on two generalized inverse eigenvalue problems for hessenberg upper triangular pencils and their application to the study of gmres convergence abstract we discuss two generalized inverse eigenvalue problems the first one for a given unreduced upper hessenberg matrix h find a nonsingular upper triangular matrix t such that all the pencils h k t k have prescribed eigenvalues where h k and t k are the leading k k principal submatrices of h and t respectively the second one for a given unitary unreduced upper hessenberg matrix q find a nonsingular upper triangular matrix t such that all the pencils t k q k have prescribed eigenvalues where t k is the leading k k principal submatrix of t and q k is the conjugate transpose of the leading k k principal submatrix of q we present the necessary and sufficient conditions for the solvability of the two problems our results lead to an alternative proof for the statement that any admissible ritz value set or admissible harmonic ritz value set is possible for the prescribed gmres residual norms here the term admissible means there are some restrictions on the sets if gmres stagnates at some iterations

optimal control of a convective boundary condition in a thermistor problem we consider the optimal control of a two dimensional steady state thermistor the problem is described by a system of two nonlinear elliptic partial differential equations with appropriate boundary conditions which model the coupling of the thermistor to its surroundings based on physical considerations an objective functional to be minimized is introduced and the convective boundary coefficient is taken as the control existence and uniqueness of the optimal control are proven to characterize this optimal control the optimality system consisting of the state and adjoint equations is derived

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

optimal control of a nonlocal thermistor problem with abc fractional time derivatives abstract we study an optimal control problem associated to a fractional nonlocal thermistor problem involving the abc atangana baleanu caputo fractional time derivative we first prove the existence and uniqueness of solution then we show that an optimal control exists moreover we obtain the optimality system that characterizes the control

convergence of the shifted qr algorithm for unitary hessenberg matrices this paper shows that for unitary hessenberg matrices the qr algorithm with an exceptional initial value modification of the wilkinson shift gives global convergence moreover the asymptotic rate of convergence is at least cubic higher than that which can be shown to be quadratic only for hermitian tridiagonal matrices under no further assumption a general mixed shift strategy with global convergence and cubic rates is also presented

remarks on the existence uniqueness and semi explicit solvability of systems of autonomous partial differential equations in divergence form with constant boundary conditions a method is proposed for reducing autonomous nonlinear boundary value problems of partial differential equations with a particular structure and constant boundary conditions to the simple problem this requires the solution of a non standard two point system of ordinary differential equations some applications are also presented for classical problems of mathematical physics

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

on investigating gmres convergence using unitary matrices for a given matrix a and right hand side b this paper investigates unitary matrices generating with some right hand sides c the same gmres residual norms as the pair a b we give characterizations of this class of unitary matrices and point out the relationship with krylov subspaces and krylov residual subspaces for the pair a b we investigate the eigenvalues of these unitary matrices in relation to the convergence behavior of gmres for the pair a b and describe the indispensable role of the eigenvector information we conclude with a formula for the gmres residual norms generated by a normal matrix b in terms of its eigenvalues and components of the right hand side c in the eigenvector basis

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

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

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

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

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

analysis of optimal control problems for the two dimensional thermistor system an optimal control problem for the thermistor system is considered first the precise mathematical problem is established and the proof of existence of the optimal solution is given with appropriate function spaces then g teaux differentiability is shown for the thermistor system with respect to control and the optimality system is obtained

optimal control of nonlocal thermistor equations we are concerned with the optimal control problem of the well known nonlocal thermistor problem i e in studying the heat transfer in the resistor device whose electrical conductivity is strongly dependent on the temperature existence of an optimal control is proved the optimality system consisting of the state system coupled with adjoint equations is derived together with a characterisation of the optimal control uniqueness of solution to the optimality system and therefore the uniqueness of the optimal control is established the last part is devoted to numerical simulations

on characters and formal degrees of discrete series of affine hecke algebras of classical types we address two fundamental questions in the representation theory of affine hecke algebras of classical types one is an inductive algorithm to compute characters of tempered modules and the other is the determination of the constants in the formal degrees of discrete series in the form conjectured by reeder j reine angew math 520 37 93 2000 the former is completely different from the lusztig shoji algorithm shoji in invent math 74 239 267 1983 lusztig in ann math 131 355 408 1990 and it is more effective in a number of cases the main idea in our proof is to introduce a new family of representations which behave like tempered modules but for which it is easier to analyze the effect of parameter specializations our proof also requires a comparison of the c theoretic results of opdam delorme slooten solleveld j inst math jussieu 3 531 648 2004 arxiv 0909 1227 int math res not 2008 adv math 220 1549 1601 2009 acta math 205 105 187 2010 and the geometric construction from kato duke math j 148 305 371 2009 am j math 133 518 553 2011 ciubotaru and kato adv math 226 1538 1590 2011

