second order linear differential equations with piecewise constant arguments subject to nonlocal boundary conditions abstract we analyze the existence of solution for the class of second order linear differential equations with functional dependence x t ax t bx t cx t dx t t t j 0 t where a b c d r t 0 and is a piecewise continuous function subject to a certain type of nonlocal mixed boundary conditions which include the value of the derivative of x at the initial point x 0

the affine polya szego principle equality cases and stability a brothers ziemer type theorem for the affine polya szego principle and a quantitative affine polya szego principle are established

t 0 mean field population dynamics approach for the random 3 satisfiability problem during the past decade phase transition phenomena in the random 3 satisfiability 3 sat problem has been intensively studied by statistical physics methods in this work we study the random 3 sat problem by the mean field first step replica symmetry broken cavity theory at the limit of temperature t 0 the reweighting parameter y of the cavity theory is allowed to approach infinity together with the inverse temperature beta with fixed ratio r y beta focusing on the system s space of satisfiable configurations we carry out extensive population dynamics simulations using the technique of importance sampling and we obtain the entropy density s r and complexity sigma r of zero energy clusters at different r values we demonstrate that the population dynamics may reach different fixed points with different types of initial conditions by knowing the trends of s r and sigma r with r we can judge whether a certain type of initial condition is appropriate at a given r value this work complements and confirms the results of several other very recent theoretical studies

a mass transportation approach to quantitative isoperimetric inequalities a sharp quantitative version of the anisotropic isoperimetric inequality is established corresponding to a stability estimate for the wulff shape of a given surface tension energy this is achieved by exploiting mass transportation theory especially gromov s proof of the isoperimetric inequality and the brenier mccann theorem a sharp quantitative version of the brunn minkowski inequality for convex sets is proved as a corollary

green s function and comparison principles for first order periodic differential equations with piecewise constant arguments in this paper we obtain the expression of the green s function related with a first order periodic differential equation with piecewise constant argument we derive comparison results for the treated linear operator by studying the sign of the obtained green s function

typical approximation performance for maximum coverage problem this study investigated the typical performance of approximation algorithms known as belief propagation the greedy algorithm and linear programming relaxation for maximum coverage problems in sparse biregular random graphs after we used the cavity method for a corresponding hard core lattice gas model results showed that two distinct thresholds of replica symmetry and its breaking exist in the typical performance threshold of belief propagation in the low density region the superiority of three algorithms in terms of a typical performance threshold is obtained by some theoretical analyses although the greedy algorithm and linear programming relaxation have the same approximation ratio in worst case performance their typical performance thresholds are mutually different indicating the importance of typical performance results of numerical simulations validate the theoretical analyses and imply further mutual relations of approximation algorithms

existence and uniqueness for anisotropic and crystalline mean curvature flows an existence and uniqueness result up to fattening for crystalline mean curvature flows with forcing and arbitrary convex mobilities is proven this is achieved by introducing a new notion of solution to the corresponding level set formulation such a solution satisfies the comparison principle and a stability property with respect to the approximation by suitably regularized problems the results are valid in any dimension and for arbitrary possibly unbounded initial closed sets the approach accounts for the possible presence of a time dependent bounded forcing term with spatial lipschitz continuity as a by product of the analysis the problem of the convergence of the almgren taylor wang minimizing movements scheme to a unique up to fattening flat flow in the case of general possibly crystalline anisotropies is settled

the sharp quantitative isoperimetric inequality a quantitative sharp form of the classical isoperimetric inequality is proved thus giving a positive answer to a conjecture by hall

nonlocal approximation of nonisotropic free discontinuity problems we prove that a class of free discontinuity problems with nonisotropic bulk and surface energy densities is approximated in the sense of gamma convergence by suitable families of nonlocal integral functionals some connections with the wulff set are also pointed out

green s function for second order periodic boundary value problems with piecewise constant arguments we obtain under suitable conditions the green s function to express the unique solution for a second order functional differential equation with periodic boundary conditions and functional dependence given by a piecewise constant function this expression is given in terms of the solutions for certain associated problems the sign of the solution is determined taking into account the sign of that green s function

