elliptic boundary value problems with λ dependent boundary conditions abstract in this paper second order elliptic boundary value problems on bounded domains r n with boundary conditions on depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework for a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure in the special case of rational nevanlinna or riesz herglotz functions on the boundary the solution operator is obtained in an explicit form in the product hilbert space l 2 l 2 m which is a natural generalization of known results on linear elliptic boundary value problems and rational boundary value problems for ordinary second order differential equations

the gl 2 c mckay correspondence in this paper we show that for any affine complete rational surface singularity the quiver of the reconstruction algebra can be determined combinatorially from the dual graph of the minimal resolution as a consequence the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph also for any finite subgroup g of rm gl 2 mathbb c it means that the endomorphism ring of the special cm mathbb c x y g modules can be used to build the dual graph of the minimal resolution of mathbb c 2 g extending mckay s observation mckay proc symp pure math 37 183 186 1980 for finite subgroups of rm sl 2 mathbb c to all finite subgroups of rm gl 2 mathbb c

exponential stability of periodic neural networks with impulsive effects and time varying delays abstract by employing young inequality and constructing suitable liapunov functions we investigate the existence and globally exponential stability of periodic neural networks with impulses and time varying delays the results extend and improve some earlier ones 1 5 12 an illustrative example and simulations are given to show the validity of the main results

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

contractions and deformations suppose that f is a projective birational morphism with at most one dimensional fibres between d dimensional varieties x and y satisfying rf ox oy consider the locus l in y over which f is not an isomorphism taking the scheme theoretic fibre c over any closed point of l we construct algebras afib and acon which prorepresent the functors of commutative deformations of c and noncommutative deformations of the reduced fibre respectively our main theorem is that the algebras acon recover l and in general the commutative deformations of neither c nor the reduced fibre can do this as the d 3 special case this proves the following contraction theorem in a neighbourhood of the point the morphism f contracts a curve without contracting a divisor if and only if the functor of noncommutative deformations of the reduced fibre is representable

existence and exponential stability of periodic solution for impulsive delay differential equations and applications abstract in this paper we consider a class of nonlinear impulsive delay differential equations by establishing an exponential estimate for delay differential inequality with impulsive initial condition and employing banach fixed point theorem we obtain several sufficient conditions ensuring the existence uniqueness and global exponential stability of a periodic solution for nonlinear impulsive delay differential equations furthermore the criteria are applied to analyze dynamical behavior of impulsive delay hopfield neural networks and the results show different behavior of impulsive system originating from one continuous system

impulsive effects on stability of cohen grossberg neural networks with variable delays in this paper a model of impulsive cohen grossberg neural networks is first formulated by establishing some impulsive differential inequalities we investigate impulsive effects on the stability of cohen grossberg neural networks with variable delays and obtain some sufficient conditions ensuring global exponential stability of the impulsive delay system our criteria not only show that the stability still remains under certain impulsive perturbations for some continuous stable neural networks but also present an approach to stabilize the unstable neural networks by utilizing impulsive effects the results extend and improve some recent works for impulsive neural networks as well as non impulsive neural networks some examples and their simulations are given for illustration of the theoretical results

dissipative eigenvalue problems for a sturm liouville operator with a singular potential in this paper we consider the sturm liouville operator d 2 dx 2 1 x on the interval a b a 0 f x f x gammaf 0 with gamma is an element of c boolean or r or by the dirichlet condition f 0 f 0 0 we also show that the corresponding operators can be obtained by norm resolvent approximation from operators where the potential 1 x is replaced by a continuous function and that their eigen and associated functions can be chosen to form a bari basis in l 2 a b

standard symmetric operators in pontryagin spaces a generalized von neumann formula and minimality of boundary coefficients certain meromorphic matrix valued functions on c r the so called boundary coefficients are characterized in terms of a standard symmetric operator s in a pontryagin space with finite not necessarily equal defect numbers a meromorphic mapping into the defect subspaces of s and a boundary mapping for s under some simple assumptions the boundary coefficients also satisfy a minimality condition it is shown that these assumptions hold if and only if for s a generalized von neumann equality is valid r 2002 elsevier science usa all rights reserved msc primary 47b50 47b25 34b07 47b32 secondary 46c20 47a06

