an explicit two step method exact for the scalar test equation y λy an explicit two step method exact for the scalar test equation y y re 0 is presented in this paper it is exponentially fitted l stable thus a stable and of order 2 with a new set of vector computations we also extend directly the method to systems of ordinary differential equations the numerical experiments demonstrate that this explicit two step method is suitable for stiff systems

a mass transportation approach to quantitative isoperimetric inequalities a sharp quantitative version of the anisotropic isoperimetric inequality is established corresponding to a stability estimate for the wulff shape of a given surface tension energy this is achieved by exploiting mass transportation theory especially gromov s proof of the isoperimetric inequality and the brenier mccann theorem a sharp quantitative version of the brunn minkowski inequality for convex sets is proved as a corollary

robust dimension free isoperimetry in gaussian space we prove the first robust dimension free isoperimetric result for the standard gaussian measure n and the corresponding boundary measure n in rn the main result in the theory of gaussian isoperimetry proven in the 1970s by sudakov and tsirelson and independently by borell states that if n a 1 2 then the surface area of a is bounded by the surface area of a half space with the same measure n a 2 1 2 our results imply in particular that if a rn satisfies n a 1 2 and n a 2 1 2 then there exists a half space b rn such that n a b clog 1 2 1 for an absolute constant c since the gaussian isoperimetric result was established only recently a robust version of the gaussian isoperimetric result was obtained by cianchi et al who showed that n a b c n for some function c n with no effective bounds compared to the results of cianchi et al our results have optimal i e no dependence on the dimension but worse dependence on

the sharp quantitative isoperimetric inequality a quantitative sharp form of the classical isoperimetric inequality is proved thus giving a positive answer to a conjecture by hall

a two sided estimate for the gaussian noise stability deficit the gaussian noise stability of a set a subset mathbb r n is defined by begin aligned mathcal s rho a mathbb p left x in a y in a right end aligned where x y are standard jointly gaussian vectors satisfying mathbb e x i y j delta ij rho borell s inequality states that for all 0 rho 1 among all sets a subset mathbb r n with a given gaussian measure the quantity mathcal s rho a is maximized when a is a half space we give a novel short proof of this fact based on stochastic calculus moreover we prove an almost tight two sided dimension free robustness estimate for this inequality by introducing a new metric to measure the distance between the set a and its corresponding half space h namely the distance between the two centroids we show that the deficit mathcal s rho h mathcal s rho a can be controlled from both below and above by essentially the same function of the distance up to logarithmic factors as a consequence we also establish the conjectured exponent in the robustness estimate proven by mossel neeman which uses the total variation distance as a metric in the limit rho rightarrow 1 we obtain an improved dimension free robustness bound for the gaussian isoperimetric inequality our estimates are also valid for a generalized version of stability where more than two correlated vectors are considered

sharp dimension free quantitative estimates for the gaussian isoperimetric inequality we provide a full quantitative version of the gaussian isoperimetric inequality the difference between the gaussian perimeter of a given set and a half space with the same mass controls the gap between the norms of the corresponding barycenters in particular it controls the gaussian measure of the symmetric difference between the set and the half space oriented so to have the barycenter in the same direction of the set our estimate is independent of the dimension sharp on the decay rate with respect to the gap and with optimal dependence on the mass

a two step explicit p stable method of high phase lag order for linear periodic ivps in this paper we present a nonlinear two step explicit p stable method of fourth algebraic order and 12th phase lag order for solving one dimensional second order linear periodic initial value problems ivps of ordinary differential equations based on a special vector arithmetic with respect to an analytic function the method can be extended to be vector applicable for multi dimensional problems directly some numerical results are reported to illustrate the efficiency of the method

faber krahn inequalities in sharp quantitative form the classical faber krahn inequality asserts that balls uniquely minimize the first eigenvalue of the dirichlet laplacian among sets with given volume in this paper we prove a sharp quantitative enhancement of this result thus confirming a conjecture by nadirashvili and bhattacharya weitsman more generally the result applies to every optimal poincare sobolev constant for the embeddings w 1 2 0 omega hookrightarrow l q omega

a two step explicit p stable method of high phase lag order for second order ivps this paper presents a two step explicit p stable method of high phase lag order for solving second order ordinary differential equations basing on a special vector operation with respect to a smooth function the method can be extended to be vector applicable in multi dimensional space

a linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing a discrete event system is a system whose behavior can be described by means of a set of time consuming activities performed according to a prescribed ordering events correspond to starting or ending some activity an analogy between linear systems and a class of discrete event systems is developed following this analogy such discrete event systems can be viewed as linear in the sense of an appropriate algebra the periodical behavior of closed discrete event systems i e involving a set of repeatedly performed activities can be totally characterized by solving an eigenvalue and eigenvector equation in this algebra this problem is numerically solved by an efficient algorithm which basically consists of finding the shortest paths from one node to all other nodes in a graph the potentiality of this approach for the performance evaluation of flexible manufacturing systems is emphasized the case of a flowshop like production process is analyzed in detail

supertropical matrix algebra iii powers of matrices and their supertropical eigenvalues abstract we investigate powers of supertropical matrices with special attention to the role of the coefficients of the supertropical characteristic polynomial especially the supertropical trace in controlling the rank of a power of a matrix this leads to a jordan type decomposition of supertropical matrices together with a supertropical eigenspace decomposition of a power of an arbitrary supertropical matrix

symmetry of minimizers of a gaussian isoperimetric problem we study an isoperimetric problem described by a functional that consists of the standard gaussian perimeter and the norm of the barycenter the second term is in competition with the perimeter balancing the mass with respect to the origin and because of that the solution is not always the half space we characterize all the minimizers of this functional when the volume is close to one by proving that the minimizer is either the half space or the symmetric strip depending on the strength of the barycenter term as a corollary we obtain that the symmetric strip is the solution of the gaussian isoperimetric problem among symmetric sets when the volume is close to one as another corollary we obtain the optimal constant in the quantitative gaussian isoperimetric inequality

