the proximal normal formula in banach space approximation by proximal normals to the clarke generalized subdifferential for a distance function generated by a nonempty closed set and the normal cone to the set generated by the proximal normals are important tools in nonsmooth analysis we give simple general versions of such formulae in infinite dimensional banach spaces which satisfy different geometrical conditions our first class of spaces with uniformly g teaux differentiable norm includes the hilbert space case and the formulae is attained through dense subsets our second class of reflexive kadec smooth spaces is the most general for which such formulae can be obtained for all nonempty closed sets in the space our technique also allows us to establish the existence of solutions for a class of optimization problems substantially extending similar work of ekeland and lebourg

clarke coderivatives of efficient point multifunctions in parametric vector optimization abstract the paper is devoted to the study of the clarke circatangent coderivatives of the efficient point multifunction of parametric vector optimization problems in banach spaces we provide inner outer estimates for evaluating the clarke circatangent coderivative of this multifunction in a broad class of conventional vector optimization problems in the presence of geometrical operator and finite and infinite functional constraints examples are given for analyzing and illustrating the obtained results

coupling eigenvalues and eigenvectors a tool for investigating the vibroacoustic behaviour of coupled vibrating systems the transmission of vibrations between coupled subsystems is treated by using coupling eigenvalues and eigenvectors it is demonstrated through a basic equation that coupling eigenvalues and eigenvectors characterize the energy exchanges between subsystems due to the coupling the coupling eigenvalue relates to the coupling strength and coupling eigenvectors to the coupling transmission path in addition in the case of several couplings a simplified method is presented in which only the prevailing modal path between subsystems is used the results obtained by using this method compare well with the reference calculation the transmission of vibrations by coupling can be calculated with a small number of variables and offers a new perspective in the range of medium frequency

subdifferential regularities of perturbed distance functions outside the target set in banach spaces abstract this paper is devoted to study the frechet and proximal regularities at points outside the target set of perturbed distance functions d s j determined by a closed subset s and a lipschitz function j also we provide some important results on the frechet proximal and clarke subdifferentials of d s j at those points in banach spaces

a graph theory approach to identify resonant and non resonant transmission paths in statistical modal energy distribution analysis statistical modal energy distribution analysis smeda extends classical statistical energy analysis sea to the mid frequency range by establishing power balance equations between modes in different subsystems this circumvents the sea requirement of modal energy equipartition and enables applying smeda to the cases of low modal overlap locally excited subsystems and to deal with complex heterogeneous subsystems as well yet widening the range of application of sea is done at a price with large models because the number of modes per subsystem can become considerable when the frequency increases therefore it would be worthwhile to have at one s disposal tools for a quick identification and ranking of the resonant and non resonant paths involved in modal energy transmission between subsystems it will be shown that previously developed graph theory algorithms for transmission path analysis tpa in sea can be adapted to smeda and prove useful for that purpose the case of airborne transmission between two cavities separated apart by homogeneous and ribbed plates will be first addressed to illustrate the potential of the graph approach a more complex case representing transmission between non contiguous cavities in a shipbuilding structure will be also presented

smoothness of the metric projection onto nonconvex bodies in hilbert spaces abstract based on a fundamental work of r b holmes from 1973 we study differentiability properties of the metric projection onto prox regular sets we show that if the set is a nonconvex body with a c p 1 smooth boundary then the projection is c p smooth near suitable open truncated normal rays which are determined only by the function of prox regularity a local version of the same result is established as well namely when the smoothness of the boundary and the prox regularity of the set are assumed only near a fixed point finally similar results are derived when the prox regular set is itself a c p 1 submanifold

relaxation of an optimal control problem involving a perturbed sweeping process we establish first in the setting of infinite dimensional hilbert space a result concerning the existence of solutions for perturbed sweeping processes whose perturbations are lipschitz single valued maps then we use this result to extend to the infinite dimensional setting a relaxation result concerning optimal control problems involving such processes

subdifferentials of a minimum time function in banach spaces in general banach space setting we study the minimum time function determined by a closed convex set k and a closed set s this function is simply the usual minkowski function of k if s is the singleton consisting of the origin in particular we show that various subdifferentials of a minimum time function are representable by virtue of corresponding normal cones of sublevel sets of the function

