the constant variation formulae for singular fractional differential systems with delay abstract this paper considers the caputo singular fractional differential systems with delay and the riemann liouville singular fractional differential systems with delay a new function is defined by the d inverse matrix and function two fundamental solutions are given the constant variation formulae for singular fractional differential systems with delay are obtained

the general vector solutions of such linear coupled matrix fractional differential equations by using kronecker structures in this paper we demonstrate how kronecker products and vector operator can be combined beautifully to solve and fit the non homogenous linear matrix fractional differential equations and such coupled linear matrix fractional differential equations the way exists which transform the given coupled matrix fractional differential equations by using kronecker structures into forms for which solutions may be readily computed some important and interesting special cases of the general system of non homogeneous linear matrix fractional differential equations of fractional order are also considered which includes the non homogenous linear fractional dynamical system with delays in control finally a brief comparison between the fractional and integer order vector solutions and some examples are given to illustrate our new approaches

large cone nonnegative matrix factorization nonnegative matrix factorization nmf has been greatly popularized by its parts based interpretation and the effective multiplicative updating rule for searching local solutions in this paper we study the problem of how to obtain an attractive local solution for nmf which not only fits the given training data well but also generalizes well on the unseen test data based on the geometric interpretation of nmf we introduce two large cone penalties for nmf and propose large cone nmf lcnmf algorithms compared with nmf lcnmf will obtain bases comprising a larger simplicial cone and therefore has three advantages 1 the empirical reconstruction error of lcnmf could mostly be smaller 2 the generalization ability of the proposed algorithm is much more powerful and 3 the obtained bases of lcnmf have a low overlapping property which enables the bases to be sparse and makes the proposed algorithms very robust experiments on synthetic and real world data sets confirm the efficiency of lcnmf

quasilinearization technique for solving nonlinear riemann liouville fractional order problems abstract in this work we deal with the quasilinearization technique for a class of nonlinear riemann liouville fractional order two point boundary value problems using quasilinearization technique we construct a monotone sequence of approximate solutions which has quadratic convergence to the unique solution of the original problem and establish the corresponding convergence estimates moreover the performance of the technique is examined through a numerical example which shows that our regularization method is available and stable

robust stability analysis of fractional order uncertain singular nonlinear system with external disturbance this paper investigates robust stability for fractional order fo singular nonlinear systems the fo system is disturbed by external uncertainty and disturbance a central analysis technique is enabled by proposing a fundamental boundedness lemma for the first time this lemma is used for robust stability analysis of fo systems especially for mittag leffler stability analysis of fo nonlinear systems more importantly how to obtain a more accurate bound is given to reduce conservative an fo proportional derivative pd controller is proposed to normalize the fo singular system furthermore a criterion for stability of the normalized fo nonlinear systems is provided by linear matrix inequalities lmis finally two illustrative simulation examples are presented to illustrate effectiveness of the proposed stability notion

asymptotic stability of q h fractional difference equations abstract asymptotic stability of linear nabla riemann liouville q h fractional difference equation is investigated in this paper a liapunov functional is constructed for the fractional difference equation the sufficient condition for the asymptotic stability of considered equations is proposed the results are illustrated with the corresponding numerical examples

calculus of variations on time scales with nabla derivatives abstract we prove a necessary optimality condition of the euler lagrange type for variational problems on time scales involving nabla derivatives of higher order the proof is done using a new and more general fundamental lemma of the calculus of variations on time scales

discriminative transfer subspace learning via low rank and sparse representation in this paper we address the problem of unsupervised domain transfer learning in which no labels are available in the target domain we use a transformation matrix to transfer both the source and target data to a common subspace where each target sample can be represented by a combination of source samples such that the samples from different domains can be well interlaced in this way the discrepancy of the source and target domains is reduced by imposing joint low rank and sparse constraints on the reconstruction coefficient matrix the global and local structures of data can be preserved to enlarge the margins between different classes as much as possible and provide more freedom to diminish the discrepancy a flexible linear classifier projection is obtained by learning a non negative label relaxation matrix that allows the strict binary label matrix to relax into a slack variable matrix our method can avoid a potentially negative transfer by using a sparse matrix to model the noise and thus is more robust to different types of noise we formulate our problem as a constrained low rankness and sparsity minimization problem and solve it by the inexact augmented lagrange multiplier method extensive experiments on various visual domain adaptation tasks show the superiority of the proposed method over the state of the art methods the matlab code of our method will be publicly available at http www yongxu org lunwen html

