p matrices are just sufficient abstract the classes of sufficient matrices and of p matrices have recently arisen in connection with the linear complementarity problem it is known that p matrices are sufficient we show that conversely every sufficient matrix is a p matrix

field theory conjecture for loop erased random walks we give evidence that the functional renormalization group frg developed to study disordered systems may provide a field theoretic description for the loop erased random walk lerw allowing to compute its fractal dimension in a systematic expansion in e 4 d up to two loop the frg agrees with rigorous bounds correctly reproduces the leading logarithmic corrections at the upper critical dimension d 4 and compares well with numerical studies we obtain the universal subleading logarithmic correction in d 4 which can be used as a further test of the conjecture

high order infeasible interior point methods for solving sufficient linear complementarity problems in this paper we develop systematically infeasible interior point methods of arbitrarily high order for solving horizontal linear complementarity problems that are sufficient in the sense ofcottle pang and venkateswaran 1989 the results apply to degenerate problems and problem shaving no strictly complementary solution variants of these methods are described that eventually avoid recentering steps and for which all components of the approximate solutions converge superlinearly at a high order and other variants which even terminate with a solution of the complementarity problem after finitely many steps

loop erased self avoiding random walk and the laplacian random walk comments that the laplacian random walk with eta 1 recently introduced by lyklema and evertsz 1986 is the same process as the loop erased self avoiding random walk analysed previously by the author 1980 some rigorous results about this process are reviewed

basis and partition identification for quadratic programming and linear complementarity problems optimal solutions of interior point algorithms for linear and quadratic programming and linear complementarity problems provide maximally complementary solutions maximally complementary solutions can be characterized by optimal partitions on the other hand the solutions provided by simplex based pivot algorithms are given in terms of complementary bases a basis identification algorithm is an algorithm which generates a complementary basis starting from any complementary solution a partition identification algorithm is an algorithm which generates a maximally complementary solution and its corresponding partition starting from any complementary solution in linear programming such algorithms were respectively proposed by megiddo in 1991 and balinski and tucker in 1969 in this paper we will present identification algorithms for quadratic programming and linear complementarity problems with sufficient matrices the presented algorithms are based on the principal pivot transform and the orthogonality property of basis tableaus

on the uniqueness of solutions to linear complementarity problems this paper characterizes the classu of all realn n matricesm for which the linear complementarity problem q m has a unique solution for all realn vectorsq interior to the conek m of vectors for which q m has any solution at all it is shown that restricting the uniqueness property to the interior ofk m is necessary because whenu the problem q m has infinitely many solutions ifq belongs to the boundary of intk m it is shown thatm must have nonnegative principal minors whenu andk m is convex finally it is shown that whenm has nonnegative principal minors only one of which is 0 andk m r n thenu andk m is a closed half space

sufficient matrices and the linear complementarity problem abstract we pose and answer two questions about solutions of the linear complementarity problem lcp the first question is concerned with the conditions on a matrix m which guarantee that for every vector q the solutions of the lcp q m are identical to the karush kuhn tucker points of the natural quadratic program associated with q m in answering this question we introduce the class of row sufficient matrices the transpose of such a matrix is what we call column sufficient the latter matrices turn out to furnish the answer to our second question which asks for the conditions on m under which the solution set of q m is convex for every q in addition to these two main results we discuss the connections of these twonew matrix classes with other well known matrix classes in linear complementarity theory

analysis of infeasible interior point paths arising with semidefinite linear complementarity problems we consider semidefinite monotone linear complementarity problems sdlcp in the space equation here n of real symmetric n n matrices equipped with the cone equation here n of all symmetric positive semidefinite matrices one may define weighted using any m equation here n as weight infeasible interior point paths by replacing the standard condition xy ri r 0 that defines the usual central path by xy yx 2 rm under some mild assumptions the most stringent is the existence of some strictly complementary solution of sdlcp these paths have a limit as r 0 and they depend analytically on all path parameters such as r and m even at the limit point r 0

a class of linear complementarity problems solvable in polynomial time we describe a condition number for the linear complementarity problem lcp which characterizes the degree of difficulty for its solution when a potential reduction algorithm is used consequently we develop a class of lcps solvable in polynomial time the result suggests that the convexity or positive semidefiniteness of the lcp may not be the basic issue that separates lcps solvable and not solvable in polynomial time

criteria for sufficient matrices abstract column sufficient row sufficient and sufficient matrices have recently arisen in connection with the linear complementarity problem we review and supplement the basic theory of these matrix classes propose new criteria for identifying them and compare these criteria with the existing ones our main mathematical tool is principal pivoting

