pulse bifurcation and transition to spatiotemporal chaos in an excitable reaction diffusion model we address the stability of solitary pulses as well as some other traveling structures near the onset of spatiotemporal chaos in a two species reaction diffusion model describing the oxidation of c

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

algebro geometric analogues of donaldson s polynomials donaldson 5 has introduced polynomials on the second homology of any simply connected closed four manifold m with b odd at least 3 which up to are invariant under orientation preserving diffeomorphisms for every c ibm there is a polynomial 7c m syma h2 s where d c 4c z2 b 1 to define it one considers j c the set which is in a natural way a manifold of gauge equivalence classes of connections on the su 2 bundle on m with c2 c antiself dual with respect to a generic metric on m there is a natural map h2 m h2 j c namely slant product with 88 where p is the universal so 3 principal bundle on m x j c one way of defining 7c m 10 is to construct a compactification of r xc carrying a fundamental class xc one proves that extends to a map h2 m h2 xc then one defines

determinantal representations of singular hypersurfaces in pn abstract a global determinantal representation of projective hypersurface x p n is a matrix whose entries are linear forms in homogeneous coordinates and whose determinant defines the hypersurface we study the properties of such representations for singular possibly reducible or non reduced hypersurfaces in particular we obtain the decomposability criteria for determinantal representations of globally reducible hypersurfaces further we classify the determinantal representations in terms of the corresponding kernel sheaves on x finally we extend the results to the case of symmetric self adjoint representations with implications to hyperbolic polynomials and the generalized lax conjecture

exact image theory for the sommerfeld half space problem part iii general formulation the general sommerfeld problem with both epsilon and mu discontinuous and a source consisting of arbitrarily oriented electric and or magnetic dipoles at the same location is considered in terms of image theory the problem is formulated with electric and magnetic fields instead of potential quantities resulting in a vector transmission line interpretation of the fourier transformed problem the image sources are seen to be located in complex space expressable in terms of a certain basic image current function which was encountered in part ii of this paper on the vertical electric dipole problem the horizontal electric magnetic dipole image is solved and found to consist of both vertical and horizontal current components the image concept is generalized to the most general three dimensional sources as a check the well known reflection coefficient method is obtained as the far field approximation of the present theory

indefinite hardy spaces on finite bordered riemann surfaces abstract it is well known that one may study hardy spaces with the domain a finite bordered riemann surface rather than the unit disk however for domains with more than one boundary component it is natural to consider besides the usual positive definite inner product on h 2 indefinite inner products obtained by picking up different signs or in the vector valued case different signature matrices when integrating over different components in this paper we obtain a necessary and sufficient condition for such an indefinite inner product to be nondegenerate and show that when this condition is satisfied we actually get a krein space furthermore each holomorphic mapping of the finite bordered riemann surface onto the unit disk which maps boundary to boundary determines an explicit isometric isomorphism between this space and a usual vector valued hardy space on the unit disk with an indefinite inner product defined by an appropriate hermitian matrix as is usual when studying hardy spaces on a multiply connected domain the elements of the space are sections of a vector bundle rather than functions the main point is to construct an appropriate extension of this bundle to the double of the finite bordered riemann surface and to use cauchy kernels for certain vector bundles on a compact riemann surface

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

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

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

the cayley hamilton theorem and inverse problems for multiparameter systems abstract we review some of the current research in multiparameter spectral theory we prove a version of the cayley hamilton theorem for multiparameter systems and list a few inverse problems for such systems some consequences of results on determinantal representations proved by dixon dickson and vinnikov for the inverse problems are discussed

complete description of determinantal representations of smooth irreducible curves abstract determinantal representations of algebraic curves are interesting in themselves and their classification is equivalent to the simultaneous classification of triples of matrices we present a complete description of determinantal representations of smooth irreducible curves over any algebraically closed field we used the notion of the class of divisors of the vector bundle corresponding to a determinantal representation we prove that two determinantal representations of a smooth curve f are equivalent if and only if the classes of divisors of the corresponding vector bundles coincide we give a precise characterization of those classes of divisors that arise from vector bundles corresponding to determinantal representations of f then we obtain a parametrization of determinantal representations of f up to equivalence by the points of the jacobian variety of f not on some exceptional subvariety in particular it follows that any smooth curve of order 3 or greater possesses an infinite number of nonequivalent determinantal representations we also specialize our results to symmetrical and self adjoint representations