existence and uniqueness of holomorphic solutions for fractional cauchy problem by employing majorant functions the existence and uniqueness of holomorphic solutions to nonlinear fractional partial differential equations the cauchy problems are introduced furthermore the analytic continuation of solutions is studied

the controllability of fractional control systems with control delay this paper discusses the controllability of fractional control systems with control delay we firstly give the solution expression for fractional control systems with control delay then give the necessary and sufficient conditions for the controllability of fractional control systems with control delay

optimal variable order fractional pid controllers for dynamical systems abstract this paper studies the design of variable order fractional proportional integral derivative vfpid controllers for linear dynamical systems for this purpose a technique to discretize fractional differential equations with variable order operators is proposed the resulting model and a particle swarm optimization algorithm are used to search for the optimal parameters of the vfpid controllers three examples illustrate the performance of the closed loop system under the action of the vfpid and compare it with the results obtained by means of the pid and fractional pid fpid furthermore time dependent parameters are also tested showing that the vfpid yields a superior performance

some remarks on g functors and the brauer morphism keywords finite group g functor restriction maps transfer maps mackey s formula frobenius reciprocity brauer morphism defect base nilpotent ideal burnside ring idempotent formula induction formula symmetric g functor character ring green ring reference ctg article 1988 003 record created on 2008 12 16 modified on 2017 05 12

elementary construction of the quiver of the mackey algebra for groups with cyclic normal p sylow subgroup through recent investigations of j thrvenaz and p j webb 4 8 the concept of mackey functors being originally introduced by j a green 3 and studied by a w m dress t yoshida h sasaki and others has regained considerable interest an important reason for this is i new interpretation of a conjecture of aiperin in terms of mackey functors in 8 using results of 7 given a commutative ring 4 by its definition it is immediate that the category of mackey functors over 4 for a finite group g is an abelian category even that it is the category of 4 representations of a quiver with relations 8 namely a mackey functor for g associates to each subgroup of g a module and moreover certain linear maps between these spaces satisfying certain relations involving g thus a mackey functor for g can be interpreted as a module over a path algebra of a quiver with relations the so called mackey algebra 6 9 the main difference to the usual representation theory of quivers and thus the main problem of interpreting mackey functors as such representations lies in the fact that the maps between these spaces do not lie in the radical of this path algebra and this path algebra is not basic in general so it is a natural problem to compute the quiver in the usual sense for the category of mackey functors for a group g in particular in the case where 4 is a field of characteristic p dividing the order of g the question of when there are up to isomorphism only finitely many indecomposable mackey functors for g led to recent investigations of thrvenaz and webb solving this problem in general and it also led to the investigations we present in this paper which presents in a special situation namely if g has a normal cyclic p sylow subgroup an explicit computation of the quiver of the now so called mackcy algebra by definition the mackey algebra is the basic algebra whose category of finitely generated

numerical simulation of the fractional bloch equations in physics and chemistry specifically in nmr nuclear magnetic resonance or mri magnetic resonance imaging or esr electron spin resonance the bloch equations are a set of macroscopic equations that are used to calculate the nuclear magnetization m m x m y m z as a function of time when relaxation times t 1 and t 2 are present recently some fractional models have been proposed for the bloch equations however effective numerical methods and supporting error analyses for the fractional bloch equation fbe are still limited in this paper the time fractional bloch equations tfbe and the anomalous fractional bloch equations afbe are considered firstly we derive an analytical solution for the tfbe with an initial condition secondly we propose an effective predictor corrector method pcm for the tfbe and the error analysis for pcm is investigated furthermore we derive an effective implicit numerical method inm for the anomalous fractional bloch equations afbe and the stability and convergence of the inm are investigated we prove that the implicit numerical method for the afbe is unconditionally stable and convergent finally we present some numerical results that support our theoretical analysis

