comparison of several iterative techniques in the solution of symmetric banded equations on a two pipe cyber 205 the effectiveness of several iterative techniques for solving matrix equations resulting from finite difference approximations to self adjoint parabolic and elliptic partial differential equations is investigated the strongly implicit procedure sip the incomplete cholesky conjugate gradient method iccg the vectorized iccg viccg the modified iccg miccg the diagonally scaled conjugate gradient method dscg the polynomial preconditioned conjugate gradient method polcg and the polynomial form of iccg piccg the comparison is made on a vector machine two pipe cyber 205 where vectorization of the code is done primarily by the vector compiler available it is found that of the methods studied polcg and miccg appear to require the least amount of cpu time an advantage of miccg over polcg is that it is less sensitive to increasing matrix size its disadvantages are that it requires an iteration parameter has a greater setup time and needs more storage than polcg

localized patterns for the quintic complex swift hohenberg equation abstract we show using numerical simulations that a variety of localized patterns arise in a model equation the quintic swift hohenberg equation with complex coefficients we demonstrate that various sizes of localized standing wave patterns are possible when the imaginary part of the complex coefficient is small localized traveling waves as well as localized standing waves with a fixed size are observed when the imaginary part is rather large we also present stable localized patterns in two spatial dimensions and study their interaction

doubly lexical orderings of matrices every matrix has a doubly lexical ordering an ordering of the rows and columns so that the row vectors are lexically or lexicographically increasing and the column vectors are lexically increasing every graph has a lexical ordering a vertex ordering making the neighbourhood matrix doubly lexical an almost linear time doubly lexical ordering algorithm is given doubly lexical orderings unify the orderings characterizing certain classes of matrices and graphs including totally balanced matrices subtree matrices and chordal graphs

analysis of accuracy of approximate simultaneous nonlinear confidence intervals on hydraulic heads in analytical and numerical test cases inaccuracies in parameter values parameterization stresses and boundary conditions of analytical solutions and numerical models of groundwater flow produce errors in simulated hydraulic heads these errors can be quantified in terms of approximate simultaneous nonlinear confidence intervals presented in the literature approximate confidence intervals can be applied in both error and sensitivity analysis and can be used prior to calibration or when calibration was accomplished by trial and error in this paper the method is expanded for use in numerical problems and the accuracy of the approximate intervals is evaluated using monte carlo runs four test cases are used to compare results in two numerical test cases the approximate simultaneous confidence intervals are generally accurate and are calculated efficiently enough to indicate that the method is useful for real world problems in one of the two analytical test cases considered the approximate intervals are too wide to be of practical interest improving the method used to approximate the uncertainty of the model inputs could make the confidence intervals useful in more problems

stable localized solutions of arbitrary length for the quintic swift hohenberg equation abstract we show that localized solutions of arbitrary length are stable over a finite parameter interval of subcritical values for the quintic swift hohenberg equation with a destabilizing cubic term this equation is thought to model a weakly hysteretic transition to stationary patterns we argue that the stabilization of the localized states of arbitrary length can be traced back to the interaction between long wavelength modulations and spatial variations on the length scale of one unit cell these results are critically compared with other known mechanisms to stabilize localized states in various situations we also discuss for which experimental systems the states predicted here could be detected including e g the stationary onset of binary fluid convection

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

secondary turing type instabilities due to strong spatial resonance we investigate the dynamics of pattern forming systems in large domains near a codimension two point corresponding to a strong spatial resonance where competing instabilities with wavenumbers in th

exact soliton solutions of the one dimensional complex swift hohenberg equation abstract using painleve analysis the hirota multi linear method and a direct ansatz technique we study analytic solutions of the 1 1 dimensional complex cubic and quintic swift hohenberg equations we consider both standard and generalized versions of these equations we have found that a number of exact solutions exist to each of these equations provided that the coefficients are constrained by certain relations the set of solutions include particular types of solitary wave solutions hole dark soliton solutions and periodic solutions in terms of elliptic jacobi functions and the weierstrass function although these solutions represent only a small subset of the large variety of possible solutions admitted by the complex cubic and quintic swift hohenberg equations those presented here are the first examples of exact analytic solutions found thus far

tractability of parameterized completion problems on chordal strongly chordal and proper interval graphs we study the parameterized complexity of three np hard graph completion problems the minimum fill in problem asks if a graph can be triangulated by adding at most k edges we develop o ck m and o k2mn f k algorithms for this problem on a graph with n vertices and m edges here f k is exponential in k and the constants hidden by the big o notation are small and do not depend on k in particular this implies that the problem is fixed parameter tractable fpt the proper interval graph completion problem motivated by molecular biology asks if a graph can be made proper interval by adding no more than k edges we show that the problem is fpt by providing a simple search tree based algorithm that solves it in o ck m time similarly we show that the parameterized version of the strongly chordal graph completion problem is fpt by giving an o ck m log n time algorithm for it all of our algorithms can actually enumerate all possible k completions within the same time bounds

