a class of label correcting methods for the k shortest paths problem in this paper we deal with the problem of finding the firstk shortest paths from a single origin node to all other nodes of a directed graph in particular we define the necessary and sufficient conditions for a set of distance label vectors on the basis of which we propose a class of methods which can be viewed as an extension of the generic label correcting method for solving the classical single origin all destinations shortest path problem the data structure used is characterized by a set ofk lists of candidate nodes and the proposed methods differ in the strategy used to select the node to be extracted at each iteration the computational results show that 1 some label correcting methods are generally much faster than the double sweep method of shier 1979 2 the most efficient node selection strategies used for solving the classical single origin all destinations shortest path problem have proved to be effective also in the case of thek shortest paths

finite element approximations of a ladyzhenskaya model for stationary incompressible viscous flow some finite element approximation procedures are presented for a model proposed by ladyzhenskaya for stationary incompressible viscous flow the approximate problems are proved to be well posed and stable under standard assumptions on the finite element families the solutions of the approximate problems converge to the solution of the original problem under minimum regularity assumptions some error estimates are derived the optimal order of accuracy is assured with or even without using exact integration rules in the approximation procedure iterative methods for solving the discrete nonlinear problems and comments on some computational experiments are provided special attention is also paid to the common properties as well as differences between the approximation procedure presented here and the approximation for the stationary navier stokes equations

lie symmetries of multidimensional difference equations a method is presented for calculating the lie point symmetries of a scalar difference equation on a two dimensional lattice the symmetry transformations act on the equations and on the lattice they take solutions into solutions and can be used to perform symmetry reduction the method generalizes the one presented in a recent publication for the case of ordinary difference equations in turn it can easily be generalized to difference systems involving an arbitrary number of dependent and independent variables

graph theory applied to noise and vibration control in statistical energy analysis models a fundamental aspect of noise and vibration control in statistical energy analysis sea models consists in first identifying and then reducing the energy flow paths between subsystems in this work it is proposed to make use of some results from graph theory to address both issues on the one hand linear and path algebras applied to adjacency matrices of sea graphs are used to determine the existence of any order paths between subsystems counting and labeling them finding extremal paths or determining the power flow contributions from groups of paths on the other hand a strategy is presented that makes use of graph cut algorithms to reduce the energy flow from a source subsystem to a receiver one modifying as few internal and coupling loss factors as possible

lie group classification of second order ordinary difference equations a group classification of invariant difference models i e difference equations and meshes is presented in the continuous limit the results go over into lie s classification of second order ordinary differential equations the discrete model is a three point one and we show that it can be invariant under lie groups of dimension 0 n 6

response variance prediction in the statistical energy analysis of built up systems in the statistical energy analysis sea of high frequency noise and vibration a complex engineering structure is represented as an assembly of subsystems the response of the system to external excitation is expressed in terms of the vibrational energy of each subsystem and these energies are found by employing the principle of power balance strictly the computed energy is an average taken over an ensemble of random structures and for many years there has been interest in extending the sea prediction to the variance of the energy a variance prediction method for a general built up structure is presented here closed form expressions for the variance are obtained in terms of the standard sea parameters and an additional set of parameters alpha k that describe the nature of the power input to each subsystem k and alpha ks that describe the nature of the coupling between subsystems k and s the theory is validated by comparison with monte carlo simulations of plate networks and structural acoustic systems

infinitely many symmetries and conservation laws for quad graph equations via the gardner method the application of the gardner method for the generation of conservation laws to all the abs equations is considered it is shown that all the necessary information for the application of the gardner method namely b cklund transformations and initial conservation laws follows from the multidimensional consistency of abs equations we also apply the gardner method to an asymmetric equation which is not included in the abs classification an analog of the gardner method for the generation of symmetries is developed and applied to the discrete korteweg de vries equation it can also be applied to all the other abs equations

a bicriterion shortest path algorithm abstract among the network models one of the more popular is the so called shortest path problem this model is used whenever it is intended to minimize a linear function which represents a distance between a predetermined pair of nodes in a given network often a single objective function is not sufficient to completely characterize some real world problems for instance in a road network two parameters as cost and time can be assigned to each arc clearly the fastest path may be too costly nevertheless the decision maker must choose one solution possibly not the best for both criteria in this paper we present an algorithm for this problem with this algorithm a special set of paths the set of pareto optimal paths is determined one objective for any pareto optimal path can not be improved without worsening the other one

framework for the a posteriori error analysis of nonconforming finite elements this paper establishes a unified framework for the a posteriori error analysis of a large class of nonconforming finite element methods the theory assures reliability and efficiency of explicit residual error estimates up to data oscillations under the conditions h1 h2 and applies to several nonconforming finite elements the crouzeix raviart triangle element the han parallelogram element the nonconforming rotated nr parallelogram element of rannacher and turek the constrained nr parallelogram element of hu and shi the p 1 element on parallelograms due to park and sheen and the dssy parallelogram element the theory is extended to include 1 irregular meshes with at most one hanging node per edge

uniformly superconvergent analysis of an efficient two grid method for nonlinear bi wave singular perturbation problem abstract the main aim of this paper is to present a two grid method for the fourth order nonlinear bi wave singular perturbation problem with low order nonconforming finite element based on the ciarlet raviart scheme the existence and uniqueness of the approximation solution are demonstrated through the brouwer fixed point theorem and the uniform superconvergent estimates in the broken h 1 norm and l 2 norm are obtained which are independent of the perturbation parameter some numerical results indicate that the proposed method is indeed an efficient algorithm

