solving systems of ivps with discontinuous derivatives numerical experiments we deal in this paper with solving ivps with singular right hand side we consider two recent algorithms of kacewicz and przybylowicz for solving systems of ivps with right hand side functions which are globally lipschitz continuous and piecewise r smooth with piecewise holder r th partial derivatives with holder exponent 0 1 the singularity hypersurface is defined by the zeros of an unknown event function we run several numerical experiments to verify the theoretical results of kacewicz and przybylowicz our tests confirm that the bounds on the error o n r can be achieved with o n function evaluations where n is a number of discretization points

two parameter degenerate sliding bifurcations in filippov systems this paper is concerned with the extension to the case of codimension 2 degenerate sliding bifurcations of the theory of sliding bifurcations in filippov systems presented in m di bernardo p kowalczyk a nordmark bifurcations of dynamical systems with sliding derivation of normal form mappings physica d 170 2002 175 205 these bifurcations were detected in experimental systems such as the dry friction oscillator and turn out to be organising centres for branches of codimension 1 sliding bifurcations the analysis is carried out for generic n dimensional piecewise smooth systems the possible degenerate scenarios are classified it is shown that several branches of codimension 1 sliding bifurcations originate from the degenerate codimension 2 points such branches are appropriately classified in the degenerate crossing sliding case a friction oscillator is used as a representative example to illustrate and confirm the theoretical derivations the importance is discussed of the unfolding of the degenerate sliding bifurcations for the development of continuation techniques

bifurcations of dynamical systems with sliding derivation of normal form mappings this paper is concerned with the analysis of so called sliding bifurcations in n dimensional piecewise smooth dynamical systems with discontinuous vector field these novel bifurcations occur when the system trajectory interacts with regions on the discontinuity set where sliding is possible the derivation of appropriate normal form maps is detailed it is shown that the leading order term in the map depends on the particular bifurcation scenario considered this is in turn related to the possible bifurcation scenarios exhibited by a periodic orbit undergoing one of the sliding bifurcations discussed in the paper a third order relay system serves as a numerical example

large deviations for increasing sequences on the plane we prove a large deviation principle with explicit rate functions for the length of the longest increasing sequence among poisson points on the plane the rate function for lower tail deviations is derived from a 1977 result of logan and shepp about young diagrams of random permutations for the upper tail we use a coupling with hammersley s particle process and convex analytic techniques along the way we obtain the rate function for the lower tail of a tagged particle in a totally asymmetric hammersley s process

sliding modes in solving convex programming problems sliding modes are used to analyze a class of dynamical systems that solve convex programming problems the analysis is carried out using concepts from the theory of differential equations with discontinuous right hand sides and lyapunov stability theory it is shown that the equilibrium points of the system coincide with the minimizers of the convex programming problem and that irrespective of the initial state of the system the state trajectory converges to the solution set of the problem the dynamic behavior of the systems is illustrated by two numerical examples

on a nonlocal extension of differentiation abstract we study an integral equation that extends the problem of anti differentiation we formulate this equation by replacing the classical derivative with a known nonlocal operator similar to those applied in fracture mechanics and nonlocal diffusion we show that this operator converges weakly to the classical derivative as a nonlocality parameter vanishes using fourier transforms we find the general solution to the integral equation we show that the nonlocal antiderivative involves an infinite dimensional set of functions in addition to an arbitrary constant however these functions converge weakly to zero as the nonlocality parameter vanishes for special types of integral kernels we show that the nonlocal antiderivative weakly converges to its classical counterpart as the nonlocality parameter vanishes

control and simulation of hybrid systems abstract we consider simulation of hybrid systems consisting of continuous time smooth systems and relays we discuss how a simulation program should detect the structural possibility of infinite fast mode switches sliding and help the user introduce new so called induced modes we also analyze the problem of how to define the dynamics on the induced modes finally we study the different possible stable sliding motion around an intersection of two transversal switching surfaces