spin glass model for the c dismantling problem c dismantling cd problem aims at finding the minimum vertex set d of a graph g v e after removing which the remaining graph will break into connected components with the size not larger than c in this paper we introduce a spin glass model with c 1 integer value states into the cd problem and then study the properties of this spin glass model by the belief propagation bp equations under the replica symmetry ansatz we give the lower bound rho c of the relative size of d with finite c on regular random graphs and erdos renyi random graphs we find rho c will decrease gradually with growing c and it converges to rho infty as c to infty the cd problem is called dismantling problem when c is a small finite fraction of v therefore rho infty is also the lower bound of the dismantling problem when v to infty to reduce the computation complexity of the bp equations taking the knowledge of the probability of a random selected vertex belonging to a remaining connected component with the size a the original bp equations can be simplified to one with only three states when c to infty the simplified bp equations are very similar to the bp equations of the feedback vertex set spin glass model h j zhou eur phys j b 86 455 2013 at last we develop two practical belief propagation guide decimation algorithms based on the original bp equations cd bpd and the simplified bp equations scd bpd to solve the cd problem on a certain graph our bpd algorithms and two other state of art heuristic algorithms are applied on various random graphs and some real world networks computation results show that the cd bpd is the best in all tested algorithms in the case of small c but considering the performance and computation consumption we recommend using scd bpd for the network with small clustering coefficient when c is large

a quantitative sobolev inequality in bv a quantitative version of the standard sobolev inequality with sharp constant for functions u in w1 1 rn or bv rn is established in terms of a distance of u from the manifold of all multiples of characteristic functions of balls inequalities involving non euclidean norms of the gradient are discussed as well

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

minimal vertex covers on finite connectivity random graphs a hard sphere lattice gas picture the minimal vertex cover or maximal independent set problem is studied on random graphs of finite connectivity analytical results are obtained by a mapping to a lattice gas of hard spheres of chemical radius 1 and they are found to be in excellent agreement with numerical simulations we give a detailed description of the replica symmetric phase including the size and entropy of the minimal vertex covers and the structure of the unfrozen component which is found to percolate at a connectivity c approximately 1 43 the replica symmetric solution breaks down at c e approximately 2 72 we give a simple one step replica symmetry broken solution and discuss the problems in the interpretation and generalization of this solution

on the integral manifold for a system of differential equations with piecewise constant argument the existence of an integral manifold for a system of differential equations with piecewise constant argument is studied

nonlocal boundary value problems for second order functional differential equations abstract we study the existence of a solution to a nonlocal boundary value problem for a class of second order functional differential equations with piecewise constant arguments the equation studied includes terms depending on the derivative as well as delay arguments in the derivative

green s function for second order differential equations with piecewise constant arguments abstract in this paper the second order nonautonomous differential equations with piecewise constant arguments x t t t x t t t t x t c t x t d t x t t t 0 t x 0 x t x 0 x t is considered we obtain the green s function to express the unique solution for the system

on the reduction principle for differential equations with piecewise constant argument of generalized type abstract in this paper we introduce a new type of differential equations with piecewise constant argument epcag more general than epca k l cooke j wiener retarded differential equations with piecewise constant delays j math anal appl 99 1984 265 297 j wiener generalized solutions of functional differential equations world scientific singapore 1993 the reduction principle v a pliss the reduction principle in the theory of the stability of motion izv akad nauk sssr ser mat 27 1964 1297 1324 in russian v a pliss integral sets of periodic systems of differential equations nauka moskow 1977 in russian is proved for epcag the structure of the set of solutions is specified we establish also the existence of global integral manifolds of quasilinear epcag in the so called critical case and investigate the stability of the zero solution

