response variability of stochastic frame structures using evolutionary field theory the integral form for the variance of the response of stochastic statically indeterminate structural systems involving the so called variability response function vrf and the spectral density function of the stochastic field modelling the uncertain system properties is established for the first time in this paper using evolutionary spectra theory the vrf is a function depending on deterministic parameters related to the geometry boundary conditions mean material properties and loading of the structural system no approximations are involved in the derivation of the integral form however a conjecture has to be made that is validated using monte carlo simulations the uncertain system property considered is the inverse of the elastic modulus flexibility closed form expressions can be derived in principle for the vrf of any statically determinate or indeterminate frame system using a flexibility based formulation alternatively a fast monte carlo simulation approach is provided to numerically evaluate the vrf it is shown in closed form and numerically that the vrf for statically indeterminate structures is a function of the standard deviation ff of the stochastic field modeling the inverse of the elastic modulus although the vrf depends on ff it appears to be independent of the functional form of the spectral density function of the stochastic field modeling the uncertain system properties for statically determinate structures the vrf is independent of ff the integral form can be used to compute the variance of the system response as well as its upper bound with minimal computational effort it also provides an excellent insight into the mechanisms controlling the response variability the upper bounds for the response variance are spectral and probability distribution free requiring knowledge of only the variance of the inverse of the elastic modulus the proposed bounds are realizable in the sense that it is possible to determine the probabilistic characteristics of the stochastic field that produces them several numerical examples are provided demonstrating the capabilities of the methodology

some exact solutions for the bending of beams with spatially stochastic stiffness to the best of the knowledge of the authors there are no exact solutions available for the bending of beams with spatially stochastic stiffness investigators are therefore utilizing various approximate techniques in the present study a new method is developed to obtain exact solutions for first and second moments of displacements for statically determinate beams that have spatially random stiffness the method is based on the full probabilistic characterization of the random stiffness so that the solutions are valid for any value of the coefficient of variation of the stiffness the deterministic governing equations and boundary conditions derived for both first and second moments allow apparently for the first time in the literature the exact solutions for the mean and covariance functions of the displacements to be determined two governing equations are uncoupled from each other and can be solved separately several exact solutions for the mean and covariance functions are obtained to illustrate the application of the method it is hoped that the exact solutions determined will serve as benchmark solutions to enable the researchers to check the accuracy of various approximate analytical and numerical techniques on the test solutions presented in this study

on the reduced signless laplacian spectrum of a degree maximal graph abstract for a simple graph g the signless laplacian of g is the matrix a g d g where a g is the adjacency matrix and d g is the diagonal matrix of vertex degrees of g the reduced signless laplacian of g is the matrix g b g where b g is the reduced adjacency matrix of g and g is the diagonal matrix whose diagonal entries are the common degrees for vertices belonging to the same neighborhood equivalence class of g a graph is said to be degree maximal if it is connected and its degree sequence is not majorized by the degree sequence of any other connected graph for a maximal graph we obtain a formula for the characteristic polynomial of its reduced signless laplacian and use the formula to derive a localization result for its reduced signless laplacian eigenvalues and to compare the signless laplacian spectral radii of two well known maximal graphs we also obtain a necessary condition for a maximal graph to have maximal signless laplacian spectral radius among all connected graphs with given numbers of vertices and edges

generalized variability response functions for two dimensional elasticity problems the ability to determine probabilistic characteristics of response quantities in structural mechanics e g displacements stresses as well as effective material properties is restricted due to lack of information on the probabilistic characteristics of the uncertain system parameters the concept of the variability response function vrf has been proposed as a means to systematically capture the effect of the stochastic spectral characteristics of uncertain system parameters modeled by homogeneous random fields on the uncertain structural response the key property of the vrf in its classical sense is its independence from the marginal probability distribution function pdf and the spectral density function sdf of the uncertain system parameters it depends only on the deterministic structural configuration and boundary conditions proofs have been provided for the existence of vrfs for linear and some nonlinear statically determinate beams for statically indeterminate structures the monte carlo based generalized variability response function gvrf methodology has been proposed recently as a generalization of the vrf concept to indeterminate linear and some nonlinear beams the methodology computes gvrfs which are analogous to vrfs for statically determinate structures and evaluates their dependence or lack thereof on the pdf and sdf of the random field thereby providing an estimate of the accuracy of the gvrf in this paper the gvrf methodology is extended to problems involving two dimensional linear continua whose stochasticity is characterized by statistically homogeneous random fields after detailing the gvrf methodology for two dimensional random fields two numerical examples are provided gvrfs are computed for the displacement response and for the effective compliance of linear plane stress systems

on the maximal index of graphs with a prescribed number of edges abstract among the graphs with a prescribed number of edges those with maximal index are determined the result confirms a conjecture of brualdi and hoffman