efficient solution of ivps with right hand sides having discontinuities on an unknown hypersurface the paper deals with the solution of systems of ivps with discontinuous right hand side functions f in our previous work we considered ivps with continuous f s having discontinuous derivatives we assumed the global lipschitz continuity of f and the analysis was heavily based on that assumption in the present paper we skip the continuity assumption we assume that f is a piecewise regular function that satisfies the lipschitz condition only in some regions of the domain by the regular function we mean a function belonging to a holder class a hypersurface where the regularity of f breaks down is not known we define an algorithm disc ivp for solving such problems and analyze its error and cost the error analysis in the presence of discontinuities of f is essentially different than that in the continuous case we show in a subclass of f s that the worst case error of the algorithm measured in the conservative supremum norm over the interval of integration is of the same order as for globally regular functions the algorithm and the analysis do not use heuristic arguments in particular we are able to bound the cost of the algorithm which leads to the bound on the e complexity of the problem the minimal cost of computing an e approximation to the solution the e complexity in the discontinuous case described above turns out to be of the same order as that in the globally regular case in the other words the solution of discontinuous problems is no more difficult than the solution of globally regular ones furthermore the algorithm disc ivp is error and cost optimal

optimal adaptive solution of piecewise regular systems of ivps with unknown switching hypersurface we study the solution of piecewise regular systems of ivps with a switching hypersurface defined by an unknown event function our aim is to design an algorithm that is not based on heuristic steps it does not use information on the event function and preserves the order known for regular systems the algorithm is based on the adaptive mesh which differs from the equidistant mesh only by a single point we provide a rigorous that is independent of heuristic arguments analysis of its error and cost

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

effective solution of discontinuous ivps using a runge kutta formula pair with interpolants an automatic technique for solving discontinuous initial value problems is developed and justified the technique is based on the use of local interpolants such as those that have been developed for use with runge kutta formula pairs numerical examples are presented to illustrate the significant improvement in efficiency and reliability that results when this technique is used with standard methods

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

event location for ordinary differential equations abstract an initial value problem for y f t y may have an associated event function g t y an event is said to occur at t when g t y t 0 we consider problems for which the definition of f t y changes at the time of an event a number of solvers locate events and restart the integration there so as to deal with the changes in f but there is little theoretical support for what is done here we prove that with reasonable assumptions about the problem and the solver the error of the numerical solution is qualitatively the same whether or not events occur numerical results obtained with a wide range of solvers confirm the theory developed here

limit cycles with chattering in relay feedback systems relay feedback has a large variety of applications in control engineering several interesting phenomena occur in simple relay systems in the paper scalar linear systems with relay feedback are analyzed it is shown that a limit cycle where part of the limit cycle consists of fast relay switchings can occur this chattering is analyzed in detail and conditions for approximating it by a sliding mode are derived a result on existence of limit cycles with chattering is given and it is shown that the limit cycles can have arbitrarily many relay switchings each period limit cycles with regular sliding modes are also discussed examples illustrate the results

a comparison of filippov sliding vector fields in codimension 2 we consider several possibilities on how to select a filippov sliding vector field on a codimension 2 singularity surface s intersection of two codimension 1 surfaces we discuss and compare several old and new approaches under the assumption that s is nodally attractive of specific interest is the selection of a smoothly varying filippov sliding vector field as a result of our analysis and experiments the best candidates of the many possibilities explored are those based on the so called barycentric coordinates in the present context one of these possibilities appear to be new

sliding motion on discontinuity surfaces of high co dimension a construction for selecting a filippov vector field in this paper we consider the issue of sliding motion in filippov systems on the intersection of two or more surfaces to this end we propose an extension of the filippov sliding vector field on manifolds of co dimension p with p 2 our model passes through the use of a multivalued sign function reformulation to justify our proposal we will restrict to cases where the sliding manifold is attractive for the case of co dimension p 2 we will distinguish between two types of attractive sliding manifold node like and spiral like the case of node like attractive manifold will be further extended to the case of p 3 finally we compare our model to other existing methodologies on some examples

