on the inverse eigenvalue problem for nonnegative matrices of order two to five abstract in this paper for a given set of numbers with special conditions we construct a nonnegative matrix such that the given set is its spectrum

nonnegative realizability with jordan structure abstract a general method is given for merging blocks in the jordan canonical form of a nonnegative matrix as a consequence results more general than any prior ones are given for the universal realizability of spectra that is spectra which are realizable by a nonnegative matrix for each possible jordan canonical form allowed by the spectrum in particular we generalize a classical result due to minc regarding positive diagonalizable matrices for example any spectrum that is diagonalizably realizable by a nonnegative matrix with mostly positive off diagonal entries is universally realizable

the inverse eigenvalue problem for nonnegative matrices abstract let a be a nonnegative matrix with spectrum 1 2 m and b be a nonnegative matrix with spectrum 1 2 n where 1 is the perron eigenvalue of b furthermore let a maximal diagonal element of a be greater than or equal to 1 in the article we construct a nonnegative matrix c with spectrum 1 2 m 2 n this construction enables us to obtain several results on how to determine new realizable lists from known realizable lists

negativity compensation in the nonnegative inverse eigenvalue problem abstract if a set of complex numbers can be partitioned as 1 s in such a way that each i is realized as the spectrum of a nonnegative matrix say a i then is trivially realized as the spectrum of the nonnegative matrix a a i in linear algebra appl 369 2003 169 it was shown that in some cases a real set can be realized even if some of the i are not realizable themselves here we systematize and extend these results in particular allowing the sets to be complex the leading idea is that one can associate to any nonrealizable set a certain negativity n and to any realizable set a certain positivity m then under appropriate conditions if m n we can conclude that is the spectrum of a nonnegative matrix additionally we prove a complex generalization of suleimanova s theorem

greedy expansions and sets with deleted digits we generalize a result of daroczy and katai on the characterization of univoque numbers with respect to a non integer base publ math debrecen 46 3 4 1995 385 by relaxing the digit alphabet to a generic set of real numbers we apply the result to derive the construction of a buchi automaton accepting all and only the greedy sequences for a given base and digit set in the appendix we prove a more general version of the fact that the expansion of an element x q q is ultimately periodic if q is a pisot number

nonnegative matrices with prescribed spectrum and elementary divisors abstract in this paper we give new sufficient conditions for the existence and construction of nonnegative matrices with prescribed elementary divisors which drastically improve and contain some of the previous known conditions we also show how to perturb complex eigenvalues of a nonnegative matrix while keeping its nonnegativity these results allow us under certain conditions to easily decide if a given list is realizable with prescribed elementary divisors

an optimally accurate discrete regularization for second order timestepping methods for navier stokes equations abstract we propose a new optimally accurate numerical regularization stabilization for a family of second order timestepping methods for the navier stokes equations nse the method combines a linear treatment of the advection term together with stabilization terms that are proportional to discrete curvature of the solutions in both velocity and pressure we rigorously prove that the entire new family of methods are unconditionally stable and o t 2 accurate the idea of curvature stabilization is new to cfd and is intended as an improvement over the commonly used speed stabilization which is only first order accurate in time and can have an adverse effect on important flow quantities such as drag coefficients numerical examples verify the predicted convergence rate and show the stabilization term clearly improves the stability and accuracy of the tested flows

the long time l2 and h1 stability of linearly extrapolated second order time stepping schemes for the 2d incompressible navier stokes equations abstract herein we present a study on the long time stability of finite element discretizations of a generalized class of semi implicit second order time stepping schemes for the 2d incompressible navier stokes equations these remarkably efficient schemes require only a single linear solve per time step through the use of a linearly extrapolated advective term our result develops a class of sufficient conditions such that if external forcing is uniformly bounded in time velocity solutions are uniformly bounded in time in both the l2 and h1 norms we provide numerical verification of these results we also demonstrate that divergence free finite elements are critical for long time h1 stability

a map of sufficient conditions for the real nonnegative inverse eigenvalue problem abstract the real nonnegative inverse eigenvalue problem rniep is the problem of determining necessary and sufficient conditions for a list of real numbers to be the spectrum of an entrywise nonnegative matrix a number of sufficient conditions for the existence of such a matrix are known in this paper in order to construct a map of sufficient conditions we compare these conditions and establish inclusion relations or independency relations between them

