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

a unifying convergence analysis of second order methods for secular equations existing numerical methods of second order are considered for a so called secular equation we give a brief description of the most important of these methods and show that all of them can be interpreted as improvements of newton s method for an equivalent problem for which newton s method exhibits convergence from any point in a given interval this interpretation unifies the convergence analysis of these methods provides convergence proofs where they were lacking and furnishes ways to construct improved methods in addition we show that some of these methods are in fact equivalent a second secular equation is also briefly considered

fuzzy star operations on an integral domain in this paper we introduce the concept of fuzzy star operations on an integral domain and show that the set of all fuzzy star operations on the integral domain forms a complete lattice we also characterize prufer domains psuedo dedekind domains generalized greatest common divisor domains and other integral domains in terms of the invertibility of certain fractionary fuzzy ideals

fuzzy semistar operations on overrings abstract in this paper we reinvestigate the notion of fuzzy semistar operation introduced in 5 we show how a fuzzy semistar operation on an integral domain r induces canonically a fuzzy semistar operation on an overring t of r and conversely how a fuzzy semistar operation on t induces canonically a semistar operation on r as an application new characterizations of prufer domains and complete description of the set of all fuzzy semistar operations of finite character on prufer domains are obtained we also characterize conducive domains

proper resolutions and gorenstein categories abstract let a be an abelian category and c an additive full subcategory of a we provide a method to construct a proper c resolution resp coproper c coresolution of one term in a short exact sequence in a from that of the other two terms by using these constructions we answer affirmatively an open question on the stability of the gorenstein category g c posed by sather wagstaff sharif and white and also prove that g c is closed under direct summands in addition we obtain some criteria for computing the c dimension and the g c dimension of an object in a

quadratically constrained least squares and quadratic problems we consider the following problem compute a vectorx such that ax b 2 min subject to the constraint x 2 a new approach to this problem based on gauss quadrature is given the method is especially well suited when the dimensions ofa are large and the matrix is sparse it is also possible to extend this technique to a constrained quadratic form for a symmetric matrixa we consider the minimization ofx t a x 2b t x subject to the constraint x 2 some numerical examples are given

fast and stable qr eigenvalue algorithms for generalized companion matrices and secular equations we introduce a class of n n structured matrices which includes three well known classes of generalized companion matrices tridiagonal plus rank one matrices comrade matrices diagonal plus rank one matrices and arrowhead matrices relying on the structure properties of we show that if a then a rq where a qr is the qr decomposition of a this allows one to implement the qr iteration for computing the eigenvalues and the eigenvectors of any a with o n arithmetic operations per iteration and with o n memory storage this iteration applied to generalized companion matrices provides new o n2 flops algorithms for computing polynomial zeros and for solving the associated rational secular equations numerical experiments confirm the effectiveness and the robustness of our approach

a new matrix free algorithm for the large scale trust region subproblem we present a new method for the large scale trust region subproblem the method is matrix free in the sense that only matrix vector products are required we recast the trust region subproblem as a parameterized eigenvalue problem and compute an optimal value for the parameter we then find the solution of the trust region subproblem from the eigenvectors associated with two of the smallest eigenvalues of the parameterized eigenvalue problem corresponding to the optimal parameter the new algorithm uses a different interpolating scheme than existing methods and introduces a unified iteration that naturally includes the so called hard case we show that the new iteration is well defined and convergent at a superlinear rate we present computational results to illustrate convergence properties and robustness of the method

spectral functions for real symmetric toeplitz matrices we derive separate spectral functions for the even and odd spectra of a real symmetric toeplitz matrix which are given by the roots of those functions these are rational functions also commonly referred to as secular functions two applications are considered spectral evolution as a function of one parameter and the computation of eigenvalues

