the classification of homogeneous cohen macaulay rings of finite representation type in this paper we classify the homogeneous cohen macaulay rings which are of finite representation type that is the cohen macaulay rings which are positively graded and generated in degree 1 with an algebraically closed field of characteristic 0 in degree 0 and which have up to isomorphisms and shifts in the grading only a finite number of indecomposable maximal cohen macaulay modules mcm modules our main contribution is to show that a homogeneous cohen macaulay ring of finite representation type and dimension 2 must have minimal multiplicity putting this together with previous results of auslander auslander reiten buchweitz greuel schreyer greuel kn6rrer and solberg we obtain

a karhunen loeve decomposition of a gaussian process generated by independent pairs of exponential random variables abstract we obtain the explicit karhunen loeve decomposition of a gaussian process generated as the limit of an empirical process based upon independent pairs of exponential random variables the orthogonal eigenfunctions of the covariance kernel have simple expressions in terms of jacobi polynomials statistical applications in extreme value and reliability theory include a cramer von mises test of bivariate independence whose null distribution and critical values are tabulated

last syzygies of 1 generic spaces abstract we consider determinantal varieties x of expected codimension defined by the maximal minors of a matrix m of linear forms representing a linear map eisenbud and popescu have conjectured that 1 generic linear maps have the property that the syzygy ideals i s of all last syzygies s of x coincide with i x we prove a geometric version of this conjecture for 1 generic linear maps the syzygy varieties syz s v i s of all last syzygies have the same support as x

krylov subspace based model reduction for a class of bilinear descriptor systems we consider model order reduction for bilinear descriptor systems using an interpolatory projection framework such nonlinear descriptor systems can be represented by a series of generalized linear descriptor systems also called subsystems by utilizing the volterra wiener approach rugh 1981 standard projection techniques for bilinear systems utilize the generalized transfer functions of these subsystems to construct an interpolating approximation however the resulting reduced order system may not match the polynomial parts of the generalized transfer functions this may result in an unbounded error in terms of h 2 or h norms in this paper we derive an explicit expression for the polynomial part of each subsystem by assuming a special structure of the bilinear system which reduces to an index 1 linear descriptor system or differential algebraic equation dae if the bilinear terms are zero this allows us to propose an interpolatory technique for bilinear daes which not only achieves interpolation but also retains the polynomial parts of the bilinear systems the approach extends the interpolatory technique for index 1 linear daes beattie and gugercin 2009 to bilinear daes numerical examples are used to illustrate the theoretical results

a krylov stability corrected coordinate stretching method to simulate wave propagation in unbounded domains the krylov subspace projection approach is a well established tool for the reducedorder modeling of dynamical systems in the time domain in this paper we address the main issues obstructing the application of this powerful approach to the time domain solution of exterior wave problems we use frequency independent perfectly matched layers to simulate the extension to infinity pure imaginary stretching functions based on zolotarev s optimal rational approximation of the square root are implemented leading to perfectly matched layers with a controlled accuracy over a complete spectral interval of interest a new krylov based solution method via stabilitycorrected operator exponents is presented which allows us to construct reduced order models roms that respect the delicate spectral properties of the original scattering problem the roms are unconditionally stable and are based on a renormalized bi lanczos algorithm we give a theoretical foundation of our method and illustrate its performance through a number of numerical examples in which we simulate two dimensional electromagnetic wave propagation in unbounded domains including a photonic waveguide example the new algorithm outperforms the conventional finitedifference time domain method for problems on large time intervals

the eigenvalues of a family of persymmetric anti tridiagonal 2 hankel matrices in this note we provide the general expression for the eigenvalues of a type of anti tridiagonal matrices

multipoint interpolation of volterra series and h2 model reduction for a family of bilinear descriptor systems abstract in this paper we investigate interpolatory model order reduction for large scale bilinear descriptor systems recently it was shown in wyatt et al 2013 for linear descriptor systems that directly extending the standard rational interpolation conditions used in h 2 optimal model reduction to descriptor systems in general yields an unbounded error in the h 2 norm this is due to the possible mismatch of the polynomial part of the original and reduced order systems this conclusion holds for nonlinear systems as well in this paper we deal with bilinear descriptor systems and aim to pay attention to the polynomial part of the bilinear descriptor system along with interpolation to this end we have shown in goyal et al 2015 how to determine the polynomial part of each subsystem of the bilinear descriptor system explicitly by assuming special structures of the system matrices considering the same structured bilinear descriptor systems in this paper we first show how to achieve multipoint interpolation of the underlying volterra series of bilinear descriptor systems while retaining the polynomial part of each subsystem of the bilinear system then we extend the interpolation based first order necessary conditions for h 2 optimality to bilinear descriptor systems and propose an iterative scheme to obtain an h 2 optimal reduced order system by means of various numerical examples we demonstrate the efficiency of the proposed model order reduction technique and compare it with reduced bilinear systems obtained by using linear irka the loewner method for bilinear systems and pod based approximations