integration of subdifferentials of nonconvex functions in nonsmooth analysis and optimization subgradients come in many different flavors e g approximate dini proximal clarke generalized see 2 5 6 12 14 191 these subgradients are important and valuable tools however many questions remain unsolved concerning the exact link between the function and its subgradients for instance can two functions not differing by an additive constant have the same subgradients in this paper we study the fundamental problem of determining functions that can be recovered up to an additive constant from the knowledge of their subgradients this integration problem is not very well understood and very few functions or classes of functions are known to be recoverable from their subgradients in section 4 we show that if the basic constraint qualification holds at r then the composition of a closed i e lowersemicontinuous proper convex function with a twice continuously differentiable mapping is determined up to an additive constant by its generalized subgradients actually in this case all above mentioned subgradients are the same beside the obvious theoretical interest of this integration problem it is our hope or perhaps our long term goal that once this problem is better understood we can then tackle the question of uniqueness of solutions to generalized differential equations involving subgradients in place of partial derivatives an example of such an equation that well deserves study is the extended hamilton jacobi equation used in optimal control see clarke 2 let us also mention that a problem similar to the integration problem is the one of determining the set valued mappings that are in fact subgradient set valued mappings uniqueness is not mandatory for a contribution to this problem see janin 8 before we look at some of the known cases where the function can be recovered from its subgradients let us look at some negative examples it is clear that not every function can be recovered up to an additive constant from its subgradients we only need to look at the following two functions 0 x50 0 x10 f x 1 x o g x 2 x o

absolute and relative path measures in a discrete system by using two analytical methods this paper analytically examines two vibration paths rank ordering methods and critically investigates several path identification issues one method is the indirect interfacial path force estimation procedure that is employed in the well known experimental transfer path analysis the other method is path disconnect scheme that has been historically utilized in industry to empirically find a dominant or defective path in this article we utilize simplified but pedagogical discrete vibratory systems to clarify the underlying principles of both methods and to assess the path rank orders our analysis is limited to a linear time invariant system with only translating motions under harmonic excitation and three distinct parallel paths with or without masses are considered alternate formulations of the interfacial path forces based on direct and indirect methods are derived the indirect yet exact interfacial path force expressions could be used to estimate time averaged dissipated power and lagrangian energy spectra in various sub systems estimations using only the driving point frequency response functions are emphasized since they would ease the experimental burden next the path disconnect method is analytically formulated and related to the path connect scheme assuming one path at a time this analysis reveals some useful relations among the path measures and the way a particular path could be connected or disconnected a laboratory experiment validates the simplified massless path model which yields asymptotic trends finally all of the path rank orders are quantified and compared using many absolute and relative measures based on interfacial force receiver motion and sub system energy relationships direction for future work is briefly discussed

a numerical criterion for uniruledness an n dimensional variety x over an algebraically closed field k is said to be uniruled if there exist an n 1 dimensional k variety w and a dominant rational map f p x w x x is called separably uniruled if f is a separable map i e k p x w is a finite separable extension of k x via f if x is uniniled then there exists a rational curve passing through a general k valued point of x and the converse is true by the countability of the components of the hilbert scheme provided that the ground field k is uncountable uniruled varieties are of importance in the classification of varieties it is conjectured that in characteristic 0 uniruledness is equivalent to k xc this article gives a numerical criterion for a projective variety to be uniruled

nef dimension of minimal models we reduce the abundance conjecture in dimension 4 to the following numerical statement if the canonical divisor k is nef and has maximal nef dimension then k is big from this point of view we classify in dimension 2 nef divisors which have maximal nef dimension but which are not big

termination of many 4 dimensional log flips we prove that any sequence of 4 dimensional log flips that begins with a klt pair x d such that k x d is numerically equivalent to an effective divisor terminates this implies termination of flips that begin with a log fano pair and termination of flips in a relative birational setting we also prove termination of directed flips with big k x d as a consequence we prove existence of minimal models of 4 dimensional dlt pairs of general type existence of 5 dimensional log flips and rationality of kodaira energy in dimension 4