automatic relevance determination in nonnegative matrix factorization with the beta divergence this paper addresses the estimation of the latent dimensionality in nonnegative matrix factorization nmf with the divergence the divergence is a family of cost functions that includes the squared euclidean distance kullback leibler kl and itakura saito is divergences as special cases learning the model order is important as it is necessary to strike the right balance between data fidelity and overfitting we propose a bayesian model based on automatic relevance determination ard in which the columns of the dictionary matrix and the rows of the activation matrix are tied together through a common scale parameter in their prior a family of majorization minimization mm algorithms is proposed for maximum a posteriori map estimation a subset of scale parameters is driven to a small lower bound in the course of inference with the effect of pruning the corresponding spurious components we demonstrate the efficacy and robustness of our algorithms by performing extensive experiments on synthetic data the swimmer dataset a music decomposition example and a stock price prediction task

on generalized linear regular delay systems in this paper we study generalized linear singular delay systems such a system is reduced to a suitable homogeneous one using the kronecker canonical form the latter is decomposed into five subsystems whose solutions are obtained moreover the form of the initial function is given so that the corresponding initial value problem is considered

integrated low rank based discriminative feature learning for recognition feature learning plays a central role in pattern recognition in recent years many representation based feature learning methods have been proposed and have achieved great success in many applications however these methods perform feature learning and subsequent classification in two separate steps which may not be optimal for recognition tasks in this paper we present a supervised low rank based approach for learning discriminative features by integrating latent low rank representation latlrr with a ridge regression based classifier our approach combines feature learning with classification so that the regulated classification error is minimized in this way the extracted features are more discriminative for the recognition tasks our approach benefits from a recent discovery on the closed form solutions to noiseless latlrr when there is noise a robust principal component analysis pca based denoising step can be added as preprocessing when the scale of a problem is large we utilize a fast randomized algorithm to speed up the computation of robust pca extensive experimental results demonstrate the effectiveness and robustness of our method

numerical algorithm to solve system of nonlinear fractional differential equations based on wavelets method and the error analysis in this paper a system of nonlinear fractional differential equations fdes are considered they have been solved by legendre wavelets method combining with its operational matrix however there are no articles about solving this system using wavelets method the main purpose of this technique is to transform the initial equations into a nonlinear system of algebraic equations which can be solved easily the convergence and error analysis are presented to show the correctness and feasibility of method proposed for solving the above mentioned problem finally the applicability and efficiency of the mentioned approach are demonstrated by three numerical examples

on a riemann liouville generalized taylor s formula in this paper a generalized taylor s formula of the kindfx j 0najx a j 1 1 tnx whereaj r x a 0 1 is established such expression is precisely the classical taylor s formula in the particular case 1 in addition detailed expressions fortn x andaj involving the riemann liouville fractional derivative and some applications are also given

controllability of second order infinite dimensional systems in the paper the approximate controllability of linear abstract second order infinite dimensional dynamical systems is considered it is proved using the frequency domain method that approximate controllability of second order system can be verified by the approximate controllability conditions for the corresponding simplified first order system general results are then applied for approximate controllability investigation of a vibratory dynamical system modeling flexible mechanical structure some special cases are also considered moreover remarks and comments on the relationships between different concepts of controllability are given the paper extends earlier results on approximate controllability of second order abstract dynamical systems

center stable manifold for planar fractional damped equations in this paper we discuss the existence of a center stable manifold for planar fractional damped equations by constructing a suitable lyapunovperron operator via giving asymptotic behavior of mittagleffler function we obtain an interesting center stable manifold theorem finally an example is provided to illustrate the result