balanced realization and model order reduction for nonlinear systems based on singular value analysis this paper discusses balanced realization and model order reduction for both continuous time and discrete time general nonlinear systems based on singular value analysis of the corresponding hankel operators singular value analysis clarifies the gain structure of a given nonlinear operator here it is proved that singular value analysis of smooth hankel operators defined on hilbert spaces can be characterized by simple equations in terms of their states a balanced realization and model order reduction procedure is derived based on it and several important properties such as stability balanced form hankel norm controllability and observability of the original system are preserved the work improves the earlier results of k fujimoto and j m a scherpen ieee trans automat control 50 2005 pp 2 18 and then continues with new balancing and model reduction results

inexact solves in interpolatory model reduction we investigate the use of inexact solves for interpolatory model reduction and consider associated perturbation effects on the underlying model reduction problem we give bounds on system perturbations induced by inexact solves and relate this to termination criteria for iterative solution methods we show that when a petrov galerkin framework is employed for the inexact solves the associated reduced order model is an exact interpolatory model for a nearby full order system thus demonstrating backward stability we also give evidence that for h2 optimal interpolation points interpolatory model reduction is robust with respect to perturbations due to inexact solves finally we demonstrate the effectiveness of direct use of inexact solves in optimal h2 approximation the result is an effective model reduction strategy that is applicable in realistically large scale settings

exterior algebra methods for the minimal resolution conjecture if r 6 r 6d 9 we show that the minimal resolution conjecture mrc fails for a general set of points in p r for almost 1 2 p r values of this strengthens the result of d eisenbud and s popescu ep1 who found a unique such for each r in the given range our proof begins like a variation of that of eisenbud and popescu but uses exterior algebra methods as explained by eisenbud g floystad and f o schreyer efs to avoid the degeneration arguments that were the most difficult part of the eisenbud popescu proof analogous techniques show that the mrc fails for linearly normal curves of degree d and genus g when d 3g 2 g 4 re proving results of schreyer m green and r lazarsfeld

balanced model reduction via the proper orthogonal decomposition a new method for performing a balanced reduction of a high order linear system is presented the technique combines the proper orthogonal decomposition and concepts from balanced realization theory the method of snapshotsisused to obtainlow rank reduced rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain theapproximationsarethenusedtoobtainabalancedreducedorder model the method is particularly effective when a small number of outputs is of interest it is demonstrated for a linearized high order system that models unsteady motion of a two dimensional airfoil computation of the exact grammians would be impractical for such a large system for this problem very accurate reducedorder models are obtained that capture the required dynamics with just three states the new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition although further development is necessary the concept also extends to nonlinear systems

a randomized balanced proper orthogonal decomposition technique abstract in this paper we consider the model reduction problem of large scale linear systems such as systems obtained through the discretization of partial differential equations we propose a randomized proper orthogonal decomposition rpod technique to obtain the reduced order model by perturbing the primal and adjoint system using gaussian white noise we show that computations required by rpod algorithm is orders of magnitude lower while its performance is much better than other state of the art algorithms we also relate the rpod algorithm to krylov subspace methods and show that it constitutes a randomized approach to computational linear algebra problems that utilize krylov subspace methods

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

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

the regularity of tor and graded betti numbers let s k x1 xn let a b be finitely generated graded s modules and let m x1 xn s we give bounds for the regularity of the local cohomology of tork a b in terms of the graded betti numbers of a and b under the assumption that dim tor1 a b 1 we apply the results to syzygies grobner bases products and powers of ideals and to the relationship of the rees and symmetric algebras for example we show that any homogeneous linearly presented m primary ideal has some power equal to a power of m and if the first n 1 2 steps of the resolution of i are linear then i 2 is a power of m

jacobian ideals of trilinear forms an application of 1 genericity be a trilinear form in r and let ja denote the ideal of r generated by all the partial derivatives of a a question that arises from the theory of hyperdeterminants see gkz page 445 is the following what can be said about the ideal ja a reason for this question emerges among other things from results which show that information on the depth of ja and more finely on the primary decomposition of ja is linked to information on the hyperdeterminant of a see bw the difficulty with hyperdeterminants whose definition makes sense only when n m p 1 is that there is no explicit formula for them however

almost gorenstein homogeneous rings and their h vectors abstract in this paper for the development of the study of almost gorenstein graded rings we discuss some relations between almost gorensteinness of cohen macaulay homogeneous rings and their h vectors concretely for a cohen macaulay homogeneous ring r we give a sufficient condition for r to be almost gorenstein in terms of the h vector of r theorem 3 1 and we also characterize almost gorenstein homogeneous domains with small socle degrees in terms of the h vector of r theorem 4 4 moreover we also provide the examples of almost gorenstein homogeneous domains arising from lattice polytopes

