comparisons of nonnegative splittings of matrices abstract the comparison of the asymptotic rates of convergence of two iteration matrices induced by two splitting of the same matrix has arisen in the work of many authors in this article some new comparison theorems for two nonnegative splittings a splitting a m n is nonnegative if m 1 exist and m 1 n is nonnegative are derived they extend the known results in the literature in addition we also point out three incorrect conditions in a paper by beauwens furthermore we give some reasonable conditions ensuring the strict inequality between the asymptotic convergence rates this also answers the open question which additional and appropriate conditions should be imposed on miller neumann splittings to obtain strict inequality finally some applications to a class of generalized aor sor and jor iterative methods whose special cases imply block also point aor sor and jor iterative methods for solving linear systems are discussed

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

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

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

splitting based on the outer inverse of matrices in this paper we introduce a conception on the splitting of a matrix a which is based on the outer inverse of the matrix a it will be evidence that splittings early are the special cases several necessary and sufficient conditions to endure the convergence of the fixed pointed system generated by the splitting are also induced

functional equations for character series associated with n n matrices l anneau a des invariants ou des traces des r matrices generiques n n possede une serie de caracteres a qui est une fonction rationnelle symetrique des variables commutantes x 1 x r si r n 2 cette serie de caracteres satisfait a une equation fonctionnelle

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

qualitative analysis of gene regulatory networks by temporal logic in this article we propose a novel formalism to model and analyse gene regulatory networks using a well established formal verification technique we model the possible behaviours of networks by logical formulae in linear temporal logic ltl by checking the satisfiability of ltl it is possible to check whether some or all behaviours satisfy a given biological property which is difficult in quantitative analyses such as the ordinary differential equation approach owing to the complexity of ltl satisfiability checking analysis of large networks is generally intractable in this method to mitigate this computational difficulty we developed two methods one is a modular checking method where we divide a network into subnetworks check them individually and then integrate them the other is an approximate analysis method in which we specify behaviours in simpler formulae which compress or expand the possible behaviours of networks in the approximate method we focused on network motifs and presented approximate specifications for them we confirmed by experiments that both methods improved the analysis of large networks we propose a novel qualitative method for analysing gene networks based on formal verification technique behaviours and properties of networks are described in temporal logic formulae by checking satisfiability of the formula we can analyse properties of the network to improve the efficiency of analysis we developed the modular and approximate method

index splitting for the drazin inverse and the singular linear system abstract an index splitting of a singular matrix a for the drazin inverse and its relative iterations for the minimal p norm solution of the singular linear system ax b b r a k k ind a are presented some convergence criteria are also given and a method is developed for constructing the index splitting of matrix a

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

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

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

division algebra coproducts of index n given a family of separable finite dimensional extensions li of a field k we construct a division algebra n2 over its center which is freely generated over k by the fields li i

irreducible components of varieties of representations the local case abstract for any positive integer d we determine the irreducible components of the varieties that parametrize the d dimensional representations of a local truncated path algebra here is a quotient k q the paths of length l 1 of a path algebra kq where k is an algebraically closed field l is a positive integer and q is the quiver with a single vertex and a finite number r of loops the components are determined in both the classical and the grassmannian settings rep d and grass d our method is to corner the components by way of a twin pair of upper semicontinuous maps from rep d to a poset consisting of sequences of semisimple modules an excerpt of the main result is as follows given a sequence s s 0 s l of semisimple modules with dim 0 l l s l d let rep s be the subvariety of rep d consisting of the points that parametrize the modules with radical layering s the radical layering of a module m is the sequence j l m j l 1 m 0 l l where j is the jacobson radical of suppose the quiver q has r 2 loops if d l 1 the variety rep d is irreducible and generically its modules are uniserial if on the other hand d l 1 then the irreducible components of rep d are the closures of the subvarieties rep s for those sequences s which satisfy the inequalities dim s l r dim s l 1 and dim s l 1 r dim s l for 0 l l generically the modules in any such component have socle layering s l s 0 as a byproduct the main result provides further installments of generic information on the modules corresponding to the irreducible components of the parametrizing varieties

