randomized divide and conquer improved path matching and packing algorithms we propose a randomized divide and conquer technique that leads to improved randomized and deterministic algorithms for np hard path matching and packing problems for the parameterized max path problem our randomized algorithm runs in time o 4 k k 2 7 m and polynomial space where m is the number of edges in the input graph improving the previous best randomized algorithm for the problem that runs in time o 5 44 k km and exponential space our randomized algorithms for the parameterized max r d matching and max r set packing problems run in time 4 r 1 k n o 1 and polynomial space improving the previous best algorithms for the problems that run in time 10 88 rk n o 1 and exponential space moreover our randomized algorithms can be derandomized to result in significantly improved deterministic algorithms for the problems and they can be extended to solve other matching and packing problems

some discrete maximum principles arising for nonlinear elliptic finite element problems the discrete maximum principle dmp is an important measure of the qualitative reliability of the applied numerical scheme for elliptic problems this paper starts with formulating simple sufficient conditions for the matrix case and for nonlinear forms in banach spaces then a dmp is derived for finite element solutions for certain nonlinear partial differential equations we address nonlinear elliptic problems with mixed boundary conditions and interface conditions allowing possibly degenerate nonlinearities and thus extending our previous results

the discrete maximum principle for linear simplicial finite element approximations of a reaction diffusion problem abstract this paper provides a sufficient condition for the discrete maximum principle for a fully discrete linear simplicial finite element discretization of a reaction diffusion problem to hold it explicitly bounds the dihedral angles and heights of simplices in the finite element partition in terms of the magnitude of the reaction coefficient and the spatial dimension as a result it can be computed how small the acute simplices should be for the discrete maximum principle to be valid numerical experiments suggest that the bound which considerably improves a similar bound in p g ciarlet p a raviart maximum principle and uniform convergence for the finite element method comput methods appl mech eng 2 1973 17 31 is in fact sharp

a geometric property of the least squares solution of linear equations abstract we derive an explicit determinantal formula for the least squares solution of an overdetermined system of linear equations from this formula it follows that the least squares solution lies in the convex hull of the solutions to the square subsystems of the original system we extend this result and prove that this geometric property holds for a more general class of problems which includes the weighted least squares and lp norm minimization problems

finite element solution of the fundamental equations of semiconductor devices in part i of the paper see zlamal 13 finite element solutions of the nonstationary semiconductor equations were constructed two fully discrete schemes were proposed one was nonlinear the other partly linear in this part of the paper we justify the nonlinear scheme we consider the case of basic boundary conditions and of constant mobilities and prove that the scheme is unconditionally stable further we show that the approximate solution extended to the whole time interval as a piecewise linear function converges in a strong norm to the weak solution of the semiconductor equations these results represent an extended and corrected version of results announced without proof in zlamal 14

convergence behavior of interior point algorithms we show that most interior point algorithms for linear programming generate a solution sequence in which every limit point satisfies the strict complementarity condition these algorithms include all path following algorithms and some potential reduction algorithms the result also holds for the monotone complementarity problem if a strict complementarity solution exists in general the limit point is a solution that maximizes the number of its nonzero components among all solutions

symplectic energy momentum preserving variational integrators the purpose of this paper is to develop variational integrators for conservative mechanical systems that are symplectic and energy and momentum conserving to do this a space time view of variational integrators is employed and time step adaptation is used to impose the constraint of conservation of energy criteria for the solvability of the time steps and some numerical examples are given

faster deterministic parameterized algorithm for k path abstract in the k path problem the input is a directed graph g and an integer k 1 and the goal is to decide whether there is a simple directed path in g with exactly k vertices we give a deterministic algorithm for k path with time complexity o 2 554 k this improves the previously best deterministic algorithm for this problem of zehavi esa 2015 whose time complexity is o 2 597 k the technique used by our algorithm can also be used to obtain faster deterministic algorithms for k tree r dimensional k matching graph motif and partial cover

dissection of the path simplex in r n into n path subsimplices abstract we review properties of acute and non obtuse simplices and of ortho simplices and path simplices dissection of path simplices is considered which leads to a new result generalization of coxeter s trisection of a path tetrahedron into three path subtetrahedra to arbitrary spatial dimension n moreover following earlier results by korotov and k i ek we show that applying this procedure recursively in the proper way leads to a self similar path simplicial refinement towards a chosen vertex of the original path simplex

averaging of directional derivatives in vertices of nonobtuse regular triangulations for a shape regular triangulation mathcal t h without obtuse angles of a bounded polygonal domain omega subset re 2 let mathcal l h be the space of continuous functions linear on the triangles from mathcal t h and h the interpolation operator from c overline omega to mathcal l h this paper is devoted to the following classical problem find a second order approximation of the derivative partial u partial z a in a direction z of a function u in c 3 overline omega in a vertex a in the form of a linear combination of the constant directional derivatives partial pi h u partial z on the triangles surrounding a an effective procedure for such an approximation is presented its error is proved to be of the size o h 2 an operator mbox w h mathcal l h longrightarrow mathcal l h times mathcal l h relating a second order approximation wh h u of nabla u to every u in c 3 overline omega is constructed and shown to be a so called recovery operator the accuracy of the presented approximation is compared with the accuracies of the local approximations by other known techniques numerically

stabilization of mehrotra s primal dual algorithm and its implementation abstract in this paper we apply a stabilization procedure proposed by kovacevic vujcic and asic to the mehrotra s primal dual interior point algorithm for linear programming transformations of the dual problem corresponding to the stabilization procedure are considered the stabilization procedure and mehrotra s algorithm are implemented in the package mathematica a number of highly degenerate test examples are used to compare the modified mehrotra s method with respect to the original one and the codes pcx lipsol and mosek we also provide numerical results on some examples from the netlib test set

