rational singularities and almost split sequences the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the finite dimen sional representations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules of the quotient singularity associated with g over the past decade almost split sequences have been playing an increasingly important role in the representation theories of finite dimensional algebras and classical orders see 5 and 3 8 for basic existence theorems in these contexts while they have been known for some time to exist in higher dimensional situations 3 it has not been at all clear how they related to singularity theory if at all the main aim of this paper is to relate almost split sequences to singularity theory by showing that the mckay quiver built from the fimte dimensional complex represen tations of a finite subgroup g of gl 2 c where c is the complex numbers is isomorphic to the ar quiver of the reflexive modules over the quotient singularity r associated with g as in the case of finite dimensional algebras the ar quiver of reflexive r modules is defined in terms of the almost split sequences of reflexive r modules in the case g c sl 2 c mckay observed that the underlying graph of the mckay quiver with the trivial module removed is isomorphic to the desingulariza tion graph of the associated singularity various explanations of this phenomenon have been given by knorrer gonzalez sprinberg verdier 6 and artin verdier 1 which along the way have established most explicitly in 1 a natural one to one correspondence between the indecomposable reflexive r modules and the nodes of the desingularization graph but why the almost split sequences describe the edge of the desingularization graph still remains to be explained an effort has been made to make this paper as self contained as possible in particular no prior knowledge of almost split sequences is required before describing the contents of the six sections of this paper we fix some notation throughout this paper g is a finite group k an algebraically closed field of characteristic not dividing the order of g and v a two dimensional k representation of g setting s k x y the k algebra of formal power series the two dimen sional representation v gives a linear action of g on s as a group of k algebra automorphisms we denote by s g the skew group ring given by this action

maximal cohen macaulay modules over the cone of an elliptic curve let r k y1 y2 y3 f f y13 y23 y33 where k is an algebraically closed field with chark 3 using atiyah bundle classification over elliptic curves we describe the matrix factorizations of the graded indecomposable reflexive r modules equivalently we describe explicitly the indecomposable bundles over the projective curve v f pk2 using the fact that over the completion r of r every reflexive module is gradable we obtain a description of the maximal cohen macaulay modules over r k y1 y2 y3 f

indecomposable cohen macaulay modules and their multiplicities the main aim of this paper is to find a large class of rings for which there are indecomposable maximal cohen macaulay modules of arbitrary high multiplicity or rank in the case of domains

the classification of homogeneous cohen macaulay rings of finite representation type in this paper we classify the homogeneous cohen macaulay rings which are of finite representation type that is the cohen macaulay rings which are positively graded and generated in degree 1 with an algebraically closed field of characteristic 0 in degree 0 and which have up to isomorphisms and shifts in the grading only a finite number of indecomposable maximal cohen macaulay modules mcm modules our main contribution is to show that a homogeneous cohen macaulay ring of finite representation type and dimension 2 must have minimal multiplicity putting this together with previous results of auslander auslander reiten buchweitz greuel schreyer greuel kn6rrer and solberg we obtain

integral mappings and the principle of local reflexivity for noncommutative l 1 spaces the operator space analogue of the strong form of the principle of local re exivity is shown to hold for any von neumann algebra predual and thus for any c algebraic dual this is in striking contrast to the situation for c algebras since for example k h does not have that property the proof uses the kaplansky density theorem together with a careful analysis of two notions of integrality for mappings of operator spaces

impulsive delay differential inequality and stability of neural networks in this article a generalized model of neural networks involving time varying delays and impulses is considered by establishing the delay differential inequality with impulsive initial conditions and using the properties of m cone and eigenspace of the spectral radius of nonnegative matrices some new sufficient conditions for global exponential stability of impulsive delay model are obtained the results extend and improve the earlier publications an example is given to illustrate the theory

boundedness and exponential stability for nonautonomous rcnns with distributed delays some sufficient conditions for the ultimate boundedness and global exponential stability of a class of nonautonomous reaction diffusion cellular neural networks rcnns with distributed delays are obtained by means of the lyapunov functional method without assuming that the activation functions f i j l are bounded the results extend and improve the earlier publications

global stability analysis in delayed cellular neural networks in this paper the author analyzes further problems of global stability for a class of cellular neural networks with delays by means of the lyapunov functional method inequalities a 2 b 2 g 2ab and a 3 b 3 c 3 g 3abc a b cg 0 analysis technique some stability criteria are obtained under more general conditions these criteria can be applied to design globally stable networks and thus have important significance in both theory and applications