numerical oscillation and non oscillation for differential equation with piecewise continuous arguments of mixed type the paper deals with the oscillation and non oscillation of the runge kutta methods for a differential equation with piecewise continuous arguments of mixed type the conditions of the oscillation and non oscillation for the runge kutta method are obtained it is proved that oscillation of the analytic solution is not preserved by the runge kutta method under any conditions the conditions under which the non oscillation of analytic solutions is preserved by the runge kutta method are obtained some numerical experiments are given

center manifolds for infinite dimensional nonautonomous differential equations abstract we study a nonlinear integral equation for a center manifold of a semilinear nonautonomous differential equation having mild solutions we assume that the linear part of the equation admits in a very general sense a decomposition into center and hyperbolic parts the center manifold is obtained directly as the graph of a fixed point for a lyapunov perron type integral operator we prove that this integral operator can be factorized as a composition of a nonlinear substitution operator and a linear integral operator the operator is formed by the green s function for the hyperbolic part and composition operators induced by a flow on the center part we formulate the usual gap condition in spectral terms and show that this condition is in fact a condition of boundedness of on corresponding spaces of differentiable functions this gives a direct proof of the existence of a smooth global center manifold

ground states of the 2d sticky disc model fine properties and n 3 4 law for the deviation from the asymptotic wulff shape we investigate ground state configurations for a general finite number n of parti cles of the heitmann radin sticky disc pair potential model in two dimensions exact energy minimizers are shown to exhibit large microscopic fluctuations about the asymptotic wulff shape which is a regular hexagon there are arbitrarily large n with ground state configura tions deviating from the nearest regular hexagon by a number of n 3 4 particles we also prove that for any n and any ground state configuration this deviation is bounded above by n 3 4 as a consequence we obtain an exact scaling law for the fluctuations about the asymptotic wulff shape in particular our results give a sharp rate of convergence to the limiting wulff shape

global exponential stability of cohen grossberg neural networks with piecewise constant argument of generalized type and impulses in this letter we consider a model of cohen grossberg neural networks with piecewise constant argument of generalized type and impulses sufficient conditions ensuring the existence and uniqueness of solutions are obtained based on constructing a new differential inequality with piecewise constant argument and impulse and using the lyapunov function method we derive sufficient conditions ensuring the global exponential stability of equilibrium point with approximate exponential convergence rate an example is given to illustrate the validity and advantage of the theoretical results

almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence abstract we give conditions for the existence uniqueness and certain stability properties of almost periodic solutions of certain hybrid delay differential equations with almost periodic time dependence

attack vulnerability of complex networks we study the response of complex networks subject to attacks on vertices and edges several existing complex network models as well as real world networks of scientific collaborations and internet traffic are numerically investigated and the network performance is quantitatively measured by the average inverse geodesic length and the size of the largest connected subgraph for each case of attacks on vertices and edges four different attacking strategies are used removals by the descending order of the degree and the betweenness centrality calculated for either the initial network or the current network during the removal procedure it is found that the removals by the recalculated degrees and betweenness centralities are often more harmful than the attack strategies based on the initial network suggesting that the network structure changes as important vertices or edges are removed furthermore the correlation between the betweenness centrality and the degree in complex networks is studied

on the almost automorphy of bounded solutions of differential equations with piecewise constant argument abstract in this paper we give sufficient spectral conditions for the almost automorphy of bounded solutions to differential equations with piecewise constant argument of the form x t a x t f t t r where a is a bounded linear operator in x and f is an x valued almost automorphic function

on periodic solutions of differential equations with piecewise constant argument the periodic quasilinear system of differential equations with small parameter and piecewise constant argument of generalized type m u akhmet integral manifolds of differential equations with piecewise constant argument of generalized type nonlinear anal tma 66 2007 367 383 m u akhmet on the reduction principle for differential equations with piecewise argument of generalized type j math anal appl 336 2007 646 663 is addressed we consider the critical case when associated linear homogeneous system admits nontrivial periodic solutions criteria of existence of periodic solutions of such equations are obtained one of the main auxiliary results of our paper is an analogue of gronwall bellman lemma for functions with piecewise constant and retarded advanced type arguments dependence of solutions on the parameter is investigated appropriate examples are given to show our results