new characterizations of row sufficient matrices in dealing with a linear complementarity problem much depends on knowing that the matrix through which the particular lcp is defined belongs to a suitable matrix class two such classes are su the so called sufficient matrices and l which were introduced in r w cottle j s pang v venkateswaran sufficient matrices and the linear complementarity problem linear algebra appl 114 115 1989 231 249 b c eaves the linear complementarity problem manage sci 17 1971 612 634 respectively in an earlier article i adler r w cottle s verma sufficient matrices belong to l math prog 106 2006 391 401 the authors proved that su is a subclass of l by definition the class su is the intersection of two distinct classes rsu the row sufficient matrices and csu the column sufficient matrices in the present work we strengthen the aforementioned inclusion by showing that all row sufficient matrices belong to l using what we call structural properties of certain matrix classes we add to the existing characterizations of rsu in r w cottle s m guu two characterizations of sufficient matrices linear algebra appl 170 1992 65 74 s m guu r w cottle on a subclass of p0 linear algebra appl 223 224 1995 325 335 h valiaho criteria for sufficient matrices linear algebra appl 233 1996 109 129 this line of inquiry was inspired by asking what must be true of a row sufficient l matrix we establish three new characterizations of rsu in terms of the matrix classes l e0 and q0 and the structural properties of sign change invariance completeness and fullness the new characterizations of rsu provide new characterizations of su by adjoining a fourth structural property we call reflectiveness

a constructive characterization of q 0 matrices with nonnegative principal minors in 2 we characterized the class of matrices with nonnegative principla minors for which the linear complementarity problem always has a solution that class is contained in the one we study here our main result gives a finitely testable set of necessary and sufficient conditions under which a matrix with nonnegative principal minors has the property that if a corresponding linear complementarity problem is feasible then it is solvable in short we constructively characterize the matrix class known asqo po

error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem consider the affine variational inequality problem it is shown that the distance to the solution set from a feasible point near the solution set can be bounded by the norm of a natural residual at that point this bound is then used to prove linear convergence of a matrix splitting algorithm for solving the symmetric case of the problem this latter result improves upon a recent result of luo and tseng that further assumes the problem to be monotone

on a new homotopy continuation trajectory for nonlinear complementarity problems most known continuation methods for p 0 complementarity problems require some restrictive assumptions such as the strictly feasible condition and the properness condition to guarantee the existence and the boundedness of certain homotopy continuation trajectory to relax such restrictions we propose a new homotopy formulation for the complementarity problem based on which a new homotopy continuation trajectory is generated for p 0 complementarity problems the most promising feature of this trajectory is the assurance of the existence and the boundedness of the trajectory under a condition that is strictly weaker than the standard ones used widely in the literature of continuation methods particularly the often assumed strictly feasible condition is not required here when applied to p complementarity problems the boundedness of the proposed trajectory turns out to be equivalent to the solvability of the problem and the entire trajectory converges to the unique least element solution provided that it exists moreover for monotone complementarity problems the whole trajectory always converges to a least 2 norm solution provided that the solution set of the problem is nonempty the results presented in this paper can serve as a theoretical basis for constructing a new path following algorithm for solving complementarity problems even for the situations where the solution set is unbounded

sufficient matrices belong to l in this paper we establish a significant matrix class inclusion that seems to have been overlooked in the literature of the linear complementarity problem we show that p the class of sufficient matrices is a subclass of l in the course of demonstrating this inclusion we introduce other new matrix classes that forge interesting new connections between known matrix classes

complementarity properties of singular m matrices abstract for a matrix a whose off diagonal entries are nonpositive its nonnegative invertibility namely that a is an invertible m matrix is equivalent to a being a p matrix which is necessary and sufficient for the unique solvability of the linear complementarity problem defined by a this in turn is equivalent to the statement that a is strictly semimonotone in this paper an analogue of this result is proved for singular symmetric z matrices this is achieved by replacing the inverse of a by the group generalized inverse and by introducing the matrix classes of strictly range semimonotonicity and range column sufficiency a recently proposed idea of p matrices plays a pivotal role some interconnections between these matrix classes are also obtained

self avoiding walks in four dimensions logarithmic corrections we present simulation results for long n or 4000 self avoiding walks in four dimensions we find definite indications of logarithmic corrections but the data are poorly described by the asymptotically leading terms detailed comparisons are presented with renormalization group flow equations derived by direct renormalization and with the results of a field theoretic calculation

loop erased self avoiding walks in two dimensions exact critical exponents and winding numbers abstract loop erased self avoiding walks lesaws are defined as walks resulting from sequential erasing of the loops of random walks the critical properties of lesaws are those of a c 2 conformal theory in two dimensions and geometrically those of subgraphs of spanning trees in this paper we derive the exact asymptotic behavior of the set of probabilities p k n that k k 2 lesaws of length n starting at neighbouring points do not intersect as well as the exact asymptotic winding angle distribution of a single two dimensional lesaw