an algorithm for the ranking of shortest paths an efficient computational implementation of a path deletion k shortest paths algorithm and a new algorithm for the same problem are presented in a path deletion k shortest paths algorithm a sequence g1 g2 gk of networks is is defined such that g1 is given network and its k th shortes path is trivially determined from the shortest path in gk in essence as soon as the shortest path in gk is determined it is excluded from gk in such a way that no new paths are formed and no more paths are deleted so far each gk two procedures are executed a shortest path algorithm and a path deletion algorithm in the presented computational implementation all the information resulting from the determination of the k th shortest path is carried throughout gk 1 gk 2 gk the new algorithm not only keeps this characteristics but also avoids the last k 1 executions of a shortest path algorithm which results in a suprising and very substantial reduction in the execution time in fact for randomly generated networks with 104 nodes and 105 arcs once the shortest the new algorithm computes the next 100 shortest paths in times of the order of 10 1 seconds to illustrate the efficiency of this algorithm comparative computational experiments are reported

symmetries of discrete dynamical systems differential difference equations of the form un fn t un 1 un un 1 are classified according to their continuous lie point symmetry groups it is shown that for nonlinear equations the symmetry group can be at most seven dimensional the integrable toda lattice is a member of this class and has a four dimensional symmetry group

estimation of acoustic source strength within a cylindrical duct by inverse methods one method for deducing the strength of an acoustic source distribution from measurements of the radiated field involves the inversion of the matrix of frequency response functions relating the field measurement points to the strengths of a number of point sources used to represent the source distribution this method uses the singular value decomposition svd as the primary analysis tool and has shown some promise in the transformation process associated with the svd the strength of the discretized point source can be simply represented by the pressure field and the inversion of corresponding matrix of the frequency response function and the transformed pressures and source distribution are related by single real numbers only i e the singular values therefore the resolution and the accuracy of the reconstruction produced by such inversion methods will be highly dependent on how these small singular values are treated during the inversion process in the present work the emphasis is placed on the description of the inverse method for the investigation of the characteristics of acoustic sources within a circular duct firstly an analytical model is developed of the internal field from a finite duct with open ends an examination is presented of the capability of the inverse method in dealing with in duct sound source problems these guide the design of an optimal sensor array which can make the condition number as small as possible for the assumed source distribution also a series of experimental demonstrations are given of the proposed in duct sensor array for producing high resolution information regarding in duct sources

on constructing solutions for the functional equation z x y n z a11x a12y a21x a22y n 1 abstract the concept of invariance factors to obtain first integrals for difference equations was presented by van horssen 2002 6 it was shown that all invariance factors have to satisfy a functional equation one of the main difficulties in finding first integrals for a system of first order difference equations is solving the aforementioned functional equation in this paper we consider a functional equation which is related to a system of two first order linear difference equations linear transformations and an adapted version of the method of separation of variables will be used to construct solutions of this functional equation the constructed solutions of this functional equation do not seem to be available in the literature

shortest path problem with forbidden paths the elementary version abstract this paper addresses the elementary shortest path problem with forbidden paths the main aim is to find the shortest paths from a single origin node to every other node of a directed graph such that the solution does not contain any path belonging to a given set i e the forbidden set it is imposed that no cycle can be included in the solution the problem at hand is formulated as a particular instance of the shortest path problem with resource constraints and two different solution approaches are defined and implemented one is a branch bound based algorithm the other is a dynamic programming approach different versions of the proposed solution strategies are developed and tested on a large set of test problems

an algorithm for ranking paths that may contain cycles abstract in this paper an algorithm is presented for determining the k best paths that may contain cycles in a directed network the basic idea behind the algorithm is quite simple once the best path has been determined it is excluded from the network in such a way that no new path is formed and no more paths are excluded this step leads to an enlarged network where all the paths but the best one can be determined the method is repeated until the desired paths have been computed the proposed algorithm can be used not only for the classical k shortest paths problem but also for ranking paths under a nonlinear objective function provided that an algorithm to determine the best path exists computational results are presented and comparisons with other approaches for the classical problem are made

optimal paths in bi attribute networks with fractional cost functions abstract an important routing problem is to determine an optimal path through a multi attribute network which minimizes a cost function of path attributes in this paper we study an optimal path problem in a bi attribute network where the cost function for path evaluation is fractional the problem can be equivalently formulated as the bi attribute rational path problem which is known to be np complete we develop an exact approach to find an optimal simple path through the network when arc attributes are non negative the approach uses some path preference structures and elimination techniques to discard from further consideration those partial paths that cannot be parts of an optimal path our extensive computational results demonstrate that the proposed method can find optimal paths for large networks in very attractive times

comparing branch and price algorithms for the multi commodity k splittable maximum flow problem abstract the multi commodity k splittable maximum flow problem consists in routing as much flow as possible through a capacitated network such that each commodity uses at most k paths and the capacities are satisfied the problem appears in telecommunications specifically when considering multi protocol label switching the problem has previously been solved to optimality through branch and price in this paper we propose two exact solution methods both based on an alternative decomposition the two methods differ in their branching strategy the first method which branches on forbidden edge sequences shows some performance difficulty due to large search trees the second method which branches on forbidden and forced edge sequences demonstrates much better performance the latter also outperforms a leading exact solution method from the literature furthermore a heuristic algorithm is presented the heuristic is fast and yields good solution values