weakly triangulated comparability graphs the class of weakly triangulated comparability graphs and their complements are generalizations of interval graphs and chordal comparability graphs we show that problems on these classes of graphs can be solved efficiently by transforming them into problems on chordal bipartite graphs we show that recognition and independent set on weakly triangulated comparability graphs can be solved in o n2 time in this manner and that the number of weakly triangulated comparability graphs is 2 theta n log 2 n we also give algorithms to compute transitive closure and transitive reduction in o n2loglogn time if the underlying undirected graph of the transitive closure is a weakly triangulated comparability graph

solving groundwater flow problems by conjugate gradient methods and the strongly implicit procedure the performance of the preconditioned conjugate gradient method with three preconditioners is compared with the strongly implicit procedure sip using a scalar computer the preconditioners considered are the incomplete cholesky iccg and the modified incomplete cholesky miccg which require the same computer storage as sip as programmed for a problem with a symmetric matrix and a polynomial preconditioner polcg which requires less computer storage than sip although polcg is usually used on vector computers it is included here because of its small storage requirements in this paper published comparisons of the solvers are evaluated all four solvers are compared for the first time and new test cases are presented to provide a more complete basis by which the solvers can be judged for typical groundwater flow problems based on nine test cases the following conclusions are reached 1 sip is actually as efficient as iccg for some of the published linear two dimensional test cases that were reportedly solved much more efficiently by iccg 2 sip is more efficient than other published comparisons would indicate when common convergence criteria are used and 3 for problems that are three dimensional nonlinear or both and for which common convergence criteria are used sip is often more efficient than iccg and is sometimes more efficient than miccg

oscillon type structures and their interaction in a swift hohenberg model abstract motivated by the observation of localized circular excitations oscillons in vertically vibrated granular layers p umbanhowar f melo h swinney nature 382 1996 793 we numerically investigate an extension of a swift hohenberg model that exhibits a subcritical transition to square patterns for sufficiently strong damping of the basic state stable oscillon structures are found the localization mechanism is quite general and is due to non adiabatic effects bound structures of oscillons of equal and opposite polarity are found with bound states of like polarity being less robust much of the phenomena are consistent with the experimental observations and suggest that oscillons are not specific to patterns in granular media or to parametrically driven systems experimental tests are suggested that would determine whether this minimal framework is sufficient to describe the phenomena

a conjugate gradient finite element model of flow for large multiaquifer systems a quasi three dimensional finite element model based on the conjugate gradient technique is developed and applied to the analysis of subsurface flow occurring in complex heterogeneous multiaquifer systems transient leakage across the aquitards is fully taken into consideration using the convolution approach originally derived by herrera 1970 the convolution integrals enhance the favorable features of the modified conjugate gradients mcg which prove to be a very efficient tool for the solution of the integrodifferential model the high performance of mcg results from two combined factors 1 the diagonal dominance of the assembled finite element matrix which is emphasized by the integral contributions 2 the essential reducibility of the matrix for a wide range of time step values which is due to the virtual decoupling of the aquifer equations the preconditioning of the conjugate gradient scheme is therefore improved and a good convergence occurs after a number of iterations smaller than n n n and n being the overall number of nodes and aquifers respectively the computational cost of the model is generally low and its relative efficiency tends to increase for small to moderate time steps and for complex multiaquifer systems where several interconnected aquifers are simulated

solving three dimensional hexahedral finite element groundwater models by preconditioned conjugate gradient methods practical aspects of three dimensional modeling of groundwater flow in heterogeneous aquifer systems are investigated using a finite element approach particular attention is given to the properties of the conductance matrix and the efficiency of the conjugate gradient method with different preconditioners diagonal scaling incomplete cholesky decomposition incomplete factorization and modified incomplete factorization it is shown that for hexahedral trilinear finite elements the resulting matrix is except for cube shaped elements never diagonally dominant which restricts the existence of several preconditioners numerical comparison of several test problems including hypothetical and field applications with different degrees of heterogeneity show that the incomplete cholesky and the incomplete factorization preconditioners if they exist are more efficient than diagonal scaling with respect to both rate of convergence and overall computing time but diagonal scaling can be considered superior because it is always possible an m matrix transformation is proposed which guarantees the existence of all preconditioners numerical comparison of the test problems shows that this technique is very effective from the resulting preconditioners the incomplete cholesky and the incomplete factorization are shown to be the most efficient but the latter is superior from the point of view of computer storage and is recommended for all practical applications