identifying optimal targets of network attack by belief propagation for a network formed by nodes and undirected links between pairs of nodes the network optimal attack problem aims at deleting a minimum number of target nodes to break the network down into many small components this problem is intrinsically related to the feedback vertex set problem that was successfully tackled by spin glass theory and an associated belief propagation guided decimation bpd algorithm zhou eur phys j b 86 455 2013 in the present work we apply the bpd algorithm which has approximately linear time complexity to the network optimal attack problem and demonstrate that it has much better performance than a recently proposed collective information algorithm morone and makse nature 524 65 2015 for different types of random networks and real world network instances the bpd guided attack scheme often induces an abrupt collapse of the whole network which may make it very difficult to defend

minimizers of anisotropic surface tensions under gravity higher dimensions via symmetrization we consider a variational model describing the shape of liquid drops and crystals under the influence of gravity resting on a horizontal surface making use of anisotropic symmetrization techniques we establish the existence convexity and symmetry of minimizers for a class of surface tensions admissible to the symmetrization procedure in the case of smooth surface tensions we obtain the uniqueness of minimizers via an ode characterization

the low temperature expansion of the wulff crystal in the 3d ising model we compute the expansion of the surface tension of the 3d random cluster model for q 1 in the limit where p goes to 1 we also compute the asymptotic shape of a plane partition of n as n goes to this same shape determines the wulff crystal to order o in the 3d ising model and more generally in the 3d random cluster model for q 1 at temperature

network dismantling abstract we study the network dismantling problem which consists of determining a minimal set of vertices in which removal leaves the network broken into connected components of subextensive size for a large class of random graphs this problem is tightly connected to the decycling problem the removal of vertices leaving the graph acyclic exploiting this connection and recent works on epidemic spreading we present precise predictions for the minimal size of a dismantling set in a large random graph with a prescribed light tailed degree distribution building on the statistical mechanics perspective we propose a three stage min sum algorithm for efficiently dismantling networks including heavy tailed ones for which the dismantling and decycling problems are not equivalent we also provide additional insights into the dismantling problem concluding that it is an intrinsically collective problem and that optimal dismantling sets cannot be viewed as a collection of individually well performing nodes

relaxation of multiple integrals in the space bv ω rp a characterisation of the surface energy density for the relaxation in v r p of the functional is obtained a lemma of de giorgi is used to modify a sequence near the boundary without increasing its total energy

on the wulff crystal in the ising model we study the phase separation phenomenon in the ising model in dimensions d 3 to this end we work in a large box with plus boundary conditions and we condition the system to have an excess amount of negative spins so that the empirical magnetization is smaller than the spontaneous magnetization m we confirm the prediction of the phenomenological theory by proving that with high probability a single droplet of the minus phase emerges surrounded by the plus phase moreover the rescaled droplet is asymptotically close to a definite deterministic shape the wulff crystal which minimizes the surface free energy in the course of the proof we establish a surface order large deviation principle for the magnetization our results are valid for temperatures t below a limit of slab thresholds t c conjectured to agree with the critical point t c moreover t should be such that there exist only two extremal translation invariant gibbs states at that temperature a property which can fail for at most countably many values and which is conjectured to be true for every t the proofs are based on the fortuin kasteleyn representation of the ising model along with coarse graining techniques to handle the emerging macroscopic objects we employ tools from geometric measure theory which provide an adequate framework for the large deviation analysis finally we propose a heuristic picture that for subcritical temperatures close enough to t c the dominant minus spin cluster of the wulff droplet permeates the entire box and has a strictly positive local density everywhere

anisotropic isoperimetric inequalities involving boundary momentum perimeter and volume abstract we consider a scale invariant functional involving the anisotropic p momentum the anisotropic perimeter and the volume we show that the wulff shape associated with the finsler norm f considered and centered at the origin is the unique minimizer of the anisotropic functional taken into consideration among all bounded convex sets

uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets given an elliptic integrand of class mathscr c 2 alpha we prove that finite unions of disjoint open wulff shapes with equal radii are the only volume constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure

bifurcation of three dimensional discontinuous cycles abstract we consider three dimensional discontinuous dynamical systems with non fixed moments of impacts existence of the center manifold is proved for the system the result is applied for the extension of the planar hopf bifurcation theorem m u akhmet perturbations and hopf bifurcation of the planar discontinuous dynamical system nonlinear analysis 60 2005 163 178 illustrative examples are constructed for the theory

on stable parametric finite element methods for the stefan problem and the mullins sekerka problem with applications to dendritic growth we introduce a parametric finite element approximation for the stefan problem with the gibbs thomson law and kinetic undercooling which mimics the underlying energy structure of the problem the proposed method is also applicable to certain quasi stationary variants such as the mullins sekerka problem in addition fully anisotropic energies are easily handled the approximation has good mesh properties leading to a well conditioned discretization even in three space dimensions several numerical computations including for dendritic growth and for snow crystal growth are presented

equilibrium shapes for planar crystals in an external field the equilibrium shape of a two dimensional crystal in a convex background potential g x is analyzed for g 0 the shape of minimum energy may be deduced from surface tension via the wulff construction but if g is not constant little is known beyond the case of a crystal sitting in a uniform field only an unpublished result of okikiolu shows each connected component of the equilibrium crystal to be convex here it will be shown that any such component minimizes energy uniquely among convex sets of its area if the wulff shape and g x are symmetric under x x it follows that the equilibrium crystal is unique convex and connected this last result leads to a new proof that convex crystals away from equilibrium remain convex as they evolve by curvature driven flow subsequent work with felix otto shows without assuming symmetry that no equilibrium crystal has more than two convex components

message passing for vertex covers constructing a minimal vertex cover of a graph can be seen as a prototype for a combinatorial optimization problem under hard constraints in this paper we develop and analyze message passing techniques namely warning and survey propagation which serve as efficient heuristic algorithms for solving these computational hard problems we show also how previously obtained results on the typical case behavior of vertex covers of random graphs can be recovered starting from the message passing equations and how they can be extended

the wulff theorem revisited the parametrized indicator measures and the brunn minkowski inequality are used to prove that the wulff set w is a minimizer for the surface energy where the density is the support function of w the support of the indicator measures associated to minimizing sequences is characterized it is shown that if w is polyhedral then minimizing sequences cannot oscillate

linearization near the integral manifold for a system of differential equations with piecewise constant argument abstract in this paper we prove a linearization result via an integral manifold for a system of differential equations with piecewise constant argument

ck centre unstable manifolds we consider the existence and smoothness of global centre unstable manifolds for finite and infinite dimensional flows or maps we show that every global centre unstable manifold can be expressed as a graph of a c k map provided that the nonlinearities are c k smooth the proofs are based on a lemma by d henry on a necessary and sufficient condition for a lipschitz map to be continuously differentiable

preservation of oscillations of the runge kutta method for equation x t ax t a1x t 1 0 the paper deals with the preservation of oscillations of the runge kutta method for equation x t ax t a 1x t 1 0 it is proved that oscillations of the analytic solution are preserved by the runge kutta method special interpolation functions of the numerical solutions are given it turns out that zeros of the interpolation function of the numerical solution converge to ones of the analytic solution with the same order of accuracy as that of the corresponding runge kutta method some numerical experiments are presented

