homological and homotopical dehn functions are different the homological and homotopical dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space the homological dehn function measures fillings of cycles by chains whereas the homotopical dehn function measures fillings of curves by disks because the two definitions involve different sorts of boundaries and fillings there is no a priori relationship between the two functions however before this work there were no known examples of finitely presented groups for which the two functions differ this paper gives such examples constructed by amalgamating a free by cyclic group with several bestvina brady groups

sign pattern matrices that admit p0 matrices abstract for sign patterns corresponding to directed or undirected cycles we identify conditions under which the patterns admit or require p 0 matrices

on inverse positivity of sub direct sums of matrices abstract in this note the authors consider the problem of inverse positivity of k subdirect sum of matrices the main results provide a solution to an open problem posed recently

sign patterns of inverse nonnegative matrices abstract we determine here the 0 sign patterns which occur among the inverses of nonsingular entrywise nonnegative matrices these results complete a sequence of work which began with the characterization of all possible sign patterns which occur among inverses of positive matrices and the characterization in this general case is more involved to state than the earlier ones

a new characterization of inverse positive matrices abstract the author provides a new characterization of inverse positive matrices using positive splittings it is shown that a matrix m is inverse positive iff the spectral radius of all positive splittings is less than one then a single positive splitting is found called a b splitting such that m is inverse positive iff it allows for a b splitting satisfying the spectral radius bound

volumes of balls in large riemannian manifolds we prove two lower bounds for the volumes of balls in a riemannian manifold if m n g is a complete riemannian manifold with lling radius at least r then it contains a ball of radius r and volume at least n r n if m n hyp is a closed hyperbolic manifold and if g is another metric on m with volume no greater than n vol m hyp then the universal cover of m g contains a unit ball with volume greater than the volume of a unit ball in hyperbolic n space

free abelian covers short loops stable length and systolic inequalities we explore the geometry of the abel jacobi map f from a closed orientable riemannian manifold x to its jacobi torus h 1 x mathbb r h 1 x mathbb z mathbb r applying m gromov s filling inequality to the typical fiber of f we prove an interpolating inequality for two flavors of shortest length invariants of loops the procedure works provided the lift of the fiber is non trivial in the homology of the maximal free abelian cover overline x classified by f we show that the finite dimensionality of the rational homology of overline x is a sufficient condition for the homological non triviality of the fiber when applied to nilmanifolds our fiberwise inequality typically gives stronger information than the filling inequality for x itself in dimension 3 we present a sufficient non vanishing condition in terms of massey products this condition holds for certain manifolds that do not fiber over their jacobi torus such as 0 framed surgeries on suitable links our systolic inequality applies to surface bundles over the circle provided the algebraic monodromy has 1 dimensional coinvariants even though the massey product invariant vanishes for some of these bundles

flexural vibration of axially loaded beams with linear or parabolic taper abstract curves are presented which enable the first five natural frequencies to be found for axially loaded tapered members with an important family of cross sections previous curves have only included axial forces in a very limited number of cases whereas here axial forces within the range of greatest practical importance can always be allowed for the present curves cover 11 combinations of end conditions three types of taper all taper ratios between 0 2 and 1 and all axial forces between p c and 0 6 p c where p c is the critical buckling load for pure compression and can be found from curves provided a list of principal nomenclature is given in appendix 2 the cross sections covered include thin circular ones of constant thickness and thin walled cross sections consisting of constant thickness flat plates which all have their breadths tapering in the same way truncated wedges of varying depth and constant width are also covered the taper is linear between the ends or linear between a maximum value at the centre and equal values at the ends or parabolic with a maximum value at the centre simple examples show how the curves can be used to obtain natural frequencies or buckling loads to an accuracy which is almost always about 1 the curves also illustrate how natural frequencies and buckling loads of thin walled members are altered by changing the amount and type of a member s taper while keeping its wall thickness and mass constant finally it is shown that for the ranges covered by the curves frequencies which separate the first five natural frequencies can be found a priori the theory used to obtain the curves was simply to divide the tapered member into sufficient uniform members to ensure convergence to the tapered result to better than plotting accuracy exact bernoulli euler dynamic stiffnesses which allowed for axial force effects were used for these uniform components in appendix 1 the effects of using exact timoshenko stiffnesses instead of the bernoulli euler ones are illustrated