a unified view on compensation criteria in the real nonnegative inverse eigenvalue problem a connection is established between the problem of characterizing all possible real spectra of entrywise nonnegative matrices the so called real nonnegative inverse eigenvalue problem and a combinatorial process consisting in repeated application of three basic manipulations on sets of real numbers given realizable sets i e sets which are spectra of some nonnegative matrix each of these three elementary transformations constructs a new realizable set this defines a special kind of realizability called c realizability and this is closely related to the idea of compensation after observing that the set of all c realizable sets is a strict subset of the set of realizable ones we show that it strictly includes in particular all sets satisfying several previously known sufficient realizability conditions in the literature furthermore the proofs of these conditions become much simpler when approached from this new point of view

a matrix completion problem over integral domains the case with 2n 3 prescribed entries abstract let 1 n n 2 be a given multiset of elements in an integral domain r and let p be a matrix of order n with at most 2 n 3 prescribed entries that belong to r under the assumption that each row each column and the diagonal of p have at least one unprescribed entry we prove that p can be completed over r to obtain a matrix a with spectrum we describe an algorithm to construct a this result is an extension to integral domains of a classical completion result by herskowitz for fields

matrix completion problems over integral domains the case with a diagonal of prescribed blocks abstract let r be an arbitrary integral domain let 1 n be a multiset of elements of r let be a permutation of 1 k let n 1 n k be positive integers such that n 1 n k n and for r 1 k let a r r n r n r we are interested in the problem of finding a block matrix q q rs r s 1 k r n n with spectrum and such that q r r a r for r 1 k cravo and silva completely characterized the existence of such a matrix when r is a field in this work we construct a solution matrix q that solves the problem when r is an integral domain with two exceptions i k 2 ii k 3 r r and n r n 2 for some r what makes this work quite unique in this area is that we consider the problem over the more general algebraic structure of integral domains which includes the important case of integers furthermore we provide an explicit and easy to implement finite step algorithm that constructs an specific solution matrix we point out that cravo and silva s proof is not constructive

universal realizability of spectra with two positive eigenvalues abstract a list of eigenvalues is said to be realizable if it is the spectrum of a nonnegative matrix diagonalizably realizable dr if it is the spectrum of a diagonalizable nonnegative matrix and universally realizable ur if there is a nonnegative matrix with this spectrum and any possible jordan canonical form allowed by the spectrum the nonnegative inverse eigenvalue problem niep asks which lists are realizable it is known that there exist spectra that are realizable but not dr we raise the question of whether dr implies ur this is known in a few cases including n 4 nonnegative spectra and suleimanova spectra we add some new classes of spectra that are ur

a nordhaus gaddum conjecture for the minimum number of distinct eigenvalues of a graph abstract we propose a nordhaus gaddum conjecture for q g the minimum number of distinct eigenvalues of a symmetric matrix corresponding to a graph g for every graph g excluding four exceptions we conjecture that q g q g c g 2 where g c is the complement of g we compute q g c for all trees and all graphs g with q g g 1 and hence we verify the conjecture for trees unicyclic graphs graphs with q g 4 and for graphs with g 7

unconditional long time stability of a velocity vorticity method for the 2d navier stokes equations we prove unconditional long time stability for a particular velocity vorticity discretization of the 2d navier stokes equations the scheme begins with a formulation that uses the lamb vector to couple the usual velocity pressure system to the vorticity dynamics equation and then discretizes with the finite element method in space and implicit explicit bdf2 in time with the vorticity equation decoupling at each time step we prove the method s vorticity and velocity are both long time stable in the l 2 l2 and h 1 h1 norms without any timestep restriction moreover our analysis avoids the use of gronwall type estimates which leads us to stability bounds with only polynomial instead of exponential dependence on the reynolds number numerical experiments are given that demonstrate the effectiveness of the method