on spectral properties of dissipative fourth order boundary value problem with a spectral parameter in the boundary condition in this article we consider dissipative fourth order boundary value problem in the lim 4 case with the spectral parameter in the boundary condition we use the maximal dissipative operator to construct a self adjoint dilation of this operator and its incoming and outgoing spectral representations then we determine the scattering matrix of dilation we also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding fourth order equation theorem on the completeness of the system of eigenvector and associated vectors of this operator are proved

vanishing of ext cluster tilting modules and finite global dimension of endomorphism rings let r be a cohen macaulay ring and m a maximal cohen macaulay r module inspired by recent striking work by iyama burban iyama keller reiten and van den bergh we study the question of when the endomorphism ring of m has finite global dimension via certain conditions about vanishing of ext modules we are able to strengthen certain results by iyama on connections between a higher dimension version of auslander correspondence and existence of non commutative crepant resolutions we also recover and extend to positive characteristics a recent theorem by burban iyama keller reiten on cluster tilting objects in the category of maximal cohen macaulay modules over reduced 1 dimensional hypersurfaces

elliptic eigenvalue problems with eigenparameter dependent boundary conditions abstract let a be a uniformly elliptic second order linear operator on a smooth bounded domain r n we study the eigenvalue problem au u subject to boundary conditions b0u b1u on where bj are linear boundary operators the problem is recast in the form a u u in a hilbert or krein space and results are given on the location and type of the spectrum full and half range completeness and regularity of critical points

singular derived categories of factorial terminalizations and maximal modification algebras abstract let x be a gorenstein normal 3 fold satisfying elf with local rings which are at worst isolated hypersurface e g terminal singularities by using the singular derived category d sg x and its idempotent completion d sg x we give necessary and sufficient categorical conditions for x to be q factorial and complete locally q factorial respectively we then relate this information to maximal modification algebras mmas introduced in 20 by showing that if an algebra is derived equivalent to x as above then x is q factorial if and only if is an mma thus all rings derived equivalent to q factorial terminalizations in dimension three are mmas as an application we extend some of the algebraic results in 6 and 14 using geometric arguments

existence and exponential stability of periodic solution for impulsive cohen grossberg neural networks with time varying delays abstract this paper is concerned with the existence and global exponential stability of periodic solution for impulsive cohen grossberg neural networks with periodic coefficients and time varying delays some novel criteria on the existence and global exponential stability of periodic solution are derived by using impulsive differential inequality technique and the banach fixed point theorem the results obtained improve and extend a few of recent literature in some situations two examples with numerical simulations are given to demonstrate the efficiency of proposed methods

three dimensional flops and noncommutative rings limburgs univ ctr dept math b 3590 diepenbeek belgium van den bergh m limburgs univ ctr dept math univ campus b 3590 diepenbeek belgium vdbergh luc ac be

the linearization of boundary eigenvalue problems and reproducing kernel hilbert spaces the boundary eigenvalue problems for the adjoint of a symmetric relation s in a hilbert space with finite not necessarily equal defect numbers which are related to the selfadjoint hilbert space extensions of s are characterized in terms of boundary coefficients and the reproducing kernel hilbert spaces they induce 2001 elsevier science inc all rights reserved

tensor products of modules and the rigidity of tor let r be a hypersurface domain that is a ring of the form s f where s is a regular local ring and f is a prime element suppose m and n are finitely generated r modules we prove two rigidity theorems on the vanishing of tor in the first theorem we assume that the regular local ring s is unramified that m r n has finite length and that dim m dim n dim r with these assumptions if tor r m n 0 for some j 0 then tor r m n 0 for all i j the second rigidity theorem states that if m r n is reflexive then tor r m n 0 for all i 1 we use these theorems to prove the following theorem valid even if s is ramified if m r n is a maximal cohen macaulay r module then both m and n are maximal cohen macaulay modules and at least one of them is free

linear relations and the kronecker canonical form abstract we show that the kronecker canonical form which is a canonical decomposition for pairs of matrices is the representation of a linear relation in a finite dimensional space this provides a new geometric view upon the kronecker canonical form each of the four entries of the kronecker canonical form has a natural meaning for the linear relation which it represents these four entries represent the jordan chains at finite eigenvalues the jordan chains at infinity the so called singular chains and the multi shift part or to state it more concisely for linear relations the kronecker canonical form is the analogue of the jordan canonical form for matrices

flops and equivalences of derived categories for threefolds with only terminal gorenstein singularities the main purpose of this paper is to show that bridgeland s moduli space of perverse point sheaves for certain flopping contractions gives the flops and the fourier mukai transform given by the birational correspondence of the flop is an equivalence between bounded derived categories