generalized fractional controller for singular systems of differential equations abstract in this article we consider a class of singular linear systems of first order and introduce a generalized fractional order feedback controller of caputo type the closed loop system in question is a singular system of differential equations having both first and fractional order derivatives we provide a comprehensive theory for the existence and uniqueness of solutions as well as for the stability of the system with inclusion of the fractional order controller an example of a singular system with a fractional order proportional integral controller as well as an example on a 3 bus power system with inclusion of a fractional order damping controller is given to illustrate our theory

robust recovery of subspace structures by low rank representation in this paper we address the subspace clustering problem given a set of data samples vectors approximately drawn from a union of multiple subspaces our goal is to cluster the samples into their respective subspaces and remove possible outliers as well to this end we propose a novel objective function named low rank representation lrr which seeks the lowest rank representation among all the candidates that can represent the data samples as linear combinations of the bases in a given dictionary it is shown that the convex program associated with lrr solves the subspace clustering problem in the following sense when the data is clean we prove that lrr exactly recovers the true subspace structures when the data are contaminated by outliers we prove that under certain conditions lrr can exactly recover the row space of the original data and detect the outlier as well for data corrupted by arbitrary sparse errors lrr can also approximately recover the row space with theoretical guarantees since the subspace membership is provably determined by the row space these further imply that lrr can perform robust subspace clustering and error correction in an efficient and effective way

on time optimal control of fractional order systems abstract fractional optimal control problem is treated from convex analytical viewpoint sufficient conditions for time optimal control similar to that of pontryagin s maximum principle are obtained for fractional order systems in the sense of riemann liouville and caputo theoretical results are supported by examples

on cauchy problem for nonlinear fractional differential equation with random discrete data abstract this paper is concerned with finding the solution u x t of the cauchy problem for nonlinear fractional elliptic equation with perturbed input data this study shows that our forward problem is severely ill posed in sense of hadamard for this ill posed problem the trigonometric of non parametric regression associated with the truncation method is applied to construct a regularized solution under prior assumptions for the exact solution the convergence rate is obtained in both l2 and h q for q 0 norm moreover the numerical example is also investigated to justify our results

graph regularized nonnegative matrix factorization for data representation matrix factorization techniques have been frequently applied in information retrieval computer vision and pattern recognition among them nonnegative matrix factorization nmf has received considerable attention due to its psychological and physiological interpretation of naturally occurring data whose representation may be parts based in the human brain on the other hand from the geometric perspective the data is usually sampled from a low dimensional manifold embedded in a high dimensional ambient space one then hopes to find a compact representation which uncovers the hidden semantics and simultaneously respects the intrinsic geometric structure in this paper we propose a novel algorithm called graph regularized nonnegative matrix factorization gnmf for this purpose in gnmf an affinity graph is constructed to encode the geometrical information and we seek a matrix factorization which respects the graph structure our empirical study shows encouraging results of the proposed algorithm in comparison to the state of the art algorithms on real world problems

the mobius transform effect in singular systems of differential equations abstract the main objective of this article is to provide a link between the solutions of an initial value problem of a linear singular system of differential equations and the solutions of its proper m systems i e systems that appear after applying the generalized mobius transform to the pencil of the original singular system prime system firstly we prove that by using the pencil of the prime system we can study the existence and uniqueness of solutions of its proper m systems moreover these solutions can be explicitly represented without resorting to any further processes of computations finally numerical examples are given to illustrate our theory

a practical formula of solutions for a family of linear non autonomous fractional nabla difference equations abstract in this article we focus on a generalized problem of linear non autonomous fractional nabla difference equations firstly we define the equations and describe how this family of problems covers other linear fractional difference equations that appear in the literature then by using matrix theory we provide a new practical formula of solutions for these type of equations finally numerical examples are given to justify our theory

convergence analysis of alternating direction method of multipliers for a family of nonconvex problems the alternating direction method of multipliers admm is widely used to solve large scale linearly constrained optimization problems convex or nonconvex in many engineering fields however there is a general lack of theoretical understanding of the algorithm when the objective function is nonconvex in this paper we analyze the convergence of the admm for solving certain nonconvex consensus and sharing problems we show that the classical admm converges to the set of stationary solutions provided that the penalty parameter in the augmented lagrangian is chosen to be sufficiently large for the sharing problems we show that the admm is convergent regardless of the number of variable blocks our analysis does not impose any assumptions on the iterates generated by the algorithm and is broadly applicable to many admm variants involving proximal update rules and various flexible block selection rules

