time dependent differential inclusions cocycle attractors and fuzzy differential equations traditional formulations of fuzzy differential equations do not reproduce the rich and varied behavior of crisp differential equations des a recent interpretation in terms of differential inclusions expressed level setwise overcomes this deficiency and opens up for profitable investigation such properties as stability attraction periodicity and the like this is especially important for investigating continuous systems which are uncertain or incompletely specified this paper studies attractors of fuzzy des in terms of cocycles and encompasses both the time dependent and autonomous cases

properties of recoverable region and semi global stabilization in recoverable region for linear systems subject to constraints this paper investigates linear systems subject to input and state constraints it is shown that the recoverable region which is the largest domain of attraction that is theoretically achievable can be semiglobally stabilized by continuous nonlinear feedbacks while satisfying the constraints moreover when trying to compute the recoverable region a reduction technique shows that we only need to compute the recoverable region for a system of lower dimension which generally leads to a considerable simplification in the computational effort

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

limits of nonlinear discrete time control systems with fast subsystems we investigate the asymptotic properties of nonlinear discrete time control systems with fast subsystems in particular we use the averaging method in order to construct a limiting system for the slow subsystem we show continuous dependence of the slow trajectories on the perturbation parameter measuring the ratio of the different step sizes as a byproduct we get an exponential formula for the reachable set of nonlinear continuous time control systems with finite control range

influences of subgrid scale dynamics on resolvable scale statistics in large eddy simulations recently the e expansion and recursive renormalization group rng theories as well as approximate inertial manifolds aim have been exploited as means of systematically modeling subgrid scales in large eddy simulations les although these theoretical approaches are rather complicated mathematically their key approximations can be investigated using direct numerical simulations dns in fact the differences among these theories can be traced to whether they retain or neglect interactions between the subgrid subgrid and subgrid resolvable scales in this paper we focus on the influence of these two interactions on the evolution of the resolvable scales in les the effecta which keeps only the interactions between the small and large scales and the effectb which on the other hand keeps only the interactions among the subgrid subgrid scales the performance of these models is analyzed using the velocity fields of the direct numerical simulations specifically our comparison is based on the analysis of the energy and enstrophy spectra as well as higher order statistics of the velocity and velocity derivatives we found that the energy spectrum and higher order statistics for the simulations with the effecta referred to hereafter as modela are in very good agreement with the filtered dns the comparison between the computations with effectb referred to hereafter as modelb and the filtered dns however is not satisfactory moreover the decorrelation between the filtered dns and modela is much slower than that of the filtered dns and modelb therefore we conclude that the modela taking into acciunt the interactions between the subgrid and resolvable scales is a faithful subgrid model for les for the range of reynolds numbers considered

superstatistics in hydrodynamic turbulence abstract superstatistics is a statistics of a statistics relevant for driven nonequilibrium systems with fluctuating intensive parameters it contains tsallis statistics as a special case we show that probability density functions of velocity differences and accelerations measured in eulerian and lagrangian turbulence experiments are well reproduced by simple superstatistics models we compare fits obtained for log normal superstatistics and 2 superstatistics tsallis statistics

differential variational inequalities this paper introduces and studies the class of differential variational inequalities dvis in a finite dimensional euclidean space the dvi provides a powerful modeling paradigm for many applied problems in which dynamics inequalities and discontinuities are present examples of such problems include constrained time dependent physical systems with unilateral constraints differential nash games and hybrid engineering systems with variable structures the dvi unifies several mathematical problem classes that include ordinary differential equations odes with smooth and discontinuous right hand sides differential algebraic equations daes dynamic complementarity systems and evolutionary variational inequalities conditions are presented under which the dvi can be converted either locally or globally to an equivalent ode with a lipschitz continuous right hand function for dvis that cannot be so converted we consider their numerical resolution via an euler time stepping procedure which involves the solution of a sequence of finite dimensional variational inequalities borrowing results from differential inclusions dis with upper semicontinuous closed and convex valued multifunctions we establish the convergence of such a procedure for solving initial value dvis we also present a class of dvis for which the theory of dis is not directly applicable and yet similar convergence can be established finally we extend the method to a boundary value dvi and provide conditions for the convergence of the method the results in this paper pertain exclusively to systems with index not exceeding two and which have absolutely continuous solutions

monotonic continuous time random walks with drift and stochastic reset events in this paper we consider a stochastic process that may experience random reset events which suddenly bring the system to the starting value and analyze the relevant statistical magnitudes we focus our attention on monotonic continuous time random walks with a constant drift the process increases between the reset events either by the effect of the random jumps or by the action of the deterministic drift as a result of all these combined factors interesting properties emerge like the existence for any drift strength of a stationary transition probability density function or the faculty of the model to reproduce power law like behavior general formulas for two extreme statistics the survival probability and the mean exit time are also derived to corroborate in an independent way the results of the paper monte carlo methods were used these numerical estimations are in full agreement with the analytical predictions