numerical analysis of arbitrarily shaped probe excited single arm printed wire antennas a general integral equation technique is described for analysis of an arbitrarily shaped single arm printed wire antenna excited through a vertical probe a unified current integral equation is formulated on the basis of dyadic green s functions and the reciprocity theorem the current distribution is obtained by using a parametric moment method in which parameter segments are adopted for the printed wire instead of the commonly employed wire length segments the radiation field solution involving both the printed antenna and vertical probe is also presented the validity of the formulation is verified by comparing the numerically obtained input impedance and radiation patterns for a linear antenna and a meander antenna with measured data a circular open loop and an archimedian spiral are investigated to illustrate the applicability of the present technique

how to count zeroes of polynomials on quadrature domains using the bezout matrix abstract classically the bezout matrix or simply bezoutian of two polynomials is used to locate the roots of the polynomial and in particular test for stability in this paper we develop the theory of bezoutians on real riemann surfaces of dividing type the main result connects the signature of the bezoutian of two real meromorphic functions to the topological data of their quotient which can be seen as the generalization of the classical cauchy index as an application we propose a method to count the number of zeroes of a polynomial in a quadrature domain using the inertia of the bezoutian we provide examples of our method in the case of simply connected quadrature domains

representations of lie algebras by non skewselfadjoint operators in hilbert space we study non skewselfadjoint representations of a finite dimensional real lie algebra gg to this end we embed a non skewselfadjoint representation of gg into a more complicated structure that we call a gg operator vessel and that is associated to an overdetermined linear conservative input state output system on the corresponding simply connected lie group gg we develop the frequency domain theory of the system in terms of representations of gg and introduce the joint characteristic function of a gg operator vessel which is the analogue of the classical notion of the characteristic function of a single non selfadjoint operator as the first non commutative example we apply the theory to the lie algebra of the ax bax b group the group of affine transformations of the line

canonical forms for symmetric skew symmetric real matrix pairs under strict equivalence and congruence a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the reals one of which is skew symmetric and the other is either symmetric or skew symmetric these reductions are by strict equivalence and by congruence over the reals or over the complex numbers and essentially complete proofs are presented the proofs are based on canonical forms attributed to jordan and kronecker some closely related results which can be derived from the canonical forms of pairs of symmetric skew symmetric real forms are also included they concern simultaneously neutral subspaces hamiltonian and skew hamiltonian matrices and canonical structures of real matrices which are selfadjoint or skew adjoint in a regular skew symmetric indefinite inner product and real matrices which are skew adjoint in a regular symmetric indefinite inner product the paper is largely expository and continues the comprehensive account of the reduction of pairs of matrices started in p lancaster l rodman canonical forms for hermitian matrix pairs under strict equivalence and congruence siam rev in press

the elementary transformation of vector bundles on regular schemes we give a generalized definition of an elementary transformation of vector bundles on regular schemes by using maximal cohen macaulay sheaves on divisors this definition is a natural extension of that given by maruyama and has a connection with that given by sumihiro by this elementary transformation we can construct up to tensoring line bundles all vector bundles from trivial bundles on nonsingular quasi projective varieties over an algebraically closed field moreover we give an application of this theory to reflexive sheaves

livsic type determinantal representations and hyperbolicity abstract hyperbolic homogeneous polynomials with real coefficients i e hyperbolic real projective hypersurfaces and their determinantal representations play a key role in the emerging field of convex algebraic geometry in this paper we consider a natural notion of hyperbolicity for a real subvariety x p d of an arbitrary codimension l with respect to a real l 1 dimensional linear subspace v p d and study its basic properties we also consider a class of determinantal representations that we call livsic type and a nice subclass of these that we call very reasonable much like in the case of hypersurfaces l 1 the existence of a definite hermitian very reasonable livsic type determinantal representation implies hyperbolicity we show that every curve admits a very reasonable livsic type determinantal representation our basic tools are cauchy kernels for line bundles and the notion of the bezoutian for two meromorphic functions on a compact riemann surface that we introduce we then proceed to show that every real curve in p d hyperbolic with respect to some real d 2 dimensional linear subspace admits a definite hermitian or even definite real symmetric very reasonable livsic type determinantal representation