parameterized and approximation algorithms for finding two disjoint matchings we first present a randomized fixed parameter algorithm for the np hard problem of deciding if there are two matchings m 1 and m 2 in a given graph g such that m 1 m 2 is a given number k the algorithm runs in o 2 k k m n expected time and can be derandomized to run in o 2 2 k 12 log 2 2 k k n m n time where n respectively m is the number of vertices respectively edges in g we then extend the algorithm to the weighted version of the problem we further present a combinatorial approximation algorithm for the np hard problem of finding two disjoint matchings in a given edge weighted graph g so that their total weight is maximized the algorithm achieves an approximation ratio close to 0 76 and runs in o m n 3 n time where is the inverse ackermann function

a monotone finite element scheme for convection diffusion equations a simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection diffusion problems in any spatial dimension by properly averaging the pde coefficients on element edges the resulting finite element stiffness matrix is an m matrix under some mild assumption for the underlying generally unstructured finite element grids as a consequence the proposed edge averaged finite element scheme is particularly interesting for the discretization of convection dominated problems this scheme admits a simple variational formulation it is easy to analyze and it is also suitable for problems with a relatively smooth flux variable some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems

stabilized galerkin approximation of convection diffusion reaction equations discrete maximum principle and convergence we analyze a nonlinear shock capturing scheme for h1 conforming piecewise affine finite element approximations of linear elliptic problems the meshes are assumed to satisfy two standard conditions a local quasi uniformity property and the xu zikatanov condition ensuring that the stiffness matrix associated with the poisson equation is an m matrix a discrete maximum principle is rigorously established in any space dimension for convection diffusion reaction problems we prove that the shock capturing finite element solution converges to that without shock capturing if the cell peclet numbers are sufficiently small moreover in the diffusion dominated regime the difference between the two finite element solutions super converges with respect to the actual approximation error numerical experiments on test problems with stiff layers confirm the sharpness of the a priori error estimates

parameterized approximation algorithms for packing problems parameterized approximation is a topic of considerable interest in the field of parameterized complexity in the past decade new color coding related techniques including the breakthrough representative sets technique have been proven extremely powerful in the design of fast parameterized algorithms furthermore packing problems which are often solved via color coding related techniques have enjoyed a race towards obtaining the fastest parameterized algorithms that solve them therefore it is natural to ask whether packing problems admit efficient parameterized approximation algorithms in this paper we answer this question affirmatively we present tradeoffs that improve the running times of algorithms for well known special cases of the 3 setk packing problem at the cost of their accuracy consider a packing problem for which there is no known algorithm with approximation ratio and a parameter k if the value of an optimal solution is at least k we seek a solution of value at least k otherwise we seek an arbitrary solution clearly if the best known parameterized algorithm that finds a solution of value t runs in time o f t for some function f we are interested in running times better than o f k our main contribution lies in the adaptation of notions fundamental to the representative sets technique to the design of approximation algorithms we introduce the definition of approximate lopsided universal sets combine partial executions of representative sets based algorithms with approximation algorithms and adapt the iterative expansion framework in the context of representative sets to the design of parameterized approximation algorithms our ideas are intuitive and may be relevant to the design of other parameterized approximation algorithms

simulation of stochastic reaction diffusion processes on unstructured meshes we model stochastic chemical systems with diffusion by a reaction diffusion master equation on a macroscopic level the governing equation is a reaction diffusion equation for the averages of the chemical species on a mesoscopic level the master equation for a well stirred chemical system is combined with a discretized brownian motion in space to obtain the reaction diffusion master equation the space is covered in our method by an unstructured mesh and the diffusion coefficients on the mesoscale are obtained from a finite element discretization of the laplace operator on the macroscale the resulting method is a flexible hybrid algorithm in that the diffusion can be handled either on the meso or on the macroscale level the accuracy and the efficiency of the method are illustrated in three numerical examples inspired by molecular biology

a study on nonnegativity preservation in finite element approximation of nagumo type nonlinear differential equations preservation of nonnengativity and boundedness in the finite element solution of nagumo type equations with general anisotropic diffusion is studied linear finite elements and the backward euler scheme are used for the spatial and temporal discretization respectively an explicit an implicit and two hybrid explicitimplicit treatments for the nonlinear reaction term are considered conditions for the mesh and the time step size are developed for the numerical solution to preserve nonnegativity and boundedness the effects of lumping of the mass matrix and the reaction term are also discussed the analysis shows that the nonlinear reaction term has significant effects on the conditions for both the mesh and the time step size numerical examples are given to demonstrate the theoretical findings

convergence of finite volume schemes for semilinear convection diffusion equations the topic of this work is the discretization of semilinear elliptic problems in two space dimensions by the cell centered finite volume method dirichlet boundary conditions are considered here a discrete poincare inequality is used and estimates on the approximate solutions are proven the convergence of the scheme without any assumption on the regularity of the exact solution is proven using some compactness results which are shown to hold for the approximate solutions

force stepping integrators in lagrangian mechanics we present a class of integration schemes for lagrangian mechanics referred to as energy stepping integrators that are momentum and energy conserving symplectic and convergent in order to achieve these properties we replace the original potential energy by a piecewise constant or terraced approximation at steps of uniform height by taking steps of diminishing height an approximating sequence of energies is generated the trajectories of the resulting approximating lagrangians can be characterized explicitly and consist of intervals of piecewise rectilinear motion we show that the energy stepping trajectories are symplectic exactly conserve all the momentum maps of the original system and subject to a transversality condition converge to trajectories of the original system when the energy step is decreased to zero these properties the excellent long term behavior of energy stepping and its automatic time step selection property are born out by selected examples of application including the dynamics of a frozen argon cluster the spinning of an elastic cube and the collision of two elastic spheres

