bounding distributions for the weight of a minimum spanning tree in stochastic networks this paper considers the problem of determining the distribution of the weight w of a minimum spanning tree for an undirected graph with edge weights that are independently distributed discrete random variables using the underlying fundamental cutsets and cycles associated with a spanning tree we are able to obtain upper and lower bounds on the distribution of w in turn these are used to establish bounds on e w our general method for deriving these bounding distributions subsumes existing approximation methods in the literature computational results indicate that the new approximation methods provide excellent bounds for some challenging test networks

strong mixed integer programming formulations for trained neural networks we present strong mixed integer programming mip formulations for high dimensional piecewise linear functions that correspond to trained neural networks these formulations can be used for a number of important tasks such as verifying that an image classification network is robust to adversarial inputs or solving decision problems where the objective function is a machine learning model we present a generic framework which may be of independent interest that provides a way to construct sharp or ideal formulations for the maximum of d affine functions over arbitrary polyhedral input domains we apply this result to derive mip formulations for a number of the most popular nonlinear operations e g relu and max pooling that are strictly stronger than other approaches from the literature we corroborate this computationally showing that our formulations are able to offer substantial improvements in solve time on verification tasks for image classification networks

probabilistic combinatorial optimization moments semidefinite programming and asymptotic bounds we address the problem of evaluating the expected optimal objective value of a 0 1 optimization problem under uncertainty in the objective coefficients the probabilistic model we consider prescribes limited marginal distribution information for the objective coefficients in the form of moments we show that for a fairly general class of marginal information a tight upper lower bound on the expected optimal objective value of a 0 1 maximization minimization problem can be computed in polynomial time if the corresponding deterministic problem is solvable in polynomial time we provide an efficiently solvable semidefinite programming formulation to compute this tight bound we also analyze the asymptotic behavior of a general class of combinatorial problems that includes the linear assignment spanning tree and traveling salesman problems under knowledge of complete marginal distributions with and without independence we calculate the limiting constants exactly

the value of side information in network flow optimization abstract consider a group of agents who seek to simultaneously traverse a graph each edge of the graph has an associated weight e g a delay and the agents seek to minimize the cumulative weight incurred by all agents as each traverses a path of the graph an edge s weight is a function of the number of agents that use that edge as well as an inherent random weight if the agents have no side information about the random component they will deterministically organize themselves so as to optimize their average performance we consider a generalization of this framework whereby the agents have access to a limited amount of shared side information about the edge weights and we study the relationship between information quantity and performance

the distribution of maximum flow with applications to multistate reliability systems this paper describes an efficient monte carlo sampling plan for estimating the distribution of maximum flow in a directed network whose arcs have random capacities such a network can be used to represent a multistate system whose multistate components capacities are subject to random deterioration the proposed sampling plan uses an easily computed a priori upper bound on the complementary distribution function to obtain an unbiased point estimator with smaller variance than the estimator obtained by crude monte carlo sampling the paper also describes procedures for interval estimation and for assessing when the sampling experiment has achieved a specified accuracy to facilitate sampling we use cumulative processes to characterize deterioration leading to the treatment of arc capacities as being multinormally distributed we describe a technique for checking the appropriateness of this model with regard to lower and upper bounds on capacity we also describe a procedure for deriving a confidence interval on the measure used to assess variance reduction an example illustrates the sampling plan and a concise summary gives all steps needed to implement the plan

piercing convex sets and the hadwiger debrunner p q problem a family of sets has the p q property if among any p members of the family some q have a nonempty intersection it is shown that for every p q d 1 there is a c c p q d such that for every family j of compact convex sets in rd which has the p q property there is a set of at most c points in rd that intersects each member of j this settles an old problem of hadwiger and debrunner

on theoretical and empirical aspects of marginal distribution choice models in this paper we study the properties of a recently proposed class of semiparametric discrete choice models referred to as the marginal distribution model mdm by optimizing over a family of joint error distributions with prescribed marginal distributions surprisingly the choice probabilities arising from the family of generalized extreme value models of which the multinomial logit model is a special case can be obtained from this approach despite the difference in assumptions on the underlying probability distributions we use this connection to develop flexible and general choice models to incorporate consumer and product level heterogeneity in both partworths and scale parameters in the choice model furthermore the extremal distributions obtained from the mdm can be used to approximate the fisher s information matrix to obtain reliable standard error estimates of the partworth parameters without having to bootstrap the method we use simulated and empirical data sets to test the performance of this approach we evaluate the performance against the classical multinomial logit mixed logit and a machine learning approach that accounts for partworth heterogeneity our numerical results indicate that mdm provides a practical semiparametric alternative to choice modeling this paper was accepted by eric bradlow special issue on business analytics

