderson to the c numerical range furthermore we give a description of the boundary generating curve of the c numerical range of certain types of nilpotent toeplitz matrices a sufficient condition for the boundary generating curve to be rational is obtained finally we explicitly compute the boundary generating curves of the numerical ranges for several concrete matrices and classify the rationality of the curves

multiplicative maps on matrices that preserve the spectrum abstract let m n denote the set of all n n matrices over the complex numbers n 2 let a n m n be either the set of all invertible matrices the set of all unitary matrices or a multiplicative semigroup containing the singular matrices theorem if a n m n is a spectrum preserving multiplicative map then there exists a matrix r in m n such that s r 1 sr for all s in a n

the numerical range of 3 3 matrices abstract let a be an n n complex matrix then the numerical range of a w a is defined to be x ax x c n x x 1 in this article a series of tests is given allowing one to determine the shape of w a for 3 3 matrices reconstruction of a up to unitary similarity from w a is also examined

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

tridiagonal forms in low dimensions abstract pati showed that every 4 4 matrix is unitarily similar to a tridiagonal matrix we give a simple proof in addition we show that in an appropriate sense there are generically precisely 12 ways to do this when the real part is diagonal it is shown that the unitary can be chosen with the form u pd where d is diagonal and p is real orthogonal however even if both real and imaginary parts are real symmetric there may be no real orthogonal matrices which tridiagonalize it on the other hand if the matrix belongs to the lie algebra sp 4 c then it can be tridiagonalized by a unitary in the symplectic group sp 2 in dimension 5 or greater there are always rank three matrices which are not tridiagonalizable

on flat portions on the boundary of the numerical range abstract the paper is devoted to matrices with flat portions on the boundary of their numerical range a constructive criterion for such portions to exist is obtained in case of tridiagonal matrices and a particular case of continuant matrices is considered as an application the cases of arbitrary 3 3 and 4 4 matrices are treated it is shown in particular that the sharp bound for the number of flat portions on the boundary of the numerical range for 4 4 matrices is four three if the matrices are assumed unitarily irreducible

strict convexity of the joint c numerical range abstract let h h 1 h 2 h m be an m tuple of n n hermitian matrices and c c 1 c 2 c n r n the joint c numerical range w c h of h is defined for x x 1 x 2 x m r m the form associated with h is defined as f h t x det ti n x 1 h 1 x 2 h 2 x m h m we prove that the convex hull of w c h is strictly convex if the roots of f h t x 0 have constant multiplicities for all x on the unit sphere in r m the result is illustrated with several examples

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

characterizations of automorphisms of operator algebras on banach spaces abstract let x be a complex banach space of dimension greater than one and denote by b x the algebra of all the bounded linear operators on x it is shown that if b x b x is a multiplicative map not assumed linear and if is sufficiently close to a linear automorphism of b x in some uniform sense then it is actually an automorphism

schur multiplicative maps preserving unitarily invariant norms characterizations are obtained for maps on real or complex matrices which preserve both the schur hadamard product and a given unitarily invariant norm

analytic torsion on manifolds with edges abstract let m g be an odd dimensional incomplete compact riemannian singular space with a simple edge singularity we study the analytic torsion on m and in particular consider how it depends on the metric g if g is an admissible edge metric we prove that the torsion zeta function is holomorphic near s 0 hence the torsion is well defined but possibly depends on g in general dimensions we prove that the torsion depends only on the asymptotic structure of g near the singular stratum of m when the dimension of the edge is odd we prove that the analytic torsion is independent of the choice of even admissible edge metrics the main tool is the construction via the methodology of geometric microlocal analysis of the heat kernel for the friedrichs extension of the hodge laplacian in all degrees in this way we obtain detailed asymptotics of this heat kernel and its trace

an extension of a theorem by cheeger and muller to spaces with isolated conical singularities the aim of this article is to extend a theorem by cheeger and muller to spaces with isolated conical singularities by generalizing the proof of bismut and zhang to the singular setting the main tools in this approach are the witten deformation and local index techniques

