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

analysis of directed networks via partial singular value decomposition and gauss quadrature abstract large scale networks arise in many applications it is often of interest to be able to identify the most important nodes of a network or to determine the ease of traveling between them we are interested in carrying out these tasks for directed networks these networks have a nonsymmetric adjacency matrix a benzi et al 6 recently proposed that these tasks can be accomplished by studying certain matrix functions such as hyperbolic cosine and sine of a t a and a a t for small to medium sized networks the required computations can be easily carried out by first computing the singular value decomposition of a however for large networks this is impractical we propose to first compute a partial singular value decomposition of a which allows us to determine a subset of nodes that contains the most important nodes or a subset of nodes between which it is easy to travel we then apply gauss quadrature to rank the nodes in these subsets several computed examples illustrate the performance of the approach proposed

subgraph centrality in complex networks we introduce a new centrality measure that characterizes the participation of each node in all subgraphs in a network smaller subgraphs are given more weight than larger ones which makes this measure appropriate for characterizing network motifs we show that the subgraph centrality c s i can be obtained mathematically from the spectra of the adjacency matrix of the network this measure is better able to discriminate the nodes of a network than alternate measures such as degree closeness betweenness and eigenvector centralities we study eight real world networks for which c s i displays useful and desirable properties such as clear ranking of nodes and scale free characteristics compared with the number of links per node the ranking introduced by c s i for the nodes in the protein interaction network of s cereviciae is more highly correlated with the lethality of individual proteins removed from the proteome

growing random sequences we consider the random sequence xn xn 1 xq with 0 where q 0 1 n 1 is chosen randomly from a probability distribution pn q when all q are chosen with equal probability i e pn q 1 n we obtain an exact solution for the mean xn and the divergence of the second moment x2n as functions of n and for 1 we examine the divergence of the mean value of xn as a function of n for the random sequences generated by power law and exponential pn q and for the non random sequence pn q q n 1

random geometric series integer sequences where each element is determined by a previous randomly chosen element are investigated analytically in particular the random geometric series xn 2xp with 0 p n 1 is studied at large n the moments grow algebraically xsn n s with s 2s 1 while the typical behaviour is xn nln 2 the probability distribution is obtained explicitly in terms of the stirling numbers of the first kind and it approaches a log normal distribution asymptotically

random fibonacci sequences solutions to the random fibonacci recurrence xn 1 xn xn 1 decrease increase exponentially xn exp n for sufficiently small large in the limits 0 and we expand the lyapunov exponent in powers of and 1 respectively for the classical case of 1 we obtain exact non perturbative results in particular an invariant measure associated with ricatti variable rn xn 1 xn is shown to exhibit plateaux around all rational r

quadrature rule based bounds for functions of adjacency matrices bounds for entries of matrix functions based on gauss type quadrature rules are applied to adjacency matrices associated with graphs this technique allows to develop inexpensive and accurate upper and lower bounds for certain quantities estrada index subgraph centrality communicability that describe properties of networks

mean value inequalities in hilbert space we establish a new mean value theorem applicable to lower semicontinuous functions on hilbert space a novel feature of the result is its multidirectionality it compares the value of a function at a point to its values on a set we then discuss some refinements and consequences of the theorem including applications to calculus flow invariance and generalized solutions to partial differential equations rtsumt on etablit un nouveau theoreme de la valeur moyenne qui s applique aux fonctions semicontinues inferieurement sur un espace de hilbert on deduit plusieurs consequences du resultat portant par exemple sur les fonctions monotones et sur les solutions generalisees des equations aux derivees partielles

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

growth and structure of stochastic sequences we introduce a class of stochastic integer sequences in these sequences every element is a sum of two previous elements at least one of which is chosen randomly the interplay between randomness and memory underlying these sequences leads to a wide variety of behaviours ranging from stretched exponential to log normal to algebraic growth interestingly the set of all possible sequence values has an intricate structure

network properties revealed through matrix functions the emerging field of network science deals with the tasks of modeling comparing and summarizing large data sets that describe complex interactions because pairwise affinity data can be stored in a two dimensional array graph theory and applied linear algebra provide extremely useful tools here we focus on the general concepts of centrality communicability and betweenness each of which quantifies important features in a network some recent work in the mathematical physics literature has shown that the exponential of a network s adjacency matrix can be used as the basis for defining and computing specific versions of these measures we introduce here a general class of measures based on matrix functions and show that a particular case involving a matrix resolvent arises naturally from graph theoretic arguments we also point out connections between these measures and the quantities typically computed when spectral methods are used for data mining tasks such as clustering and ordering we finish with computational examples showing the new matrix resolvent version applied to real networks