simplicial powers of graphs in a graph a vertex is simplicial if its neighborhood is a clique for an integer k 1 a graph g v g e g is the k simplicial power of a graph h v h e h h a root graph of g if v g is the set of all simplicial vertices of h and for all distinct vertices x and y in v g xy e g if and only if the distance in h between x and y is at most k this concept generalizes k leaf powers introduced by nishimura ragde and thilikos which were motivated by the search for underlying phylogenetic trees k leaf powers are the k simplicial powers of trees recently a lot of work has been done on k leaf powers and their roots as well as on their variants phylogenetic roots and steiner roots for k 5 k leaf powers can be recognized in linear time and for k 4 structural characterizations are known for k 6 the recognition and characterization problems of k leaf powers are still open since trees and block graphs i e connected graphs whose blocks are cliques have very similar metric properties it is natural to study k simplicial powers of block graphs we show that leaf powers of trees and simplicial powers of block graphs are closely related and we study simplicial powers of other graph classes containing all trees such as ptolemaic graphs and strongly chordal graphs

localized patterns and hole solutions in one dimensional extended systems abstract the existence stability properties dynamical evolution and bifurcation diagram of localized patterns and hole solutions in one dimensional extended systems are studied from the point of view of front interactions an adequate envelope equation is derived from a prototype model that exhibits these particle like solutions this equation allows us to obtain an analytical expression for the front interaction which is in good agreement with numerical simulations

additive sparse spanners for graphs with bounded length of largest induced cycle in this paper we show that every chordal graph with n vertices and m edges admits an additive 4 spanner with at most 2n 2 edges and an additive 3 spanner with at most o n log n edges this significantly improves results of peleg and schaffer from graph spanners j graph theory 13 1989 99 116 our spanners are additive and easier to construct an additive 4 spanner can be constructed in linear time while an additive 3 spanner is constructable in o m log n time furthermore our method can be extended to graphs with largest induced cycles of length k any such graph admits an additive k 1 spanner with at most 2n 2 edges which is constructable in o nk m time

simultaneous influence of additive and multiplicative noise on stationary dissipative solitons we investigate the simultaneous influence of spatially homogeneous multiplicative noise as well as of spatially correlated additive noise on the formation of localized patterns in the framework of the cubic quintic complex ginzburg landau equation depending on the ratio between the strength of additive and multiplicative noise we find a number of distinctly different types of behavior including explosions collapse filling in and spatio temporal disorder as well as intermittent behavior of all types listed techniques used to analyze the results include snapshots x t plots and plots of the spatially and temporally averaged amplitude as a function of the strength of multiplicative noise while keeping the strength of additive noise fixed typically 50 realizations are used for averaging to obtain the corresponding data points in these diagrams for the widths of these distribution as a function of additive noise we obtain a linear decrease in the limit of fairly large but fixed values of the multiplicative noise to summarize our findings concisely we show three dimensional plots of the mean pattern amplitude and the generalized susceptibility as a function of the strengths of additive and multiplicative noise we critically compare the results of our investigations with those obtained in the two limiting cases of purely additive and of purely multiplicative noise

instabilities and spatio temporal chaos in hexagon patterns with rotation abstract the dynamics of hexagon patterns in rotating systems are investigated within the framework of modified swift hohenberg equations that can be considered as simple models for rotating convection with broken up down symmetry e g non boussinesq rayleigh benard or marangoni convection in the weakly nonlinear regime a linear stability analysis of the hexagons reveals long and short wave instabilities which can be steady or oscillatory the oscillatory short wave instabilities can lead to stable hexagon patterns that are periodically modulated in space and time or to a state of spatio temporal chaos with a fourier spectrum that precesses on average in time the chaotic state can exhibit bistability with the steady hexagon pattern there exist regimes in which the steady hexagon patterns are unstable at all wave numbers

a comparison of the incomplete cholesky conjugate gradient method with the strongly implicit method as applied to the solution of two dimensional groundwater flow equations this paper compares the efficiency of the strongly implicit procedure sip and the incomplete cholesky conjugate gradient method iccg applied to the solution of the finite difference approximating equations for groundwater flow results for five isotropic two dimensional test problems are presented three are linear confined aquifer problems and two are nonlinear water table aquifer problems in all but one of the test problems the aquifer was considered to be nonhomogeneous both sip and iccg as applied to water table aquifer problems make use of iteration parameters these were varied for each of the test problems to reduce the amount of computational work needed to find a solution iccg was usually substantially more efficient than sip when applied to the confined aquifer test problems for the water table aquifer test problems sip and iccg performed equally well