on linearizations of the quadratic two parameter eigenvalue problems abstract we present several transformations that can be used to solve the quadratic two parameter eigenvalue problem qmep by formulating an associated linear multiparameter eigenvalue problem two of these transformations are generalizations of the well known linearization of the quadratic eigenvalue problem and linearize the qmep as a singular two parameter eigenvalue problem the third replaces all nonlinear terms by new variables and adds new equations for their relations the qmep is thus transformed into a nonsingular five parameter eigenvalue problem the advantage of these transformations is that they enable one to solve the qmep using existing numerical methods for multiparameter eigenvalue problems we also consider several special cases of the qmep where some matrix coefficients are zero

real separated algebraic curves quadrature domains ahlfors type functions and operator theory abstract the aim of this paper is to inter relate several algebraic and analytic objects such as real type algebraic curves quadrature domains functions on them and rational matrix functions with special properties and some objects from operator theory such as vector toeplitz operators and subnormal operators our tools come from operator theory but some of our results have purely algebraic formulation we make use of xia s theory of subnormal operators and of the previous results by the author in this direction we also correct in section 5 some inaccuracies in the works of d v yakubovich subnormal operators of finite type i xia s model and real algebraic curves in c 2 rev mat iberoamericana 14 1998 95 115 d v yakubovich subnormal operators of finite type ii structure theorems rev mat iberoamericana 14 1998 623 681 by the author

noncommutative convexity arises from linear matrix inequalities this paper concerns polynomials in g noncommutative variables x x1 xg inverses of such polynomials and more generally noncommutative rational expressions with real coefficients which are formally symmetric and analytic near 0 the focus is on rational expressions r r x which are matrix convex near 0 i e those rational expressions r for which there is an 0 such that if x x1 xg is a g tuple of n n symmetric matrices satisfying in x12 xg2 is positive definite and y is also then the symmetric matrix tr x 1 t r y r tx 1 t y is positive semidefinite for all numbers t 0 t 1 this article gives a complete classification of matrix convex rational expressions see theorem 3 3 by representing such r in terms of a symmetric linear pencil l x id jajxj 0d 100 1 r 0 in the noncommuting variables xj where aj are symmetric d d matrices namely for a real number r is a schur complement of the linear pencil l moreover given a matrix convex r the set consisting of g tuples x of n n symmetric matrices 0 1 x r x iis negative definite has component containing 0 which is the same as the negativity set 0 2 x l x is negative definite for l conditions like l x is negative definite are known as linear matrix inequalities lmis in the engineering literature and arguably the main advance in linear systems theory in the 1990s was the introduction of lmi techniques in this language what we have shown in 0 1 vs 0 2 is that the set of solutions to a convex matrix inequality with noncommutative unknowns is the same as the set of solutions to some lmi in many engineering systems problems convexity would have all of the advantages of lmis indeed convexity guarantees that solutions are global and convexity bodes well for reliability of the numerics since lmis have a structure which is seemingly much more rigid than convexity there is the continual hope that a convexity based theory will be more far reaching than lmis but will it there are two natural situations one where the unknowns are scalars and one where the unknowns are matrices appearing in formulas which respect matrix multiplication these latter problems mathematically yield expressions with noncommutative unknowns and they arise in engineering systems problems which are dimensionless in the sense that they scale automatically with dimension as do most of the classics of control theory that is the case we study here and the result stated above suggests the surprising conclusion that for dimensionless systems problems convexity offers no greater generality than lmis indeed the result proves this for a class of model problems furthermore we show that existing algorithms together with algorithms described here construct the lmis above which are equivalent to the matrix inequalities based on the given matrix convex rational function r in a very different direction we prove that a symmetric polynomial p in g noncommutative symmetric variables has a symmetric determinantal representation namely there are symmetric matrices a0 ag in srd d with a0 invertible such that 0 3 detp x det a0 la x for each x a g tuple of symmetric n n matrices of course taking n 1 implies immediately that a commuting variables polynomial p on rg has a symmetric determinantal representation for g 2 much stronger commutative results can be obtained using tools of algebraic geometry but these do not seem to generalize to the higher dimensional case on the other hand a nonsymmetric commutative determinantal representation for any g is due to valiant universality of determinant in algebraic complexity theory our determinantal representation theorem is a bi product of the theory of systems realizations of noncommutative rational functions and can be read independently of much of the rest of the paper while the notion of noncommutative rational functions is standard the equivalence relation we use on rational expressions in our construction based on evaluating rational expressions on matrices is new and gives a new approach to noncommutative rational functions