a new look at proper orthogonal decomposition we investigate some basic properties of the proper orthogonal decomposition pod method as it is applied to data compression and model reduction of finite dimensional nonlinear systems first we provide an analysis of the errors involved in solving a nonlinear ode initial value problem using a pod reduced order model then we study the effects of small perturbations in the ensemble of data from which the pod reduced order model is constructed on the reduced order model we explain why in some applications this sensitivity is a concern while in others it is not we also provide an analysis of computational complexity of solving an ode initial value problem and study the computational savings obtained by using a pod reduced order model we provide several examples to illustrate our theoretical results

a framework for the solution of the generalized realization problem in this paper we present a novel way of constructing generalized state space representations e a b c d of interpolants matching tangential interpolation data the loewner and shifted loewner matrices are the key tools in this approach

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

model order reduction for parametrized nonlinear hyperbolic problems as an application to uncertainty quantification abstract in this work we present a model order reduction mor technique for hyperbolic conservation laws with applications in uncertainty quantification uq the problem consists of a parametrized time dependent hyperbolic system of equations where the parameters affect the initial conditions and the fluxes in a non linear way the procedure utilized to reduce the order is a combination of a greedy algorithm in the parameter space a proper orthogonal decomposition pod in time and empirical interpolation method eim to deal with non linearities drohmann 2012 we provide under some hypothesis an error bound for the reduced solution with respect to the high order one the algorithm shows small errors and savings of the computational time up to 90 in the uq simulations which are performed to validate the algorithm

symmetry preserving degenerations of the generic symmetric matrix abstract one considers certain degenerations of the generic symmetric matrix over a field k of characteristic zero and the main structures related to the determinant f of the matrix such as the ideal generated by its partial derivatives the polar map defined by these derivatives and its image v f the hessian matrix the ideal and the map given by the cofactors and the dual variety of v f a complete description of these structures is obtained

linearly presented perfect ideals of codimension 2 in three variables abstract the goal of this paper is the fine structure of the ideals in the title with emphasis on the properties of the associated rees algebra and the special fiber the watershed between the present approach and some of the previous work in the literature is that here one does not assume that the ideals in question satisfy the usual generic properties one exception is a recent work of n p h lan which inspired the present work here we recover and extend his work we strongly focus on the behavior of the ideals of minors of the corresponding so called hilbert burch matrix and on conjugation features of the latter we apply the results to three important models linearly presented ideals of plane fat points reciprocal ideals of hyperplane arrangements and linearly presented monomial ideals

scrollar syzygies of general canonical curves with genus le 8 we prove that for a general canonical curve c c z 9 1 of genus g the space of g 5 2 th last scrollar syzygies is isomorphic to the brill noether locus c 1 g 2 2 schreyer has conjectured that these scrollar syzygies span the space of all g 5 2 th last syzygies of c using mukai varieties we prove this conjecture for genus 6 7 and 8

a structured quasi arnoldi procedure for model order reduction of second order systems abstract existing krylov subspace based structure preserving model order reduction methods for the second order systems proceed in two stages the first stage is to generate a basis matrix of the underlying krylov subspace the second stage is to employ an explicit subspace projection to obtain a reduced order model with a moment matching property an open problem is how to avoid explicit projection so that it will be efficient for truly large scale systems in addition it is also desired that a structure preserving reduced system of order n matches maximum 2 n moments in this paper we propose a new procedure to compute a so called structured quasi arnoldi sqa decomposition once the sqa decomposition is computed a structure preserving reduced order model can be defined immediately from the decomposition without the explicit subspace projection furthermore the reduced model of order n matches maximum 2 n moments numerical examples demonstrate that the transpose free sqa based reduced model is compatible with the two sided structure preserving explicit projection methods and is more accurate than the one sided structure preserving explicit projection methods due to the higher number of matched moments

linear system approximation via covariance equivalent realizations abstract a truncation technique for model reduction to simultaneously match a specified number of output covariance derivatives is described these covariance equivalent realizations which match q 1 covariance derivatives are called q covers in general q covers are not unique the additional freedom is used herein so that the q cover obtained also matches q markov parameters the truncation technique uses a form of the observability matrices of the full order system to determine a priori the order required of the reduced order model to match a specified number of output covariance derivatives and markov parameters the resulting realization is shown to be independent of the basis of the complete model to within a unitary transformation stability conditions for the reduced order model are also described and the relationships are established between stochastically equivalent realizations and the q covers