a pinching theorem for homotopy spheres the length excess of a triangle measures how much the triangle inequality fails to be an equality this notion was first studied seriously in ag we say that a bounded metric space x x d has excess 5 e if there are points p q e x such that d p x d x q d p q e for all x ex the excess e x of x is the infimum of all e 0 where x has excess e notice that the standard spheres have excess o in general a closed riemannian manifold m has excess 0 if and only if there is a point p e m whose cutlocus c p is a single point q em such a manifold is clearly a twisted sphere i e it is the union of two discs glued together along their common boundary conversely according to a result of weinstein b appendix c 4 any twisted sphere carries a riemannian metric for which there is a point whose cut locus is another point with this in mind we are interested in closed riemannian manifolds with small excess however without any further geometric restrictions small excess has no topological significance see problem 8 though in fact any closed n manifold m can be given a riemannian metric withdiam m 1c and arbitrarily small excess simply make sure that all topological complexities of m i e the complement of a topological n disc are contained in a sufficiently small metric ball and what remains is metrically close to the complement of a small ball in the unit sphere sn 1 our main result is theorem a given k e r d v 0 and an integer n 2 there is an t e n k d v such that any closed riemannian n mani old m with sectional curvature km k diamm d volm v and e m e is a homotopy sphere

free vibration analysis of a cylindrical shell circular plate system with general coupling and various boundary conditions abstract the free vibration of a structure consisting of a finite circular cylindrical shell closed at one end by a circular plate is analyzed in this paper emphasis is given to the characterization of structural coupling and boundary conditions these are incorporated into the model by means of continuous distributions of springs along the shell and the plate interface a general formulation based on a variational principle is used this formulation allows a wide spectrum of boundary conditions and coupling conditions between the shell and the plate an issue the importance of which is clearly shown by a literature review very good accuracy of the method is demonstrated by solving test problems for lower order modes of the plate and of the shell for which some results are available in the literature comparisons are also made using finite element analysis on a plate ended shell showing that the proposed approach is a convenient effecient and accurate one for determining the modal behavior of such a complex structural system other numerical results are then presented for a shell rigidly attached to a plate to illustrate the coupling phenomena between the shell and the plate it is shown that there exist three different types of modes for this combined structure they are termed plate controlled modes shell controlled modes and strongly coupled modes respectively it is also shown that each type is closely related to the modal character of each of uncoupled elements

hard and soft packing radius theorems singular spaces are playing an increasingly important role in riemannian geometry in particular this is true of the so called alexandrov spaces i e finite hausdorff dimensional complete inner metric spaces with a lower curvature bound in the local triangle comparison sense interest in alexandrov spaces is largely explained by the fact that the natural process of taking gromov hausdorff limits is closed in alexandrov geometry but not in riemannian geometry this simple observation has become a powerful tool in riemannian geometry see for example fy gp2 or gpw and pe in this paper we use a new means to obtain results in riemannian geometry by studying alexandrov spaces the operation of forming spherical suspensions of positively curved spaces an operation which is also closed in alexandrov geometry but not in riemannian geometry will be applied to prove a differentiable sphere theorem which is unlike all previous solutions to the following basic problem

on the use of orthogonal polynomials in the rayleigh ritz method for the study of the flexural vibration and buckling of isotropic and orthotropic rectangular plates series composed of recently proposed orthogonal polynomial functions are used in the rayleigh ritz method to generate results for a number of flexural vibration and buckling problems for rectangular isotropic and orthotropic plates it is demonstrated that the approach yields excellent results not only for natural frequencies and buckling loads but also for the more sensitive mode shape bending moment and shear force distribution problem

the free vibration of a type of tapered beams this paper studies the vibrational characteristics of tapered beams with continuously varying rectangular cross section of depth and breadth proportional to xs and xt respectively where both s and t are arbitrary real numbers for a truncated beam and arbitrary positive numbers for a sharp ended beam and x is the axial co ordinate measured from the sharp end of the beam the bernoulli euler theory of bending is used to describe the motion of the beam a new set of beam functions are developed as the admissible functions which are the complete solution of the tapered beam under an arbitrary static load expanded into a taylor series the eigenfrequency equation is obtained by the rayleigh ritz method the accuracy is assured from the convergency and comparison studies the effect of the location of the taylor series expanding point on the convergency is discussed the analysis shows that the present approach is convergent for arbitrary truncation factor by taking the midpoint of the beam as the expanding point of the taylor series numerical results are tabulated for three different tapered beams with various boundary conditions and truncation factors it is shown that the eigenfrequencies can be obtained with high accuracy by using only a small number of terms of the static beam functions 2000 elsevier science s a all rights reserved