conducive integral domains this article introduces the concept of a conducive domain that is an integral domain each of whose overrings apart from the quotient field has nonzero conductor the principal examples of conducive domains are all d m constructions and all pseudo valuation domains several characterizations are obtained notably that a domain r is conducive if and only if r has a valuation overring with nonzero conductor it is proved that if r is a conducive domain but not a field then spec r is pinched at a prime p such that the set of primes within p is linearly ordered the converse is shown for r a prufer domain in which case p prp consequences include pullback characterizations of the seminormal resp prufer conducive domains special attention is paid to the class of conducive g domains with attendant interplay between conducive and the property of having a maximum overring

solving the trust region subproblem by a generalized eigenvalue problem the state of the art algorithms for solving the trust region subproblem trs are based on an iterative process involving solutions of many linear systems eigenvalue problems subspace optimization or line search steps a relatively underappreciated fact due to gander golub and von matt linear algebra appl 114 1989 pp 815 839 is that trss can be solved by one generalized eigenvalue problem with no outer iterations in this paper we rediscover this fact and discover its great practicality which exhibits good performance both in accuracy and efficiency moreover we generalize the approach in various directions namely by allowing for an ellipsoidal constraint dealing with the so called hard case and obtaining approximate solutions efficiently when high accuracy is unnecessary we demonstrate that the resulting algorithm is a general purpose trs solver effective both for dense and large sparse problems including the so called hard case our algorithm is easy to implement its essence i

the acyclic closure of an exact category and its triangulation abstract for any exact category a with splitting idempotents a maximal exact category t a containing a as a biresolving subcategory is constructed important types of exact categories including n tilting torsion classes categories of cohen macaulay modules over a cohen macaulay order or categories of gorenstein projectives are shown to be of the form t a the quotient category t a a in the sense of grothendieck always exists and carries a triangulated structure more generally it is proved that any biresolving subcategory a of an exact category m gives a triangulated localization m a for example the unbounded derived category d a of an exact category a is obtained directly with no passage through a homotopy category of complexes as applications some recent developments related to gorenstein projectivity non commutative crepant resolutions singularity categories and cohen macaulay representations are extended and improved in the new framework for example the concept of non commutative resolution of a noetherian frobenius category is extended to arbitrary exact categories which leads to an overarching connection with representation dimension of exact categories and n tilting

strong duality in nonconvex quadratic optimization with two quadratic constraints we consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints when the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions we then develop a connection between the image of the real and complex spaces under a quadratic mapping which together with the results in the complex case lead to a condition that ensures strong duality in the real setting preliminary numerical simulations suggest that for random instances of the extended trust region subproblem the sufficient condition is satisfied with a high probability furthermore we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems finally we discuss an application of our results to robust least squares problems

iterative algorithm for joint zero diagonalization with application in blind source separation a new iterative algorithm for the nonunitary joint zero diagonalization of a set of matrices is proposed for blind source separation applications on one hand since the zero diagonalizer of the proposed algorithm is constructed iteratively by successive multiplications of an invertible matrix the singular solutions that occur in the existing nonunitary iterative algorithms are naturally avoided on the other hand compared to the algebraic method for joint zero diagonalization the proposed algorithm requires fewer matrices to be zero diagonalized to yield even better performance the extension of the algorithm to the complex and nonsquare mixing cases is also addressed numerical simulations on both synthetic data and blind source separation using time frequency distributions illustrate the performance of the algorithm and provide a comparison to the leading joint zero diagonalization schemes

a d c optimization algorithm for solving the trust region subproblem this paper is devoted to difference of convex functions d c optimization d c duality local and global optimality conditions in d c programming the d c algorithm dca and its application to solving the trust region problem the dca is an iterative method that is quite different from well known related algorithms thanks to the particular structure of the trust region problem the dca is very simple requiring only matrix vector products and in practice converges to the global solution the inexpensive implicitly restarted lanczos method of sorensen is used to check the optimality of solutions provided by the dca when a nonglobal solution is found a simple numerical procedure is introduced both to find a feasible point having a smaller objective value and to restart the dca at this point it is shown that in the nonconvex case the dca converges to the global solution of the trust region problem using only matrix vector products and requiring at most 2m 2 restarts where m is the number of distinct negative eigenvalues of the coefficient matrix that defines the problem numerical simulations establish the robustness and efficiency of the dca compared to standard related methods especially for large scale problems