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

on the correlation functions of the vector bundle generalization of the bc system it is shown that the determinants of the correlation functions of the generalized bc system introduced recently are given as pullbacks of the non abelian theta divisor

self replicating pulses and sierpinski gaskets in excitable media in our previous papers we have shown by computer simulations that a sierpinski gasket pattern appears in a bonhoeffer van der pol type reaction diffusion system in this paper we show another class of regular self similar structure which is found in four different excitable reaction diffusion systems this result strongly implies that the existence of the self similar spatiotemporal evolution is universal in excitable reaction diffusion media

intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a riemann surface let n and d be coprime positive integers and define m n d to be the moduli space of semi stable holomorphic vector bundles of rank n degree d and fixed determinant on a compact riemann surface e this moduli space is a compact kifhler manifold which has been studied from many different points of view for more than three decades see for instance narasimhan and seshadri 41 the subject of this article is the characterization of the intersection pairings in the cohomology ring1 h m n d a set of generators of this ring was described by atiyah and bott in their seminal 1982 paper 2 on the yangmills equations on riemann surfaces where in addition inductive formulas for the betti numbers of m n d obtained earlier using number theoretic methods 13 25 were rederived by poincare duality knowledge of the intersection pairings between products of these generators or equivalently knowledge of the evaluation on the fundamental class of products of the generators completely determines the structure of the cohomology ring in 1991 donaldson 15 and thaddeus 47 gave formulas for the intersection pairings between products of these generators in h m 2 1 in terms of bernoulli numbers then using physical methods witten 50 found formulas for generating functions from which could be extracted the intersection pairings between products of these generators in h m n d for general rank n these generalized his rigorously proved formulas 49 for the symplectic volume of m n d for instance the symplectic volume of m 2 1 is given by

polymorphic and regular localized activity structures in a two dimensional two component reaction diffusion lattice with complex threshold excitation abstract space time dynamics of the system modeling collective behaviour of electrically coupled nonlinear units is investigated the dynamics of a local cell is described by the fitzhugh nagumo system with complex threshold excitation it is shown that such a system supports formation of two distinct kinds of stable two dimensional spatially localized moving structures without any external stabilizing actions these are regular and polymorphic structures the regular structures preserve their shape and velocity under propagation while the shape and velocity as well as other integral characteristics of polymorphic structures show rather complex temporal behaviour both kinds of structures represent novel sorts of spatially temporal patterns which have not been observed before in typical two component reaction diffusion type systems it is demonstrated that there exist two types of regular structures single and bound states and three types of polymorphic structures periodic quasiperiodic and even chaotic ones the partition of the parameter plane into regions corresponding to the existence of these different types of structures is carried out high multistability of regular structures is indicated the interaction of regular structures is investigated the correspondence between the structures and trajectories in multidimensional phase space associated with the system is given bifurcation mechanisms leading to the loss of stability of regular structures as well as to a transition from one type of polymorphic structure to another are indicated the mechanisms of formation of regular and polymorphic structures are discussed

factorisation of generalised theta functions i we prove a version of factorisation relating the space of sections of theta bundles on the moduli spaces of parabolic rank 2 vector bundles on curves of genusg andg 1

the level rank duality for non abelian theta functions we prove that the strange duality conjecture of beauville donagi tu holds for all curves we establish first a more extended level rank duality interesting in its own right from which the standard level rank duality follows by restriction

an algebraic geometry study of the b c system with arbitrary twist fields and arbitrary statistics we present an analysis of the generalb c system including the system on a compact riemann surface of arbitrary genusg 0 by postulating that its correlation functions should only have the singularities imposed by the operator product expansion ope of the system studying a very in fact optimally general form of theb c system we prove rigorously that the standard practice of eliminating zero modes and even the standard lagrangian follow from the analyticity structure dictated by the ope alone we extend the analysis to consider the most general case of the presence of twist e g spin fields we then determine all the possible correlation functions of theb c system with statistics unspecified compatible with the ope on imposing fermi and bose statistics we obtain the correlation functions of the fermionicb c and systems respectively

spiking patterns emerging from wave instabilities in a one dimensional neural lattice the dynamics of a one dimensional lattice chain of electrically coupled neurons modeled by the fitzhugh nagumo excitable system with modified nonlinearity is investigated we have found that for certain conditions the lattice exhibits a countable set of pulselike wave solutions the analysis of homoclinic and heteroclinic bifurcations is given corresponding bifurcation sets have the shapes of spirals twisting to the same center the appearance of chaotic spiking patterns emerging from wave instabilities is discussed

