an approximability result of the multi vehicle scheduling problem on a path with release and handling times in this paper we consider a scheduling problem of vehicles on a path g with n vertices and n 1 edges there are m identical vehicles each vertex in g has exactly one job any of the n jobs must be processed by some vehicle each job has a release time and a handling time with the edges symmetric travel times are associated the problem asks to find an optimal schedule of the m vehicles that minimizes the maximum completion time of all the jobs the problem is known to be np hard for any fixed m 2 in this paper we show that the problem with a fixed m admits a polynomial time approximation scheme our algorithm can be extended to the case where g is a tree so that a polynomial time approximation scheme is obtained if m and the number of leaves in g are fixed

on fiedler and parter vertices of acyclic matrices let a be a real symmetric matrix and let be a real number the algebraic multiplicity of as an eigenvalue of a is denoted by ma and the principal submatrix of a obtained by deleting row and column i from a is denoted by a i if ma i ma resp ma i ma then index i is said to be a fiedler vertex resp a parter vertex of a for in this paper we provide geometric characterizations of fiedler and parter vertices of acyclic matrices and give a geometric proof for the parter wiener theorem in c r johnson a leal duarte c m saiago the parter wiener theorem refinement and generalization siam j matrix anal appl 25 2003 352 361 furthermore we describe a structure of an acyclic matrix in terms of fiedler and parter vertices which enables us to construct an acyclic matrix of a desired form according to the locations of fiedler and parter vertices

a time domain numerical modeling of two dimensional wave propagation in porous media with frequency dependent dynamic permeability an explicit finite difference scheme is presented for solving the two dimensional biot equations of poroelasticity across the full range of frequencies the key difficulty is to discretize the johnson koplik dashen jkd model which describes the viscous dissipations in the pores indeed the time domain version of biot jkd model involves order 1 2 fractional derivatives which amount to a time convolution product to avoid storing the past values of the solution a diffusive representation of fractional derivatives is used the convolution kernel is replaced by a finite number of memory variables that satisfy local in time ordinary differential equations the coefficients of the diffusive representation follow from an optimization procedure of the dispersion relation then various methods of scientific computing are applied the propagative part of the equations is discretized using a fourth order finite difference scheme whereas the diffusive part is solved exactly an immersed interface method is implemented to discretize the geometry on a cartesian grid and also to discretize the jump conditions at interfaces numerical experiments are proposed in various realistic configurations

on the possible multiplicities of the eigenvalues of a hermitian matrix whose graph is a tree we consider the general problem of determining which lists of multiplicities for the eigenvalues occur among hermitian matrices the graph of whose off diagonal entries is a given tree several restrictions are cited and a construction strategy is given together these are sufficient to characterize all lists for each tree in two infinite classes the double paths and generalized stars and to tabulate all lists for trees on fewer than nine vertices such tables should be useful for formulating and dispelling general conjectures

a new solution algorithm for improving performance of ant colony optimization this study proposes an improved solution algorithm using ant colony optimization aco for finding global optimum for any given test functions the procedure of the aco algorithms simulates the decision making processes of ant colonies as they forage for food and is similar to other artificial intelligent techniques such as tabu search simulated annealing and genetic algorithms aco algorithms can be used as a tool for optimizing continuous and discrete mathematical functions the proposed algorithm is based on each ant searches only around the best solution of the previous iteration with b the proposed algorithm is called as acorses an abbreviation of aco reduced search space b is proposed for improving aco s solution performance to reach global optimum fairly quickly the acorses is tested on fourteen mathematical test functions taken from literature and encouraging results were obtained the performance of acorses is compared with other optimization methods the results showed that the acorses performs better than other optimization algorithms available in literature in terms of minimum values of objective functions and number of iterations