an efron stein inequality for nonsymmetric statistics on etablit l analogue d un resultat d efron et stein 1981 que l on demontre a l aide d une technique d espace de hilbert introduite par vitale 1984

event driven integration of linear structural dynamics models under unilateral elastic constraints abstract this paper proposes an algorithm for the numerical simulation of linear structural dynamics problems under unilateral elastic constraints i e constraints with a linear force displacement characteristic whenever active the presented procedure relies on an event driven strategy for the handling of the contact constraints in combination with one step schemes dedicated to the time integration of the second order equations of motion efficiency of the procedure follows from the use of cubic hermite interpolation to continuously extend the normal gap functions that reflect the openings of the contact interfaces robustness follows from the proper handling of complex numerical situations e g numerical grazing or discontinuity sticking through appropriate algorithm structure and numerical implementation and integration stability is guaranteed by the very nature of the algorithm and that of the one step integration scheme following a detailed coverage of the integration procedure and the countermeasures to the expected numerical difficulties three application examples are treated for illustration purposes a matlab implementation of the procedure is provided online download and usage information are given in appendix a

a survey of numerical methods for ivps of odes with discontinuous right hand side this work is dedicated to the memory of donato trigiante who has been the first teacher of numerical analysis of the second author the authors remember donato as a generous teacher always ready to discuss with his students able to give them profound and interesting suggestions here we present a survey of numerical methods for differential systems with discontinuous right hand side in particular we will review methods where the discontinuities are detected by using an event function so called event driven methods and methods where the discontinuities are located by controlling the local errors so called time stepping methods particular attention will be devoted to discontinuous systems of filippov s type where sliding behavior on the discontinuity surface is allowed

analysis of a model for precipitation and dissolution coupled with a darcy flux abstract in this paper we deal with the numerical analysis of an upscaled model of a reactive flow in a porous medium which describes the transport of solutes undergoing precipitation and dissolution leading to the formation degradation of crystals inside the porous matrix the model is defined at the darcy scale and it is coupled to a darcy flow characterized by a permeability field that changes in space and time according to the precipitated crystal concentration the model involves a non linear multi valued reaction term which is treated exactly by solving an inclusion problem for the solutes and the crystals dynamics we consider a weak formulation for the coupled system of equations expressed in a dual mixed form for the darcy field and in a primal form for the solutes and the precipitate and show its well posedness without resorting to regularization of the reaction term convergence to the weak solution is proved for its finite element approximation we perform numerical experiments to study the behavior of the system and to assess the effectiveness of the proposed discretization strategy in particular we show that a method that captures the discontinuity yields sharper dissolution fronts with respect to methods that regularize the discontinuous term

sliding motion in filippov differential systems theoretical results and a computational approach in this work we discuss some theoretical and numerical aspects of solving differential equations with discontinuous right hand sides of filippov type in particular i we propose second order corrections to the theory of filippov ii we provide a systematic and nonambiguous way to define the vector field on the intersection of several surfaces of discontinuity and iii we propose and implement a numerical method to approximate a trajectory of systems with discontinuous right hand sides and illustrate its performance on a few examples

an analysis of a class of neural networks for solving linear programming problems a class of neural networks that solve linear programming problems is analyzed the neural networks considered are modeled by dynamic gradient systems that are constructed using a parametric family of exact nondifferentiable penalty functions it is proved that for a given linear programming problem and sufficiently large penalty parameters any trajectory of the neural network converges in finite time to its solution set for the analysis lyapunov type theorems are developed for finite time convergence of nonsmooth sliding mode dynamic systems to invariant sets the results are illustrated via numerical simulation examples

