a bound on the spectral radius of graphs with e edges the spectral radius p a of the adjacency matrix a of a graph g with e edges satisfiesp a i l equalityoccnrsifandonlyif e i andgisa disjoint union of the complete graph k and isolated vertices 0

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

approximate convexity and submonotonicity abstract it is shown that a locally lipschitz function is approximately convex if and only if its clarke subdifferential is a submonotone operator consequently in finite dimensions the class of locally lipschitz approximately convex functions coincides with the class of lower c 1 functions directional approximate convexity is introduced and shown to be a natural extension of the class of lower c 1 functions in infinite dimensions the following characterization is established a multivalued operator is maximal cyclically submonotone if and only if it coincides with the clarke subdifferential of a locally lipschitz directionally approximately convex function which is unique up to a constant furthermore it is shown that in asplund spaces every regular function is generically approximately convex

dini derivative and a characterization for lipschitz and convex functions on riemannian manifolds dini derivatives in riemannian manifold settings are studied in this paper in addition a characterization for lipschitz and convex functions defined on riemannian manifolds and sufficient optimality conditions for constraint optimization problems in terms of the dini derivative are given

subdifferential representation formula and subdifferential criteria for the behavior of nonsmooth functions abstract several kinds of behaviors of extended real valued lower semicontinuous functions are known to be equivalent to certain appropriate conditions in terms of the clarke subdifferential the paper provides a systematic study showing that any such condition with the clarke subdifferential is valid if and only if it holds with any operator representing the clarke subdifferential like in the subdifferential proximal formula

perturbed optimal control problems we examine a family of problems obtained by perturbing infinite dimensionally the dynamics of an optimal control problem a formula is derived for the generalized gradient of the associated value function one which specializes to yield for instance information about ordinary directional derivatives several examples are discussed

integration of subdifferentials of lower semicontinuous functions on banach spaces abstract this paper studies the lower semicontinuous functions on banach spaces whose subdifferentials are not disassociated from several conditions are given to ensure that two lower semicontinuous functions with the same subdifferential are equal up to an additive constant

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

on the spectral radius of 0 1 matrices we determine the maximum spectral radius for 0 1 matrices with k2 andk2 1 1 s respectively and for symmetric 0 1 matrices with zero trace and e k21 s graphs with e edges in all cases equality is characterized

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

upper bounds on the spectral radius of book free and or k2 l free graphs abstract the spectral radius g of a graph g is the largest eigenvalue of its adjacency matrix let b k denote a book with k pages in this paper we generalize a result of lu et al m lu h liu f tian a new upper bound for the spectral radius of graphs with girth at least 5 linear algebra appl 414 2006 512 516 on the upper bound for the spectral radius of connected graphs with girth at least 5 to connected b k 1 k 2 l 1 free graphs g of order n with maximum degree as follows g k l k l 2 4 4 l n 1 2 with equality if and only if g is a strongly regular graph with parameters k l this implies sharp upper bounds for book free or k 2 l free graphs

prox regular functions in hilbert spaces this paper studies the prox regularity concept for functions in the general context of hilbert space in particular a subdifferential characterization is established as well as several other properties it is also shown that the moreau envelopes of such functions are continuously differentiable

approximate mean value theorem for upper subderivatives the purpose of this paper is to give a mean value theorem for a lower semicontinuous function f on a banach space using the upper subderivatives defined by rockafellar l as will be shown by example section 4 it may happen that the subdifferential af x is empty for all x in the closed line segment a 61 where a and b belong to x therefore we cannot formulate the mean value theorem in such a form as that obtained in 2 and 3 for other generalized derivatives we prove among other factors that there exists a sequence xk converging to some point x e a 61 x b such that the sets df xj are nonempty and

convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities in this paper we analyze and discuss the well posedness of two new variants of the so called sweeping process introduced by moreau in the early 70s moreau in sem anal convexe montpellier 1971 with motivation in plasticity theory the first new variant is concerned with the perturbation of the normal cone to the moving convex subset c t c t supposed to have a bounded variation by a lipschitz mapping under some assumptions on the data we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation the second variant for which a large analysis is made concerns a first order sweeping process with velocity in the moving set c t c t this class of problems subsumes as a particular case the evolution variational inequalities widely used in applied mathematics and unilateral mechanics duvaut and lions in inequalities in mechanics and physics springer berlin 1976 assuming that the moving subset c t c t has a continuous variation for every t in 0 t t 0 t with c 0 c 0 bounded we show that the problem has at least a lipschitz continuous solution the well posedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator we also discuss some applications of the sweeping process to the study of vector hysteresis operators in the elastoplastic model krej i in eur j appl math 2 281 292 1991 to the planning procedure in mathematical economy henry in j math anal appl 41 179 186 1973 and cornet in j math anal appl 96 130 147 1983 and to nonregular electrical circuits containing nonsmooth electronic devices like diodes acary et al nonsmooth modeling and simulation for switched circuits lecture notes in electrical engineering springer new york 2011 the theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof our methodology is based only on tools from convex analysis like other papers in this collection we show in this presentation how elegant modern convex analysis was influenced by moreau s seminal work