some results of the degenerate fractional differential system with delay an analytic study on linear systems of degenerate fractional differential equations with constant coefficients is presented we discuss the existence and uniqueness of solutions for the initial value problem of linear degenerate fractional differential systems the exponential estimation of the degenerate fractional differential system with delay and sufficient conditions for the finite time stability for the system are obtained finally an example is provided to illustrate the effectiveness of the presented analytical approaches

caputo and related fractional derivatives in singular systems abstract by using the caputo c fractional derivative and two recently defined alternative versions of this derivative the caputo fabrizio cf and the atangana baleanu ab fractional derivative firstly we focus on singular linear systems of fractional differential equations with constant coefficients that can be non square matrices or square singular we study existence of solutions and provide formulas for the case that there do exist solutions then we study the existence of unique solution for given initial conditions several numerical examples are given to justify our theory

a note on property of the mittag leffler function abstract recently the authors have found in some publications that the following property 0 1 of mittag leffler function is taken for granted and used to derive other properties 0 1 e a t s e a t e a s t s 0 where a is a real constant and 0 in this note it is proved that the above property is unavailable unless 1 or a 0 moreover a new equality on e a t is developed whose limit state as 1 is just the property 0 1

nuclear norm based matrix regression with applications to face recognition with occlusion and illumination changes recently regression analysis has become a popular tool for face recognition most existing regression methods use the one dimensional pixel based error model which characterizes the representation error individually pixel by pixel and thus neglects the two dimensional structure of the error image we observe that occlusion and illumination changes generally lead approximately to a low rank error image in order to make use of this low rank structural information this paper presents a two dimensional image matrix based error model namely nuclear norm based matrix regression nmr for face representation and classification nmr uses the minimal nuclear norm of representation error image as a criterion and the alternating direction method of multipliers admm to calculate the regression coefficients we further develop a fast admm algorithm to solve the approximate nmr model and show it has a quadratic rate of convergence we experiment using five popular face image databases the extended yale b ar eurecom multi pie and frgc experimental results demonstrate the performance advantage of nmr over the state of the art regression based methods for face recognition in the presence of occlusion and illumination variations

bayesian optimal control for a non autonomous stochastic discrete time system the main objective of this article is to develop bayesian optimal control for a class of non autonomous linear stochastic discrete time systems by taking into consideration that the disturbances in the system are given by a random vector with components belonging to an exponential family with a natural parameter we prove that the bayes control is the solution of a linear system of algebraic equations for the case that this linear system is singular we apply optimization techniques to gain the bayesian optimal control furthermore we extend these results to generalized linear stochastic systems of difference equations and provide the bayesian optimal control for the case where the coefficients of this type of systems are non square matrices

robust transfer metric learning for image classification metric learning has attracted increasing attention due to its critical role in image analysis and classification conventional metric learning always assumes that the training and test data are sampled from the same or similar distribution however to build an effective distance metric we need abundant supervised knowledge i e side label information which is generally inaccessible in practice because of the expensive labeling cost in this paper we develop a robust transfer metric learning rtml framework to effectively assist the unlabeled target learning by transferring the knowledge from the well labeled source domain specifically rtml exploits knowledge transfer to mitigate the domain shift in two directions i e sample space and feature space in the sample space domain wise and class wise adaption schemes are adopted to bridge the gap of marginal and conditional distribution disparities across two domains in the feature space our metric is built in a marginalized denoising fashion and low rank constraint which make it more robust to tackle noisy data in reality furthermore we design an explicit rank constraint regularizer to replace the rank minimization np hard problem to guide the low rank metric learning experimental results on several standard benchmarks demonstrate the effectiveness of our proposed rtml by comparing it with the state of the art transfer learning and metric learning algorithms