a solution condition for complementarity problems with an application to spatial price equilibrium a sufficient condition is developed for the existence of solutions to complementarity problems which is applicable to all continuous functions in addition it is shown to imply a number of simpler conditions including the well known coerciveness condition finally this condition is applied to an existence problem in spatial price equilibrium where it is shown to yield not only a more general result but also a simpler method of proof

integer solution for linear complementarity problem we consider the problem of finding an integer solution to a linear complementarity problem we introduce the class i of matrices for which the corresponding linear complementarity problem has an integer complementary solution for every vector q for which it has a solution strong principal unimodularity forms a sufficient condition for inclusion in the class i it is also shown to be necessary for some well known classes of matrices including the class of positive semidefinite matrices this is used in deriving necessary and sufficient conditions for a convex quadratic program to have an integer optimal solution for all b and c for which it has an optimal solution characterizations are also derived for some other well known classes of matrices in linear complementarity to belong to the class i in the end a peeling algorithm for finding integer solution for linear complementarity problems is given and related results are derived

invasion percolation and global optimization invasion bond percolation ibp is mapped exactly into prim close quote s algorithm for finding the shortest spanning tree of a weighted random graph exploring this mapping which is valid for arbitrary dimensions and lattices we introduce a new ibp model that belongs to the same universality class as ibp and generates the minimal energy tree spanning the ibp cluster copyright ital 1996 the american physical society

theory of minimum spanning trees i mean field theory and strongly disordered spin glass model the minimum spanning tree mst is a combinatorial optimization problem given a connected graph with a real weight cost on each edge find the spanning tree that minimizes the sum of the total cost of the occupied edges we consider the random mst in which the edge costs are quenched independent random variables there is a strongly disordered spin glass model due to newman and stein phys rev lett 72 2286 1994 which maps precisely onto the random mst we study scaling properties of random msts using a relation between kruskal s greedy algorithm for finding the mst and bond percolation we solve the random mst problem on the bethe lattice bl with appropriate wired boundary conditions and calculate the fractal dimension d 6 of the connected components viewed as a mean field theory the result implies that on a lattice in euclidean space of dimension d there are of order w d ensuremath d large connected components of the random mst inside a window of size w and that d d c d 6 is a critical dimension this differs from the value 8 suggested by newman and stein we also critique the original argument for 8 and provide an improved scaling argument that again yields d c 6 the result implies that the strongly disordered spin glass model has many ground states for dg6 and only of order one below six the results for msts also apply on the poisson weighted infinite tree which is a mean field approach to the continuum model of msts in euclidean space and is a limit of the bl in a companion paper we develop an ensuremath epsilon 6 ensuremath d expansion for the random mst on critical percolation clusters

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

on singular n0 matrices and the class q abstract the class of n and n 0 matrices arises in the theory of global univalence of functions multivariate analysis and in complementarity theory in this paper we study singular n 0 matrices with q property

on hidden z matrices and the linear complementarity problem abstract in this paper we explore various matrix theoretic aspects of the hidden z class and demonstrate how the concept of principal pivot transform can be effectively used to extend many existing results in fact we revisit various results obtained for hidden z class by mangasarian cottle and pang in context of solving linear complementarity problems as linear programs we identify hidden z matrices of special category and discuss the number of solutions of the associated linear complementarity problems we also present game theoretic interpretation of various results related to hidden z class and obtain proofs following the game theoretic approach of raghavan for a subclass of z matrices

on the transpose of a pseudomonotone matrix and the lcp abstract in the study of linear complementary problem it is known that pseudomonotone matrices belong to the class p 0 q 0 in this note we show that under certain conditions such as invertibility or normality the transpose of a pseudomonotone matrix belongs to the class q 0

criss cross methods a fresh view on pivot algorithms criss cross methods are pivot algorithms that solve linear programming problems in one phase starting with any basic solution the first finite criss cross method was invented by chang terlaky and wang independently unlike the simplex method that follows a monotonic edge path on the feasible region the trace of a criss cross method is neither monotonic with respect to the objective function nor feasibility preserving the main purpose of this paper is to present mathematical ideas and proof techniques behind finite criss cross pivot methods a recent result on the existence of a short admissible pivot path to an optimal basis is given indicating shortest pivot paths from any basis might be indeed short for criss cross type algorithms the origins and the history of criss cross methods are also touched upon