on of truncated polynomial rings group homology spectral sequences are used to investigate k of truncated polynomial rings if f is a finite field of odd characteristic we show that relative k of the pair formula presented which has been identified by van der kallen and stienstra is isomorphic to formula presented when q is sufficiently large we also show that formula presented and is isomorphic to the associated k group modulo an elementary abelian 2 group

the spectral turan problem about graphs with no 6 cycle abstract a spectral version of extremal graph theory problem is as follows what is the maximum spectral radius of an h free graph of order n nikiforov posed a conjecture on spectral extremal value of cycles in the past decade much attention has been paid to this conjecture and other questions on spectral extremal graphs in order to approach to nikiforov s conjecture we provide a new conjecture on spectral extremal value involving fans and wheels moreover we show it is true for k 2

partial extensions of attouch s theorem with applications to proto derivatives of subgradient mappings attouch s theorem which gives on a reflexive banach space the equivalence between the mosco epi convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients has many important applications in convex optimization in particular generalized derivatives have been defined in terms of the epi convergence or graph convergence of certain difference quotient mappings and attouch s theorem has been used to relate these various generalized derivatives these relations can then be used to study the stability of the solution mapping associated with a parameterized family of optimization problems we prove in a hilbert space several partial extensions of attouch s theorem to functions more general than convex these functions are called primal lower nice furthermore we use our extensions to derive a relationship between the second order epi derivatives of primal lower nice functions and the proto derivative of their associated subgradient mappings

a bound on the spectral radius of graphs abstract let g be a simple connected graph with n vertices and m edges let a be the adjacency matrix of g the spectral radius a of a satisfies a 2m n 1 with equality if and only if g is isomorphic to one of the following two graphs a the star k1 n 1 b the complete graph kn

the proximal normal formula in hilbert space on considere la formule normale proximale dans un espace de hilbert on presente une demonstration simple du theoreme de borwein strojwas

integrability of subdifferentials of certain bivariate functions abstract we study integrability properties of bivariate functions defined on the product of banach spaces

on the signless laplacian spectral radius of irregular graphs abstract let g be a connected irregular graph in this paper we give lower and upper bounds for the signless laplacian radius q 1 of g we show that q 1 4 e n 2 2 n and q 1 2 1 n d 1 4 hold where d is the diameter of g

a new upper bound for the spectral radius of graphs with girth at least 5 let g be a simple connected graph with v g v1 v 2 v n and girth at least 5 let be the maximum degree of g in this paper we present a new sharp upper bound for the spectral radius of g as follows

necessary and sufficient conditions for nonsmooth mathematical programs with equilibrium constraints in this paper we consider a mathematical program with equilibrium constraints mpec formulated as a mathematical program with complementarity constraints then we derive a necessary optimality result for nonsmooth mpec on any asplund space also under generalized convexity assumptions we establish sufficient optimality conditions for this program in banach spaces

an extension of attouch s theorem and its application to second order epi differentiation of convexly composite functions in 1977 hedy attouch established that a sequence of closed proper convex functions epi converges to a convex function if and only if the graphs of the subdierentials converge in the mosco sense to the subdifferential of the limiting function and roughly speaking there is a condition that fixes the constant of integration we show that the theorem is valid if instead one considers functions that are the composition of a closed proper convex function with a twice continuously differentiable mapping in addition a constraint qualification is imposed

sharp upper bounds on the spectral radius of graphs abstract let g be a simple connected graph with n vertices m edges and degree sequence d 1 d 2 d n the spectral radius g of graph g is the largest eigenvalue of its adjacency matrix in this paper we present some sharp upper bounds of the spectral radius in terms of the degree sequence of graphs

generalized hessian properties of regularized nonsmooth functions we take up the question of second order expansions for a class of functions of importance in optimization namely moreau envelope regularizations of nonsmooth functions f it is shown that when f is prox regular which includes convex functions and the extended real valued functions representing problems of nonlinear programming the many second order properties that can be formulated around the existence and stability of expansions of the envelopes of f or of their gradient mappings are linked by surprisingly extensive lists of equivalences with each other and with generalized differentiation properties of f itself this clarifies the circumstances conducive to developing computational methods based on envelope functions such as second order approximations in nonsmooth optimization and variants of the proximal point algorithm the results establish that generalized second order expansions of moreau envelopes at least can be counted on in most situations of interest in finite dimensional optimization

bounds on graph eigenvalues i abstract we prove three results about the spectral radius g of a graph g a let t r n be the r partite turan graph of order n if g is a k r 1 free graph of order n then g t r n unless g t r n b for most irregular graphs g of order n and size m g 2 m n 1 2 m 2 n c let 0 k l if g is a graph of order n with no k 2 k k 1 and no k 2 l 1 then g min g k l 1 k l 1 2 4 l n 1 2

derivatives with support and applications a new concept of derivative is introduced reflecting the global rather than the local behavior of a given function basic properties and calculus rules for this derivative are provided then they are applied to establish some characterizations of convex functions and some optimality conditions in mathematical programming