enumerative coding for grassmannian space the grassmannian space gq n k is the set of all k dimensional subspaces of the vector space fqn recently codes in the grassmannian have found an application in network coding the main goal of this paper is to present efficient enumerative encoding and decoding techniques for the grassmannian these coding techniques are based on two different orders for the grassmannian induced by different representations of k dimensional subspaces of fqn one enumerative coding method is based on a ferrers diagram representation and on an order for gq n k based on this representation the complexity of this enumerative coding is o k5 2 n k 5 2 digit operations another order of the grassmannian is based on a combination of an identifying vector and a reduced row echelon form representation of subspaces the complexity of the enumerative coding based on this order is o nk n k log n log log n digit operations a combination of the two methods reduces the complexity on average by a constant factor

controllability of impulsive neutral stochastic functional differential inclusions with infinite delay this paper deals with the controllability of a class of impulsive neutral stochastic functional differential inclusions with infinite delay in an abstract space sufficient conditions for the controllability are derived with the help of the fixed point theorem for discontinuous multi valued operators due to dhage an example is provided to illustrate the obtained theory

iterative learning control of inhomogeneous distributed parameter systems frequency domain design and analysis abstract this paper aims to construct a design and analysis framework for iterative learning control of linear inhomogeneous distributed parameter systems lidpss which may be hyperbolic parabolic or elliptic and include many important physical processes such as diffusion vibration heat conduction and wave propagation as special cases owing to the system model characteristics lidpss are first reformulated into a matrix form in the laplace transform domain then through the determination of a fundamental matrix the transfer function of lidps is precisely evaluated in a closed form the derived transfer function provides the direct input output relationship of the lidps and thus facilitates the consequent ilc design and convergence analysis in the frequency domain the proposed control design scheme is able to deal with parametric and non parametric uncertainties and make full use of the process repetition while avoid any simplification or discretization for the 3d dynamics of lidps in the time space and iteration domains in the end two illustrative processes are addressed to demonstrate the efficacy of the proposed iterative learning control scheme

a survey on iterative learning control for nonlinear systems in this article we review the recent advances in iterative learning control ilc for nonlinear dynamic systems in the research field of ilc two categories of system nonlinearities are considered namely the global lipschitz continuous glc functions and local lipschitz continuous llc functions ilc for glc systems is widely studied and analysed using contraction mapping approach and the focus of recent exploration moves to application problems though a number of theoretical issues remain open ilc for llc systems is currently a hot area and the recent research focuses on ilc design and analysis by means of lyapunov approach the objectives of this article are to introduce recent development and advances in nonlinear ilc schemes highlight their effectiveness and limitations as well as discuss the directions for further exploration of nonlinear ilc

maximum rank array codes and their application to crisscross error correction a mu n n k array code c over a field f is a k dimensional linear space of n n matrices over f such that every nonzero matrix in c has rank or mu it is first shown that the dimension of such array codes must satisfy the singleton like bound k

more on the decoder error probability for reed solomon codes a combinatorial technique similar to the principle of inclusion and exclusion is used to obtain an exact formula for p sub e u the decoder error probability for reed solomon codes the p sub e u for the 255 223 reed solomon code used by nasa and for the 31 15 reed solomon code jtids code are calculated using the exact formula and are observed to approach the qs of the codes rapidly as u gets large an upper bound for the expression mod p sub e u q 1 mod is derived and shown to decrease nearly exponentially as u increases

constant rank codes and their connection to constant dimension codes constant dimension codes have recently received attention due to their significance to error control in noncoherent random linear network coding what the maximal cardinality of any constant dimension code with finite dimension and minimum distance is and how to construct the optimal constant dimension code or codes that achieves the maximal cardinality both remain open research problems in this paper we introduce a new approach to solving these two problems we first establish a connection between constant rank codes and constant dimension codes via this connection we show that optimal constant dimension codes correspond to optimal constant rank codes over matrices with sufficiently many rows as such the two aforementioned problems are equivalent to determining the maximum cardinality of constant rank codes and to constructing optimal constant rank codes respectively to this end we then derive bounds on the maximum cardinality of a constant rank code with a given minimum rank distance propose explicit constructions of optimal or asymptotically optimal constant rank codes and establish asymptotic bounds on the maximum rate of a constant rank code