kinetic models for the trading of goods in this paper we introduce kinetic equations for the evolution of the probability distribution of two goods among a huge population of agents the leading idea is to describe the trading of these goods by means of some fundamental rules in price theory in particular by using cobb douglas utility functions for the binary exchange and the edgeworth box for the description of the common exchange area in which utility is increasing for both agents this leads to a boltzmann type equation in which the post interaction variables depend in a nonlinear way from the pre interaction ones other models will be derived by suitably linearizing this boltzmann equation in presence of uncertainty in the exchanges it is shown that the solution to some of the linearized kinetic equations develop pareto tails where the pareto index depends on the ratio between the gain and the variance of the uncertainty in particular the result holds true for the solution of a drift diffusion equation of fokker planck type obtained from the linear boltzmann equation as the limit of quasi invariant trades

weak disorder in fibonacci sequences we study how weak disorder affects the growth of the fibonacci series we introduce a family of stochastic sequences that grow by the normal fibonacci recursion with probability 1 but follow a different recursion rule with a small probability we focus on the weak disorder limit and obtain the lyapunov exponent that characterizes the typical growth of the sequence elements using perturbation theory the limiting distribution for the ratio of consecutive sequence elements is obtained as well a number of variations to the basic fibonacci recursion including shift doubling and copying are considered

random fibonacci sequences and the number 1 13198824 begin abstract for the familiar fibonacci sequence defined by f 1 f 2 1 and f n f n 1 f n 2 for n 2 f n increases exponentially with n at a rate given by the golden ratio 1 sqrt 5 2 1 61803398 ldots but for a simple modification with both additions and subtractions the it random fibonacci sequences defined by t 1 t 2 1 and for n 2 t n pm t n 1 pm t n 2 where each pm sign is independent and either or with probability 1 2 it is not even obvious if abs t n should increase with n our main result is that begin equation sqrt n abs t n rightarrow 1 13198824 ldots text as n rightarrow infty end equation with probability 1 finding the number 1 13198824 ldots involves the theory of random matrix products stern brocot division of the real line a fractal like measure a computer calculation and a rounding error analysis to validate the computer calculation end abstract

convergence analysis of an augmented lagrangian method for mathematical programs with complementarity constraints abstract in this paper a mathematical program with complementarity constraints mpcc is reformulated as a nonsmooth constrained optimization problem by using the fischer burmeister function an augmented proximal lagrangian method is applied to tackle the resulting constrained optimization problem the augmented lagrangian problems are in general nonsmooth we derive first and second order optimality conditions for the augmented lagrangian problems using an approximate smooth variational principle and establish that the limit point of a sequence of points that satisfy the second order necessary optimality conditions of the augmented lagrangian problems is a strongly stationary point of the original mpcc if the limit point is feasible to mpcc and the linear independence constraint qualification for mpcc and the upper level strict complementarity condition hold at the limit point

proximal normal cone analysis on smooth banach spaces and applications unlike the frechet penot clarke and other order one notions of normal cones the notion of proximal normal cone describes variational behavior of order two and most of results on proximal normal cone are established in the hilbert space framework a possible reason is that in a general banach space such as in the classical banach spaces l p and l p with 1 leq p 2 the proximal normal cone of a smooth subset a at every point of a may consist of the zero element only nevertheless on the other hand we show in this paper that for a fairly big class of smooth banach spaces including the classical banach spaces l p and l p with 2 leq p infty the consideration of proximal normal cones provides a lot of useful information especially for leading to satisfactory treatment for constrained optimization problems some of our results are new even in the hilbert space case

error estimates and evaluation of matrix functions via the faber transform the need to evaluate expressions of the form f a or f a b where f is a nonlinear function a is a large sparse n times n matrix and b is an n vector arises in many applications this paper describes how the faber transform applied to the field of values of a can be used to determine improved error bounds for popular polynomial approximation methods based on the arnoldi process applications of the faber transform to rational approximation methods and in particular to the rational arnoldi process also are discussed

history dependent random processes ulam has defined a history dependent random sequence by the recursion x n 1 x n x u n where u n n 1 is a sequence of independent random variables with u n uniformly distributed on 1 n and x 1 1 we introduce a new class of continuous time history dependent random processes regulated by poisson processes the simplest of these a univariate process regulated by a homogeneous poisson process replicates in continuous time the essential properties of ulam s sequence and greatly facilitates its analysis we consider several generalizations and extensions of this including bivariate and multivariate coupled history dependent processes and cases when the dependence on the past is not uniform the analysis of the discrete time formulations of these models would be at the very least an extremely formidable project but we determine the asymptotic growth rates of their means and higher moments with relative ease