constructing new realisable lists from old in the niep abstract given a list of complex numbers 1 2 n we say that is realisable if is the spectrum of some entrywise nonnegative matrix the nonnegative inverse eigenvalue problem or niep is the problem of characterising all realisable lists given a realisable list 2 3 m where is the perron eigenvalue and 2 is real we find families of lists 1 2 n for which 1 2 n 3 4 m is realisable in addition given a realisable list i i 4 5 m where is the perron eigenvalue and and are real we find families of lists 1 2 3 4 for which 1 2 3 4 4 5 m is realisable

the nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial ebl digraphs the nonnegative inverse eigenvalue problem niep is given a family of complex numbers 1 n find necessary and sufficient conditions for the existence of a nonnegative matrix a of order n with spectrum loewy and london r loewy d london a note on the inverse eigenvalue problems for nonnegative matrices linear and multilinear algebra 6 1978 83 90 resolved it for n 3 and for n 4 when the spectrum is real in our way of handling the niep we focus our attention on the coefficients of the characteristic polynomial of a thus the niep that we consider is given k1 k2 kn real numbers find necessary and sufficient conditions for the existence of a nonnegative matrix a of order n with characteristic polynomial xn k1xn 1 k2xn 2 kn the coefficients of the characteristic polynomial are closely related to the cyclic structure of the weighted digraph with adjacency matrix a we introduce a special type of digraph structure that we shall call ebl in which this relation is specially simple we give some results that show the interest of ebl structures we completely solve the niep from the coefficients of the characteristic polynomial for n 4 we also solve a special case of the niep for n 2p 1 with k1 kp 1 0 and p 2

elgenvalues of nonnegative matrices abstract in a partial continuation of work by fiedler some spectral properties of symmetric nonnegative matrices are extended to general nonnegative matrices

existence and construction of nonnegative matrices with prescribed spectrum abstract we consider the following inverse spectrum problem for nonnegative matrices given a set of real numbers 1 2 n find necessary and sufficient conditions for the existence of an n n nonnegative matrix a with spectrum in particular by the use of a relevant theorem of brauer we obtain new simple sufficient conditions for the problem to have a solution moreover we can always construct a solution matrix which is nonnegative generalized stochastic

a new family of stable mixed finite elements for the 3d stokes equations a natural mixed element approach for the stokes equations in the velocity pressure formulation would approximate the velocity by continuous piecewise polynomials and would approximate the pressure by discontinuous piecewise polynomials of one degree lower however many such elements are unstable in 2d and 3d this paper is devoted to proving that the mixed finite elements of this p k p k 1 type when k 3 satisfy the stability condition the babuska brezzi inequality on macro tetrahedra meshes where each big tetrahedron is subdivided into four subtetrahedra this type of mesh simplifies the implementation since it has no restrictions on the initial mesh the new element also suits the multigrid method

nonnegative realization of spectra having negative real parts abstract the nonnegative inverse eigenvalue problem niep is the problem of determining necessary and sufficient conditions for a list of complex numbers to be the spectrum of a nonnegative matrix in this paper the problem is completely solved in the case when all numbers in the given list except for one the perron eigenvalue have real parts smaller than or equal to zero

realizability criterion for the symmetric nonnegative inverse eigenvalue problem abstract let 1 2 n a set of real numbers the real nonnegative inverse eigenvalue problem rniep is the problem of determining necessary and sufficient conditions in order that be the spectrum of an entrywise nonnegative n n matrix if there exists a nonnegative matrix a with spectrum we say that is realized by a many realizability criteria for the existence of such a matrix a are known this paper shows that a realizability criterion given by the author which contains both kellogg s realizability criterion and borobia s realizability criterion is sufficient for the existence of an n n symmetric nonnegative matrix with prescribed spectrum

on nonnegative matrices with prescribed eigenvalues and diagonal entries abstract we consider the problem of the existence and construction of nonnegative matrices with prescribed spectrum and diagonal entries necessary and sufficient conditions have been obtained for n 3 by perfect and fiedler in the cases nonnegative and nonnegative symmetric respectively for n 4 they obtained sufficient conditions many partial results about the problem have been published by several authors mainly by smigoc this is a long standing unsolved inverse problem but also necessary to apply a perturbation result due to r rado which has played an important role in the study of nonnegative inverse eigenvalue and inverse elementary divisors problems distinct versions of rado s result have been also obtained for certain structured matrices to apply rado s result and its different versions we need to guarantee the existence of an r r r n nonnegative structured matrix with prescribed spectrum and diagonal entries this is an important motivation for this work here we prove new sufficient conditions for n 4 which extend and strictly contain the perfect and fiedler conditions our results generate an algorithmic procedure to construct a solution matrix