curves on threefolds and a conjecture of griffiths harris we prove that any arithmetically gorenstein curve on a smooth general hypersurface x subset mathbb p 4 of degree at least 6 is a complete intersection this gives a characterisation of complete intersection curves on general type hypersurfaces in mathbb p 4 we also verify that certain 1 cycles on a general quintic hypersurface are non trivial elements of the griffiths group

operator p compact mappings abstract we introduce the classes of operator p compact mappings and completely right p nuclear operators which are natural extensions to the operator space framework of their corresponding banach operator ideals we relate these two classes define natural operator space structures and study several properties of these ideals we show that the class of operator compact mappings in fact coincides with a notion already introduced by webster in the nineties in a very different language this allows us to provide an operator space structure to webster s class

rank 3 arithmetically cohen macaulay bundles over hypersurfaces abstract let x be a smooth projective hypersurface of dimension 5 and let e be an arithmetically cohen macaulay bundle on x of any rank we prove that e splits as a direct sum of line bundles if and only if h i x 2 e 0 for i 1 2 3 4 as a corollary this result proves a conjecture of buchweitz greuel and schreyer for the case of rank 3 arithmetically cohen macaulay bundles

the griffiths group of a general calabi yau threefold is not finitely generated j 2k 1 x h 2k 1 x c f k h 2k 1 x h 2k 1 x z where f k h 2k 1 x is the set ofclasses representable by a closed form in f k a 2k 1 x that is which is locally ofthe form i j i j dzi dzj with i j 2k 1 and i k griffiths 9 has defined the abel jacobi map k k x j 2k 1 x where k x is the group ofcodimension k algebraic cycles homologous to zero on x using the identification j 2k 1 x f n k 1 h 2n 2k 1 x h2n 2k 1 x z n dim x

classifying thick subcategories of the stable category of cohen macaulay modules abstract various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics in this paper as a higher dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite p group due to benson carlson and rickard we consider classifying thick subcategories of the stable category of cohen macaulay modules over a gorenstein local ring the main result of this paper yields a complete classification of the thick subcategories of the stable category of cohen macaulay modules over a local hypersurface in terms of specialization closed subsets of the prime ideal spectrum of the ring which are contained in its singular locus

homogeneous acm bundles on a grassmannian abstract in this paper we characterize homogeneous arithmetically cohen macaulay acm for short bundles on grassmannians gr k n and we construct irreducible families of indecomposable acm bundles on a grassmannian gr k n of arbitrary high rank and dimension as a consequence we prove that all grassmann varieties gr k n are of wild representation type unless the projective spaces gr 0 n and gr n 1 n and the hyperquadric gr 1 3 in p 5 which are of finite representation type

robust stability analysis for interval cohen grossberg neural networks with unknown time varying delays in this paper robust stability problems for interval cohen grossberg neural networks with unknown time varying delays are investigated using linear matrix inequality m matrix theory and halanay inequality techniques new sufficient conditions independent of time varying delays are derived to guarantee the uniqueness and the global robust stability of the equilibrium point of interval cohen grossberg neural networks with time varying delays all these results have no restriction on the rate of change of the time varying delays compared to some existing results these new criteria are less conservative and are more convenient to check two numerical examples are used to show the effectiveness of the present results

acm sheaves of pure rank two on reducible hyperquadrics abstract we classify a special type of arithmetically cohen macaulay sheaves of rank two on reducible and reduced quadric hypersurfaces as a consequence we show that a reducible and reduced quadric surface is of wild type

improved stability criteria for a class of neural networks with variable delays and impulsive perturbations this paper further investigates the problem of stability analysis for a kind of neural networks with variable delays and nonlinear impulses with the help of m matrix the homeomorphism theory some effective inequalities and analysis techniques certain novel criteria on the existence uniqueness and exponential stability of the equilibrium point have been established moreover these criteria possess adjustable real parameters which extend and improve many existing results in the literature in the end two numerical examples are provided to illustrate the validation of the theoretical results