γ convergence of quadratic functionals with oscillating linear perturbations abstract we explore the convergence of quadratic functionals perturbed by linear terms with oscillatory coefficients in particular we are interested in understanding the interaction between the oscillations of the quadratic part and the linear term we also look at this issue from the perspective of the associated equations of optimality as in homogenization

multiscale convergence and reiterated homogenisation this paper generalises the notion of two scale convergence to the case of multiple separated scales of periodic oscillations it allows us to introduce a multi scale convergence method for the reiterated homogenisation of partial differential equations with oscillating coefficients this new method is applied to a model problem with a finite or infinite number of microscopic scales namely the homogenisation of the heat equation in a composite material finally it is generalised to handle the homogenisation of the neumann problem in a perforated domain

on the decoder error probability of bounded rank distance decoders for maximum rankdistance codes in this correspondence we first introduce the concept of elementary linear subspace which has similar properties to those of a set of coordinates we then use elementary linear subspaces to derive properties of maximum rank distance mrd codes that parallel those of maximum distance separable codes using these properties we show that for mrd codes with error correction capability the decoder error probability of bounded rank distance decoders decreases exponentially with based on the assumption that all errors with the same rank are equally likely

a matroid framework for noncoherent random network communications models for noncoherent error control in random linear network coding rlnc and store and forward saf have been recently proposed in this paper we model different types of random network communications as the transmission of flats of matroids this novel framework encompasses rlnc and saf and allows us to introduce a novel protocol referred to as random affine network coding ranc based on affine combinations of packets although the models previously proposed for rlnc and saf only consider error control using our framework we first evaluate and compare the performance of different network protocols in the error free case we define and determine the rate average delay and throughput of such protocols and we also investigate the possibilities of partial decoding before the entire message is received we thus show that ranc outperforms rlnc in terms of data rate and throughput thanks to a more efficient encoding of messages into packets second we model the possible alterations of a message by the network as an operator channel which generalizes the channels proposed for rlnc and saf error control is thus reduced to a coding theoretic problem on flats of a matroid where two distinct metrics can be used for error correction we study the maximum cardinality of codes on flats in general and codes for error correction in ranc in particular we finally design a class of nearly optimal codes for ranc based on rank metric codes for which we propose a low complexity decoding algorithm the gain of ranc over rlnc is thus preserved with no additional cost in terms of complexity

packing and covering properties of subspace codes for error control in random linear network coding codes in the projective space and codes in the grassmannian over a finite field referred to as subspace codes and constant dimension codes cdcs respectively have been proposed for error control in random linear network coding for subspace codes and cdcs a subspace metric was introduced to correct both errors and erasures and an injection metric was proposed to correct adversarial errors in this paper we investigate the packing and covering properties of subspace codes with both metrics we first determine some fundamental geometric properties of the projective space with both metrics using these properties we then derive bounds on the cardinalities of packing and covering subspace codes and determine the asymptotic rates of optimal packing and optimal covering subspace codes with both metrics our results not only provide guiding principles for the code design for error control in random linear network coding but also illustrate the difference between the two metrics from a geometric perspective in particular our results show that optimal packing cdcs are optimal packing subspace codes up to a scalar for both metrics if and only if their dimension is half of their length up to rounding in this case cdcs suffer from only limited rate loss as opposed to subspace codes with the same minimum distance we also show that optimal covering cdcs can be used to construct asymptotically optimal covering subspace codes with the injection metric only

applying the random variable transformation method to solve a class of random linear differential equation with discrete delay abstract we randomize the following class of linear differential equations with delay x t a x t b x t t 0 and initial condition x t g t t 0 by assuming that coefficients a and b are random variables and the initial condition g t is a stochastic process we consider two cases depending on the functional form of the stochastic process g t and then we solve from a probabilistic point of view both random initial value problems by determining explicit expressions to the first probability density function f x t of the corresponding solution stochastic processes afterwards we establish sufficient conditions on the involved random input parameters in order to guarantee that f x t converges as 0 to the first probability density function say f x t of the corresponding associated random linear problem without delay 0 the paper concludes with several numerical experiments illustrating our theoretical findings