large eddy simulation without filter an large eddy simulation les formalism based on sampling operators instead of filters is developed the major advantage of this approach is that sampling operators commute with the product and their application to nonlinear terms is not at the origin of any closure problem in absence of filters that smooth out the small scale structures in the flow the discretization errors in the les are expected to be important they must be modelled the possible confusion between modelling and discretization errors is however avoided since these two effects are identical in the present formalism a generalized dynamic procedure is proposed for sampling based les which allows for model parameter optimization and does not require a detailed analysis of the discretization error in addition to its interesting mathematical properties for les the velocity obtained by a spatial sampling is much closer to experimental probe data than the filtered velocity field

variational collision integrator for polymer chains the numerical simulation of many particle systems e g in molecular dynamics often involves constraints of various forms we present a symplectic integrator for mechanical systems with holonomic bilateral and unilateral contact constraints the latter being in the form of a non penetration condition the scheme is based on a discrete variant of hamilton s principle in which both the discrete trajectory and the unknown collision time are varied cf r fetecau j marsden m ortiz m west nonsmooth lagrangian mechanics and variational collision integrators siam j appl dyn syst 2 2003 381 416 as a consequence the collision event enters the discrete equations of motion as an unknown that has to be computed on the fly whenever a collision is imminent the additional bilateral constraints are efficiently dealt with employing a discrete null space reduction including a projection and a local reparametrisation step which considerably reduces the number of unknowns and improves the condition number during each time step as compared to a standard treatment with lagrange multipliers we illustrate the numerical scheme with a simple example from polymer dynamics a linear chain of beads and test it against other standard numerical schemes for collision problems

refined similarity hypothesis using three dimensional local averages the refined similarity hypotheses of kolmogorov regarded as an important ingredient of intermittent turbulence has been tested in the past using one dimensional data and plausible surrogates of energy dissipation we employ data from direct numerical simulations at the microscale reynolds number r 650 on a periodic box of 4096 3 grid points to test the hypotheses using three dimensional averages in particular we study the small scale properties of the stochastic variable v u r re r 1 3 where u r is the longitudinal velocity increment and e r is the dissipation rate averaged over a three dimensional volume of linear size r we show that v is universal in the inertial subrange in the dissipation range the statistics of v are shown to depend solely on a local reynolds number

asymptotic freedom non gaussian perturbation theory and the application of renormalization group theory to isotropic turbulence a model equation with an extended range of asymptotic freedom is used as the basis of a non gaussian perturbation expansion in powers of the control parameter of the conditional average re expansion in the local reynolds number allows the systematic rederivation of an earlier heuristic theory w d mccomb and a g watt phys rev lett 65 3281 1990

order results for implicit runge kutta methods applied to stiff systems an extensive order analysis is given for the local and global errors of certain classes of runge kutta methods applied to general nonlinear stiff initial value problems it turns out that there are three order levels for linear problems y ay the classical orders apply for general nonlinear problems however two types of order reductions occur there a distinction has to be made between stiff problems where only f y is large and stiff problems where also some of the other derivatives f t f tt f yy cdots are large for these two classes of problems order results can be guaranteed which differ only by one power of the stepsize parameter h the order levels for both classes of problems are considerably lower than the respective conventional orders these order results are optimal in the sense that higher orders would contradict numerical observations made with particular initial value problems

a c density theorem for differential inclusions with lipschitz continuous right hand sides abstract by definition solutions of differential inclusions are absolutely continuous functions with l 1 derivatives we prove that at least for a class of lipschitz continuous right hand sides the c solutions are dense in the set of all solutions with respect to the supremum norm

on the universality of the kolmogorov constant in numerical simulations of turbulence motivated by a recent survey of experimental data k r sreenivasan phys fluids 2778 1995 we examine data on the kolmogorov spectrum constant in numerical simulations of isotropic turbulence using results both from previous studies and from new direct numerical simulations over a range of reynolds numbers up to 240 on the taylor scale at grid resolutions up to 5123 it is noted that in addition to k 5 3 scaling identification of a true inertial range requires spectral isotropy in the same wavenumber range we found that a plateau in the compensated three dimensional energy spectrum at k eta 0 1 0 2 commonly used to infer the kolmogorov constant from the compensated three dimensional energy spectrum actually does not represent proper inertial range behavior rather a proper if still approximate inertial range emerges at k eta 0 02 0 05 when r sub sub sub sub sub sub sub sub 0 53 for c 1 62 in excellent agreement with experiments however the one and three dimensional estimates are not fully consistent because of departures due to numerical and statistical limitations from isotropy of the computed spectra at low wavenumbers the inertial scaling of structure functions in physical space is briefly addressed since dns is still restricted to moderate reynolds numbers an accurate evaluation of the kolmogorov constant is very difficult we focus on providing new insights on the interpretation of kolmogorov 1941 similarity in the dns literature and do not consider issues pertaining to the refined similarity hypotheses of kolmogorov

new perspectives in turbulence scaling laws asymptotics and intermittency intermittency a basic property of fully developed turbulent flow decreases with growing viscosity therefore classical relationships obtained in the limit of vanishing viscosity must be corrected when the reynolds number is finite but large these corrections are the main subject of the present paper they lead to a new scaling law for wall bounded turbulence which is of key importance in engineering and to a reinterpretation of the kolmogorov obukhov scaling for the local structure of turbulence which has been of paramount interest in both theory and applications the background of these results is reviewed in similarity methods in the statistical theory of vortex motion and in intermediate asymptotics and relevant experimental data are summarized