trisecting an orthoscheme a tetrahedron having two right angles at each of two vertices was investigated by lobachevsky who called it a pyramid schlafli who called it an orthoscheme wythoff who called it double rectangular and schoute who called its theory polygonometry there is a simple procedure for dissecting such a tetrahedron into three smaller orthoschemes the two cutting planes meet three of the four faces which are right angled triangles along lines which can easily be described when the tetrahedron is unfolded so as to put all the faces in one plane the arrangement of lines suggests an interesting theorem of absolute geometry when a particular spherical orthoscheme of known volume is dissected into three pieces and the volumes of these smaller orthoschemes are expressed as definite integrals the result is a peculiar identity which has not been verified directly there is a one parameter family of orthoschemes for which the three smaller orthoschemes are all congruent the euclidean member of this family turns out to be related to a very simply frieze pattern of integers

failure of the discrete maximum principle for an elliptic finite element problem there has been a long standing question of whether certain mesh restrictions are required for a maximum condition to hold for the discrete equations arising from a finite element approximation of an elliptic problem this is related to knowing whether the discrete green s function is positive for triangular meshes allowing sufficiently good approximation of h 1 functions we study this question for the poisson problem in two dimensions discretized via the galerkin method with continuous piecewise linears we give examples which show that in general the answer is negative and furthermore we extend the number of cases where it is known to be positive our techniques utilize some new results about discrete green s functions that are of independent interest

edge disjoint packing of stars and cycles abstract we study the parameterized complexity of two graph packing problems edge disjoint k packing of s stars and edge disjoint k packing of s cycles with respect to the choice of parameters we show that although the two problems are fpt with both k and s as parameters they are unlikely to be fixed parameter tractable when parameterized by only k or only s in terms of kernelization complexity we show that edge disjoint k packing of s stars has a kernel with size polynomial in both k and s but in contrast unless np conp poly edge disjoint k packing of s cycles does not have a kernel with size polynomial in both k and s and moreover does not have a kernel with size polynomial in s for any fixed k we also show that edge disjoint k packing of 4 cycles admits a 96 k 2 kernel in general graphs and a 96 k kernel in planar graphs

modified cholesky factorizations in interior point algorithms for linear programming we investigate a modified cholesky algorithm typical of those used in most interior point codes for linear programming cholesky based interior point codes are popular for three reasons their implementation requires only minimal changes to standard sparse cholesky algorithms allowing us to take full advantage of software written by specialists in that area they tend to be more efficient than competing approaches that use alternative factorizations and they perform robustly on most practical problems yielding good interior point steps even when the coefficient matrix of the main linear system to be solved for the step components is ill conditioned we investigate this surprisingly robust performance by using analytical tools from matrix perturbation theory and error analysis illustrating our results with computational experiments finally we point out the potential limitations of this approach

on nonobtuse simplicial partitions this paper surveys some results on acute and nonobtuse simplices and associated spatial partitions these partitions are relevant in numerical mathematics including piecewise polynomial approximation theory and the finite element method special attention is paid to a basic type of nonobtuse simplices called path simplices the generalization of right triangles to higher dimensions in addition to applications in numerical mathematics we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics

tight bounds on angle sums of nonobtuse simplices it is widely known that the sum of the angles of a triangle equals two right angles far less known are the answers to similar questions for tetrahedra and higher dimensional simplices in this paper we review some of these less known results and look at them from a different point of view then we continue to derive tight bounds on the dihedral angle sums for the subclass of nonobtuse simplices all the dihedral angles of such simplices are less than or equal to right they have several important applications brandts et al 2009 the main conclusion is that when the spatial dimension n is even the range of dihedral angle sums of nonobtuse simplices is n times smaller than the corresponding range for arbitrary simplices when n is odd it is n 1 times smaller

variational integrators and the newmark algorithm for conservative and dissipative mechanical systems the purpose of this work is twofold first we demonstrate analytically that the classical newmark family as well as related integration algorithms are variational in the sense of the veselov formulation of discrete mechanics such variational algorithms are well known to be symplectic and momentum preserving and to often have excellent global energy behavior this analytical result is veried through numerical examples and is believed to be one of the primary reasons that this class of algorithms performs so well second we develop algorithms for mechanical systems with forcing and in particular for dissipative systems in this case we develop integrators that are based on a discretization of the lagrange d alembert principle as well as on a variational formulation of dissipation it is demonstrated that these types of structured integrators have good numerical behavior in terms of obtaining the correct amounts by which the energy changes over the integration run

discontinuous variational time integrators for complex multibody collisions the objective of the present work is to formulate a new class of discontinuous variational time integrators that allow the system to adopt two possibly different configurations at each sampling time tk representing predictor and corrector configurations of the system the resulting sequence of configuration pairs then represents a discontinuous or non classical trajectory continuous or classical trajectories are recovered simply by enforcing a continuity constraint at all times in particular in systems subject to one sided contact constraints simulated via discontinuous variational time integrators the predictor configuration is not required to satisfy the one sided constraints whereas the corrector configuration is obtained by a closest point projection cpp onto the admissible set the resulting trajectories are generally discontinuous or non classical but are expected to converge to classical or continuous solutions for decreasing time steps we account for dissipation including friction by means of a discrete lagrange d alembert principle and make extensive use of the spacetime formalism in order to ensure exact energy conservation in conservative systems and the right rate of energy decay in dissipative systems the structure range and scope of the discontinuous variational time integrators and their accuracy characteristics are illustrated by means of examples of application concerned with rigid multibody dynamics

updating the inverse of a matrix the sherman morrison woodbury formulas relate the inverse of a matrix after a small rank perturbation to the inverse of the original matrix the history of these fomulas is presented and various applications to statistics networks structural analysis asymptotic analysis optimization and partial differential equations are discussed the sherman morrison woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix this paper surveys the history of these formulas and we examine some applications where these formulas are helpful