constrained inverse regression for incorporating prior information inverse regression methods facilitate dimension reduction analyses of high dimensional data by extracting a small number of factors that are linear combinations of the original predictor variables but the estimated factors may not lend themselves readily to interpretation consistent with prior information our approach to solving this problem is to first incorporate prior information via theory or data driven constraints on model parameters and then apply the proposed method constrained inverse regression cir to extract factors that satisfy the constraints we provide chi squared and t tests to assess the significance of each factor and its estimated coefficients and we also generalize cir to other inverse regression methods in situations where both dimension reduction and factor interpretation are important finally we investigate cir s small sample performance test data driven constraints and present a marketing example to illustrate its use in discovering meaningful factors that influence th

an auslander type result for gorenstein projective modules an artin algebra a is said to be cm finite if there are only finitely many up to isomorphisms indecomposable finitely generated gorenstein projective a modules we prove that for a gorenstein artin algebra it is cm finite if and only if every its gorenstein projective module is a direct sum of finitely generated gorenstein projective modules this is an analogue of auslander s theorem on algebras of finite representation type m auslander a functorial approach to representation theory in representations of algebras workshop notes of the third internat conference in lecture notes in math vol 944 springer verlag berlin 1982 pp 105 179 m auslander representation theory of artin algebras ii comm algebra 1974 269 310

optimization based stability analysis of structures under unilateral constraints this paper discusses an optimization based technique for determining the stability of a given equilibrium point of the unilaterally constrained structural system which is subjected to the static load we deal with the three problems in mechanics sharing the common mathematical properties i structures containing no compression cables ii frictionless contacts and iii elastic plastic trusses with non negative hardening it is shown that the stability of a given equilibrium point of these structures can be determined by solving a maximization problem of a convex function over a convex set on the basis of the difference of convex functions optimization we propose an algorithm to solve the stability determination problem at each iteration of which a second order cone programming problem is to be solved the problems presented are solved for various structures to determine the stability of given equilibrium points copyright 2008 john wiley sons ltd

a linear time algorithm for trust region problems we consider the fundamental problem of minimizing a general quadratic function over an ellipsoidal domain also known as the trust region sub problem we give the first provable linear time in the number of non zero entries of the input algorithm for approximately solving this problem specifically our algorithm returns an epsilon approximate solution in runtime of order n sqrt epsilon where n is the number of non zero entries in the input this matches the runtime of nesterov s accelerated gradient descent suitable for the special case in which the quadratic objective is convex and the runtime of the lanczos method which is applicable when the problem is purely quadratic

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

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

the derived category of an exact category there is some confusion in the literature regarding the derived category of an exact category 8 thomason gives a satisfactory treatment of the bounded derived category in ct 1 11 6 see also appendix a the result is that provided all weakly split epimorphisms in 8 are admissible the bounded derived category may be defined as usual although thomason does not say it the categories d 6 d d may also be defined in the same way the definition of the unbounded derived category is more difficult and is very poorly treated in the literature for instance in bbd this derived category is only defined provided every morphism in 6 admits a kernel see bbd 1 1 41 this is very unsatisfactory since very few exact categories satisfy the condition in this article we will show that d b may be defined whenever 6 is saturated karoubian in thomason s terminology an exact category is saturated if every idempotent splits i e 6 contains all direct summands of its objects for the purpose of comparing with thomason s result the bounded derived category is defined for more 8 s all that is required to define db is that whenever an idempotent e a a factors as a f ba a with fog lb then e is split thomason calls such idempotents weakly split and we will honor his notation we will show here that these constructions are in some sense best possible remark 1 8 for d 6 remarks 1 9 and 1 10 for d d d b db 6 there are two reasons why i wrote this note one is to correct the misconceptions in the literature but more importantly the proof of the key result lemma 1 2 depends on an important characterization of tpaisse subcategories due to rickard and this article is intended to highlight rickard s criterion a full triangulated subcategory y of a triangulated

radical embeddings and representation dimension given a representation finite algebra b and a subalgebra a of b such that the jacobson radicals of a and b coincide we prove that the representation dimension of a is at most three by a result of igusa and todorov this implies that the finitistic dimension of a is finite