block matrices and guo s index for block circulant matrices with circulant blocks abstract in this paper we deal with circulant and partitioned into n by n circulant blocks matrices and introduce spectral results concerning this class of matrices the problem of finding lists of complex numbers corresponding to a set of eigenvalues of a nonnegative block matrix with circulant blocks is treated along the paper we call realizable list if its elements are the eigenvalues of a nonnegative matrix the guo s index 0 of a realizable list is the minimum spectral radius such that the list up to the initial spectral radius together with 0 is realizable the guo s index of block circulant matrices with circulant blocks is obtained and in consequence necessary and sufficient conditions concerning the niep nonnegative inverse eigenvalue problem for the realizability of some spectra are given

a characterization of trace zero nonnegative 5 5 matrices let 1 2 3 4 5 be a list of complex numbers with 1 2 3 4 5 0 necessary and sufficient conditions for the existence of an entry wise nonnegative 5 5 matrix a with spectrum are presented

existence and construction of nonnegative matrices with complex spectrum abstract the following inverse spectrum problem for nonnegative matrices is considered given a set of complex numbers 1 2 n find necessary and sufficient conditions for the existence of an n n nonnegative matrix a with spectrum our work is motivated by a relevant theoretical result of guo wuwen linear algebra appl 266 1997 261 theorem 2 1 there exists a real parameter 0 max 2 j n j such that the problem has a solution if and only if 1 0 in particular we discuss how to compute 0 and the solution matrix a for certain class of matrices a sufficient condition for the problem to have a solution is also derived

nonnegative matrices with prescribed extremal singular values we consider the problem of constructing nonnegative matrices with prescribed extremal singular values in particular given 2n 1 real numbers s 1 j and s j j j 1 n we construct an nxn nonnegative bidiagonal matrix b and an nxn nonnegative semi bordered diagonal matrix c such that s 1 j and s j j are respectively the minimal and the maximal singular values of certain submatrices b j and c j of b and c respectively by using a singular value perturbation result we also construct an nxn nonnegative matrix with prescribed singular values s 1 s n

on the long time stability of the implicit euler scheme for the two dimensional navier stokes equations in this paper we study the stability for all positive time of the fully implicit euler scheme for the two dimensional navier stokes equations more precisely we consider the time discretization scheme and with the aid of the discrete gronwall lemma and the discrete uniform gronwall lemma we prove that the numerical scheme is stable

realizable lists via the spectra of block matrices abstract in this paper we present spectral results for matrices partitioned into higher order blocks where not all blocks are necessarily square using these results sufficient conditions on a given list to be the list of eigenvalues of a nonnegative matrix are obtained and the corresponding matrix is constructed in particular spectral results for symmetric matrices are derived a result on guo perturbations for permutative matrices on given lists is generalized

realizable lists via the spectra of structured matrices abstract a square matrix of order n with n 2 is called a permutative matrix or permutative when all its rows up to the first one are permutations of precisely its first row in this paper the spectra of a class of permutative matrices are studied in particular spectral results for matrices partitioned into 2 by 2 symmetric blocks are presented and using these results sufficient conditions on a given list to be the list of eigenvalues of a nonnegative permutative matrix are obtained and the corresponding permutative matrices are constructed guo perturbations on given lists are exhibited

beta expansions natural extensions and multiple tilings associated with pisot units from the works of rauzy and thurston we know how to construct multiple tilings of some euclidean space using the conjugates of a pisot unit beta and the greedy beta transformation in this paper we consider different transformations generating expansions in base beta including cases where the associated subshift is not sofic under certain mild conditions we show that they give multiple tilings we also give a necessary and sufficient condition for the tiling property generalizing the weak finiteness property w for greedy beta expansions remarkably the symmetric beta transformation does not satisfy this condition when beta is the smallest pisot number or the tribonacci number this means that the pisot conjecture on tilings cannot be extended to the symmetric beta transformation closely related to these multiple tilings are natural extensions of the transformations which have many nice properties they are invariant under the lebesgue measure under certain conditions they provide markov partitions of the torus they characterize the numbers with purely periodic expansion and they allow determining any digit in an expansion without knowing the other digits

