the boundary of the irreducible components for invariant subspace varieties given partitions alpha beta gamma the short exact sequences begin aligned 0 longrightarrow n alpha longrightarrow n beta longrightarrow n gamma longrightarrow 0 end aligned of nilpotent linear operators of jordan types alpha beta gamma respectively define a constructible subset mathbb v alpha gamma beta of an affine variety geometrically the varieties mathbb v alpha gamma beta are of particular interest as they occur naturally and since they typically consist of several irreducible components in fact each littlewood richardson tableau vargamma of shape alpha beta gamma contributes one irreducible component overline mathbb v vargamma we consider the partial order vargamma le mathsf boundary widetilde vargamma on lr tableaux which is the transitive closure of the relation given by mathbb v widetilde vargamma cap overline mathbb v vargamma ne emptyset in this paper we compare the boundary relation with partial orders given by algebraic combinatorial and geometric conditions it is known that in the case where the parts of alpha are at most two all those partial orders are equivalent we prove that those partial orders are also equivalent in the case where beta setminus gamma is a horizontal and vertical strip moreover we discuss how the orders differ in general

statistical mechanics and phase transitions in clustering a new approach to clustering based on statistical physics is presented the problem is formulated as fuzzy clustering and the association probability distribution is obtained by maximizing the entropy at a given average variance the corresponding lagrange multiplier is related to the temperature and motivates a deterministic annealing process where the free energy is minimized at each temperature critical temperatures are derived for phase transitions when existing clusters split it is a hierarchical clustering estimating the most probable cluster parameters at various average variances

the closure of the set of periodic modules over a concealed canonical algebra is regular in codimension one abstract let be a concealed canonical algebra and d the dimension vector of a module which is periodic respect to the action of the auslander reiten translation in the paper we investigate the union of the closures of the orbits of the periodic modules of dimension vector d we show that this set is closed and regular in codimension one

a graduated assignment algorithm for graph matching a graduated assignment algorithm for graph matching is presented which is fast and accurate even in the presence of high noise by combining graduated nonconvexity two way assignment constraints and sparsity large improvements in accuracy and speed are achieved its low order computational complexity o lm where l and m are the number of links in the two graphs and robustness in the presence of noise offer advantages over traditional combinatorial approaches the algorithm not restricted to any special class of graph is applied to subgraph isomorphism weighted graph matching and attributed relational graph matching to illustrate the performance of the algorithm attributed relational graphs derived from objects are matched then results from twenty five thousand experiments conducted on 100 mode random graphs of varying types graphs with only zero one links weighted graphs and graphs with node attributes and multiple link types are reported no comparable results have been reported by any other graph matching algorithm before in the research literature twenty five hundred control experiments are conducted using a relaxation labeling algorithm and large improvements in accuracy are demonstrated

structured perturbation of the brunovsky form a particular case we study the variation of the feedback invariants of a complex rectangular n x n m matrix when we make small additive perturbations on the last m columns in the particular case when the square n x n submatrix is similar to a block diagonal matrix with two blocks in the diagonal associated with the controllable part and the noncontrollable part of the pair a b respectively

sylvester matrix equation for matrix pencils abstract we study a homogeneous linear matrix equation system related to the strict equivalence of matrix pencils we obtain the dimension of the vector space of its solutions in terms of the invariants of the strict equivalence we give a characterization of the strict equivalence of matrix pencils by rank tests and we extend roth s criterion for the corresponding nonhomogeneous system

degenerations in a class of matrix varieties and prinjective modules abstract let k be an algebraically closed field and let i be a finite partially ordered set to any vector v n i we associate the irreducible variety mat v i with an action of the algebraic group g v i see section 6 the main aim of this paper is to give a description of degenerations of orbits of this action in terms of homomorphisms and extensions of prinjective ki modules

the possible dimensions of the higher secant varieties let x c p be an irreducible closed subvariety of projective space the kth higher secant variety of x denoted x is the closure of the union of all linear spaces spanned by k points of x let a ix dim x dim x l it is known that the sequence ty x is nonincreasing and that there may be at most one 1 in such a sequence we prove that these two conditions are also sufficient conditions for an arbitrary sequence a of nonnegative integers to be the fl x s for an irreducible variety x to this end we exhibit the ideals of the higher secant varities of rational normal scrolls as determinantal ideals this allows us to compute their dimensions showing that there is in fact always a rational normal scroll x such that ak x a for any sequence a satisfying the two conditions above adlandsvik al a2 and flenner and vogel f v much of this work has centered on the behavior of the differences ak x dimx dimx 1 of the dimensions of consecutive higher secant varieties it is well known that this sequence is nonincreasing and that there is at most one 1 in the sequence palatini claimed moreover that once ak x 1 were counterexamples a2 we answer here the question of what sequences actually appear as the ai x s of some variety x some of the guiding examples of varieties with degenerate higher secant varieties come from rational normal scrolls explicit equations for the scrolls themselves have been known for some time cf acgh we give a determi nantal presentation of the ideals of their higher secant varieties generalizing a result for rational normal scrolls in w this presentation enables us to compute the dimensions of all the higher secant varieties of a rational normal scroll and gives examples of varieties x with a prescribed nonincreasing ak x with at most one 1

normality of orbit closures for directing modules over tame algebras abstract we show that the orbit closures for directing modules over tame algebras are normal and cohen macaulay the proof is based on degenerations to normal toric varieties

compressed algebras artin algebras having given socle degrees and maximal length j emsalem and the author showed in 18 that a general polynomialf of degree j in the ring sp k yl yr has jr rl 1 linearly independent partial derivates of order i for i 0 1 t j 2 here we generalize the proof to show that the various partial derivates of s polynomials of specified degrees are as independent as possible given the room available using this result we construct and describe the varieties g e and z e parametrizing the graded and nongraded compressed algebra quotients a r i of the power series ring r k xl sxr having given socle type e these algebras are artin algebras having maximal length dimk a possible given the embedding degree r and given the socle type sequence e els s es where ei is the number of generators of the dual module a of a having degree i the variety z e is locally closed irreducible and is a bundle over g e fibred by affine spaces an whose dimension is known we consider the compressed algebras a new class of interesting algebras and a source of examples many of them are nonsmoothable have no deformation to k k for dimension reasons for some choices of the sequence e d buchsbaum d eisenbud and the author have shown that the graded compressed algebras of socle type e have almost linear minimal resolutions over r with ranks and degrees determined by e other examples have given type e dimk socle a and are defined by an ideal i with certain given numbers of generators in r

multidimensional scaling by iterative majorization using radial basis functions abstract this paper considers the use of radial basis functions for modelling the non linear transformation of a data set obtained by a multidimensional scaling analysis this approach has two advantages over conventional nonmetric multidimensional scaling it reduces the number of parameters to estimate and it provides a transformation that may be used on an unseen test set a scheme based on iterative majorization is proposed for obtaining the parameters of the network

classifying representations by way of grassmannians let a be a finite dimensional algebra over an algebraically closed field criteria are given which characterize existence of a fine or coarse moduli space classifying up to isomorphism the representations of a with fixed dimension d and fixed squarefree top t next to providing a complete theoretical picture some of these equivalent conditions are readily checkable from quiver and relations of a in the case of existence of a moduli space unexpectedly frequent in light of the stringency of fine classification this space is always projective and in fact arises as a closed subvariety brassy of a classical grassmannian even when the full moduli problem fails to be solvable the variety grass t d is seen to have distinctive properties recommending it as a substitute for a moduli space as an application a characterization of the algebras having only finitely many representations with fixed simple top is obtained in this case of finite local representation type at a given simple t the radical layering j l m j l 1 m l 0 is shown to be a classifying invariant for the modules with top t this relies on the following general fact obtained as a byproduct proper degenerations of a local module m never have the same radical layering as m

the change of similarity invariants under row perturbations the problem of the possible similarity classes of all the matrices obtained by small perturbations on some rows of a given complex square matrix is under consideration some necessary conditions on these similarity classes are provided the near converse problem is also studied under what conditions can a given collection of polynomials or partitions be the invariant factors or the weyr characteristic of a matrix obtained by perturbing some rows of a given matrix a complete answer is provided in some particular cases the obtained conditions can all be derived from the previously mentioned necessary conditions

on open orbits and their complements 1 1 we fix an algebraically closed field k and a quiver q without oriented cycles whose vertex set q we identify with 1 n for a in the set q of arrows of q we denote by ta and ha the tail and the head of a respectively a representation x of q consists of a family x i in qo of finite dimensional k vector spaces and a family x a x ta x ha a e q1 of k linear maps a morphism f x y between two representations of q is a family f i x i y i in qo of k linear maps satisfying f ha ox a y a of ta for all arrows a we denote the category of representations of q by mod q it is an abelian category of global dimension at most 1 we will denote the only possibly non trivial extension group ext by ext the dimension vector of a representation x of q is the vector dim x dim x l dim x n in n and the dimension of x is the natural number dim x dim x 1 dim x n the support of x is the full subquiver of q whose vertices are i x i zo for a vector d d d e n we define r q d to be the set of representations x of q such that x i k for ie qo thus r q d is the product r q d n hom kdru kdhe

generic well posedness of nonconvex optimal control problems the tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions in our previous work a generic existence and uniqueness result with respect to variations of the integrand of the integral functional without the convexity condition was established for a class of optimal control problems satisfying the cesari growth condition in this paper we extend this generic existence and uniqueness result to a class of optimal control problems in which the right hand side of differential equations is also subject to variations

top stable degenerations of finite dimensional representations i abstractgiven a finite dimensional representation m of a finite dimensional algebra two hierarchies of degenerations of m are analyzed in the context of their natural orders the poset of those degenerations of m which share the top m jm with m here j denotes the radical of the algebra and the sub poset of those which share the full radical layering gl j l m j l 1 mgr l 0 with m in particular the article addresses the existence of proper top stable or layer stable degenerations more generally it addresses the sizes of the corresponding posets including bounds on the lengths of saturated chains as well as structure and classification

generic complete eigenstructures for sets of matrix polynomials with bounded rank and degree abstract the set pol d r m n of m n complex matrix polynomials of grade d and normal rank at most r in a complex d 1 m n dimensional space is studied for r 1 min m n 1 we show that pol d r m n is the union of the closures of the r d 1 sets of matrix polynomials with rank r degree exactly d and explicitly described complete eigenstructures in addition for the full rank rectangular polynomials i e r min m n and m n we show that pol d r m n coincides with the closure of a single set of the polynomials with rank r degree exactly d and the described complete eigenstructure these complete eigenstructures correspond to generic m n matrix polynomials of grade d and rank at most r

the complexity of minimal degenerations for representation finite selfinjective algebras abstract minimal degenerations of modules over all but one exceptional class of representation finite selfinjective algebras are given by extensions g zwara colloq math 75 1998 91 this means in terms of the complexity of degenerations that these minimal degenerations have complexity one the minimal degenerations of modules for the exceptional class of representation finite selfinjective algebras are studied by r aehle j london math soc 66 2002 73 in this paper it is shown that these degenerations have complexity less or equal to two

applications of littlewood richardson tableaux to computing generic extension of semisimple invariant subspaces of nilpotent linear operators abstract the main aim of the paper is to present a combinatorial algorithm that applying littlewood richardson tableaux with entries equal to 1 computes generic extensions of semisimple invariant subspaces of nilpotent linear operators moreover we discuss geometric properties of generic extensions and their connections with combinatorics

stratification of full rank polynomial matrices we show that perturbations of polynomial matrices of full normal rank can be analyzed viathe study of perturbations of companion form linearizations of such polynomial matrices it is proved that a

regularity in codimension one of orbit closures in module varieties let md k denote the space of d d matrices with coefficients in an algebraically closed field k let x be an orbit closure in the product md k t equipped with the action of the general linear group gld k by simultaneous conjugation we show that x is regular at any point y such that the orbit of y has codimension one in x the proof uses mainly the representation theory of associative algebras

change of the congruence canonical form of 2 by 2 and 3 by 3 matrices under perturbations and bundles of matrices under congruence we construct the hasse diagrams g2 and g3 for the closure ordering on the sets of congruence classes of 2 2 and 3 3 complex matrices in other words we construct two directed graphs whose vert

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

almost split sequences in subcategories let r be a commutative artin ring and let i be an r algebra which is a finitely generated r module in 5 auslander and reiten introduced the notion of an almost split sequence in mod a the category of finitely generated modules over a and the existence and uniqueness of such short exact sequences were established there the theory of almost split sequences developed further and similar results for certain subcategories of mod i were proved by bautista and martinez 8 and roggenkamp 9 the main purpose of this paper is to develop a more general theory for subcategories of mod i having almost split sequences which has the previous examples as special cases notions closely related to that of almost split sequences are those of minimal left and right almost split morphisms these notions were introduced in 61 for mod i and further developed in 7 for subcategories of mod ii this paper is based upon 7 and we therefore use the same notations and conventions as used there we now give some definitions and then proceed to describe the content of the paper section by section let c be a subcategory of mod ii then a morphism g b c in c is said to be a right almost split morphism in c if i g is not a splittable epimorphism and ii whenever there is a nonsplittable epimorphism h c c in c there exists an h c b such that gh h dually a morphismfi a f b in c is said to be left almost split if i f is not a splittable monomorphism and ii whenever there is a nonsplittable monomorphism h a a there exists an h b a such that h f h we say that c has right almost split morphisms if for all indecomposable objects

generic well posedness of optimal control problems without convexity assumptions the tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions in a j zaslavski nonlinear anal to appear a generic existence and uniqueness result with respect to variations of the integrand of the integral functional without the convexity condition was established for a class of optimal control problems satisfying the cesari growth condition in this paper we extend the generic existence and uniqueness result in a j zaslavski nonlinear anal to appear to a class of optimal control problems in which constraint maps are also subject to variations the main result of the paper is obtained as a realization of a variational principle extending the variational principle introduced in a d ioffe and a j zaslavski siam j control optim 38 2000 pp 566 581

generic well posedness of variational problems without convexity assumptions the tonelli existence theorem in the calculus of variations and its subsequent modifications were established for integrands f which satisfy convexity and growth conditions in our previous work a generic well posedness result with respect to variations of the integrand of the integral functional without the convexity condition was established for a class of optimal control problems satisfying the cesari growth condition in this paper we extend this generic well posedness result to two classes of variational problems in which the values at the end points are also subject to variations the main results of the paper are obtained as realizations of a general variational principle

on degenerations between preprojective modules over wild quivers abstract we study minimal degenerations between preprojective modules over wild quivers asymptotic properties of such degenerations are studied with respect to codimension and numbers of indecomposable direct summands we provide families of minimal disjoint degenerations of arbitrary codimension for almost all wild quivers and show that no such examples exist in the remaining cases

on degenerations and extensions of finite dimensional modules abstract we derive a cancellation theorem for degenerations of modules that says in particular that projective or injective common direct summands can always be neglected combining the cancellation result with the existence of almost split sequences we characterize the orbit closure of a module living on preprojective components by the fact that the dimension of the homomorphism space to any other module does not decrease for representation directed algebras whence in particular for path algebras of dynkin quivers we provide an alternative proof which shows in addition that any minimal degenerationnofmcomes from an exact sequence with middle termmwhose end terms add up ton by a careful examination the same is true for degenerations of matrix pencils having used so far the existence of certain extensions to obtain degenerations we then turn the tables and use degenerations to produce a lot of interesting short exact sequences in particular we show that any non simple indecomposable over a tame quiver is an extension of an indecomposable and a simple

canonical form associated with the problem of perturbation of one column of a controllable pair abstract let a b be a completely controllable matrix pair when we consider the problem of characterizing the controllability indices of all the matrix pairs obtained by small perturbations on one column of b a new equivalence relation arises in a natural way this equivalence relation is a kind of partial feedback equivalence as a consequence the controllability indices are invariant but they do not form a complete system of invariants we have found new invariants for this equivalence relation and we have obtained a canonical form in terms of these invariants

miniversal deformations of matrices under congruence and reducing transformations arnold 1971 1 constructed a miniversal deformation of a square complex matrix under similarity that is a simple normal form to which not only a given square matrix a but all matrices b close to it can be reduced by similarity transformations that smoothly depend on the entries of b we give miniversal deformations of matrices of sesquilinear forms that is of square complex matrices under congruence and construct an analytic reducing transformation to a miniversal deformation analogous results for matrices under congruence were obtained by dmytryshyn futorny and sergeichuk 2012 11

on minimal disjoint degenerations for preprojective representations of quivers we derive a root test for degenerations as described in the title in the case of dynkin quivers this leads to a conceptual proof of the fact that the codimension of a minimal disjoint degeneration is always one for euclidean quivers it enables us to show a periodic behaviour this reduces the classification of all these degenerations to a finite problem that we have solved with the aid of a computer it turns out that the codimensions are bounded by two somewhat surprisingly the regular and preinjective modules play an essential role in our proofs

linear degenerations of flag varieties partial flags defining equations and group actions we continue generalize and expand our study of linear degenerations of flag varieties from cerulli irelli et al math z 287 1 2 615 654 2017 we realize partial flag varieties as quiver grassmannians for equi oriented type a quivers and construct linear degenerations by varying the corresponding quiver representation we prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration the former is the partial analogue of the mf degenerate flag variety and the latter coincides with the partial pbw degenerate flag variety we compute the generating function of the number of orbits in the flat irreducible locus and study the natural family of line bundles on the degenerations from the flat irreducible locus we also describe explicitly the reduced scheme structure on these degenerations and conjecture that similar results hold for the whole flat locus finally we prove an analogue of the borel weil theorem for the flat irreducible locus

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

vector quantization with complexity costs vector quantization is a data compression method by which a set of data points is encoded by a reduced set of reference vectors the codebook a vector quantization strategy is discussed that jointly optimizes distortion errors and the codebook complexity thereby determining the size of the codebook a maximum entropy estimation of the cost function yields an optimal number of reference vectors their positions and their assignment probabilities the dependence of the codebook density on the data density for different complexity functions is investigated in the limit of asymptotic quantization levels how different complexity measures influence the efficiency of vector quantizers is studied for the task of image compression the wavelet coefficients of gray level images are quantized and the reconstruction error is measured the approach establishes a unifying framework for different quantization methods like k means clustering and its fuzzy version entropy constrained vector quantization or topological feature maps and competitive neural networks

experiments in projection and clustering by simulated annealing simulated annealing is a stochastic relaxation algorithm which has been used successfully to optimize functions of many variables this paper analyzes the simulated annealing algorithm when applied to the minimization of functions from two common problems encountered in exploratory pattern analysis projection and clustering the projection is a nonlinear mapping of patterns in high dimension to two dimensions the simulated annealing mapping is compared to gradient descent minimization of the same objective function as well as eigenvector projection the simulated annealing clustering is compared to a k means algorithm empirical results show that simulated annealing can produce results as good as those obtained by conventional methods but are impractical for small data sets because of the high computational cost simulated annealing does in the case of the mapping problem yield a better optimization and better retained structure for large data sets containing tight gaussian clusters

desingularization of quiver grassmannians for dynkin quivers abstract a desingularization of arbitrary quiver grassmannians for representations of dynkin quivers is constructed in terms of quiver grassmannians for an algebra derived equivalent to the auslander algebra of the quiver

perturbation theory of matrix pencils through miniversal deformations abstract we give new proofs of several known results about perturbations of matrix pencils in particular we give a direct and constructive proof of andrzej pokrzywa s theorem 1983 in which the closure of the orbit of each kronecker canonical matrix pencil is described in terms of inequalities for invariants of matrix pencils a more abstract description is given by klaus bongartz 1996 by methods of representation theory we formulate and prove pokrzywa s theorem in terms of successive replacements of direct summands in a kronecker canonical pencil first we show that it is sufficient to prove pokrzywa s theorem in two cases for matrices under similarity and for each matrix pencil p q that is a direct sum of two indecomposable pencils then we calculate the kronecker canonical form of pencils that are close to p q in fact the kronecker canonical form is calculated for only those pencils that belong to a miniversal deformation of p q this is sufficient since all pencils in a neighborhood of p q are reduced to them by a smooth strict equivalence transformation

multiple flag varieties of finite type abstract we classify all products of flag varieties with finitely many orbits under the diagonal action of the general linear group we also classify the orbits in each case and construct explicit representatives

degenerations over a singularities and construction of degenerations over commutative rings abstract we give a necessary condition of degeneration via matrix representations and consider degenerations of indecomposable cohen macaulay modules over hypersurface singularities of type a we also provide a method to construct degenerations of finitely generated modules over commutative rings

tight monomials and the monomial basis property abstract we generalize a criterion for tight monomials of quantum enveloping algebras associated with symmetric generalized cartan matrices and a monomial basis property of those associated with symmetric classical cartan matrices to their respective symmetrizable case we then link the two by establishing that a tight monomial is necessarily a monomial defined by a weakly distinguished word as an application we develop an algorithm to compute all tight monomials in the rank 2 dynkin case

variational principles and well posedness in optimization and calculus of variations the concluding result of the paper states that variational problems are generically solvable and even well posed in a strong sense without the convexity and growth conditions always present in individual existence theorems this and some other generic well posedness theorems are obtained as realizations of a general variational principle extending the variational principle of deville godefroy zizler

a generalized majorization method for least souares multidimensional scaling of pseudodistances that may be negative the usual convergence proof of the smacof algorithm model for least squares multidimensional scaling critically depends on the assumption of nonnegativity of the quantities to be fitted called the pseudodistances when this assumption is violated erratic convergence behavior is known to occur three types of circumstances in which some of the pseudodistances may become negative are outlined nonmetric multidimensional scaling with normalization on the variance metric multidimensional scaling including an additive constant and multidimensional scaling under the city block distance model a generalization of the smacof method is proposed to resolve the difficulty that is based on the same rationale frequently involved in robust fitting with least absolute residuals

the bruhat chevalley order of parabolic group actions in general linear groups and degeneration for δ filtered modules abstract let k be an algebraically closed field and v a finite dimensional k space let gl v be the general linear group of v and p a parabolic subgroup of gl v now p acts on its unipotent radical p u and on p u lie p u the lie algebra of p u via the adjoint action more generally we consider the action of p on the l th member of the descending central series of p u denoted by p l u all instances when p acts on p l u for l 0 with a finite number of orbits are known in this note we give a complete combinatorial description of the closure relation on the set of p orbits on p l u i e the bruhat chevalley order for every finite case there is a canonical bijection between the set of p orbits on p l u and the set of isomorphism classes of filtered modules of a particular dimension vector e of a certain quasi hereditary algebra a t l these isomorphism classes in turn are given by the orbits of a reductive group g e on the variety r e of all a t l modules with filtration and dimension vector e the subcategory of a t l mod of all filtered a t l modules of dimension vector e is denoted by f e in our chief result theorem 1 1 we show that provided there is only a finite number of isomorphism classes of indecomposable modules in f the following three posets coincide 1 the bruhat chevalley order on the set of p orbits on p l u 2 the bruhat chevalley order on the set of g e orbits on r e 3 the poset opposite to the so called hom order on the set of isomorphism classes of f e the advantage of this hom order is that it is given purely by discrete invariants and that it can be computed explicitly for any given finite case theorem 1 1 then in turn allows us to explicitly determine the closure relations for the p orbits on p l u with the aid of this hom order we present some examples of hasse diagrams in an appendix

data visualization by multimensional scaling a deterministic annealing approach abstract multidimensional scaling addresses the problem how proximity data can be faithfully visualized as points in a low dimensional euclidean space the quality of a data embedding is measured by a stress function which compares proximity values with euclidean distances of the respective points the corresponding minimization problem is non convex and sensitive to local minima we present a novel deterministic annealing algorithm for the frequently used objective sstress and for sammon mapping derived in the framework of maximum entropy estimation experimental results demonstrate the superiority of our optimization technique compared to conventional gradient descent methods

irreducible components of module varieties an example abstract we determine the irreducible components of the variety of d dimensional modules over the algebra k 2 2 q with q k for any d and we describe any intersection of irreducible components as the zero set of some ideal we show that any such intersection is irreducible to our knowledge this is the first time irreducible components are described by equations for modules over an algebra which is neither representation finite nor hereditary

on perturbations and the equivalence orbit of a matrix pencil abstract two theorems on the canonical kronecker form of a perturbed matrix pencil and the characterization of the closure of the set of all matrix pencils with a fixed kronecker canonical form are given

nonoccurrence of the lavrentiev phenomenon for many optimal control problems in this paper we study nonoccurrence of the lavrentiev phenomenon for a large class of nonconvex optimal control problems identified with the corresponding complete metric space of integrands mathcal m which satisfy a growth condition common in the literature and are lipschitzian on bounded sets we establish that for most elements of mathcal m in the sense of baire category the infimum on the full admissible class of trajectory control pairs is equal to the infimum on a subclass of trajectory control pairs whose controls are bounded by a certain constant

spaces of hankel matrices over finite fields abstract let f be a finite field for each 1 n we construct a 2 n dimensional linear space h n of n n hankel matrices over f such that rank a for all 0 a h n it follows that if w is a linear space of hankel matrices over f such that rank a for all a w then dim w

an explicit description of the irreducible components of the set of matrix pencils with bounded normal rank abstract the set of m n singular matrix pencils with normal rank at most r is an algebraic set with r 1 irreducible components these components are the closure of the orbits under strict equivalence of r 1 matrix pencils which are in kronecker canonical form in this paper we provide a new explicit description of each of these irreducible components which is a parametrization of each component therefore one can explicitly construct any pencil in each of these components the new description of each of these irreducible components consists of the sum of r rank 1 matrix pencils namely a column polynomial vector of degree at most 1 times a row polynomial vector of degree at most 1 where we impose one of these two vectors to have degree zero the number of row vectors with zero degree determines each irreducible component

on degenerations of cohen macaulay modules abstract in relation to degenerations of modules we introduce several partial orders on the set of isomorphism classes of finitely generated modules over a noetherian commutative local ring our main theorem says that under several special conditions any degenerations of maximal cohen macaulay modules are essentially obtained by the degenerations of auslander reiten sequences

the algebraic structure of pencils and block toeplitz matrices abstract we restate several results on the kronecker structure obtainable after a given matrix pencil is affected by an arbitrarily small perturbation and state some new results on the kronecker structure after a new row is appended or a column deleted the results are simple consequences of a theory of majorization for semi infinite integer sequences the bounds after a new row is appended correspond to bounding the controllability indices after adding a single input to a linear time invariant dynamical system we illustrate the known relations between the kronecker indices the segre characteristics and the nullities of certain block toeplitz matrices constructed from the original pencils and discuss how those nullities change when the pencil is affected by a perturbation or a new row

convex potentials and their conjugates in analog mean field optimization this paper deals with the problem of mapping hybrid i e both discrete and continuous constrained optimization problems onto analog networks the saddle point paradigm of mean field methods in statistical physics provides a systematic procedure for finding such a mapping via the notion of effective energy specifically it is shown that within this paradigm to each closed bounded constraint set is associated a smooth convex potential function using the conjugate or the legendre fenchel transform of the convex potential the effective energy can be transformed to yield a cost function that is a natural generalization of the analog hopfield energy descent dynamics and deterministic annealing can then be used to find the global minimum of the original minimization problem when the conjugate is hard to compute explicitly it is shown that a minimax dynamics similar to that of arrow and hurwicz in lagrangian optimization can be used to find the saddle points of the effective energy as an illustration of its wide applicability the effective energy framework is used to derive hopfield like energy functions and descent dynamics for two classes of networks previously considered in the literature winner take all networks and rotor networks even when the cost function of the original optimization problem is not quadratic

on minimal disjoint degenerations of modules over tame path algebras abstract for representations of tame quivers the degenerations are controlled by the dimensions of various homomorphism spaces furthermore there is no proper degeneration to an indecomposable therefore up to common direct summands any minimal degeneration from m to n is induced by a short exact sequence 0 u m v 0 with indecomposable ends that add up to n we study these building blocs of degenerations and we prove that the codimensions are bounded by two therefore a quiver is dynkin resp euclidean resp wild iff the codimension of the building blocs is one resp bounded by two resp unbounded we explain also that for tame quivers the complete classification of all the building blocs is a finite problem that can be solved with the help of a computer

stable degenerations of cohen macaulay modules abstract as a stable analogue of degenerations we introduce the notion of stable degenerations for cohen macaulay modules over a gorenstein local algebra we shall give several necessary and or sufficient conditions for the stable degeneration these conditions will be helpful to see when a cohen macaulay module degenerates to another

hankel determinantal rings have rational singularities abstract hankel determinantal rings i e determinantal rings defined by minors of hankel matrices of indeterminates arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves they may also be viewed as linear specializations of generic determinantal rings we prove that over fields of characteristic zero hankel determinantal rings have rational singularities in the case of positive prime characteristic we prove that they are f pure independent of the characteristic we give a complete description of the divisor class groups of these rings and show that each divisor class group element is the class of a maximal cohen macaulay module

geometry of directing modules over tame algebras throughout the paper k denotes a fixed algebraically closed field by an algebra we mean a finite dimensional k algebra associative with an identity and by a module a finite dimensional left a module the class of algebras may be divided into two disjoint subclasses one class consists of tame algebras for which the indecomposable modules occur in each dimension in a finite number of discrete and a finite number of one parameter families the second class is formed by the wild algebras whose representation theory is as complicated as the study of finite dimensional vector spaces together with two noncommuting endomorphisms for which the classification up to isomorphism is a well known unsolved problem hence we can realistically hope to describe modules only for tame algebras given an algebra a and a nonnegative vector d in the grothendieck group k a of a it is an interesting task to study the 0 affine variety mod d of a modules of dimension vector d and the action a of the corresponding product g d of general linear groups for example we may ask when the variety mod d is irreducible smooth complete a intersection gorenstein cohen macaulay normal the main aim of this paper is to describe the geometry of module varieties mod d for the dimension vectors d of arbitrary directing moda w x ules over tame algebras recall that following 20 an indecomposable a module m is said to be directing if it does not belong to a cycle m a m a a m a m of nonzero nonisomorphisms between inde1 r composable a modules directing modules have played an important role

linear degenerations of flag varieties linear degenerate flag varieties are degenerations of flag varieties as quiver grassmannians for type a flag varieties we obtain characterizations of flatness irreducibility and normality of these degenerations via rank tuples some of them are shown to be isomorphic to schubert varieties and can be realized as highest weight orbits of partially degenerate lie algebras generalizing the corresponding results on degenerate flag varieties to study normality cell decompositions of quiver grassmannians are constructed in a wider context of equioriented quivers of type a

similarity and block similarity we present some equivalent characterizations of the block similarity of matrix pairs with an application to a characterization of the similarity of two square matrices also a survey of some generalizations of properties of the invariants of similarity to the invariants of block similarity is presented

minimal singularities in orbit closures of matrix pencils abstract we show that all singularities occurring in minimal degenerations of matrix pencils are cohen macaulay and regular in codimension 1

geometry of regular modules over canonical algebras we classify canonical algebras such that for every dimension vector of a regular module the corresponding module variety is normal respectively a complete intersection we also prove that for the dimension vectors of regular modules normality is equivalent to irreducibility