discretization of ideal fluid dynamics in the eulerian representation a hamiltonian discretization of one dimensional compressible fluid dynamics is made possible by analyzing the properties of semidirect product lie algebras associated to representations of the lie algebra of vector fields in r

on the complexity of nonoverlapping multivariate marginal bounds for probabilistic combinatorial optimization problems given a combinatorial optimization problem with an arbitrary partition of the set of random objective coefficients we evaluate the tightest possible bound on the expected optimal value for joint distributions consistent with the given multivariate marginals of the subsets in the partition for univariate marginals this bound was first proposed by meilijson and nadas meilijson i a nadas 1979 convex majorization with an application to the length of critical path j appl probab 16 3 671 677 we generalize the bound to nonoverlapping multivariate marginals using multiple choice integer programming new instances of polynomial time computable bounds are identified for discrete distributions for the problem of selecting up to m items out of a set of n items of maximum total weight the multivariate marginal bound is shown to be computable in polynomial time when the size of each subset in the partition is o log n for an activity on arc pert network the partition is naturally defined by subsets of incoming arcs into nodes the multivariate marginal bound on expected project duration is shown to be computable in time polynomial in the maximum number of scenarios for any subset and the size of the network as an application a polynomial time solvable two stage stochastic program for project crashing is identified an important feature of the bound developed in this paper is that it is exactly achievable by a joint distribution unlike many of the existing bounds

exponentially derived switching schemes for inviscid flow abstract a class of exponential schemes used for singular perturbation problems is taken and finite difference schemes for inviscid flow with shocks is derived from them in particular exponential schemes are formulated for steady viscous flow in a variable area duct using both a one equation model with the physical viscous terms and the euler equations with artificial viscosity upon taking the limit as the viscosity coefficient goes to zero exponentially derived switching eds schemes are obtained which switch the direction of finite differencing based upon characteristic directions of the reduced problem for the euler equations some of the eds schemes can be identified as flux vector splitting the split coefficient matrix method and a scheme of huang some aspects of uniqueness of finite difference solutions are discussed

implicit explicit residual smoothing for the multidimensional euler and navier stokes equations in this paper an implicit explicit version of the concept of residual smoothing in several space dimensions will be introduced this version will improve damping properties and robustness considerably numerical examples are presented showing reduction of the cpu time needed to compute steady state solutions for inviscid and viscid cases as well as for subsonic and transonic cases

boundedly simple groups of automorphisms of trees abstract a group is boundedly simple if for some constant n every nontrivial conjugacy class generates the whole group in n steps for a large class of trees tits proved simplicity of a canonical subgroup of the automorphism group which is generated by pointwise stabilizers of edges we prove that only for uniform subdivisions of biregular trees are such groups boundedly simple in fact these groups are 8 boundedly simple as a consequence we prove that if g is boundedly simple and g acts by automorphisms on a tree then g fixes some vertex of a or stabilizes some end of a or the smallest nonempty g invariant subtree of a is a uniform subdivision of a biregular tree

on the dimension of cat 0 spaces where mapping class groups act whenever the mapping class group of a closed orientable surface of genus g acts by semisimple isometries on a complete cat 0 space of dimension less than g it fixes a point

the intersection of a matroid and an oriented matroid abstract we prove the following matroid intersection theorem let m be a matroid with rank function and let o be an oriented matroid of rank r both defined on the same ground set v and satisfying v r if every subset s v with v s r contains a positive circuit of o then there is a positive circuit of o which is independent in m this contains imre barany s colorful caratheodory theorem as a special case the proof uses topological methods and combines the folkman lawrence representation theorem with a generalization of kalai and meshulam s topological colorful helly theorem

a probabilistic model for minmax regret in combinatorial optimization in this paper we propose a new probabilistic model for minimizing the anticipated regret in combinatorial optimization problems with distributional uncertainty in the objective coefficients the interval uncertainty representation of data is supplemented with information on the marginal distributions as a decision criterion we minimize the worst case conditional value at risk of regret the proposed model includes the interval data minmax regret model as a special case for the class of combinatorial optimization problems with a compact convex hull representation a polynomial sized mixed integer linear program is formulated when a the range and mean are known and b the range mean and mean absolute deviation are known and a mixed integer second order cone program is formulated when c the range mean and standard deviation are known for the subset selection problem of choosing k elements of maximum total weight out of a set of n elements the probabilistic regret model is shown to be solvable in polynomial time in the instances a and b above this extends the current known polynomial complexity result for minmax regret subset selection with range information only

stochastic bounds on distributions of optimal value functions with applications to pert network flows and reliability in many classical combinatorial optimization problems including critical and shortest paths maximum flow and network reliability the introduction of uncertainty considerably complicates the calculation of system performance in fact in these contexts computing system performance exactly can often be an impossible task therefore obtaining stochastic bounds on the system s performance becomes an attractive and useful alternative this paper studies several stochastic bounds that are applicable to these contexts and to a broader set of problems that can be described by the general combinatorial concepts of clutters and blocking clutters we begin our discussion by defining these unifying concepts and illustrating their specialization in several problem contexts