free vibration analysis of a type of tapered beams by using adomian decomposition method abstract the adomian decomposition method adm is employed in this paper to investigate the free vibrations of tapered euler bernoulli beams with a continuously exponential variation of width and a constant thickness along the length under various boundary conditions based on adm the governing differential equation for the tapered beam becomes a recursive algebraic equation by using the boundary condition equations the dimensionless natural frequencies and corresponding mode shapes can be easily obtained simultaneously the computed results for different boundary conditions and non uniformity ratios are presented the accuracy is assured from the convergence and comparison published results

free vibration analysis of a uniform cantilever beam with point masses by an analytical and numerical combined method abstract the natural frequencies and mode shapes of a uniform cantilever beam carrying any number of concentrated masses were determined by using an analytical and numerical combined method anc method the eigenvalue equation was first derived analytically by using the expansion theorem and then the eigenvalues and eigenvectors were calculated numerically in comparison with the general finite element method fem the anc method has the advantage that there is no necessity to derive the property matrices of each beam element and then to develop the overall ones of the entire beam in comparison with the transfer matrix method tmm the anc method requires no tedious matrix multiplication so that some computer time may be saved the anc method is also better than the pure analytical closed form method since instead of a few special cases a uniform beam carrying any number of point masses of various magnitudes or distributions along the beam length can be easily solved besides by using the anc method there is no difficulty in taking more modes in the mode superposition equation and hence more higher mode natural frequencies of better accuracy may be obtained

sub direct sums and positivity classes of matrices abstract it is well known that a direct sum is positive semidefinite if and only if each of the direct summands is positive semidefinite in fact it is also known that this statement remains true if positive semidefinite is replaced with doubly nonnegative completely positive totally nonnegative m matrix and p matrix etc for each of these classes we consider corresponding questions for a more general sum of two matrices of which the direct sum and ordinary sum are special cases

on complete manifolds of nonnegative kth ricci curvature in this paper we establish some vanishing and finiteness theorems for the topological type of complete open riemannian manifolds under certain positivity conditions for curvature key tools are comparison techniques and morse theory of busemann and distance functions

on the filling radius of positively curved alexandrov spaces it was shown by f wilhelm that gromov s filling radius of any positively curved closed riemannian manifolds are less than that of the round sphere unless they are isometric to each other in this short paper we adapt his proof to see that the same is true for any positively curved closed alexandrov spaces as well

on the eigenvalues of some tridiagonal matrices a solution is given for a problem on eigenvalues of some symmetric tridiagonal matrices suggested by william trench the method presented can be generalizable to other problems

flowers on riemannian manifolds in this paper we will present two upper bounds for the length of a smallest flower shaped geodesic net in terms of the volume and the diameter of a manifold minimal geodesic nets are critical points of the length functional on the space of graphs immersed into a riemannian manifold let mn be a closed riemannian manifold of dimension n we prove that there exists a minimal geodesic net that consists of one vertex and at most 2n 1 geodesic loops based at that vertex of total length 2n d where d is the diameter of mn we also show that there exists a minimal geodesic net that consists of one vertex and at most 3 n 1 2 loops of total length leq2 n 1 2 3 n 1 3 fill rad m n leq2 n 1 frac 5 2 3 n 1 3 n 1 n n vol m n frac 1 n where fill rad mn denotes the filling radius and vol mn denotes the volume of mn

characterizations of sign patterns of inverse positive matrices abstract a result of johnson leighton and robinson characterizing sign patterns of real matrices with nonzero entries whose inverses are entrywise positive is generalized the restriction to matrices with nonzero entries is removed and additional five equivalent conditions are established one of them using a graph theoretical approach expedites a simple criterion for recognition of such sign patterns