an algorithm for constructing nonnegative matrices with prescribed real eigenvalues provided with the real spectrum this paper presents a numerical procedure based on the induction principle to solve two types of inverse eigenvalue problems one for nonnegative matrices and another for symmetric nonnegative matrices as an immediate application our approach can offer not only the sufficient condition to solve inverse eigenvalue problems for nonnegative or symmetric nonnegative matrices but also a numerical way to solve inverse eigenvalue problems for stochastic matrices numerical examples are presented to show the capacity of our method

on the universal realizability problem abstract we say that a list 1 2 n of complex numbers is realizable if it is the spectrum of an entrywise nonnegative matrix a we say that is universally realizable ur if is realizable for each possible jordan canonical form allowed by the universal realizability problem of spectra urp has been solved for certain kind of real and complex spectra mainly spectra in the left half plane in this paper we look for extending the kind of spectra that are ur by given new sufficient conditions for the urp to have a solution in particular we show that some realizability criteria for the real and symmetric nonnegative inverse eigenvalue problem are realizability criteria for the urp as well

on the nonnegative eigenvalue problem abstract 1 1 n is the spectrum of a stochastic matrix of order n 1 if and only if there exists some real matrix b of order n with spectrum 1 n and some n simplex s r n containing the origin such that bs s this result appears implicitly in ciarlet s work and it provides us with a geometrical tool to obtain sufficient conditions for the nonnegative eigenvalue problem we employ it here to generalize the sufficient conditions given by kellogg

the nonnegative inverse eigenvalue problem abstract let 1 n be a list of complex numbers the nonnegative inverse eigenvalue problem niep is the problem of determining necessary and sufficient conditions in order that be the spectrum of an entrywise nonnegative n n matrix our purpose is to give an overview of the niep its history modern results and subproblems with particular emphasis on symmetric realizing matrices a substantial bibliography is included

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

on nonnegative matrices similar to positive matrices abstract let nn denote the set of those 1 2 n c n such that there exists a nonnegative matrix with perron root equal to one and spectrum 1 2 n we prove that nn is star shaped with respect to 1 0 0 and that 1 2 n n n is on the boundary of nn if and only if 1 2 n is not the spectrum of any positive matrix as a consequence attention is given to the problem of determining which nonnegative matrices are similar to positive ones more precisely we address the question of which pattern matrices p satisfy that any nonnegative matrix with pattern p is similar to a positive matrix some partial results are obtained among them that any irreducible nonnegative matrix with a positive line is similar to a positive matrix which allow us to give a complete solution to the case of 3 by 3 matrices

the real nonnegative inverse eigenvalue problem is np hard abstract a list of complex numbers is realizable if it is the spectrum of a nonnegative matrix in 1949 suleimanova posed the nonnegative inverse eigenvalue problem niep the problem of determining which lists of complex numbers are realizable the version for reals of the niep rniep asks for realizable lists of real numbers in the literature there are many sufficient conditions or criteria for lists of real numbers to be realizable we will present an unified account of these criteria then we will see that the decision problem associated to the rniep is np hard and we will study the complexity for the decision problems associated to known criteria

long time stability of a classical efficient scheme for two dimensional navier stokes equations this paper considers the long time stability property of a popular semi implicit scheme for the two dimensional incompressible navier stokes equations in a periodic box that treats the viscous term implicitly and the nonlinear advection term explicitly we consider both the semidiscrete discrete in time but continuous in space and fully discrete schemes with either fourier galerkin spectral or fourier pseudospectral collocation methods we prove that in all cases the scheme is long time stable provided that the timestep is sufficiently small the long time stability in the l 2 and h 1 norms further leads to the convergence of the global attractors and invariant measures of the scheme to those of the navier stokes equations at vanishing timestep