time domain numerical modeling of wave propagation in 2d heterogeneous porous media this paper deals with the numerical modeling of wave propagation in porous media described by biot s theory the viscous efforts between the fluid and the elastic skeleton are assumed to be a linear function of the relative velocity which is valid in the low frequency range the coexistence of propagating fast compressional wave and shear wave and of a diffusive slow compressional wave makes numerical modeling tricky to avoid restrictions on the time step the biot s system is splitted into two parts the propagative part is discretized by a fourth order ader scheme while the diffusive part is solved analytically near the material interfaces a space time mesh refinement is implemented to capture the small spatial scales related to the slow compressional wave the jump conditions along the interfaces are discretized by an immersed interface method numerical experiments and comparisons with exact solutions confirm the accuracy of the numerical modeling the efficiency of the approach is illustrated by simulations of multiple scattering

all pairs suffice for a p set abstract a p set of a symmetric matrix a is a set of indices such that the nullity of the matrix obtained from a by removing rows and columns indexed by is more than the nullity of a it is known that each subset of a p set is a p set it is also known that a set of indices such that each singleton subset is a p set need not be a p set this note shows that if all pairs of vertices of a set with at least two elements are p sets then the set is a p set

the minimum number of eigenvalues of multiplicity one in a diagonalizable matrix over a field whose graph is a tree abstract it is known that an n by n hermitian matrix n 2 whose graph is a tree necessarily has at least two eigenvalues the largest and smallest in particular with multiplicity 1 recently much of the multiplicity theory for eigenvalues of hermitian matrices whose graph is a tree has been generalized to geometric multiplicities of eigenvalues of matrices over a general field whose graph is a tree however the two 1 s fact does not generalize here we give circumstances under which there are two 1 s and give several examples without two 1 s that limit our positive results

equivalent acoustic impedance model part 2 analytical approximation abstract in the context of the prediction of noise levels in vibroacoustic systems numerical models or analytical models can be developed generally numerical models are adapted to the low and medium frequency ranges and analytical models to the medium and high frequency ranges for analytical models a classical approximation consists of modelling the multilayer system by an equivalent acoustic impedance this paper deals with a multilayer system consisting of a porous medium inserted between two thin plates part 1 of this paper is devoted to the experiments performed and to the development of a probabilistic algebraic model for the equivalent acoustic impedance in the present part 2 an analytical method is constructed for this multilayer system this method consists of introducing the unbounded medium in the plane directions x1 and x2 while the medium is bounded in the x3 direction a two dimensional space fourier transform introducing the wave vector co ordinates k1 and k2 is used for a given frequency and for k1 and k2 fixed the boundary value problem in x3 consists of 12 differential equations in x3 whose coefficients depend on k1 and k2 with boundary conditions this system of equations is solved by using adapted algebraic calculations by inverse fourier transform with respect to k1 and k2 the equivalent acoustic impedance is deduced the method which is proposed is not usual finally a comparison of this analytical approach is compared with the experimental results

computation of minimal rank and path cover number for certain graphs for a given undirected graph g the minimum rank of g is defined to be the smallest possible rank over all real symmetric matrices a whose i j th entry is non zero whenever i j and i j is an edge in g the path cover number of g is the minimum number of vertexdisjoint paths occurring as induced subgraphs of g that cover all the vertices of g for trees the relationship between minimum rank and path cover number is completely understood however for non trees only sporadic results are known we derive formulae for the minimum rank and path cover number for graphs obtained from edge sums and formulae for minimum rank of vertex sums of graphs in addition we examine previously identified special types of vertices and attempt to unify the theory behind them

inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree the case of generalized stars and double generalized stars we characterize the possible lists of ordered multiplicities among matrices whose graph is a generalized star a tree in which at most one vertex has degree greater than 2 or a double generalized star here the inverse eigenvalue problem iep for symmetric matrices whose graph is a generalized star is settled the answer is consistent with a conjecture that determination of the possible ordered multiplicities is equivalent to the iep for a given tree moreover a key spectral feature of the iep in the case of generalized stars is shown to characterize them among trees