on the switch markov chain for perfect matchings we study a simple markov chain the switch chain on the set of all perfect matchings in a bipartite graph this markov chain was proposed by diaconis graham and holmes as a possible approach to a sampling problem arising in statistics we ask for which hereditary classes of graphs is the markov chain ergodic and for which is it rapidly mixing we provide a precise answer to the ergodicity question and close bounds on the mixing question we show for the first time that the mixing time of the switch chain is polynomial in the case of monotone graphs a class that includes examples of interest in the statistical setting

conjugate gradient type methods and preconditioning in this paper we consider various iterative methods for the numerical solution of very large sparse linear systems of equations which arise in the discretization of partial differential equations as the performance of the methods generally depends on the characteristics of the problems to be solved a judicious choice between the methods will require knowledge about the system the aim of this paper is to review the properties of the various iteration methods in order to assist the user in making a deliberate selection

analytical approach to localized structures in a simple reaction diffusion system we study from an analytical point of view a simple reaction diffusion model which admits stable oscillating localized structures as a consequence of the coexistence between a stable limit cycle and a stable fixed point using a generalized matching approach we are able to find approximate analytical expressions for localized oscillating structures in this reaction diffusion model capturing all the essential ingredients of these breathing particlelike solutions

influence of dirichlet boundary conditions on dissipative solitons in the cubic quintic complex ginzburg landau equation we investigate the influence of dirichlet boundary conditions on various types of localized solutions of the cubic quintic complex ginzburg landau equation as it arises as an envelope equation near the weakly inverted onset of traveling waves we find that various types of nonmoving pulses and holes can accommodate dirichlet boundary conditions by having for holes two halves of a ensuremath pi hole at each end of the box moving pulses of fixed shape as they arise for periodic boundary conditions are replaced by a nonmoving asymmetric pulse which has half a ensuremath pi hole at the end of the box in the original moving direction to guarantee that dirichlet boundary conditions are met moving breathing pulses as they arise for periodic boundary conditions propagate toward one end of the container and stop moving while the breathing persists indefinitely finally breathing and moving holes are replaced by two nonbreathing half ensuremath pi holes at each end of the container and one hump in the bulk

intermittency in the transition to turbulence it is commonly known that the intermittent transition from laminar to turbulent flow in pipes occurs because at intermediate values of a prescribed pressure drop a purely laminar flow offers too little resistance but a fully turbulent one offers too much we propose a phenomenological model of the flow which is able to explain this in a quantitative way through a hysteretic transition between laminar and turbulent states characterized by a disturbance amplitude variable that satisfies a natural type of evolution equation the form of this equation is motivated by physical observations and derived by an averaging procedure and we show that it naturally predicts disturbances having the characteristics of slugs and puffs the model predicts oscillations similar to those which occur in intermittency in pipe flow but it also predicts that stationary biphasic states can occur in sufficiently short pipes

unstructured additive schwarz conjugate gradient method for elliptic problems with highly discontinuous coefficients this paper concerns the iterative solution of symmetric elliptic problems with piecewise constant coefficients in two or three space dimensions discretized by linear finite element methods on unstructured triangular or tetrahedral meshes the effect of the discontinuous coefficients is studied by first postulating that there are d fixed regions of the domain where the coefficient takes constant positive values a a1 ad and then considering certain positive coefficient sequences a m in which some of these values approach 0 or infty as m rightarrow infty we consider the performance of additive schwarz domain decomposition preconditioners constructed from local solves on automatically generated subdomains together with a global solve on some coarser grid assuming no relationship between the regions on which the coefficient function is constant and either the subdomains or the coarse grid we show that the preconditioned conjugate gradient method converges in both the energy and the euclidean norms with a number of iterations which grows only logarithmically in the size of the maximum jump cal j m max a k m a l m k l 1 ldots d as m rightarrow infty the result is obtained by a careful analysis of the preconditioned matrix it cannot be obtained by the usual procedure of estimating condition numbers a simple example is given in which the condition number of both the original and the preconditioned stiffness matrices degrade linearly in cal j m recent results of chan smith and zou have shown that by using this preconditioner together with the conjugate gradient method the number of iterations can be bounded independently of the mesh diameter provided the subdomains have overlap commensurate with the size of the coarse mesh our results now show that this method is also highly resilient to discontinuous coefficients even if no attention is paid to the coefficient discontinuity in the construction of the solver