subsmooth sets functional characterizations and related concepts prox regularity of a set poliquin rockafellar thibault 2000 or its global version proximal smoothness clarke stern wolenski 1995 plays an important role in variational analysis not only because it is associated with some fundamental properties as the local continuous differentiability of the function dist c or the local uniqueness of the projection mapping but also because in the case where c is the epigraph of a locally lipschitz function it is equivalent to the weak convexity lower c 2 property of the function in this paper we provide an adapted geometrical concept called sub smoothness which permits an epigraphic characterization of the approximate convex functions or lower c 1 property subsmooth sets turn out to be naturally situated between the classes of prox regular and of nearly radial sets this latter class has been recently introduced by lewis in 2002 we hereby relate it to the mifflin semismooth functions

prox regular sets and epigraphs in uniformly convex banach spaces various regularities and other properties we continue the study of prox regular sets that we began in a previous work in the setting of uniformly convex banach spaces endowed with a norm both uniformly smooth and uniformly convex e g l p w m p spaces we prove normal and tangential regularity properties for these sets and in particular the equality between mordukhovich and proximal normal cones we also compare in this setting the proximal normal cone with different holderian normal cones depending on the power types s q of moduli of smoothness and convexity of the norm in the case of sets that are epigraphs of functions we show that j primal lower regular functions have prox regular epigraphs and we compare these functions with poliquin s primal lower nice functions depending on the power types s q of the moduli the preservation of prox regularity of the intersection of finitely many sets and of the inverse image is obtained under a calmness assumption a conical derivative formula for the metric projection mapping of prox regular sets is also established among other results of the paper it is proved that the attouch wets convergence preserves the uniform r prox regularity property and that the metric projection mapping is in some sense continuous with respect to this convergence for such sets

graph theory applied to noise and vibration control in statistical energy analysis models a fundamental aspect of noise and vibration control in statistical energy analysis sea models consists in first identifying and then reducing the energy flow paths between subsystems in this work it is proposed to make use of some results from graph theory to address both issues on the one hand linear and path algebras applied to adjacency matrices of sea graphs are used to determine the existence of any order paths between subsystems counting and labeling them finding extremal paths or determining the power flow contributions from groups of paths on the other hand a strategy is presented that makes use of graph cut algorithms to reduce the energy flow from a source subsystem to a receiver one modifying as few internal and coupling loss factors as possible

stability of tri canonical curves of genus two we completely classify tri canonically embedded curves of genus two that are chow semistable and identify the moduli space of them with the compact moduli space of binary sextics this moduli space is the log canonical model for the pair big overline m 2 alpha delta 0 frac 1 alpha 2 delta 1 frac 1 2 xi big for 7 10 alpha le 9 11 whose log canonical divisor pulls back to k overline m 2 alpha delta on the moduli stack

infimal convolution and optimal time control problem iii minimal time projection set we continue the investigation of a general minimal time problem with a convex constant dynamics and a lower semicontinuous extended real valued target function defined on a banach space in this paper we obtain an explicit description of the minimal time projection set in terms of the legendre fenchel conjugate function and provide some sufficient conditions for this set to be nonempty the single valuedness and lipschitz property of the minimal time projection are also investigated

subdifferentials of a minimal time function in normed spaces abstract in a general normed vector space we study the minimal time function determined by a differential inclusion where the set valued mapping involved has constant values of a bounded closed convex set u and by a closed target set s we show that proximal and frechet subdifferentials of a minimal time function are representable by virtue of corresponding normal cones of sublevel sets of the function and level or suplevel sets of the support function of u the known results in the literature require the set u to have the origin as an interior point or u be compact in particular if the set u is the unit closed ball the results obtained reduce to the subdifferential of the distance function defined by s

a note on lattice simplices and toric varieties in this note we answer the question on terminal singularities of toric 4 folds raised by k matsuki in 1 p 857 we thank david cox for suggesting this problem to us a nonsingular integer d x d matrix a with column vectors ai a2 ad is called primitive if the d simplex conv 0 a a a contains no lattice points except its vertices given any rational matrix or vector x we write den x for the smallest positive integer v such that v x is an integer matrix we abbreviate ed 1 1 1 e zd