time discretized variational formulation of non smooth frictional contact the present work extends the non smooth contact class of algorithms introduced by kane et al to the case of friction the formulation specifically addresses contact geometries e g involving multiple collisions between tightly packed non smooth bodies for which neither normals nor gap functions can be properly defined a key aspect of the approach is that the incremental displacements follow from a minimum principle the objective function comprises terms which account for inertia strain energy contact friction and external forcing the euler lagrange equations corresponding to this extended variational principle are shown to be consistent with the equations of motion of solids in frictional contact in addition to its value as a basis for formulating numerical algorithms the variational framework offers theoretical advantages as regards the selection of trajectories in cases of non uniqueness we present numerical and analytical examples which demonstrate the good momentum and energy conservation characteristics of the numerical algorithms as well as the ability of the approach to account for stick and slip conditions

discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions one of the most important problems in numerical simulations is the preservation of qualitative properties of solutions of the mathematical models by computed approximations for problems of elliptic type one of the basic properties is the continuous maximum principle in our work we present several variants of the maximum principles and their discrete counterparts for scalar second order nonlinear elliptic problems with mixed boundary conditions the problems considered are numerically solved by the continuous piecewise linear finite element approximations built on simplicial meshes sufficient conditions providing the validity of the corresponding discrete maximum principles are presented geometrically they mean that the employed meshes have to be of acute or nonobtuse type depending of the type of the problem finally some examples of real life problems where the preservation of maximum principles plays an important role are presented

aspects of backward error analysis of numerical odes abstract in this paper some preliminary cases are studied to see how the idea of backward error analysis carries over to numerical solution of ordinary differential equations the interest here is in one step methods and infinite time intervals

the life span of backward error analysis for numerical integrators backward error analysis is a useful tool for the study of numerical approximations to ordinary differential equations the numerical solution is formally interpreted as the exact solution of a perturbed differential equation given as a formal and usually divergent series in powers of the step size for a rigorous analysis this series has to be truncated in this article we study the influence of this truncation to the difference between the numerical solution and the exact solution of the perturbed differential equation results on the long time behaviour of numerical solutions are obtained in this way we present applications to the numerical phase portrait near hyperbolic equilibrium points to asymptotically stable periodic orbits and hopf bifurcation and to energy conservation and approximation of invariant tori in hamiltonian systems

acute type refinements of tetrahedral partitions of polyhedral domains we present a new technique to perform refinements on acute type tetrahedral partitions of a polyhedral domain provided that the center of the circumscribed sphere around each tetrahedron belongs to the tetrahedron the resulting family of partitions is of acute type thus all the tetrahedra satisfy the maximum angle condition both these properties are highly desirable in finite element analysis

weakened acute type condition for tetrahedral triangulations and the discrete maximum principle we prove that a discrete maximum principle holds for continuous piecewise linear finite element approximations for the poisson equation with the dirichlet boundary condition also under a condition of the existence of some obtuse internal angles between faces of terahedra of triangulations of a given space domain this result represents a weakened form of the acute type condition for the three dimensional case

time integrators based on approximate discontinuous hamiltonians summary we introduce a class of time integration algorithms for finite dimensional mechanical systems whose hamiltonians are separable by partitioning the system s configuration space to construct an approximate potential energy we define an approximate discontinuous hamiltonian adh whose resulting equations of motion can be solved exactly the resulting integrators are symplectic and precisely conserve the approximate energy which by design is always close to the exact one we then propose two adh algorithms for finite element discretizations of nonlinear elastic bodies these result in two classes of explicit asynchronous time integrators that are scalable and because they conserve the approximate hamiltonian could be considered to be unconditionally stable in some circumstances in addition these integrators can naturally incorporate frictionless contact conditions we discuss the momentum conservation properties of the resulting methods and demonstrate their performance with several problems such as rotating bodies and multiple collisions of bodies with rigid boundaries copyright 2011 john wiley sons ltd

on the maximum angle condition for linear tetrahedral elements synge s maximum angle condition for triangular elements is generalized to tetrahedral elements for the generalized condition it is proved that tetrahedra may degenerate in a certain way and the error of the standard linear interpolation remains o h in the w p 1 omega norm for sufficiently smooth functions and p in 1 infty

reversible long term integration with variable stepsizes the numerical integration of reversible dynamical systems is considered a backward analysis for variable stepsize one step methods is developed and it is shown that the numerical solution of a symmetric one step method implemented with a reversible stepsize strategy is formally equal to the exact solution of a perturbed differential equation which again is reversible this explains geometrical properties of the numerical flow such as the nearby preservation of invariants in a second part the efficiency of symmetric implicit runge kutta methods linear error growth when applied to integrable systems is compared with explicit nonsymmetric integrators quadratic error growth

backward error analysis for numerical integrators backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods this is true in particular for the integration of hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time such results are typically based on two aspects of backward error analysis i it can be shown that the modified vector fields have some qualitative properties which they share with the given problem and ii an estimate is given for the difference between the best interpolating vector field and the numerical method these aspects have been investigated recently for example by benettin and giorgilli in j statist phys 74 1994 pp 1117 1143 by hairer in ann numer math 1 1994 pp 107 132 and by hairer and lubich in numer math 76 1997 pp 441 462 in this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit taylor series expansion of the numerical method and the corresponding flow maps as in the above cited works as an application we discuss the long time integration of chaotic hamiltonian systems and the approximation of time averages along numerically computed trajectories

