lower bounds for the perron root of a nonnegative matrix abstract new lower bounds frequently better than previous bounds and readily computable for the perron root r a of a nonnegative matrix a are obtained the theoretical comparison of new bounds with the known ones is supplemented with two numerical examples

existence of maximal surfaces in asymptotically flat spacetimes we prove the existence of maximal surfaces in asymptotically flat spacetime satisfying an interior condition this uses a priori estimates which can also be applied to prescribed mean curvature surfaces in cosmological spacetimes and the dirichlet problem

a sequence of lower bounds for the spectral radius of nonnegatfve matrices abstract we present a sequence of progressively better lower bounds for the s pectral radius of a nonnegative matrix each element of the sequence is a function of the spectral radius of the geometric symmetrization of a power of the matrix as defined by schwenk each element provides a better approximation to the spectral radius than the elements in the sequences provided by other authors

existence and extendibility of rotationally symmetric graphs with a prescribed higher mean curvature function in euclidean and minkowski spaces abstract in this paper we investigate the existence of rotationally symmetric entire graphs resp entire spacelike graphs with prescribed k th mean curvature function in euclidean space r n 1 resp minkowski spacetime l n 1 as a previous step we analyze the associated homogeneous dirichlet problem on a ball which is not elliptic for k 1 and then we prove that it is possible to extend the solutions moreover a sufficient condition for uniqueness is given in both cases

multiple positive radial solutions for a dirichlet problem involving the mean curvature operator in minkowski space abstract we study the dirichlet problem with mean curvature operator in minkowski space div v 1 v 2 x v q 0 in b r v 0 on b r where 0 is a parameter q 1 r 0 0 r is continuous strictly positive on 0 and b r x r n x r using upper and lower solutions and leray schauder degree type arguments we prove that there exists 0 such that the problem has zero at least one or at least two positive radial solutions according to 0 or moreover is strictly decreasing with respect to r

a lower bound for the spectral radius of a digraph abstract we show that the spectral radius d of a digraph d with n vertices and c 2 closed walks of length 2 satisfies d c 2 n moreover equality occurs if and only if d is the symmetric digraph associated to a c 2 n regular graph plus some arcs that do not belong to cycles as an application of this result we construct new sharp upper bounds for the low energy of a digraph which extends koolen and moulton bounds of the energy to digraphs

positive solutions of the dirichlet problem for the prescribed mean curvature equation in minkowski space abstract we prove the existence of multiple positive solutions of the dirichlet problem for the prescribed mean curvature equation in minkowski space div u 1 u 2 f x u u in u 0 on here is a bounded regular domain in r n and the function f f x s is either sublinear or superlinear or sub superlinear near s 0 the proof combines topological and variational methods

on the speed rate of convergence of solutions to conservation laws with nonlinear diffusions abstract in this paper we analyze the long time behavior of solutions to conservation laws with nonlinear diffusion terms of different types saturating diffusions monotone and non monotone and singular nonlinear diffusions are considered in particular the cases of mean curvature type diffusions both in the euclidean space and in lorentz minkowski space enter in our framework after dealing with existence and stability of monotone steady states in a bounded interval of the real line with dirichlet boundary conditions we discuss the speed rate of convergence to the asymptotic limit as t finally in the particular case of a burgers flux function we show that the solutions exhibit the phenomenon of metastability

sharp bounds for the largest eigenvalue of the signless laplacian of a graph let g be a simple connected graph with n vertices and m edges denote the degree of vertex vi by d vi the matrix q g d g a g is called the signless laplacian of g where d g diag d v1 d v2 d vn and a g denote the diagonal matrix of vertex degrees and the adjacency matrix of g respectively let q1 g be the largest eigenvalue of q g in this paper we first present two sharp upper bounds for q1 g involving the maximum degree and the minimum degree of the vertices of g and give a new proving method on another sharp upper bound for q1 g then we present three sharp lower bounds for q1 g involving the maximum degree and the minimum degree of the vertices of g moreover we determine all extremal graphs which attain these sharp bounds