general representations for wavefield modeling and inversion in geophysics acoustic electromagnetic elastodynamic poroelastic and electroseismic waves are all governed by a unified matrix vector wave equation the matrices in this equation obey the same symmetry properties for each of these wave phenomena this implies that the wave vectors for each of these phenomena obey the same reciprocity theorems by substituting green s matrices in these reciprocity theorems unified wavefield representations are obtained analogous to the well known acoustic wavefield representations these unified representations find applications in geophysical modeling migration inversion multiple elimination and interferometry

lower bounds for minimum semidefinite rank from orthogonal removal and chordal supergraphs abstract the minimum semidefinite rank msr of a graph is the minimum rank among positive semidefinite matrices with the given graph the os number is a useful lower bound for msr which arises by considering ordered vertex sets with some connectivity properties in this paper we develop two new interpretations of the os number we first show that os number is also equal to the maximum number of vertices which can be orthogonally removed from a graph under certain nondegeneracy conditions our second interpretation of the os number is as the maximum possible rank of chordal supergraphs who exhibit a notion of connectivity we call isolation preserving these interpretations not only give insight into the os number but also allow us to prove some new results for example we show that msr g g 2 if and only if os g gzsfnc 2

sets of parter vertices which are parter sets abstract given an hermitian matrix whose graph is a tree having a multiple eigenvalue the parter wiener theorem guarantees the existence of principal submatrices for which the multiplicity of increases the vertices of the tree whose removal gives rise to these principal submatrices are called weak parter vertices and with some additional conditions are called parter vertices a set of k parter vertices whose removal increases the multiplicity of by k is called parter set as observed by several authors a set of parter vertices is not necessarily a parter set in this paper we prove that if a is a symmetric matrix whose graph is a tree and is an eigenvalue of a whose multiplicity does not exceed 3 then every set of parter vertices for relative to a is also a parter set

minimum rank matrices with prescribed graph abstract we study properties of real symmetric matrices with prescribed graph and lowest possible rank we concentrate on the case where the graph is a tree and give an algorithm for computing the minimum possible rank in this case a more general problem is also considered

interface conditions for biot s equations of poroelasticity interface conditions at a boundary between two porous media are derived directly from biot s equations of poroelasticity by replacing the discontinuity surface with a thin transition layer in which the properties of the medium change rapidly yet continuously and then taking the limit as the thickness of the transition layer approaches zero the interface conditions obtained in this way the well known open pore conditions are shown to be the only ones that are fully consistent with the validity of biot s equations throughout the poroelastic continuum including surfaces across which the medium properties are discontinuous but partially blocked or completely impermeable interfaces exist these may be looked upon as the case of a thin layer with its permeability taken to be proportional to the layer thickness again in the limit as layer thickness approaches zero this approach can serve as a simple recipe for modeling such an interface in any heterogeneous numerical scheme for poroelastic media

questions conjectures and data about multiplicity lists for trees abstract we review and discuss a number of questions and conjectures about multiplicity lists occurring among real symmetric matrices whose graph is a tree our investigation is aided by a new electronic database containing all multiplicity lists for trees on fewer than 12 vertices some questions and conjectures are familiar and some are new and new information is given about several

linearly independent vertices and minimum semidefinite rank we study the minimum semidefinite rank of a graph using vector representations of the graph and of certain subgraphs we present a sufficient condition for when the vectors corresponding to a set of vertices of a graph must be linearly independent in any vector representation of that graph and conjecture that the resulting graph invariant is equal to minimum semidefinite rank rotation of vector representations by a unitary matrix allows us to find the minimum semidefinite rank of the join of two graphs we also improve upon previous results concerning the effect on minimum semidefinite rank of the removal of a vertex

noncrossing partitions noncrossing graphs and q permanental equations abstract we first prove a few simple results to illustrate some algebraic combinatorial features of the q permanent this is followed by a characterization of a noncrossing permutation in terms of the numbers of inversions of its cycles then we use a family of derivative formulas for the q permanent of a square matrix a to characterize several structures of noncrossing kind each such formula f characterizes a set d f of digraphs in the sense that d d f iff f is valid for all matrices a with digraph d in this way we characterize among others digraphs with noncrossing permutation subdigraphs noncrossing graphs noncrossing forests we use the derivative formulas to prove two particular cases of a conjecture on the q monotonicity of the q permanent of a hermitian positive definite matrix