fiber dynamics in turbulent flows general modeling framework the paper at hand deals with the modeling of turbulence effects on the dynamics of a long slender elastic fiber independent of the choice of the drag model a general aerodynamic force concept is derived on the basis of the velocity field for the randomly fluctuating component of the flow its construction as a centered differentiable gaussian field complies thereby with the requirements of the stochastic k epsilon turbulence model and kolmogorov s universal equilibrium theory on local isotropy

the use of explicit filters in large eddy simulation abstract explicit filtering is considered as a means of controlling the numerical errors that result when finite difference methods are used in large eddy simulation les the notion that the finite difference expressions themselves act as an effective filter is shown to be false for three dimensional simulations performed on nonuniform meshes for consistency the nonlinear terms in the navier stokes equations should be filtered explicitly at each time step in order to insure that the spectral content of the solution remains fixed at the desired filter level the explicit filtering operation allows a separation between the filter size and the mesh spacing and can be used to control the impact of the numerical errors numerical tests of the explicit filtering approach in turbulent channel flow are used to investigate the effectiveness of the explicit filtering approach and to assess its associated cost explicit filtering is shown to improve the computed results but this improvement comes at a rather high computational cost the explicitly filtered approach is also compared with straightforward mesh refinement as an alternative means of improving the computed results mesh refinement is also seen to increase the accuracy of the simulation but some traces of numerical error appear to persist in the solution

the asymptotic and oscillatory behavior of neutral difference inclusions in this paper the asymptotic and oscillatory properties of solutions to the first order neutral difference inclusion of the email protected x n r nx n t p nf x n s 1 x n s 2 x n s s n 0 1 2 are discussed where f r s 2 r 1 is a compact convex upper semicontinuous multifunction some examples are given to illustrate main theorems

second order discrete approximations to strongly convex differential inclusions we present a runge kutta type scheme for discretization of strongly convex differential inclusions the accuracy of the approximation to the set of trajectories of the differential inclusion on a given interval is proven to be of second order with respect to the step length

spectral content of fractional brownian motion with stochastic reset we analyse the power spectral density psd s t f with t being the observation time and f the frequency of a fractional brownian motion fbm with an arbitrary hurst index undergoing a stochastic resetting to the origin at a constant rate r the resetting process introduced some time ago as an example of an efficient optimisable search algorithm to this end we first derive an exact expression for the covariance function of an arbitrary not necessarily a fbm process with a reset expressing it through the covariance function of the parental process without a reset which yields the desired result for the fbm in a particular case we then use this result to compute exactly the power spectral density for fbm for all frequency f the asymptotic large frequency f behaviour of the psd turns out to be distinctly different for sub and super diffusive fbms we show that for large f the psd has a power law tail where the exponent for sub diffusive fbm while for all thus somewhat unexpectedly the exponent in the superdiffusive case h 1 2 sticks to its brownian value and does not depend on h

construction of commutative filters for les on unstructured meshes a method of constructing discrete filters for large eddy simulation of turbulent flows on unstructured meshes is presented the commutation error between differentiation and filtering can be made arbitrarily small with these filters the filtering method is applied to various test cases to demonstrate commutation an extension to three dimensions and implementation into an unstructured solver for les are discussed

explicit small scale velocity simulation for high re turbulent flows a new approach two level simulation tls is developed based on the explicit reconstruction of the small scale velocity by solving the small scale governing equations on the domain with reduced dimension representing a collection of one dimensional lines embedded in the three dimensional flow domain a coupled system of equations that is not based on an eddy viscosity hypothesis is derived based on the decomposition of flow variables into the large scale and the small scale components without introducing the concept of filtering simplified treatment of the small scale equations is proposed based on neglecting the small scale advective and dissipative terms in the line orthogonal directions fully coupled simulation of the forced isotropic turbulence at re l 65 114 demonstrates potential benefits and challenges of the tls approach for high re turbulent flows using coarse grids

nonperturbative renormalization group study of the stochastic navier stokes equation we study the renormalization group flow of the average action of the stochastic navier stokes equation with power law forcing using galilean invariance we introduce a nonperturbative approximation adapted to the zero frequency sector of the theory in the parametric range of the holder exponent 4 2e of the forcing where real space local interactions are relevant in any spatial dimension d we observe the convergence of the resulting renormalization group flow to a unique fixed point which yields a kinetic energy spectrum scaling in agreement with canonical dimension analysis kolmogorov s 5 3 law is thus recovered for e 2 as also predicted by perturbative renormalization at variance with the perturbative prediction the 5 3 law emerges in the presence of a saturation in the e dependence of the scaling dimension of the eddy diffusivity at e 3 2 when according to perturbative renormalization the velocity field becomes infrared relevant