a linear programming based algorithm for the signed separation of non smooth convex bodies a subdifferentiable global contact detection algorithm the supporting separating hyperplane ssh algorithm based on the signed distance between supporting hyperplanes of two convex sets is developed it is shown that for polyhedral sets the ssh algorithm may be evaluated as a linear program and that this linear program is always feasible and always subdifferentiable with respect to the configuration variables which define the constraint matrix this is true regardless of whether the program is primal degenerate dual degenerate or both the subgradient of the ssh linear program always lies in the normal cone of the closest admissible configuration to an inadmissible contact configuration in particular if a contact surface exists the subgradient of the ssh linear program is orthogonal to the contact surface as required of contact reactions this property of the algorithm is particularly important in modeling stiff systems rigid bodies and tightly packed or jammed systems

stable finite elements for problems with wild coefficients we consider solving an elliptic boundary value problem in the case that the coefficients vary by many orders of magnitude over the domain a linear finite element method is used it is shown that the standard method for solving the resulting linear equations in finite precision arithmetic can give an arbitrarily inaccurate answer because of ill conditioning in the stiffness matrix a new method for solving the linear equations is proposed this method is based on a mixed formulation and gives a numerically accurate answer independent of the variation in the coefficients the numerical error in the soution of the linear system for the new method is shown to depend on the aspect ratio of the triangulation

simulation of stochastic diffusion via first exit times in molecular biology it is of interest to simulate diffusion stochastically in the mesoscopic model we partition a biological cell into unstructured subvolumes in each subvolume the number of molecules is recorded at each time step and molecules can jump between neighboring subvolumes to model diffusion the jump rates can be computed by discretizing the diffusion equation on that unstructured mesh if the mesh is of poor quality due to a complicated cell geometry standard discretization methods can generate negative jump coefficients which no longer allows the interpretation as the probability to jump between the subvolumes we propose a method based on the mean first exit time of a molecule from a subvolume which guarantees positive jump coefficients two approaches to exit times a global and a local one are presented and tested in simulations on meshes of different quality in two and three dimensions

looking at the stars the problem of packing k vertex disjoint copies of a graph h into another graph g is np complete if h has more than two vertices in some connected component in the framework of parameterized complexity we analyze a particular family of instances of this problem namely the packing of stars we give a quadratic kernel for packing k copies of h k1 s when we consider the special case of s 2 i e h being a star with two leaves we give a linear kernel and an algorithm running in time o 25 301kk2 5 n3

the k distinct language parameterized automata constructions abstract in this paper we pioneer a study of parameterized automata constructions for languages related to the design of parameterized algorithms we focus on the k distinct language l k k defined as the set of words of length k over an alphabet whose symbols are all distinct this language is implicitly related to several breakthrough techniques developed during the last two decades to design parameterized algorithms for fundamental problems such as k path and r dimensional k matching building upon the color coding divide and color and narrow sieves techniques we obtain the following automata constructions for l k we develop non deterministic automata nfas of sizes 4 k o k n o 1 and 2 e k o k n o 1 where the latter satisfies a bounded ambiguity property relevant to approximate counting as well as a non deterministic xor automaton nxa of size 2 k n o 1 where n we show that our constructions can be used to develop both deterministic and randomized algorithms for k path r dimensional k matching and module motif in a natural manner considering also their approximate counting variants our framework is modular and consists of two parts designing an automaton for k distinct and designing a problem specific automaton as well as an algorithm for deciding whether the intersection automaton s language is empty or for counting the number of accepting paths in it

arbitrary level hanging nodes for adaptive hp fem approximations in 3d in this paper we discuss constrained approximation with arbitrary level hanging nodes in adaptive higher order finite element methods hp fem for three dimensional problems this technique enables using highly irregular meshes and it greatly simplifies the design of adaptive algorithms as it prevents refinements from propagating recursively through the finite element mesh the technique makes it possible to design efficient adaptive algorithms for purely hexahedral meshes we present a detailed mathematical description of the method and illustrate it with numerical examples

acceleration of material dominated calculations via phase space simplicial subdivision and interpolation we develop an acceleration method for material dominated calculations based on phase space simplicial interpolation of the relevant material response functions this process of interpolation constitutes an approximation scheme by which an exact material response function is replaced by a sequence of approximating response functions the terms in the sequence are increasingly accurate thus ensuring the convergence of the overall solution the acceleration ratio depends on the dimensionality the complexity of the deformation the time step size and the fineness of the phase space interpolation we ascertain these trade offs analytically and by recourse to selected numerical tests the numerical examples with piecewise quadratic interpolation in phase space confirm the analytical estimates

an algebraic discrete maximum principle in hilbert space with applications to nonlinear cooperative elliptic systems discrete maximum principles are derived for finite element discretizations of nonlinear elliptic systyems with cooperative and weakly diagonally dominant coupling the results are achieved via an algebraic discrete maximum principle organized in a hilbert space framework and in the case of simplicial elements are obtained under weakened acute type conditions for the finite element fe meshes

efficient computation of representative families with applications in parameterized and exact algorithms let m e i be a matroid and let s ls1 str be a family of subsets of e of size p a subfamily s s is q representative for s if for every set y e of size at most q if there is a set x s disjoint from y with x y i then there is a set x s disjoint from y with x y i by the classic result of bollobas in a uniform matroid every family of sets of size p has a q representative family with at most p qp sets in his famous two families theorem from 1977 lovasz proved that the same bound also holds for any matroid representable over a field f we give an efficient construction of a q representative family of size at most p qp in time bounded by a polynomial in p qp t and the time required for field operations we demonstrate how the efficient construction of representative families can be a powerful tool for designing single exponential parameterized and exact exponential time algorithms the applications of our approach include the following in the l ong d irected c ycle problem the input is a directed n vertex graph g and the positive integer k the task is to find a directed cycle of length at least k in g if such a cycle exists as a consequence of our 6 75k o k no 1 time algorithm we have that a directed cycle of length at least log n if such a cycle exists can be found in polynomial time in the m inimum e quivalent g raph meg problem we are seeking a spanning subdigraph d of a given n vertex digraph d with as few arcs as possible in which the reachability relation is the same as in the original digraph d we provide an alternative proof of the recent results for algorithms on graphs of bounded treewidth showing that many connectivity problems such as h amiltonian c ycle or s teiner t ree can be solved in time 2o t n on n vertex graphs of treewidth at most t for the special case of uniform matroids on n elements we give a faster algorithm to compute a representative family we use this algorithm to provide the fastest known deterministic parameterized algorithms for k p ath k t ree and more generally k s ubgraph i somorphism where the k vertex pattern graph is of constant treewidth