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

triadic categories without localization triadic categories arise as homotopy categories m cm of 2 termed complexes over a category cm of representations of a finite dimensional algebra a classical order or any higher dimensional cohen macaulay order they are close to triangulated categories with the main difference that triangles are replaced by 4 termed complexes triads which are functorial in addition the triadic category m cm is exact i e it has a distinguished class of short exact sequences with the usual properties we give a new characterization of triadic categories which makes no use of localization as a consequence the triadic structure of m cm can be derived from the exact structure of m m cm

globally solving the trust region subproblem using simple first order methods we consider the trust region subproblem which is given by a minimization of a quadratic not necessarily convex function over the euclidean ball based on the well known second order necessary and

numerical solution of a secular equation a method is proposed for the solution of a secular equation arising in modified symmetric eigenvalue problems and in several other areas this equation has singularities which make the application of standard root finding methods difficult in order to solve the equation a class of transformations of variables is considered which transform the equation into one for which newton s method converges from any point in a certain given interval in addition the form of the transformed equation suggests a convergence accelerating modification of newton s method the same ideas are applied to the secant method and numerical results are presented

kernelized sorting object matching is a fundamental operation in data analysis it typically requires the definition of a similarity measure between the classes of objects to be matched instead we develop an approach which is able to perform matching by requiring a similarity measure only within each of the classes this is achieved by maximizing the dependency between matched pairs of observations by means of the hilbert schmidt independence criterion this problem can be cast as one of maximizing a quadratic assignment problem with special structure and we present a simple algorithm for finding a locally optimal solution

triangle singularities ade chains and weighted projective lines abstract we study the singularity category of the triangle singularity f x a y b z c or s k x y z f attached to the weighted projective line x given by the weight triple a b c by investigating the stable category vect x of vector bundles on x obtained from the category of vector bundles by factoring out all the line bundles we show that vect x is fractional calabi yau whose cy dimension is a function of the euler characteristic of x and establish the existence of a tilting object having the shape of an a 1 b 1 c 1 cuboid the weight types 2 a b yield an explanation of happel seidel symmetry for a class of important nakayama algebras further the weight sequence 2 3 p corresponds to an ade chain the e n chain extrapolating the exceptional dynkin cases e 6 e 7 and e 8 to a whole sequence of triangulated categories

solving inverse cone constrained eigenvalue problems we compare various algorithms for constructing a matrix of order n whose pareto spectrum contains a prescribed set lambda lambda 1 ldots lambda p of reals in order to avoid overdetermination one assumes that p does not exceed n 2 the inverse pareto eigenvalue problem under consideration is formulated as an underdetermined system of nonlinear equations we also address the issue of computing lorentz spectra and solving inverse lorentz eigenvalue problems

segmentation given partial grouping constraints we consider data clustering problems where partial grouping is known a priori we formulate such biased grouping problems as a constrained optimization problem where structural properties of the data define the goodness of a grouping and partial grouping cues define the feasibility of a grouping we enforce grouping smoothness and fairness on labeled data points so that sparse partial grouping information can be effectively propagated to the unlabeled data considering the normalized cuts criterion in particular our formulation leads to a constrained eigenvalue problem by generalizing the rayleigh ritz theorem to projected matrices we find the global optimum in the relaxed continuous domain by eigendecomposition from which a near global optimum to the discrete labeling problem can be obtained effectively we apply our method to real image segmentation problems where partial grouping priors can often be derived based on a crude spatial attentional map that binds places with common salient features or focuses on expected object locations we demonstrate not only that it is possible to integrate both image structures and priors in a single grouping process but also that objects can be segregated from the background without specific object knowledge