conditional mode elimination and scale invariant dissipation in isotropic turbulence abstract we show that a conditional average based on a limit in which the fourier modes of the turbulent velocity field with wavenumber k k c where k c is an arbitrary cutoff wavenumber are held constant can be used to separate the nonlinear coupling to high k modes into coherent and random parts with the latter rigorously determining the net energy transfer in addition we show that three symmetry breaking terms which are generated by the conditional average of the navier stokes equation do not contribute to the equation for the energy dissipation two of these terms vanish identically under unconditional averaging and wavenumber integration respectively and the remaining one vanishes in the limit of asymptotic freedom when calculated by a quasi stochastic estimate from the high k momentum equation if the cutoff k c is chosen to be large enough then the conditionally averaged high k equation can be solved perturbatively in terms of the local reynolds number which is less than unity on this basis an expression for the renormalized dissipation rate is obtained as an expansion in a parameter which is equal to the square of the local reynolds number a recursive calculation is made of the renormalized dissipation rate in which the expansion parameter reaches a maximum value of 0 16 at the fixed point it is also shown that a previous markovian approximation can be replaced by an exact summation to consistent order in perturbation theory

dynamical properties of hybrid systems simulators this paper analyzes the dynamical properties of a class of hybrid systems simulators a hybrid system is a dynamical system with a state that can both flow and jump its simulator attempts to generate its solutions approximately the paper presents mild regularity conditions on the hybrid system and its simulator to guarantee that simulated solutions are close to actual solutions on compact hybrid time intervals and that asymptotically stable compact sets are preserved in a semiglobal practical sense under simulation in fact it is established that asymptotically stable compact sets are continuous in the integration step size parameter of the simulator that is as the step size of the simulator converges to zero the asymptotically stable set observed in simulations approaches the asymptotically stable compact set of the true hybrid system examples are used to illustrate concepts and results

efficient solution of ordinary differential equations with a parametric lexicographic linear program embedded this work analyzes the initial value problem in ordinary differential equations with a parametric lexicographic linear program lp embedded the lp is said to be embedded since the dynamics depend on the solution of the lp which is in turn parameterized by the dynamic states this problem formulation finds application in dynamic flux balance analysis which serves as a modeling framework for industrial fermentation reactions it is shown that the problem formulation can be intractable numerically which arises from the fact that the lp induces an effective domain that may not be open a numerical method is developed which reformulates the system so that it is defined on an open set the result is a system of semi explicit index one differential algebraic equations which can be solved with efficient and accurate methods it is shown that this method addresses many of the issues stemming from the original problem s intractability the application of the method to examples of industrial fermentation processes demonstrates its effectiveness and efficiency

mathematical and physical constraints on large eddy simulation of turbulence recent progress in the theoretical foundations of large eddy simulation is reviewed most of the work reported is motivated by conceptual difficulties encountered in applying the large eddy simulation method to inhomogeneous complex geometry flows among the topics covered are the problem of the lack of commutation between filtering and derivative operators for inhomogeneous flows the issues of enforcing symmetry and realizability conditions in subgrid modeling and the problem of unacceptably high numerical errors in large eddy simulation implementations with finite difference methods

a formal derivation of the local energy transfer let theory of homogeneous turbulence a statistical closure of the navier stokes hierarchy which leads to equations for the two point two time covariance of the velocity field for stationary homogeneous isotropic turbulence is presented it is a generalisation of the self consistent field method due to edwards 1964 for the stationary single time velocity covariance the probability distribution functional p u t is obtained in the form of a series from the liouville equation by means of a perturbation expansion about a gaussian distribution which is chosen to give the exact two point two time covariance the triple moment is calculated in terms of an ensemble averaged infinitesimal velocity field propagator and shown to yield the edwards result as a special case the use of a gaussian zero order distribution has been found to justify the introduction of a fluctuation response relation which is in accord with modern dynamical theories in a sense this work completes the analogy drawn by edwards between turbulence and brownian motion originally edwards had shown that the noise input was determined by the correlation of the velocity field with the externally applied stirring forces but was unable to determine the system response now we find that the system response is determined by the correlation of the velocity field with internal quasi entropic forces this analysis is valid to all orders of perturbation theory and allows the recovery of the local energy transfer let theory which had previously been derived by more heuristical methods the let theory is known to be in good agreement with experimental results it is also unique among two point statistical closures in displaying an acceptable i e non markovian relationship between the transfer spectrum and the system response in accordance with experimental results as a result of the latter property it is compatible with the kolmogorov k41 spectral phenomenology

on stability of sets for sampled data nonlinear inclusions via their approximate discrete time models and summability criteria this paper consists of two main parts in the first part we provide a framework for stabilization of arbitrary not necessarily compact closed sets for sampled data nonlinear differential inclusions via their approximate discrete time models we generalize d nesic a r teel and p v kokotovic systems control lett 38 1999 pp 259 270 theorem 1 in several different directions we consider stabilization of arbitrary closed sets plants described as sampled data differential inclusions and arbitrary dynamic controllers in the form of difference inclusions our result does not require the knowledge of a lyapunov function for the approximate model which is a standing assumption in d nesic and a r teel ieee trans automat control 49 2004 pp 1103 1122 and d nesic a r teel and p v kokotovic systems control lett 38 1999 pp 259 270 theorem 2 we present checkable conditions that one can use to conclude semiglobal asymptotic stability or global exponential stability ges of the sampled data system via appropriate properties of its approximate discrete time model in the second part we present sufficient conditions for stability of parameterized difference inclusions that involve various summability criteria on trajectories of the system to conclude global asymptotic stability gas or ges and they represent discrete time counterparts of results given in a r teel e panteley and a loria math control signals systems 15 2002 pp 177 201 these summability criteria are not lyapunov based and they are tailored to be used within our above mentioned framework for stabilization of sampled data differential inclusions via their approximate discrete time models we believe that these tools will be a useful addition to the toolbox for controller design for sampled data nonlinear systems via their approximate discrete time models