nested bounds for the perron root of a nonnegative matrix abstract it is well known that for a nonnegative matrix a the smallest row sum r a and the largest row sum r a provide lower and upper bounds respectively for the perron root of a these bounds are generalized for a partitioned nonnegative matrix a the new bounds are better than r a and r a and they can be further improved by a refinement of the partition known monotonicity and convergence properties of r a and r a are generalized for the new bounds

the convexity of the level sets of maximal strictly space like hypersurfaces defined on 2 dimensional space forms abstract for the maximal strictly space like hypersurface defined on 2 dimensional space forms we derive some geometrical properties including the regularity and the strict convexity and the curvature description of the level sets

nonconstant positive radial solutions for neumann problem involving the mean extrinsic curvature operator abstract let b be the unit ball in r n with n 2 let f c 1 0 r f 0 0 f for some 0 f s s for s 0 f s s for s and f k r where k r is the k th radial eigenvalue of i in the unit ball with neumann boundary condition we use the unilateral global bifurcation theorem to show the existence of nonconstant positive radial solutions of the quasilinear neumann problem div u 1 u 2 u f u in b u 0 on b

holomorphic representation of constant mean curvature surfaces in minkowski space consequences of non compactness in loop group methods abstract we give an infinite dimensional generalized weierstrass representation for spacelike constant mean curvature cmc surfaces in minkowski 3 space r 2 1 the formulation is analogous to that given by dorfmeister pedit and wu for cmc surfaces in euclidean space replacing the group su 2 with su 1 1 the non compactness of the latter group however means that the iwasawa decomposition of the loop group used to construct the surfaces is not global we prove that it is defined on an open dense subset after doubling the size of the real form su 1 1 and prove several results concerning the behavior of the surface as the boundary of this open set is encountered we then use the generalized weierstrass representation to create and classify new examples of spacelike cmc surfaces in r 2 1 in particular we classify surfaces of revolution and surfaces with screw motion symmetry as well as studying another class of surfaces for which the metric is rotationally invariant

the dirichlet problem for monge ampere equations in non convex domains and spacelike hypersurfaces of constant gauss curvature in this paper we extend the well known results on the existence and regularity of solutions of the dirichlet problem for monge ampere equations in a strictly convex domain to an arbitrary smooth bounded domain in r as well as in a general riemannian manifold we prove for the nondegenerate case that a sufficient and necessary condition for the classical solvability is the existence of a subsolution for the totally degenerate case we show that the solution is in c11 q if the given boundary data extends to a locally strictly convex c2 function ori q as an application we prove some existence results for spacelike hypersurfaces of constant gauss kronecker curvature in minkowski space spanning a prescribed boundary

positive rotationally symmetric solutions for a dirichlet problem involving the higher mean curvature operator in minkowski space abstract in this paper we investigate the global behavior of positive rotationally symmetric solutions for dirichlet problem involving the k th mean curvature operator in minkowski space l n 1 which is not elliptic for k 1 the proofs of our main results are based upon bifurcation techniques

entire spacelike translating solitons in minkowski space abstract in this paper we study entire spacelike translating solitons in minkowski space by constructing convex spacelike solutions to 1 3 in bounded convex domains we obtain many entire smooth convex strictly spacelike translating solitons by prescribing boundary data at infinity

convexity of the solutions to the constant mean curvature spacelike surface equation in the lorentz minkowski space abstract we prove that a spacelike graph of constant mean curvature h 0 in the 3 dimensional lorentz minkowski space over a bounded domain with pseudo elliptic boundary is strictly convex by a pseudo elliptic curve we mean a closed and planar curve which intersects any branch of any hyperbola at most at five points we also provide an example that shows that we cannot remove the assumption on the boundary being a pseudo elliptic curve