skew group algebras in the representation theory of artin algebras then the corresponding crossed product algebra i g or i x g for short has an elements cgieg gi a li i addition is componentwise and multiplication is given by g a g and g g2 y g g2 m in this paper we assume that the values of y lie in the center z a of i hence the action of g on i is given by a group homomorphism g p aut i and 3 can be left out in the special case that y is the trivial map we write 1g instead of i g and the elements as c c li gi ag is then called a skew group ring there is a lot of literature on skew group algebras and crossed product algebras and on the relationship with the ring ac whose elements are those elements of a left fixed by g much work has been done on which properties of li are inherited by n g or ac some of the work on the relationship between these rings has its roots in trying to develop a galois theory for noncommutative rings we refer to 3 7 13 15 19 21 23 25 27 281 and their references in this paper we study these constructions when i is an artin algebra and g a finite group of order n such that n is invertible in 1 under these assumptions the construction preserves central properties of interest in the

analysis of third order methods for secular equations third order numerical methods are analyzed for secular equations these equations arise in several matrix problems and numerical linear algebra applications a closer look at an existing method shows that it can be considered as a classical method for an equivalent problem this not only leads to other third order methods it also provides the means for a unifying convergence analysis of these methods and for their comparisons finally we consider approximated versions of the aforementioned methods

three results on frobenius categories this paper consists of three results on frobenius categories 1 we give sufficient conditions on when a factor category of a frobenius category is still a frobenius category 2 we show that any frobenius category is equivalent to an extension closed exact subcategory of the frobenius category formed by cohen macaulay modules over some additive category this is an analogue of gabriel quillen s embedding theorem for frobenius categories 3 we show that under certain conditions an exact category with enough projective and enough injective objects allows a natural new exact structure with which the given category becomes a frobenius category several applications of the results are discussed

a semidefinite framework for trust region subproblems with applications to large scale minimization primal dual pairs of semidefinite programs provide a general framework for the theory and algorithms for the trust region subproblem trs this latter problem consists in minimizing a general quadratic function subject to a convex quadratic constraint and therefore it is a generalization of the minimum eigenvalue problem the importance of trs is due to the fact that it provides the step in trust region minimization algorithms the semidefinite framework is studied as an interesting instance of semidefinite programming as well as a tool for viewing known algorithms and deriving new algorithms for trs in particular a dual simplex type method is studied that solves trs as a parametric eigenvalue problem this method uses the lanczos algorithm for the smallest eigenvalue as a black box therefore the essential cost of the algorithm is the matrix vector multiplication and thus sparsity can be exploited a primal simplex type method provides steps for the so called hard case extensive numerical tests for large sparse problems are discussed these tests show that the cost of the algorithm is 1 n times the cost of finding a minimum eigenvalue using the lanczos algorithm where 0 n 1 is a fraction which decreases as the dimension increases

absolute relative and tate cohomology of modules of finite gorenstein dimension we study finitely generated modules m over a ring r noetherian on both sides if m has finite gorenstein dimension mbox g dim rm in the sense of auslander and bridger then it determines two other cohomology theories besides the one given by the absolute cohomology functors rm ext n r m relative cohomology functors rm ext n mathcal g m are defined for all non negative integers n they treat the modules of gorenstein dimension 0 as projectives and vanish for n mbox g dim rm tate cohomology functors widehat rm ext n r m are defined for all integers n all groups widehat rm ext n r m n vanish if m or n has finite projective dimension comparison morphisms varepsilon mathcal g n colon rm ext n mathcal g m to rm ext n r m and varepsilon r n colon rm ext n r m to widehat rm ext n r m link these functors we give a self contained treatment of modules of finite g dimension establish basic properties of relative and tate cohomology and embed the comparison morphisms into a canonical long exact sequence 0 to rm ext 1 mathcal g m to cdots to rm ext n mathcal g m to rm ext n r m to widehat rm ext n r m to rm ext n 1 mathcal g m to cdots we show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings 2000 mathematical subject classification 16e05 13h10 18g25