p _1 nonconforming finite elements on triangulations into triangles and quadrilaterals the p 1 nonconforming finite element is introduced for arbitrary triangulations into quadrilaterals and triangles of multiple connected lipschitz domains an explicit a priori analysis for the combination of the park sheen and the crouzeix raviart nonconforming finite element methods is given for second order elliptic pdes with inhomogeneous dirichlet boundary conditions

the existence of lie symmetries for first order analytic discrete dynamical systems if a first order discrete dynamical system dds possesses a continuous symmetry the system is known to be linearizable in this paper we show that a first order analytic dds always possesses a symmetry in a neighbourhood of a hyperbolic fixed point

affine linear and d4 symmetric lattice equations symmetry analysis and reductions we consider lattice equations on which are autonomous affine linear and possess the symmetries of the square some basic properties of equations of this type are derived as well as a sufficient linearization condition and a conservation law a systematic analysis of the lie point and the generalized three and five point symmetries is presented it leads to the generic form of the symmetry generators of all the equations in this class which satisfy a certain non degeneracy condition finally symmetry reductions of certain lattice equations to discrete analogs of the painleve equations are considered

note on the complexity of the shortest path models for column generation in vrptw in this note we prove that the relaxation approach in designing the subproblem of pricing out only the feasible routes for the set partition formulation of the vrptw is justified on complexity grounds that is the first dynamic programming model presented in m desrochers j desrosiers and m solomon 1992 that is able to price out all feasible routes is np hard in the strong sense

solving the shortest path tour problem in this paper we study the shortest path tour problem in which a shortest path from a given origin node to a given destination node must be found in a directed graph with non negative arc lengths such path needs to cross a sequence of node subsets that are given in a fixed order the subsets are disjoint and may be different sized a polynomial time reduction of the problem to a classical shortest path problem over a modified digraph is described and two solution methods based on the above reduction and dynamic programming respectively are proposed and compared with the state of the art solving procedure the proposed methods are tested on existing datasets for this problem and on a large class of new benchmark instances the computational experience shows that both the proposed methods exhibit a consistent improved performance in terms of computational time with respect to the existing solution method

symmetries and first integrals of ordinary difference equations this paper describes a new symmetry based approach to solving a given ordinary difference equation by studying the local structure of the set of solutions we derive a systematic method for determining one parameter lie groups of symmetries in closed form such groups can be used to achieve successive reductions of order if there are enough symmetries the difference equation can be completely solved several examples are used to illustrate the technique for transitive and intransitive symmetry groups it is also shown that every linear second order ordinary difference equation has a lie algebra of symmetry generators that is isomorphic to sl 3 the paper concludes with a systematic method for constructing first integrals directly which can be used even if no symmetries are known

method of measuring transmission paths abstract a theoretical explanation and experimental proof are presented of a method for localizing and evaluating the transmission paths of any signal in a black box among a set of points previously defined in it the signal should behave linearly and the system should be able to receive external excitations separately at each of its points such excitations need not be the signal under study but they should be linearly related to it also presented are the equations that once the transmission paths have been determined allow the evaluation of the excitations which act on the system

error estimates of two level finite element method for smagorinsky model two level finite element method for simulating smagorinsky model in large eddy simulation is investigated in this two level algorithm a linearized discrete problem is solved by using the variational multiscale method on the coarse mesh corresponding to newton linearization method the linearized smagorinsky model is solved on the fine mesh the error estimates derived in this paper implies that by choosing appropriate mesh sizes and the radius of the spatial filter used in smagorinsky model the two level method provides the optimal convergence rates for the velocity in h1 norm and the pressure in l2 norm meanwhile the two different numerical experiments are given to support the optimal convergence rates and the high efficiency of two level algorithm

symmetry reduction of ordinary finite difference equations using moving frames the technique of equivariant moving frames is incorporated into the classical symmetry reduction method of ordinary finite difference equations using the recurrence relations for the finite difference invariants computations are performed symbolically without relying on the coordinate expressions of the canonical variables and the difference invariants

traffic assignment model with fuzzy level of travel demand an efficient algorithm based on quasi logit formulas abstract the place of fuzzy concepts in traffic assignment ta models has been studied in recent literature keeping fuzzy level of travel demand in mind we propose a new ta model in which the travel costs of links are depended on their congestion from the results of such fuzzy ta model network planners are able to estimate the number of travelers on network links by using zero one variables the proposed model is transformed into a crisp mixed integer problem with respect to path flow variables in order to produce the logit flows from this problem damberg et al algorithm is modified then the level of certainty is maximized and perceived travel delays are minimized for a fixed certainty degree the obtained solution which is named the fuzzy equilibrium flow satisfies a quasi logit formula similar to ordinary expression of the logit route choice model eventually we examine the quality of different path enumeration techniques in the proposed model

invariant submanifold flows given a lie group acting on a manifold our aim is to analyze the evolution of differential invariants under invariant submanifold flows the constructions are based on the equivariant method of moving frames and the induced invariant variational bicomplex applications to integrable soliton dynamics and to the evolution of differential invariant signatures used in equivalence problems and object recognition and symmetry detection in images are discussed