structure preserving tangential interpolation for model reduction of port hamiltonian systems port hamiltonian systems result from port based network modeling of physical systems and are an important example of passive state space systems in this paper we develop a framework for model reduction of large scale multi input multi output port hamiltonian systems via tangential rational interpolation the resulting reduced model is a rational tangential interpolant that retains the port hamiltonian structure hence it remains passive we introduce an h 2 inspired algorithm for effective choice of interpolation points and tangent directions and present several numerical examples illustrating its effectiveness

gale duality and free resolutions of ideals of points what is the shape of the free resolution of the ideal of a general set of points in p this question is central to the programme of connecting the geometry of point sets in projective space with the structure of the free resolutions of their ideals there is a lower bound for the resolution computable from the known hilbert function and it seemed natural to conjecture that this lower bound would be achieved this is the minimal resolution conjecture lorenzini 1987 1993 although the conjecture has been shown to hold in many cases three examples discovered computationally by frank olaf schreyer in 1993 strongly suggested that it would fail in general in this paper we shall describe a novel structure inside the free resolution of a set of points which accounts for the failure and provides a counterexample in p for every r 6 r 6 9 we begin by reviewing the conjecture and its status consider a set of c points in the projective r space over a eld k say c pk let s k x0 xr let ic be the homogeneous ideal of c and let sc denote the homogeneous coordinate ring of c let

an adaptive strategy for the bivariate solution of finite element problems using multivariate nested pade approximants most engineering applications involving solutions by numerical methods are dependent on several parameters whose impact on the solution may significantly vary from one to the other at times an ev

on chevalley groups acting on projective planes in io hcring introduced the concept of a strongly irreducible collineation group let n be a projective plane k a collineation group of n k is said to be strongly irreducible iff k fixes no points lines triangles or subplanes of n if k is a finite group acting strongly irreducible on 71 generated by perspectivities then hering shows in 10 that k is either an extension of a 3 group with a subgroup of the automorphismgroup or there is a normal subgroup g in k g a nonabelian finite simple group such that k aut g the aim of this paper is to prove the following theorem

pade approximants and the modal connection towards increased robustness for fast parametric sweeps summary in order to increase the robustness of a pade based approximation of parametric solutions to finite element problems an a priori estimate of the poles is proposed the resulting original approach is shown to allow for a straightforward efficient subsequent pade based expansion of the solution vector components overcoming some of the current convergence and robustness limitations in particular this enables for the intervals of approximation to be chosen a priori in direct connection with a given choice of pade approximants the choice of these approximants as shown in the present work is theoretically supported by the montessus de ballore theorem concerning the convergence of a series of approximants with fixed denominator degrees key features and originality of the proposed approach are i a component wise expansion which allows to specifically target subsets of the solution field and ii the a priori simultaneous choice of the pade approximants and their associated interval of convergence for an effective and more robust approximation an academic acoustic case study a structural acoustic application and a larger acoustic problem are presented in order to demonstrate the potential of the approach proposed this article is protected by copyright all rights reserved

fast frequency sweeps with many forcing vectors through adaptive interpolatory model order reduction summary the goal of this work is to examine the efficacy of interpolatory model order reduction on frequency sweep problems with many forcing vectors the adaptive method proposed relies on the implicit interpolatory properties of subspace projection with basis vectors spanning the forced response of the system and its derivatives the algorithm is similar to a recently proposed adaptive scheme in that it determines both interpolation location and order within the frequency domain of interest the bounds of efficiency of the approach as the number of forcing vectors grows are explored through the use of rough floating operation counts this in turn guides choices made in the adaptive algorithm including a new technique that prevents excessive subspace growth by capitalizing on subspace direction coupling in order to demonstrate the algorithm s utility a series of frequency sweep problems drawn from the field of structural acoustics is analyzed it is demonstrated that increased order of interpolation can actually degrade the efficiency of the algorithm as the number of forcing vectors grows but that this can be limited by the subspace size throttling procedure developed here copyright 2014 john wiley sons ltd

interpolatory projection methods for structure preserving model reduction abstract we present a framework for interpolatory model reduction that treats systems having a generalized coprime factorization c s s 1 b s d this includes rational krylov based interpolation methods as a special case the broader framework allows retention of special structure in reduced models such as symmetry second and higher order structure state constraints internal delays and infinite dimensional subsystems two numerical examples are provided for systems associated with viscoelastic dynamics and with an internal state delay that demonstrate the effectiveness and flexibility of the approach