further flexural vibration curves for axially loaded beams with linear or parabolic taper curves are presented which enable the first five natural frequencies to be found for axially loaded tapered members with an important family of cross sections these cross sections complement those of an earlier paper such that the curves in the two papers cover all of the most common types of tapered member the curves presented cover eleven combinations of end conditions three types of taper all taper ratios between 0 3 and 1 and all axial forces between pc and 0 6pc where pc is the critical buckling load for pure compression and can be found from curves provided the results cover truncated wedges which have their depth and width varying identically so that all cross sections are similar in the sense that similar triangles are the cross sections covered also include thin circular ones with thickness proportional to diameter and thin walled cross sections consisting of flat plates which all have their breadths and thicknesses tapering identically the taper is linear between the ends or linear between a maximum value at the centre and equal values at the ends or parabolic with a maximum value at the centre the curves can be used to obtain natural frequencies or buckling loads to an accuracy which is almost always about 1 the curves also illustrate how natural frequencies and buckling loads are altered by changing the amount and type of a member s taper while keeping its mass constant finally it is hown that for the ranges covered by the curves frequencies which separate the first five natural frequencies can be found a priori the theory used to obtain the curves was simply to divide the tapered member into sufficient uniform members to ensure convergence to the tapered result to better than plotting accuracy exact bernoulli euler dynamic stiffnesses which allowed for axial force effects were used for these uniform components illustrative results show the effects of using exact timoshenko stiffnesses instead of the bernoulli euler ones

a note on the vibrations of restrained beams and rods with point masses abstract this paper is concerned with the approximate determination of the fundamental frequency and first mode shape of a beam or rod to which springs and point masses are attached after formulating the general eigenvalue problem which forms the basis of the analysis the theory is applied to three examples and it is shown that the results are quite satisfactory

sign pattern matrices that admit m n p or inverse m matrices in this paper we identify the sign pattern matrices that occur among the n matrices the p matrices and the m matrices we also address to the class of inverse m matrices and the related admissibility of sign pattern matrices problem

gromov hyperbolic spaces and the sharp isoperimetric constant in this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is gromov hyperbolic as a particular consequence we obtain that euclidean space is a borderline case for gromov hyperbolicity in terms of the isoperimetric function we prove similar results for the linear filling radius inequality our results strengthen and generalize theorems of gromov papasoglu and others

inverse h matrices abstract the notion of inverse m matrices has been quite well studied and many of their properties have been documented in the literature in this article a proposal to consider the larger class of inverse h matrices is made first certain special types of real matrices are identified as inverse h matrices while these matrices are not inverse m matrices in general their definitions are inspired by matrices that are known to be inverse m matrices in the rest of the article motivated by results that hold for inverse m matrices the endeavor will be to attempt at proving corresponding results for inverse h matrices it is shown that for the full class of inverse h matrices many of these extensions are false nevertheless two subclasses of real inverse h matrices are identified where these generalizations are shown to hold

sign properties of metzler matrices with applications abstract several results about sign properties of metzler matrices are obtained it is first established that checking the sign stability of a metzler sign matrix can be either characterized in terms of the hurwitz stability of the unit sign matrix in the corresponding qualitative class or in terms the negativity of the diagonal elements of the metzler sign matrix and the acyclicity of the associated directed graph similar results are obtained for the case of metzler block matrices and metzler mixed matrices the latter being a class of metzler matrices containing both sign and real type entries the problem of assessing the sign stability of the convex hull of a finite and summable family of metzler matrices is also solved and a necessary and sufficient condition for the existence of common lyapunov functions for all the matrices in the convex hull is obtained the concept of sign stability is then generalized to the concept of ker b sign stability a problem that arises in the analysis of certain jump markov processes a sufficient condition for the ker b sign stability of metzler sign matrices is obtained and formulated using inverses of sign matrices and the concept of l matrices several applications of the results are discussed in the last section

convergence theory of restricted multiplicative schwarz methods convergence results for the restricted multiplicative schwarz rms method the multiplicative version of the restricted additive schwarz ras method for the solution of linear systems of the form ax b are provided an algebraic approach is used to prove convergence results for nonsymmetric m matrices several comparison theorems are also established these theorems compare the asymptotic rate of convergence with respect to the amount of overlap the exactness of the subdomain solver and the number of domains moreover comparison theorems are given between the rms and ras methods as well as between the rms and the classical multiplicative schwarz method

on the analysis of laterally vibrating slender beams subject to various complicating effects the rayleigh ritz method with orthogonally generated polynomials as admissible functions is used to obtain a simple analysis of the free vibration problem of slender beams subject to various complicating effects these effects are the presence of concentrated masses and or concentrated moments of inertia intermediate supports axial loading and non uniformity of cross section such may exist separately or in any combination numerical results are presented for several example beams illustrating the accuracy and versatility of the approach