a comparison of optimization based approaches for a model computational aerodynamics design problem abstract the objective of this paper is to compare three optimization based methods for solving aerodynamic design problems we use the euler equations for one dimensional duct flow as a model problem the optimization methods are i the black box method with finite difference gradients ii a modification where gradients are found by an algorithm based on the implicit function theorem and iii an all atonce method where the flow and design variables are simultaneously altered the three methods are applied to the model problem and compared for efficiency robustness and implementation difficulty we also show that the black box implicit gradient method is equivalent to applying the variational or optimal control approach to design optimization directly to the discretized analysis problem rather than to the continuous problem as is usually done the black box method with implicit gradients seems to provide a good compromise of features and can be retrofitted to most existing analysis codes to turn them into design codes although the all at once method was found to be less robust than the black box methods when it succeeded it was considerably more efficient

the constrained bottleneck problem in networks we consider problems on networks that are captured by two performance measures one performance measure is any general cost function of a solution the other is a bottleneck measure that describes the worst maximum cost component of the solution the paper contains algorithms to solve three problems in one problem we minimize the bottleneck subject to a constraint on the generalized cost in the second problem we minimize the generalized cost subject to a constraint on the bottleneck in the third problem we consider the two criteria simultaneously and find all the pareto optimum solutions the major result is that the introduction of the bottleneck measure changes the complexity of the original general cost problem by a factor which is at most linear in the number of links

dynamical approach study of spurious steady state numerical solutions of nonlinear differential equations i the dynamics of time discretization and its implications for algorithm development in computational fluid dynamics abstract the goal of this paper is to utilize the theory of nonlinear dynamics approach to investigate the possible sources of errors and slow convergence and nonconvergence of steady state numerical solutions when using the time dependent approach for nonlinear hyperbolic and parabolic partial differential equations terms this interdisciplinary research belongs to a subset of a new field of study in numerical analysis sometimes referred to as the dynamics of numerics and the numerics of dynamics at the present time this new interdisciplinary topic is still the property of an isolated discipline with all too little effort spent in pointing out an underlying generality that could make it adaptable to diverse fields of applications this is the first of a series of research papers under the same topic our hope is to reach researchers in the fields of computational fluid dynamics cfd and in particular hypersonic and combustion related cfd by simple examples in which the exact solutions of the governing equations are known the application of the apparently straightforward numerical technique to genuinely nonlinear problems can be shown to lead to incorrect or misleading results one striking phenomenon is that with the same initial data the continuum and its discretized counterpart can asymptotically approach different stable solutions this behavior is especially important for employing a time dependent approach to the steady state since the initial data are usually not known and a freestream condition or an intelligent guess for the initial conditions is often used with the unique property of the different dependence of the solution on initial data for the partial differential equation and the discretized counterpart it is not easy to delineate the true physics from numerical artifacts when numerical methods are the sole source of solution procedure for the continuum part i concentrates on the dynamical behavior of time discretization for scalar nonlinear ordinary differential equations in order to motivate this new yet unconventional approach to algorithm development in cfd and to serve as an introduction for parts 11 and iii of the same series of research papers

adaptive harmonic balance solutions to euler s equation a new adaptive harmonic balance method is presented and applied to a variety of subsonic and supersonic onedimensional e owe elds containing strong moving shocks the adaptive method augments the frequency content in each cell as required to capture the local e ow physics augmentation automatically adjusts with grid density resulting in lower frequency content on coarse grids that cannot adequately resolve high frequency terms a study of the effect of augmentation thresholds increments and scheduling on the performance and accuracy of the adaptive method is presented and optimal parameters identie ed when optimal parameters are used the new adaptive harmonic balancemethod produces solutions equivalent to a nonadapted harmonic balance solution but with up to a 50 reduction in run time

helly numbers of acyclic families the helly number of a family of sets with empty intersection is the size of its largest inclusion wise minimal sub family with empty intersection let f be a finite family of open subsets of an arbitrary locally arc wise connected topological space assume that for every sub family g f the intersection of the elements of g has at most r connected components each of which is a q homology cell we show that the helly number of f is at most r d 1 where d is the smallest integer j such that every open set of has trivial q homology in dimension j and higher in particular drd d this bound is best possible we also prove a stronger theorem where the intersection of small sub families may have more than r connected components each possibly with nontrivial homology in low dimension as an application we obtain several explicit bounds on helly numbers in geometric transversal theory for which only ad hoc geometric proofs were previously known in certain cases the bound we obtain is better than what was previously known in fact our proof bounds the leray number of the nerves of the families under consideration and thus also yields under similar assumptions a fractional helly theorem a p q theorem and the existence of small weak nets