bifurcation of nodal radial solutions for a prescribed mean curvature problem on an exterior domain abstract we investigate the existence and multiplicity of nodal radial solutions for a prescribed mean curvature problem on the exterior domain of a ball using the global bifurcation theory moreover we establish the asymptotic behavior of the solutions on the subcontinua which bifurcate from a trivial branch all results are obtained by considering a radially equivalent second order problem with a singular weight

parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes spacelike hypersurfaces of prescribed mean curvature in cosmological spacetimes are constructed as asymptotic limits of a geometric evolution equation in particular an alternative constructive proof is given for the existence of maximal and constant mean curvature slices

the fuzzy saw and fuzzy topsis procedures based on ordered fuzzy numbers a new fuzzy simple additive weighting technique fsaw based on ordered fuzzy numbers is presented a new fuzzy topsis ftopsis based on ordered fuzzy numbers is presented the fuzzy saw and fuzzy topsis methods effectively handles the lack of precision in the available information and gives better results the fuzzy saw and fuzzy topsis methods can deal with the representation of the information in a more direct and adequate way when we are unable to express it precisely by a score from an ordinal scale the fuzzy saw and fuzzy topsis are rational and understandable and operates on linguistic values which are sometimes more adequate in decision making in this paper two new techniques linguistic fuzzy simple additive weighting fsaw and linguistic fuzzy technique for order of preference by similarity to the ideal solution ftopsis based on ordered fuzzy numbers ofn are presented although the linguistic fuzzy approach is discussed in the literature there is no research on the applications of ofn to fuzzy multi criteria decision making problems the proposed methods extend the fsaw and ftopsis techniques since decision data are often vague and imprecise linguistic variables represented by ordered fuzzy numbers are used to rate the alternatives with respect to each criterion we have provided the theoretical foundations of this approach as well as numerical examples to illustrate its applications the evaluations of alternatives are given by linguistic expressions in which linguistic terms have to be quantified on the previously determined scale of values in particular we focus on the problem of extending the scale of values used in evaluating the criteria to take into account intermediate values such as more than2 or less than3 together with 2 and 3

a lower bound for the perron root of a nonnegative matrix ii abstract this paper is a continuation of an earlier one by the author we obtain a new lower bound for the perron root r a of a nonnegative n n matrix a we show that for a with at least one principal submatrix of order two with different diagonal entries and with at least one positive off diagonal entry the bound is at least as good as the one from 7 and we determine a class of nonnegative matrices for which it is essentially better we also show that not every nonnegative square matrix is similar to a nonnegative square matrix with identical diagonal entries

entire spacelike hypersurfaces of constant gaus curvature in minkowski space we prove existence and stability of smooth entire strictly convex spacelike hypersurfaces of prescribed gaus curvature in minkowski space the proof is based on barrier constructions and local a priori estimates

the best possible lower bound for the perron root using traces abstract let a be an n n matrix with real eigenvalues wolkowicz and styan presented bounds for the eigenvalues using only n tr a and tr a 2 we show that their lower bound for the largest eigenvalue works also as a lower bound for the perron root of a if a is nonnegative and its eigenvalues are not necessarily real we also show that this bound is optimal under certain conditions finally we solve completely the problem to find the optimal lower bound for the perron root using only n tr a and tr a 2

global structure of radial positive solutions for a prescribed mean curvature problem in a ball abstract in this paper we are concerned with the global structure of radial positive solutions of boundary value problem div n v f x v 0 in b r v 0 on b r where n y y 1 y 2 y r n is a positive parameter b r x r n x r and denotes the euclidean norm in r n all results depending on the behavior of nonlinear term f near 0 are obtained by using global bifurcation techniques

moving centre monotonicity formulae for minimal submanifolds and related equations abstract monotonicity formulae play a crucial role for many geometric pdes especially for their regularity theories for minimal submanifolds in a euclidean ball the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball then its area is at least that of the equatorial disk recently brendle and hung proved a sharp area bound for minimal submanifolds when the prescribed point is not the centre of the ball which resolved a conjecture of alexander hoffman and osserman their proof involves asymptotic analysis of an ingeniously chosen vector field and the divergence theorem in this article we prove a sharp moving centre monotonicity formula for minimal submanifolds which implies the aforementioned area bound we also describe similar moving centre monotonicity formulae for stationary p harmonic maps mean curvature flow and the harmonic map heat flow