localized states in the generalized swift hohenberg equation the swift hohenberg equation with quadratic and cubic nonlinearities exhibits a remarkable wealth of stable spatially localized states the presence of these states is related to a phenomenon called homoclinic snaking numerical computations are used to illustrate the changes in the localized solution as it grows in spatial extent and to determine the stability properties of the resulting states the evolution of the localized states once they lose stability is illustrated using direct simulations in time i introduction ever since the observation that the subcritical complex ginzburg landau equation exhibits stable spatially localized states 1 there has been considerable interest in the properties of these states the presence of these states has important consequences for other systems described by partial differential equations on the line since the ginzburg landau equation describes the behavior of such systems near bifurcation from the trivial state of the system specifically the complex ginzburg landau equation describes the evolution of a long wavelength oscillatory instability as well as oscillatory instabilities at finite wavelength in systems with broken reflection symmetry in contrast near a steady state bifurcation with finite wavelength the evolution of the instability is described by the real ginzburg landau equation and this equation possesses only unstable spatially localized states it is of interest therefore to examine what happens to these unstable states at larger amplitude where the real ginzburg landau equation no longer provides an adequate description of the system in this paper we show that the localized states can become stable at such amplitudes and indeed that there is a large multiplicity of coexisting stable localized states under very general conditions we are able to relate the existence of these states to a phenomenon sometimes called homoclinic snaking that is well known from the theory of reversible systems with 1 1 resonance and use this theory to construct a large number of such states the stability properties of these states are also determined and the evolution of nonstationary localized states is studied by numerical integration in time it is an interesting fact that closely related phenomena have already been described in several areas involving pattern formation the theory was originally developed in the context of water waves where localized states have been studied by moving into a reference frame of the waves and converting the problem into an ordinary differential equation ode the resulting localized states are called solitary waves and in some cases turn out to be solitons kirchgassner 2 has pioneered a successful approach to this type of problem that led to a number of advances in this area specifically the ode is viewed as a dynamical system in space and localized states are sought as homoclinic orbits connecting the trivial state to itself whether such orbits are possible depends in part on the stability properties of the trivial state eigenvalues with positive real part indicate that a nontrivial state can grow from x while eigenvalues with negative real part indicate that such a state may return under appropriate conditions back to the trivial state as x the spectrum of the linearization about the trivial state is influenced by spatial symmetries of the system in many cases and in particular in the case considered here the ode is reversible as a result the bifurcations that are encountered as a parameter is increased are nongeneric in the present case the spatial dynamics of the system near the trivial state turn out to be described by the reversible 1 1 resonance the unfolding of this resonance has been worked out in detail by iooss and peroueme 3 and can be used to understand the appearance of a variety of homoclinic orbits in this system and hence of localized states with different spatial structure

global feedback control for pattern forming systems global feedback control of pattern formation in a wide class of systems described by the swift hohenberg sh equation is investigated theoretically by means of stability analysis and numerical simulations two cases are considered i feedback control of the competition between hexagon and roll patterns described by a supercritical sh equation and ii the use of feedback control to suppress the blowup in a system described by a subcritical sh equation in case i it is shown that feedback control can change the hexagon and roll stability regions in the parameter space as well as cause a transition from up to down hexagons and stabilize a skewed mixed mode hexagonal pattern in case ii it is demonstrated that feedback control can suppress blowup and lead to the formation of spatially localized patterns in the weakly nonlinear regime the effects of a delayed feedback are also investigated for both cases and it is shown that delay can induce temporal oscillations as well as blowup

standard graded vertex cover algebras cycles and leaves the aim of this paper is to characterize simplicial complexes which have standard graded vertex cover algebras this property has several nice consequences for the squarefree monomial ideals defining these algebras it turns out that such simplicial complexes are closely related to a range of hypergraphs which generalize bipartite graphs and trees these relationships allow us to obtain very general results on standard graded vertex cover algebras which cover previous major results on rees algebras of squarefree monomial ideals

an efficient estimate based on fft in topological verification method in this paper localized patterns of the quintic swift hohenberg equation are studied a numerical verification method with the conley index theory developed in zgliczynski and mischaikow rigorous numerics for partial differential equations the kuramoto sivashinsky equation found comput math 1 2001 255 288 is used in order to prove these patterns a new technique to efficiently obtain estimates for nonlinear terms is presented the key idea is based on the pseudo spectral method it is shown that this technique is inevitable for the verification of the localized patterns

finite element algorithms for simulating three dimensional groundwater flow and solute transport in multilayer systems a finite element formulation is developed for the simulation of groundwater flow and solute transport in multilayer systems of several aquifers and aquitards this formulation is general flexibile and capable of taking full advantage of the nature of flow in such multilayer systems a fully three dimensional spatial representation can be performed for certain aquifers or for the entire flow system if needed those parts of the system that require three dimensional spatial discretization are handled effectively by combining two dimensional basis functions in the x y space and one dimensional basis functions in the z space furthermore the formulation has a desirable option to perform one dimensional representation of flow and transport in aquitards and areal representation of flow and transport in certain aquifers that do not require the fully three dimensional discretization when the one dimensional representation of an aquitard is used coupling of adjacent aquifer and aquitard layers is handled using a convolution integral approach a general solution strategy is also developed to allow systematic time stepping and cost effective matrix handling schemes for those parts of the system that require three dimensional discretization an algorithm referred to as the alternate sublayer and line sweep procedure is presented for decomposing three dimensional matrix equations this algorithm can accomodate several thousand nodal unknowns without requiring excessive core storage and cpu time four examples of transient flow and transport problems are provided these examples show verification of numerical results and demonstrate that the present finite element models are far more cost effective than earlier three dimensional models that are based on the conventional galerkin approach and direct matrix solution algorithms