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

irreducible components of varieties of representations the acyclic case the goals of this article are as follows 1 to determine the irreducible components of the affine varieties mathbf rep mathbf d lambda parametrizing the representations with dimension vector mathbf d where lambda traces a major class of finite dimensional algebras 2 to generically describe the representations encoded by the components the target class consists of those truncated path algebras lambda over an algebraically closed field k which are based on a quiver q without oriented cycles the main result characterizes the irreducible components of mathbf rep mathbf d lambda in representation theoretic terms and provides a means of listing them from quiver and loewy length of lambda combined with existing theory this classification moreover yields an array of generic features of the modules parametrized by the components such as generic minimal projective presentations generic sub and quotient modules etc our second principal result pins down the generic socle series of the modules in the components it does so for more general lambda in fact the information on truncated path algebras of acyclic quivers supplements the theory available in the special case where lambda kq filling in generic data on the mathbf d dimensional representations of q with any fixed loewy length

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

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

computing generalized inverses of matrices by iterative methods based on splittings of matrices in this paper we study the iterative methods for computing the generalized inverses of the form a 2 t s we prove the result that if a has the splitting a m n and if the generalized inverses a 2 t s and m 2 t s exist then the iterationx j 1 m 2 t snx j m 2 t sconverges to a 2 t s for any x 0 if and only ift t s s email protected 2 t sn 1 or equivalently m 2 t email protected 2 t sn 1 the splitting a m n is called t s splitting of a if m satisfies email protected s c m where t and s are subspaces of c n and c m respectively with dimt dims we also present a specific method to construct convergent t s splitting of a apply these two results to the most of commonly used generalized inverses such as a a d a d w a h k we can get criteria of convergence for the corresponding iterations many old and new splittings and specific choice of the related convergent splittings

bio pepa a framework for the modelling and analysis of biological systems in this work we present bio pepa a process algebra for the modelling and the analysis of biochemical networks it is a modification of pepa originally defined for the performance analysis of computer systems in order to handle some features of biological models such as stoichiometry and the use of general kinetic laws bio pepa may be seen as an intermediate formal compositional representation of biological systems on which different kinds of analyses can be carried out bio pepa is enriched with some notions of equivalence specifically the isomorphism and strong bisimulation for pepa have been considered and extended to our language finally we show the translation of a biological model into the new language and we report some analysis results

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

grobner bases for the rings of special orthogonal and 2 2 matrix invariants abstract we present a grobner basis for the ideal of relations among the standard generators of the algebra of invariants of the special orthogonal group acting on k tuples of vectors the cases of so 3 and so 4 are interpreted in terms of the algebras of invariants and semi invariants of k tuples of 2 2 matrices in particular we present in an explicit form a grobner basis for the 2 2 matrix invariants finally we use a sagbi basis to show that the algebra of so 2 invariants is a koszul algebra

universal denominators of hilbert series the denominator of the hilbert series of a finitely generated r module m does not always divide the denominator of the hilbert series of r for this reason we define the universal denominator the universal denominator of a module m is the least common multiple of the denominators of the hilbert series of all submodules of m the universal denominator behaves nicely with respect to short exact sequences and tensor products it also has interesting geometric interpretations formulas are given for the universal denominator for rings of invariants dixmier gave a conjectural formula for the denominator of the hilbert series of invariants of binary forms we show that the universal denominator is actually equal to dixmier s formula in that case

relative invariants and subgeneric orbits of quivers of finite and tame type the notion of a quiver was introduced by gabriel in 2 we are dealing with the determination of relative invariants of representation varieties attached to quivers to be more precise let us first recall the basic concepts a quiver q consists of two finite sets qo the set of points and q1 the set of arrows and that for all e q1 there is assigned its starting point e qo and its end point e e qo let k be an algebraically closed field a representation of q is given by finite dimensional k vectorspaces v for l i e qo and k linear maps a v v for all e q1 it is well known that the representations of q form an abelian category usually denoted by q and that q is of finite type those where q admits only a finite number of indecomposable objects iff the underlying graph of q is a dynkin diagram and that q is of tame type iff the underlying graph of q is a euclidean diagram let v vi be a representation of q then the dimension type of v is this is a positive integral vector in qn defined as dim v dim k vi i s qo