on the griffiths group of the cubic sevenfold for x a smooth complex projective variety one of the important questions in algebraic geometry is the existence and the behaviour of non trivial subvarieties on x i e subvarieties that are not the intersection of x with a hypersurface in general the approach is as follows we let zd x be the free abelian group generated by the irreducible subvarieties of codimension d in x and introduce various equivalence relation on zd x the problem is then to understand the quotients of zd x under these relations the first and most obvious is homological equivalence but there are other interesting relations namely algebraic and rational equivalence the so called classical cases when d 1 divisors or d dimx 0 cycles are well understood but little is known in general in these cases homological and algebraic equivalence coincide and the group of cycles algebraically equivalent to zero modulo rational equivalence is isomorphic to an abelian variety one of the first result for general d is due to griffiths 8 who showed that homological and algebraic equivalence do not coincide if d 1 dimx the group of cycles on x of codimension d homologous to zero modulo algebraic equivalence is called the d th griffiths group of x and is denoted by gd x we will mainly consider the q vector space g6 x gd x q or equivalently the griffiths group modulo torsion clemens 5 then showed that g6 x can have infinite dimension over q clemens example is given by a general quintic hypersurface in p4 and this opened the way to the construction of other examples see 11 21 131 17

a remark on the approximation of p compact operators by finite rank operators abstract the banach operator ideal k p of p compact operators was introduced in p d sinha a k karn compact operators whose adjoints factor through subspaces of l p studia math 150 2002 17 33 let p 1 and let x be a closed subspace of an l p space we show that x has the approximation property if and only if for every banach space y the linear space f y x of finite rank operators is k p norm dense in k p y x i e x has the k p approximation property

the dimension of the category of maximal cohen macaulay modules over cohen macaulay local rings of dimension one abstract the dimension of triangulated categories was introduced by rouquier in 2008 5 as an analogue of this concept dao and takahashi defined the dimension of subcategories of abelian categories in 2014 3 the dimension of categories is an important invariant for studying the structure of both the category and the ground ring when it is the category of modules for example the dimension of the category of maximal cohen macaulay modules is zero if the ground ring has finite cohen macaulay representation type in this paper we focus on the category of maximal cohen macaulay modules over a hypersurface of dimension one

global exponential stability for shunting inhibitory cnns with delays in this paper the global exponential stability for shunting inhibitory cellular neural networks sicnns with delays is studied by constructing suitable lyapunov functionals and applying some critical analysis techniques the sufficient conditions guaranteeing the network s global exponential stability are obtained our results impose less restrictive conditions than those in the references moreover an example is given to illustrate the feasibility of the conditions in our results

on the noether lefschetz theorem and some remarks on codimension two cycles here of general moduli means that there is a countable union v of subvarieties of the space pn of surfaces of degree d in p3 such that the statement pie s z holds for s p n v noether it would seem stated this theorem but never completely proved it instead he gave a plausibility argument based on the following construction 1 let g be the hilbert scheme of curves of degree n and genus 9 in p3 l gr3 d l ip n the space of surfaces of degree d in p3 and zc t the incidence correspondence s o a c s c z s

the hilbert kunz functions of two dimensional rings of type ade abstract we compute the hilbert kunz functions of two dimensional rings of type ade by using representations of their indecomposable maximal cohen macaulay modules in terms of matrix factorizations and as first syzygy modules of homogeneous ideals

the dual space of l x y τp and the p approximation property we establish a representation of the dual space of l x y the space of bounded linear operators from a banach space x into a banach space y endowed with the topology p of uniform convergence on p compact subsets of x we apply this representation and solve the duality problem for the p approximation property p ap that is if the dual space x has the p ap then so does x however the converse does not hold in general we show that given 2 2p p 2 this subspace is the davie space in lq davie 1973 5 which does not have the approximation property it follows that for every 2 p there exists a banach space yp such that it has the p ap but its dual space yp fails to have the p ap we study the relation of the p ap with the denseness of finite rank operators in the topology p finally we introduce the p compact approximation property p cap and show for every 2 p that the davie space in c0 fails to have the p cap and also that a variant of the willis space willis 1992 17 has the p cap but it fails to have the p ap

the representation type of segre varieties abstract the goal of this work is to prove that all segre varieties n 1 n s p n n i 1 s n i 1 1 unless the quadric surface in p 3 support families of arbitrarily large dimension and rank of simple ulrich and hence acm vector bundles therefore they are of wild representation type

stability analysis on delayed cellular neural networks in this paper we study the problem of stability for the delayed cellular neural networks dcnn by making use of the lyapunov functional method the method of variation of constants and combining with the method of inequality analysis some criterions of global exponential stability and global asympototic stability for delayed cellular neural networks are established these criterions can be used to design globally stable networks and thus have important significance in both theory and applications