chern classes and characteristic cycles of determinantal varieties abstract let k be an algebraically closed field of characteristic 0 for m n we define m n k to be the set of m n matrices over k with kernel dimension k this is a projective subvariety of p m n 1 and is called the generic determinantal variety in most cases m n k is singular with singular locus m n k 1 in this paper we give explicit formulas computing the chern mather class c m and the chern schwartz macpherson class c s m of m n k as classes in the projective space we also obtain formulas for the conormal cycles and the characteristic cycles of these varieties and for their generic euclidean distance degree further when k c we prove that the characteristic cycle of the intersection cohomology sheaf of a determinantal variety agrees with its conormal cycle and hence is irreducible our formulas are based on calculations of degrees of certain chern classes of the universal bundles over the grassmannian for some small values of m n k we use macaulay2 to exhibit examples of the chern mather classes the chern schwartz macpherson classes and the classes of characteristic cycles of m n k on the basis of explicit computations in low dimensions we formulate conjectures concerning the effectivity of the classes and the vanishing of specific terms in the chern schwartz macpherson classes of the largest strata m n k m n k 1 the irreducibility of the characteristic cycle of the intersection cohomology sheaf follows from the kashiwara dubson s microlocal index theorem a study of the tjurina transform of m n k and the recent computation of the local euler obstruction of m n k

the projective geometry of the gale transform abstract the gale transform an involution on sets of points in projective space appears in a multitude of guises and in subjects as diverse as optimization coding theory theta functions and recently in our proof that certain general sets of points fail to satisfy the minimal free resolution conjecture see eisenbud and popescu 1999 invent math 136 419 449 in this paper we reexamine the gale transform in the light of modern algebraic geometry we give a more general definition in the context of finite locally gorenstein subschemes we put in modern form a number of the more remarkable examples discovered in the past and we add new constructions and connections to other areas of algebraic geometry we generalize goppa s theorem in coding theory and we give new applications to castelnuovo theory we also give references to classical and modern sources

entropy stable reduced order modeling of nonlinear conservation laws abstract reduced order models of nonlinear conservation laws in fluid dynamics do not typically inherit stability properties of the full order model we introduce projection based hyper reduced models of nonlinear conservation laws which are globally conservative and inherit a semi discrete entropy inequality independently of the choice of basis and choice of parameters

structure preserving galerkin pod reduced order modeling of hamiltonian systems abstract the proper orthogonal decomposition reduced order model pod rom has been widely used as a computationally efficient surrogate model in large scale numerical simulations of complex systems however when it is applied to a hamiltonian system a naive application of the pod method can destroy the hamiltonian structure in the reduced order model in this paper we develop a new reduced order modeling approach for hamiltonian systems which modifies the galerkin projection based pod rom so that the appropriate hamiltonian structure is preserved since the pod truncation can degrade the approximation of the hamiltonian function we propose to use a pod basis from shifted snapshots to improve the approximation to the hamiltonian function we further derive a rigorous a priori error estimate for the structure preserving rom and demonstrate its effectiveness in several numerical examples this approach can be readily extended to dissipative hamiltonian systems port hamiltonian systems etc

a matrix of linear forms which is annihilated by a vector of indeterminates abstract let r k t 1 t f be a standard graded polynomial ring over the field k and be an f g matrix of linear forms from r where 1 g f assume t 1 t f is 0 and that grade i g is exactly one short of the maximum possible grade we resolve r r i g prove that r has a g linear resolution record explicit formulas for the h vector and multiplicity of r and prove that if f g is even then the ideal i g is unmixed furthermore if f g is odd then we identify an explicit generating set for the unmixed part i g unm of i g resolve r i g unm and record explicit formulas for the h vector of r i g unm the rings r i g and r i g unm automatically have the same multiplicity these results have applications to the study of the blow up algebras associated to linearly presented grade three gorenstein ideals

symbolic powers of perfect ideals of codimension 2 and birational maps abstract this work is about symbolic powers of codimension two perfect ideals in a standard polynomial ring over a field assuming that the entries of the corresponding presentation matrix are general linear forms the main contribution of the present approach is the use of the birational theory underlying the nature of the ideal and the details of a deep interlacing between generators of its symbolic powers and the inversion factors stemming from the inverse map to the birational map defined by the linear system spanned by the generators of this ideal a full description of the corresponding symbolic rees algebra is given in some cases one application is an affirmative solution of a conjecture of eisenbud mazur in 11 section 2

solution of 3d time domain electromagnetic problems using optimal subspace projection abstracttime domain problems for controlled source electromagnetic exploration require accurate discretization of the solution for multiple spacial and temporal scales therefore forward simulation using conventional computational methods becomes computationally expensive even without accounting for induced polarization ip effects these effects create another complication caused by the presence of a convolution integral in the time domain maxwell system we suggested a novel fast and robust algorithm to solve the 3d time domain electromagnetic em problems that can be considered as a generalization of the spectral lanczos decomposition method the new method also allowed us to incorporate the ip effects without significant cost increase the discretized large scale maxwell system was projected onto a small subspace consisting of the laplace domain solutions the so called parameter dependent krylov subspace for an optimally chosen set of laplace parameters the projected system preserved stabilit