on the comparison of some realizability criteria for the real nonnegative inverse eigenvalue problem abstract a result by brauer shows how to modify one single eigenvalue of a matrix via a rank one perturbation without changing any of the remaining eigenvalues this together with the properties of real matrices with constant row sums was exploited by the authors in a previous work in connection with the nonnegative inverse eigenvalue problem obtaining conditions which are sufficient for the existence of an entrywise nonnegative matrix with prescribed spectrum in this work we make use of brauer s theorem again to show that most of the previous results giving sufficient conditions for the real nonnegative inverse eigenvalue problem can be derived by using brauer s theorem moreover the technique is constructive and there is an algorithmic procedure to construct a matrix realizing the spectrum in particular we show that if either kellogg s realizability criterion or borobia s realizability criterion is satisfied then soto s realizability criterion is also satisfied none of the converses are true thus the condition given by soto appears to be the most general sufficient condition so far for the real nonnegative inverse eigenvalue problem

isospectral flow method for nonnegative inverse eigenvalue problem with prescribed structure the nonnegative inverse eigenvalue problem is that given a family of complex numbers l l 1 l n find a nonnegative matrix of order n with spectrum l this problem is difficult and remains unsolved partially in this paper we focus on its generalization that the reconstructed nonnegative matrices should have some prescribed entries it is easy to see that this new problem will come back to the common nonnegative inverse eigenvalue problem if there is no constraint of the locations of entries a numerical isospectral flow method which is developed by hybridizing the optimization theory and steepest descent method is used to study the reconstruction moreover an error estimate of the numerical iteration for ordinary differential equations on the matrix manifold is presented after that a numerical method for the nonnegative symmetric inverse eigenvalue problem with prescribed entries and its error estimate are considered finally the approaches are verified by the numerical test results

an inverse eigenvalue problem for symmetric and normal matrices abstract some new sufficient conditions are found for n real numbers to be the spectrum of some n n symmetric nonnegative matrix and for n complex numbers to be the spectrum of some n n normal nonnegative matrix

on universal realizability of spectra abstract a list 1 2 n of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix the list is said to be universally realizable ur if it is the spectrum of a nonnegative matrix for each possible jordan canonical form allowed by it is well known that an n n nonnegative matrix a is co spectral to a nonnegative matrix b with constant row sums in this paper we extend the co spectrality between a and b to a similarity between a and b when the perron eigenvalue is simple we also show that if 0 and 1 2 n is ur then 1 2 n is also ur we give counter examples for the cases 1 2 n is ur implies 1 2 3 n is ur and 1 2 are ur implies 1 2 is ur

the maximum of the minimal multiplicity of eigenvalues of symmetric matrices whose pattern is constrained by a graph abstract in this paper we introduce a parameter mm g defined as the maximum over the minimal multiplicities of eigenvalues among all symmetric matrices corresponding to a graph g we develop basic properties of mm g and compute it for several families of graphs

connecting sufficient conditions for the symmetric nonnegative inverse eigenvalue problem abstract we say that a list of real numbers is symmetrically realisable if it is the spectrum of some entrywise nonnegative symmetric matrix the symmetric nonnegative inverse eigenvalue problem sniep is the problem of characterising all symmetrically realisable lists in this paper we present a recursive method for constructing symmetrically realisable lists the properties of the realisable family we obtain allow us to make several novel connections between a number of sufficient conditions developed over forty years starting with the work of fiedler in 1974 we show that essentially all previously known sufficient conditions are either contained in or equivalent to the family we are introducing

an efficient second order in time scheme for approximating long time statistical properties of the two dimensional navier stokes equations we investigate the long time behavior of the following efficient second order in time scheme for the 2d navier stokes equations in a periodic box begin array ll frac 3 omega n 1 4 omega n omega n 1 2k nabla perp 2 psi n psi n 1 cdot nabla 2 omega n omega n 1 nu delta omega n 1 f n 1 quad delta psi n omega n end array the scheme is a combination of a 2nd order in time backward differentiation and a particular explicit adams bashforth treatment of the advection term therefore only a linear constant coefficient poisson solver is needed at each time step we prove uniform in time bounds on this scheme in dot l 2 dot h 1 per and dot h 2 per provided that the time step is sufficiently small these time uniform estimates further lead to the convergence of long time statistics stationary statistical properties of the scheme to that of the nse itself at vanishing time step