energy transfer for high frequencies in built up structures this paper is concerned with the prediction of vibrations at high frequencies in built up structures in the high frequency range the dynamical behaviour of structures is driven by energy transfer rather than modal aspects the description of systems in terms of energy is thus well suited and leads to closed form equations of an integral type on energy variables the numerical solution of these equations is achieved with appropriate software called ceres and the results are compared with experimental measurements on a multi plate structure

on the reciprocity relationship between direct field radiation and diffuse reverberant loading this analysis is concerned with the derivation of a diffuse field reciprocity relationship between the diffuse field excitation of a connection to a structural or acoustic subsystem and the radiation impedance of the connection such a relationship has been derived previously for connections described by a single degree of freedom in the present work it is shown that the diffuse field reciprocity relationship also arises when describing the ensemble average response of connections to structural or acoustic subsystems with uncertain boundaries furthermore it is shown that the existing diffuse field reciprocity relationship can be extended to encompass connections that possess an arbitrary number of degrees of freedom the present work has application to i the calculation of the diffuse field response of structural acoustic systems modeled by finite elements boundary elements and infinite elements ii the general calculation of the coupling loss factors employed in statistical energy analysis sea and iii the derivation of an alternative analysis method for describing the dynamic interactions of coupled subsystems with uncertain boundaries a generalized boundary approach to sea

solving biobjective combinatorial max ordering problems by ranking methods and a two phases approach abstract in this paper we propose a new method to solve biobjective combinatorial optimization problems of the max ordering type the method is based on the two phases method and ranking algorithms to efficiently construct k best solutions for the underlying single objective combinatorial problem we show that the method overcomes some of the difficulties of procedures proposed earlier we illustrate this by an example and discuss the difficulties in extending it to more than two objectives

the tricriterion shortest path problem with at least two bottleneck objective functions abstract the focus of this paper is on the tricriterion shortest path problem where two objective functions are of the bottleneck type for example minmax or maxmin the third objective function may be of the same kind or we may consider for example minsum or maxprod let p n be the complexity of a classical single objective algorithm responsible for this third function where n is the number of nodes and m be the number of arcs of the graph an o m 2 p n algorithm is presented that can generate the minimal complete set of pareto optimal solutions finding the maximal complete set is also possible optimality proofs are given and extensions for several special cases are presented computational experience for a set of randomly generated problems is reported

the simplest nonconforming mixed finite element method for linear elasticity in the symmetric formulation on n rectangular grids abstract a family of mixed finite elements is proposed for solving the first order system of linear elasticity equations in any space dimension where the stress field is approximated by symmetric finite element tensors this family of elements has a perfect matching between the stress and the displacement the discrete spaces for the normal stress i i the shear stress i j and the displacement u i are span 1 x i span 1 x i x j and span 1 respectively on rectangular grids in particular the definition remains the same for all space dimensions as a result of these choices the theoretical analysis is independent of the spatial dimension as well in 1d the element is nothing else but the 1d raviart thomas element which is the only conforming element in this family in 2d and higher dimensions they are new elements but of the minimal degrees of freedom the total degrees of freedom per element are 2 plus 1 in 1d 7 plus 2 in 2d and 15 plus 3 in 3d these elements are the simplest element for any space dimension the well posedness condition and the optimal a priori error estimate of the family of finite elements are proved numerical tests in 2d and 3d are presented to show a superiority of the new elements over others as a superconvergence is exhibited and proved

symmetry adapted moving mesh schemes for the nonlinear schrodinger equation in this paper we consider symmetry preserving difference schemes for the nonlinear schrodinger equation where n is the number of space dimensions this equation describes one dimensional waves in n space dimensions in many physical situations including phenomena in plasma physics and nonlinear optics we will consider the nonintegrable case n 2 for which the equation admits solutions that blow up in a finite time and construct discretizations based upon moving mesh schemes that have the same lie group properties and lagrangian structures as the continuous counterpart

nonconforming mixed finite element approximation to the stationary navier stokes equations on anisotropic meshes abstract the main aim of this paper is to study the error estimates of a rectangular nonconforming finite element for the stationary navier stokes equations under anisotropic meshes that is the nonconforming rectangular element is taken as approximation space for the velocity and the piecewise constant element for the pressure the convergence analysis is presented and the optimal error estimates both in a broken h 1 norm for the velocity and in an l 2 norm for the pressure are derived on anisotropic meshes

conservation laws for integrable difference equations this paper deals with conservation laws for all integrable difference equations that belong to the famous adler bobenko suris classification all inequivalent three point conservation laws are found as are three five point conservation laws for each equation we also describe a method of generating conservation laws from known ones this method can be used to generate higher order conservation laws from those that are listed here

a hybrid method for the vibration analysis of complex structural acoustic systems a new method is presented for the analysis of complex dynamic systems which is based on partitioning the system degrees of freedom into a global set and a local set the global equations of motion are formulated and solved in a standard deterministic manner although due account is taken of the presence of the local degrees of freedom via an approach which is analogous to fuzzy structure theory the local equations of motion are formulated and solved by using statistical energy analysis sea with due account being taken of power input from the global degrees of freedom the method can be considered to encompass a number of existing analysis techniques and to form a flexible framework within which a number of detailed modeling strategies can be devised a simple rod system provides an initial application and it is shown that the method yields very good results from low to high frequency excitation