spin glass phase transitions and minimum energy of the random feedback vertex set problem a feedback vertex set fvs of an undirected graph contains vertices from every cycle of this graph constructing a fvs of sufficiently small cardinality is very difficult in the worst cases but for random graphs this problem can be efficiently solved by converting it into an appropriate spin glass model h j zhou eur phys j b 86 455 2013 in the present work we study the spin glass phase transitions and the minimum energy density of the random fvs problem by the first step replica symmetry breaking 1rsb mean field theory for both regular random graphs and erdos renyi graphs we determine the inverse temperature beta 1 at which the replica symmetric mean field theory loses its local stability the inverse temperature beta d of the dynamical clustering phase transition and the inverse temperature beta s of the static condensation phase transition these critical inverse temperatures all change with the mean vertex degree in a nonmonotonic way and beta d is distinct from beta s for regular random graphs of vertex degrees k 60 while beta d are identical to beta s for erdos renyi graphs at least up to mean vertex degree c 512 we then derive the zero temperature limit of the 1rsb theory and use it to compute the minimum fvs cardinality

minimal contagious sets in random regular graphs the bootstrap percolation or threshold model is a dynamic process modelling the propagation of an epidemic on a graph where inactive vertices become active if their number of active neighbours reach some threshold we study an optimization problem related to it namely the determination of the minimal number of active sites in an initial configuration that leads to the activation of the whole graph under this dynamics with and without a constraint on the time needed for the complete activation this problem encompasses in special cases many extremal characteristics of graphs like their independence decycling or domination number and can also be seen as a packing problem of repulsive particles we use the cavity method including the effects of replica symmetry breaking an heuristic technique of statistical mechanics many predictions of which have been confirmed rigorously in the recent years we have obtained in this way several quantitative conjectures on the size of minimal contagious sets in large random regular graphs the most striking being that 5 regular random graph with a threshold of activation of 3 resp 6 regular with threshold 4 have contagious sets containing a fraction 1 6 resp 1 4 of the total number of vertices equivalently these numbers are the minimal fraction of vertices that have to be removed from a 5 regular resp 6 regular random graph to destroy its 3 core we also investigated survey propagation like algorithmic procedures for solving this optimization problem on single instances of random regular graphs

a uniqueness proof for the wulff theorem the wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of small elastic crystals it consists in minimising the generally anisotropic surface energy among sets of given volume a solution of this problem is given by a geometric construction due to wulff in the class of sets of finite perimeter this was first shown by j e taylor who using methods of geometric measure theory also proved uniqueness here a more analytic uniqueness proof is presented the main ingredient is a sharpened version of the brunn minkowski inequality

parametric approximation of isotropic and anisotropic elastic flow for closed and open curves deckelnick and dziuk math comput 78 266 645 671 2009 proved a stability bound for a continuous in time semidiscrete parametric finite element approximation of the elastic flow of closed curves in mathbb r d d geq2 we extend these ideas in considering an alternative finite element approximation of the same flow that retains some of the features of the formulations in barrett et al j comput phys 222 1 441 462 2007 siam j sci comput 31 1 225 253 2008 ima j numer anal 30 1 4 60 2010 in particular an equidistribution mesh property for this new approximation we obtain also a stability bound for a continuous in time semidiscrete scheme apart from the isotropic situation we also consider the case of an anisotropic elastic energy in addition to the evolution of closed curves we also consider the isotropic and anisotropic elastic flow of a single open curve in the plane and in higher codimension that satisfies various boundary conditions

existence of compactly supported global minimisers for the interaction energy the existence of compactly supported global minimisers for continuum models of particles interacting through a potential is shown under almost optimal hypotheses the main assumption on the potential is that it is catastrophic or not h stable which is the complementary assumption to that in classical results on thermodynamic limits in statistical mechanics the proof is based on a uniform control on the local mass around each point of the support of a global minimiser together with an estimate on the size of the gaps it may have the class of potentials for which we prove the existence of global minimisers includes power law potentials and for some range of parameters morse potentials widely used in applications we also show that the support of local minimisers is compact under suitable assumptions