on systems of linear fractional differential equations with constant coefficients abstract this paper deals with the study of linear systems of fractional differential equations such as the following system 1 y a x y b x where y is the riemann liouville or the caputo fractional derivative of order 0 2 a x a 11 x a 1 n x a n 1 x a nn x b x b 1 x b n x are matrices of known real functions in a way analogous to the usual case we show how a generalized matrix exponential function and certain fractional green function in connection with the mittag leffler type functions would allow us to obtain an explicit representation of the general solution to the system 1 when a is a constant matrix

dimensionality dependent generalization bounds for k dimensional coding schemes the k dimensional coding schemes refer to a collection of methods that attempt to represent data using a set of representative k dimensional vectors and include nonnegative matrix factorization dictionary learning sparse coding k means clustering and vector quantization as special cases previous generalization bounds for the reconstruction error of the k dimensional coding schemes are mainly dimensionality independent a major advantage of these bounds is that they can be used to analyze the generalization error when data are mapped into an infinite or high dimensional feature space however many applications use finite dimensional data features can we obtain dimensionality dependent generalization bounds for k dimensional coding schemes that are tighter than dimensionality independent bounds when data are in a finite dimensional feature space yes in this letter we address this problem and derive a dimensionality dependent generalization bound for k dimensional coding schemes by bounding the covering number of the loss function class induced by the reconstruction error the bound is of order where m is the dimension of features k is the number of the columns in the linear implementation of coding schemes and n is the size of sample when n is finite and when n is infinite we show that our bound can be tighter than previous results because it avoids inducing the worst case upper bound on k of the loss function the proposed generalization bound is also applied to some specific coding schemes to demonstrate that the dimensionality dependent bound is an indispensable complement to the dimensionality independent generalization bounds

structure constrained low rank representation benefiting from its effectiveness in subspace segmentation low rank representation lrr and its variations have many applications in computer vision and pattern recognition such as motion segmentation image segmentation saliency detection and semisupervised learning it is known that the standard lrr can only work well under the assumption that all the subspaces are independent however this assumption cannot be guaranteed in real world problems this paper addresses this problem and provides an extension of lrr named structure constrained lrr sc lrr to analyze the structure of multiple disjoint subspaces which is more general for real vision data we prove that the relationship of multiple linear disjoint subspaces can be exactly revealed by sc lrr with a predefined weight matrix as a nontrivial byproduct we also illustrate that sc lrr can be applied for semisupervised learning the experimental results on different types of vision problems demonstrate the effectiveness of our proposed method

comparison theorems and distributions of solutions to uncertain fractional difference equations abstract this paper primarily focuses on the distributions of the solutions for uncertain fractional difference equations by comparison theorems first comparison theorems are obtained for the fractional difference equations involving riemann liouville type then the concepts of symmetrical uncertain variable and path to an uncertain fractional difference equation are introduced moreover the relations between the solutions for the uncertain fractional difference equations with symmetrical uncertain variables and their paths are established and verified by the obtained comparison theorems finally the numerical results of the uncertainty distributions of solutions for the uncertain fractional difference equations are proposed considering the relations between their solutions and paths

algorithm dependent generalization bounds for multi task learning often tasks are collected for multi task learning mtl because they share similar feature structures based on this observation in this paper we present novel algorithm dependent generalization bounds for mtl by exploiting the notion of algorithmic stability we focus on the performance of one particular task and the average performance over multiple tasks by analyzing the generalization ability of a common parameter that is shared in mtl when focusing on one particular task with the help of a mild assumption on the feature structures we interpret the function of the other tasks as a regularizer that produces a specific inductive bias the algorithm for learning the common parameter as well as the predictor is thereby uniformly stable with respect to the domain of the particular task and has a generalization bound with a fast convergence rate of order mathcal o 1 n where n is the sample size of the particular task when focusing on the average performance over multiple tasks we prove that a similar inductive bias exists under certain conditions on the feature structures thus the corresponding algorithm for learning the common parameter is also uniformly stable with respect to the domains of the multiple tasks and its generalization bound is of the order mathcal o 1 t where t is the number of tasks these theoretical analyses naturally show that the similarity of feature structures in mtl will lead to specific regularizations for predicting which enables the learning algorithms to generalize fast and correctly from a few examples