positive periodic solutions to an indefinite minkowski curvature equation abstract we investigate the existence non existence multiplicity of positive periodic solutions both harmonic i e t periodic and subharmonic i e kt periodic for some integer k 2 to the equation u 1 u 2 a t g u 0 where 0 is a parameter a t is a t periodic sign changing weight function and g 0 0 is a continuous function having superlinear growth at zero in particular we prove that for both g u u p with p 1 and g u u p 1 u p q with 0 q 1 p the equation has no positive t periodic solutions for close to zero and two positive t periodic solutions a small one and a large one for large enough moreover in both cases the small t periodic solution is surrounded by a family of positive subharmonic solutions with arbitrarily large minimal period the proof of the existence of t periodic solutions relies on a recent extension of mawhin s coincidence degree theory for locally compact operators in product of banach spaces while subharmonic solutions are found by an application of the poincare birkhoff fixed point theorem after a careful asymptotic analysis of the t periodic solutions for

classification and evolution of bifurcation curves for the one dimensional minkowski curvature problem and its applications abstract in this paper we study the classification and evolution of bifurcation curves of positive solutions for the one dimensional minkowski curvature problem u 1 u 2 f u in l l u l u l 0 where l 0 f c 0 c 2 0 and f u 0 for u 0 furthermore we show that for sufficiently large l 0 the bifurcation curve s l may have arbitrarily many turning points finally we apply these results to obtain the global bifurcation diagrams for ambrosetti brezis cerami problem liouville bratu gelfand problem and perturbed gelfand problem with the minkowski curvature operator respectively moreover we will make two lists which show the different properties of bifurcation curves for minkowski curvature problems corresponding semilinear problems and corresponding prescribed curvature problems

tight bounds on the spectral radius of asymmetric nonnegative matrices abstract for an arbitrary asymmetric nonnegative n n matrix a we identify a pair of symmetric matrices whose largest eigenvalues bound the spectral radius of a furthermore we show that these bounding matrices are best possible by characterizing matrices a which attain equality with either the upper or the lower bounding matrix the lower bound may be extended to some matrices with negative entries provided they have no negative cycles

global bifurcation of solutions of the mean curvature spacelike equation in certain friedmann lemaitre robertson walker spacetimes abstract we study the existence of spacelike graphs for the prescribed mean curvature equation in the friedmann lemaitre robertson walker flrw spacetime by using a conformal change of variable this problem is translated into an equivalent problem in the lorentz minkowski spacetime then by using rabinowitz s global bifurcation method we obtain the existence and multiplicity of positive solutions for this equation with 0 dirichlet boundary condition on a ball moreover the global structure of the positive solution set is studied

prescribed mean curvature graphs with neumann boundary conditions in some flrw spacetimes abstract we identify a family of friedmann lemaitre robertson walker flrw spacetimes such that the radially symmetric prescribed curvature problem with neumann boundary condition is solvable on a ball of small radius such family contains some examples of interest in cosmology

light likeness of boundaries of extremal surfaces of mixed type in physical space times abstract in this paper we prove the light likeness of boundaries of smooth extremal surfaces of mixed type in general physical space time r 1 n n 1 in particular we improve gu s theorem on the light likeness of boundaries of extremal surfaces in r 1 2 and prove the light likeness of boundaries of smooth extremal surfaces of mixed type in general physical space times as a consequence we show that a curve moving in a physical space time keeps its like property and the boundary only exists when its world sheet at the initial time has light like points this implies that any extremal surface of mixed type is generated by an initial curve of mixed type