geodesic loops and periodic geodesics on a riemannian manifold diffeomorphic to s 3 let m n be a closed 2 connected riemannian manifold such that 3 m n 0 in this paper we prove that either there exists a periodic geodesic on m n of length 6d where d is the diameter of m n or at each point p m n there exists a geodesic loop of length 2d

n matrix completion problem abstract an n n matrix is called an n matrix if all principal minors are negative in this paper we are interested in n matrix completion problems that is when a partial n matrix has an n matrix completion in general a combinatorially or non combinatorially symmetric partial n matrix does not have an n matrix completion here we prove that a combinatorially symmetric partial n matrix has an n matrix completion if the graph of its specified entries is a 1 chordal graph we also prove that there exists an n matrix completion for a partial n matrix whose associated graph is an undirected cycle

exact bernoulli euler dynamic stiffness matrix for a range of tapered beams bernoulli euler theory and bessel functions are used to obtain explicit expressions for the exact dynamic stiffnesses for axial torsional and flexural vibrations of any beam which is tapered such that a varies as yn and gj and i both vary as y n 2 where a gj and i have their usual meanings y cx l 1 c is a constant such that c 1 l is the length of the beam and x is the distance from one end of the beam numerical checks give better than seven figure agreement with the stiffnesses obtained by extrapolation from stepped beams with 400 and 500 uniform elements a procedure is given for calculating the number of natural frequencies exceeded by any trial frequency when the ends of the member are clamped this enables an existing algorithm to be used to obtain the natural frequencies of structures which contain tapered members

on the free transverse vibration of annular and circular thin sectorial plates subject to certain complicating effects abstract the rayleigh ritz method with orthogonally generated polynomials as admissible functions is used to obtain an eigenvalue equation for the free transverse vibration of thin annular sectorial plates circular sectorial plates are treated with this equation by permitting the inner radius to become negligibly small the plates may be of isotropic or polar orthotropic material of uniform or radially and or circumferentially varying thickness and may include interior circumferential or radial line supports numerical values for the natural frequency parameters of several different plates are presented and compared with previously published values where available

transverse vibrations of a rotating uniform cantilever beam with tip mass as predicted by using beam characteristic orthogonal polynomials in the rayleigh ritz method abstract natural frequencies and mode shapes of a rotating uniform cantilever beam with a tip mass are studied by using beam characteristic orthogonal polynomials in the rayleigh ritz method the set of orthogonal polynomials which satisfy the geometrical boundary conditions are generated by using the gram schmidt process the results are compared with those obtained by the myklestad method the extended galerkin method and finite element methods the variation of natural frequencies with the speed of rotation is plotted for several parameter combinations such as setting angle tip mass moment of inertia of tip mass etc mode shapes at different rotational speeds are also plotted use of orthogonal polynomials for the deflection shapes enables the computation of higher natural frequencies of any order to be accomplished without facing any numerical difficulties which is not the case when arbitrary polynomial expressions are used

a note on transverse vibrations of continuous beams subject to an axial force and carrying concentrated masses abstract polynomial co ordinate functions are used in order to obtain an approximate solution to the title problem the algorithmic procedure is quite simple leads to adequate engineering accuracy and can be easily extended to the case of several intermediate supports

volume growth of some uniformly contractible riemannian manifolds it is shown that if a uniformly contractible riemannian n manifold m g is k quasi isometric to an n dimensional normed space v n cdot k 1 then liminf r rightarrow infty frac vol g ball r r n omega n geq frac 1 k 2n where n is the volume of the unit euclidean ball in particular if m is uniformly contractible and d gh m d g v n cdot infty then m has at least euclidean volume growth this corollary covers an earlier result by burago and ivanov our results are motivated by a volume growth theorem contained in gromov s book gromov in progress in mathematics vol 152 birkhauser boston 1999 p 256 we give a detailed proof of this theorem using the same argument we also derive a generalization of the theorem which is pointed out by gromov