the resolution of the generic residual intersection of a complete intersection the concept of residual intersection introduced by artin and nagata l in 1972 is a fruitful generalization of linkage as the following two examples attest let i be a strongly cohen macaulay ideal in a cohenmacaulay local ring r if z satisfies the condition g then huneke 6 proposition 4 31 has proved that the extended rees algebra r it t 1 is defined by an ideal which is obtained from i by way of residual intersection since the extended rees algebra is a deformation of both the rees algebra r it and the associated graded algebra gz i it contains considerable information about the analytic properties of i the definitions of residual intersection strongly cohen macaulay and g may be found in section 4 huneke 6 theorem 4 11 has also shown that the ideal i generated by the maximal order minors of a generic n x m matrix is a residual intersection of a generic codimension two cohen macaulay ideal since j is rather poorly behaved with respect to linkage s it is promising that it is close to a well understood ideal once we weaken the tie from linkage to residual intersection

numerical solutions of the thermistor equations abstract this paper examines heat conduction in a thermistor used as a current surge regulator the problem consists of coupled nonlinear nonlocal parabolic initial boundary value problems simplifying assumptions are made which lead to two different problems each of which consists of a one space dimensional nonlocal parabolic initial boundary value problem numerical methods for the approximate solutions of both the steady state and the transient problems are described and the results of the numerical experiments are presented

the superlinear convergence behaviour of gmres abstract gmres is a rather popular iterative method for the solution of nonsingular nonsymmetric linear systems it is well known that gmres often has a so called superlinear convergence behaviour i e the rate of convergence seems to improve as the iteration proceeds for the conjugate gradients method this phenomenon has been related to a modest degree of convergence of the ritz values it has been observed in experiments that for gmres too changes in the convergence behaviour seem to be related to the convergence of ritz values in this paper we prove that as soon as eigenvalues of the original operator are sufficiently well approximated by ritz values gmres from then on converges at least as fast as for a related system in which these eigenvalues and their eigenvector components are missing

the qr algorithm for unitary hessenberg matrices abstract let h be an n n unitary right hessenberg matrix with positive subdiagonal elements using what we call the schur parameterization of h we show how one step of the shifted qr algorithm for h can be carried out in o n arithmetic operations coupled with the shift strategy of eberlein and huang 3 this will permit computation of the spectrum of h to machine precision in o n 2 operations one potential application is the computation of gauss szego quadrature formulas 12 given the associated schur parameters 7 the weights can also be computed by direct analogy with 6

non validity of index law in fractional calculus a fractional differential operator with markovian and non markovian properties abstract we presented an analysis of evolutions equations generated by three fractional derivatives namely the riemann liouville caputo fabrizio and the atangana baleanu fractional derivatives for each evolution equation we presented the exact solution for time variable and studied the semigroup principle the riemann liouville fractional operator verifies the semigroup principle but the associate evolution equation does not the caputo fabrizio fractional derivative does not satisfy the semigroup principle but surprisingly the exact solution satisfies very well all the principle of semigroup however the atangana baleanu for small time is the stretched exponential derivative which does not satisfy the semigroup as operators for a large time the atangana baleanu derivative is the same with riemann liouville fractional derivative thus satisfies semigroup principle as an operator the solution of the associated evolution equation does not satisfy the semigroup principle as riemann liouville with the connection between semigroup theory and the markovian processes we found out that the atangana baleanu fractional derivative has at the same time markovian and non markovian processes we concluded that the fractional differential operator does not need to satisfy the semigroup properties as they portray the memory effects which are not always markovian we presented the exact solutions of some evolutions equation using the laplace transform in addition to this we presented the numerical solution of a nonlinear equation and show that the model with the atangana baleanu fractional derivative has random walk for small time we also observed that the mittag leffler function is a good filter than the exponential and power law functions which makes the atangana baleanu fractional derivatives powerful mathematical tools to model complex real world problems

relative beilinson monad and direct image for families of coherent sheaves the higher direct image complex of a coherent sheaf or finite complex of coherent sheaves under a projective morphism is a fundamental construction that can be defined via a cech complex or an injective resolution both inherently infinite constructions using exterior algebras and relative versions of theorems of beilinson and bernstein gel fand gel fand we give an alternate description in finite terms using this description we can characterize the generic complex over the variety of finite free complexes of a given shape as the direct image of an easily described vector bundle we can also give explicit descriptions of the loci in the base spaces of flat families of sheaves in which some cohomological conditions are satisfied for example the loci where vector bundles on projective space split in a certain way or the loci where a projective morphism has higher dimensional fibers our approach is so explicit that it yields an algorithm suited for computer algebra systems