on maximal surfaces in asymptotically flat space times existenc of maximal and almost maximal hypersurfaces in asymptotically flat space times is established under boundary conditions weaker than those considered previously we show in particular that every vacuum evolution of asymptotically flat data for the einstein equations can be foliated by slices maximal outside a spatially compact set and that every strictly stationary asymptotically flat space time can be foliated by maximal hypersurfaces amongst other uniqueness results we show that maximal hypersurfaces can be used to partially fix an asymptotic poincare group

a decreasing sequence of upper bounds for the perron root abstract we obtain a decreasing sequence of upper bounds for the perron root of a nonnegative matrix these upper bounds are new and readily computable for any nonnegative matrix with zero trace the mentioned sequence converges to the perron root we apply these results to the adjacency matrix of a graph

perron root bounding for nonnegative persymmetric matrices the eigenvalue problem for a symmetric persymmetric matrix can be reduced to two symmetric eigenvalue problems of lower order in this paper we find in which of these problems the perron root of a nonnegative symmetric persymmetric matrix lies this is applied to bound the perron root of such class of matrices

on the uniqueness of the helicoid and enneper s surface in the lorentz minkowski space mathbb r _ 1 3

periodic maximal surfaces in the lorentz minkowski space mathbb l 3 a maximal surface mathcal s with isolated singularities in a complete flat lorentzian 3 manifold mathcal n is said to be entire if it lifts to a periodic entire multigraph sim mathcal s in mathbb l 3 in addition mathcal s is called of finite type if it has finite topology finitely many singular points and sim mathcal s is a finitely sheeted multigraph complete or proper maximal immersions with isolated singularities in mathcal n are entire and entire embedded maximal surfaces in mathcal n with a finite number of singularities are of finite type we classify complete flat lorentzian 3 manifolds carrying entire maximal surfaces of finite type and deal with the topology weierstrass representation and asymptotic behavior of this kind of surfaces finally we construct new examples of periodic entire embedded maximal surfaces in mathbb l 3 with fundamental piece having finitely many singularities

maximal surfaces and the universal teichmuller space we show that any element of the universal teichmuller space is realized by a unique minimal lagrangian diffeomorphism from the hyperbolic plane to itself the proof uses maximal surfaces in the 3 dimensional anti de sitter space we show that in adsn 1 any subset e of the boundary at infinity which is the boundary at infinity of a space like hypersurface bounds a maximal space like hypersurface in ads3 if e is the graph of a quasi symmetric homeomorphism then this maximal surface is unique and it has negative sectional curvature as a by product we find a simple characterization of quasi symmetric homeomorphisms of the circle in terms of 3 dimensional projective geometry

rotationally symmetric spacelike translating solitons for the mean curvature flow in minkowski space abstract the rotationally symmetric spacelike translating solitons for the mcf in the minkowski space are considered in this paper to be specific the complete classification for such translating solitons according to their rotation axes which are timelike spacelike and lightlike is provided in particular the explicit parametrization of the rotationally symmetric spacelike hypersurface of the lightlike rotation axis with the prescribed mean curvature is obtained

on the umbilicity of complete constant mean curvature spacelike hypersurfaces in this paper we show that a complete spacelike hypersurface immersed with constant mean curvature either in the de sitter space or in the anti de sitter space must be totally umbilical provided that its gauss mapping has some suitable behavior in particular we use an extension of hopf s maximum principle due to yau in order to give new positive answers to goddard s conjecture

spacelike hypersurfaces with prescribed boundary values and mean curvature we consider the boundary value problem for the mean curvature operator in minkowski space and give necessary and sufficient conditions for the existence of smooth strictly spacelike solutions our main results hold for non constant mean curvature and make no assumptions about the smoothness of the boundary or boundary data

positive radial solutions for dirichlet problems with mean curvature operators in minkowski space abstract in this paper by using leray schauder degree arguments and critical point theory for convex lower semicontinuous perturbations of c 1 functionals we obtain existence of classical positive radial solutions for dirichlet problems of type div v 1 v 2 f x v 0 in b r v 0 on b r here b r x r n x r and f 0 r 0 r is a continuous function which is positive on 0 r 0