d orientation sequences for continuous functions and nonlinear complementarity problems we introduce the new concept of d orientation sequence for continuous functions it is shown that if there does not exist a d orientation sequence for a continuous function then the corresponding complementarity problem cp has a solution we believe that such a result characterizes an intrinsic property of cps as the concept of exceptional family of elements the notion of d orientation sequence of a function is also a powerful tool for investigating the existence theorems of cps we use this new tool to establish among other things a new existence result for a class of p mapping cps

asymptotic analysis of a computational method for time and frequency dependent radiative transfer we consider a time dependent energy dependent nonlinear radiative transfer problem in which opacities are large o 1 and interior sources are small o an asymptotic analysis of this problem as 0 leads to the equilibrium diffusion equation in the interior of the system along with boundary conditions and initial conditions for this equation we apply the same asymptotic analysis to a discrete version of the problem in which the frequency variable is discretized by the multigroup method the direction variable by the discrete ordinates method the time variable by the fully implicit method and the spatial variable by a subcell balance method we find that as 0 the discrete solution satisfies a robust discretized version of the correct equilibrium diffusion equation with boundary conditions and initial conditions that are remarkably accurate the analysis thus predicts that if a spatial grid is chosen that resolves interior temperature gradients then the numerical method obtains an accurate solution in the interior of the system even though the optical thickness of the spatial cells tends to and boundary layers in the transport solution are not resolved we go a step further to analyze problems that are optically thin at some photon frequencies but thick at others and show that once again the discrete solution is remarkably accurate we present numerical results that verify these and other predictions of the analyses

asymptotic diffusion limit of cell temperature discretisation schemes for thermal radiation transport abstract this paper attempts to unify the asymptotic diffusion limit analysis of thermal radiation transport schemes for a linear discontinuous representation of the material temperature reconstructed from cell centred temperature unknowns in a process known as source tilting the asymptotic limits of both monte carlo continuous in space and deterministic approaches based on linear discontinuous finite elements for solving the transport equation are investigated in slab geometry the resulting discrete diffusion equations are found to have nonphysical terms that are proportional to any cell edge discontinuity in the temperature representation based on this analysis it is possible to design accurate schemes for representing the material temperature for coupling thermal radiation transport codes to a cell centred representation of internal energy favoured by ale arbitrary lagrange eulerian hydrodynamics schemes

theory of minimum spanning trees ii exact graphical methods and perturbation expansion at the percolation threshold continuing the program begun by the authors in a previous paper we develop an exact low density expansion for the random minimum spanning tree mst on a finite graph and use it to develop a continuum perturbation expansion for the mst on critical percolation clusters in space dimension d the perturbation expansion is proved to be renormalizable in d 6 dimensions we consider the fractal dimension d sub p of paths on the latter mst our previous results lead us to predict that d sub p 2 for d d sub c 6 using a renormalization group approach we confirm the result for d 6 and calculate d sub p to first order in epsilon 6 d for d 6 using the connection with critical percolation with the result d sub p 2 epsilon 7 o epsilon sup 2

on the complexity of computing the handicap of a sufficient matrix the class of sufficient matrices is important in the study of the linear complementarity problem lcp some interior point methods ipm s for lcp s with sufficient data matrices have complexity polynomial in the bit size of the matrix and its handicap in this paper we show that the handicap of a sufficient matrix may be exponential in its bit size implying that the known complexity bounds of interior point methods are not polynomial in the input size of the lcp problem we also introduce a semidefinite programming based heuristic that provides a finite upper bond on the handicap for the sub class of mathcal p matrices where all principal minors are positive

critical exponent for the loop erased self avoiding walk by monte carlo methods a monte carlo simulation was performed for loop erased self avoiding walks lesaw to ascertain the exponentv for the z2 and z3 lattices the estimated values were 2v 1 600 0 006 in two dimensions and 2v 1 232 0 008 in three dimensions leading to the conjecturev 4 5 for the two dimensional lesaw these results add to existing evidence that the loop erased self avoiding walks are not in the same universality class as self avoiding walks

red green blue model we experimentally study the red green blue model which is a system of loops obtained by superimposing three dimer coverings on offset hexagonal lattices we find that when the boundary conditions are flat the red green blue loops are closely related to stochastic loewner evolution with parameter ensuremath kappa 4 mathrm sle 4 and double dimer loops which are the loops formed by superimposing two dimer coverings of the cartesian lattice but we also find that the red green blue loops are more tightly nested than the double dimer loops we also investigate the two dimensional minimum spanning tree and find that it is not conformally invariant