semiconvergence criteria of iterations and extrapolated iterations and constructive methods of semiconvergent iteration matrices in this paper we simultaneously study iterations and extrapolated iterations for the consistent rectangular or sigular linear systems 1 1 ax b in the case that a is rectangular let a m n be a subproper splitting of a and let the generalized inverse m t s 1 2 exist and in the case that a is singular let a m n is a usual splitting of a consider the iteration 2 3 x j 1 gx j c where g m t s 1 2 nandc m t s 1 2 b if a is rectangular and g m 1n and c m 1b if a is singular and the extrapolated iteration 2 19 x j 1 g wx email protected where email protected r w 0 1 and 2 20 g w email protected email protected we establish the semiconcovergence criteria of the iterations 2 3 and 2 19 and present the efficient and convenient construction methods of the semiconvergent iteration matrices g and g w and the choices of the extrapolated parameter w

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 translations of temporal logic of actions into monadic second order logic abstract the temporal logic of action introduced by lamport 4 for specifying the behavior of concurrent systems is compared with monadic second order logic which is accepted as an universal formalism for specifying temporal behaviors the consequences of lamport s decision to combine in the existential quantifier of tla both the standard existential quantifier and the non logical closure under stuttering are investigated a continuous time interpretation is provided for tla and it is argued that this interpretation is more appropriate than the standard discrete time interpretation also some decidability problems are investigated

indecomposable vector bundles and stable higgs bundles over smooth projective curves we prove that the number of geometrically indecomposable vector bundles of xed rank r and degree d over a smooth projective curve x dened over a nite eld is given by a polynomial depending only on r d and the genus g of x in the weil numbers of x we provide a closed formula expressed in terms of generating series for this polynomial we also show that the same polynomial computes the number of points of the moduli space of stable higgs bundles of rank r and degree d over x this entails a closed formula for the poincar e polynomial of the moduli spaces of stable higgs bundles over a compact riemann surface and hence also for the poincar e polynomials of the twisted character varieties for the groups gl r

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

semiconvergence of nonnegative splittings for singular matrices in this paper we discuss semiconvergence of the matrix splitting methods for solving singular linear systems the concepts that a splitting of a matrix is regular or nonnegative are generalized and we introduce the terminologies that a splitting is quasi regular or quasi nonnegative the equivalent conditions for the semiconvergence are proved comparison theorem on convergence factors for two different quasi nonnegative splittings is presented as an application the semiconvergence of the power method for solving the markov chain is derived the monotone convergence of the quasi nonnegative splittings is proved that is for some initial guess the iterative sequence generated by the iterative method introduced by a quasi nonnegative splitting converges towards a solution of the system from below or from above

der kanonische modul eines invariantenrings abstract we calculate the canonical module of a ring of invariants r of a reductive group acting on an affine variety from that we derive a criterion for the gorenstein property to hold for r in case r is the ring of invariants of a graded ring we obtain lower and upper bounds for the degree of the generating function of r

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

canonical forms of two by two by two matrices we determine the canonical forms of two by two by two matrices in order to do so we present a collection of covariants that separates the canonical forms the covariants also fulfill a number of identities moreover we show that these canonical forms and covariants are related to those of the binary cubic and skew symmetric tensors of step 3 in the exterior algebra ext u where dim u 6

the equations for the moduli space of n points on the line a central question in invariant theory is that of determining the relations among invariants geometric invariant theory git quotients come with a natural ample line bundle and hence often a natural projective embedding this question translates to determining the equations of the moduli space under this embedding this article deals with one of the most classical quotients the space of ordered points on the projective line we show that under any weighting of the points this quotient is cut out scheme theoretically by a particularly simple set of quadric relations with the single exception of the segre cubic threefold the space of six points with equal weight we also show that the ideal of relations is generated in degree at most four and we give an explicit description of the generators if all the weights are even e g in the case of equal weight for odd n then we show that the ideal of relations is generated by quadrics