numerical enclosure for multiple eigenvalues of an hermitian matrix whose graph is a tree abstract in this paper we consider a numerical enclosure method for multiple eigenvalues of an hermitian matrix whose graph is a tree if an hermitian matrix a whose graph is a tree has multiple eigenvalues it has the property that matrices which are associated with some branches in the undirected graph of a have the same eigenvalues by using this property and interlacing inequalities for hermitian matrices we show an enclosure method for multiple eigenvalues of an hermitian matrix whose graph is a tree since we do not generally know whether a given matrix has exactly a multiple eigenvalue from approximate computations we use the property of interlacing inequalities to enclose some eigenvalues including multiplicities in this process we only use the enclosure of simple eigenvalues to enclose a multiple eigenvalue by using a computer and interval arithmetic

the inverse inertia problem for graphs cut vertices trees and a counterexample abstract let g be an undirected graph on n vertices and let s g be the set of all real symmetric n n matrices whose nonzero off diagonal entries occur in exactly the positions corresponding to the edges of g the inverse inertia problem for g asks which inertias can be attained by a matrix in s g we give a complete answer to this question for trees in terms of a new family of graph parameters the maximal disconnection numbers of a graph we also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs finally we give an example of a graph that is not inertia balanced which settles an open problem from the october 2006 aim workshop on spectra of families of matrices described by graphs digraphs and sign patterns we also determine some restrictions on the inertia set of any graph

on the minimum rank of the join of graphs and decomposable graphs abstract for a given undirected graph g the minimum rank of g is defined to be the smallest possible rank over all real symmetric matrices a whose i j th entry is nonzero whenever i j and i j is an edge in g in this work we consider joins and unions of graphs and characterize the minimum rank of such graphs in the case of balanced inertia several consequences are provided for decomposable graphs also known as cographs

on the relative position of multiple eigenvalues in the spectrum of an hermitian matrix with a given graph for hermitian matrices whose graph is a given tree the relationships among vertex degrees multiple eigenvalues and the relative position of the underlying eigenvalue in the ordered spectrum are discussed in detail in the process certain aspects of special vertices whose removal results in an increase in multiplicity are investigated

on the number of p vertices of some graphs abstract we provide positive answers to some open questions presented recently by kim and shader on a continuity like property of the p vertices of nonsingular matrices whose graph is a path a criterion for matrices associated with more general trees to have at most n 1 p vertices is established the cases of the cycles and stars are also analyzed several algorithms for generating matrices with a given number of p vertices are proposed

numerical modeling of 1d transient poroelastic waves in the low frequency range propagation of transient mechanical waves in porous media is numerically investigated in 1d the framework is the linear biot model with frequency independent coefficients the coexistence of a propagating fast wave and a diffusive slow wave makes numerical modeling tricky a method combining three numerical tools is proposed a fourth order ader scheme with time splitting to deal with the time marching a space time mesh refinement to account for the small scale evolution of the slow wave and an interface method to enforce the jump conditions at interfaces comparisons with analytical solutions confirm the validity of this approach

spectral multiplicity and splitting results for a class of qualitative matrices abstract this paper presents some results concerning the location and multiplicity of eigenvalues of sign symmetric matrices whose associated graphs are trees in particular it extends previous spectral multiplicity and splitting results proved by others

the change in multiplicity of an eigenvalue due to adding or removing edges abstract we investigate the change in the multiplicities of the eigenvalues of a hermitian matrix with a simple graph g when edges are inserted into g or removed from g we focus upon cases in which the multiplicity of the eigenvalue does not change due to inserting or removing edges incident to a vertex furthermore we show how the change in the multiplicities of the eigenvalues occur when two disjoint graphs are connected with one edge based upon the status of the vertices that are connected lastly we give the possible classifications of cut edges in a graph and characterize the occurrence of each possible status