largest j simplices in d cubes some relatives of the hadamard maximum determinant problem abstract this paper studies the computationally difficult problem of finding a largest j dimensional simplex in a given d dimensional cube the case in which j d is of special interest for it is equivalent to the hadamard maximum determinant problem it has been solved for infinitely many values of d but not for d 14 the subcase in which j d 3 mod 4 subsumes the famous problem on the existence of hadamard matrices the known results for the case j d are here summarized and used but the main focus is on fixed small values of j when j 1 the problem is trivial and when j 2 or j 3 it is here solved completely i e for all d beyond that the results are fragmentary but numerous and they lead to several attractive conjectures some other problems involving simplices in cubes are mentioned and the relationship of largest simplices to d optimal weighing designs is discussed

polynomial time recognition of squares of ptolemaic graphs and 3 sun free split graphs the square of a graph g denoted g 2 is obtained from g by putting an edge between two distinct vertices whenever their distance is two then g is called a square root of g 2 deciding whether a given graph has a square root is known to be np complete even if the root is required to be a chordal graph or even a split graph we present a polynomial time algorithm that decides whether a given graph has a ptolemaic square root if such a root exists our algorithm computes one with a minimum number of edges in the second part of our paper we give a characterization of the graphs that admit a 3 sun free split square root this characterization yields a polynomial time algorithm to decide whether a given graph has such a root and if so to compute one

three partition refinement algorithms we present improved partition refinement algorithms for three problems lexicographic sorting relational coarsest partition and double lexical ordering our double lexical ordering algorithm uses a new efficient method for unmerging two sorted sets

hardness results and approximation algorithms for weighted paired domination in graphs let g v e be a simple graph without isolated vertices a vertex set s v is a paired dominating set if every vertex in v s has a neighbor in s and the induced subgraph g s has a perfect matching in this paper we investigate the approximation hardness of paired domination in graphs for weighted paired domination an approximation algorithm in general graphs and an exact dynamic programming style algorithm in trees are also given

snakes and ladders in an inhomogeneous neural field model continuous neural field models with inhomogeneous synaptic connectivities are known to support traveling fronts as well as stable bumps of localized activity we analyze stationary localized structures in a neural field model with periodic modulation of the synaptic connectivity kernel and find that they are arranged in a snakes and ladders bifurcation structure in the case of heaviside firing rates we construct analytically symmetric and asymmetric states and hence derive closed form expressions for the corresponding bifurcation diagrams we show that the ideas proposed by beck and co workers to analyze snaking solutions to the swift hohenberg equation remain valid for the neural field model even though the corresponding spatial dynamical formulation is non autonomous we investigate how the modulation amplitude affects the bifurcation structure and compare numerical calculations for steep sigmoidal firing rates with analytic predictions valid in the heaviside limit

on the eigenvalue distribution of a class of preconditioning methods a class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationax b wherea is a symmetric positive definite matrix the methods are based on an incomplete factorization of the matrixa and include both pointwise and blockwise factorization we study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofc 1a wherec is the preconditioning matrix we also show graphic representations of the eigenvalues and present numerical tests of the methods

polynomial preconditioners for conjugate gradient calculations dubois greenbaum and rodrigue proposed using a truncated neumann series as an approximation to the inverse of a matrix a for the purpose of preconditioning conjugate gradient iterative approximations to ax b if we assume that a has been symmetrically scaled to have unit diagonal and is thus of the form i g then the neumann series is a power series in g with unit coefficients the incomplete inverse was thought of as a replacement of the incomplete cholesky decomposition suggested by meijerink and van der vorst in the family of methods iccg n the motivation for the replacement was the desire to have a preconditioned conjugate gradient method which only involved vector operations and which utilized long vectors we here suggest parameterizing the incomplete inverse to form a preconditioning matrix whose inverse is a polynomial in g we then show how to select the parameters to minimize the condition number of the product of the polynomial and i g theoretically the resulting algorithm