on global exponential stability of cellular neural networks with lipschitz continuous activation function and variable delays for a class of cellular neural networks cnns with variable delays this paper presents some new sufficient conditions for guaranteeing the global exponential stability these conditions are derived by using lyapunov functional method and combining with the inequality 3abc 0 technique furthermore these stability conditions depend only on the network s coefficients and are totally independent of the delays the results which do not require the cloning template to be symmetric are easy to use in network design compared with existing results our results are shown to be superior to other ones numerical examples are given to demonstrate the effectiveness of our results

delay dependent robust h control of uncertain linear systems with time varying delays abstract this paper proposes an optimization based approach to the robust h control problem of uncertain continuous or discrete time linear time invariant systems with different time varying delays in the state vector and control input of the dynamic equation and controlled output sufficient delay dependent conditions are derived for the control stabilization problem where both the size of the time varying delay and the size of its time derivative in the continuous time case play a crucial role for the closed loop stability with a guaranteed h performance index the solutions that are proposed for the h control problem are similar or less conservative when compared to other recent approaches

on a theorem of castelnuovo and the equations defining space curves introduction 491 w notat ion and conventions 493 w a regularity theorem 494 w a rationality theorem 498 w the existence of secant lines 501 w open problems 505 references 506

a construction of gorenstein projective modules abstract we determine all the gorenstein projective modules over the t 2 extension of a gorenstein algebra and over a m 0 b where a and b are self injective algebras and m is an a b bimodule with m a and m b projective using this we obtain a criterion for the cm finiteness of the t 2 extensions of cm finite gorenstein algebras as an application we get non trivial examples of cm finite gorenstein algebras

gorenstein singularity categories abstract the aim of this paper is to introduce gorenstein singularity category d g p s g b a as the verdier quotient of the gorenstein derived category d g p b a by the triangulated subcategory k b gp a the main result shows that if a is cm contravariantly finite abelian category then the gorenstein singularity category d g p s g b a is triangle equivalent to the gorenstein defect category d defect b a in the sense of 7 we point out several classes of algebras with non zero singularity categories but their gorenstein singularity categories become zero we also prove that a gorenstein derived equivalence between cm contravariantly finite abelian categories with infinite direct sums induces a gorenstein singular equivalence

dynamical behaviors analysis of memristor based fractional order complex valued neural networks with time delay abstract the robust stability for memristor based fractional order complex valued neural networks fcvnns with time delay is investigated here in complex plane by using the lyapunov method and under the sense of filippov solutions the existence of unique equilibrium and globally asymptotic stability for such neural networks nns have been obtained when the nonlinear complex valued activation functions could be split into two real and imaginary parts moreover locally asymptotic stability for such neural nns have been proposed when the nonlinear complex activation functions are bounded lastly three numerical examples are given to confirm the efficiency of theorems

fuzzy hyperbolic neural network with time varying delays in this paper the formula of fuzzy basis functions with time varying delays dfbf is firstly given based on the dfbf a novel fuzzy hyperbolic neural network with time varying delays dfhnn and its approximation properties are presented respectively by constructing an appropriate lyapunov krasovskii functional the stability of dfhnn is discussed and some new criteria concerning the global exponential stability of dfhnn are proposed finally three numerical examples are given to demonstrate the effectiveness of the obtained results

tensor products of operator spaces abstract in this paper we lay the foundations for a systematic study of tensor products of subspaces of c algebras to accomplish this various notions of duality are introduced and employed elementary proofs of the complete injectivity of the haagerup norm and of the extension theorem for completely bounded maps are given pisier s gamma norms are examined and found to be special cases of the haagerup norm we identify the greatest operator space cross norm and show that the spatial tensor norm is the least operator space cross norm in an appropriate sense indeed most of the elementary theory of banach space tensor norms generalizes to the category of operator spaces