efficient computation of the bounds of continuous time imprecise markov chains abstract when the initial distribution and transition rates for a continuous time markov chain are not known precisely robust methods are needed to study the evolution of the process in time to avoid judgements based on unwarranted precision we follow the ideas successfully applied in the study of discrete time model to build a framework of imprecise markov chains in continuous time the imprecision in the distributions over the set of states is modelled with upper and lower expectation functionals which equivalently represent sets of probability distributions uncertainty in transitions is modelled with sets of transition rates compatible with available information the kolmogorov s backward equation is then generalised into the form of a generalised differential equation with generalised derivatives and set valued maps the upper and lower expectation functionals corresponding to imprecise distributions at given times are determined by the maximal and minimal solutions of these equations the second part of the paper is devoted to numerical methods for approximating the boundary solutions the methods are based on discretisation of the time interval a uniform and adaptive grid discretisations are examined the latter is computationally much more efficient than the former one but is not applicable on every interval therefore to achieve maximal efficiency a combination of the methods is used

discrete approximations relaxation and optimization of one sided lipschitzian differential inclusions in hilbert spaces we study discrete approximations of nonconvex differential inclusions in hilbert spaces and dynamic optimization optimal control problems involving such differential inclusions and their discrete approximations the underlying feature of the problems under consideration is a modified one sided lipschitz condition imposed on the right hand side i e on the velocity sets of the differential inclusion which is a significant improvement of the conventional lipschitz continuity our main attention is paid to establishing efficient conditions that ensure the strong approximation in the w1 p norm as p 1 of feasible trajectories for the one sided lipschitzian differential inclusions under consideration by those for their discrete approximations and also the strong convergence of optimal solutions to the corresponding dynamic optimization problems under discrete approximations to proceed with the latter issue we derive a new extension of the bogolyubov type relaxation density theorem to the case of differential inclusions satisfying the modified one sided lipschitzian condition all the results obtained are new not only in the infinite dimensional hilbert space framework but also in finite dimensional spaces

a framework for stabilization of nonlinear sampled data systems based on their approximate discrete time models a unified framework for design of stabilizing controllers for sampled data differential inclusions via their approximate discrete time models is presented both fixed and fast sampling are considered in each case sufficient conditions are presented which guarantee that the controller that stabilizes a family of approximate discrete time plant models also stabilizes the exact discrete time plant model for sufficiently small integration and or sampling periods previous results in the literature are extended to cover 1 continuous time plants modeled as differential inclusions 2 general approximate discrete time plant models 3 dynamical discontinuous controllers modeled as difference inclusions and 4 stability with respect to closed arbitrary not necessarily compact sets

elimination of turbulence modes using a conditional average with asymptotic freedom the part of the nonlinear term in the navier stokes equation which represents coupling to the small scale modes may be averaged out by introducing a weak conditional average with asymptotic freedom in wavenumber a residual deterministic part while important for individual realizations makes a negligible contribution to the renormalization of the dissipation rate this is because the full ensemble average needed to establish the energy balance relaxes the constraint on the conditional average

diffusion with stochastic resetting at power law times what happens when a continuously evolving stochastic process is interrupted with large changes at random intervals ensuremath tau distributed as a power law ensuremath sim ensuremath tau ensuremath 1 ensuremath alpha phantom rule 0 28em 0ex ensuremath alpha g0 modeling the stochastic process by diffusion and the large changes as abrupt resets to the initial condition we obtain exact closed form expressions for both static and dynamic quantities while accounting for strong correlations implied by a power law our results show that the resulting dynamics exhibits a spectrum of rich long time behavior from an ever spreading spatial distribution for ensuremath alpha l1 to one that is time independent for ensuremath alpha g1 the dynamics has strong consequences on the time to reach a distant target for the first time we specifically show that there exists an optimal ensuremath alpha that minimizes the mean time to reach the target thereby offering a step towards a viable strategy to locate targets in a crowded environment

the basic equations for the large eddy simulation of turbulent flows in complex geometry abstract the equations for large eddy simulation les of turbulent flows are derived by applying a filtering operation to the navier stokes equations les of inhomogeneous turbulent flows requires the use of filters with variable filter width the use of such filters invalidates the standard derivation of the basic equations for the filtered fields since the filtering operation in general does not commute with the operation of differentiation in this paper an alternate definition of the filtering operation based on the mapping function of the nonuniform grid is introduced it is shown that with this modified definition the filtering and differentiation operations commute up to an error which is second order in the filter width it is also shown that the commutation error can be expressed in terms of the filtered field and its derivatives as an asymptotic series in the square of the filter width these results are then applied to the navier stokes equations to derive the basic equations satisfied by the filtered fields