on γ convergence in divergence free fields through young measures the explicit characterization of the limit energy density of sequences of general non periodic functionals defined on divergence free fields is achieved by means of the div young measure associated with relevant sequences of functions through a minimization problem

multiscale asymptotic method for maxwell s equations in composite materials in this paper we discuss the multiscale analysis of maxwell s equations in composite materials with a periodic microstructure the new contributions in this paper are the determination of higher order correctors and the explicit convergence rate for the approximate solutions see theorem 2 3 consequently we present the multiscale finite element method and derive the convergence result see theorem 4 1 the numerical results demonstrate that higher order correctors are essential for solving maxwell s equations in composite materials

brief paper iterative learning control for uncertain systems robust monotonic convergence analysis in this paper we present a novel robust monotonic convergence rmc analysis approach for finite time interval iterative learning control ilc for uncertain systems for that purpose a finite time interval model for uncertain systems is introduced this model is subsequently used in an rmc analysis based on m analysis as a result we can handle additive and multiplicative uncertainty models in the rmc problem formulation analyze rmc of linear time invariant mimo systems controlled by any linear trial invariant ilc controller and formulate additional straightforward rmc conditions for ilc controlled systems to illustrate the derived results we analyze the rmc properties of linear quadratic lq norm optimal ilc

maximum rank distance codes as space time codes the critical design criterion for space time codes in asymptotically good channels is the minimum rank between codeword pairs rank codes are a two dimensional matrix code construction where by the rank is the metric of merit we look at the application of rank codes to space time code design in particular we provide construction methods of full rank codes over different complex signal constellations for arbitrary numbers of antennas and codeword periods we also derive a singleton type bound on the rate of a code for the rank metric and we show that rank codes satisfy this bound with equality

reiterated homogenization of the vector potential formulation of a magnetostatic problem in anisotropic composite media abstract we present an explicit characterization of the effective coefficients of a family of boundary value problems with multiscale periodic oscillatory coefficients which correspond to the vector potential formulation of a magnetostatic problem in anisotropic composite media with periodic microstructures moreover we study the convergence of sequences of multiscale periodic integral functionals depending on the curl of divergence free fields applying the properties of multiscale young measures associated with sequences of divergence free fields

controllability of fractional evolution nonlocal impulsive quasilinear delay integro differential systems in this work the controllability result of a class of fractional evolution nonlocal impulsive quasilinear delay integro differential systems in a banach space has been established by using the theory of fractional calculus fixed point technique and also we introduced a new concept called a u resolvent family as an application that illustrates the abstract results an example is given

coding for errors and erasures in random network coding the problem of error control in random linear network coding is considered a ldquononcoherentrdquo or ldquochannel obliviousrdquo model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic motivated by the property that linear network coding is vector space preserving information transmission is modeled as the injection into the network of a basis for a vector space v and the collection by the receiver of a basis for a vector space u a metric on the projective geometry associated with the packet space is introduced and it is shown that a minimum distance decoder for this metric achieves correct decoding if the dimension of the space v capu is sufficiently large if the dimension of each codeword is restricted to a fixed integer the code forms a subset of a finite field grassmannian or equivalently a subset of the vertices of the corresponding grassmann graph sphere packing and sphere covering bounds as well as a generalization of the singleton bound are provided for such codes finally a reed solomon like code construction related to gabidulin s construction of maximum rank distance codes is described and a sudan style ldquolist 1rdquo minimum distance decoding algorithm is provided

packing and covering properties of rank metric codes this paper investigates packing and covering properties of codes with the rank metric first we investigate packing properties of rank metric codes then we study sphere covering properties of rank metric codes derive bounds on their parameters and investigate their asymptotic covering properties

a class of boundary value problems for first order impulsive integro differential equations with deviating arguments this paper is concerned with a class of boundary value problems for nonlinear mixed impulsive integro differential equations with deviating arguments we establish a new comparison principle and use the method of upper and lower solutions together with the monotone iterative technique under suitable conditions we obtain the existence results of extremal solutions for the problems an example is also given to illustrate our results