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

affine planes admitting a collineation group with a transitive action on the line at infinity abstract the affine planes admitting a collineation group with a transitive action on the line at infinity are investigated and an essentially complete classification is achieved when the group is almost simple a rather deep result is also obtained when the action on the line at infinity is faithful and primitive

quantum entanglement and geometry of determinantal varieties quantum entanglement was first recognized as a feature of quantum mechanics in the famous paper of einstein podolsky and rosen recently it has been realized that quantum entanglement is a key ingredient in quantum computation quantum communication and quantum cryptography in this paper we introduce algebraic sets which are determinantal varieties in the complex projective spaces or the products of complex projective spaces for the mixed states on bipartite or multipartite quantum systems as their invariants under local unitary transformations these invariants are naturally arised from the physical consideration of measuring mixed states by separable pure states our construction has applications in the following important topics in quantum information theory 1 separability criterion it is proved that the algebraic sets must be a union of the linear subspaces if the mixed states are separable 2 simulation of hamiltonians it is proved that the simulation of semipositive hamiltonians of the same rank implies the projective isomorphisms of the corresponding algebraic sets 3 construction of bound entangled mixed states examples of the entangled mixed states which are invariant under partial transpositions thus ppt bound entanglement are constructed systematically from our new separability criterion

resultants and chow forms via exterior syzygies let w be a vector space of dimension n 1 over a field k the chow divisor of a k dimensional variety x in p p w is the hypersurface in the grassmannian gk 1 of planes of codimension k 1 in p whose points over the algebraic closure of k are the planes that meet x the chow form of x is the defining equation of the chow divisor for example the resultant of k 1 forms of degree e in k 1 variables is the chow form of p embedded by the e th veronese mapping in p with n k e k 1 more generally the chow divisor of a k cycle i ni vi on projective space is defined to be i nidi where di is the chow divisor of vi the chow divisor of a sheaf f with k dimensional support is the chow divisor of the associated k cycle of f in this paper we will give a new expression for the chow divisor and apply it to give explicit formulas in many new cases starting with a sheaf f on p we use exterior algebra methods to define a canonical and effectively computable chow complex of f on each grassmannian of planes in p if f has k dimensional support we show that the chow form of f is the determinant of the chow complex of f on the grassmannian of planes of codimension k 1 the beilinson monad of f beilinson 1978 is the chow complex of f on the grassmannian of 0 planes that is on p itself in particular we are able to give explicit determinantal and pfaffian formulas for resultants in some cases where no polynomial formulas were known for example the horrocks mumford bundle gives rise to polynomial formulas for the resultant of five homogeneous forms of degrees 4 6 or 8 in five variables the easiest of our new formulas to write down is for the resultant of 3 quadratic forms in three variables the chow form of the veronese surface in p using the tangent bundle of p conclude that it can be written in bezout form described below as the pfaffian of the matrix 0 245 345 135 045 035 145 235

counterexamples to additivity of minimum output p renyi entropy for p close to 0 complementing recent progress on the additivity conjecture of quantum information theory showing that the minimum output p renyi entropies of channels are not generally additive for p 1 we demonstrate here by a careful random selection argument that also at p 0 and consequently for sufficiently small p there exist counterexamples

the shifted proper orthogonal decomposition a mode decomposition for multiple transport phenomena transport dominated phenomena provide a challenge for common mode based model reduction approaches we present a model reduction method which is suited for these kinds of systems it extends the proper orthogonal decomposition pod by introducing time dependent shifts of the snapshot matrix the approach called shifted proper orthogonal decomposition spod features a determination of the multiple transport velocities and a separation of these one and two dimensional test examples reveal the good performance of the spod for transport dominated phenomena and its superiority in comparison to the pod

the projective dimension of codimension two algebras presented by quadrics abstract motivated by a question of stillman we find a sharp upper bound for the projective dimension of ideals of height two generated by quadrics in a polynomial ring with arbitrary large number of variables we prove that ideals generated by n quadrics define cyclic modules with projective dimension at most 2 n 2 we refine this bound according to the multiplicity of the ideal we ask whether tight upper bounds for the projective dimension of ideals generated by quadrics can be expressed only in terms of their height and number of minimal generators

an adaptive scheme for a class of interpolatory model reduction methods for frequency response problems frequency sweep problems arise in many structural dynamic acoustic and structural acoustic applications in each case they incur the evaluation of a frequency response function for a typically large number of frequencies because each function evaluation requires the solution of an often large scale system of equations frequency sweep problems are computationally intensive interpolatory model order reduction is a powerful tool for reducing their cost however the performance of this tool depends on the location and number of the interpolation frequency points it also depends on the number of consecutive frequency derivatives of the response function that are matched at each frequency point so far these two choices have been made in the literature in a heuristic manner in contrast this paper proposes an automatic adaptive strategy based on monitoring the euclidean norm of the relative residual associated with the function to be evaluated over the frequency band of interest more specifically the number of interpolation points and the number of matched frequency derivatives are adaptively increased until the global euclidean norm of the relative residual is reduced below a user specified tolerance the robustness accuracy and computational efficiency of this adaptive strategy are highlighted with the solution of several frequency sweep problems associated with large scale structural dynamic acoustic and structural acoustic finite element models copyright 2012 john wiley sons ltd