nonconforming generalized multiscale finite element methods a framework is introduced for nonconforming multiscale approach based on gmsfem generalized multiscale finite element method snapshot spaces are constructed for each macro scale block the snapshot spaces can be based on either conforming or nonconforming elements with suitable dimension reduction offline spaces are constructed moment function spaces are then introduced to impose continuity among the local offline spaces which results in nonconforming gmsfe spaces oversampling and randomized boundary condition strategies are considered steps for the nonconforming gmsfem are given explicitly error estimates are derived numerical results are presented to support the efficiency of the proposed approach we propose a framework for the nonconforming generalized multiscale methods with explicit steps concretely given to illustrate our methodology our approach is simple and noble in the sense that it is free of parameters which are essential in other existing discontinuous galerkin methods oversampling ideas enhance approximation quality of our approach our method performs well to resolve highly heterogeneous media we provide several numerical examples to confirm the efficiency of the proposed method

a subspace of the dssy nonconforming quadrilateral finite element space for the stokes equations in this paper we propose a subspace of the dssy nonconforming quadrilateral finite element space the product of this space together with the piecewise constant space can be used for approximating the velocity and pressure variables respectively in solving stokes problems more precisely this space consists of the p 1 nonconforming quadrilateral finite element space augmented by macro bubble functions based on the dssy nonconforming quadrilateral space under a hood taylor type assumption on meshes it is shown that the pair satisfies the discrete inf sup condition using a boundedness estimate of an interpolation operator based on edge integrals numerical results are presented

a nonconforming finite element method for the stationary smagorinsky model abstract in this paper we focus on a low order nonconforming finite element method fem for the stationary smagorinsky model the velocity and pressure are approximated by the constrained nonconforming rotated q1 element cn q 1 r o t and piecewise constant element respectively optimal error estimates of the velocity in the broken h1 norm and l2 norm and the pressure in the l2 norm are derived by some nonlinear analysis techniques and aubin nitsche duality argument the supercloseness and superconvergent results are also obtained under some reasonable regularity assumptions finally a numerical example is implemented to confirm our theoretical analysis

two and three index formulations of the minimum cost multicommodity k splittable flow problem abstract the multicommodity flow problem mcfp considers the efficient routing of commodities from their origins to their destinations subject to capacity restrictions and edge costs baier et al g baier e kohler m skutella on the k splittable flow problem in 10th annual european symposium on algorithms 2002 101 113 introduced the maximum flow multicommodity k splittable flow problem mckfp where each commodity may use at most k paths between its origin and its destination this paper studies the np hard minimum cost multicommodity k splittable flow problem mcmckfp in which a given flow of commodities has to be satisfied at the lowest possible cost the problem has applications in transportation problems where a number of commodities must be routed using a limited number of distinct transportation units for each commodity based on a three index formulation by truffot et al j truffot c duhamel p mahey branch and price pour le probleme du multiflot k separable de cout minimal in limos umr 6158 cnrs roadef 05 2005 we present a new two index formulation for the problem and solve both formulations through branch and price the three index algorithm by truffot et al is improved by introducing a simple heuristic method to reach a feasible solution by eliminating some symmetry a novel branching strategy for the two index formulation is presented forbidding subpaths in the branching children though the proposed heuristic for the three index algorithm improves its performance the three index algorithm is still outperformed by the two index algorithm both with respect to running time and to the number of solved test instances

a new local stabilized nonconforming finite element method for solving stationary navier stokes equations in this paper we study a new local stabilized nonconforming finite element method based on two local gauss integrals for solving the stationary navier stokes equations this nonconforming method utilizes the lowest equal order pair of mixed finite elements i e ncp 1 p 1 error estimates of optimal order are obtained and numerical results agreeing with these estimates are demonstrated numerical comparisons with other mixed finite element methods for solving the navier stokes equations are also presented to show the better performance of the present method

a new approach of superconvergence analysis for two dimensional time fractional diffusion equation abstract in this paper a new approach of superconvergent estimate of bilinear finite element is established for two dimensional time fractional diffusion equation under fully discrete scheme the novelty of this approach is the combination technique of the interpolation and ritz projection as well as the superclose estimate in h 1 norm between them which avoids the difficulty of constructing a postprocessing operator for ritz projection operator and reduces the regularity requirement of the exact solution at the same time three numerical examples are carried out to verify the theoretical analysis

conservation laws of partial difference equations with two independent variables this paper introduces a technique for obtaining the conservation laws of a given scalar partial difference equation with two independent variables unlike methods that are based on nother s theorem this new technique does not use symmetries neither does it require the difference equation to have any special structure such as a lagrangian hamiltonian or multisymplectic formulation instead it uses a discrete analogue of the variational complex

a new second order nonconforming mixed finite element scheme for the stationary stokes and navier stokes equations abstract in this paper a new stable nonconforming mixed finite element scheme with second order accuracy is proposed for the stationary stokes and navier stokes equations in which a new nonconforming rectangular element is taken for approximating space the velocity and the bilinear element for the pressure the optimal error estimates for the approximation of both the velocity and the pressure in l 2 norm are established as well as one in a broken h 1 norm for the velocity numerical experiments are given which are consistent with our theoretical analysis

approximation by quadrilateral finite elements we consider the approximation properties of finite element spaces on quadrilateral meshes the finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element it is known that for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l p and order r in w 1 p is that the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r we show by means of a counterexample that this latter condition is also necessary as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