gorenstein derived categories abstract gorenstein derived categories are defined and the relation with the usual derived categories is given the bounded gorenstein derived categories of gorenstein rings and of finite dimensional algebras are explicitly described via the homotopy categories of gorenstein projective modules and some applications are obtained gorenstein derived equivalences between cm finite gorenstein algebras are discussed

deformations of rational surface singularities and reflexive modules with an application to flops abstract blowing up a rational surface singularity in a reflexive module gives a any partial resolution dominated by the minimal resolution the main theorem shows how deformations of the pair singularity module relates to deformations of the corresponding pair of partial resolution and locally free strict transform and to deformations of the underlying spaces the results imply some recent conjectures on small resolutions and flops

finite gorenstein representation type implies simple singularity let r be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive r modules we prove that if this set is finite then either it has exactly one element represented by the rank 1 free module or r is gorenstein and an isolated singularity if r is complete then it is even a simple hypersurface singularity the crux of our proof is to argue that if the residue field has a totally reflexive cover then r is gorenstein or every totally reflexive r module is free

variational analysis of an extended eigenvalue problem abstract for a symmetric matrix b rn n and a vector a rn the maximal extended eigenvalue a max x r n s t b i x a x t x 1 is known to arise in optimality conditions for the mathematical programming problemp a given by max x t b x 2 a t x x t x 1 as well as in extended rayleigh ritz type results pertaining to the one parameter family of nonsymmetric border perturbations ofb given by a t a b a t a t nonsmooth analysis is employed in order to describe the function with special emphasis on its sensitivity near the origin further connections with p a are drawn

doubly constrained robust capon beamformer with ellipsoidal uncertainty sets the doubly constrained robust dcr capon beamformer with a spherical uncertainty set was introduced and studied by stoica and wang here we consider the generalized dcr problem gdcr in which the uncertainty set is an ellipsoid rather than a sphere although as noted previously by stoica and wang this problem is nonconvex and appears to be intractable we show that it can be solved efficiently in fact we prove that the gdcr beamformer can be obtained as a solution to a convex optimization problem to this end we first derive a strong duality result for nonconvex quadratic programs with two quadratic constraints over the complex domain specializing the results to our context leads to a semidefinite programming formulation of the gdcr beamformer

radicals of fuzzy ideals abstract there are several natural ways to fuzzify the concepts of prime ideal primary ideal the radical of an ideal and the primary representation of an ideal in a commutative ring in this paper we introduce some new approaches and compare them with known ones our main goal is to unify the various possible fuzzy primary representations of a fuzzy ideal to prepare the way for the study of fuzzy nonlinear systems of equations

a constrained eigenvalue problem in this paper we consider the following mathematical and computational problem given the quantities a n m by n m matrix symmetric n 0 n n m by m matrix with full rank t vector of dimension m with nt t 1 determine an x such that operatorname x t ax min subject to the constraints n t x t i x t x 1 ii variants of this problem occur in many applications 1 5 7 8 11 the problem has been studied previously when t 0 the null vector cf 4 6 when t 0 then the problem becomes more complicated

a numerical comparison of methods for solving secular equations numerical comparisons are carried out by the most prominent methods both second and higher order for solving secular equations these equations arise in several matrix problems and numerical linear algebra applications in addition to the comparisons we propose a new starting point our results show a clear advantage to higher order methods