geometric aspects of the symmetric inverse m matrix problem abstract we investigate the symmetric inverse m matrix problem from a geometric perspective the central question in this geometric context is which conditions on the k dimensional facets of an n simplex s guarantee that s has no obtuse dihedral angles the simplest of such conditions is that if all triangular facets of s are equilateral then s is regular and thus nonobtuse first we study the properties of an n simplex s whose k facets are all nonobtuse and generalize some classical results by fiedler 14 we prove that if all n 1 facets of an n simplex s are nonobtuse each makes at most one obtuse dihedral angle with another facet this helps to identify a special type of tetrahedron which we will call sub orthocentric with the property that if all tetrahedral facets of s are sub orthocentric then s is nonobtuse rephrased in the language of linear algebra this constitutes a purely geometric proof of the fact that each symmetric ultrametric matrix 28 is the inverse of a weakly diagonally dominant m matrix the geometric proof provides valuable insights that supplement the discrete measure theoretic 24 and the linear algebraic 25 proof the review papers 20 21 support our believe that the linear algebraic perspective on the inverse m matrix problem dominates the literature the geometric perspective however connects sign properties of entries of inverses of a symmetric positive definite matrix to the dihedral angle properties of an underlying simplex and enables an explicit visualization of how these angles and signs can be manipulated this will serve to formulate purely geometric conditions on the k facets of an n simplex s that may render s nonobtuse also for k 3 for this we generalize the class of sub orthocentric tetrahedra that gives rise to the class of ultrametric matrices to sub orthocentric simplices that define symmetric positive definite matrices a with special types of k k principal submatrices for k 3 each sub orthocentric simplices is nonobtuse and we conjecture that any simplex with sub orthocentric facets only is sub orthocentric itself along the way several additional new concepts will be introduced such as vertex gramians simplicial matrix classes the dual hull and the sub orthocentric set of a nonobtuse simplex and nonblocking matrices these concepts may also be of use in a different linear algebraic setting in particular in the context if completely positive matrices for which we will prove some auxiliary results as well

analysis and design of jump coefficients in discrete stochastic diffusion models in computational systems biology the mesoscopic model of reaction diffusion kinetics is described by a continuous time discrete space markov process to simulate diffusion stochastically the jump coefficients are obtained by a discretization of the diffusion equation using unstructured meshes to represent complicated geometries may lead to negative coefficients when using piecewise linear finite elements several methods have been proposed to modify the coefficients to enforce the nonnegativity needed in the stochastic setting in this paper we present a method to quantify the error introduced by that change we interpret the modified discretization matrix as the exact finite element discretization of a perturbed equation the forward error the error between the analytical solutions to the original and the perturbed equations is bounded by the backward error the error between the diffusion of the two equations we present a backward analysis algorithm to compute the diffusion coefficient from a given discretization matrix the analysis suggests a new way of deriving nonnegative jump coefficients that minimizes the backward error the theory is tested in numerical experiments indicating that the new method is superior and also minimizes the forward error

parameterized algorithms for edge biclique and related problems abstract maximum edge biclique and related problems have wide applications in management science bioinformatics etc in this paper we study parameterized algorithms for parameterized edge biclique problem parameterized edge biclique packing problem parameterized biclique edge deletion problem and parameterized bipartite biclique clustering problem for the parameterized edge biclique problem the current best result is of running time o 2 k and we give a parameterized algorithm of running time o n k 2 5 k k where k is the parameter and n is the number of vertices in the given graph for the parameterized edge biclique packing problem based on randomized divide and conquer technique a parameterized algorithm of running time o k k l o g k 4 2 k 1 t is given where k is the parameter and t is the number of bicliques in the solution we study the parameterized biclique edge deletion problem on bipartite graphs and general graphs and give parameterized algorithms of running time o 2 k and o 3 k respectively for the parameterized bipartite biclique clustering problem based on modular decomposition method a kernel of size o k 2 and a parameterized algorithm of running time o 2 42 k n m are presented where k is the parameter n is the number of vertices and m is the number of edges in the given graph

local nonobtuse tetrahedral refinements around an edge in this note we show how to generate and conformly refine nonobtuse tetrahedral meshes locally around and towards an edge so that all dihedral angles of all resulting tetrahedra remain nonobtuse the proposed technique can be used e g for a numerical treatment of solution singularities and also for various mesh adaptivity procedures near the reentrant corners of cylindric type 3d domains

on the generalization of the synge křižek maximum angle condition for d simplices abstract in this note we present a generalization of the maximum angle condition proposed by j l synge in 1957 and m k i ek in 1992 for triangular and tetrahedral elements respectively for the case of higher dimensional simplicial finite elements its relations to the other angle type conditions commonly used in interpolation theory and in finite element analysis are discussed

discrete maximum principle and adequate discretizations of linear parabolic problems in this paper we analyze the connections between the different qualitative properties of numerical solutions of linear parabolic problems with dirichlet type boundary condition first we formulate the qualitative properties for the differential equations and shed light on their relations then we show how the well known discretization schemes can be written in the form of a one step iterative process we give necessary and sufficient conditions of the main qualitative properties of these iterations we apply the results to the finite difference and galerkin finite element solutions of linear parabolic problems in our main result we show that the nonnegativity preservation property is equivalent to the maximum minimum principle and they imply the maximum norm contractivity in one two and three dimensions we list sufficient a priori conditions that ensure the required qualitative properties finally we demonstrate the above results on numerical examples