persistence contractivity and global stability in logistic equations with piecewise constant delays abstract we establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium n 1 a i 0 m b i of the following differential equation with piecewise constant arguments dn t dt n t r t 1 an t i 0 m b i n n i n t n 0 1 2 n 0 n 0 0 and n j n j 0 j 1 2 m where r t is a nonnegative continuous function on 0 r t 0 i 0mbi 0 bi 0 i 0 1 2 m and a i 0mbi 0 these new conditions depend on a b0 and i 1mbi and hence these are other type conditions than those given by so and yu hokkaido math j 24 1995 269 286 and others in particular in the case m 0 and r t r 0 we offer necessary and sufficient conditions for the persistence and contractivity of solutions we also investigate the following differential equation with nonlinear delay terms dx t dt x t r t 1 ax t g x t x t 1 x t m t 0 x k k 0 0 k m and 0 0 where r t is a nonnegative continuous function on 0 r t 0 1 ax g x x x 0 has a unique solution x 0 and g x0 x1 xm c1 0 0 0

method of lyapunov functions for differential equations with piecewise constant delay we address differential equations with piecewise constant argument of generalized type 5 8 and investigate their stability with the second lyapunov method despite the fact that these equations include delay stability conditions are merely given in terms of lyapunov functions that is no functionals are used several examples one of which considers the logistic equation are discussed to illustrate the development of the theory some of the results were announced at the 14th international congress on computational and applied mathematics iccam2009 antalya turkey in 2009

on quasi periodic solutions of differential equations with piecewise constant argument we study the existence of quasi periodic solutions to differential equations with piecewise constant argument epca for short it is shown that epca with periodic perturbations possess a quasi periodic solution and no periodic solution the appearance of quasi periodic rather than periodic solutions is due to the piecewise constant argument this new phenomenon illustrates a crucial difference between ode and epca the results are extended to nonlinear equations

stability in cellular neural networks with a piecewise constant argument in this paper by using the concept of differential equations with piecewise constant arguments of generalized type 1 4 a model of cellular neural networks cnns 5 6 is developed the lyapunov razumikhin technique is applied to find sufficient conditions for the uniform asymptotic stability of equilibria global exponential stability is investigated by means of lyapunov functions an example with numerical simulations is worked out to illustrate the results

finite valued feedback laws and piecewise classical solutions abstract often in engineering literature we find control systems in which the open loop inputs are piecewise constant and take values in a finite set such open loop inputs cause the system to have fairly regular solutions on the other hand when acting in closed loop feedback laws taking values in a finite set may not be reinterpreted as open loop inputs of the considered type in fact pathological behaviours such as the accumulation of discontinuities may appear zeno phenomenon we give some conditions which can be used as tools for building finite valued feedback laws not causing such pathological behaviours

integral manifolds of differential equations with piecewise constant argument of generalized type abstract in this paper we introduce a general type of differential equations with piecewise constant argument epcag the existence of global integral manifolds of the quasilinear epcag is established when the associated linear homogeneous system has an exponential dichotomy the smoothness of the manifolds is investigated the existence of bounded and periodic solutions is considered a new technique of investigation of equations with piecewise argument based on an integral representation formula is proposed appropriate illustrating examples are given

stability analysis on the finite temperature replica symmetric and first step replica symmetry broken cavity solutions of the random vertex cover problem the vertex cover problem is a prototypical hard combinatorial optimization problem it was studied in recent years by physicists using the cavity method of statistical mechanics in this paper the stability of the finite temperature replica symmetric rs and the first step replica symmetry broken 1rsb cavity solutions of the vertex cover problem on random regular graphs of finite vertex degree k are analyzed by population dynamics simulations we found that 1 the lowest temperature for the rs solution to be stable t rs k is not a monotonic function of k 2 at relatively large connectivity k and temperature t slightly below the dynamic transition temperature t d k the 1rsb solutions with small but non negative complexity values are stable and 3 the dynamical transition temperature t d and kauzmann temperature t k is equal to each other similar results are obtained on random poissonian graphs

differential equations with state dependent piecewise constant argument abstract a new class of differential equations with state dependent piecewise constant argument is introduced it is an extension of systems with piecewise constant argument fundamental theoretical results for the equations the existence and uniqueness of solutions the existence of periodic solutions and the stability of the zero solution are obtained appropriate examples are constructed