distribution of sizes of erased loops of loop erased random walks in two and three dimensions we show that in the loop erased random walk problem the exponent characterizing probability distribution of areas of erased loops is superuniversal in d dimensions the probability that the erased loop has an area a varies as a 2 for large a independent of d for 2 d 4 we estimate the exponents characterizing the distribution of perimeters and areas of erased loops in d 2 and 3 by large scale monte carlo simulations our estimate of the fractal dimension z in two dimensions is consistent with the known exact value 5 4 in three dimensions we get z 1 6183 0 0004 the exponent for the distribution of durations of avalanche in the three dimensional abelian sandpile model is determined from this by using scaling relations

invasion percolation a new form of percolation theory a new kind of percolation problem is described which differs from ordinary percolation theory in that it automatically finds the critical points of the system the model is motivated by the problem of one fluid displacing another from a porous medium under the action of capillary forces but in principle it may be applied to any kind of invasion process which proceeds along a path at least resistance the name invasion percolation is proposed for this new process similarities to and differences from ordinary percolation theory are discussed

determining the handicap of a sufficient matrix abstract any linear complementarity problem with a sufficient matrix can be solved by means of the unified interior point method the complexity bound of the method is the better the smaller the so called handicap of the matrix is we propose a method for determining the handicap of a sufficient matrix and show that a sufficient matrix and its transpose have the same handicap

an alternative theorem for generalized variational inequalities and solvability of nonlinear quasi p m complementarity problems in this paper an alternative theorem and hence a sufficient solution condition is established for generalized variational inequality problems the concept of exceptional family for generalized variational inequality is introduced this concept is general enough to include as special cases the notions of exceptional family of elements and the d orientation sequence for continuous functions particularly we apply the alternative theorem for investigating the solvability of the nonlinear complementarity problems with so called quasi p m maps which are broad enough to encompass the quasi monotone maps and p maps as the special cases an existence theorem for this class of complementarity problems is established which significantly generalizes several previous existence results in the literature

scaling of waves in the bak tang wiesenfeld sandpile model n with aen ae0 n a similar ansatzes should according to this view hold for the number of topplings s which difiers from a because sites can topple more than once in an avalanche and for the radius and duration of avalanches but recent inves tigations 10 12 show that the two dimensional btw model may be characterized by a multifractal behavior where difierent moments of a are governed by exponents aen which are not linear in n and are indeed not related to each other for difierent n difierent reasons for this have been proposed in 10 11 and in 12 notice that multi fractality of avalanches can be proven for certain variants of the 1 d sandpile model 14 deviations from pure power laws had been seen already in early simulations but were usually interpreted as fl nite size efiects due to avalanches which touch the bound ary of the lattice to illustrate that this is most likely not true and that there is a real problem with simple scaling we show in fig 1 the ratio ps x l pa x l of the integrated distributions ps x l r 1 x dx 0 ps x 0 l for d 2 and for difierent values of l in these simula tions cylindrical boundary conditions were used open at y l 2 and periodic at x 0 l and data were col lected only for avalanches starting at y 0 in this way we hope to have minimized boundary efiects also since we do not make separate flts to ps x l and pa x l we have none of the uncertainties inherent in such flts due

asymptotic analysis of the spatial discretization of radiation absorption and re emission in implicit monte carlo we perform an asymptotic analysis of the spatial discretization of radiation absorption and re emission in implicit monte carlo imc a monte carlo technique for simulating nonlinear radiative transfer specifically we examine the approximation of absorption and re emission by a spatially continuous artificial scattering process and either a piecewise constant or piecewise linear emission source within each spatial cell we consider three asymptotic scalings representing i a time step that resolves the mean free time ii a courant limit on the time step size and iii a fixed time step that does not depend on any asymptotic scaling for the piecewise constant approximation we show that only the third scaling results in a valid discretization of the proper diffusion equation which implies that imc may generate inaccurate solutions with optically large spatial cells if time steps are refined however we also demonstrate that for a certain class of problems the piecewise linear approximation yields an appropriate discretized diffusion equation under all three scalings we therefore expect imc to produce accurate solutions for a wider range of time step sizes when the piecewise linear instead of piecewise constant discretization is employed we demonstrate the validity of our analysis with a set of numerical examples

scaling and universality in continuous length combinatorial optimization we consider combinatorial optimization problems defined over random ensembles and study how solution cost increases when the optimal solution undergoes a small perturbation delta for the minimum spanning tree the increase in cost scales as delta2 for the minimum matching and traveling salesman problems in dimension d 2 the increase scales as delta3 this is observed in monte carlo simulations in d 2 3 4 and in theoretical analysis of a mean field model we speculate that the scaling exponent could serve to classify combinatorial optimization problems of this general kind into a small number of distinct categories similar to universality classes in statistical physics