a family of space time connecting discretization schemes with local impact detection for elastodynamic contact problems abstract we derive formulate and analyze a new family of discretization schemes for elastodynamic contact problems which implicitly resolve the individual impact times for each node on the contact interface within our approach information from the space discrete setting is incorporated into the time discretization by means of pointwise chosen parameters for the time discretization scheme the members of this family can be interpreted as modified newmark schemes thus making them easily understandable and implementable we prove that for certain parameter choices the algorithms are dissipative methods further as our analysis and our numerical experiments show a special solution dependent choice of parameters leads to a new space time connecting discretization with a highly stable behavior of displacements velocities and boundary stresses at the contact interface

finite element analysis of nonsmooth contact this work develops robust contact algorithms capable of dealing with complex contact situations involving several bodies with corners amongst the mathematical tools we bring to bear on the problem is nonsmooth analysis following clarke f h clarke optimization and nonsmooth analysis john wiley and sons new york 1983 we specifically address contact geometries for which both the use of normals and gap functions have difficulties and therefore precludes the application of most contact algorithms proposed to date such situations arise in applications such as fragmentation where angular fragments undergo complex collision sequences before they scatter we demonstrate the robustness and versatility of the nonsmooth contact algorithms developed in this paper with the aid of selected two and three dimensional applications

discrete maximum principle for fe solutions of the diffusion reaction problem on prismatic meshes in this paper we analyse the discrete maximum principle dmp for a stationary diffusion reaction problem solved by means of prismatic finite elements we derive geometric conditions on the shape parameters of the prismatic partitions which guarantee validity of the dmp the presented numerical tests show the sharpness of the obtained conditions

on continuous and discrete maximum principles for elliptic problems with the third boundary condition in this work we present and discuss some continuous and discrete maximum principles for linear elliptic problems of the second order with the third boundary condition almost never addressed to in the available literature in this context solved by finite element and finite difference methods numerical tests are given

the maximum angle condition is not necessary for convergence of the finite element method we show that the famous maximum angle condition in the finite element analysis is not necessary to achieve the optimal convergence rate when simplicial finite elements are used to solve elliptic problems this condition is only sufficient in fact finite element approximations may converge even though some dihedral angles of simplicial elements tend to

a 2 o k n algorithm for k cycle in minor closed graph families abstract let c be a proper minor closed family of graphs we present a randomized algorithm that given a graph g c with n vertices finds a simple cycle of size k in g if exists in 2 o k n time the algorithm applies to both directed and undirected graphs in previous linear time algorithms for this problem the runtime dependence on k is super exponential the algorithm can be derandomized yielding a 2 o k n log n time algorithm

on 0 1 polytopes with nonobtuse triangulations recently brandts et al 2013 5 studied 0 1 triangulations of the unit n cube in with simplices that only have nonobtuse dihedral angles an example is the standard triangulation into n simplices it is proved in 5 that for each n 3 there is essentially only one other nonobtuse 0 1 triangulation of in here we will outline an investigation into 0 1 triangulations of other 0 1 polytopes with simplices that only have nonobtuse dihedral angles as far as we know this is the only source that combines both concepts 0 1 polytopes and nonobtuse 0 1 triangulations in particular we investigate nonobtuse 0 1 triangulations of 0 1 polytopes in i3 and i4

the development of variable step symplectic integrators with application to the two body problem the authors develop and test variable step symplectic runge kutta nystrom algorithms for the integration of hamiltonian systems of ordinary differential equations numerical experiments suggest that for symplectic formulae moving from constant to variable stepsizes results in a marked decrease in efficiency on the other hand symplectic formulae with constant stepsizes may outperform available standard nonsymplectic variable step codes for the model situation consisting in the long time integration of the two body problem our experimental findings are backed by theoretical analysis

the optimal refinement strategy for 3 d simplicial meshes 3 d mesh refinement strategies are the object of interest in the present paper the dependence of finite element solutions on various refinement procedures is discussed a new refinement strategy which is optimal with respect to the measure of degeneracy for all canonical domains is obtained comparisons with known refinement techniques are made some isoparametric examples are presented

variational time integrators the purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest in particular the conservation properties of both synchronous and asynchronous variational integrators avis are discussed in detail we present selected numerical examples which demonstrate the excellent accuracy conservation and convergence characteristics of avis in these tests avis are found to result in substantial speed ups at equal accuracy relative to explicit newmark a mathematical proof of convergence of the avis is also presented in this paper finally we develop the subject of horizontal variations and configurational forces in discrete dynamics this theory leads to exact path independent characterizations of the configurational forces acting on discrete systems notable examples are the configurational forces acting on material nodes in a finite element discretisation and the j integral at the tip of a crack in a finite element mesh

well centered triangulation meshes composed of well centered simplices have nice orthogonal dual meshes the dual voronoi diagram this is useful for certain numerical algorithms that prefer such primal dual mesh pairs we prove that well centered meshes also have optimality properties and relationships to delaunay and minmax angle triangulations we present an iterative algorithm that seeks to transform a given triangulation in two or three dimensions into a well centered one by minimizing a cost function and moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed the cost function is a direct result of a new characterization of well centeredness in arbitrary dimensions that we present ours is the first optimization based heuristic for well centeredness and the first one that applies in both two and three dimensions we show the results of applying our algorithm to small and large two dimensional meshes some with a complex boundary and obtain a well centered tetrahedralization of the cube we also show numerical evidence that our algorithm preserves gradation and that it improves the maximum and minimum angles of acute triangulations created by the best known previous method