on the variance of the optimal alignments score for binary random words and an asymmetric scoring function we investigate the order of the variance of the optimal alignments oa score of two independent iid binary random words having the same length the letters are equiprobable but the scoring function is such that one letter has a larger score than the other in this setting we prove that the order of variance is linear in the common length oas constitute a generalization of longest common subsequences they can be represented as optimal paths in a two dimensional last passage percolation setting with dependent weights

analysis and generic properties of gene regulatory networks with graded response functions the mass of new genomic data has lead to a growing interest in gene regulatory models describing the regulatory aspects of gene activity in these models the rates of change of gene product concentrations are expressed as a sum of regulatory switches in terms of sums of products of sigmoid functions mimicking complex logical on off functions turning gene activity on and off the customary way to analyse such equations is to replace the sigmoid functions by step functions and disregard models with autoregulation this leads to discontinuous equations of motion but in models without autoregulation continuous solutions can easily be obtained with effective autoregulation this simple approach breaks down as real regulators have finite gain and autoregulation is ubiquitous in biological systems we propose a generalised gene regulatory model framework admitting autoregulation and graded sigmoid functions with different steepnesses using singular perturbation analysis methods we derive a set of reformulated continuous equations for the limit solution when the sigmoids approach step functions and show that the solution for steep sigmoids approaches this limit solution uniformly in a finite time interval a conspicuous feature of the limit solution is so called sliding motionduring which the solution slides along a threshold hyperplane or the intersection of such hyperplanes a particular mapping leads to a dual picture of the flow where the fast parts are magnified at the expense of the slow parts combining the two dual pictures a simple and powerful method to analyse the flow is obtained we show that for steep sigmoid functions the flow may be highly sensitive to the relative steepnesses of the sigmoids this has important consequences when stochastic effects are taken into account also it implies that the observation by glass and

quasi selective ultrafilters and asymptotic numerosities abstract we isolate a new class of ultrafilters on n called quasi selective because they are intermediate between selective ultrafilters and p points under the continuum hypothesis these three classes are distinct the existence of quasi selective ultrafilters is equivalent to the existence of asymptotic numerosities for all sets of tuples a n k such numerosities are hypernatural numbers that generalize finite cardinalities to countable point sets most notably they maintain the structure of ordered semiring and in a precise sense they allow for a natural extension of asymptotic density to all sets of tuples of natural numbers

linear complementarity systems we introduce a new class of dynamical systems called linear complementarity systems the time evolution of these systems consists of a series of continuous phases separated by events which cause a change in dynamics and possibly a jump in the state vector the occurrence of events is governed by certain inequalities similar to those appearing in the linear complementarity problem of mathematical programming the framework we describe is suitable for certain situations in which both differential equations and inequalities play a role for instance in mechanics electrical networks piecewise linear systems and dynamic optimization we present a precise definition of the solution concept of linear complementarity systems and give sufficient conditions for existence and uniqueness of solutions

on embedding certain partial orders into the p points under rudin keisler and tukey reducibility the study of the global structure of ultrafilters on the natural numbers with respect to the quasi orders of rudin keisler and rudin blass reducibility was initiated in the 1970s by blass keisler kunen and rudin in a 1973 paper blass studied the special class of p points under the quasiordering of rudin keisler reducibility he asked what partially ordered sets can be embedded into the p points when the p points are equipped with this ordering this question is of most interest under some hypothesis that guarantees the existence of many p points such as martin s axiom for centered posets in his 1973 paper he showed under this assumption that both 1 and the reals can be embedded analogous results were obtained later for the coarser notion of tukey reducibility we prove in this paper that martin s axiom for centered posets implies that the boolean algebra p fin equipped with its natural partial order can be embedded into the p points both under rudin keisler and tukey reducibility consequently the continuum hypothesis implies that every partial order of size at most continuum embeds into the p points under both notions of reducibility