inviscid and viscous stationary waves of gas flow through contracting expanding nozzles abstract in this work we study steady states of one dimensional viscous isentropic compressible flows through a contracting expanding nozzle treating the viscosity coefficient as a singular parameter the steady state problem can be viewed as a singularly perturbed system for a contracting expanding nozzle a complete classification of steady states is given and the existence of viscous profiles is established via the geometric singular perturbation theory particularly interesting is the existence of a maximal sub to super transonic wave and its role in the formation of other complicated transonic waves consisting of a sub to super portion

persistency model and its applications in choice modeling given a discrete maximization problem with a linear objective function where the coefficients are chosen randomly from a distribution we would like to evaluate the expected optimal value and the marginal distribution of the optimal solution we call this the persistency problem for a discrete optimization problem under uncertain objective and the marginal probability mass function of the optimal solution is named the persistence value in general this is a difficult problem to solve even if the distribution of the objective coefficient is well specified in this paper we solve a subclass of this problem when the distribution is assumed to belong to the class of distributions defined by given marginal distributions or given marginal moment conditions under this model we show that the persistency problem maximizing the expected objective value over the set of distributions can be solved via a concave maximization model the persistency model solved using this formulation can be used to obtain important qualitative insights to the behavior of stochastic discrete optimization problems we demonstrate how the approach can be used to obtain insights to problems in discrete choice modeling using a set of survey data from a transport choice modeling study we calibrate the random utility model with choice probabilities obtained from the persistency model numerical results suggest that our persistency model is capable of obtaining estimates that perform as well if not better than classical methods such as logit and cross nested logit models we can also use the persistency model to obtain choice probability estimates for more complex choice problems we illustrate this on a stochastic knapsack problem which is essentially a discrete choice problem under budget constraint

split domain harmonic balance solutions to burger s equation for large amplitude disturbances a new split domain harmonic balance approach is presented the split domain approach combines the conventional multidomain harmonic balance approach with a split operator technique in a unique way to solve periodic unsteady e ow problems efe ciently the new technique is applied to burger s equation to obtain solutions for two large amplitude periodic boundary conditions a single frequency sine wave and a simulated wake function solutions containing strong moving discontinuities are obtained with fourier series containing up to 48 frequencies for various grid densities the split domain harmonic balance solutions are compared with conventional time accurate solutions the differences between the two are found to be asymptotic with respect to the number of fourierfrequenciesincluded in addition theharmonicbalanceapproachwasfound to besensitiveto grid density

uniqueness of steady state solutions for difference equations on overlapping grids the uniqueness of continuous and discontinuous steady state solutions is studied for difference approximations of hyperbolic problems on a strip with overlapping grids in the continuous case an easily verifiable uniqueness criterion is obtained from a normal mode analysis this criterion is applied to examine the uniqueness of commonly used interface treatments in the discontinuous case a shock is assumed to be contained in the overlap analysis for this case is developed on a linear model equation most of the results are further confirmed by computing smooth and shocked flows in a divergent duct the present theory can also be applied to analyze uniqueness for more general boundary value problems

viscous standing asymptotic states of isentropic compressible flows through a nozzle in this work we consider a viscous regularization of a well known one dimensional model for isentropic viscous compressible flows through a nozzle for the existence and multiplicity of standing asymptotic states for a certain type of ducts a complete analysis in a framework of dynamical systems is provided as an application of the geometric singular perturbation theory we show that all standing asymptotic states admit viscous profiles

implicit conservative schemes for the euler equations a new approach to a characteristic modeling scheme is presented that does not require the governing equations to be written in characteristi c variables or the flux terms to be split into positive and negative parts the method is based on the observation that for certain finite volume schemes the upwind influence can be accounted for through the conditions applied at the boundaries of individual cells instead of flux differencing the method is developed and applied to the one dimensional nozzle flow equations for subsonic transonic and supersonic flows and is then extended to two dimensions using an approximate factorization for the problem of a shock wave reflecting from a flat plate the present solution algorithm reduces the computational work per time step by 40 of that required by standard central difference implicit calculations i introduction n umerical methods that attempt to model the physics of the differential equations being approximated by properly accounting for the directions in which signals are propagated can be called characteristic modeling schemes the standard approach to characteristic modeling schemes is through the implementation of upwind schemes see refs 1 7 to devise an upwind scheme the primative fluxes are split into contributions whose jacobians have eigenvalues that are either 0 or 0 respectively at every point on the numerical mesh these fluxes are then differenced either upwind or downwind of that point