an adaptive spectral galerkin stochastic finite element method using variability response functions a methodology is proposed in this paper to construct an adaptive sparse polynomial chaos pc expansion of the response of stochastic systems whose input parameters are independent random variables modeled as random fields the proposed methodology utilizes the concept of variability response function vrf in order to compute an a priori low cost estimate of the spatial distribution of the second order error of the response as a function of the number of terms used in the truncated karhunen lo eve kl expansion this way the influence of the response variance to the spectral content correlation structure of the random input is taken into account through a spatial variation of the truncated kl terms the criterion for selecting the number of kl terms at different parts of the structure is the uniformity of the spatial distribution of the second order error this way a significantly reduced number of pc coefficients with respect to classical pc expansion is required in order to reach a uniformly distributed target second order error this results in an increase of sparsity of the coefficient matrix of the corresponding linear system of equations leading to an enhancement of the computational efficiency of the spectral stochastic finite element method ssfem copyright c 2014 john wiley sons ltd

degree maximal graphs are laplacian integral abstract an integer sequence d d 1 d 2 d n is graphic if there is a graph whose degree sequence is d a graph g is maximal if its degree sequence is majorized by no other graphic sequence the laplacian matrix of g is l g d g a g where d g is the diagonal matrix of vertex degrees and a g is the 0 1 adjacency matrix the article contains an explicit algorithmic construction of all maximal graphs from which it follows e g that apart from 0 the laplacian spectrum of a maximal graph is the conjugate of its degree sequence

the greedy strategy for optimizing the perron eigenvalue we address the problems of minimizing and of maximizing the spectral radius over a compact family of non negative matrices those problems being hard in general can be efficiently solved for some special families we consider the so called product families where each matrix is composed of rows chosen independently from given sets a recently introduced greedy method works very fast however it is applicable mostly for strictly positive matrices for sparse matrices it often diverges and gives a wrong answer we present the selective greedy method that works equally well for all non negative product families including sparse ones for this method we prove a quadratic rate of convergence and demonstrate its efficiency in numerical examples the numerical examples are realised for two cases finite uncertainty sets and polyhedral uncertainty sets given by systems of linear inequalities in dimensions up to 2000 the matrices with minimal maximal spectral radii in product families are found within a few iterations applications to dynamical systems and to the graph theory are considered

maximal graphs and graphs with maximal spectral radius abstract a maximal graph is a connected graph with degree sequence not majorized by the degree sequence of any other graph relationships between three different sequences of integers that describe maximal graphs are given recursive and explicit forms of the characteristic polynomial of the adjacency matrix of a maximal graph are derived a correspondence between maximal graphs and connected graphs with stepwise adjacency matrices shows that among all connected graphs with n vertices and e edges the graph with maximal spectral radius is a maximal graph such maximal graphs are identified for certain values of n and e

the method of layer potentials for the heat equation in lipschitz cylinders pour l equation de la chaleur dans des cylindres de lipschitz on etudie des problemes aux valeurs limites d initiales avec conditions de neumann et de dirichlet

flexibility based upper bounds on the response variability of simple beams spectral and probability distribution free upper bounds on the response variability of both statically determinate and indeterminate beams are established in the present paper based on exact closed form analytic expressions derived for the variance of the response displacement a conjecture has to be made in the case of statically indeterminate beams in order to establish these bounds the conjecture is supported through an argument postulating the existence of an integral form for the variance of the response displacement and through a brute force optimization procedure providing numerical validation such bounds require knowledge of only the variance of the stochastic field modeling the inverse of the elastic modulus and are realizable in the sense that it is possible to fully determine the probabilistic characteristics of the stochastic field modeling the inverse of the elastic modulus that produces them furthermore it is possible to fully determine also the corresponding stochastic field modeling the elastic modulus that produces these bounds these spectral and probability distribution free bounds can also be computed numerically using a so called fast monte carlo simulation procedure that does not require a closed form analytic expression for the response displacement making this approach much more general numerical examples are provided involving a statically determinate and a statically indeterminate beam

the least eigenvalue of graphs with given connectivity abstract let g be a simple graph and a g be the adjacency matrix of g the eigenvalues of g are those of a g in this paper we characterize the graphs with the minimal least eigenvalue among all graphs of fixed order with given vertex connectivity or edge connectivity

spectral simplex method we develop an iterative optimization method for finding the maximal and minimal spectral radius of a matrix over a compact set of nonnegative matrices we consider matrix sets with product structure i e all rows are chosen independently from given compact sets row uncertainty sets if all the uncertainty sets are finite or polyhedral the algorithm finds the matrix with maximal minimal spectral radius within a few iterations it is proved that the algorithm avoids cycling and terminates within finite time the proofs are based on spectral properties of rank one corrections of nonnegative matrices the practical efficiency is demonstrated in numerical examples and statistics in dimensions up to 500 some generalizations to non polyhedral uncertainty sets including euclidean balls are derived finally we consider applications to spectral graph theory mathematical economics dynamical systems and difference equations