proximal analysis in reflexive smooth banach spaces abstract in the present paper we introduce and study a new proximal normal cone in reflexive banach spaces in terms of a generalized projection operator two new variants of generalized proximal subdifferentials are also introduced in reflexive smooth banach spaces the density theorem for both proximal subdifferentials has been proved in p uniformly convex and q uniformly smooth banach spaces various important properties and applications of our concepts are also proved

on the algebras of bps states we define an algebra on the space of bps states in theories with extended supersymmetry we show that the algebra of perturbative bps states in toroidal compactification of the heterotic string is closely related to a generalized kac moody algebra we use d brane theory to compare the formulation of rr charged bps algebras in type ii compactification with the requirements of string string duality and find that the rr charged bps states should be regarded as cohomology classes on moduli spaces of coherent sheaves the equivalence of the algebra of bps states in heterotic iia dual pairs elucidates certain results and conjectures of nakajima and gritsenko nikulin on geometrically defined algebras and furthermore suggests nontrivial generalizations of these algebras in particular to any calabi yau 3 fold there are two canonically associated algebras exchanged by mirror symmetry

the quantification of structure borne transmission paths by inverse methods part 1 improved singular value rejection methods structure borne sound from installed machinery is often transmitted into a receiver structure via many connection points and several co ordinate directions at each of them in order to quantify the contributions from the various connection points the operational forces at the interfaces or an equivalent set of forces at some other locations should be determined these forces may be combined with measured transfer functions to determine their contributions to the sound at the receiver locations inverse methods are becoming widely used in which a matrix of measured accelerances is inverted at each frequency and used with operational acceleration data to find the forces due to poor conditioning of this matrix however the results can often be unreliable in this paper using both simulations and measurements an assessment is made of the success and failure of various strategies for dealing with the problems of ill conditioning in particular over determination and singular value rejection in each case the test structure is a rectangular plate and a wide frequency range is covered to include regions of both low and high modal overlap critical for the rejection of singular values is a suitable threshold it is established that previously used thresholds based on estimates of error in either accelerances or operational responses cannot be used universally an alternative approach is developed in which the accelerance matrix is perturbed by a different amount for each sample of the operational responses based on this approach a more robust strategy is proposed which takes account simultaneously of the effect of errors in both the accelerances and operational responses

theoretical mathematics toward a cultural synthesis of mathematics and theoretical physics is speculative mathematics dangerous recent interactions between physics and mathematics pose the question with some force traditional mathematical norms discourage speculation but it is the fabric of theoretical physics in practice there can be benefits but there can also be unpleasant and destructive consequences serious caution is required and the issue should be considered before rather than after obvious damage occurs with the hazards carefully in mind we propose a framework that should allow a healthy and positive role for speculation

degeneration of surfaces with trivial canonical bundle the object of this note is to prove the theorem stated below the main idea of the proof and much of the detail are already to be found in v i kulikov s important paper k our contribution to the proof and the differences with kulikov s paper are discussed later in this introduction theorem let wl x d be a semistable degeneration of surfaces such that

a note on regularity of sets and of distance functions in banach space abstract we show that in every infinite dimensional banach space there is a compact set c and x in c such that c is regular at x in the sense of nonsmooth analysis but such that the associated distance function d c is not regular at x and neither is d c

chern numbers for singular varieties and elliptic homology a fundamental goal of algebraic geometry is to do for singular varieties whatever we can do for smooth ones intersection homology for example directly produces groups associated to any variety which have almost all the properties of the usual homology groups of a smooth variety minimal model theory suggests the possibility of working more indirectly by relating any singular variety to a variety which is smooth or nearly so here we use ideas from minimal model theory to define some characteristic numbers for singular varieties generalizing the chern numbers of a smooth variety this was suggested by goresky and macpherson as a next natural problem after the definition of intersection homology 11 we find that only a subspace of the chern numbers can be defined for singular varieties a convenient way to describe this subspace is to say that a smooth variety has a fundamental class in complex bordism whereas a singular variety can at most have a fundamental class in a weaker homology theory elliptic homology we use this idea to give an algebro geometric definition of elliptic homology complex bordism modulo flops equals elliptic homology this paper was inspired by some questions asked by jack morava the descriptions of elliptic homology given by gerald hohn 13 were also an important influence thanks to dave bayer mike stillman john stembridge sheldon katz and stein stromme for their computer algebra programs macaulay sf and schubert which helped in guessing the right answer