a gauge invariant discretization on simplicial grids of the schrodinger eigenvalue problem in an electromagnetic field we propose a method to compute approximate eigenpairs of the schrodinger operator on a bounded domain in the presence of an electromagnetic field formulated for simplicial meshes the method combines techniques from lattice gauge theory and finite elements retaining the discrete gauge invariance of the former but allowing for noncongruent space elements as in the latter the error of the method is studied in the framework of strang s variational crimes comparing with a standard galerkin approach when the meshes are quasi uniform and satisfy the discrete maximum principle we prove that for a smooth electromagnetic field the crime is of the order of the mesh width h for a coulomb potential it is of order h log h and for a general finite energy electromagnetic field it is of order h 1 2

on scaled projections and pseudoinverses abstract let x be a matrix of full column rank and let d be a diagonal matrix with positive diagonal elements the weighted pseudoinverse defined by x d x t dx 1 x t d and the associated oblique projection p d xx d arise in many applications in this paper we show that the norms of both matrices are bounded by numbers that are independent of d

sign preserving of principal eigenfunctions in p1 finite element approximation of eigenvalue problems of second order elliptic operators this paper is concerned with the p1 finite element approximation of the eigenvalue problem of second order elliptic operators subject to the dirichlet boundary condition the focus is on the preservation of the basic properties of the principal eigenvalue and eigenfunctions of the continuous problem it is shown that when the stiffness matrix is an irreducible m matrix the discrete eigenvalue problem maintains almost all of the basic properties such as the smallest eigenvalue being real and simple and the corresponding eigenfunctions being either positive or negative inside the physical domain mesh conditions leading to such a stiffness matrix are also studied a sufficient condition is that the mesh is simplicial interiorly connected and acute when measured in the metric specified by the inverse of the diffusion matrix the acute requirement can be replaced by the delaunay condition in two dimensions numerical results show that when the stiffness matrix is not an m matrix a finite element approximation can be structurally different from the continuous eigenvalue problem the eigenfunctions corresponding to the smallest eigenvalue can change sign over the physical domain and the smallest eigenvalue in modulus can even be complex for the case with nonsymmetric operators

asynchronous variational integrators we describe a class of asynchronous variational integrators avi for nonlinear elastodynamics the avis are characterized by the following distinguishing attributes i the algorithms permit the selection of independent time steps in each element and the local time steps need not bear an integral relation to each other ii the algorithms derive from a spacetime form of a discrete version of hamilton s principle as a consequence of this variational structure the algorithms conserve local energy and momenta exactly subject to solvability of the local time steps numerical tests reveal that even when local energy balance is not enforced exactly the global and local energy behavior of the avis is quite remarkable a property which can probably be traced to the symplectic nature of the algorithm

a nonconforming finite element method for the biot s consolidation model in poroelasticity a stable finite element scheme that avoids pressure oscillations for a three field biot s model in poroelasticity is considered the involved variables are the displacements fluid flux darcy velocity and the pore pressure and they are discretized by using the lowest possible approximation order crouzeix raviart finite elements for the displacements lowest order raviart thomas nedelec elements for the darcy velocity and piecewise constant approximation for the pressure mass lumping technique is introduced for the raviart thomas nedelec elements in order to eliminate the darcy velocity and therefore reduce the computational cost we show convergence of the discrete scheme which is implicit in time and use these types of elements in space with and without mass lumping finally numerical experiments illustrate the convergence of the method and show its effectiveness to avoid spurious pressure oscillations when mass lumping for the raviart thomas nedelec elements is used

triangulations of cayley and tutte polytopes cayley polytopes were defined recently as convex hulls of cayley compositions introduced by cayley in 1857 in this paper we resolve braun s conjecture which expresses the volume of cayley polytopes in terms of the number of connected graphs we extend this result to a two variable deformations which we call tutte polytopes the volume of the latter is given via an evaluation of the tutte polynomial of the complete graph our approach is based on an explicit triangulation of the cayley and tutte polytope we prove that simplices in the triangulations correspond to labeled trees and forests the heart of the proof is a direct bijection based on the neighbors first search graph traversal algorithm r esum

hopdm version 2 12 a fast lp solver based on a primal dual interior point method hopdm is an implementation of the primal dual interior point method for solving large scale linear programming lp problems hopdm stands for higher order primal dual method its newest version 2 12 of may 1995 is a robust and efficient code that compares favourably with today s commercial lp solvers

global and local refinement techniques yielding nonobtuse tetrahedral partitions preservation of basic qualitative properties for example the maximum principle ofthe solution of partial differential equations by its finite element approximations is an important goal in mathematical modelling and simulation nonobtuse tetrahedral partitions and linear finite elements guarantee the validity of the discrete analogues of the maximum principle for a wide class of parabolic and elliptic problems in order to get more accurate approximation we often need to refine the used partitions globally or locally in this paper we first propose two variants of global refinement techniques which produce nonobtuse face to face tetrahedral partitions second we present a new local refinement technique which generates nonobtuse face to face tetrahedral partitions in a neighbourhood of a given vertex

lagrange constraints for transient finite element surface contact a new approach to enforce surface contact conditions in transient non linear finite element problems is developed in this paper the method is based on the lagrange multiplier concept and is compatible with explicit time integration operators compatibility with explicit operators is established by referencing lagrange multipliers one time increment ahead of associated surface contact displacement constraints however the method is not purely explicit because a coupled system of equations must be solved to obtain the lagrange multipliers an important development herein is the formulation of a highly efficient method to solve the lagrange multiplier equations the equation solving strategy is a modified gauss seidel method in which non linear surface contact force conditions are enforced during iteration the new surface contact method presented has two significant advantages over the widely accepted penalty function method surface contact conditions are satisfied more precisely and the method does not adversely affect the numerical stability of explicit integration transient finite element analysis results are presented for problems involving impact and sliding with friction a brief review of the classical lagrange multiplier method with implicit integration is also included