remarks on fractional derivatives in this paper we further discuss the properties of three kinds of fractional derivatives the grunwald letnikov derivative the riemann liouville derivative and the caputo derivative especially we compare the riemann liouville derivative with the caputo derivative and sequential property of the caputo derivative is also derived which is helpful in translating the higher fractional order differential systems into lower ones besides we also compare the riemann liouville derivative and the caputo derivative with the classical derivative

learning fast low rank projection for image classification rooted in a basic hypothesis that a data matrix is strictly drawn from some independent subspaces the low rank representation lrr model and its variations have been successfully applied in various image classification tasks however this hypothesis is very strict to the lrr model as it cannot always be guaranteed in real images moreover the hypothesis also prevents the sub dictionaries of different subspaces from collaboratively representing an image fortunately in supervised image classification low rank signal can be extracted from the independent label subspaces ils instead of the independent image subspaces iis therefore this paper proposes a projective low rank representation plr model by directly training a projective function to approximate the lrr derived from the labels to the best of our knowledge plr is the first attempt to use the ils hypothesis to relax the rigorous iis hypothesis in the lrr models we further prove a low rank effect that the representations learned by plr have high intraclass similarities and large interclass differences which are beneficial to the classification tasks the effectiveness of our proposed approach is validated by the experimental results on three databases

on stable manifolds for planar fractional differential equations in this paper we establish a local stable manifold theorem near a hyperbolic equilibrium point for planar fractional differential equations the construction of this stable manifold is based on the associated lyapunov perron operator an example is provided to illustrate the result

numerical solution of fractional integro differential equations by a hybrid collocation method fractional integro differential equations have been recently solved by many methods such as adomian decomposition method differential transform method collocation method and taylor expansion approach in this paper a hybrid collocation method is used which combines a non polynomial collocation used on the first subinterval and graded piecewise polynomial collocation used on the rest of the interval a theoretical analysis for the convergence order of the method is presented some numerical examples are given which confirm the theoretical results

on non homogeneous singular systems of fractional nabla difference equations in this article we study the initial value problem of a class of non homogeneous singular systems of fractional nabla difference equations whose coefficients are constant matrices by taking into consideration the cases that the matrices are square with the leading coefficient singular non square and square with a matrix pencil which has an identically zero determinant we provide necessary and sufficient conditions for the existence and uniqueness of solutions more analytically we study the conditions under which the system has unique infinite and no solutions furthermore we provide a formula for the case of the unique solution finally numerical examples are given to justify our theory

modeling with fractional difference equations abstract in this paper we develop some basics of discrete fractional calculus such as leibniz rule and summation by parts formula we define simplest discrete fractional calculus of variations problem and derive euler lagrange equation we introduce and solve gompertz fractional difference equation for tumor growth models

perturbation and robust stability of autonomous singular linear matrix difference equations in the perturbation theory of linear matrix difference equations it is well known that the theory of finite and infinite elementary divisors of regular matrix pencils is complicated by the fact that arbitrarily small perturbations of the pencil can cause them to disappear in this paper the perturbation theory of complex weierstrass canonical form for regular matrix pencils is investigated by using matrix pencil theory and the weierstrass canonical form of the pencil we obtain bounds for the finite elementary divisors of a perturbed pencil moreover we study robust stability of a class of linear matrix difference equations of first and higher order whose coefficients are square constant matrices

defect theory for maximal ideals and simple functors keywords green functor source green correspondence maximal ideal finite group fixed point algebra defect theory for g algebras defect group restriction transfer conjugation defect pair functorial ideal simple g functors reference ctg article 1991 002doi 10 1016 0021 8693 91 90168 8 record created on 2008 12 16 modified on 2017 05 12