descent direction algorithm with multicommodity flow problem for signal optimization and traffic assignment jointly network managers wish to optimize control parameters such as signal setting which are very related to the traffic assignment models on the other hand traffic assignment patterns as an important instrument for predicting the amount of flow on network links are dependent to control decisions according to the significance of this concept some important papers about this mutually relation are reviewed in this paper then we implement a nonlinear algorithm on a minimal cost multicommodity flow mcmf problem to optimize some control policies subject to optimal flows although we take signal times into account but this approach has a far more reaching application in urban network control and design we employ a hybrid intelligent algorithm integrating decent direction algorithm and an interior point algorithm in a mutually consistent scheme for obtaining optimal signals and equilibrium flows an example is given to illustrate the effectiveness of our scheme

internal external and generalized symmetries backlund s theorem which characterizes contact transformations is generalized to give an analogous characterization of internal symmetries of systems of differential equations for a wide class of systems of differential equations we prove that every internal symmetry comes from a first order generalized symmetry and conversely every first order generalized symmetry satisfying certain explicit contact conditions determines an internal symmetry we analyze the contact conditions in detail deducing powerful necessary conditions for a system of differential equations that admit genuine internal symmetries i e ones which do not come from classical external symmetries applications include a direct proof that both the internal symmetry group and the first order generalized symmetries of a remarkable differential equation due to hilbert and cartan are the noncompact real form of the exceptional simple lie group g2

exploring trust region method for the solution of logit based stochastic user equilibrium problem abstract in this research paper we explored using the trust region method to solve the logit based sue problem we proposed a modified trust region newton mtrn algorithm for this problem when solving the trust region sue subproblem we showed that applying the well known steihaug toint method is inappropriate since it may make the convergence rate of the major iteration very slow in the early stage of the computation to overcome this drawback a modified steihaug toint method was proposed we proved the convergence of our mtrn algorithm and showed its convergence rate is superlinear for the implication of our algorithm we proposed an important principle on how to select the basic route for each od pair we indicated that it is a crucial principle to accelerate the convergence rate of the minor iteration i e trust region subproblem solving iteration in this study other implication issues for the sue problem are also considered including the computation of the trial step and the strategy to ensure strict feasibility iteration point we compared the mtrn algorithm with the gradient projection gp algorithm on the sioux falls network some results of numerical analysis are also reported

a cheapest nonconforming rectangular finite element for the stationary stokes problem abstract we introduce a least degrees of freedom nonconforming finite element pair to approximate the stokes equations based on quadrilateral meshes the finite element space for the velocity field is composed of the p 1 nonconforming quadrilateral element plus only one additional global dssy douglas santos sheen ye or rt rannacher turek type of bubble function the pressure field is approximated by the piecewise constant function we show that the discrete problem is nonsingular and the element pair satisfies a weak discrete inf sup condition several numerical examples are shown to confirm the efficiency and reliability of the proposed method

lie group formalism for difference equations the methods of lie group analysis of differential equations are generalized so as to provide an infinitesimal formalism for calculating symmetries of difference equations several examples are analysed one of them being a nonlinear difference equation for the linear equations the symmetry algebra of the discrete equation is found to be isomorphic to that of its continuous limit

solving some lexicographic multi objective combinatorial problems abstract the lexicographic multi objective optimization problem r lmop with r sum or bottleneck objectives optimizes a first objective then as far as a choice remains a second one then a third one and so on a general solution scheme from literature is based on scaling for the case that only sum criteria are involved the complexity is o rlogn t n m with t n m the complexity of optimization on combinatorial sum problem instances with m arcs and n nodes for cases with at least one bottleneck criterion the complexity is at least o rnlog2n t n m we describe a solution scheme that is suited to solve the shortest path r lmop the minimum spanning tree r lmop and the linear assignment r lmop the complexity is in all cases o r t n m

finding the k shortest paths we give algorithms for finding the k shortest paths not required to be simple connecting a pair of vertices in a digraph our algorithms output an implicit representation of these paths in a digraph with n vertices and m edges in time o m n log n k we can also find the k shortest paths from a given source s to each vertex in the graph in total time o m n log n kn we describe applications to dynamic programming problems including the knapsack problem sequence alignment maximum inscribed polygons and genealogical relationship discovery

validity diagrams of statistical energy analysis abstract this paper is concerned with the validity domain of statistical energy analysis sea which is defined in terms of four criteria the mode count n and the modal overlap m must be high the normalized attenuation factor m and the coupling strength must be small the application of dimensional analysis on the governing equations of plates gives the space of dimensionless parameters in which the validity domain of sea must be delimited this domain is discussed on the basis of geometry of the surfaces delimiting it the diagrams of validity of sea are introduced and discussed a numerical simulation on a couple of rectangular plates coupled along one edge illustrates the theoretical approach

a multi modal approach to the location of a rapid transit line abstract the location of a rapid transit line rtl represents a very complex decision problem because of the large number of decision makers unquantifiable criteria and uncertain data in this context operational research can help in the design process by providing tools to generate and assess alternative solutions for this purpose two bicriterion mathematical programming models the maximum coverage shortest path model and the median shortest path model have been developed in the past in this paper a new bicriterion model which can evaluate in a more realistic way the attractivity of an rtl is introduced to calculate an estimation of the non inferior solution set of the problem a procedure based on a k shortest path algorithm was developed this approach was applied to a well known sample problem and the results are discussed and compared with those obtained using a median shortest path model