fuzzy semistar operations on integral domains in this paper we introduce the notion of fuzzy semistar operation on an integral domain and relate it to the existing notion of fuzzy star operation we show that the set of all fuzzy semistar operations on the integral domain is a complete lattice we also characterize when an overring of a domain is a prufer domain and when an overring is a flat module over the domain in terms of a fuzzy semistar operation

on the solution of the tikhonov regularization of the total least squares problem total least squares tls is a method for treating an overdetermined system of linear equations bf a bf x approx bf b where both the matrix bf a and the vector bf b are contaminated by noise tikhonov regularization of the tls trtls leads to an optimization problem of minimizing the sum of fractional quadratic and quadratic functions as such the problem is nonconvex we show how to reduce the problem to a single variable minimization of a function mathcal g over a closed interval computing a value and a derivative of mathcal g consists of solving a single trust region subproblem for the special case of regularization with a squared euclidean norm we show that mathcal g is unimodal and provide an alternative algorithm which requires only one spectral decomposition a numerical example is given to illustrate the effectiveness of our method

a second order cone based approach for solving the trust region subproblem and its variants we study the trust region subproblem trs of minimizing a nonconvex quadratic function over the unit ball with additional conic constraints despite having a nonconvex objective it is known that the classical trs and a number of its variants are polynomial time solvable in this paper we follow a second order cone soc based approach to derive an exact convex reformulation of the trs under a structural condition on the conic constraint our structural condition is immediately satisfied when there are no additional conic constraints and it generalizes several such conditions studied in the literature as a result our study highlights an explicit connection between the classical nonconvex trs and smooth convex quadratic minimization which allows for the application of cheap iterative methods such as nesterov s accelerated gradient descent to the trs furthermore under slightly stronger conditions we give a low complexity characterization of the convex hull of the epigraph of the nonconvex quadrati

fuzzy semistar operations of finite character on integral domains this paper is devoted to the construction of examples of fuzzy semistar operations on integral domains a special case of interest is the fuzzy semistar operation of finite character we show that the set of fuzzy semistar operations of finite character on an integral domain is a complete lattice we also prove that the fuzzy semistar operation on an integral domain defined by a finite collection of overrings and union preserving fuzzy semistar operations of finite character on the overrings is of finite character an example is constructed to prove that if the collection of overrings is infinite the corresponding fuzzy semistar operation may not be of finite character we describe the set of fuzzy semistar operation of finite character on valuation domains

some extensions of theorems of knorrer and herzog popescu abstract a construction due to knorrer shows that if n is a maximal cohen macaulay module over a hypersurface defined by f y 2 then the first syzygy of n y n decomposes as the direct sum of n and its own first syzygy this was extended by herzog popescu to hypersurfaces f y n replacing n y n by n y n 1 n we show in the same setting as herzog popescu that the first syzygy of n y k n is always an extension of n by its first syzygy and moreover that this extension has useful approximation properties we give two applications first we construct a ring over which every finitely generated module has an eventually 2 periodic projective resolution prompting us to call it a non commutative hypersurface ring second we give upper bounds on the dimension of the stable module category a k a the singularity category of a hypersurface defined by a polynomial of the form x 1 a 1 x d a d

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

topology trivialization and large deviations for the minimum in the simplest random optimization finding the global minimum of a cost function given by the sum of a quadratic and a linear form in n real variables over n 1 dimensional sphere is one of the simplest yet paradigmatic problems in optimization theory known as the trust region subproblem or constraint least square problem when both terms in the cost function are random this amounts to studying the ground state energy of the simplest spherical spin glass in a random magnetic field we first identify and study two distinct large n scaling regimes in which the linear term magnetic field leads to a gradual topology trivialization i e reduction in the total number mathcal n tot of critical stationary points in the cost function landscape in the first regime mathcal n tot remains of the order n and the cost function energy has generically two almost degenerate minima with the tracy widom tw statistics in the second regime the number of critical points is of the order of unity with a finite probability for a single minimum in that case the mean total number of extrema minima and maxima of the cost function is given by the laplace transform of the tw density and the distribution of the global minimum energy is expected to take a universal scaling form generalizing the tw law though the full form of that distribution is not yet known to us one of its far tails can be inferred from the large deviation theory for the global minimum in the rest of the paper we show how to use the replica method to obtain the probability density of the minimum energy in the large deviation approximation by finding both the rate function and the leading pre exponential factor