hammersley s interacting particle process and longest increasing subsequences in a famous paper 8 hammersley investigated the lengthl n of the longest increasing subsequence of a randomn permutation implicit in that paper is a certain one dimensional continuous space interacting particle process by studying a hydrodynamical limit for hammersley s process we show by fairly soft arguments that limn 1 2 el n 2 this is a known result but previous proofs 14 11 relied on hard analysis of combinatorial asymptotics

a model for the hepatic glucose metabolism based on hill and step functions we propose a system of first order ordinary differential equations to describe and understand the physiological mechanisms of the interplay between plasma glucose and insulin and their behaviors in diabetes the proposed model is based on hill and step functions which are used to simulate the switch like behavior that occurs in metabolic regulatory variables when some of the threshold parameters are approached a simplified piecewise linear system is also proposed to study the possible equilibria and solutions and used to introduce simple theoretical control mechanisms representing the action of an artificial pancreas and regulating exogenous insulin

standard deviation of the longest common subsequence let l n be the length of the longest common subsequence of two independent i i d sequences of bernoulli variables of length n we prove that the order of the standard deviation of l n is vn provided the parameter of the bernoulli variables is small enough this validates waterman s conjecture in this situation philos trans r soc lond ser b 344 1994 383 390 the order conjectured by chvatal and sankoff j appl probab 12 1975 306 315 however is different

optimal solution of a class of non autonomous initial value problems with unknown singularities in this paper we are interested in a rigorous analysis of the solution of a class of initial value problems with singularities we consider scalar non autonomous ivps with separated variables and one unknown singularity in each variable which leads to four unknown events in the two dimensional space many algorithms proposed in the literature for singular ivps are practically oriented they do not avoid heuristic arguments and are often checked for efficiency by numerical experiments we design an adaptive algorithm with no heuristic steps for solving the considered problems and provide rigorous bounds on the error we show that in spite of the presence of singularities the algorithm preserves the optimal error known for the regular case lower bounds on the error of any algorithm in the case of a larger number of singularities are also discussed

souslin partitions of products of finite sets abstract to every infinite sequence of positive integers m m i i we associate two fields of sets a field cl m of subsets of and a field pcl m of subsets of their relevance to ramsey theory is based on the fact that for every cl m measurable partition c 2 there is a sequence h i i with hi mi such that c is constant on i hi similarly for every pcl m measurable partition c 2 there is h and a sequence h i i of sets with hi and hi mi such that c is constant on i hi h in di prisco et al j combin theory ser a 93 2001 333 combinatorica to appear it is shown that cl m and pcl m are fields that contain all closed sets and therefore all borel subsets of their corresponding domains we show here that they are in fact closed under souslin operation and that under suitable assumptions they contain all reasonably definable subsets of their corresponding domains these results are then used to show that the classical partition relation is not equivalent to its polarized version solving thus a long standing problem in this area see j symbolic logic 58 1998 860 notas de logica matematica vol 39 universidad nacional del sur bahia blanca argentina 1994 pp 89 94

fast switches in relay feedback systems relays are common in automatic control systems even linear systems with relay feedback are however far from fully understood new results are given about the behavior of these systems via a state space approach it is proved that there exist multiple fast switches if and only if the sign of the first nonvanishing markov parameter of the linear system is positive fast switches are shown to occur as part of stable limit cycles an analysis is developed for these limit cycles that illustrates how they can be predicted

near coherence of filters ii applications to operator ideals the stone cech remainder of a half line order ideals of sequences and slenderness of groups the set theoretic principle of near coherence of filters ncf is lnown to be neither provable nor refutable from the usual axioms of set theory we show that ncf is equivalent to the following statements among others 1 the ideal of compact operators on hilbert space is not the sum of two smaller ideals 2 the stone tech remainder of a half line has only one composant this was first proved by j mioduszewski 3 the partial ordering of slenderness classes of abelian groups minus its top element is directed upward and in fact has a top element thus 11 these statements are also consistent and independent