an aggregate label setting policy for the multi objective shortest path problem we consider label setting algorithms for the multi objective shortest path problem with any number of sum and bottleneck objectives we propose a weighted sum aggregate ordering of the labels specifically tailored to combine sum and bottleneck objectives we show that the aggregate order leads to a consistent reduction of solution times up to two thirds with respect to the classical lexicographic order

the constrained shortest path problem with stochastic correlated link travel times abstract this paper investigates the constrained shortest path problem in a transportation network where the link travel times are assumed to be random variables defined on the basis of joint probability mass functions a 0 1 integer programming model is formulated to find the least expected travel time paths besides the flow balance and side constraints the unique link selection constraint is particularly introduced to guarantee that only an optimal path can be finally generated then a lagrangian relaxation solution approach is provided to relax the hard constraints and decompose the relaxed model into two parts an algorithmic framework integrating the sub gradient algorithm label correcting algorithm and k shortest path algorithm is designed to minimize the gap between the upper and lower bounds to find near optimal solutions an extension that considers the joint probability mass functions of link travel times varying with time is discussed we decompose the time dependent problem into three subproblems and solve it by the modified algorithmic framework computational tests are conducted on different scales of transportation networks experimental results in a large scale network indicate that the proposed algorithm can quickly find the high quality solutions with small relative gaps demonstrating the effectiveness of the proposed approaches

interchanging parameters and integrals in dynamical systems the mapping case we consider dynamical systems with discrete time maps that possess one or more integrals depending upon parameters we show that integrals can be used to replace parameters in the original map so as to construct a different map with different integrals we also highlight a process of reparametrization that can be used to increase the number of parameters in the original map prior to using integrals to replace them properties of the original map and the new map are compared the theory is motivated by and illustrated with examples of a three dimensional trace map and some four dimensional maps previously shown to be integrable

a graph theory approach to identify resonant and non resonant transmission paths in statistical modal energy distribution analysis statistical modal energy distribution analysis smeda extends classical statistical energy analysis sea to the mid frequency range by establishing power balance equations between modes in different subsystems this circumvents the sea requirement of modal energy equipartition and enables applying smeda to the cases of low modal overlap locally excited subsystems and to deal with complex heterogeneous subsystems as well yet widening the range of application of sea is done at a price with large models because the number of modes per subsystem can become considerable when the frequency increases therefore it would be worthwhile to have at one s disposal tools for a quick identification and ranking of the resonant and non resonant paths involved in modal energy transmission between subsystems it will be shown that previously developed graph theory algorithms for transmission path analysis tpa in sea can be adapted to smeda and prove useful for that purpose the case of airborne transmission between two cavities separated apart by homogeneous and ribbed plates will be first addressed to illustrate the potential of the graph approach a more complex case representing transmission between non contiguous cavities in a shipbuilding structure will be also presented

invariant conservation law preserving discretizations of linear and nonlinear wave equations symmetry and conservation law preserving finite difference discretizations are obtained for linear and nonlinear one dimensional wave equations on five and nine point stencils using the theory of lie point symmetries of difference equations and the discrete direct multiplier method of conservation law construction in particular for the linear wave equation an explicit five point scheme is presented that preserves the discrete analogs of its basic geometric point symmetries and six of the corresponding conservation laws for a class of nonlinear wave equations arising in hyperelasticity a nine point implicit scheme is constructed preserving four point symmetries and three local conservation laws other discretizations of the nonlinear wave equations preserving different subsets of conservation laws are discussed

invariantization of the crank nicolson method for burgers equation abstract in this article a geometric technique to construct numerical schemes for partial differential equations pdes that inherit lie symmetries is proposed the moving frame method enables one to adjust the numerical schemes in a geometric manner and systematically construct proper invariant versions of them to illustrate the method we study invariantization of the crank nicolson scheme for burgers equation with careful choice of normalization equations the invariantized schemes are shown to surpass the standard scheme successfully removing numerical oscillation around sharp transition layers

the adjoint equation method for constructing first integrals of difference equations a new method for finding first integrals of discrete equations is presented it can be used for discrete equations which do not possess a variational lagrangian or hamiltonian formulation the method is based on a newly established identity which links symmetries of the underlying discrete equations solutions of the discrete adjoint equations and first integrals the method is applied to an invariant mapping and to discretizations of second order and third order ordinary differential equations in examples the set of independent first integrals makes it possible to find the general solution of the discrete equations

infinitely many conservation laws for the discrete kdv equation rasin and hydon 2007 j phys a math theor 40 12763 73 suggested a way to construct an infinite number of conservation laws for the discrete kdv equation dkdv by repeated application of a certain symmetry to a known conservation law it was not decided however whether the resulting conservation laws were distinct and nontrivial in this paper we obtain the following results 1 we give an alternative method to construct an infinite number of conservation laws using a discrete version of the gardner transformation 2 we give a direct proof that the conservation laws obtained by the method of rasin and hydon are indeed distinct and nontrivial 3 we consider a continuum limit in which the dkdv equation becomes a first order eikonal equation in this limit the two sets of conservation laws become the same and are evidently distinct and nontrivial this proves the nontriviality of the conservation laws constructed by the gardner method and gives an alternative proof of the nontriviality of the conservation laws constructed by the method of rasin and hydon