solving inverse eigenvalue problems via householder and rank one matrices abstract a brief and practical algorithm is introduced to solve symmetric inverse eigenvalue problems which we call hrou algorithm the algorithm is based on householder transformations and rank one updating we give some basic properties and the computational amount and develop sensitivity analysis of hrou algorithm furthermore we develop hrou algorithm into a multi level and adaptive one named mlahrou to solve symmetric nonnegative inverse eigenvalue problems new sufficient conditions to ensure symmetric nonnegative matrices and symmetric m matrices are given many numerical examples are given to verify our theory compare with existing results and show the efficiency of our algorithms

spectra of principal submatrices of nonnegative matrices abstract let a be an n n nonnegative matrix with the spectrum 1 2 n and let a 1 be an m m principal submatrix of a with the spectrum 1 2 m in this paper we present some cases where the realizability of 1 2 m 1 2 s implies the realizability of 1 2 n 1 2 s and consider the question whether this holds in general in particular we show that the list 1 2 n 1 2 m is realizable

graphs that allow all the eigenvalue multiplicities to be even abstract let g be an undirected graph on n vertices and let s g be the set of all n n real symmetric matrices whose nonzero off diagonal entries occur in exactly the positions corresponding to the edges of g the inverse eigenvalue problem for a graph g is a problem of determining all possible lists that can occur as the lists of eigenvalues of matrices in s g this question is in general hard to answer and several variations were studied most notably the minimum rank problem in this paper we introduce the problem of determining for which graphs g there exists a matrix in s g whose characteristic polynomial is a square i e the multiplicities of all its eigenvalues are even we solve this question for several families of graphs

quasi periodic β expansions and cut languages abstract motivated by the analysis of neural net models between integer and rational weights we introduce a so called cut language over a real digit alphabet which contains finite expansions i e base representations of the numbers less than a given threshold we say that an infinite expansion is eventually quasi periodic if its tail sequence formed by the numbers whose representations are obtained by removing leading digits contains an infinite constant subsequence we prove that a cut language is regular iff its threshold is a quasi periodic number whose all expansions are eventually quasi periodic by showing that altogether they have a finite number of tail values for algebraic bases we prove that there is an eventually quasi periodic expansion with an infinite number of tail values iff there is a conjugate of on the unit circle for transcendental combined with algebraic digits a expansion is eventually quasi periodic iff it has a finite number of tail values for a pisot base and digits from the smallest field extension q over rational numbers including we show that any number from q is quasi periodic in addition we achieve a dichotomy that a cut language is either regular or non context free and we show that any cut language with rational parameters is context sensitive

unconditional stability and long term behavior of transient algorithms for the incompressible navier stokes and euler equations abstract this paper examines the long term behavior dissipativity and unconditional nonlinear stability properties of time integration algorithms for the incompressible navier stokes equations including both direct schemes and fractional step projection methods the algorithms are termed nonlinear unconditionally stable if in the absence of a forcing term but for arbitrary initial conditions the computed kinetic energy decreases for arbitrary step sizes i e l 2 stability such an a priori stability estimate is a characteristic feature of the navier stokes system similarly the algorithms are said to exhibit asymptotic t independent long term dissipative behavior if as the continuum flow generated by the navier stokes equations the algorithmic flow also exhibits an absorbing set and a maximal attractor both direct and fractional step methods are described which inherit these fundamental properties of the navier stokes system for any time step size moreover a subclass of algorithms which retain these strong notions of nonlinear stability and long term dissipative behavior is identified which in addition has the remarkable property of being linear within the time step the key implication is that no iterations are required to compute the solution within a time step a specific member of this class is shown to possess the same absorbing set as the continuum navier stokes system a second order accurate unconditionally stable method is also identified which retains the property of linearity within a time step numerical analysis and computational aspects involved in the implementation of these methods are addressed in detail and illustrated in representative numerical simulations

construction of nonnegative symmetric matrices with given spectrum abstract let 1 n be the spectrum of a nonnegative symmetric matrix a with the perron eigenvalue 1 a diagonal entry c and let 1 m be the spectrum of a nonnegative symmetric matrix b with the perron eigenvalue 1 we show how to construct a nonnegative symmetric matrix c with the spectrum 1 max 0 1 c 2 n 2 m