weak regularity of functions and sets in asplund spaces abstract in this paper we study a new concept of weak regularity of functions and sets in asplund spaces we show that this notion includes prox regular functions functions whose subdifferential is weakly submonotone and amenable functions in infinite dimension we establish also that weak regularity is equivalent to mordukhovich regularity in finite dimension finally we give characterizations of the weak regularity of epi lipschitzian sets in terms of their local representations

limiting subdifferentials of perturbed distance functions in banach spaces abstract we explore in arbitrary banach spaces the frechet type e subdifferentials and the limiting subdifferentials for the perturbed distance function d s j determined by a closed subset s and a lower semicontinuous function j defined on s in particular upper and lower estimates for the frechet type e subdifferentials and for the limiting subdifferentials are provided in terms of the corresponding subdifferentials of the sum of the associated functions j and s

linear regularity and ϕ regularity of nonconvex sets in this paper we discuss some sufficient conditions for the linear regularity and bounded linear regularity and their variations of finitely many closed not necessarily convex sets in a normed vector space the accompanying necessary conditions are also given in the setting of asplund spaces

bv solutions of nonconvex sweeping process differential inclusion with perturbation abstract this paper is devoted to the study of differential inclusions particularly discontinuous perturbed sweeping processes in the infinite dimensional setting on the one hand the sets involved are assumed to be prox regular and to have a variation given by a function which is of bounded variation and right continuous on the other hand the perturbation satisfies a linear growth condition with respect to a fixed compact subset finally the case where the sets move in an absolutely continuous way is recovered as a consequence

mukai flops and deformations of symplectic resolutions we prove that two projective symplectic resolutions of open image in new window are connected by mukai flops in codimension 2 for a finite sub group g sp 2n it is also shown that two projective symplectic resolutions of open image in new window are deformation equivalent

ascending chain condition for log canonical thresholds and termination of log flips we prove that the ascending chain condition acc for log canonical lc thresholds in dimension d existence of log flips in dimension d and the log minimal model program lmmp in dimension d 1 imply termination of any sequence of log flips starting with a d dimensional effective lc pair and also imply termination of flops in dimension d in particular the latter terminations in dimension 4 follow from the alexeev borisov conjecture in dimension 3

softness sleekness and regularity properties in nonsmooth analysis abstract we study continuity properties of tangent and normal cones to a closed subset of a banach space such a study can be seen as a nonsmooth set theoretic analogue of the study of continuously differentiable functions we give particular attention to the case where different concepts coincide a desirable feature such properties are able to offer we identify a wide class of sets and functions for which this property occurs

termination of flops for 4 folds the purpose of this note is to prove the termination of flops for 4 folds i e the second part of the so called flop conjecture in dimension 4 which asserts that there is no infinite sequence of flops with respect to a fixed reference divisor for projective 4 folds having only terminal singularities our idea for the proof stems from crossbreeding the ideas of kollar in ko3 where he proves the termination of flops for 3 folds with those of kawamata in kmm where he shows the termination of flips for 4 folds we should mention that the notion of difficulty due to shokurov si plays the most basic role in the ar

a numerical scheme for a class of sweeping processes the aim of this paper is to study a whole class of first order differential inclusions which fit into the framework of perturbed sweeping process by uniformly prox regular sets after obtaining well posedness results we propose a numerical scheme based on a prediction correction algorithm and we prove its convergence finally we apply these results to a problem coming from the modelling of crowd motion