shortest path problems with partial information models and algorithms for detecting dominance abstract in this work we focus on partial information models for the well known shortest path problem where we consider multiple instances of values for the parameters that determine the cost of each arc this allows coping with the uncertainty about the future the imprecision of data the arbitrariness of some options the evolving values of the decision makers dms and or the multiplicity of dms in group decision making this paper proves some results and presents detailed algorithms to identify the set of non dominated paths a concept from decision theory under partial information we first address problems with a finite set of instances then problems with a general eventually not discrete set of instances and finally we study a particular case of the latter which complies with a condition that may hold in some situations to deal with these partial information problems we propose a new use for existing multicriteria algorithms based on the ranking of shortest paths

a computational improvement for a shortest paths ranking algorithm abstract in this short note we present a computational improvement for the shortest paths ranking algorithm due to some of the authors according to its computational complexity this new version of the algorithm entails a very significant reduction in its execution time and an even larger reduction in the required computer memory space computational experiments are reported

the shortest path problem with forbidden paths we consider a variant of the constrained shortest path problem where the constraints come from a set of forbidden paths arc sequences that cannot be part of any feasible solution two solution approaches are proposed for this variant the first uses aho and corasick s keyword matching algorithm to filter paths produced by a k shortest paths algorithm the second generalizes martins deviation path approach for the k shortest paths problem by merging the original graph with a state graph derived from aho and corasick s algorithm like martins approach the second method amounts to a polynomial reduction of the shortest path problem with forbidden paths to a classic shortest path problem its significant advantage over the first approach is that it allows considering forbidden paths in more general shortest path problems such as the shortest path problem with resource constraints

lie symmetries of finite difference equations discretizations of the helmholtz heat and wave equations on uniform lattices are considered in various space time dimensions the symmetry properties of these finite difference equations are determined and it is found that they retain the same lie symmetry algebras as their continuum limits solutions with definite transformation properties are obtained identities and formulas for these functions are then derived using the symmetry algebra

linear finite element methods for planar linear elasticity a linear nonconforming conforming displacement finite element method for the pure displacement pure traction problem in two dimensional linear elasticity for a homogeneous isotropic elastic material is considered in the case of a convex polygonal configuration domain error estimates in the energy l sup 2 norm are obtained the convergence rate does not deteriorate for nearly incompressible material furthermore the convergence analysis does not rely on the theory of saddle point problems 22 refs

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

conservation laws for nqc type difference equations this paper presents a classification of all three point conservation laws for a large class of integrable difference equations that has been described by nijhoff quispel and capel we show that every nonlinear equation from this class has at least two nontrivial conservation laws most of the conservation laws that are found are new

on a special class of bicriterion path problems abstract in this paper we are concerned with a special class of bicriterion path problems for the considered class of bicriterion problems at least one of the objectives is type maxmin for the other one besides an objective type maxmin also a minsum and a minratio type objective are considered two algorithms are presented for the considered class of bicriterion path problems the first one only determines the minimal complete set of nondominated paths and the second one determines the entire set of nondominated paths both algorithms can be used for any type of bicriterion path problems since one of the objectives is type maxmin and an algorithm exists to determine the best path for the other objective computational statistics for the three types of considered bicriterion path problems are also presented

p1 nonconforming quadrilateral finite element methods for second order elliptic problems a p1 nonconforming quadrilateral finite element is introduced for second order elliptic problems in two dimensions unlike the usual quadrilateral nonconforming finite elements which contain quadratic polynomials or polynomials of degree greater than 2 our element consists of only piecewise linear polynomials that are continuous at the midpoints of edges one of the benefits of using our element is convenience in using rectangular or quadrilateral meshes with the least degrees of freedom among the nonconforming quadrilateral elements an optimal rate of convergence is obtained also a nonparametric reference scheme is introduced in order to systematically compute stiffness and mass matrices on each quadrilateral an extension of the p1 nonconforming element to three dimensions is also given finally several numerical results are reported to confirm the effective nature of the proposed new element

estimation of sea coupling loss factors using a dual formulation and fem modal information part ii numerical applications numerical applications of the approach presented in the first of these two companion papers is proposed in the cases of coupled beams and coupled plates results are compared with other calculations clf calculation for coupled beams is achieved with analytical modal description the influence of thickness ratio and damping about the quality of the prediction are presented and discussed clf determination with fem modal description are then applied to the case of two thin plates coupled in an l shape this case permit to achieve comparison with other calculation and will be representative of the applicability of the method to more complex structures as industrial ones a very good agreement is noticed that validate the calculation of clf with the present method

extension of the statistical modal energy distribution analysis for estimating energy density in coupled subsystems abstract the present article deals with an extension of the statistical modal energy distribution analysis smeda method to estimate kinetic and potential energy density in coupled subsystems the smeda method uses the modal bases of uncoupled subsystems and focuses on the modal energies rather than the global energies of subsystems such as sea statistical energy analysis this method permits extending sea to subsystems with low modal overlap or to localized excitations as it does not assume the existence of modal energy equipartition we demonstrate that by using the modal energies of subsystems computed by smeda it is possible to estimate energy distribution in subsystems this approach has the same advantages of standard sea as it uses very short calculations to analyze damping effects the estimation of energy distribution from smeda is applied to an academic case and an industrial example

symmetries and conditional symmetries of differential difference equations two different methods of finding lie point symmetries of differential difference equations are presented and applied to the two dimensional toda lattice continuous symmetries are combined with discrete ones to obtain various reductions to lower dimensional equations in particular to differential equations of the delay type the concept of conditional symmetries is extended from purely differential to differential difference equations and shown to incorporate backlund transformations