on global exponential stability for impulsive cellular neural networks with time varying delays in this paper the problem of global exponential stability for cellular neural networks cnns with time varying delays and fixed moments of impulsive effect is studied a new sufficient condition has been presented ensuring the global exponential stability of the equilibrium points by using piecewise continuous lyapunov functions and the razumikhin technique combined with young s inequality the results established here extend those given previously 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

almost sure and moment asymptotic boundedness of stochastic delay differential systems abstract the purpose of this article is to investigate the almost sure and moment asymptotic boundedness for a class of stochastic delay differential systems using stochastic analysis techniques a generalized non autonomous l operator delay differential inequality including cross item is presented based on the obtained inequality some sufficient verifiable criteria for the almost sure and moment asymptotic boundedness of the considered systems are derived analytically two numerical examples are given to illustrate the effectiveness of our theoretical results

on kreĭn s formula abstract kre n s formula provides a parametrization of the generalized resolvents and straus extensions of a closed symmetric operator with equal possibly infinite defect numbers in a hilbert space in terms of nevanlinna families in a parameter space the aim of this note is to give a simple complete analytical proof of kre n s formula

generalized resolvents and the boundary value problems for hermitian operators with gaps a hermitian operator a with gaps j j 1 j m is studied the self adjoint extensions which put exactly kj eigenvalues into each gap j j in particular for kj 0 1 j m the extensions preserving the gaps are described in terms of boundary conditions the generalized resolvents of the extensions with the indicated properties are described also a solvability criterion and description of all the solutions of the hamburger moment problem with supports in r j 1m j j are obtained in terms of the nevanlinna matrix

remarks on non commutative crepant resolutions of complete intersections abstract we study obstructions to existence of non commutative crepant resolutions in the sense of van den bergh over local complete intersections

flops and clusters in the homological minimal model programme suppose that f x rightarrow mathrm spec r is a minimal model of a complete local gorenstein 3 fold where the fibres of f are at most one dimensional so by van den bergh duke math j 122 3 423 455 2004 there is a noncommutative ring lambda derived equivalent to x for any collection of curves above the origin we show that this collection contracts to a point without contracting a divisor if and only if a certain factor of lambda is finite dimensional improving a result of donovan and wemyss contractions and deformations arxiv 1511 00406 we further show that the mutation functor of iyama and wemyss invent math 197 3 521 586 2014 6 is functorially isomorphic to the inverse of the bridgeland chen flop functor in the case when the factor of lambda is finite dimensional these results then allow us to jump between all the minimal models of mathrm spec r in an algorithmic way without having to compute the geometry at each stage we call this process the homological mmp this has several applications in git approaches to derived categories and also to birational geometry first using mutation we are able to compute the full git chamber structure by passing to surfaces we say precisely which chambers give the distinct minimal models and also say which walls give flops and which do not enabling us to prove the craw ishii conjecture in this setting second we are able to precisely count the number of minimal models and also give bounds for both the maximum and the minimum numbers of minimal models based only on the dual graph enriched with scheme theoretic multiplicity third we prove a bijective correspondence between maximal modifying r module generators and minimal models and for each such pair in this correspondence give a further correspondence linking the endomorphism ring and the geometry this lifts the auslander mckay correspondence to dimension three

eigenvalues of discrete sturm liouville problems with eigenparameter dependent boundary conditions abstract we consider the discrete right definite sturm liouville problems p t 1 y t 1 q t y t m t y t t 1 t z a 0 b 0 y 0 c 0 d 0 y 0 a 1 b 1 y t 1 c 1 d 1 y t 1 where 1 t z 1 2 t m t 0 for t 1 t z 1 i i 0 where i a i d i b i c i for i 0 1 we obtain the existence of the eigenvalues the sign changing times of the eigenfunctions and the interlacing results of the eigenvalues of the above problem the dirichlet problem and the neumann problem

on the finiteness of the derived equivalence classes of some stable endomorphism rings we prove that the stable endomorphism rings of rigid objects in a suitable frobenius category have only finitely many basic algebras in their derived equivalence class and that these are precisely the stable endomorphism rings of objects obtained by iterated mutation the main application is to the homological minimal model programme for a 3 fold flopping contraction f x rightarrow mathrm spec r where x has only gorenstein terminal singularities there is an associated finite dimensional algebra a text con known as the contraction algebra as a corollary of our main result there are only finitely many basic algebras in the derived equivalence class of a text con and these are precisely the contraction algebras of maps obtained by a sequence of iterated flops from f this provides evidence towards a key conjecture in the area