the cohen macaulay type of cohen macaulay rings let r be an integrally closed complete local commutative noetherian cohen macaulay domain with maximal ideal m and r m k an algebraically closed field we abbreviate cohen macaulay to cm and denote by cm r the category of finitely generated cm modules in this paper we deal with the question of when r is of finite cm type that is has only a linite number of indecomposable cm modules if dim r 2 and r is a c algebra with r m the complex numbers then the r of finite cm type are exactly the fixed rings x y where x y are indeterminates and g c gl 2 c is a finite group acting on x y 17 3 151 for hypersurfaces in characteristic zero the r of finite cm type are exactly the simple hypersurface singularities in the sense of arnold 18 133 so in this case there is a nice connection with algebraic geometry when dim r 3 and r is not a hypersurface we know only two examples of finite cm type and they both have dimension 3 these are r kccxo x x y y iliwo xoy x y x y x yc and r2 k x y z lz where the generator of z acts by sending each variable to its negative and the characteristic of k is different from 2 each of these rings belongs naturally to a larger class r is a scroll of type 2 1 we define more generally a scroll of type m m see 141 and show that when dim r 2 3 then all other scrolls except type 1 l which is a hypersurface are of infinite cm type r is among the rings kccj xjig where the order of g c gl n k is invertible in k and n 2 3 and is the only one of finite cm type these are the main results of this paper to prove that the scroll of type 2 1 is of finite cm type we use the 1 oool 8708 89 7 50

arithmetically cohen macaulay bundles on complete intersection varieties of sufficiently high multidegree recently it has been proved that any arithmetically cohen macaulay acm bundle of rank two on a general smooth hypersurface of degree at least three and dimension at least four is a sum of line bundles when the dimension of the hypersurface is three a similar result is true provided the degree of the hypersurface is at least six we extend these results to complete intersection subvarieties by proving that any acm bundle of rank two on a general smooth complete intersection subvariety of sufficiently high multi degree and dimension at least four splits we also obtain partial results in the case of threefolds

globally exponential stability conditions for cellular neural networks with time varying delays in this paper the problems of global exponential stability for cellular neural networks cnn with time varying delays are studied several sufficient conditions guaranteeing the network s global exponential stability are established these results can easily be used to design and verify globally stable networks furthermore the results presented here are independent of the form of specific delays and have important significance in both theory and applications

on α nuclear operators with applications to vector valued function spaces abstract let x be a banach space and let y be a closed subspace of a banach space z let be a tensor norm our main result is as follows assume that x or z has the approximation property if there is a bounded linear extension operator from y to z then any bounded linear operator t x y is nuclear whenever t is nuclear from x to z using this result we characterize the space c p x respectively uc p x of continuous functions from a compact hausdorff space into a banach space x whose range is p compact respectively unconditionally p compact

noether lefschetz locus for surfaces we generalize m green s explicit noether lefschetz theorem to the family of smooth complete intersection surfaces in the higher dimensional projective spaces moreover we give a new proof of the density theorem due to c ciliberto j harris and r miranda 5

super stretched and graded countable cohen macaulay type abstract we define what it means for a cohen macaulay ring to be super stretched and show that cohen macaulay rings of graded countable cohen macaulay type are super stretched we use this result to show that rings of graded countable cohen macaulay type and positive dimension have possible h vectors 1 1 n or 1 n 1 further one dimensional standard graded gorenstein rings of graded countable type are shown to be hypersurfaces this result is not known in higher dimensions in the non gorenstein case rings of graded countable cohen macaulay type of dimension larger than 2 are shown to be of minimal multiplicity

rank 3 acm bundles on general hypersurfaces in p5 abstract we prove that a general hypersurface in p 5 of degree d 3 does not support an indecomposable rank 3 arithmetically cohen macaulay acm bundle this settles the base case of a generic version of a conjecture of buchweitz greuel and schreyer

on global exponential stability of delayed cellular neural networks with time varying delays a new sufficient condition has been presented ensuring the global exponential stability of cellular neural networks with time varying delays by using an approach based on delay differential inequality combining with young inequality the results established here extend those earlier given in the literature compared with the method of lyapunov functionals as in most previous studies our method is simpler and more effective for stability analysis

deformations of maximal cohen macaulay modules let r m be a complete cohen macaulay isolated singularity over a field k which is either perfect or k k p 0 according to dieterich di and yo ch 6 see also pol pr or more generally chp there exists a system of parameters x xl xr of r such that the base change functor r x rdefines an injection v preserving the indecomposability from the set of isomorphism classes of maximal cohen macaulay r modules to the set of isomorphism classes of r x modules in this case the ideal generated by x is called after di a reduction ideal trying to describe the image of v we noticed in po2 that a finitely generated r x module is in imv if and only if it has the form xl x p for a finitely generated r2 r x 2 x module satisfying