vibration of an euler bernoulli beam of constant depth and with linearly varying breadth abstract the transverse vibration of a homogeneous euler bernoulli beam of constant depth and linearly varying breadth is investigated in this paper a direct analytical solution is presented for the mode shape equation the constraints imposed by the boundary conditions yield the frequency equation the four ideal boundary conditions considered are clamped pinned sliding and free for a complete beam sharp ended it is a necessary condition that the sharp end remain free and as a result the frequency equations for clamped pinned sliding or free boundary conditions at the large end are the determinant of a 2 2 matrix equated to zero a truncated beam is considered as part of a hypothetical complete beam the frequency equation for truncated beam is the determinant of a 4 4 matrix equated to zero the roots of the frequency equations are determined by an iterative procedure the first four natural frequencies of complete beam and the first three frequencies of truncated beams for 16 combinations of the ideal boundary conditions for various truncation factors are presented the results are benchmarks and are therefore presented in tabular form to preserve the accuracy to six figures

on the lateral vibration of thin annular and circular composite plates subject to certain complicating effects abstract series comprising orthogonally generated polynomial functions are used in the rayleigh ritz method to obtain an eigenvalue equation for the flexural vibration of thin flat annular and circular plates the plates may be of isotropic or polar orthotropic material of uniform or radially varying thickness and may pass over intermediate concentric ring supports numerical results are presented for a number of plates illustrating the versatility and accuracy of the approach

two positive nontrivial solutions for a class of semilinear elliptic variational systems abstract we obtain existence and localization results of positive nontrivial solutions for a class of semilinear elliptic variational systems the proof is based on variants of schechter s localized critical point theorems for hilbert spaces not identified to their duals and on the technique of inverse positive matrices the leray schauder boundary condition is also involved

classical jacobi polynomials closed form solutions for transverse vibrations this paper deals with transverse vibrations of nonuniform homogeneous beams and plates classes of beams and axisymmetrical circular plates whose boundary value problems of free transverse vibrations and free transverse axisymmetrical vibrations respectively can be reduced to an eigenvalue singular problem singularities occur at both ends of orthogonal polynomials are reported exact natural frequencies and jacobi polynomials as exact mode shapes which result directly from eigenvalues and eigenfunctions of eigenvalue singular problems of classical orthogonal polynomials are reported for these classes the above classes of beams and plates hereafter called jacobi classes are given by geometry and boundary conditions the geometry consists of parabolic thickness variation with respect to the axial coordinate for beams and with respect to the radius for plates beams belonging to this class have either one or two sharp ends singularities along with certain boundary conditions plates have zero thickness at zero and outer radii the boundary value problems associated with plates and beams of two sharp ends are free boundary problems two other boundary value problems hinged free and sliding free are reported for beams with one sharp end also exact natural frequencies and mode shapes for uniformly rotating beams with hinged free boundary are reported

inverse m matrices abstract this is an attempt at a comprehensive expository study of those nonnegative matrices which happen to be inverses of m matrices and is aimed at an audience conversant with basic ideas of matrix theory a theme is the parallels anddifferences between the class of m matrices and the class of inverse m matrices among the primary tools used are diagonal multiplications the neumann expansion and the form of the inverse of a partitioned matrix some of the results seem not to have appeared before and several unsolved problems are mentioned

systoles des surfaces plates singulieres de genre deux the extremal metrics for the isosystolic problem on surfaces of genus two are studied in this paper we show that contrary to e calabi s conjecture in genus three no flat metric with conical singularities is extremal for this problem in genus two

some inheritance properties for complementary basic matrices abstract we say that the product of a row vector and a column vector is intrinsic if there is at most one non zero product of corresponding coordinates analogously we speak about intrinsic product of two or more matrices as well as about intrinsic factorizations of matrices since all entries of the intrinsic product are products of entries of the multiplied matrices there is no addition the class of complementary basic matrices cb matrices was recently introduced as matrices if of order n a g i 1 g i 2 g i n 1 where i 1 i 2 i n 1 is a permutation of 1 2 n 1 and the matrices g k k 1 n 1 have the form g k i k 1 c k i n k 1 for some 2 2 matrices c k it was observed that 1 independently of the permutation all such matrices with given c i s have the same spectrum though they do not form a similarity class 2 the classical companion matrix belongs to the class of cb matrices m fiedler a note on companion matrices linear algebra appl 372 2003 325 331 9 3 the multiplication of the g i s is intrinsic we explore connections between the 2 2 matrices c k and the resulting cb matrix g k in particular which properties are inherited from the c k to the g k we consider two situations for the ordinary real cb matrices and for the corresponding sign pattern matrices

