lipschitz regularity for minima without strict convexity of the lagrangian abstract we give in a non smooth setting some conditions under which some of the minimizers of f u x d x g x u x d x among the functions in w 1 1 that lie between two lipschitz functions are lipschitz we weaken the usual strict convexity assumption in showing that if just the faces of the epigraph of a convex function f r n r are bounded and the boundary datum u 0 satisfies a generalization of the bounded slope condition introduced by a cellina then the minima of f u x d x on u 0 w 0 1 1 whenever they exist are lipschitz a relaxation result follows

the existence of solutions to variational problems of slow growth abstract we consider the existence of solutions in the space w 1 1 to the problem minimize l v x d x on w 0 1 1 where l is of slow linear or at most quadratic growth we present a necessary and sufficient condition in order that for any smooth boundary datum and for any bounded with smooth boundary the minimum problem be solvable

percolation theory and fragmentation measures in social networks abstract we study the statistical properties of a recently proposed social networks measure of fragmentation f after removal of a fraction q of nodes or links from the network the measure f is defined as the ratio of the number of pairs of nodes that are not connected in the fragmented network to the total number of pairs in the original fully connected network we compare this measure with the one traditionally used in percolation theory p the fraction of nodes in the largest cluster relative to the total number of nodes using both analytical and numerical methods we study erd s renyi er and scale free sf networks under various node removal strategies we find that for a network obtained after removal of a fraction q of nodes above criticality p 1 f 1 2 for fixed p and close to criticality we show that 1 f better reflects the actual fragmentation for a given p 1 f has a broad distribution and thus one can improve significantly the fragmentation of the network we also study and compare the fragmentation measure f and the percolation measure p for a real national social network of workplaces linked by the households of the employees and find similar results

integral representation of abstract functionals of autonomous type in this work we extend the results of amar et al to a family of abstract functionals of autonomous type satisfying suitable locality and additivity properties and general integral growth conditions of superlinear type we single out a condition which is necessary and sufficient in order for a functional of this class to admit an integral representation and sufficient to have an integral representation for its lower semicontinuous envelope we also show that the integrand f x q satisfies some nice regularity properties in the q variable in particular a convexity type property along lines by adapting the reparametrization techniques introduced in an earlier work by the author to the case at issue we then prove that the family of integral functionals associated to integrands of this kind do meet the condition mentioned above in particular it is closed by vargamma convergence

the classical problem of the calculus of variations in the autonomous case relaxation and lipschitzianity of solutions we consider the problem of minimizing b a l x t x t dt x a a x b b under the assumption that the lagrangian l is continuous and satisfies a growth assumption that does not imply superlinear growth we provide a result on the relaxation of the functional and show that a solution to the minimum problem is lipschitzian

nonlocal dispersal cooperative systems acceleration propagation among species abstract we focus on the influences of nonlocal dispersal kernels on the spatial propagation in nonlocal dispersal cooperative systems with initial values having nonempty compact supports it is well known that the solution of a monostable nonlocal dispersal scalar equation spreads at a finite speed when the kernel is thin tailed and propagates by accelerating when the kernel has a heavy tail the fat tailed however in such systems we find that one species can propagate by accelerating although its dispersal kernel is thin tailed which is a new and interesting phenomenon more precisely we show that the spatial propagation of every species at large time is mainly determined by the tail of the maximum of their dispersal kernels which further implies that their spatial propagation is accelerated by each other because of the cooperation relation between them this gives us a new understanding of the cooperation relation in the spatial propagation of nonlocal dispersal cooperative systems

existence and lipschitz regularity of solutions to bolza problems in optimal control in this paper we investigate the existence and lipschitz continuity of optimal trajectories for the autonomous bolza problem in control theory the main feature of our results is that they relax the usual fast growth condition for the lagrangian furthermore we show that optimal solutions do satisfy the maximum principle

lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands the study of regularity properties for minimizers of problems of the form min u u dt u e iv q r u a a u b b 1 1 has been undertaken for a long time in calculus of variations and a number of results are by now available on this subject when f t s z is a smooth coercive function problem 1 1 was considered by tonelli see 18 19 who proved that every minimizer u t of the problem 1 1 is locally lipschitz on a relatively open set of full measure in a b under some additional assumptions on f it is also possible to prove global regularity theorems see for instance cesari 3 chap 2 sect 61 in recent years much attention has been devoted to problems of the form l l on the one hand in weakening the hypotheses of tonelli s regularity results and on the other in finding counterexamples of nonregular minimizers for coercive functionals with respect to this last subject we recall that it has been shown see ball and mizel l and clarke and vinter 6 that there exist polynomial functions f t s z such that for suitable c 0 and p

necessary conditions and non existence results for autonomous nonconvex variational problems abstract classical one dimensional autonomous lagrange problems are considered in absence of any smoothness convexity or coercivity condition on the energy density we prove a dubois reymond type necessary condition expressed as a differential inclusion involving the subdifferential of convex analysis as a consequence a non existence result is obtained

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

regularity properties of solutions to the basic problem in the calculus of variations this paper concerns the basic problem in the calculus of variations minimize a functional j defined by j x j l t x t x t dt a over a class of arcs x whose values at a and b have been specified existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs conditions which must be strengthened in order that the standard necessary conditions apply the question arises what necessary conditions hold merely under hypotheses of existence theory say the classical tonelli conditions it is shown that given a solution x there exists a relatively open subset q of a b of full measure on which x is locally lipschitz and satisfies a form of the euler lagrange equation the main theorem of which this is a corollary can also be used in conjunction with various classes of additional hypotheses to deduce the global smoothness of solutions three such classes are identified and results of bernstein tonelli and morrey are extended one of these classes is of a novel nature and its study implies the new result that when l is independent of t the solution has essentially bounded derivative

another theorem of classical solvability in small for one dimensional variational problems in this paper we suggest a direct method for studying local minimizers of one dimensional variational problems which naturally complements the classical local theory this method allows us both to recover facts of the classical local theory and to resolve a number of problems which were previously unreachable the basis of these results is a regularity theory a priori estimates and compactness in c1 for solutions of obstacle problems with sufficiently close obstacles in these problems we establish that solutions exist and inherit regularity of the obstacles even under assumptions on integrands that are much weaker than those required in the classical local theory

existence of minimizers for nonconvex noncoercive simple integrals we consider the problem of minimizing autonomous simple integrals such as min left int 0 t f left x t x prime t right dt colon text x in ac 0 t x 0 x 0 x t x t right tag cal p where f mathbb r times mathbb r to 0 infty is a possibly nonconvex function with either superlinear or slow growth at infinity assuming that the relaxed problem cal p ast ast obtained from cal p by replacing f with its convex envelope f with respect to the derivative variable x prime admits a solution we prove attainment for cal p under mild regularity and growth assumptions on f and f we discuss various instances of growth conditions on f that yield solutions to the corresponding relaxed problem cal p ast ast and we present examples that show that the hypotheses on f and f considered here for attainment are essentially sharp

relaxation and regularity in the calculus of variations in this work we prove that if l t u is a continuous function in t and u borel measurable in with bounded non convex pieces in then any absolutely continuous solution u to the variational problem min abl t u t u t dt u w01 1 a b is quasi regular in the sense of tonelli i e u is locally lipschitz on an open set of full measure of a b under the further assumption that either l is lipschitz continuous in u locally uniformly in but not necessarily in t or l is invariant under a group of c1 transformations as in the noether s theorem without one of those further assumptions the solution could be not regular as shown by a recent example in gratwick and preiss 2010 13 our result is then optimal in this sense moreover we improve the standard hypothesis used so far in buttazzo et al 1998 1 clarke and vinter 1985 5 6 csornyei et al 2008 7 tonelli 1915 15 which have been the lipschitz continuity of l in u locally uniform in and t and some growth condition in we also show that the relaxed and the original problem have the same solutions without assuming any of the two further assumptions above this extends a result in mariconda and treu 2004 14 to the non autonomous case

reparametrizations and approximate values of integrals of the calculus of variations abstract we prove an approximation result that implies the non occurrence of the lavrentiev phenomenon

normality of the maximum principle for nonconvex constrained bolza problems we consider a bolza optimal control problem with state constraints it is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle in general the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls in the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate cf a v arutyanov s m aseev investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints siam j control optim 35 1997 930 952 f rampazzo r b vinter a theorem on existence of neighbouring trajectories satisfying a state constraint with applications to optimal control ima 16 4 1999 335 351 f rampazzo r b vinter degenerate optimal control problems with state constraints siam j control optim 39 4 2000 989 1007 our aim is to show that actually conditions of a similar nature guarantee normality of every nondegenerate maximum principle in particular we allow the initial condition to be fixed and the state constraints to be nonsmooth to prove normality we use j yorke type linearization of control systems and show the existence of a solution to a linearized control system satisfying new state constraints defined in turn by linearization of the original set of constraints along an extremal trajectory

robustness in semantic networks based on cliques abstract here we present a study on how the structure of semantic networks based on cliques specifically article titles behaves when vertex removal strategies i e random and uniform vertex removal rur highest degree vertex removal hdr and highest intermediation centrality vertex removal hicr are applied to this type of network we propose a method for calculation of the average size of the small components and we identify the existence of a fraction f p where the topological structure of the network changes semantic networks based on cliques maintain the small world phenomenon when subjected to rur hdr and hicr for fractions of removed vertices less than or equal to f p

one dimensional variational problems whose minimizers do not satisfy the euler lagrange equation in this paper we consider the problem of minimizing i u text int a b text f x u x u x dx 1 1 in the set a of absolutely continuous function u a b satisfying the end conditions u left a right text alpha text quad u left b right text beta text 1 2 where and are given constant in 1 1 a b is a finite interval denotes frac d dx and the integrand f f x u p is assumed to be smooth nonnegative and to satisfy the regularity condition f pp text 0 text 1 3

a new variational inequality in the calculus of variations and lipschitz regularity of minimizers abstract we consider a problem of the calculus of variations of the form p minimize i x a b t x t x t d t x a x b subject to x w 1 m a b r n x t c a e x t t a b where a b r n r n r is borel measurable c is a cone is a nonempty subset of r n and is an arbitrary extended valued function we prove that if t t x satisfies a nonsmooth version of cesari s condition s then any local minimizer of p in the ac norm is lipschitz whenever is coercive the proof is obtained via a new variational inequality formulated here that holds under the extended condition s just borel measurability if is autonomous for a e t in a b w t x t x t v v t x t x t p t v 1 v 0 where p t is an absolutely continuous arc the derivative of which belongs to a suitable subdifferential of x t x t a e on a b the proof of the directional weierstrass type condition w is based on clarke s nonsmooth recent versions of the maximum principle

existence of vector minimizers for nonconvex 1 dim integrals with almost convex lagrangian this paper proves new results of existence of minimizers for the nonconvex integral abl x x dt among the ac functions x a b rn with x a a x b b our lagrangian l is e g lsc with superlinear growth assuming values freely we replace convexity by almost convexity a hypothesis which in the radial superlinear case l s f s is automatically satisfied provided f s is convex at zero

necessary and sufficient conditions for optimality of nonconvex noncoercive autonomous variational problems with constraints we consider the classical autonomous constrained variational problem of minimization of a b v t v t dt in the class v w 1 1 a b v a v b v t 0 a e in a b where a x 0 r is a lower semicontinuous nonnegative integrand which can be nonsmooth nonconvex and noncoercive we prove a necessary and sufficient condition for the optimality of a trajectory v 0 in the form of a dubois reymond inclusion involving the subdifferential of convex analysis moreover we also provide a relaxation result and necessary and sufficient conditions for the existence of the minimum expressed in terms of an upper limitation for the assigned mean slope 0 b a applications to various noncoercive variational problems are also included