convergence of upwind schemes for a stationary shock a nonlinear first order boundary value problem with discontinuous solutions is considered it arises in the study of gasflow through a duct and allows in general for multiple solutions new convergence results for three difference schemes are presented and the sharpness of numerical layers is established for the eo scheme stability of a physically correct solution with respect to time evolution is shown

risk taking in stochastic pert networks abstract this paper studies a pert problem which incorporates project managers risk taking behavior in stochastic environments in this stochastic pert network which we refer to as riskpert activity durations are nonnegative random variables and a project manager uses a disutility function to evaluate each path and chooses that path which maximizes his expected disutility as an optimal or critical path this path as opposed to the classical critical path is used as a basis to analyze the pert network and to estimate the probability of completion time the general riskpert is difficult to solve however special cases when activity durations are statistically independent and disutility functions are linear exponential quadratic or linear exponential are solvable exactly the illustrative examples demonstrate that riskpert captures managers risk taking behavior and provides results that are more realistic than those of the classical pert models furthermore this paper extends some of the riskpert models to include multi dimensional pert networks

low dimensional representations of matrix groups and group actions on cat 0 spaces and manifolds abstract we study low dimensional representations rigidity problem of matrix groups over general rings by considering group actions on cat 0 spaces spheres and acyclic manifolds

the value of side information in shortest path optimization consider an agent who seeks to traverse the shortest path in a graph having random edge weights if the agent has no side information about the realizations of the edge weights it should simply take the path of least average length we consider a generalization of this framework whereby the agent has access to a limited amount of side information about the edge weights ahead of choosing a path we define a measure for information quantity provide bounds on the agent s performance relative to the amount of side information it receives and present algorithms for optimizing side information the results are based on a new graph characterization tied to shortest path optimization

colourful linear programming and its relatives we consider the following colourful generalization of linear programming given sets of points s1 sk rd referred to as colours and a point b rd decide whether there is a colourfult s1 sk such that b convt and if there is one find it linear programming is obtained by taking k d 1 and s1 sd 1 if k d 1 and b i 1d 1 convsi then a solution always exists we describe an efficient iterative approximation algorithm for this problem that finds a colourful t whose convex hull contains a point e close to b and analyze its real arithmetic and turing time complexities in contrast we show that colourful linear programming is strongly np complete we consider a class of linear algebraic relatives of colourful linear programming and give a computational complexity classification of the related decision and counting problems that arise we also introduce and discuss the complexity of a hierarchy of w1 w2 matroid basis nonbasis problems and give an application of colourful linear programming to the algorithmic problems of tverberg s theorem in combinatorial geometry

adaptive harmonic balance method for nonlinear time periodic flows an adaptive harmonic balance approach is developed and applied to a supersonic subsonic diverging nozzle subject to unsteady periodic outflow conditions the adaptive method minimizes the computational work required to obtain the harmonic balance solution by augmenting the frequency content in each cell as required to capture the local flow physics augmentation automatically adjusts with grid density resulting in lower frequency content on coarse grids the new adaptive harmonic balance method produces solutions equivalent to a non adapted harmonic balance solutions but with up to an 86 reduction in computational effort

constant access systems a general framework for greedy optimization on stochastic networks we consider network optimization problems in which the weights of the edges are random variables we develop conditions on the combinatorial structure of the problem which guarantee that the objective function value is a first passage time in an appropriately constructed continuous time markov chain the arc weights must be distributed exponentially the method of solution of the deterministic problem must be greedy in a general sense and the accumulation of objective function value during the greedy procedure must occur at a constant rate we call these structures constant access systems after the third property examples of constant access systems include the shortest path system the longest path system the time until disconnection in a network of failing components and some bottleneck optimization problems for each system we give the distribution of the objective function the distribution of the solution of the problem and the probability that a given arc is a member of the optimal solution we

at infinity of finite dimensional cat 0 spaces we show that any filtering family of closed convex subsets of a finite dimensional cat 0 space x has a non empty intersection in the visual bordification x over bar x boolean or partial derivative x using this fact several results known for proper cat 0 spaces may be extended to finite dimensional spaces including the existence of canonical fixed points at infinity for parabolic isometries algebraic and geometric restrictions on amenable group actions and geometric superrigidity for non elementary actions of irreducible uniform lattices in products of locally compact groups

implicit methods of second order accuracy for the euler equations this paper presents a study of noniterative implicit finite difference methods for hyperbolic systems new space centered methods involving two time levels are considered an analysis is made of various properties such as solvability stability dissipation dispersion and efficient solution of the algebraic systems the computation of shock waves with a cfl number larger than the unity is discussed unconditional stability results are proved for a case having several space variables accurate solutions of the euler equations are obtained that offer major reductions in computing costs over explicit methods