high rank linear syzygies on low rank quadrics we study the linear syzygies of a homogeneous ideal i subseteq s rm sym bf k v focussing on the graded betti numbers b i i 1 rm dim bf k rm tor i s i bf k i 1 for a variety x and divisor d with v h 0 d what conditions on d ensure that b i i 1 ne 0 eisenbud has shown that a decomposition d sim a b such that a and b have at least two sections gives rise to determinantal equations and corresponding syzygies in i x and conjectured that if i 2 is generated by quadrics of rank le 4 then the last nonvanishing b i i 1 is a consequence of such equations we describe obstructions to this conjecture and prove a variant the obstructions arise from toric specializations of the rees algebra of koszul cycles and we give an explicit construction of toric varieties with minimal linear syzygies of arbitrarily high rank this gives one answer to a question posed by eisenbud and koh

error estimation in reduced basis method for systems with time varying and nonlinear boundary conditions many physical phenomena such as mass transport and heat transfer are modeled by systems of partial differential equations with time varying and nonlinear boundary conditions control inputs and disturbances typically affect the system dynamics at the boundaries and a correct numerical implementation of boundary conditions is therefore crucial however numerical simulations of high order discretized partial differential equations are often too computationally expensive for real time and many query analysis for this reason model complexity reduction is essential in this paper it is shown that the classical reduced basis method is unable to incorporate time varying and nonlinear boundary conditions to address this issue it is shown that by using a modified surrogate formulation of the reduced basis ansatz combined with a feedback interconnection and a input related term the effects of the boundary conditions are accurately described in the reduced order model the results are compared with the classical reduced basis method unlike the classical method the modified ansatz incorporates boundary conditions without generating unphysical results at the boundaries moreover a new approximation of the bound and a new estimate for the error induced by model reduction are introduced the effectiveness of the error measures is studied through simulation case studies and a comparison with existing error bounds and estimates is provided the proposed approximate error bound gives a finite bound of the actual error unlike existing error bounds that grow exponentially over time finally the proposed error estimate is more accurate than existing error estimates

a cyclic low rank smith method for large sparse lyapunov equations in this paper we present the cyclic low rank smith method which is an iterative method for the computation of low rank approximations to the solution of large sparse stable lyapunov equations it is based on a generalization of the classical smith method and profits by the usual low rank property of the right hand side matrix the requirements of the method are moderate with respect to both computational cost and memory furthermore we propose a heuristic for determining a set of suboptimal alternating direction implicit adi shift parameters this heuristic which is based on a pair of arnoldi processes does not require any a priori knowledge on the spectrum of the coefficient matrix of the lyapunov equation numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the adi parameters

projection free approximate balanced truncation of large unstable systems in this article we show that the projection free snapshot based balanced truncation method can be applied directly to unstable systems we prove that even for unstable systems the unmodified balanced proper orthogonal decomposition algorithm theoretically yields a converged transformation that balances the gramians including the unstable subspace we then apply the method to a spatially developing unstable system and show that it results in reduced order models of similar quality to the ones obtained with existing methods due to the unbounded growth of unstable modes a practical restriction on the final impulse response simulation time appears which can be adjusted depending on the desired order of the reduced order model recommendations are given to further reduce the cost of the method if the system is large and to improve the performance of the method if it does not yield acceptable results in its unmodified form finally the method is applied to the linearized flow around a cylinder at re 100 to show that it actually is able to accurately reproduce impulse responses for more realistic unstable large scale systems in practice the well established approximate balanced truncation numerical framework therefore can be safely applied to unstable systems without any modifications additionally balanced reduced order models can readily be obtained even for large systems where the computational cost of existing methods is prohibitive

straightening law and powers of determinantal ideals of hankel matrices in this paper we establish a standard monomial theory for generic hankel matrices by a generic hankel matrix we mean a matrix y yij with yij x i j 1 where the x i are indeterminates over a field k we use this structure to determine the symbolic powers and the primary decomposition of the powers of the determinantal ideals it of y further we prove that the symbolic and ordinary rees algebras of it are cohen macaulay normal domains the first standard monomial theory was developed by hodge h to study the homogeneous coordinate ring of the grassmannian variety later standard monomial theories were established for generic matrices by doubilet rota and stein drs and for generic symmetric and generic skew symmetric matrices by de concini and procesi dp these are all examples of algebras with straightening law asl for short over a poset or over a doset the abstract notion of asl was introduced and developed by eisenbud e1 and by de concini eisenbud and procesi dep2 see also bv these structures turned out to be an extremely powerful tool in studying determinantal rings and ideals arising from the above mentioned generic matrices let now x1 xn be indeterminates over an arbitrary field k for j 1 n we denote by xj the j n 1 j hankel matrix with entries x1 xn that is