a mckay correspondence for reflection groups we construct a noncommutative desingularization of the discriminant of a finite reflection group g as a quotient of the skew group ring a s g if g is generated by order 2 reflections then this quotient identifies with the endomorphism ring of the reflection arrangement a g viewed as a module over the coordinate ring sg of the discriminant of g this yields in particular a correspondence between the nontrivial irreducible representations of g to certain maximal cohen macaulay modules over the coordinate ring sg these maximal cohen macaulay modules are precisely the nonisomorphic direct summands of the coordinate ring of the reflection arrangement a g viewed as a module over sg we identify some of the corresponding matrix factorizations namely the so called logarithmic co residues of the discriminant

the spectrum of a modified linear pencil abstract suppose the spectrum of a symmetric definite linear pencil is known this paper addresses the question of what can be said about the spectrum when scalar multiples of a rank one update are added to each matrix in the pencil the secular equation for this problem is derived and from it a certain separation property is found which gives insight into the connection between the eigenvalues before and after modification in the context of structural dynamics the result characterises the behaviour of a finite dimensional vibrating system undergoing mass and stiffness modifications the result also leads to applications such as a divide and conquer algorithm for the eigenvalues of the modified system so called matrix tearing and spectral shifting an illustrative example is also given

on algebras of finite cohen macaulay type abstract we study artin algebras and commutative noetherian complete local rings r in connection with the following decomposition property of gorenstein projective modules any gorenstein projective module is a direct sum of finitely generated modules we show that the class of algebras enjoying coincides with the class of virtually gorenstein algebras of finite cohen macaulay type introduced in beligiannis and reiten 2007 27 beligiannis 2005 24 thus we solve the problem stated in chen 2008 33 this is proved by characterizing when a resolving subcategory is of finite representation type in terms of decomposition properties of its closure under filtered colimits thus generalizing a classical result of auslander 1976 9 and ringel and tachikawa 1974 63 in the commutative case if r admits a non free finitely generated gorenstein projective module then we show that r is of finite cohen macaulay type iff r is gorenstein and satisfies we also generalize a result of yoshino 2005 68 by characterizing when finitely generated modules without extensions with the ring are gorenstein projective finally we study the stable relative auslander algebra of a virtually gorenstein algebra of finite cohen macaulay type and under the presence of a cluster tilting object we give descriptions of the stable category of gorenstein projective modules in terms of the cluster category associated to the quiver of the stable relative auslander algebra in this setting we show that the cluster category is invariant under derived equivalences

fractionary fuzzy ideals and dedekind domains we show that a finite valued fractionary fuzzy ideal of r with invertible level ideals has a finite minimal generating set as a fuzzy r submodule we also characterize dedekind domains in terms of the factorization of fuzzy ideals as products of prime fuzzy ideals and also in terms of the invertibility of certain fractionary fuzzy ideals we also examine the factorization of fractionary fuzzy ideals

eigenvalue based algorithm and analysis for nonconvex qcqp with one constraint a nonconvex quadratically constrained quadratic programming qcqp with one constraint is usually solved via a dual sdp problem or more s algorithm based on iteratively solving linear systems in this work we introduce an algorithm for qcqp that requires finding just one eigenpair of a generalized eigenvalue problem and involves no outer iterations other than the usually black box iterations for computing the eigenpair numerical experiments illustrate the efficiency and accuracy of our algorithm we also analyze the qcqp solution extensively including difficult cases and show that the canonical form of a matrix pair gives a complete classification of the qcqp in terms of boundedness and attainability and explain how to obtain a global solution whenever it exists