generalized ray singer conjecture i a manifold with a smooth boundary this paper is devoted to a proof of a generalized ray singer conjecture for a manifold with boundary the dirichlet and the neumann boundary conditions are independently given on each connected component of the boundary and the transmission boundary condition is given on the interior boundary the ray singer conjecture rs claims that for a closed manifold the combinatorial and the analytic torsion norms on the determinant of the cohomology are equal for a manifold with boundary the ratio between the analytic torsion and the combinatorial torsion is computed some new general properties of the ray singer analytic torsion are found the proof does not use any computation of eigenvalues and their asymptotic expansions or explicit expressions for the analytic torsions of any special classes of manifolds

spectrum preserving linear maps abstract let x and y be banach spaces we show that a spectrum preserving surjective linear map from b x to b y is either of the form t ata 1 for an isomorphism a of x onto y or the form t bt b 1 for an isomorphism b of x onto y

multiplicative preservers of higher dimensional numerical ranges abstract let h be a complex separable hilbert space and k be a positive integer smaller than the dimension of h in this paper we characterize multiplicative maps b h b h that preserve the k dimensional numerical ranges no linearity surjectivity or continuity of is assumed it is shown that such a map is a c isomorphism

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

normal forms for orthogonal similarity classes of skew symmetric matrices let f be an algebraically closed field of characteristic different from 2 define the orthogonal group on f as the group of n by n matrices x over f such that xx in where x is the transpose of x and in the identity matrix we show that every nonsingular n by n skew symmetric matrix over f is orthogonally similar to a bidiagonal skew symmetric matrix in the singular case one has to allow some 4 diagonal blocks as well if further the characteristic is 0 we construct the normal form for the on f similarity classes of skew symmetric matrices in this case the known normal forms as presented in the well known book by gantmacher are quite different finally we study some related varieties of matrices we prove that the variety of normalized nilpotent n by n bidiagonal matrices for n 2s 1 is irreducible of dimension s as a consequence the skew symmetric nilpotent n by n bidiagonal matrices are shown to form a variety of pure dimension s

multiplicative preservers on semigroups of matrices we characterize multiplicative maps on semigroups of square matrices satisfying p p for matrix sets p such as rank k idempotent matrices totally nonnegative matrices p0 matrices m0 matrices positive semidefinite matrices hermitian matrices normal matrices and contractions we also characterize multiplicative maps satisfying g x x for various functions g on square matrices such as the spectrum spectral radius numerical range numerical radius and matrix norms

the analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds of finite volume abstract in this paper we study the regularized analytic torsion of finite volume hyperbolic manifolds we consider sequences of coverings x i of a fixed hyperbolic orbifold x 0 our main result is that for certain sequences of coverings and strongly acyclic flat bundles the analytic torsion divided by the index of the covering converges to the l 2 torsion our results apply to certain sequences of arithmetic groups in particular to sequences of principal congruence subgroups of so 0 d 1 z and to sequences of principal congruence subgroups or hecke subgroups of bianchi groups

elliptical higher rank numerical range of some toeplitz matrices abstract the higher rank numerical range is described for a class of matrices which happen to be unitarily reducible to direct sums of at most 2 by 2 blocks in particular conditions are established under which tridiagonal matrices have elliptical rank k numerical ranges

canonical forms for hermitian matrix pairs under strict equivalence and congruence after a brief historical review and an account of the canonical forms attributed to jordan and kronecker a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the complex numbers and over the reals these reductions are by strict equivalence and by congruence and essentially complete proofs are presented some closely related results which can be derived from the canonical forms are also included they concern simultaneous diagonalization a new criterion for the existence of positive definite linear combinations of a pair of hermitian matrices and the canonical structures of matrices which are self adjoint in an indefinite inner product

analytic and topological torsion for manifolds with boundary and symmetry letg be a finite group acting on a riemannian manifoldm by isometries we introduce analytic torsion an m m1 v r z repr g pl torsion pl m m1 v k1 rg poincare torsion pd m m1 v k1 rg and euler characteristic m m1 v repr g for m the disjoint union of m1 and m2 and v an equivariant coefficient system the analytic torsionis defined in terms of the spectrum of the laplace operator the pl torsionis based on the cellular chain complex and poincare torsionmeasures the failure of equivariant poincare duality in the pl setting which does hold in the analytic context denote by repr g the subgroup of repr g generated by the irreducible representations of real or complex type we define an isomorphism