algorithmic aspects of upper paired domination in graphs abstract a set d of vertices in a graph g is a paired dominating set of g if every vertex of g is adjacent to a vertex in d and the subgraph induced by d contains a perfect matching not necessarily as an induced subgraph a paired dominating set of g is minimal if no proper subset of it is a paired dominating set of g the upper paired domination number of g denoted by pr g is the maximum cardinality of a minimal paired dominating set of g in upper pds it is required to compute a minimal paired dominating set with cardinality pr g for a given graph g in this paper we show that upper pds cannot be approximated within a factor of n 1 e for any e 0 unless p np and upper pds is apx complete for bipartite graphs of maximum degree 4 on the positive side we show that upper pds can be approximated within o factor for graphs with maximum degree we also show that upper pds is solvable in polynomial time for threshold graphs chain graphs and proper interval graphs

complexity of distance paired domination problem in graphs suppose g v e is a simple graph and k is a fixed positive integer a subset d v is a distancek dominating set of g if for every u v there exists a vertex v d such that d g u v k where d g u v is the distance between u and v in g a set d v is a distancek paired dominating set of g if d is a distance k dominating set and the induced subgraph g d contains a perfect matching given a graph g v e and a fixed integer k 0 the min distancek paired dom set problem is to find a minimum cardinality distance k paired dominating set of g in this paper we show that the decision version of min distancek paired dom set is np complete for undirected path graphs this strengthens the complexity of decision version of min distancek paired dom set problem in chordal graphs we show that for a given graph g unless np dtime n o l o g l o g n min distancek paired dom set problem cannot be approximated within a factor of 1 e lnn for any e 0 where n is the number of vertices in g we also show that min distancek paired dom set problem is apx complete for graphs with degree bounded by 3 on the positive side we present a linear time algorithm to compute the minimum cardinality of a distance k paired dominating set of a strongly chordal graph g if a strong elimination ordering of g is provided we show that for a given graph g min distancek paired dom set problem can be approximated with an approximation factor of 1 ln2 k ln d g where d g denotes the maximum degree of g

guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems abstract this paper presents incomplete decompositions for various types of matrices as they occur in the implicit discretisation of practical problems a review is given of methods for the usual five point discretisation of a self adjoint elliptic second order partial differential equation in two dimensions on a square the matrices which occur in this type of problem are symmetric m matrices of very regular structure the convergence behaviour of the different decompositions for this case is demonstrated by numerical experiments the paper also gives decompositions for the following type of matrices i symmetric m matrices of a different structure ii symmetric positive definite matrices iii non symmetric matrices

the convergence behaviour of iterative methods on severely stretched grids in this paper we examine the dramatic influence that a severe stretching of finite difference grids can have on the convergence behaviour of iterative methods for the most important classes of iterative methods this phenomenon is considered for a simple model problem with various boundary conditions and an exponential grid it is shown that grid compression near 2 neumann boundary or in the centre can make the convergence of some methods extremely slow whereas grid compression near a dirichlet boundary can be very advantageous more theoretical insight is obtained by analysing the spectrum of the jacob matrix for one and two dimensional problems several bounds on dominant eigenvalues of this matrix are given the final conclusions are also applicable to problems with a variable diffusion coefficient and convection diffusion equations solved by central difference schemes

stable localized solutions in nonlinear optics with large dissipation abstract we study the spatio temporal behavior of the electric field inside a laser with saturable absorber in the good cavity limit we find that the underlying nonlinear partial differential equation admits stable spatially localized solutions for both the one and the two dimensional case we critically examine similarities and differences with the localized solutions found recently near a weakly inverted bifurcation to traveling waves

asymptotics of cellular buckling close to the maxwell load we study the deformation of an elastic strut on a nonlinear winkler foundation subjected to an axial compressive load p using multi scale analysis and numerical methods we describe the localized cellular post buckled state of the system when p is removed from the critical load p 2 the solutions and their modulation frequencies differ significantly from those predicted by weakly nonlinear analysis very close to p 2 in particular when p approaches the maxwell load pm the localized solutions approach a large amplitude heteroclinic connection between an unbuckled solution and a periodic solution an asymptotic description of pm in terms of the system parameters is given the agreement between the numerical calculations and the asymptotic approximations is striking

on the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs golumbic kaplan and shamir in their paper m c golumbic h kaplan r shamir graph sandwich problems j algorithms 19 1995 449 473 on graph sandwich problems published in 1995 left the status of sandwich problems for strongly chordal graphs and chordal bipartite graphs open we prove that the sandwich problem for strongly chordal graphs is np complete we also give some comments on the computational complexity of the sandwich problem for chordal bipartite graphs