on complete manifolds with nonnegative ricci curvature complete open riemannian manifolds mn g with nonnegative sectional curvature are well understood the basic results are toponogov s splitting theorem and the soul theorem cg1 the splitting theorem has been extended to manifolds of nonnegative ricci curvature cg2 on the other hand the soul theorem does not extend even topologically according to recent examples in gm2 a different method to construct manifolds which carry a metric with ric 0 but no metric with nonnegative sectional curvature has been given by l berard bergery bb this leads to the question cf also y1 is there any finiteness result for complete riemannian manifolds with ric 0 the answer is certainly affirmative in the low dimensional special cases n 2 where all notions of curvature coincide and n 3 where nonnegative ricci curvature has been studied by means of stable minimal surfaces msy sy on the other hand j p sha and d g yang shy have constructed complete manifolds with strictly positive ricci curvature in higher dimensions for example they can choose the underlying space to be r4 x s3 with infinitely many copies of s3 x cp 2 attached to it by surgery cf also shy 1 it is therefore clear that any finiteness result for arbitrary dimensions requires additional assumptions the purpose of this paper is to establish the following main result

some results about inverse positive matrices abstract a nonsingular real matrix a is said to be inverse positive if all the elements of its inverse are nonnegative this class of matrices contains the m matrices from which inherit some of their properties and applications especially in economy and in the description of iterative methods for solving nonlinear systems in this paper we present some new characterizations for inverse positive matrices and we analyze when this concept is preserved by the sub direct sum of matrices

matrices with zero line sums and maximal rank abstract an n n sign pattern admits a matrix with zero line sums and rank n 1 if and only if it is fully indecomposable and every arc of an associated directed bipartite graph lies on a circuit this proves a conjecture of fiedler and grone made in the study of sign patterns of inverse positive matrices

collapsing and pinching under a lower curvature bound in this paper we are concerned with collapsing phenomena and pinching problems of riemannian manifolds whose sectional curvatures are uniformly bounded from below for a positive integer n and for d 0 let x be the set of compact riemannian n manifolds m with sectional curvatures km 1 and diameters diam m d by the weak compactness theorem of gromov glp x is relatively compact in the set of compact metric spaces with respect to the hausdorff distance furthermore gromov proved in g2 that the sum of betti numbers of any element in x is uniformly bounded in terms of the given constants these results suggest that x is an object of study on which we could develop some geometry and topology in fact when volumes are uniformly bounded away from zero the following results are known let v be the subset of x with volume vol m v grove and petersen gp1 proved that the set of homotopy types of elements in 4 v is finite related results are in y2 for pinching problems otsu shiohama and the author osy y3 obtained some differential sphere theorems in the class x v according to a recent paper gpw if n 3 4 then v contains at most finitely many diffeomorphism types let mi i 1 2 be a convergent sequence in a and x be the limit of them we say that mi collapses to x if the hausdorff dimension of x is less than n by the volume condition no collapsing occurs in 4 v from this point of view it is quite natural to ask what collapsing phenomena occur and what types of pinching theorems hold in the class x in this situation it seems difficult to determine the singularities of x some examples are given in section 1 in this paper we study collapsing phenomena in the case when x is a riemannian manifold and establish a pinching theorem for a riemannian manifold m we denote by inj m the injectivity radius of m when n is written in place of x and the metric of n is normalized our

volume of minimal hypersurfaces in manifolds with nonnegative ricci curvature we establish a min max estimate on the volume width of a closed riemannian manifold with nonnegative ricci curvature more precisely we show that every closed riemannian manifold with nonnegative ricci curvature admits a pl morse function whose level set volume is bounded in terms of the volume of the manifold as a consequence of this sweep out estimate there exists an embedded closed possibly singular minimal hypersurface whose volume is bounded in terms of the volume of the manifold

on the spread of positively curved alexandrov spaces it was proved by f wilhelm that gromov s filling radius of closed positively curved manifolds with a uniform lower bound on sectional curvature attains the maximum with the round sphere recently the author proved that this is also the case for closed finite dimensional alexandrov spaces with a positive lower curvature bound these were proved as a corollary of a comparison theorem for the invariant called spread of those spaces in this paper we extend the latter result to infinite dimensional alexandrov spaces

lower bounds on the energy of maps on demontre que pour la metrique standard sur rp n l application identite est minimisante de l energie dans sa classe d homotopie