applications of a brauer theorem in the nonnegative inverse eigenvalue problem abstract a result by brauer which shows how to modify one single eigenvalue of a matrix without changing any of the remaining eigenvalues plays a relevant role in the study of the nonnegative inverse eigenvalue problem niep perfect in a long time ignored paper 1955 presents an extension of this result which shows how to modify r eigenvalues of a matrix of order n r n via a rank r perturbation without changing any of the n r remaining eigenvalues by using this extension perfect gives a realizability criterion for the real niep which is not contained in soto s realizability criterion in this work by extending perfect s result we give a new realizability criterion for the real niep which contains soto s criterion thus this new realizability criterion appears to be the most general sufficient condition for the real niep so far we also contribute to the solution of the symmetric niep for n 5

divergence free finite elements on tetrahedral grids for k 6 it was shown two decades ago that the p k p k 1 mixed element on triangular grids approximating the velocity by the continuous p k piecewise polynomials and the pressure by the discontinuous p k 1 piecewise polynomials is stable for all k 4 provided the grids are free of a nearly singular vertex the problem with the method in 3d was posted then and remains open the problem is solved partially in this work it is shown that the p k p k 1 element is stable and of optimal order in approximation on a family of uniform tetrahedral grids for all k 6 the analysis is to be generalized to non uniform grids when we can deal with the complicity of 3d geometry for the divergence free elements the finite element spaces for the pressure can be avoided in computation if a classic iterated penalty method is applied the finite element solutions for the pressure are computed as byproducts from the iterate solutions for the velocity numerical tests are provided

guo perturbation for symmetric nonnegative circulant matrices abstract guo w guo eigenvalues of nonnegative matrices linear algebra appl 266 1997 261 270 sets the question if the list 1 2 n is symmetrically realizable that is is the spectrum of a symmetric nonnegative matrix and t 0 whether or not the list t 1 t 2 t 3 n is also symmetrically realizable in this paper we give an affirmative answer to this question in the case that the realizing matrix is circulant or left circulant we also give a necessary and sufficient condition for to be the spectrum of a nonnegative left circulant matrix

an inverse eigenvalue problem for nonnegative matrices abstract this paper gives a solution to an inverse eigenvalue problem of nonnegative matrices which was proposed by miroslav fiedler

a characterization of trace zero bisymmetric nonnegative 5 5 matrices abstract let 1 2 3 4 5 1 be real numbers such that i 1 5 i 0 in 16 o spector proved that a necessary and sufficient condition for 1 2 3 4 5 to be the eigenvalues of a traceless symmetric nonnegative 5 5 matrix is 2 5 0 and i 1 5 i 3 0 in this article we show that this condition is also a necessary and sufficient condition for 1 2 3 4 5 to be the spectrum of a traceless bisymmetric nonnegative 5 5 matrix

long term stability estimates and existence of a global attractor in a finite element approximation of the navier stokes equations with numerical subgrid scale modeling variational multiscale methods lead to stable finite element approximations of the navier stokes equations dealing with both the indefinite nature of the system pressure stability and the velocity stability loss for high reynolds numbers these methods enrich the galerkin formulation with a subgrid component that is modeled in fact the effect of the subgrid scale on the captured scales has been proved to dissipate the proper amount of energy needed to approximate the correct energy spectrum thus they also act as effective large eddy simulation turbulence models and allow one to compute flows without the need to capture all the scales in the system in this article we consider a dynamic subgrid model that enforces the subgrid component to be orthogonal to the finite element space in the l 2 sense we analyze the long term behavior of the algorithm proving the existence of appropriate absorbing sets and a compact global attractor the improvements with respect to a finite element galerkin approximation are the long term estimates for the subgrid component which are translated to effective pressure and velocity stability thus the stabilization introduced by the subgrid model into the finite element problem does not deteriorate for infinite time intervals of computation

finite element approximation of the nonstationary navier stokes problem part ii stability of solutions and error estimates uniform in time in this paper assumptions about the stability of a solution are introduced in the numerical analysis of the nonstationary navier stokes problem for the purpose of extending local a priori error estimates and local a posteriori error estimates globally in time