iterative learning control with initial state learning for fractional order nonlinear systems this paper presents a p type iterative learning control ilc scheme with initial state learning for a class of a 0 a 1 fractional order nonlinear systems by introducing the l norm and using a generalized gronwall inequality the sufficient condition for the robust convergence of the tracking errors with respect to initial positioning errors under p type ilc is obtained based on this convergence condition the learning gain of the initial learning and input learning updating law can be determined unlike the existing methods the ilc scheme will not fix the initial value on the expected condition at the beginning of each iteration finally the validity of the methods are verified by a numerical example

an iterative learning control approach for linear systems with randomly varying trial lengths this technical note addresses an iterative learning control ilc design problem for discrete time linear systems where the trial lengths could be randomly varying in the iteration domain an ilc scheme with an iteration average operator is introduced for tracking tasks with non uniform trial lengths which thus mitigates the requirement on classic ilc that all trial lengths must be identical in addition the identical initialization condition can be absolutely removed the learning convergence condition of ilc in mathematical expectation is derived through rigorous analysis as a result the proposed ilc scheme is applicable to more practical systems in the end two illustrative examples are presented to demonstrate the performance and the effectiveness of the averaging ilc scheme for both time invariant and time varying linear systems

a rank metric approach to error control in random network coding the problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of rotter and kschischang a large class of constant dimension subspace codes is investigated it is shown that codes in this class can be easily constructed from rank metric codes while preserving their distance properties moreover it is shown that minimum distance decoding of such subspace codes can be reformulated as a generalized decoding problem for rank metric codes where partial information about the error is available this partial information may be in the form of erasures knowledge of an error location but not its value and deviations knowledge of an error value but not its location taking erasures and deviations into account when they occur strictly increases the error correction capability of a code if mu erasures and delta deviations occur then errors of rank t can always be corrected provided that 2t les d 1 mu delta where d is the minimum rank distance of the code for gabidulin codes an important family of maximum rank distance codes an efficient decoding algorithm is proposed that can properly exploit erasures and deviations in a network coding application where n packets of length m over f q are transmitted the complexity of the decoding algorithm is given by o dm operations in an extension field f qn

an average operator based pd type iterative learning control for variable initial state error this note studies the effect of variable initial state error in iterative learning control ilc systems and proposes a new ilc algorithm based on an average operator then it is shown that when the proposed algorithm is applied to linear time invariant lti systems the effect of the initial state error can be exactly estimated under a specific condition while the existing algorithms guarantee only the boundness of the error or the convergence from stochastic point of view to show the validity of the proposed algorithm a numerical example is given

multiscale young measures in almost periodic homogenization and applications we prove the existence of multiscale young measures associated with almost periodic homogenization we give applications of this tool in the homogenization of nonlinear partial differential equations with an almost periodic structure such as scalar conservation laws nonlinear transport equations hamilton jacobi equations and fully nonlinear elliptic equations motivated by the application in nonlinear transport equations we also prove basic results on flows generated by lipschitz almost periodic vector fields which are of interest in their own in our analysis an important role is played by the so called bohr compactification mathbb g n of mathbb r n this is a natural parameter space for the young measures our homogenization results provide also the asymptotic behavior for the whole set of mathbb g n translates of the solutions which is in the spirit of recent studies on the homogenization of stationary ergodic processes

exponential stability of the exact solutions and the numerical solutions for a class of linear impulsive delay differential equations this paper is concerned with exponential stability of a class of linear impulsive delay differential equations iddes exponential stability of this kind of equations is studied by the properties of delay differential equations ddes without impulsive perturbations when different delay differential equations ddes without impulsive perturbations are chosen different sufficient conditions for exponential stability of the linear impulsive delay differential equations iddes are provided numerical methods for this kind of equations are constructed the convergence and exponential stability of the numerical solutions are studied and some experiments are given

on the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths abstract in this paper a new class of stochastic impulsive differential equations involving bernoulli distribution is introduced for tracking the random discontinuous trajectory a modified tracking error associated with a piecewise continuous variable by zero order holder is defined in the sequel a new random ilc scheme by adopting global and local iteration average operators is designed too sufficient conditions to guarantee the convergence of modified tracking error are obtained by using the tools of mathematical analysis via an impulsive gronwall inequality finally two illustrative examples are presented to demonstrate the performance and the effectiveness of the averaging ilc scheme to track the random discontinuous trajectory