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

hodge dual operators and model algebras for rational representations of the general linear group abstract in this paper we construct a family of algebras each of whose members is a multiplicity free sum of irreducible rational representations of the general linear group gl n c we then use the properties of a generalized version of the hodge dual operator to determine an explicit basis for each of these algebras and by restriction we obtain an explicit basis for each of the irreducible rational representations of gl n c our results cleanly extends classical algebro combinatorial results on polynomial representations of gl n c to rational representations

gmres with deflated restarting a modification is given of the gmres iterative method for nonsymmetric systems of linear equations the new method deflates eigenvalues using wu and simon s thick restarting approach siam j matrix anal appl 22 2000 pp 602 616 it has the efficiency of implicit restarting but is simpler and does not have the same numerical concerns the deflation of small eigenvalues can greatly improve the convergence of restarted gmres also it is demonstrated that using harmonic ritz vectors is important because then the whole subspace is a krylov subspace that contains certain important smaller subspaces

optimal control for the thermistor problem this paper is concerned with the state constrained optimal control of the two dimensional thermistor problem a quasi linear coupled system of a parabolic and elliptic pde with mixed boundary conditions this system models the heating of a conducting material by means of direct current existence uniqueness and continuity for the state system are derived by employing maximal elliptic and parabolic regularity by similar arguments the linearized state system is discussed while the adjoint system involving measures is investigated using a duality argument these results allow us to derive first order necessary conditions for the optimal control problem

codimension 1 orbits and semi invariants for the representations of an equioriented graph of type _ on considere le diagramme de dynkin d n equioriente et la variete hom v 1 v 3 i 2 n hom v i v i 1 v j un espace vectoriel sur k sur lequel agit le groupe g i 1 n gl v i on determine l orbite maximale et les orbites de codimension 1 de cette action

a survey on fuzzy fractional differential and optimal control nonlocal evolution equations abstract we survey some representative results on fuzzy fractional differential equations controllability approximate controllability optimal control and optimal feedback control for several different kinds of fractional evolution equations optimality and relaxation of multiple control problems described by nonlinear fractional differential equations with nonlocal control conditions in banach spaces are considered

on the conormal bundle of the determinantal variety call w such variety if we choose bases b v a a b u v i b for each module we can identify w with the variety of pairs of matrices y x with entries in r x being an m x n matrix y an ii x m matrix such that x y omx y x onxm now let a be the coordinate ring of the affine space a of dimension 2 m x n i e the polynomial ring r x yji i l m j i n and

complexes which arise from a matrix and a vector resolutions of divisors on certain varieties of complexes abstract consider the polynomial ring r r0 x y where r0 is a normal domain and x1 g and yg f are matrices of indeterminates the r ideal j i1 xy imin f g y defines a variety of complexes over r0 the divisor class group of r j is isomorphic to cl r0 z i where i is an ideal of r j generated by appropriately chosen lower order minors of y we produce the minimal r free resolution of i i for all integers i 1 if f is greater than or equal to g then j is a generic residual intersection of the generic grade g complete intersection i1 x the resolutions that we produce in this case are in many ways analogous to resolutions of divisors on generic residual intersections of grade two perfect ideals or grade three gorenstein ideals

on the minimal free resolution of the universal ring for resolutions of length two abstract hochster established the existence of a commutative noetherian ring c and a universal resolution u of the form 0 c e c f c g 0 such that for any commutative noetherian ring s and any resolution v equal to 0 s e s f s g 0 there exists a unique ring homomorphism c s with v u c s in the present paper we assume that f e g and we find the minimal resolution of c z q by free b modules where b is a polynomial ring over the field of rational numbers the modules of the resolution are described in terms of schur functors the graded strands of the differential are described in terms of pieri maps

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

optimal control problem for stochastic evolution equations in hilbert spaces in this article we consider an optimal control problem in which the controlled state dynamics is governed by a stochastic evolution equation in hilbert spaces and the cost functional has a quadratic growth the existence and uniqueness of the optimal control are obtained by the means of an associated backward stochastic differential equations with a quadratic growth and an unbounded terminal value as an application an optimal control of stochastic partial differential equations with dynamical boundary conditions is also given to illustrate our results

arithmetic cohen macaulayness and arithmetic normality for schubert varieties let g be a semisimple simply connected chevalley group over a field k let t be a maximal torus b a borel subgroup d t and w the weyl group of g relative to t let p be a maximal parabolic subgroup of classical type cf 7 8 i e a maximal parabolic subgroup such that if w is the associated fundamental weight then i w av i 1 2 w c a is 2 for all roots ae being a w invariant scalar product on x t q where x t character group of t for t e w wp let x t denote the schubert variety in g p associated to t namely x t is the zariski closure btp mod p in g p of the bruhat cell endowed with the canonical reduced structure for the canonical projective embedding g p in p h g p l let rt denote the homogeneous co ordinate ring of x t then the main result of this paper is given by

geometric aspects of iterated matrix multiplication abstract this paper studies geometric properties of the iterated matrix multiplication polynomial and the hypersurface that it defines we focus on geometric aspects that may be relevant for complexity theory such as the symmetry group of the polynomial the dual variety and the jacobian loci of the hypersurface that are computed with the aid of representation theory of quivers

rigid ideals by deforming quadratic letterplace ideals abstract we compute the deformation space of quadratic letterplace ideals l 2 p of finite posets p when its hasse diagram is a rooted tree these deformations are unobstructed the deformed family has a polynomial ring as the base ring the ideal j 2 p defining the full family of deformations is a rigid ideal and we compute it explicitly in simple example cases j 2 p is the ideal of maximal minors of a generic matrix the pfaffians of a skew symmetric matrix and a ladder determinantal ideal