positive radial solutions for multiparameter dirichlet systems with mean curvature operator in minkowski space and lane emden type nonlinearities abstract we deal with dirichlet systems involving the mean curvature operator in minkowski space m w div w 1 w 2 in a ball in r n using the fixed point index and the lower and upper solutions method we first obtain the existence of positive solutions for a class of differential systems with a singular laplacian subjected to homogeneous mixed boundary conditions the main application of the general results concerns dirichlet systems with lane emden type nonlinearities in a superlinear case depending on two parameters 1 2 so we prove that for such a system there exists a continuous curve which separates the first quadrant in two disjoint unbounded open sets o 1 and o 2 such that the system has zero at least one or at least two radial positive solutions according to 1 2 o 1 1 2 or 1 2 o 2

harmonic maps and classical surface theory in minkowski 3 space harmonic maps from a surface s with nondegenerate prescribed and induced metrics are characterized showing that holomorphic quadratic differentials play the same role for harmonic maps from a surface with indefinite prescribed metric as they do in the riemannian case moreover holomorphic quadratic differentials are shown to arise as naturallv on surfaces of constant h or k in m3 as on their counterparts in e3 the connection between the sine gordon sinh gordon and cosh gordon equations and harmonic maps is explained various local and global results are established for surfaces in m3 with constant h or constant k 0 in particular the gauss map of a spacelike or timelike surface in m3 is shown to be harmonic if and only if h is constant also k is shown to assume values arbitrarilv close to h2 on any entire spacelike surface in m3 with constant h except on a hyperbolic cylinder

inequalities for the perron root related to levinger s theorem abstract for the perron roots of square nonnegative matrices a b and a d 1btd where d is a diagonal matrix with positive diagonal entries the inequality a d 1btd a b is proved under the assumption that a and b have a common unordered pair of nonorthogonal right and left perron vectors the case of equality is analyzed the above inequality generalizes the inequality a 1 bt a 1 b proved under stronger assumptions by bapat and implies a generalization of levinger s theorem on the monotonicity of the perron root of a weighted arithmetic mean of a nonnegative matrix and its transpose also for the perron root a c d 1atd c c 1 0 c of a weighted entrywise geometric mean of a and d 1atd where a ij and o denotes the hadamard product the monotonicity property dual to that asserted by generalized levinger s theorem is established

the space of complete embedded maximal surfaces with isolated singularities in the 3 dimensional lorentz minkowski space we prove that a complete embedded maximal surface in 3 dx12 dx22 dx32 with a finite number of singularities is an entire maximal graph with conelike singularities over any spacelike plane and so it is asymptotic to a spacelike plane or a half catenoid we show that the moduli space of entire maximal graphs over x3 0 in with n 1 2 singular points and vertical limit normal vector at infinity is a 3n 4 dimensional differentiable manifold the convergence in means the one of conformal structures and weierstrass data and it is equivalent to the uniform convergence of graphs on compact subsets of x3 0 moreover the position of the singular points in 3 and the logarithmic growth at infinity can be used as global analytical coordinates with the same underlying topology we also introduce the moduli space of marked graphs with n 1 singular points a mark in a graph is an ordering of its singularities which is a n 1 sheeted covering of we prove that identifying marked graphs differing by translations rotations about a vertical axis homotheties or symmetries about a horizontal plane the corresponding quotient space is an analytic manifold of dimension 3n 1 this manifold can be identified with a spinorial bundle associated to the moduli space of weierstrass data of graphs in

extensions of the topsis for group decision making under fuzzy environment the aim of this paper is to extend the topsis to the fuzzy environment owing to vague concepts frequently represented in decision data the crisp value are inadequate to model real life situations in this paper the rating of each alternative and the weight of each criterion are described by linguistic terms which can be expressed in triangular fuzzy numbers then a vertex method is proposed to calculate the distance between two triangular fuzzy numbers according to the concept of the topsis a closeness coefficient is defined to determine the ranking order of all alternatives by calculating the distances to both the fuzzy positive ideal solution fpis and fuzzy negative ideal solution fnis simultaneously finally an example is shown to highlight the procedure of the proposed method at the end of this paper