the principal pivoting method revisited the principal pivoting method ppm for the linear complementarity problem lcp is shown to be applicable to the class of lcps involving the newly identified class of sufficient matrices

higher correlations universal distributions and finite size scaling in the field theory of depinning recently we constructed a renormalizable field theory up to two loops for the quasi static depinning of elastic manifolds in a disordered environment here we explore further properties of the theory we show how higher correlation functions of the displacement field can be computed drastic simplifications occur unveiling much simpler diagrammatic rules than anticipated this is applied to the universal scaled width distribution the expansion in d 4 epsilon predicts that the scaled distribution coincides to the lowest orders with the one for a gaussian theory with propagator g q 1 q d 2 zeta zeta being the roughness exponent the deviations from this gaussian result are small and involve higher correlation functions which are computed here for different boundary conditions other universal quantities are defined and evaluated we perform a general analysis of the stability of the fixed point we find that the correction to scaling exponent is omega epsilon and not epsilon 3 as used in the analysis of some simulations a more detailed study of the upper critical dimension is given where the roughness of interfaces grows as a power of a logarithm instead of a pure power

more on positive subdefinite matrices and the linear complementarity problem abstract in this article we consider positive subdefinite matrices psbd recently studied by j p crouzeix et al siam j matrix anal appl 22 2000 66 and show that linear complementarity problems with psbd matrices of rank 2 are processable by lemke s algorithm and that a psbd matrix of rank 2 belongs to the class of sufficient matrices introduced by r w cottle et al linear algebra appl 114 115 1989 231 we also show that if a matrix a is a sum of a merely positive subdefinite copositive plus matrix and a copositive matrix and a feasibility condition is satisfied then lemke s algorithm solves lcp q a this supplements the results of jones and evers

traveling salesman problem conformal invariance and dense polymers we propose that the statistics of the optimal tour in the planar random euclidean traveling salesman problem is conformally invariant on large scales this is exhibited in the power law behavior of the probabilities for the tour to zigzag repeatedly between two regions and in subleading corrections to the length of the tour the universality class should be the same as for dense polymers and minimal spanning trees the conjectures for the length of the tour on a cylinder are tested numerically

two characterizations of sufficient matrices abstract two characterizations are given for the class of sufficient matrices defined by cottle pang and venkateswaran the first is a direct translation of the definition into linear programming terms the second can be thought of as a generalization of a theorem of t d parsons on p matrices

scaling of loop erased walks in 2 to 4 dimensions we simulate loop erased random walks on simple hyper cubic lattices of dimensions 2 3 and 4 these simulations were mainly motivated to test recent two loop renormalization group predictions for logarithmic corrections in d 4 simulations in lower dimensions were done for completeness and in order to test the algorithm in d 2 we verify with high precision the prediction d 5 4 where the number of steps n after erasure scales with the number n of steps before erasure as n nd 2 in d 3 we again find a power law but with an exponent different from the one found in the most precise previous simulations d 1 6236 0 0004 finally we see clear deviations from the naive scaling n n in d 4 while they agree only qualitatively with the leading logarithmic corrections predicted by several authors their agreement with the two loop prediction is nearly perfect

on a global projection type error bound for the linear complementarity problem abstract we consider a projection type error bound for the linear complementarity problem involving a matrix m and vector q first we show that the mangasarian ren sufficient condition on m for this error bound to hold globally for all q such that the problem is solvable is also necessary second we derive necessary and sufficient conditions on m and q for this error bound to hold globally thus extending a result of luo and tseng which in addition assumes m is psd plus third we derive necessary and sufficient conditions on m for this error bound to hold globally independent of q these conditions are closely related to m being lipschitzian we also discuss application of our result to the affine variational inequality problem

choosing a spanning tree for the integer lattice uniformly consider the nearest neighbor graph for the integer lattice zd in d dimensions for a large finite piece of it consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees as the piece gets larger this approaches a limiting measure on the set of spanning graphs for zd this is shown to be a tree if and only if d 4 in this case the tree has only one topological end that is there are no doubly infinite paths when d 2 5 the spanning forest has infinitely many components almost surely with each component having one or two topological ends

a fully polynomial time approximation algorithm for computing a stationary point of the general linear complementarity problem we apply a potential reduction algorithm to solve the general linear complementarity problem glcp minimize xty subject to ax by cz q and x y z 0 we show that the algorithm is a fully polynomial time approximation scheme fptas for computing an e approximate stationary point of the glcp note that there are some glcps in which every stationary point is a solution xty 0 these include the lcps with row sufficient matrices we also show that the algorithm is a polynomial time algorithm for a special class of glcps