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

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

tensor product decompositions and open orbits in multiple flag varieties abstract for a connected semisimple algebraic group g we consider some special infinite series of tensor products of simple g modules whose g fixed point spaces are at most one dimensional we prove that their existence is closely related to the existence of open g orbits in multiple flag varieties and address the problem of classifying such series

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

iterative methods for computing the generalized inverses a 2 t s of a matrix a abstract in this paper the iterative methods for computing the generalized inverse a t s 2 the unique matrix x such that xax x r x t and n x s are established and the iterative methods for computing many important generalized inverses of matrices such as a a mn a d a a d w a l 1 a l and ap l are constructed

unitary and euclidean representations of a quiver abstract a unitary euclidean representation of a quiver is given by assigning to each vertex a unitary euclidean vector space and to each arrow a linear mapping of the corresponding vector spaces we recall an algorithm for reducing the matrices of a unitary representation to canonical form give a certain description of the representations of canonical form and reduce the problem of classifying euclidean representations to the problem of classifying unitary representations we also describe the set of dimensions of all indecomposable unitary euclidean representations of a quiver and establish the number of parameters in an indecomposable unitary representation of a given dimension

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 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

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

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

the complexity of propositional linear temporal logics the complexity of satisfiability and determination of truth in a particular finite structure are considered for different propositional linear temporal logics it is shown that these problems are np complete for the logic with f and are pspace complete for the logics with f x with u with u s x operators and for the extended logic with regular operators given by wolper

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

preconditionings and splittings for rectangular systems recently eiermann marek and niethammer have shown how to applysemiiterative methods to a fixed point systemx tx c which isinconsistent or in whichthe powers of the fixed point operator t have no limit to obtain iterative methods which converge to some approximate solution to the fixed point system in view of their results we consider here stipulations on apreconditioning qax qb of the systemax b and separately on asplitting a m n which lead to fixed point systems such that with the aid of a semiiterative method the iterative scheme converges to a weighted moore penrose solution to the systemax b we show in several ways that to obtain a meaningful limit point from a semiiterative method requires less restrictions on the splittings or the reconditionings than those which have been required in the classical picard iterative method see e g the works of berman and plemmons berman and neumann and tanabe we pay special attention to the case when the weighted moore penrose solution which is sought is the minimal norm least squares solution toax b

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

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

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

temporal logic analysis of gene networks under parameter uncertainty the lack of precise numerical information for the values of biological parameters severely limits the development and analysis of models of genetic regulatory networks to deal with this problem we propose a method for the analysis of genetic regulatory networks under parameter uncertainty we consider models based on piecewise multiaffine differential equations dynamical properties expressed in temporal logic and intervals for the values of uncertain parameters the problem is then either to guarantee that the system satisfies the expected properties for every possible parameter value the corresponding parameter set is then called valid or to find valid subsets of a given parameter set the proposed method uses discrete abstractions and model checking and allows for efficient search of the parameter space however the abstraction process creates spurious behaviors in the abstract systems along which time does not progress consequently the verification of liveness properties expressing that something will eventually happen and implicitly assuming progress of time often fails a solution to this second problem is proposed using the notion of transient regions this approach has been implemented in a tool for robust verification of gene networks and applied to the tuning of a synthetic network built in e coli

reasoning about infinite computations we investigate extensions of temporal logic by connectives defined by finite automata on infinite words we consider three different logics corresponding to three different types of acceptance conditions finite looping and repeating for the automata it turns out however that these logics all have the same expressive power and that their decision problems are all pspace complete we also investigate connectives defined by alternating automata and show that they do not increase the expressive power of the logic or the complexity of the decision problem