some monotonicity properties of schur powers of matrices and related inequalities abstract a series of inequalities are developed relating the spectral radius a b of the schur product a b of two nonnegative matrices a and b with those of a a and b b yielding a b a a b b 1 2 as a corollary it is proved that the spectral radius of the schur powers r a r a r a a a r factors satisfies 1 r log r is decreasing while 1 r 1 log r is increasing the latter provided a is a stochastic matrix the entropy of a finite stationary markov chain is identified with d r dr r 1 a number of majorization comparisons for the spectral radius of schur powers is given

on the electrostatic born infeld equation with extended charges in this paper we deal with the electrostatic born infeld equation left begin array ll operatorname div left displaystyle frac nabla phi sqrt 1 nabla phi 2 right rho quad in mathbb r n displaystyle lim x to infty phi x 0 end array right quad quad quad quad mathcal bi where rho is an assigned extended charge density we are interested in the existence and uniqueness of the potential phi and finiteness of the energy of the electrostatic field nabla phi we first relax the problem and treat it with the direct method of the calculus of variations for a broad class of charge densities assuming rho is radially distributed we recover the weak formulation of mathcal bi and the regularity of the solution of the poisson equation under the same smoothness assumptions in the case of a locally bounded charge we also recover the weak formulation without assuming any symmetry the solution is even classical if rho is smooth then we analyze the case where the density rho is a superposition of point charges and discuss the results in kiessling commun math phys 314 509 523 2012 other models are discussed as for instance a system arising from the coupling of the nonlinear klein gordon equation with the born infeld theory

inequalities of rayleigh quotients and bounds on the spectral radius of nonnegative symmetric matrices abstract given a square nonnegative symmetric matrix a the rayleigh quotient of a nonnegative vector u under a is given by q a u u t au u t u we show that qa u au is not less than qa u where denotes coordinatewise square roots and is the hadamard product but that qa au may be smaller than qa u further we examine issues of convergence

a fuzzy multi criteria decision making model by associating technique for order preference by similarity to ideal solution with relative preference relation generally classical multi criteria decision making mcdm methods were extended to encompass uncertainty and vagueness of messages under fuzzy environment for solving decision making problems especially for technique for order preference by similarity to ideal solution topsis in the fuzzy extension of topsis fuzzy numbers comparison and aggregation based on fuzzy preference relation are important issues to compute distance values between alternatives and ideal or anti ideal solution or rank feasible alternatives because lots of messages are reserved by fuzzy preference relation however fuzzy preference relation on pair wise comparison is commonly too complex to calculate to avoid the drawback we use a relative preference relation improved from fuzzy preference relation in the fuzzy extension of topsis for computing distance values between alternatives and ideal or anti ideal solution or obtaining relative closeness coefficients of alternatives thus the relative preference relation on fuzzy numbers will be associated with topsis under fuzzy environment to develop a fuzzy multi criteria decision making fmcdm model through the association above fmcdm problems can be easily solved by the model further we compare the proposed model with other methods to demonstrate the model s feasibility and rationality

bounds for the spectral radius and the largest singular value abstract we derive an increasing sequence of lower bounds for the spectral radius of a matrix with real spectrum and progressively improved bounds for the largest singular value of a complex matrix we also find estimates for the rank of normal matrices with real spectrum and for the rank of normal nonnegative matrices including some sufficient condition for such matrices to be invertible

adding one handle to half plane layers in this paper we build properly embedded singly periodic minimal surfaces which have infinite total curvature in the quotient by the period these surfaces are constructed by adding a handle to the toroidal half plane layers defined by h karcher the technics that use is to solve a jenkins serrin problem over a strip domain and to consider the conjugate minimal surface to the graph