on the dimension of subspaces with bounded schmidt rank we consider the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states this is motivated in part by results of hayden et al e print arxiv quant ph 0407049 commun math phys 265 95 2006 which show that in large d d dimensional systems there exist random subspaces of dimension almost d2 all of whose states have entropy of entanglement at least logd o 1 it is also a generalization of results on the dimension of completely entangled subspaces which have connections with the construction of unextendible product bases here we take as entanglement measure the schmidt rank and determine for every pair of local dimensions da and db and every r the largest dimension of a subspace consisting only of entangled states of schmidt rank r or larger this exact answer is a significant improvement on the best bounds that can be obtained using the random subspace techniques in hayden et al we also determine the converse the l

a survey of projection based model reduction methods for parametric dynamical systems numerical simulation of large scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena however the inherent large scale nature of the models often leads to unmanageable demands on computational resources model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate yet accurately represent the original large scale system behavior model reduction of linear nonparametric dynamical systems has reached a considerable level of maturity as reflected by several survey papers and books however parametric model reduction has emerged only more recently as an important and vibrant research area with several recent advances making a survey paper timely thus this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods parametric model reduction targets the broad class of problems for wh

some linear syzygy conjectures in this paper we introduce four conjectures refining a very useful result of mark green s on the linear part of the free resolution of a graded module over a polynomial ring we prove a number of special cases of the conjectures including all cases pertaining to modules over a polynomial ring in at most four variables in a later paper we will give a complete classification of the nets of skew symmetric transformations on a sdimensional vector space over an algebraically closed field from which we deduce that our conjectures are true in five variables as well

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

exact algorithms for linear matrix inequalities let a x a 0 x 1a 1 cdots x na n be a linear matrix or pencil generated by given symmetric matrices a 0 a 1 ldots a n of size m with rational entries the set of real vectors x such that the pencil is positive semidefinite is a convex semialgebraic set called spectrahedron described by a linear matrix inequality we design an exact algorithm that up to genericity assumptions on the input matrices computes an exact algebraic representation of at least one point in the spectrahedron or decides that it is empty the algorithm does not assume the existence of an interior point and the computed point minimizes the rank of the pencil on the spectrahedron the degree d of the algebraic representation of the point coincides experimentally with the algebraic degree of a generic semidefinite program associated to the pencil we provide explicit bounds for the complexity of our algorithm proving that the maximum number of arithmetic operations that are performed is essentially quadratic in a mu

projection based reduced order models for a cut finite element method in parametrized domains abstract this work presents a reduced order modeling technique built on a high fidelity embedded mesh finite element method such methods and in particular the cutfem method are attractive in the generation of projection based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains and in general to handle topological changes the combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems avoiding remeshing as well as the reference domain formulation often used in the reduced order modeling for boundary fitted finite element formulations the resulting novel methodology is presented on linear elliptic and stokes problems together with several test cases to assess its capability the role of a proper extension and transport of embedded solutions to a common background is analyzed in detail

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

an improved algorithm for balanced pod through an analytic treatment of impulse response tails we present a modification of the balanced proper orthogonal decomposition balanced pod algorithm for systems with simple impulse response tails in this new method we use dynamic mode decomposition dmd to estimate the slowly decaying eigenvectors that dominate the long time behavior of the direct and adjoint impulse responses this is done using a new low memory variant of the dmd algorithm appropriate for large datasets we then formulate analytic expressions for the contribution of these eigenvectors to the controllability and observability gramians these contributions can be accounted for in the balanced pod algorithm by simply appending the impulse response snapshot matrices direct and adjoint respectively with particular linear combinations of the slow eigenvectors aside from these additions to the snapshot matrices the algorithm remains unchanged by treating the tails analytically we eliminate the need to run long impulse response simulations lowering storage requirements and speeding up ensuing computations to demonstrate its effectiveness we apply this method to two examples the linearized complex ginzburg landau equation and the two dimensional fluid flow past a cylinder as expected reduced order models computed using an analytic tail match or exceed the accuracy of those computed using the standard balanced pod procedure at a fraction of the cost

dimension reduction of large scale second order dynamical systems via a second order arnoldi method a structure preserving dimension reduction algorithm for large scale second order dynamical systems is presented it is a projection method based on a second order krylov subspace a second order arnoldi soar method is used to generate an orthonormal basis of the projection subspace the reduced system not only preserves the second order structure but also has the same order of approximation as the standard arnoldi based krylov subspace method via linearization the superior numerical properties of the soar based method are demonstrated by examples from structural dynamics and microelectromechanical systems