energy transfer and dissipation in forced isotropic turbulence a model for the reynolds number dependence of the dimensionless dissipation rate ce was derived from the dimensionless karman howarth equation resulting in ce ce c rl o 1 rl where rl is the integral scale reynolds number the coefficients c and ce arise from asymptotic expansions of the dimensionless second and third order structure functions this theoretical work was supplemented by direct numerical simulations dns of forced isotropic turbulence for integral scale reynolds numbers up to rl 5875 r 435 which were used to establish that the decay of dimensionless dissipation with increasing reynolds number took the form of a power law rln with exponent value n 1 000 0 009 and that this decay of ce was actually due to the increase in the taylor surrogate u3 l the model equation was fitted to data from the dns which resulted in the value c 18 9 1 3 and in an asymptotic value for ce in the infinite reynolds number limit of ce 0 468 0 006

convergence of the forward euler method for nonconvex differential inclusions the convergence of reachable sets for nonconvex differential inclusions is considered when the right hand side in the differential inclusion is a compact valued lipschitz continuous set valued function it is shown that the convergence in hausdorff distance of reachable sets for a forward euler discretization is linear in the time step

differential mixed variational inequalities in finite dimensional spaces in this paper a differential vector variational inequality is introduced and studied in finite dimensional euclidean spaces the existence of a caratheodory weak solution for the differential vector variational inequality is presented under some suitable conditions furthermore the upper semicontinuity and the lower semicontinuity of the solution sets for the differential variational inequality are established when both the mapping and the constraint set are perturbed by two different parameters

a large eddy simulation method for dgsem using non linearly optimized relaxation filters abstract in this paper we apply a specifically designed dissipative spatial filter as sub grid scale model within the increasingly popular discontinuous galerkin methods and the closely related flux reconstruction high order methods for large eddy simulation the parameters of the filter kernel are optimized with data obtained from direct numerical simulation that is filtered and used as a ground truth to fit the overall kinetic energy and dissipation rate over time the optimization is carried out for polynomial degree 3 to 10 the optimal kernels are rigorously tested in the limit of infinite reynolds number flows hit and taylor green vortex flow additionally a brief extension to plane turbulent channel flow is given

convergence order of a numerical scheme for sweeping process in j venel numer math 118 2011 pp 367 400 an implementable algorithm was introduced to compute discrete solutions of sweeping processes i e specific first order differential inclusions the convergence of this numerical scheme was proved thanks to compactness arguments here we establish that this algorithm is of order frac 1 2 the considered sweeping process involves a set valued map given by a finite number of inequality constraints the proof rests on a metric qualification condition between the sets associated to each constraint

the exponential formula for the reachable set of a lipschitz differential inclusion the main goal of this paper is to prove a formula for the reachable set of a lipschitz differential inclusion with convex values the formula involves a kuratowski limit of sets that resembles a standard approach of defining the exponential of a matrix this explains the title the proof of the main theorem partially relies on a c 1 approximation result due to filippov for which a new proof is given a new approach of characterizing the value function associated with a mayer optimal control problem is given as an application

on the spectrum of fractional brownian motions fractional brownian motions fbms provide useful models for a number of physical phenomena whose empirical spectra obey power laws of fractional order however due to the nonstationary nature of these processes the precise meaning of such spectra remains generally unclear two complementary approaches are proposed which are intended to clarify this point the first one based on a time frequency analysis takes into account the nonstationary nature of fbm and puts emphasis on time averaged measurements the second one based on a time scale analysis is matched to self similarity properties of fbm and reveals an underlying stationary structure relative to each time scaling

classical and strong convexity of sublevel sets and application to attainable sets of nonlinear systems necessary and sufficient conditions for convexity and strong convexity respectively of connected sublevel sets that are defined by finitely many real valued c 1 1 maps are presented a novel characterization of strongly convex sets in terms of the so called local quadratic support is proved the results concerning strong convexity are used to derive sufficient conditions for attainable sets of continuous time nonlinear systems to be strongly convex an application of these conditions is a novel method to overapproximate attainable sets when strong convexity is present

differential inclusions with stable subinclusions wherexer te to t andf r x to t g r is a set valued mapping with compact and convex images an absolutely continuous function x m is a solution of 1 if it satisfies 1 for a e t e to t thus a differential inclusion generalizes an ordinary differential equation by permitting a set valued right hand side differential equations containing either control uncertain parameters or discontinuity in the right hand side can be transformed into the framework of 1 l 21 a well known and very useful result due to filippov 3 estimates the uniform distance from an arbitrary absolutely continuous function x to the set of solutions of 1 starting from x t by means of constant times the deviation

incremental unknowns multilevel methods and the numerical simulation of turbulence abstract the purpose of this monograph is to describe the dynamic multilevel dml methodology applied to the numerical simulation of turbulence the general setting of the navier stokes equations is recalled and also a number of basic notions on the statistical theory of turbulence the practical limitations of direct numerical simulation dns and the needs for modeling are emphasized the objective of this article is to discuss the modeling and the numerical simulation of turbulent flows by multilevel methods related to the concept of approximate inertial manifolds aim this mathematical concept stemming from the dynamical systems theory is briefly presented aim are based on a decomposition of the velocity field into small and large scale components they give a slaving law of the small scales as a function of the large scales the novel aspect of the work presented here is the time adaptative dynamical implementation of these multilevel methods indeed the dml methodology is based on the properties of this decomposition of the velocity field and on numerical arguments numerical analysis of multilevel methods for a simple system is proposed dml methods with some numerical and physical justifications are described for homogeneous turbulence the numerical results obtained are discussed