3 3 matrices with a flat portion on the boundary of the numerical range the set of unitarily irreducible complex 3 3 matrices whose numerical range has a flat portion is explicitly described it is proved that the set is connected and real analytic properties of the set are studied connectedness of the sets of unitarily irreducible 3 3 matrices having two other possible shapes of their numerical ranges is proved as well

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

when is the numerical range of a nilpotent matrix circular the problem formulated in the title is investigated the case of nilpotent matrices of size at most 4 allows a unitary treatment the numerical range of a nilpotent matrix m of size at most 4 is circular if and only if the traces trm m 2 and trm m 3 are null the situation becomes more complicated as soon as the size is 5 the conditions under which a 5x5 nilpotent matrix has circular numerical range are thoroughly discussed

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

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

spectrum preserving additive maps abstract let x and y be complex banach spaces we show that a spectrum preserving surjective additive map from b x to b y is either of the form f ata 1 for a linear isomorphism a of x onto y or of the form t bt b 1 for a linear isomorphism b of x onto y

some multiplicative preservers on b h abstract in this paper we describe the form of those continuous multiplicative maps on b h h being a separable complex hilbert space of dimension not less than 3 which preserve the rank or the corank furthermore we characterize those continuous presup semigroup endomorphisms of b h which are spectrum non increasing

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

relative zeta functions relative determinants and scattering theory we use the method of zeta function regularization to regularize the ratio det a det a0 of the determinants of two elliptic self adjoint operators a a0 satisfying certain natural assumptions this is of interest especially if the regularized determinants of the individual operators don t exist as for example in the case of elliptic operators on a noncompact manifold

sets of matrices with given joint numerical range abstract the following question is addressed to what extent the n tuple of m m hermitian matrices is determined by its joint numerical range the cases m 2 n arbitrary and m n 3 are considered in detail

moving zeros among matrices abstract we investigate the zero patterns that can be created by unitary similarity in a given matrix and the zero patterns that can be created by simultaneous unitary similarity in a given sequence of matrices the latter framework allows a simultaneous hessenberg formulation of pati s tridiagonal result for 4 4 matrices this formulation appears to be a strengthening of pati s theorem our work depends at several points on the simplified proof of pati s result by davidson and djokovic the hessenberg approach allows us to work with ordinary similarity and suggests an extension from the complex to arbitrary algebraically closed fields this extension is achieved and related results for 5 5 and larger matrices are formulated and proved

flaw detection in sandwich plates based on time harmonic response using genetic algorithm an inverse identification technique is proposed to assess the presence location size and degree of flaw in the core layer of sandwich plates the time harmonic response of the sandwich plates is used as the input for inverse analysis the responses of the plates in forward analysis are predicted using finite element method fem based on mindlin plate theory the flaw is characterized by a set of parameters the genetic algorithm ga is applied to search the parameters to improve the efficiency of ga in searching for the true solution a two stage searching method is proposed numerical examples are given to illustrate the proposed technique

structural damage detection based on covariance of covariance matrix with general white noise excitation abstract covariance of the auto cross covariance matrix based method is studied for the damage identification of a structure with illustrations on its advantages and limitations the original method is extended for structures under direct white noise excitations the auto cross covariance function of the measured acceleration and its corresponding derivatives are formulated analytically and the method is modified in two new strategies to enable successful identification with much fewer sensors numerical examples are adopted to illustrate the improved method and the effects of sampling frequency and sampling duration are discussed results show that the covariance of covariance calculated from responses of higher order modes of a structure play an important role to the accurate identification of local damage in a structure