linear stability of the sub to super inviscid transonic stationary wave for gas flow in a nozzle of varying area abstract in this work we consider the linear stability of the sub to super inviscid transonic stationary wave of a one dimensional model of isentropic compressible flows through a nozzle of varying area this sub to super inviscid transonic stationary wave is newly founded by the authors using the geometric singular perturbation theory the main result of this work is to show that the sub to super inviscid transonic stationary wave is physically relevant in the sense that it is l linearly stable on any bounded space interval as long as its velocity is greater than 1 2 of the sound speed

a topological colorful helly theorem abstract let f 1 f d 1 be d 1 families of convex sets in r d the colorful helly theorem see discrete math 40 1982 141 asserts that if i 1 d 1 f i for all choices of f 1 f 1 f d 1 f d 1 then there exists an 1 i d 1 such that f f i f our main result is both a topological and a matroidal extension of the colorful helly theorem a simplicial complex x is d leray if h i y q 0 for all induced subcomplexes y x and i d theorem letxbe ad leray complex on the vertex setv suppose m is a matroidal complex on the same vertex setvwith rank function ifm xthen there exists a simplex xsuch that v d

existence and uniqueness for an exponentially derived switching scheme an existence and uniqueness result for a certain finite difference scheme applied to an equation governing steady one dimensional inviscid shocked flow in a variable area duct is presented

cat 1 spaces divergence groups and their commensurators a cat 1 space is a metric geodesic space in which every geodesic triangle is thinner than its associated comparison triangle in the hyperbolic plane b briha gr the cat 1 property is one among many possible generalizations to singular spaces of the notion of negative curvature important examples of cat 1 spaces include riemannian manifolds of sectional curvature k 1 and their convex subsets b g s metric trees and piecewise hyperbolic cell complexes mou da hag be 1 be 2 b br in this paper we establish certain superrigidity results for isometric actions of a group on a cat 1 space in the following two settings a the group is a subgroup of a locally compact group g with comg where g is a sufficiently large discrete subgroup and comg g g g 1 g and share a subgroup of finite index is the commensurator of in g b the group is an irreducible lattice in g n 1g k where each g is a semisimple algebraic group defined over a local field k the issues addressed in this paper are motivated on one hand by earlier work of g a margulis ma dealing with the linear representation theory of where in case a g is a semisimple group and g a lattice and on the other hand by the results of lubotzky mozes and zimmer l m z concerning isometric actions of on trees where comg g is the group of automorphisms of a regular tree and g is a lattice our approach to establishing superrigidity results is based on ergodic theoretic methods developed by margulis ma zi 3 a c b in this context the following notion of boundary of a locally compact group will be useful let b be a standard borel space on which acts by borel automorphisms preserving a finite measure class

a fractional helly theorem for convex lattice sets abstract a set of the form c z d where c r d is convex and z d denotes the integer lattice is called a convex lattice set it is known that the helly number of d dimensional convex lattice sets is 2d we prove that the fractional helly number is only d 1 for every d and every 0 1 there exists 0 such that whenever f1 fn are convex lattice sets in z d such that i i f i for at least n d 1 index sets i 1 2 n of size d 1 then there exists a lattice point common to at least n of the fi this implies a p d 1 theorem for every p d 1 that is if f is a finite family of convex lattice sets in z d such that among every p sets of f some d 1 intersect then f has a transversal of size bounded by a function of d and p

boundary approximations for implicit schemes for one dimensional inviscid equations of gasdynamics the applicability to practical calculations of recent theoretical developments in the stability analysis of difference approximations is examined for initial boundary value problems of the hyperbolic type for the numerical experiments the one dimensional inviscid gasdynamic equations in conservation law form are selected a class of implicit schemes based on linear multistep methods for ordinary differential equations is chosen and the use of space or space time extrapolations as implicit or explicit boundary schemes is emphasized some numerical examples with various inflow outflow conditions highlight the commonly discussed issues explicit vs implicit boundary schemes and unconditionally stable schemes hen finite difference schemes are used to solve initial boundary value problems for the equations of fluid dynamics it is well known that most methods require more boundary conditions than those required by the governing partial differential equations these additional boundary conditions for the finite difference equations are often called numerical boundary conditions the numerical boundary conditions cannot be imposed arbitrarily but are determined in general using interior information for example by extrapolation or uncentered approximations in this paper any numerical procedure used to provide a numerical boundary condition will be called a boundary scheme whatever schemes are used for the numerical boundary conditions it is a common practice to assume that the boundary scheme has a local effect and will not affect the solution globally during the early 1970s kreiss 1 2 osher 3 gustafsson et al 4 varah 5 and gustafsson6 published a series of papers establishing methods for checking the stability and accuracy of difference approximations with boundary schemes included since then further progress has been made in the theory of linear difference approximations for initial boundary value problems of the hyperbolic and parabolic type 7 12 because improper treatment of the boundary conditions can lead to instability and inaccuracy even though we start with a stable interior scheme i e scheme for the interior points it is appropriate to adopt an approach that includes the stability and accuracy of the combined interior and boundary schemes surveys of recent developments and extensive bibliographies are included in papers by coughran13 and yee 14 the purpose of this paper is to examine the applicability to practical calculations for nonlinear gasdynamic problems of recent theoretical stability analyses of implicit difference approximations for initial boundary value problems of the hyperbolic type as numerical computations have progressed the use of the conservative form of the gasdynamic equations has gained popularity for physical reasons it is sometimes desirable to specify boundary conditions in the nonconservative variables and to compute with conservative variables in the interior we will consider the additional complications introduced by this procedure