dynamical behavior of impulsive and periodic cohen grossberg neural networks abstract this paper investigates the existence and global stability of the periodic solution x t to cohen grossberg neural networks with periodic coefficients and impulses by using comparison results for impulsive differential equations and the method of lyapunov we describe the asymptotic behavior of all solutions in addition we give an explicit formula for the rate of exponential decay at infinity of the euclidean norm x t x t where x t is any solution of our model such a formula involves the jumps and the average of a suitable periodic function depending on the other parameters of the neural networks

stability of artificial neural networks with impulses sufficient conditions are obtained for the existence and asymptotic stability of a unique equilibrium of a hopfield type neural network with lipschitzian activation functions without assuming their boundedness monotonicity or differentiability and subjected to impulsive state displacements at fixed instants of time both the continuous time and their corresponding discrete time networks are considered the sufficient conditions of the discrete time network do not restrict the step size appearing in the discretization process and these conditions approach as the step size tends to zero those of the conditions of the continuous time networks the sufficient conditions are in terms of the parameters of the network only and are easy to verify also when the impulsive jumps are absent the results reduce to those of the non impulsive systems

boundary relations and their weyl families the concepts of boundary relations and the corresponding weyl families are introduced let s be a closed symmetric linear operator or more generally a closed symmetric relation in a hilbert space h let h be an auxiliary hilbert space let graphics and let jh be defined analogously a unitary relation g from the krein space h 2 j h to the kre in space h 2 j h is called a boundary relation for the adjoint s if ker gamma s the corresponding weyl family m lambda is de ned as the family of images of the defect subspaces n over cap lambda lambda is an element of c r under gamma here gamma need not be surjective and is even allowed to be multi valued while this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex hilbert space h and the class of unitary relations gamma h 2 j h h 2 j h it also generalizes the notion of so called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems moreover these new notions yield for instance the following realization theorem every h valued maximal dissipative for lambda is an element of c holomorphic family of linear relations is the weyl family of a boundary relation which is unique up to unitary equivalence if certain minimality conditions are satisfied further connections between analytic and spectral theoretical properties of weyl families and geometric properties of boundary relations are investigated and some applications are given

existence and global asymptotic stability of positive periodic solution of delayed cohen grossberg neural networks abstract in this paper a class of periodic cohen grossberg neural networks with discrete and distributed time varying delays is considered by an extension of the lyapunov krasovskii functional method a novel criterion for the existence and uniqueness and global asymptotic stability of positive periodic solution is derived in terms of m matrix without any restriction on uniform positiveness of the amplification functions comparison and illustrative examples are given to illustrate the effectiveness of the obtained results

operators without eigenvalues in finite dimensional vector spaces abstract we introduce the concept of a canonical subspace of c d z and among other results prove the following statements an operator in a finite dimensional vector space has no eigenvalues if and only if it is similar to the operator of multiplication by the independent variable on a canonical subspace of c d z an operator in a finite dimensional pontryagin space is symmetric and has no eigenvalues if and only if it is isomorphic to the operator of multiplication by the independent variable in a canonical subspace of c d z with an inner product determined by a full matrix polynomial nevanlinna kernel

global exponential stability for impulsive cellular neural networks with time varying delays abstract in this paper we study the problem of global exponential stability for cellular neural networks cnn with time varying delays and fixed moments of impulsive effect we establish several stability criteria by employing lyapunov functions and the razumikhin technique

an analytic characterization of the eigenvalues of self adjoint extensions let a be a self adjoint extension in k of a fixed symmetric operator a in k k an analytic characterization of the eigenvalues of a is given in terms of the q function and the parameter function in the krein naimark formula here k and k are krein spaces and it is assumed that a locally has the same spectral properties as a self adjoint operator in a pontryagin space the general results are applied to a class of boundary value problems with dependent boundary conditions

on the reproducing kernel of a pontryagin space of vector valued polynomials abstract we give necessary and sufficient conditions under which the reproducing kernel of a pontryagin space of d 1 vector polynomials is determined by a generalized nevanlinna pair of d d matrix polynomials

extinction in nonautonomous t periodic competitive lotka volterra system a nonautonomous t periodic competitive lotka volterra system of n species is considered it is shown that if the coefficients are t periodic continuous and satisfy certain inequalities then any solution with strictly positive initial conditions has the property that all but one of its components vanish while the remaining component approaches the canonical solution of a certain logistic differential equation