fixed set iterations for relaxed lipschitz multimaps abstract a dynamical system described by an autonomous differential inclusion with a right hand side satisfying a relaxed lipschitz condition as well as its euler approximations are studied we investigate the asymptotic properties of the solutions and of the attainable sets it is shown that the system has a strongly flow invariant set or a fixed set that is a set such that each trajectory starting from it does not leave it this set is also an attractor i e it attracts the continuous and the discrete euler trajectories as the time tends to infinity we give estimates of the rate of attraction an algorithm for approximating the fixed set by the attainable sets of the discrete system is also presented

optimal stochastic restart renders fluctuations in first passage times universal stochastic restart may drastically reduce the expected run time of a computer algorithm expedite the completion of a complex search process or increase the turnover rate of an enzymatic reaction these diverse first passage time fpt processes seem to have very little in common but it is actually quite the other way around here we show that the relative standard deviation associated with the fpt of an optimally restarted process i e one that is restarted at a constant nonzero rate which brings the mean fpt to a minimum is always unity we interpret further generalize and discuss this finding and the implications arising from it

second order runge kutta approximations in control constrained optimal control in this paper we analyze second order runge kutta approximations to a nonlinear optimal control problem with control constraints if the optimal control has a derivative of bounded variation and a coercivity condition holds we show that for a special class of runge kutta schemes the error in the discrete approximating control is o h2 where h is the mesh spacing

a general class of commutative filters for les in complex geometries a class of filters for large eddy simulations of turbulent inhomogeneous flows is presented a general set of rules for constructing discrete filters in complex geometry is given and examples of such filters are presented with these filters the commutation error between numerical differentiation and filtering can be made arbitrarily small allowing for derivation of a consistent set of equations for the large scale field the application of such filters for explicit filtering in large eddy simulations and the issue of boundary conditions for the filtered field are also discussed

aging wiener khinchin theorem and critical exponents of 1 f β noise the power spectrum of a stationary process may be calculated in terms of the autocorrelation function using the wiener khinchin theorem we here generalize the wiener khinchin theorem for nonstationary processes and introduce a time dependent power spectrum ensuremath langle s t m ensuremath omega ensuremath rangle where t m is the measurement time for processes with an aging autocorrelation function of the form ensuremath langle i t i t ensuremath tau ensuremath rangle t mathrm ensuremath upsilon ensuremath phi mathrm ea ensuremath tau t where ensuremath phi mathrm ea x is a nonanalytic function when x is small we find aging 1 f ensuremath beta noise aging 1 f ensuremath beta noise is characterized by five critical exponents we derive the relations between the scaled autocorrelation function and these exponents we show that our definition of the time dependent spectrum retains its interpretation as a density of fourier modes and discuss the relation to the apparent infrared divergence of 1 f ensuremath beta noise we illustrate our results for blinking quantum dot models single file diffusion and brownian motion in a logarithmic potential

on the discretization of switched linear systems abstract a bilinear single control system which can be viewed as a control formulation of a linear switched system is considered the control is restricted to take values either i in 0 1 switched system or ii in 0 1 relaxed system in more practical considerations the control is often allowed to change only at the points of a given time net with a step length h the paper investigates what is the approximation error in terms of the reachable set in the two cases i and ii the error estimates that follow directly from known results are of order h and h respectively in the present paper estimations of order h and h 1 5 are proved in a constructive way the second one makes use of the effect of non accumulation of errors established earlier by the second author

fast numerical simulations of 2d turbulence using a dynamic model for subfilter motions we present numerical simulations of 2d turbulent flow using a new model for the subfilter scales which are computed using a dynamic equation linking the subfilter scales with the resolved velocity this equation is not postulated but derived from the constitutive equations under the assumption that the non linear interactions of subfilter scales between themselves are small compared to their distortions by the resolved scales such an assumption results in a linear stochastic equation for the subfilter scales which can be numerically solved by a decomposition of the subfilter scales into localized wave packets the wave packets are randomly produced by the smallest of the resolved scales they are further transported by the resolved scale velocity and they have wavenumbers and amplitudes which evolve according to the resolved strain performance of our model is compared with direct numerical simulations of decaying and forced turbulence for the same resolution numerical simulations using our model allow for a significant reduction of the computational time of the order of 100 in the case we consider and allow to achieve of significantly larger reynolds number than the direct method

anomalous diffusion under stochastic resettings a general approach we present a general formulation of the resetting problem which is valid for any distribution of resetting intervals and arbitrary underlying processes we show that in such a general case a stationary distribution may exist even if the reset free process is not stationary as well as a significant decreasing in the mean first passage time we apply the general formalism to anomalous diffusion processes which allow simple and explicit expressions for poissonian resetting events

second order discrete approximation to linear differential inclusions an approximation to a linear differential inclusion by means of n stage single step discrete inclusions is presented which is of second order accuracy with respect to n approximations of this type with higher order of accuracy are shown not to exist in general the result is applied for a second order discretization of control constrained optimal control problems and approximate feedback design making use of dynamic programming