multi depot vehicle scheduling problems with time windows and waiting costs the multi depot vehicle scheduling problem with time windows mdvsptw consists of scheduling a fleet of vehicles to cover a set of tasks at minimum cost each task is restricted to begin within a prescribed time interval and vehicles are supplied by different depots the problem is formulated as an integer nonlinear multi commodity network flow model with time variables and is solved using a column generation approach embedded in a branch and bound framework this paper breaks new ground by considering costs on exact waiting times between two consecutive tasks instead of minimal waiting times this new and more realistic cost structure gives rise to a nonlinear objective function in the model optimal and heuristic versions of the algorithm have been extensively tested on randomly generated urban bus scheduling problem ubsp and freight transport scheduling problem ftsp the results show that such a general solution methodology outperforms specialized algorithms when minimal waiting costs are used and can efficiently treat the case with exact waiting costs

heuristic approaches for solving transit vehicle scheduling problem with route and fueling time constraints abstract electric bus scheduling problem can be defined as vehicle scheduling problem with route and fueling time constraints vsprftc every vehicle s travel miles route time after charging is limited thus the vehicle must be recharged after taking several trips and the minimal charging time fueling time must be satisfied a multiple ant colony algorithm aca was presented to solve vsprftc based on aca used to solve traveling salesman problem tsp a new metaheuristic approach inspired by the foraging behavior of real colonies of ants the vsprftc considered in this paper minimizes a multiple hierarchical objective function the first objective is to minimize the number of tours or vehicles and the second is to minimize the total deadhead time new improvement of aca as well as detailed operating steps was provided on the basis of former algorithm then in order to settle contradiction between accelerating convergence and avoiding prematurity or stagnation improvement on route construction rule and pheromone updating rule was adopted a group feasible trip sets blocks had been produced after the process of applying aca in dealing with the fueling time constraint a bipartite graphic model and its optimization algorithm are developed for trip set connecting in a hub and spoke network system to minimize the number of vehicle required the maximum matching of the bipartite graph is obtained by calculating the maximum inflow with the ford fulkerson algorithm at last an example was analyzed to demonstrate the correctness of the application of this algorithm it proved to be more efficient and robust in solving this problem

dynamic poroelasticity of thinly layered structures abstract compressional seismic p waves propagating in poroelastic fluid saturated laminated sediments are strongly affected by the medium heterogeneity here simple analytical expressions for the p wave phase velocity and attenuation coefficient are derived both are functions of frequency and statistical medium parameters such as correlation lengths and variances the theoretical results are compared with results from numerical simulations and show good agreement in heterogeneous media impedance fluctuations lead to poroelastic scattering variations of the layer compressibilities cause inter layer flow a 1 d macroscopic local flow from the seismic frequency range 10 100 hz up to ultrasonic frequencies attenuation due to heterogeneity is strongly enhanced compared to homogeneous biot models the new theory automatically includes different asymptotic approximations such as poroelastic backus averaging in the quasi static and the no flow limit geometrical optics and intermediate frequency ranges

maximal p sets of matrices whose graph is a tree abstract the graph of a real n n symmetric matrix a g a is the undirected graph on n vertices whose edges correspond precisely with the nonzero off diagonal entries of a let m a 0 denote the multiplicity of 0 as an eigenvalue of a it is possible that the multiplicity of 0 as an eigenvalue of an n 1 n 1 principal submatrix a n 1 of a is larger than m a 0 when this occurs the vertex in g a corresponding to the row and column of a that was deleted to form a n 1 is called a p vertex of a if there is a list of principal submatrices of a a a n 1 a n 2 a n k each one being obtained from the matrix preceding it in the list by deleting a row and column such that m a 0 m a n 1 0 m a n k 0 the set of vertices in g a corresponding to the rows and columns of a that were deleted to form a n k is a called a p set of a let a be a matrix whose graph is a tree i e a connected graph with no cycles it is known that a set of p vertices of a is not necessarily a p set of a a main result of this paper shows that if every subset of size 2 of a set is a p set of a then the entire set is a p set of a this and other results are used to determine the maximal p sets and thus all of the p sets of a