ranking hubs and authorities using matrix functions the notions of subgraph centrality and communicability based on the exponential of the adjacency matrix of the underlying graph have been effectively used in the analysis of undirected networks in this paper we propose an extension of these measures to directed networks and we apply them to the problem of ranking hubs and authorities the extension is achieved by bipartization i e the directed network is mapped onto a bipartite undirected network with twice as many nodes in order to obtain a network with a symmetric adjacency matrix we explicitly determine the exponential of this adjacency matrix in terms of the adjacency matrix of the original directed network and we give an interpretation of centrality and communicability in this new context leading to a technique for ranking hubs and authorities the matrix exponential method for computing hubs and authorities is compared to the well known hits algorithm both on small artificial examples and on more realistic real world networks a few other ranking algorithms are also discussed and compared with our technique the use of gaussian quadrature rules for calculating hub and authority scores is discussed

growth and decay of random fibonacci sequences for 0 they grow exponentially by formulating the problem as a markov chain involving random matrix products and computing its invariant measure a fractal the lyapunov constant limn x n 1 n is determined numerically for a wide range of values and its dependence on is observed to be non smooth the limit is defined in the almost sure sense this generalizes recent work of viswanath who proved 1 1 131 988 24 by a simple rescaling these results also apply to the more general random recurrence x n 1 x n x n 1 for fixed and these random recurrence relations have links with many fields including ergodic theory dynamical systems heavy tailed statistics spectral theory continued fractions and condensed matter physics

reconstruction of the clarke subdifferential by the lasry lions regularizations abstract we prove that the clarke subdifferential of a locally lipschitz function with a growth condition defined on a hilbert space can be represented by the derivatives of its lasry lions regularizations we complete a result of j benoist showing a similar representation for the clarke subdifferential of a lipschitz function in an arbitrary banach space by the clarke subdifferentials of its lasry lions regularizations

differentiability of lipschitz functions on banach spaces abstract the main result of this note says that if the norm of a banach space e is differentiable frechet gateaux or in some intermediate sense away from the origin then every locally lipschitz function defined on an open subset g of e is differentiable in the same sense at every point of some subset of g dense in g as the main corollary we prove that every lipschitz function on an asplund space is frechet differentiable on a dense set moreover in all these results the mean value theorem for the points of differentiability which is equivalent to the possibility of computing the clarke subdifferential from the points of differentiability is shown to hold

decay rates in probability metrics towards homogeneous cooling states for the inelastic maxwell model we quantify the long time behavior of solutions to the nonlinear boltzmann equation for spatially uniform freely cooling inelastic maxwell molecules by means of the contraction property of a suitable metric in the set of probability measures existence uniqueness and precise estimates of overpopulated high energy tails of the self similar profile proved in ref 9 are revisited and derived from this new liapunov functional for general initial conditions the solutions of the boltzmann equation are then proved to converge with computable rate as t to the self similar solution in this distance which metrizes the weak convergence of measures moreover we can relate this fourier distance to the euclidean wasserstein distance or tanaka functional proving also its exponential convergence towards the homogeneous cooling states the findings are relevant in the understanding of the conjecture formulated by ernst and brito in refs 15 16 and complement and improve recent studies on the same problem of bobylev and cercignani 9 and bobylev cercignani and one of the authors 11

the resistance of randomly grown trees copyright 2011 iop publishing ltd this is a preprint version of the published article which can be accessed from the link below

second order sufficient conditions for error bounds in banach spaces recently huang and ng presented second order sufficient conditions for error bounds of continuous and ga teaux differentiable functions in banach spaces wu and ye dropped the assumption of huang and ng on ga teaux differentiability but required the space to be a hilbert space we carry on this research in two directions first we extend wu and ye s result to some non hilbert spaces second same as huang and ng we work on banach spaces but provide different second order sufficient conditions that may allow the function to be non ga teaux differentiable

analysis of directed networks via the matrix exponential abstract the matrix exponential has been identified as a useful tool for the analysis of undirected networks with sound theoretical justifications for its ability to model important aspects of a given network its use for directed networks however is less developed and has been less successful so far in this article we discuss some methods to identify important nodes in a directed network using the matrix exponential taking into account that the notion of importance changes whether we consider the influence of a given node along the edge directions downstream influence or how it is influenced by directed paths that point to it upstream influence in addition we introduce a family of importance measures based on counting walks that are allowed to reverse their direction a limited number of times thus capturing relationships arising from influencing the same nodes or being influenced by the same nodes without sacrificing information about edge direction these measures provide information about branch points

approximately convex functions and approximately monotonic operators abstract we present characterizations of some generalized convexity properties of functions with the help of a general subdifferential we stress the case of lower semicontinuous functions we also study the important case of marginal functions and we provide representation results

pagerank beyond the web google s pagerank method was developed to evaluate the importance of web pages via their link structure the mathematics of pagerank however are entirely general and apply to any graph or network in any domain thus pagerank is now regularly used in bibliometrics social and information network analysis and for link prediction and recommendation it s even used for systems analysis of road networks as well as biology chemistry neuroscience and physics we ll see the mathematics and ideas that unite these diverse applications