localized periodic patterns for the non symmetric generalized swift hohenberg equation abstract a new asymptotic multiple scale expansion is used to derive envelope equations for localized spatially periodic patterns in the context of the generalized swift hohenberg equation an analysis of this envelope equation results in parametric conditions for localized patterns furthermore it yields corrections for wave number selection which are an order of magnitude larger for asymmetric nonlinearities than for the symmetric case the analytical results are compared with numerical computations which demonstrate that the condition for localized patterns coincides with vanishing hamiltonian and lagrangian for periodic solutions one striking feature of the choice of scaling parameters is that the derived condition for localized patterns agrees with the numerical results for a significant range of parameters which are an o 1 distance from the bifurcation thus providing a novel approach for studying these localized patterns

front motion metastability and subcritical bifurcations in hydrodynamics metastability occurs whenever under given external conditions as defined by the reynolds number in hydrodynamics more than one linearly stable solution may exist in variational problems the preferred state is usually claimed to be the one with the lowest energy but in hydrodynamics and except for very special conditions no energy functional exists from an analogy between amplitude and reaction diffusion equations it is argued that the border separating in space two possible solutions of the flow equations as turbulent and laminar in pipe flows or boundary layers moves with a constant mean velocity depending on the control parameter the sign of this velocity allows to decide which state is stable and which is metastable possible physical implications of these ideas for various problems of hydrodynamics are discussed

continuation of localised coherent structures in nonlocal neural field equations we study localised activity patterns in neural field equations posed on the euclidean plane such models are commonly used to describe the coarse grained activity of large ensembles of cortical neurons in a spatially continuous way we employ matrix free newton krylov solvers and perform numerical continuation of localised patterns directly on the integral form of the equation this opens up the possibility to study systems whose synaptic kernel does not lead to an equivalent pde formulation we present a numerical bifurcation study of localised states and show that the proposed models support patterns of activity with varying spatial extent through the mechanism of homoclinic snaking the regular organisation of these patterns is due to spatial interactions at a specific scale associated with the separation of excitation peaks in the chosen connectivity function the results presented form a basis for the general study of localised cortical activity with inputs and more specifically for investigating the localised spread of orientation selective activity that has been observed in the primary visual cortex with local visual input

distance paired domination problems on subclasses of chordal graphs let g v e be a graph without isolated vertices for a positive integer k a set s v is a k distance paired dominating set if each vertex in v s is within distance k of a vertex in s and the subgraph induced by s contains a perfect matching in this paper we present two linear time algorithms to find a minimum cardinality k distance paired dominating set in interval graphs and block graphs which are two subclasses of chordal graphs in addition we present a characterization of trees with unique minimum k distance paired dominating set

saturated flow in a single fracture evaluation of the reynolds equation in measured aperture fields fracture transmissivity and detailed aperture fields are measured in analog fractures specially designed to evaluate the utility of the reynolds equation the authors employ a light transmission technique with well defined accuracy approximately 1 error to measure aperture fields at high spatial resolution approximately 0 015 cm a hele shaw cell is used to confirm the approach by demonstrating agreement between experimental transmissivity simulated transmissivity on the measured aperture field and the parallel plate law in the two rough walled analog fractures considered the discrepancy between the experimental and numerical estimates of fracture transmissivity was sufficiently large approximately 22 47 to exclude numerical and experimental errors 2 as a source they conclude that the three dimensional character of the flow field is important for fully describing fluid flow in the two rough walled fractures considered and that the approach of depth averaging inherent in the formulation of the reynolds equation is inadequate they also explore the effects of spatial resolution aperture measurement technique and alternative definitions for link transmissivities in the finite difference formulation including some that contain corrections for tortuosity perpendicular to the mean fracture plane and stokes flow various formulations for link transmissivity are shown to converge at high resolution approximately 1 5 the spatial correlation length in the smoothly varying fracture at coarser resolutions the solution becomes increasingly sensitive to definition of link transmissivity and measurement technique aperture measurements that integrate over individual grid blocks were less sensitive to measurement scale and definition of link transmissivity than point sampling techniques

exponential asymptotics of localised patterns and snaking bifurcation diagrams abstract localised patterns emerging from a subcritical modulation instability are analysed by carrying the multiple scales analysis beyond all orders the model studied is the swift hohenberg equation of nonlinear optics which is equivalent to the classical swift hohenberg equation with a quadratic and a cubic nonlinearity applying the asymptotic technique away from the maxwell point first it is shown how exponentially small terms determine the phase of the fast spatial oscillation with respect to their slow sech type amplitude in the vicinity of the maxwell point the beyond all orders calculation yields the pinning range of parameters where stable stationary fronts connect the homogeneous and periodic states the full bifurcation diagram for localised patterns is then computed analytically including snake and ladder bifurcation curves this last step requires the matching of the periodic oscillation in the middle of a localised pattern both with an up and a down front to this end a third super slow spatial scale needs to be introduced in which fronts appear as boundary layers in addition the location of the maxwell point and the oscillation wave number of localised patterns are required to fourth order accuracy in the oscillation amplitude