new vector forms of elemental functions with taylor series in many practical applications one often need to use different elementary function in order to approximate the solution of the problem under consideration in this paper some new vector forms of elemental functions with taylor series are developed with the aid of a new set of vector calculations

levy gromov s isoperimetric inequality for an infinite dimensional diffusion generator we establish by simple semigroup arguments a levy gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the gaussian measure this produces in particular a new proof of the gaussian isoperimetric inequality this isoperimetric inequality strengthens the classical logarithmic sobolev inequality in this context a local version for the heat kernel measures is also proved which may then be extended into an isoperimetric inequality for the wiener measure on the paths of a riemannian manifold with bounded ricci curvature

on the stability of a two step method for solving periodic ivps in appl math math 138 2003 435 we presented a nonlinear explicit method for solving periodic initial value problems which is p stable in the sense of lambert and watson however it is not sufficient to investigate the stability behavior of the method based on the scalar test equation y email protected after introducing a new test system and extending the definitions of the interval of periodicity and the phase lag we discuss the stability property and the phase lag of the method in this note

a sixth order a stable explicit one step method for stiff systems this paper presents a sixth order explicit one step method which is proved to be a stable the numerical experiments demonstrate that the new method is suitable for stiff systems

on the isoperimetric deficit in gauss space we prove a sharp quantitative version of the isoperimetric inequality in the space bbb r n endowed with the gaussian measure

minimality via second variation for a nonlocal isoperimetric problem we discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts we show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and in fact satisfy a quantitative isoperimetric inequality with respect to sets that are l 1 close the link with local minimizers for the diffuse interface ohta kawasaki energy is also discussed as a byproduct of the quantitative estimate we get new results concerning periodic local minimizers of the area functional and a proof via second variation of the sharp quantitative isoperimetric inequality in the standard euclidean case as a further application we address the global and local minimality of certain lamellar configurations

a two step explicit p stable method for solving second order initial value problems the conceptions of periodicity interval and p stability were introduced by lambert and watson j inst math appl 18 1976 189 in connection with multistep methods for second order initial value problems this paper presents a new two step explicit p stable method of order two for solving initial value problems of second order ordinary differential equations based on a special vector operation the method can be extended to be vector applicable

tropical plactic algebra the cloaktic monoid and semigroup representations abstract a new tropical plactic algebra is introduced in which the knuth relations are inferred from the underlying semiring arithmetic encapsulating the ubiquitous plactic monoid p n this algebra manifests a natural framework for accommodating representations of p n and equivalently of young tableaux and its moderate coarsening the cloaktic monoid k n and the co cloaktic monoid the faithful linear representations of k n and by tropical matrices which constitute a tropical plactic algebra are shown to provide linear representations of the plactic monoid to this end the paper develops a special type of configuration tableaux corresponding bijectively to semi standard young tableaux these special tableaux allow a systematic encoding of combinatorial properties in numerical algebraic ways including algorithmic benefits the interplay between these algebraic combinatorial structures establishes a profound machinery for exploring semigroup attributes in particular satisfying of semigroup identities this machinery is utilized here to prove that k n and admit all the semigroup identities satisfied by n n triangular tropical matrices which holds also for p 3

the vector form of a sixth order a stable explicit one step method for stiff problems abstract a sixth order a stable explicit one step method for stiff ordinary differential equations is extended directly to solve systems of equations some defects of the component form of this method are avoided to perform these a new set of vector computations are introduced some numerical experiments are presented to show its superiority

quantitative isoperimetry a la levy gromov on a riemannian manifold with a positive lower bound on the ricci tensor the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of a round sphere of suitable radius the deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise these results are actually obtained in the more general context of possibly non smooth metric measure spaces with curvature dimension conditions through a quantitative analysis of the transport rays decompositions obtained by the localization method

an isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in gauss space we prove an isoperimetric inequality on the discrete cube which is the precise analog of a logarithmic inequality due to talagrand as a consequence the gaussian isoperimetric inequality is derived

a class of two step explicit methods for periodic ivps in this paper we present a class of non linear explicit second order methods for solving one dimensional periodic initial value problems ivps which are p stable in the sense of lambert and watson and of high order phase lag error for multi dimensional problems we introduce a special vector operation such that our methods can be extended to be vector applicable directly however it is not sufficient to investigate the stability behavior of the vector form of the methods based on the scalar test equation y l 2y after introducing a new test system and extending the definitions of the interval of periodicity and the phase lag order we discuss the stability property and the phase lag of the vector applicable methods the methods are p stable and of high phase lag for one frequency problems for multi frequency problems methods are of only second phase lag order and may have not a primary interval of periodicity some numerical results illustrate the conclusions

a selection principle for the sharp quantitative isoperimetric inequality we introduce a new variational method for the study of isoperimetric inequalities with quantitative terms the method is general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter two notable applications are presented first we give a new proof of the sharp quantitative isoperimetric inequality in mathbb r n second we positively answer a conjecture by hall concerning the best constant for the quantitative isoperimetric inequality in mathbb r 2 in the small asymmetry regime

supertropical polynomials and resultants abstract this paper a continuation of izhakian and rowen in press 5 involves a closer study of polynomials over supertropical semirings and their version of tropical geometry we introduce the concept of relatively prime polynomials in one indeterminate and resultants with the aid of some topology polynomials in one indeterminant are seen to be relatively prime iff they do not have a common tangible root iff their resultant is tangible applying various morphisms of supertropical varieties leads to a supertropical version of bezout s theorem