analysis of mean and mean square response of general linear stochastic finite element systems a general finite element based formulation is presented for the analysis of the mean and mean square response of stochastic structural systems whose material properties are described by random fields a flexibility based formulation is followed that does not involve any approximations closed formed integral expressions for the mean and mean square value of the response displacement of statically indeterminate stochastic structures are introduced in this paper these integral expressions involve the spectral density function of a stochastic field modeling the uncertain material properties and two new quantities named the mean response function mrf and the variability response function for the mean square response vrf1 the mrf and the vrf1 have many similarities with the classical variability response function vrf used to estimate the variance of the response displacement the existence of the mrf and the vrf1 depends on a conjecture that is validated numerically using a brute force monte carlo simulation approach a so called finite element method based fast monte carlo simulation procedure fem fmcs is introduced for the accurate and efficient numerical evaluation of the mrf and the vrf1 this methodology is established for the analysis of stochastic beam frame structures as well as for more general stochastic finite element systems i e plane stress and shell problems numerical examples are provided including a statically indeterminate beam a plane stress problem and a shell structure the mrf and vrf1 are used to perform sensitivity parametric analyses with respect to various probabilistic characteristics involved in the problem i e correlation distance standard deviation and to establish realizable upper bounds on the mean value of the response displacement throughout this paper the vrf is computed in parallel to the mrf and the vrf1 it should be noted that although the concept of the vrf has been introduced in an earlier work this is the first time that a flexibility based approach has been used to estimate the vrf for two dimensional problems

on the maximal index of connected graphs abstract let h n e denote the set of all connected graphs having n vertices and e edges the graphs in h n n k with maximal index are determined for k of form r 2 1 and n arbitrary

connected graphs with maximal q index the one dominating vertex case abstract by the signless laplacian of a simple graph g we mean the matrix q g d g a g where a g d g denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of g for every pair of positive integers n k it is proved that if 3 k n 3 then h n k the graph obtained from the star k 1 n 1 by joining a vertex of degree 1 to k 1 other vertices of degree 1 is the unique connected graph that maximizes the largest signless laplacian eigenvalue over all connected graphs with n vertices and n k edges

unoriented laplacian maximizing graphs are degree maximal abstract a connected graph is said to be unoriented laplacian maximizing if the spectral radius of its unoriented laplacian matrix attains the maximum among all connected graphs with the same number of vertices and the same number of edges a graph is said to be threshold maximal if its degree sequence is not majorized by the degree sequence of any other graph and in addition the graph is connected it is proved that an unoriented laplacian maximizing graph is maximal and also that there are precisely two unoriented laplacian maximizing graphs of a given order and with nullity 3 our treatment depends on the following known characterization a graph g is threshold maximal if and only if for every pair of vertices u v of g the sets n u v n v u where n u denotes the neighbor set of u in g are comparable with respect to the inclusion relation and in addition the graph is connected a conjecture about graphs that maximize the unoriented laplacian matrix among all graphs with the same number of vertices and the same number of edges is also posed

graphs with maximal signless laplacian spectral radius abstract by the signless laplacian of a simple graph g we mean the matrix q g d g a g where a g d g denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of g it is known that connected graphs g that maximize the signless laplacian spectral radius q g over all connected graphs with given numbers of vertices and edges are degree maximal for a maximal graph g with n vertices and r distinct vertex degrees r r 1 1 it is proved that q g q h for some maximal graph h with n 1 respectively n vertices and the same number of edges as g if either g has precisely two dominating vertices or there exists an integer i 2 i r 2 respectively if there exist positive integers i l with l 2 i r 2 such that i r 1 i n 1 respectively i r 1 i l r l 1 graphs that maximize q g over the class of graphs with m edges and m k vertices for k 0 1 2 3 are completely determined

spectral stochastic finite element analysis for laminated composite plates abstract a spectral stochastic finite element formulation with consideration of multi layer effect and spatial variability of material properties is developed for probabilistic analysis of laminated composite plates the material properties of each lamina are modeled as a set of random fields and represented by the karhunen loeve expansion the expansion is incorporated into a spatial discretization in accordance with a standard finite element procedure based on the first order shear deformation theory a spectral expansion with use of polynomial chaos is employed to represent the stochastic nodal displacements in terms of standard normal random variables a preconditioning matrix is then proposed for the solution of spectral stochastic finite element equations with use of a preconditioning conjugate gradient technique the various statistics of interest of the system responses are finally obtained by means of the coefficients of the spectral expansion the numerical accuracy and the computational efficiency of the method are demonstrated by comparison with monte carlo simulation

hardy spaces and the neumann problem in l p for laplace s equation in lipschitz domains on donne des resultats optimaux pour la resolubilite du probleme de neumann dans des domaines de lipschitz a donnees dans l p on obtient des resultats du point extreme correspondant pour des espaces de hardy