grothendieckʼs nuclear operator theorem revisited with an application to p null sequences abstract let x and y be banach spaces and let be a tensor norm the principal result is the following theorem if either x or y has the approximation property then each nuclear operator t x y such that t y x can be approximated in the nuclear norm by finite rank operators of type x y in the special case of grothendieck nuclear operators the theorem provides a strengthening for the classical theorem on the nuclearity of operators with a nuclear adjoint the hypotheses about the approximation property are essential the main application yields an affirmative answer to c pineiro j m delgado p convergent sequences and banach spaces in which p compact sets are q compact proc amer math soc 139 2011 957 967 for p 1 a sequence x n x is p null if and only if lim x n 0 and x n is relatively p compact in x

stability bifurcation and global existence of a hopf bifurcating periodic solution for a class of three neuron delayed network models in this paper a system of three delay differential equations representing a hopfield type general model for three neurons with two way bidirectional time delayed connections between the neurons and time delayed self connection from each neuron to itself is studied delay independent and delay dependent sufficient conditions for linear stability instability and the occurrence of a hopf bifurcation about the trivial equilibrium are addressed the partition of the resulting parametric space into regions of stability instability and hopf bifurcation in the absence of self connection is realized to extend the local hopf branches for large delay values a particular bidirectional delayed tri neuron model without self connection is investigated sufficient conditions for global existence of multiple non constant periodic solutions are obtained for such a model using the global hopf bifurcation theorem for functional differential equations due to j wu and the bendixson criterion for higher dimensional ordinary differential equations due to li and muldowney and following the approach developed by wei and li

cycle map for strictly decomposable cycles we introduce a class of cycles called nondegenerate strictly decomposable cycles and show that the image of each cycle in this class under the refined cycle map to an extension group in the derived category of arithmetic mixed hodge structures does not vanish this class contains certain cycles in the kernel of the abel jacobi map the construction gives a refinement of nori s argument in the case of a self product of a curve as an application we show that a higher cycle which is not annihilated by the reduced higher abel jacobi map produces uncountably many indecomposable higher cycles on the product with a variety having a nonzero global 1 form

brauer thrall for totally reflexive modules let r be a commutative noetherian local ring that is not gorenstein it is known that the category of totally reflexive modules over r is representation infinite provided that it contains a non free module the main goal of this paper is to understand how complex the category of totally reflexive modules can be in this situation local rings r m with m3 0 are commonly regarded as the structurally simplest rings to admit diverse categorical and homological characteristics for such rings we obtain conclusive results about the category of totally reflexive modules modeled on the brauer thrall conjectures starting from a non free cyclic totally reflexive module we construct a family of indecomposable totally reflexive r modules that contains for every n n a module that is minimally generated by n elements moreover if the residue field r m is algebraically closed then we construct for every n n an infinite family of indecomposable and pairwise non isomorphic totally reflexive r modules each of which is minimally generated by n elements the modules in both families have periodic minimal free resolutions of period at most 2

maurey s factorization theory for operator spaces we prove an operator space version of maurey s theorem which claims that every absolutely p 1 summing map on c k is automatically absolutely q summing for q p our results imply in particular that every completely bounded map from b h with values in pisier s operator space oh is completely p summing for p 2 this fails for p 2 as applications we obtain eigenvalue estimates for translation invariant maps defined on the von neumann algebra vn g associated with a discrete group g we also develop a notion of cotype which is compatible with factorization results on noncommutative lp spaces

delay dependent robust dissipativity conditions for delayed neural networks with random uncertainties this paper deals with the problem of the robust dissipativity analysis for delayed neural networks with randomly occurring uncertainties the randomly occurring uncertainties under consideration are assumed to follow certain mutually uncorrelated bernoulli distributed white noise sequences by using reciprocally convex approach combined with an extended wirtinger inequality some delay dependent conditions for the concerned neural networks to be stochastically strictly q s r dissipative are established finally two numerical examples are given to illustrate the reduced conservatism and effectiveness of our proposed approach

the banach ideal of a compact operators and related approximation properties abstract we use the notion of a compact sets determined by an operator ideal a introduced by carl and stephani 1984 to show that many known results of certain approximation properties and several ideals of compact operators can be systematically studied under this framework for banach operator ideals a we introduce a way to measure the size of a compact sets and use it to give a norm on k a the ideal of a compact operators then we study two types of approximation properties determined by a compact sets we focus our attention on an approximation property which makes use of the norm defined on k a this notion fits the definition of the a approximation property recently introduced by oja 2012 with k a instead of a we exemplify the power of the carl stephani theory and the geometric structure introduced here by appealing to some recent developments on p compactness