analysis of difference schemes for a stationary shock problem we consider the limit function u of the solutions u varepsilon of a singularly perturbed boundary value problem in general u will be discontinuous two difference approximations of the boundary value problem are studied and the discrete values for varepsilon 0 h to 0 are compared with u error estimates of order o h and o h 2 are proved and the sharpness of numerical layers is established

asymptotic theory of greedy approximations to minimal k point random graphs let spl chi sub n x sub 1 x sub n be an independent and identically distributed i i d sample having multivariate distribution p we derive almost sure a s limits for the power weighted edge weight function of greedy approximations to a class of minimal graphs spanning k of the n samples the class includes minimal k point graphs constructed by the partitioning method of ravi sundaram marathe rosenkrantz and ravi see proc 5th annu acm siam symp discrete algorithms arlington va p 546 55 1994 where the edge weight function satisfies the quasi additive property of redmond and yukich see ann appl probab vol 4 no 4 p 1057 73 1994 in particular this includes greedy approximations to the k point minimal spanning tree k mst steiner tree k st and the traveling salesman problem k tsp an expression for the influence function of the minimal weight function is given which characterizes the asymptotic sensitivity of the graph weight to perturbations in the underlying distribution the influence function takes a form which indicates that the k point minimal graph in d 1 dimensions has robustness properties in r sup d which are analogous to those of rank order statistics in one dimension a direct result of our theory is that the log weight of the k point minimal graph is a consistent nonparametric estimate of the renyi entropy of the distribution p possible applications of this work include analysis of random communication network topologies estimation of the mixing coefficient in spl epsiv contaminated mixture models outlier discrimination and rejection clustering and pattern recognition robust nonparametric regression two sample matching and image registration

generalized semi markov processes antimatroid structure and second order properties a generalized semi markov scheme models the structure of a discrete event system such as a network of queues by studying combinatorial and geometric representations of schemes we find conditions for second order properties convexity concavity sub supermodularity of their event epochs and event counting processes a scheme generates a language of feasible strings of events we show that monotonicity of the event epochs is equivalent to this language forming an antimatroid with repetition this connection gives rise to a rich combinatorial structure and serves as a starting point for other properties for example by strengthening the antimatroid condition we give several equivalent characterizations of the convexity of event epochs within a scheme all of these correspond in slightly different ways to making a certain score space a lattice to closing an ordinary antimatroid under intersections we also establish second order properties across schemes tied together through a synchronization mechanism a geometric view based on the score space facilitates verification of these properties in certain queueing systems

nonlinear singular perturbation problems and one sided difference schemes we consider the boundary value problem varepsilon y a y y b x y f x 1 leqq x leqq 1 for y 1 and y 1 given and b x 0 equiv 0 b y x y geqq 0 we construct a simple difference scheme approximating varepsilon u xx a u u x b x u f x u t for t geqq 0 1 leqq x leqq 1 with the same boundary conditions and an arbitrary initial guess as the numerical artificial time approaches infinity we prove convergence to a unique steady state solution as the mesh size delta x and the parameter varepsilon approach zero in any order we prove convergence to the solution of the corresponding o d e we use one sided differences to approximate the transport term the method applies equally well to multidimensional analogues we present numerical results verifying the theory

one sided difference approximations for nonlinear conservation laws we analyze one sided or upwind finite difference approximations to hyperbolic partial differential equations and in particular nonlinear conservation laws second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions we show that no such stable approximation of order higher than two is possible these one sided schemes have desirable properties for shock calculations we show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance new and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles existence uniqueness and stability of discrete shocks numerical examples are given

steady shock tracking and newton s method applied to one dimensional duct flow abstract a new computational approach combining shock tracking and newton s method is applied to steady one dimensional flow through a variable area duct on a transformed computational grid where the shock is fixed the physical shock location appears explicitly as an unknown in a set of finite difference equations and is coupled to the other unknowns the space time characteristics for the associated time dependent problem are used in formulating the boundary conditions the resulting system is solved by newton s method the computed results agree very well with an exact solution and the newton iterates converge rapidly in comparison to some explicit shock capturing time asymptotic methods