equivalence between the abelian sandpile model and the q 0 limit of the potts model abstract we establish an equivalence between the undirected abelian sandpile model and the q 0 limit of the q state potts model the equivalence is valid for arbitrary finite graphs two dimensional abelian sandpile models thus correspond to a conformal field theory with central charge c 2 the equivalence also gives a monte carlo algorithm to generate random spanning trees we study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the abelian sandpile model the average number of sites burnt upto time t varies at ta in two dimensions our numerically determined value of a agrees with the theoretical prediction a 8 5 we relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes

on the equivalence of nonlinear complementarity problems and least element problems strictly pseudomonotonez maps operating on banach lattices are considered equivalence of complementarity problems and least element problems is established under certain regularity and growth conditions this extends a recent result by riddell 1981 for strictly monotonez maps to the pseudomonotone case some other problems equivalent to the above are discussed as well

on a subclass of p0 abstract in 1966 fiedler and ptak wrote the first systematic investigation of the matrix class p 0 consisting of all real square matrices with nonnegative principal minors in this note we focus on a particular subclass of p 0 that has arisen within the field of linear complementarity theory namely the class su of sufficient matrices our principal result is that su contains another class p we conjecture that the latter two classes are in fact the same

on the classes of fully copositive and fully semimonotone matrices abstract in this paper we consider the class c 0 f of fully copositive and the class e 0 f of fully semimonotone matrices we show that c 0 f matrices with positive diagonal entries are column sufficient we settle a conjecture made by murthy and parthasarathy to the effect that a c 0 f q 0 matrix is positive semidefinite by providing a counterexample we finally consider e 0 f matrices introduced by cottle and stone math program 27 1983 191 213 and partially address stone s conjecture to the effect that e 0 f q 0 p 0 by showing that e 0 f d c matrices are p 0 where d c is the doverspike class of matrices

pc matrices and the linear complementarity problem abstract we introduce a new matrix class pc which consists of those matrices m for which the solution set of the corresponding linear complementarity problem is connected for every q rn we consider lemke s pivotal method from the perspective of piecewise linear homotopies and normal maps and show that lemke s method processes all matrices in pc q0 we further investigate the relationship of the class pc to other known matrix classes and show that column sufficient matrices are a subclass of pc as are 2 2 p0 matrices

a linear discontinuous spatial differencing scheme forsnradiative transfer calculations various types of linear discontinuous spatial differencing schemes have been developed for thesn discrete ordinates equations approximating the linear boltzmann transport equation it has been shown through an asymptotic analysis that the 1d slab geometry lumped linear discontinuous scheme not only goes over to a convergent and robust differencing of the diffusion equation in the monoenergetic thick diffusion limit but it also yields the correct interior solution even when boundary layers are left unresolved by the spatial mesh in the present work we generalize this scheme to obtain a 1d slab geometry lumped linear discontinuous scheme for the nonlinear radiative transfer equation and the associated material temperature equation we present a full nonlinear energy dependent asymptotic analysis of the behavior of this scheme in the thick equilibrium diffusion limit we find that this scheme goes over to a convergent and robust differencing of the equilibrium diffusion equation on the interior of the mesh but it does not yield the exact interior solution when boundary layers are left unresolved by the spatial mesh nevertheless the interior solution obtained with spatially unresolved boundary layers is always well behaved and fairly accurate computational results are presented which test the predictions of our asymptotic analysis and demonstrate the efficiency of our solution technique

the scaling limit of loop erased random walk in three dimensions we show that the scaling limit exists and is invariant under dilations and rotations we give some tools that might be useful to show universality

the linear complimentarity problem sufficient matrices and the criss cross method specially structured linear complementarity problems lcps and their solution by the criss cross method are examined the criss cross method is known to be finite for lcps with positive semidefinite bisymmetric matrices and with p matrices it is also a simple finite algorithm for oriented matroid programming problems recently cottle pang and venkateswaran identified the class of column row sufficient matrices they showed that sufficient matrices are a common generalization of p and psd matrices cottle also showed that the principal pivoting method with a clever modification can be applied to row sufficient lcps in this paper the finiteness of the criss cross method for sufficient lcps is proved further it is shown that a matrix is sufficient if and only if the criss cross method processes all the lcps defined by this matrix and all the lcps defined by the transpose of this matrix and any parameter vector

exponents and bounds for uniform spanning trees in d dimensions uniform spanning trees are a statistical model obtained by taking the set of all spanning trees on a given graph such as a portion of a cubic lattice in d dimensions with equal probability for each distinct tree some properties of such trees can be obtained in terms of the laplacian matrix on the graph by using grassmann integrals we use this to obtain exact exponents that bound those for the power law decay of the probability that k distinct branches of the tree pass close to each of two distinct points as the size of the lattice tends to infinity