on the reciprocity relationship between direct field radiation and diffuse reverberant loading this analysis is concerned with the derivation of a diffuse field reciprocity relationship between the diffuse field excitation of a connection to a structural or acoustic subsystem and the radiation impedance of the connection such a relationship has been derived previously for connections described by a single degree of freedom in the present work it is shown that the diffuse field reciprocity relationship also arises when describing the ensemble average response of connections to structural or acoustic subsystems with uncertain boundaries furthermore it is shown that the existing diffuse field reciprocity relationship can be extended to encompass connections that possess an arbitrary number of degrees of freedom the present work has application to i the calculation of the diffuse field response of structural acoustic systems modeled by finite elements boundary elements and infinite elements ii the general calculation of the coupling loss factors employed in statistical energy analysis sea and iii the derivation of an alternative analysis method for describing the dynamic interactions of coupled subsystems with uncertain boundaries a generalized boundary approach to sea

subdifferentiation of integral functionals we examine how the subdifferentials of nonconvex integral functionals can be deduced from the subdifferentials of the corresponding integrand or at least be estimated with the help of them in fact assuming some regularity properties of the integrands we obtain exact expressions for the subdifferentials of the integral functionals we draw some consequences in terms of duality for such integral functionals extending in this way the early work of rockafellar to the nonconvex case

a simple min cut algorithm we present an algorithm for finding the minimum cut of an undirected edge weighted graph it is simple in every respect it has a short and compact description is easy to implement and has a surprisingly simple proof of correctness its runtime matches that of the fastest algorithm known the runtime analysis is straightforward in contrast to nearly all approaches so far the algorithm uses no flow techniques roughly speaking the algorithm consists of about v nearly identical phases each of which is a maximum adjacency search

local differentiability of distance functions recently clarke stern and wolenski characterized in a hilbert space the closed subsets c for which the distance function dc is continuously differentiable everywhere on an open tube of uniform thickness around c here a corresponding local theory is developed for the property of dc being continuously differentiable outside of c on some neighborhood of a point x c this is shown to be equivalent to the prox regularity of c at x which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation additional characterizations are provided in terms of dc being locally of class c 1 or such that dc 2 is convex around x for some 0 prox regularity of c at x corresponds further to the normal cone mapping nc having a hypomonotone truncation around x and leads to a formula for pc by way of nc the local theory also yields new insights on the global level of the clarke stern wolenski results and on a property of sets introduced by shapiro as well as on the concept of sets with positive reach considered by federer in the finite dimensional setting

sweeping process with regular and nonregular sets abstract differential inclusions involving the normal cone to a moving set are investigated a special attention is paid to the sweeping process associated with sets for which no regularity assumption is required

method of measuring transmission paths abstract a theoretical explanation and experimental proof are presented of a method for localizing and evaluating the transmission paths of any signal in a black box among a set of points previously defined in it the signal should behave linearly and the system should be able to receive external excitations separately at each of its points such excitations need not be the signal under study but they should be linearly related to it also presented are the equations that once the transmission paths have been determined allow the evaluation of the excitations which act on the system

the quantification of structure borne transmission paths by inverse methods part 2 use of regularization techniques the inversion of an ill conditioned matrix of measured data lies at the heart of procedures for the quantification of structure borne sources and transmission paths in an earlier paper the use of over determination singular value decomposition and the rejection of small singular values was discussed in the present paper alternative techniques for regularizing the matrix inversion are considered such techniques have been used in the field of digital image processing and more recently in relation to nearfield acoustic holography the application to structure borne sound transmission involves matrices which vary much more with frequency and from one element to another in this study tikhonov regularization is used with the ordinary cross validation method for selecting the regularization parameter an iterative inversion technique is also studied here a form of cross validation is developed allowing an optimum value of the iteration parameter to be selected simulations are carried out using a rectangular plate structure to assess the relative merits of these techniques experiments are also performed to validate the results both techniques are found to give considerably improved results compared to singular value rejection

birational geometry of terminal quartic 3 folds i in this paper we study the birational geometry of certain examples of mildly singular quartic 3 folds a quartic 3 fold is a special case of a fano variety that is a variety x with ample anticanonical sheaf ox kx nonsingular fano 3 folds have been studied extensively examples studied so far fall within two classes either x is close to being rational and it then has very many biregularly distinct birational models as a fano 3 fold or at the other extreme x has a unique model in this paper we construct examples of singular quartic 3 folds with exactly two birational