diffusion with optimal resetting we consider the mean time to absorption by an absorbing target of a diffusive particle with the addition of a process whereby the particle is reset to its initial position with rate r we consider several generalisations of the model of m r evans and s n majumdar 2011 diffusion with stochastic resetting phys rev lett 106 160601 i a space dependent resetting rate r x ii resetting to a random position z drawn from a resetting distribution p z iii a spatial distribution for the absorbing target pt x as an example of i we show that the introduction of a non resetting window around the initial position can reduce the mean time to absorption provided that the intial position is sufficiently far from the target we address the problem of optimal resetting that is minimising the mean time to absorption for a given target distribution for an exponentially decaying target distribution centred at the origin we show that a transition in the optimal resetting distribution occurs as the target distribution narrows

random walk with random resetting to the maximum position we study analytically a simple random walk model on a one dimensional lattice where at each time step the walker resets to the maximum of the already visited positions to the rightmost visited site with a probability r and with probability 1 r it undergoes symmetric random walk i e it hops to one of its neighboring sites with equal probability 1 r 2 for r 0 it reduces to a standard random walk whose typical distance grows as n for large n in the presence of a nonzero resetting rate 0 r 1 we find that both the average maximum and the average position grow ballistically for large n with a common speed v r moreover the fluctuations around their respective averages grow diffusively again with the same diffusion coefficient d r we compute v r and d r explicitly we also show that the probability distribution of the difference between the maximum and the location of the walker becomes stationary as n however the approach to this stationary distribution is accompanied by a dynamical phase transition characterized by a weakly singular large deviation function we also show that r 0 is a special critical point for which the growth laws are different from the r 0 case and we calculate the exact crossover functions that interpolate between the critical r 0 and the off critical r 0 behavior for finite but large n

subdifferential analysis of differential inclusions via discretization abstract the framework of differential inclusions encompasses modern optimal control and the calculus of variations necessary optimality conditions in the literature identify potentially optimal paths but do not show how to perturb paths to optimality we first look at the corresponding discretized inclusions estimating the subdifferential dependence of the optimal value in terms of the endpoints of the feasible paths our approach is to first estimate the coderivative of the reachable map the discretized nonsmooth euler lagrange and transversality conditions follow as a corollary we obtain corresponding results for differential inclusions by passing discretized inclusions to the limit

difference methods for differential inclusions a survey the main objective of this survey is to study convergence properties of difference methods applied to differential inclusions it presents in a unified way a number of results scattered in the li

asymptotic behaviour of nonautonomous difference inclusions abstract nonautonomous difference inclusions are formulated as difference inclusion cocycles which generalize set valued semigroups corresponding to autonomous difference inclusions and the existence of a difference inclusion cocycle attractor is established under the assumption of upper semi continuity of the cocycle mappings if the difference inclusion cocycle has an absorbing set several illustrative examples are provided

dynamical transition in the temporal relaxation of stochastic processes under resetting a stochastic process when subject to resetting to its initial condition at a constant rate generically reaches a nonequilibrium steady state we study analytically how the steady state is approached in time and find an unusual relaxation mechanism in these systems we show that as time progresses an inner core region around the resetting point reaches the steady state while the region outside the core is still transient the boundaries of the core region grow with time as power laws at late times with new exponents alternatively at a fixed spatial point the system undergoes a dynamical transition from the transient to the steady state at a characteristic space dependent timescale t x we calculate analytically in several examples the large deviation function associated with this spatiotemporal fluctuation and show that generically it has a second order discontinuity at a pair of critical points characterizing the edges of the inner core these singularities act as separatrices between typical and atypical trajectories our results are verified in the numerical simulations of several models such as simple diffusion and fluctuating one dimensional interfaces

large deviations for markov processes with resetting markov processes restarted or reset at random times to a fixed state or region in space have been actively studied recently in connection with random searches foraging and population dynamics here we study the large deviations of time additive functions or observables of markov processes with resetting by deriving a renewal formula linking generating functions with and without resetting we are able to obtain the rate function of such observables characterizing the likelihood of their fluctuations in the long time limit we consider as an illustration the large deviations of the area of the ornstein uhlenbeck process with resetting other applications involving diffusions random walks and jump processes with resetting or catastrophes are discussed

models and simulations of dynamic frictional contact of a beam abstract we investigate a mathematical model for the dynamic thermomechanical behavior of a viscoelastic beam that is in frictional contact with a rigid moving surface friction is modeled by a version of coulomb s law with slip dependent coefficient of friction taking into account the frictional heat generation we prove the existence and uniqueness of the weak solution describe an algorithm for the numerical solutions and present results of numerical simulations including the frequency distribution of the noise generated by the stick slip motion we also show that when the surface moves too fast there are no steady solutions and therefore the system in thermally unstable

study of a maximal monotone model with a delay term we present a model involving a friction term and a delay term by using results of existence and uniqueness for maximal monotone differential inclusions we give theoretical results for this model an implicit euler numerical scheme and results related to order of convergence are also provided from both a theoretical and a numerical point of view