asymptotic equilibrium diffusion analysis of time dependent monte carlo methods for grey radiative transfer the equations of nonlinear time dependent radiative transfer are known to yield the equilibrium diffusion equation as the leading order solution of an asymptotic analysis when the mean free path and mean free time of a photon become small we apply this same analysis to the fleck cummings carter forest and n kaoua monte carlo approximations for grey frequency independent radiative transfer although monte carlo simulation usually does not require the discretizations found in deterministic transport techniques monte carlo methods for radiative transfer require a time discretization due to the nonlinearities of the problem if an asymptotic analysis of the equations used by a particular monte carlo method yields an accurate time discretized version of the equilibrium diffusion equation the method should generate accurate solutions if a time discretization is chosen that resolves temperature changes even if the time steps are much larger than the mean free time of a photon this analysis is of interest because in many radiative transfer problems it is a practical necessity to use time steps that are large compared to a mean free time our asymptotic analysis shows that i the n kaoua method has the equilibrium diffusion limit ii the carter forest method has the equilibrium diffusion limit if the material temperature change during a time step is small and iii the fleck cummings method does not have the equilibrium diffusion limit we include numerical results that verify our theoretical predictions

asymptotic diffusion limit of the symbolic monte carlo method for the transport equation we use asymptotic analysis to study the diffusion limit of the symbolic implicit monte carlo simc method for the transport equation for standard simc with piecewise constant basis functions we demonstrate mathematically that the solution converges to the solution of a wrong diffusion equation nevertheless a simple extension to piecewise linear basis functions enables to obtain the correct solution we present numerical examples which illustrate the analysis

loop erased self avoiding random walk in two and three dimensions if n is the position of the self avoiding random walk in d obtained by erasing loops from simple random walk then it is proved that the mean square displacemente n 2 grows at least as fast as the flory predictions for the usual saw i e at least as fast asn3 2 ford 2 andn6 5 ford 3 in particular if the mean square displacement of the usual saw grows liken1 18 ind 3 as expected then the loop erased process is in a different universality class

principal pivot transforms of some classes of matrices in this paper we study and characterize various classes of matrices that are defined based on principal pivot transforms we show that matrices in these classes have nonnegative principal minors

loop erased random walk as a spin system observable the determination of the hausdorff dimension of the scaling limit of loop erased random walk is closely related to the study of the one point function of loop erased random walk i e the probability a loop erased random walk passes through a given vertex recent work in the theoretical physics literature has investigated the hausdorff dimension of loop erased random walk in three dimensions by applying field theory techniques to study spin systems that heuristically encode the one point function of loop erased random walk inspired by this we introduce two different spin systems whose correlation functions can be rigorously shown to encode the one point function of loop erased random walk

minimum spanning trees and random resistor networks in d dimensions we consider minimum cost spanning trees both in lattice and euclidean models in d dimensions for the cost of the optimum tree in a box of size l we show that there is a correction of order l ensuremath theta where ensuremath theta ensuremath leqslant 0 is a universal d dependent exponent there is a similar form for the change in optimum cost under a change in boundary condition at nonzero temperature t there is a crossover length ensuremath xi ensuremath sim t ensuremath ensuremath nu such that on length scales larger than ensuremath xi the behavior becomes that of uniform spanning trees there is a scaling relation ensuremath theta ensuremath 1 ensuremath nu and we provide several arguments that show that ensuremath nu and ensuremath 1 ensuremath theta both equal ensuremath nu mathrm perc the correlation length exponent for ordinary percolation in the same dimension d in all dimensions d ensuremath geqslant 1 the arguments all rely on the close relation of kruskal s greedy algorithm for the minimum spanning tree percolation and for some arguments random resistor networks the scaling of the entropy and free energy at small nonzero t and hence of the number of near optimal solutions is also discussed we suggest that the steiner tree problem is in the same universality class as the minimum spanning tree in all dimensions as is the traveling salesman problem in two dimensions hence all will have the same value of ensuremath theta ensuremath 3 4 in two dimensions

notes on sufficient matrices abstract we give some constructive characterizations for sufficient matrices and present some matrix classes whose elements are identified as column sufficient matrices in particular we give the structure of column row sufficient matrices which leads to efficient testing methods for column row sufficiency

minimum spanning trees on random networks we show that the geometry of minimum spanning trees mst on random graphs is universal because of this geometric universality we are able to characterize the energy of mst using a scaling distribution p epsilon found using uniform disorder we show that the mst energy for other disorder distributions is simply related to p epsilon we discuss the relationship to invasion percolation to the directed polymer in a random media to uniform spanning trees and also the implications for the broader issue of universality in disordered systems