the structure of linear relations in euclidean spaces the structure of a linear relation multivalued operator in a euclidean space is completely determined a linear relation can be written as a direct sum of three relations of different classes which are jordan relations completely singular relations and multishifts all three classes of relations are characterized in terms of the spectrum and their chain structure which leads to a generalization of the classical jordan canonical form

a structure theorem for reproducing kernel pontryagin spaces we illustrate a relationship between reproducing kernel spaces and orthogonal polynomials via a general structure theorem the christofell darboux formula emerges as a limit case

existence and global attractivity of almost periodic solution for cellular neural network with distributed delays this paper is devoted to the existence and global attractivity of almost periodic solution for a class of cellular neural network with distributed delays and variable coefficients some sufficient conditions ensuring the existence and global attractivity of almost periodic solution are derived by employing banach fixed point theory and using differential inequality technique the results allow for the consideration of all unbounded neuron signal functions but not necessarily surjective thus these conditions obtained have highly important significance in designs and applications of the networks we extend and improve previously known results an example is also worked out to demonstrate the advantages of our results

maximal modifications and auslander reiten duality for non isolated singularities we first generalize classical auslander reiten duality for isolated singularities to cover singularities with a one dimensional singular locus we then define the notion of ct modules for non isolated singularities and we show that these are intimately related to noncommutative crepant resolutions nccrs when r has isolated singularities ct modules recover the classical notion of cluster tilting modules but in general the two concepts differ then wanting to generalize the notion of nccrs to cover partial resolutions of operatorname spec r in the main body of this paper we introduce a theory of modifying and maximal modifying modules under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three dimensional gorenstein rings are shown to be derived equivalent we then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory our mutation works in arbitrary dimension and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian

on bezoutians of nonsquare matrix polynomials and inversion of matrices with nonsquare blocks abstract the notion of bezoutian of nonsquare matrix polynomials is defined it is used to establish formulae for the inverse of square matrices partitioned into nonsquare blocks in particular a generalization of the gohberg semencul formula for the inverse of a generalized toeplitz matrix is proved

the theory of v bezoutians and its applications abstract some propositions of the theory of v bezoutians are presented as an application the following problem is considered to construct an appropriate pontryagin space in which a given matrix boundary value problem containing the eigenvalue parameter in the boundary conditions is self adjoint the solution of this problem is given in three special cases with the help of v bezoutians of matrix polynomials entering the boundary conditions

sturm liouville problems with boundary conditions rationally dependent on the eigenparameter ii necessary and sufficient conditions are given for two sequences n and n to be the eigenvalues and norming constants of a sturm liouville boundary value problem y qy y y 0 cos y 0 sin and y 1 f y 1 where f is a rational function of herglotz nevanlinna type it is also proved that q and f are uniquely determined by the sequences n and n

reduction of triangulated categories and maximal modification algebras for ca_n singularities in this paper we define and study triangulated categories in which the homspaces have krull dimension at most one over some base ring hence they have a natural 2 step filtration and each factor of the filtration satisfies some calabi yau type property if c is such a category we say that c is calabi yau with dim c 1 we extend the notion of calabi yau reduction to this setting and prove general results which are an analogue of known results in cluster theory such categories appear naturally in the setting of gorenstein singularities in dimension three as the stable categories cm r of cohen macaulay modules we explain the connection between calabi yau reduction of cm r and both partial crepant resolutions and q factorial terminalizations of spec r and we show under quite general assumptions that calabi yau reductions exist in the remainder of the paper we focus on complete local can singularities r by using a purely algebraic argument based on calabi yau reduction of cm r we give a complete classification of maximal modifying modules in terms of the symmetric group generalizing and strengthening results in 7 10 where we do not need any restriction on the ground field we also describe the mutation of modifying modules at an arbitrary not necessarily indecomposable direct summand as a corollary when k d c we obtain many autoequivalences of the derived category of the q factorial terminalizations of spec r

on the exponential stability and periodic solutions of delayed cellular neural networks a set of criteria is presented for the global exponential stability and the existence of periodic solutions of delayed cellular neural networks dcnns by constructing suitable lyapunov functionals introducing many parameters and combining with the elementary inequality technique these criteria have important leading significance in the design and applications of globally stable dcnns and periodic oscillatory dcnns in addition earlier results are extended and improved other results are contained two examples are given to illustrate the theory