tverberg s theorem is 50 years old a survey this survey presents an overview of the advances around tverberg s theorem focusing on the last two decades we discuss the topological linear algebraic and combinatorial aspects of tverberg s theorem and its applications the survey contains several open problems and conjectures 2018 american mathematical society

on the solution of convection diffusion boundary value problems using equidistributed grids the effect of using grid adaptation on the numerical solution of model convection diffusion equations with a conservation form is studied the grid adaptation technique studied is based on moving a fixed number of mesh points to equidistribute a generalization of the arc length of the solution in particular a parameter dependent monitor function is introduced which incorporates fixed meshes approximate arc length equidistribution and equidistribution of the absolute value of the solution in a single framework thus the resulting numerical method is a coupled nonlinear system of equations for the mesh spacings and the nodal values a class of singularly perturbed problems including burgers s equation in the limit of small viscosity is studied singular perturbation and bifurcation techniques are used to analyze the solution of the discretized equations and numerical results are compared with the results from the analysis computation of the bifurcation diagram of the system is performed numerically using a continuation method and the results are used to illustrate the theory it is shown that equidistribution does not remove spurious solutions present on a fixed mesh and that furthermore the spurious solutions can be stable for an appropriate moving mesh method

approximations for the random minimal spanning tree with application to network provisioning this paper considers the problem of determining the mean and distribution of the length of a minimal spanning tree mst on an undirected graph whose arc lengths are independently distributed random variables we obtain bounds and approximations for the mst length and show that our upper bound is much tighter than the naive bound obtained by computing the mst length of the deterministic graph with the respective means as arc lengths we analyze the asymptotic properties of our approximations and establish conditions under which our bounds are asymptotically optimal we apply these results to a network provisioning problem and show that the relative error induced by using our approximations tends to zero as the graph grows large

group actions and helly s theorem abstract we describe a connection between the combinatorics of generators for certain groups and the combinatorics of helly s 1913 theorem on convex sets we use this connection to prove fixed point theorems for actions of these groups on nonpositively curved metric spaces these results are encoded in a property that we introduce called property fa r which reduces to serre s property fa when r 1 the method applies to s arithmetic groups in higher q rank to simplex reflection groups including some nonarithmetic ones and to higher rank chevalley groups over polynomial and other rings for example sl n z x 1 x d n 2

a conservative piecewise steady difference scheme for transonic nozzle flow abstract we construct a conservative scheme which approximates gas flow through a duct by discontinuity waves rarefaction waves and steady waves analytical studies on the interaction and stability of these nonlinear elementary waves are used to determine the evolution of the state variables the scheme is consistent admissible and reduces to the godunov scheme when the duct is uniform numerical results show that the scheme is stable and tends to a stable steady flow it also compares favorably with a fractional godunov scheme

calculation of shocked one dimensional flows on abruptly changing grids by mathematical programming abstract the steady state inviscid and nearly inviscid burgers equations and the euler equations for steady state shocked flow in a quasi 1 dimensional nozzle are discretized by cell centered finite differences on the cells of an arbitrarily spaced grid inviscid and viscous terms are approximated to second order by 2 point schemes and 4 point schemes respectively there results an overdetermined system of nonlinear algebraic equations this system is solved by a mathematical programming procedure that minimizes a weighted sum of the absolute values of the residuals the l 1 norm of the vector of residuals in this algorithm which is nonconservative no upwinding switches arbitrary constants or heuristic quantities are used the artificial viscosity used to solve the inviscid burgers equation is small 10 15 u no artificial viscosity of any kind is used for the euler equations the numerical solutions of both the viscous and the inviscid problems are accurate and nonoscillatory on grids with abrupt refinements in mesh length by factors as high as 10 4 shocks are invariably captured in one cell and this cell is rarely more than three cells away from the cell in which the physical shock occurs

optimization problems on graphs with independent random edge weights optimization problems on complete graphs with edge weights drawn independently from a fixed distribution are considered several methods for analyzing these problems are discussed including greedy methods applications of boole s inequality and exploitation of relationships with results about random unweighted graphs these techniques are illustrated in the case in which the edge weights are drawn from a normal distribution in particular we investigate the expected behavior of the minimum weight clique on k vertices we describe the asymptotic behavior in probability and or almost surely of the random variable which describes the optimum we also discuss the asymptotic behavior of its mean finally techniques are demonstrated by which we may determine an asymptotic description of the behavior of a greedy algorithm for this problem

solving a class of stochastic minimization problems this work gives a methodology for analyzing a class of discrete minimization problems with random element weights the minimum weight solution is shown to be an absorbing state in a markov chain while the distribution of weight of the minimum weight element is shown to be of phase type we then present two sided bounds for matroids with nbue distributed weights as well as for weights with bounded positive hazard rates we illustrate our method using a realistic military communications problem