computational error bounds for multiple or nearly multiple eigenvalues abstract in this paper bounds for clusters of eigenvalues of non selfadjoint matrices are investigated we describe a method for the computation of rigorous error bounds for multiple or nearly multiple eigenvalues and for a basis of the corresponding invariant subspaces the input matrix may be real or complex dense or sparse the method is based on a quadratically convergent newton like method it includes the case of defective eigenvalues uncertain input matrices and the generalized eigenvalue problem computational results show that verified bounds are still computed even if other eigenvalues or clusters are nearby the eigenvalues under consideration

the structure of matrices with a maximum multiplicity eigenvalue there is remarkable and distinctive structure among hermitian matrices whose graph is a given tree t and that have an eigenvalue of multiplicity that is a maximum for t among such structure we give several new results 1 no vertex of t may be neutral 2 neutral vertices may occur if the largest multiplicity is less than the maximum 3 every parter vertex has at least two downer branches 4 removal of a parter vertex changes the status of no other vertex and 5 every set of parter vertices forms a parter set statements 3 4 and 5 are also not generally true when the multiplicity is less than the maximum some of our results are used to give further insights into prior results and both the review of necessary background and the development of new structural lemmas may be of independent interest

graphs that allow all the eigenvalue multiplicities to be even abstract let g be an undirected graph on n vertices and let s g be the set of all n n real symmetric matrices whose nonzero off diagonal entries occur in exactly the positions corresponding to the edges of g the inverse eigenvalue problem for a graph g is a problem of determining all possible lists that can occur as the lists of eigenvalues of matrices in s g this question is in general hard to answer and several variations were studied most notably the minimum rank problem in this paper we introduce the problem of determining for which graphs g there exists a matrix in s g whose characteristic polynomial is a square i e the multiplicities of all its eigenvalues are even we solve this question for several families of graphs

wave simulation in 2d heterogeneous transversely isotropic porous media with fractional attenuation a cartesian grid approach a time domain numerical modeling of transversely isotropic biot poroelastic waves is proposed in two dimensions the viscous dissipation occurring in the pores is described using the dynamic permeability model developed by johnson koplik dashen jkd some of the coefficients in the biot jkd model are proportional to the square root of the frequency in the time domain these coefficients introduce shifted fractional derivatives of order 1 2 involving a convolution product based on a diffusive representation the convolution kernel is replaced by a finite number of memory variables that satisfy local in time ordinary differential equations resulting in the biot da diffusive approximation model the properties of both the biot jkd and the biot da models are analyzed hyperbolicity decrease of energy dispersion to determine the coefficients of the diffusive approximation two approaches are analyzed gaussian quadratures and optimization methods in the frequency range of interest the nonlinear optimization is shown to be the better way of determination a splitting strategy is then applied to approximate numerically the biot da equations the propagative part is discretized using a fourth order ader scheme on a cartesian grid whereas the diffusive part is solved exactly an immersed interface method is implemented to take into account heterogeneous media on a cartesian grid and to discretize the jump conditions at interfaces numerical experiments are presented comparisons with analytical solutions show the efficiency and the accuracy of the approach and some numerical experiments are performed to investigate wave phenomena in complex media such as multiple scattering across a set of random scatterers

vehicle routing problem with time windows and a limited number of vehicles abstract this paper introduces a variant of the vehicle routing problem with time windows where a limited number of vehicles is given m vrptw under this scenario a feasible solution is one that may contain either unserved customers and or relaxed time windows we provide a computable upper bound to the problem to solve the problem we propose a tabu search approach characterized by a holding list and a mechanism to force dense packing within a route we also allow time windows to be relaxed by introducing the notion of penalty for lateness in our approach customer jobs are inserted based on a hierarchical objective function that captures multiple objectives computational results on benchmark problems show that our approach yields solutions that are competitive to best published results on vrptw on m vrptw instances experiments show that our approach produces solutions that are very close to computed upper bounds moreover as the number of vehicles decreases the routes become more densely packed monotically this shows that our approach is good from both the optimality as well as stability point of view

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached