some algorithms for semi invariants of quivers abstract we present some theorems and algorithms for calculating perpendicular categories and locally semi simple decompositions we implemented a computer program tetiva based on these algorithms and we offer this program for everybody s use

on the estimation of optical flow relations between different approaches and some new results abstract a common basis is suggested for the optical flow estimation approaches of nagel 1983 haralick and lee 1983 and tretiak and pastor 1984 based on a discussion of these approaches an exact solution for the system of partial differential equations proposed by horn and schunck 1981 is given at gray value corners and extrema the insight gained by this solution results in a modification of the oriented smoothness approach of nagel 1983 which thereby becomes considerably simpler in addition the optical flow estimation approach of hildreth 1983 1984 can be shown to represent a kind of special case of this modified oriented smoothness approach in a more direct manner than discussed in nagel 1984

geometric characterizations of the representation type of hereditary algebras and of canonical algebras abstract we show that a finite connected quiver q with no oriented cycles is tame if and only if for each dimension vector d and each integral weight of q the moduli space m q d s s of semi stable d dimensional representations of q is just a projective space in order to prove this we show that the tame quivers are precisely those whose weight spaces of semi invariants satisfy a certain log concavity property furthermore we characterize the tame quivers as being those quivers q with the property that for each schur root d of q the field of rational invariants k rep q d gl d is isomorphic to k or k t next we extend this latter description to canonical algebras more precisely we show that a canonical algebra is tame if and only if for each generic root d of and each indecomposable irreducible component c of rep d the field of rational invariants k c gl d is isomorphic to k or k t along the way we establish a general reduction technique for studying fields of rational invariants on schur irreducible components of representation varieties

pixelwise adaptive blind optical flow assuming nonstationary statistics we address some of the major issues in optical flow within a new framework assuming nonstationary statistics for the motion field and for the errors problems addressed include the preservation of discontinuities model data errors outliers confidence measures and performance evaluation in solving these problems we assume that the statistics of the motion field and the errors are not only spatially varying but also unknown we thus derive a blind adaptive technique based on generalized cross validation for estimating an independent regularization parameter for each pixel our formulation is pixelwise and combines existing first and second order constraints with a new second order temporal constraint we derive a new confidence measure for an adaptive rejection of erroneous and outlying motion vectors and compare our results to other techniques in the literature a new performance measure is also derived for estimating the signal to noise ratio for real sequences when the ground truth is unknown

geometry and the zero sets of semi invariants for homogeneous modules over canonical algebras abstract we characterize the canonical algebras such that for all dimension vectors of homogeneous modules the corresponding module varieties are complete intersections respectively normal we also investigate the sets of common zeros of semi invariants of non zero degree in important cases in particular we show that for sufficiently big vectors they are complete intersections and calculate the number of their irreducible components

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

hilbert series and degree bounds for matrix semi invariants abstract we study the ring r n m of invariants for the left right action of sl n sl n on m tuples of n n complex matrices we show that r 3 m is generated by invariants of degree 309 for all m then we use a combinatorial description of the invariants to show that r n m cannot be generated by invariants of degree n 2 for large m we also compute the hilbert series for several cases

representation of occurrences for road vehicle traffic our 3d model based computer vision subsystem extracts vehicle trajectories from monocular digitized videos recording road vehicles in inner city traffic steps are documented which import these quantitative geometrical results into a conceptual representation based on a fuzzy metric temporal horn logic fmthl see k h schafer unscharfe zeitlogische modellierung von situationen und handlungen in bildfolgenauswertung und robotik dissertation 1996 the facts created by this import step can be understood as verb phrases which describe elementary actions of vehicles in image sequences of road traffic scenes the current contribution suggests a complete conceptual representation of elementary vehicle actions and reports results obtained by an implementation of this approach from real world traffic videos

koregulare und aquidimensionale darstellungen zusammenfassung the aim of this note is to give a classification of all irreducible representations v of a connected semisimple algebraic group g such that either the quotient v g is smooth or the quotient map v v g has equidimensional fibres it follows by the classification that if the quotient map is equidimensional then the quotient v g is smooth

the zero set of semi invariants for extended dynkin quivers we show that the set of common zeros z q d of all semi invariants vanishing at 0 on the variety rep q d of all representations with dimension vector d of an extended dynkin quiver q under the group gl d is a complete intersection if d is big enough in case rep q d does not contain an open gl d orbit which is the case not considered so far the number of irreducible components of z q d grows with d except if q is an oriented cycle

toward a symbolic representation of intensity changes in images the symbolic representation of gray value variations is studied with emphasis on the gradient of the image function the goal is to relate the results of this analysis to the structure of the picture which is determined by the physics of the image generation process candidates for contour points are the maximal magnitudes of the gray value gradient for different scales in the direction of the gradient based on the output of such a bank of gradient filters a procedure is proposed to select automatically a suitable scale and with that the size of the right convolution kernel the application of poorly adapted filters which make the exact localization of gray value corners or t x and y junctions more difficult is thus avoided possible gaps at such junctions are discussed for images of real scenes and possibilities for the closure of some of these gaps are demonstrated when the extrema of the magnitudes of the gray value gradients are used

comutations underlying the measuremnt of visual motion abstract the organization of movement in a changing image provides a valuable source of information for analyzing the environment in terms of objects their motion in space and their three dimensional structure a description of this movement is not provided to our visual system directly however it must be inferred from the pattern of changing intensity that reaches the eye this paper examines the problem of motion measurement which we formulate as the computation of an instantaneous two dimensional velocity field initial motion measurements occur at the location of significant intensity changes as suggested by marr and ullman 1 these measurements provide only one component of velocity and must be integrated to compute the two dimensional velocity field a fundamental problem for this integration is that motion is not determined uniquely from the changing image an additional constraint of smoothness of the velocity field is formulated based on the physical assumption that surfaces are generally smooth allowing the computation of a unique solution a theoretical analysis of the conditions under which this computation yields the correct velocity field suggests that the solution is physically plausible empirical studies show the results to be consistent with human motion perception

on the littlewood richardson polynomials we prove the equivalence of several descriptions of generators of rings of semiinvariants of quivers due to domokos and zubkov schofield and van den bergh and our earlier work we also show that the dimensions of semi invariants of weights n depend polynomially on n 2002 elsevier science usa all rights reserved

git cones and quivers in this work we improve results of ressayre in geometric invariant theory and generalized eigenvalue problem ii pp 1 25 2008 ressayre in ann inst fourier 180 389 441 2010 on git cones associated to the action of a reductive group g on a projective variety x these results are applied to give a short proof of the derksen weyman theorem that parametrizes bijectively the faces of a rational cone associated to any quiver without oriented cycles an important example of such a cone is the horn cone

quiver generalization of a conjecture of king tollu and toumazet abstract stretching the parameters of a littlewood richardson coefficient of value 2 by a factor of n results in a coefficient of value n 1 12 9 19 we give a geometric proof of a generalization for representations of quivers

a functorial construction of moduli of sheaves we show how natural functors from the category of coherent sheaves on a projective scheme to categories of kronecker modules can be used to construct moduli spaces of semistable sheaves this construction simplifies or clarifies technical aspects of existing constructions and yields new simpler definitions of theta functions about which more complete results can be proved

module varieties over canonical algebras abstract the main purpose of this paper is the study of module varieties over the class of canonical algebras providing a rich source of examples of varieties with interesting properties our main tool is a stratification of module varieties which was recently introduced by richmond this stratification does not require a precise knowledge of the module category if it is finite then it provides a method to classify irreducible components we determine the canonical algebras for which this stratification is finite in this case we describe the algorithm for calculating the dimension of the variety and the number of irreducible components of maximal dimension for an infinite family of examples we give easy combinatorial criteria for irreducibility cohen macaulay and normality

general representations of quivers we investigate general representations of a quiver q there are two main questions that we shall address the first which was raised by kac is to find the dimension vectors of the summands of a general representation of dimension vector a the second is to find the dimension vectors of the subrepresentations of a general representation of dimension vector a these two questions are closely related one consequence of our investigation is to find an algorithm to compute the canonical decomposition of an arbitrary dimension vector in terms of the euler form of the quiver

semi invariants of gentle algebras by deformation method and sphericity abstract we investigate the algebras of semi invariants for gentle algebras applying so called deformation method and using our observation that varieties of complexes and circular complexes are spherical this yields an explicit description of generators and relations for the algebras of semi invariants

semi direct products of lie algebras and covariants abstract let q be a connected algebraic group with lie algebra q symmetric invariants of q i e the q invariants in the symmetric algebra s q of q is a first approximation to the understanding of the coadjoint action q q and coadjoint q orbits in this article we study a class of non reductive lie algebras where the description of the symmetric invariants is possible and the coadjoint representation has a number of nice invariant theoretic properties if g is a semisimple group with lie algebra g and v is g module then we define q to be the semi direct product of g and v then we are interested in the case where the generic isotropy group for the g action on v is reductive and commutative it turns out that in this case symmetric invariants of q can be constructed via certain g equivariant maps from g to v covariants

detection of intensity changes with subpixel accuracy using laplacian gaussian masks we present a system that takes a gray level image as input locates edges with subpixel accuracy and links them into lines edges are detected by finding zero crossings in the convolution of the image with laplacian of gaussian log masks the implementation differs markedly from m i t s as we decompose our masks exactly into a sum of two separable filters instead of the usual approximation by a difference of two gaussians dog subpixel accuracy is obtained through the use of the facet model 1 we also note that the zero crossings obtained from the full resolution image using a space constant for the gaussian and those obtained from the 1 n resolution image with 1 n pixel accuracy and a space constant of n for the gaussian are very similar but the processing times are very different finally these edges are grouped into lines using the technique described in 2

semi invariants of quivers and saturation for littlewood richardson coefficients q is defined in the space of all weights by one homogeneous linear equation and by a finite set of homogeneous linear inequalities in particular the set q is saturated i e if n q then also q these results when applied to a special quiver q tn n n and to a special dimension vector show that the gln module v appears in v v if and only if the partitions and satisfy an explicit set of inequalities this gives new proofs of the results of klyachko 7 3 and knutson and tao 8 the proof is based on another general result about semi invariants of quivers theorem 1 in the paper 10 schofield defined a semi invariant cw for each indecomposable representation w of q we show that the semi invariants of this type span each weight space in si q this seems to be a fundamental fact connecting semi invariants and modules in a direct way given this fact the results on sets of weights follow at once from the results in another paper of schofield 11

a theoretical framework for convex regularizers in pde based computation of image motion many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy typically these energy functionals consist of two terms a data term which requires e g that a brightness constancy assumption holds and a regularizer that encourages global or piecewise smoothness of the flow field in this paper we present a systematic classification of rotation invariant convex regularizers by exploring their connection to diffusion filters for multichannel images this taxonomy provides a unifying framework for data driven and flow driven isotropic and anisotropic as well as spatial and spatio temporal regularizers while some of these techniques are classic methods from the literature others are derived here for the first time we prove that all these methods are well posed they posses a unique solution that depends in a continuous way on the initial data an interesting structural relation between isotropic and anisotropic flow driven regularizers is identified and a design criterion is proposed for constructing anisotropic flow driven regularizers in a simple and direct way from isotropic ones its use is illustrated by several examples

semi invariants for quivers with relations for a given quiver q and a dimension vector let si q be the ring of semi invariants for dimensional representations schofield defined in s for any dimensional representation v with 0 a semi invariant c si q we recall that we proved in dw that the ring of semi invariants si q is generated by such schofield semi invariants in this note we generalize the results of dw to quivers with relations in the case when the base field has characteristic zero we make this assumption throughout the paper let q be a quiver and let i be an ideal generated by admissible relations the variety rep q i of representations of q satisfying the relations from i does not have to be irreducible we are interested in the generators of rings of semiinvariants on the irreducible components of rep q i it is clear that the ring of semi invariants on such components will be generated by the restrictions of the semi invariants c where v is a representation of q however such description is not very useful we might have a case when the quiver q with relations i is of a finite or tame type but q is wild in such case one would still like to describe the semi invariants of q i explicitly but the description of indecomposable representations of q is impossible therefore it

reliable estimation of dense optical flow fields with large displacements in this paper we show that a classic optical flow technique by nagel and enkelmann 1986 ieee trans pattern anal mach intell vol 8 pp 565 593 can be regarded as an early anisotropic diffusion method with a diffusion tensor we introduce three improvements into the model formulation that i avoid inconsistencies caused by centering the brightness term and the smoothness term in different images ii use a linear scale space focusing strategy from coarse to fine scales for avoiding convergence to physically irrelevant local minima and iii create an energy functional that is invariant under linear brightness changes applying a gradient descent method to the resulting energy functional leads to a system of diffusion reaction equations we prove that this system has a unique solution under realistic assumptions on the initial data and we present an efficient linear implicit numerical scheme in detail our method creates flow fields with 100 density over the entire image domain it is robust under a large range of parameter variations and it can recover displacement fields that are far beyond the typical one pixel limits which are characteristic for many differential methods for determining optical flow we show that it performs better than the optical flow methods with 100 density that are evaluated by barron et al 1994 int j comput vision vol 12 pp 43 47 our software is available from the internet

motion field and optical flow qualitative properties it is shown that the motion field the 2 d vector field which is the perspective projection on the image plane of the 3 d velocity field of a moving scene and the optical flow defined as the estimate of the motion field which can be derived from the first order variation of the image brightness pattern are in general different unless special conditions are satisfied therefore dense optical flow is often ill suited for computing structure from motion and for reconstructing the 3 d velocity field by algorithms which require a locally accurate estimate of the motion field a different use of the optical flow is suggested it is shown that the smoothed optical flow and the motion field can be interpreted as vector fields tangent to flows of planar dynamical systems stable qualitative properties of the motion field which give useful informations about the 3 d velocity field and the 3 d structure of the scene usually can be obtained from the optical flow the idea is supported by results from the theory of structural stability of dynamical systems

on the variety of complexes publisher summary this chapter focuses on the characteristic free study of the buchsbaum eisenbud varieties of complexes in detail with the use of young diagrams and young tableaux to give an explicit basis for buchsbaum and eisenbud have studied the varieties of complexes from the point of view of resolutions and have proven the results for the variety w k 1 a young tableau is a filling of the boxes of a young diagram with integers out of 1 2 and is called the shape of the young tableau the chapter presents a proposition that states that ach element in can be written as a linear combination of standard multitableaux

invariant theory of canonical algebras abstract based on the first fundamental theorem of classical invariant theory we present a reduction technique for computing relative invariants for quivers with relations this is applied to the invariant theory of canonical algebras and yields an explicit construction of the moduli spaces together with the quotient morphisms from the corresponding representation spaces for families of modules with a fixed dimension vector belonging to the central sincere separating subcategory by means of a tilting process we extend these results to the invariant theory of concealed canonical algebras thus covering the cases of tame hereditary tame concealed and tubular algebras respectively our approach yields in particular a uniform treatment to an essential part of the invariant theory of extended dynkin quivers a topic popular over the years but stretches far beyond since also concealed canonical algebras of tubular or wild representation type are covered

classification of multiplicity free symplectic representations abstract let g be a connected reductive group acting on a finite dimensional vector space v assume that v is equipped with a g invariant symplectic form then the ring o v of polynomial functions becomes a poisson algebra the ring o v g of invariants is a sub poisson algebra we call v multiplicity free if o v g is poisson commutative i e if f g 0 for all invariants f and g alternatively g also acts on the weyl algebra w v and v is multiplicity free if and only if the subalgebra w v g of invariants is commutative in this paper we classify all multiplicity free symplectic representations

product of diagonal elements of matrices abstract let a c n n for 1 k n let w k a k i 1 uau ii uu i when k 1 the set reduces to the classical numerical range w a of a in this note we investigate the geometrical properties such as convexity star shapedness and simple connectedness of wk a furthermore it is shown that if wk a has some special boundary points then a is unitarily reducible counterexamples for a conjecture on wn a due to marcus are given

optical flow with an intensity weighted smoothing matching algorithms which use image structure such as edges and junctions cannot perform well on unstructured images for such images global gradient based methods may be more appropriate some of the assumptions involved in applying discretized conservation equations such as the optical flow equation are discussed and an intensity weighted method of estimating dense displacement fields is presented which attempts to avoid some of the problems of the standard methods the only derivative required is the intensity gradient and the method includes procedures for automatic parameter evaluation the algorithm is expected to perform better than conventional algorithms on images without strong texture as is demonstrated on the test set of images

tripartite to bipartite entanglement transformation by stochastic local operations and classical communication and the structure of matrix spaces we study the problem of transforming a tripartite pure state to a bipartite one using stochastic local operations and classical communication slocc it is known that the tripartite to bipartite slocc convertibility is characterized by the maximal schmidt rank of the given tripartite state i e the largest schmidt rank over those bipartite states lying in the support of the reduced density operator in this paper we further study this problem and exhibit novel results in both multi copy and asymptotic settings utilizing powerful results from the structure of matrix spaces in the multi copy regime we observe that the maximal schmidt rank is strictly super multiplicative i e the maximal schmidt rank of the tensor product of two tripartite pure states can be strictly larger than the product of their maximal schmidt ranks we then provide a full characterization of those tripartite states whose maximal schmidt rank is strictly super multiplicative when taking tensor product with itself notice that such tripartite states admit strict advantages in tripartite to bipartite slocc transformation when multiple copies are provided in the asymptotic setting we focus on determining the tripartite to bipartite slocc entanglement transformation rate computing this rate turns out to be equivalent to computing the asymptotic maximal schmidt rank of the tripartite state defined as the regularization of its maximal schmidt rank despite the difficulty caused by the super multiplicative property we provide explicit formulas for evaluating the asymptotic maximal schmidt ranks of two important families of tripartite pure states by resorting to certain results of the structure of matrix spaces including the study of matrix semi invariants these formulas turn out to be powerful enough to give a sufficient and necessary condition to determine whether a given tripartite pure state can be transformed to the bipartite maximally entangled state under slocc in the asymptotic setting applying the recent progress on the non commutative rank problem we can verify this condition in deterministic polynomial time

semi stable subcategories for euclidean quivers in this paper we study the semi stable subcategories of the category of representations of a euclidean quiver and the possible intersections of these subcategories contrary to the dynkin case we find out that the intersection of semi stable subcategories may not be semi stable however only a finite number of exceptions occur and we give a description of these subcategories moreover one can attach a simplicial complex to any acyclic quiver q and this simplicial complex allows one to completely determine the canonical decomposition of any dimension vector this simplicial complex has a nice description in the euclidean case it is described using an arrangement of convex pieces of hyperplanes each piece being indexed by a real schur root or a set of quasi simple objects

on the hu hurley tam conjecture concerning the generalized numerical range abstract suppose m and n are integers such that 1 m n and h is a subgroup of the symmetric group s m of degree m define the generalized matrix function associated with the principal character of the group h on an m m matrix b b ij by d h b h j 1 m b j j and define the generalized numerical range of an n n matrix a associated with d h by w h a d h v av v is n m such that v v i m it is known that w h a is convex if m 1 or if m n 2 hu hurley and tam made the following conjecture suppose h s m 2 m n with m n 2 2 let a m n be a normal matrix then w h a is convex if and only if a is a multiple of a hermitian matrix in this note counterexamples are given to show that the conjecture is not true when m some techniques are developed to show that the conjecture is valid under more restrictive assumptions

perpendicular categories null cones and dense orbits abstract let q be a quiver k an algebraically closed field and d a prehomogeneous dimension vector for q with canonical decomposition d i 1 r i dim t i we denote by z q d the set of common zeros of all semi invariants in k rep q d of positive degree we study the relations between the existence of a dense gl d orbit in the null cone z q d and the position of the representations t 1 t r in the auslander reiten quiver associated to q

symmetric quivers invariant theory and saturation theorems for the classical groups abstract let g denote either a special orthogonal group or a symplectic group defined over the complex numbers we prove the following saturation result for g given dominant weights 1 r such that the tensor product v n 1 v n r contains nonzero g invariants for some n 1 we show that the tensor product v 2 1 v 2 r also contains nonzero g invariants this extends results of kapovich millson and belkale kumar and complements similar results for the general linear group due to knutson tao and derksen weyman our techniques involve the invariant theory of quivers equipped with an involution and the generic representation theory of certain quivers with relations

estimation of optical flow based on higher order spatiotemporal derivatives in interlaced and non interlaced image sequences abstract this contribution 1 investigates local differential techniques for estimating optical flow and its derivatives based on the brightness change constraint by using the tensor calculus representation we build the taylor expansion of the gray value derivatives as well as of the optical flow in a spatiotemporal neighborhood such a formulation provides a unifying framework for all existing local differential approaches and allows to derive new systems of equations for the estimation of the optical flow and of its derivatives we also tested various optical flow estimation approaches on real image sequences recorded by a calibrated camera which was fixed on the arm of a robot by moving the arm of the robot along a precisely defined trajectory we can determine the true displacement rate of scene surface elements projected into the image plane and compare it quantitatively with the results of different optical flow estimators since the optical flow estimators are based on gray value derivatives of up to fourth order we were forced to develop modified gaussian derivative filters to obtain acceptable estimates for the derivatives further we show quantitatively that these filters contribute to a much more robust optical flow estimation in addition successive lines of tv cameras have an offset in time due to the interlace technique we demonstrate the adaptation of filter kernels for estimating higher order spatiotemporal derivatives in interlaced image sequences

product of two diagonal entries of a 3 by 3 normal matrix abstract the set of product of two diagonal entries of 3 by 3 normal matrices unitarily similar to a diagonal matrix is a triangle if the diagonal entries form an acute angled triangle inscribed to the unit circle

variational optical flow computation in real time this paper investigates the usefulness of bidirectional multigrid methods for variational optical flow computations although these numerical schemes are among the fastest methods for solving equation systems they are rarely applied in the field of computer vision we demonstrate how to employ those numerical methods for the treatment of variational optical flow formulations and show that the efficiency of this approach even allows for real time performance on standard pcs as a representative for variational optic flow methods we consider the recently introduced combined local global method it can be considered as a noise robust generalization of the horn and schunck technique we present a decoupled as well as a coupled version of the classical gau spl szlig seidel solver and we develop several multigrid implementations based on a discretization coarse grid approximation in contrast with standard bidirectional multigrid algorithms we take advantage of intergrid transfer operators that allow for nondyadic grid hierarchies as a consequence no restrictions concerning the image size or the number of traversed levels have to be imposed in the experimental section we juxtapose the developed multigrid schemes and demonstrate their superior performance when compared to unidirectional multigrid methods and nonhierachical solvers for the well known 316 spl times 252 yosemite sequence we succeeded in computing the complete set of dense flow fields in three quarters of a second on a 3 06 ghz pentium4 pc this corresponds to a frame rate of 18 flow fields per second which outperforms the widely used gau spl szlig seidel method by almost three orders of magnitude

generic modules for gentle algebras abstract in this article we construct the generic modules in each irreducible component of variety of dimensional modules of a triangular gentle algebra the construction is completely combinatorial and allows for determination of canonical decomposition of irreducible components as well as calculation of the dimension of the higher self extension spaces for generic modules

quivers long exact sequences and horn type inequalities we give necessary and sufficient inequalities for the existence of long exact sequences of m finite abelian p groups with fixed isomorphism types this problem is related to some generalized littlewood richardson coefficients that we define in this paper we also show how this problem is related to eigenvalues of hermitian matrices satisfying certain in equalities when m 3 we recover the horn type inequalities that solve the saturation conjecture for littlewood richardson coefficients and horn s conjecture

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

poincare series of semi invariants of 2 2 matrices abstract the poincare series of the algebra of sl 2 c sl 2 c invariants of m tuples of 2 2 matrices is presented both as a rational function and as a series of schur functions we show that this algebra of invariants is generated by the determinants the mixed discriminants and the discriminants of 2 2 matrices consequences on invariants of three dimensional matrices of the shape 2 2 m are discussed for arbitrary n 2 we prove an explicit functional equation for the poincare series of the sl n c sl n c invariants of m tuples of n n matrices

the accuracy of the computation of optical flow and of the recovery of motion parameters the accuracy and the dependence on parameters of a general scheme for the analysis of time varying image sequences are discussed the approach is able to produce vector fields from which it is possible to recover 3 d motion parameters such as time to collision and angular velocity the numerical stability of the computed optical flow and the dependence of the recovery of 3 d motion parameters on spatial and temporal filtering is investigated by considering optical flows computed on subsampled images or along single scanlines it is also possible to recover 3 d motion parameters from reduced optical flows an adequate estimate of time to collision can be obtained from sequences of images with spatial resolution reduced to 128 128 pixels or from sequences of single scanlines passing near the focus of expansion the use of kalman filtering increases the accuracy and the robustness of the estimation of motion parameters the proposed approach seems to be able to provide not only a theoretical background but also practical tools that are adequate for the analysis of time varying image sequences

multiserial and special multiserial algebras and their representations abstract in this paper we study multiserial and special multiserial algebras these algebras are a natural generalization of biserial and special biserial algebras to algebras of wild representation type we define a module to be multiserial if its radical is the sum of uniserial modules whose pairwise intersection is either 0 or a simple module we show that all finitely generated modules over a special multiserial algebra are multiserial in particular this implies that in analogy to special biserial algebras being biserial special multiserial algebras are multiserial we then show that the class of symmetric special multiserial algebras coincides with the class of brauer configuration algebras where the latter are a generalization of brauer graph algebras we end by showing that any symmetric algebra with radical cube zero is special multiserial and so in particular it is a brauer configuration algebra

non commutative resolutions of quotient singularities for reductive groups in this paper we generalize standard results about non commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups we show in particular that quotient singularities for reductive groups always have non commutative resolutions in an appropriate sense moreover we exhibit a large class of such singularities which have twisted non commutative crepant resolutions we discuss a number of examples both new and old that can be treated using our methods notably we prove that twisted non commutative crepant resolutions exist in previously unknown cases for determinantal varieties of symmetric and skew symmetric matrices in contrast to almost all prior results in this area our techniques are algebraic and do not depend on knowing a commutative resolution of the singularity

tame quivers semi invariants and complete intersections abstract let q be a tame connected quiver and d a prehomogeneous dimension vector for q then the set of common zeros of all non constant semi invariants for d dimensional representations of q is not too far from being a complete intersection in fact there is a bound 4 on the difference between the number of generators for an ideal defining the zero set and its codimension

invariants of quivers under the action of classical groups abstract we consider a generalization of representations of quivers that can be derived from the ordinary representations of quivers by considering a product of arbitrary classical linear groups instead of a product of the general linear groups and by considering the dual action of groups on vertex vector spaces together with the usual action a generating system for the corresponding algebra of invariants is found in particular semi invariants of supermixed representations of quivers are established as a consequence a generating system for the algebra of so n invariants of several matrices is constructed over a field of characteristic different from 2 the proof uses the reduction to semi invariants of mixed representations and the decomposition formula that generalizes amitsur s formula for the determinant

left symmetric algebras for mathfrak gl n we study the classification problem for left symmetric algebras with commutation lie algebra gr n in characteristic 0 the problem is equivalent to the classification of etale affine representations of g n algebraic invariant theory is used to characterize those modules for the algebraic group sl n which belong to affine etale representations of gr n from the classification of these modules we obtain the solution of the classification problem for gl n as another application of our approach we exhibit left symmetric algebra structures on certain reductive lie algebras with a one dimensional center and a non simple semisimple ideal

on a constraint equation for the estimation of displacement rates in image sequences the commonly used constraint equation delta g sup t u g sub t 0 for the estimation of optical flow can only be justified by assumptions that are in general far too restrictive for image sequences of real world scenes b g schunck 1985 86 recently argued that a constraint equation for the estimation of what he called image flow has to include a term containing the divergence of this image flow without presenting however a stringent derivation based on perspective projection from 3 d scene space the present author derives a constraint equation based on a combination of perspective projection and notions from differential geometry in addition he demonstrates the quantitive effects of taking into account radiometric considerations based on the use of lambertian reflection properties and isotropic illumination in scene space

scalar extensions of derived categories and non fourier mukai functors abstract orlov s famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a fourier mukai functor this result has been extended by lunts and orlov to include functors from perfect complexes to quasi coherent complexes in this paper we show that the latter extension is false without the full faithfulness hypothesis our results are based on the properties of scalar extensions of derived categories whose investigation was started by pawel sosna and the first author

the minimal model program for the hilbert scheme of points on p2 and bridgeland stability abstract in this paper we study the birational geometry of the hilbert scheme p 2 n of n points on p 2 we discuss the stable base locus decomposition of the effective cone and the corresponding birational models we give modular interpretations to the models in terms of moduli spaces of bridgeland semi stable objects we construct these moduli spaces as moduli spaces of quiver representations using g i t and thus show that they are projective there is a precise correspondence between wall crossings in the bridgeland stability manifold and wall crossings between mori cones for n 9 we explicitly determine the walls in both interpretations and describe the corresponding flips and divisorial contractions

the b functions of semi invariants of quivers abstract in this paper we compute b functions or bernstein sato polynomials of various semi invariants of quivers the main tool is an explicit relation for the b functions between semi invariants that correspond to each other under reflection functors or castling transforms this enables us to compute recursively the b functions for all dynkin quivers and extended dynkin quivers with prehomogeneous dimension vectors

image interpolation using enhanced multiresolution critical point filters there are increasing demands for image interpolation in various fields such as virtual reality computer animation and video transmission the critical point filters cpf we have proposed previously enable completely automatic matching of two images in our previous method however it has taken a long time to compute the optimal parameter values by iterative searches this paper proposes an improved algorithm called the enhanced critical point filters ecpf where the parameters are stably computed using together the inverse mapping from the destination to the source it takes only a second to match images whose size is 64 64 the algorithm is also improved in its precision by directly handling color images while the previous algorithm has taken only the intensity value into account we apply ecpf to keyframe interpolation of video sequences we also apply ecpf to interpolating two different views of an object to generate any intermediate views in this case there are many constraints that can be used to determine the mapping between the images we propose a method to use such constraints to improve the accuracy of the mappings as the results image based pseudo 3d models are easily created from a set of views without any special devices

eigenvalues of majorized hermitian matrices and littlewood richardson coefficients abstract answering a question raised by s friedland we show that the possible eigenvalues of hermitian matrices or compact operators a b and c with c a b are given by the same inequalities as in klyachko s theorem for the case where c a b except that the equality corresponding to tr c tr a tr b is replaced by the inequality corresponding to tr c tr a tr b the possible types of finitely generated torsion modules a b and c over a discrete valuation ring such that there is an exact sequence b c a are characterized by the same inequalities

constructing coherently g invariant modules abstract let g be a reductive group acting on a path algebra kq as automorphisms we assume that g admits a graded polynomial representation theory and the action is polynomial we describe the quiver q g of the smash product algebra k q k m g where m g is the associated algebraic monoid of g we use q g representations to construct coherently g invariant modules of q as an application we construct algebraic semi invariants on the quiver representation spaces from those g invariant modules

on the invariant theory for acyclic gentle algebras in this paper we show that the fields of rational invariants over the irreducible components of the module varieties for an acyclic gentle algebra are purely transcendental extensions along the way we exhibit for such fields of rational invariants a transcendence basis in terms of schofield s determinantal semi invariants we also show that the moduli spaces of modules over regular irreducible component are just a products of projective spaces contents

moduli spaces for representations of concealed canonical algebras abstract continuing m domokos and h lenzing j algebra 228 2000 738 762 we investigate the relative invariants and moduli spaces introduced in a d king quart j math oxford ser 2 45 1994 515 530 which are related to a separating family of stable tubes in the module category of a concealed canonical algebra the class of concealed canonical algebras consists of representation infinite algebras of tame or wild representation type and contains in particular the representation theory of extended dynkin quivers we present a uniform characteristic free approach avoiding case by case analysis we introduce the notion of admissible weight for such algebras these are the weights i e elements of the dual of the grothendieck group of the module category attached to the infinite moduli spaces for families of modules from the separating subcategory by using an explicit description of the semigroup of admissible weights the corresponding notion of semistability is analyzed and is related to the formation of perpendicular categories we construct relative invariants belonging to admissible weights and prove that the constructed system is complete in the case of canonical algebras it turns out that for any concealed canonical algebra and for any admissible weight all the corresponding moduli spaces are isomorphic to some projective space moreover any such moduli space can be naturally identified with the moduli space belonging to a special weight called the rank as a consequence we show that in the case of a tame concealed algebra any infinite moduli space for families of modules is a projective space and all fields of rational invariants on irreducible components of representation spaces are purely transcendental

polynomial degree bounds for matrix semi invariants abstract we study the left right action of sl n sl n on m tuples of n n matrices with entries in an infinite field k we show that invariants of degree n 2 n define the null cone consequently invariants of degree n 6 generate the ring of invariants if char k 0 we also prove that for m 0 invariants of degree at least n n 1 are required to define the null cone we generalize our results to matrix invariants of m tuples of p q matrices and to rings of semi invariants for quivers for the proofs we use new techniques such as the regularity lemma by ivanyos qiao and subrahmanyam and the concavity property of the tensor blow ups of matrix spaces we will discuss several applications to algebraic complexity theory such as a deterministic polynomial time algorithm for non commutative rational identity testing and the existence of small division free formulas for non commutative polynomials

some explicit solutions of the additive deligne simpson problem and their applications abstract in this paper we construct three infinite series and two extra triples e8 and e 8 of complex matrices b c and a b c of special spectral types associated to simpson s classification in amer math soc proc 1 1992 157 and magyar et al classification in adv math 141 1999 97 this enables us to construct fuchsian systems of differential equations which generalize the hypergeometric equation of gauss riemann in a sense they are the closest relatives of the famous equation because their triples of spectral flags have finitely many orbits for the diagonal action of the general linear group in the space of solutions in all the cases except for e8 we also explicitly construct scalar products such that a b and c are self adjoint with respect to them in the context of fuchsian systems these scalar products become monodromy invariant complex symmetric bilinear forms in the spaces of solutions when the eigenvalues of a b and c are real the matrices and the scalar products become real as well we find inequalities on the eigenvalues of a b and c which make the scalar products positive definite as proved by klyachko spectra of three hermitian or real symmetric matrices b c and a b c form a polyhedral convex cone in the space of triple spectra he also gave a recursive algorithm to generate inequalities describing the cone the inequalities we obtain describe non recursively some faces of the klyachko cone

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

multiple constraints to compute optical flow the computation of the optical flow field from an image sequence requires the definition of constraints on the temporal change of image features in this paper we consider the implications of using multiple constraints in the computational schema in the first step it is shown that differential constraints correspond to an implicit feature tracking therefore the best results either in terms of measurement accuracy and speed in the computation are obtained by selecting and applying the constraints which are best tuned to the particular image feature under consideration considering also multiple image points not only allows us to obtain a locally better estimate of the velocity field but also to detect erroneous measurements due to discontinuities in the velocity field moreover by hypothesizing a constant acceleration motion model also the derivatives of the optical flow are computed several experiments are presented from real image sequences

on locally semi simple representations of quivers abstract in this paper we solve a problem raised by victor kac in 7 on locally semi simple quiver representations specifically we show that an acyclic quiver q is of tame representation type if and only if every representation of q with a semi simple ring of endomorphisms is locally semi simple

mapping cones in the bounded derived category of a gentle algebra abstract in this article we describe the triangulated structure of the bounded derived category of a gentle algebra by describing the triangles induced by the morphisms between indecomposable objects in a basis of their hom space

minimal generating set for semi invariants of quivers of dimension two abstract a minimal by inclusion generating set for the algebra of semi invariants of a quiver of dimension 2 2 is established over an infinite field of arbitrary characteristic the mentioned generating set consists of the determinants of generic matrices and the traces of tree paths of pairwise different multidegrees where in the case of characteristic different from two we take only admissible paths as a consequence we describe relations modulo decomposable semi invariants

generating invariant rings of quivers in arbitrary characteristic abstract it is well known that the ring of polynomial invariants of a reductive group is finitely generated however it is difficult to give strong upper bounds on the degrees of the generators especially over fields of positive characteristic in this paper we make use of the theory of good filtrations along with recent results on the null cone to provide polynomial bounds for matrix semi invariants in arbitrary characteristic and consequently for matrix invariants our results generalize to invariants and semi invariants of quivers

schur functors and schur complexes the study of schur functors has a relatively long history its main impetus derived from representation theory originally in characteristic zero over the years however with the development of modular representations and algebraic geometry over fields of positive characteristic the need for a theory of universal polynomial functors increased and since the mid 1960s approaches to a characteristic free treatment of schur functors have been developing see for instance the recent book of green 1 l in which the treatments by carter and lusztig 6 higman 12 and towber 20 among others are discussed our own interest in such a treatment was awakened by the work of lascoux 141 on resolutions of determinantal ideals although his thesis treated only the characteristic zero case it suggested that a general and elementary theory of schur functors could be developed using only the rudiments of multilinear algebra involving the hopf algebra structures of the symmetric exterior and divided power algebras moreover this elementary development admitted of a natural generalization to the idea of schur complexes whose usefulness was demonstrated for instance in our construction of a universal minimal

cluster fans stability conditions and domains of semi invariants we show that the cone of finite stability conditions of a quiver q without oriented cycles has a fan covering given by the dual of the cluster fan of q along the way we give new proofs of schofield s results 1991 on perpendicular categories from our results we recover igusa orr todorov weyman s theorem on cluster complexes and domains of semi invariants for dynkin quivers for arbitrary quivers we also give a description of the domains of semi invariants labeled by real schur roots in terms of quiver exceptional sets

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