heteroclinic contours and self replicated solitary waves in a reaction diffusion lattice with complex threshold excitation abstract the space time dynamics of the network system modeling collective behavior of electrically coupled nonlinear cells is investigated the dynamics of a local cell is described by the fitzhugh nagumo system with complex threshold excitation heteroclinic orbits defining traveling wave front solutions are investigated in a moving frame system a heteroclinic contour formed by separatrix manifolds of two saddle foci is found in the phase space the existence of such structure indicates the appearance of complex wave patterns in the network such solutions have been confirmed and analyzed numerically complex homoclinic orbits found in the neighborhood of the heteroclinic contour define the propagation of composite pulse excitations that can be self replicated in collisions leading to the appearance of complex wave patterns

elementary transformations of pfaffian representations of plane curves abstract let c be a smooth curve in p 2 given by an equation f 0 of degree d in this paper we consider elementary transformations of linear pfaffian representations of c elementary transformations can be interpreted as actions on a rank 2 vector bundle on c with canonical determinant and no sections which corresponds to the cokernel of a pfaffian representation every two pfaffian representations of c can be bridged by a finite sequence of elementary transformations pfaffian representations and elementary transformations are constructed explicitly for a smooth quartic applications to aronhold bundles and theta characteristics are given

the bc system of higher rank revisited the bc system of higher rank introduced recently is examined further it is shown that the correlation functions are connected with certain non abelian functions and it is discussed how quasi determinants arise

finite dimensional de branges spaces on riemann surfaces abstract we study certain finite dimensional reproducing kernel indefinite inner product spaces of multiplicative half order differentials on a compact real riemann surface these spaces are analogues of the spaces introduced by l de branges when the riemann sphere is replaced by a compact real riemann surface of a higher genus in de branges theory an important role is played by resolvent like difference quotient operators r here we introduce generalized difference quotient operators r y for any non constant meromorphic function y on the riemann surface the spaces we study are invariant under generalized difference quotient operators and can be characterized as finite dimensional indefinite inner product spaces invariant under two operators r y 1 i and r y 2 2 where y 1 and y 2 generate the field of meromorphic functions on the riemann surface which satisfy a supplementary identity analogous to the de branges identity for difference quotients just as the classical de branges spaces and difference quotient operators appear in the operator model theory for a single nonselfadjoint or nonunitary operator the spaces we consider and generalized difference quotient operators appear in the model theory for commuting nonselfadjoint operators with finite nonhermitian ranks

the stringy e function of the moduli space of rank 2 bundles over a riemann surface of genus 3 we compute the stringy e function or the motivic integral of the moduli space of rank 2 bundles over a riemann surface of genus 3 in doing so we answer a question of batyrev about the stringy e functions of the git quotients of linear representations

evaluation of sommerfeld integrals using chebyshev decomposition antenna analysis sommerfeld integrals ensue when one uses a spectral domain transformation to calculate fields of a dipole source in a homogeneous space or in a layered medium with planar boundaries a new and efficient method of evaluating these integrals is presented this method still converge in the case of unbounded homogeneous media or when source and observer are both on the boundary itself as in microstrip lines the integration is based on decomposition of the integrand into chebyshev polynomials numerical results are given and compared with published literature

the strange duality conjecture for generic curves let sux r be the moduli space of semi stable vector bundles of rank r with trivial determinant over a connected smooth projective algebraic curve x of genus g 1 over c recall that a vector bundle e on x is called semi stable if for any subbundle v deg v rk v deg e rk e points of sux r correspond to isomorphism classes of semi stable rank r vector bundles with trivial determinant up to an equivalence relation for any line bundle l of degree g 1 onx define l e sux r h e l 1 this turns out be a non zero cartier divisor whose associated line bundle l o l does not depend upon l it is known that l generates the picard group of sux r for this and the precise definition of l in terms of determinant of cohomology see dn let u x k be the moduli space of semi stable rank k and degree k g 1 bundles on x recall that on u x k there is a canonical non zero theta cartier divisor k whose underlying set is f u x k h x f 0 put m o k consider the natural map k r sux r u x k u x kr given by tensor product from the theorem of the square it follows that k rm is isomorphic to l m the canonical element kr h0 u x kr m and the kunneth theorem give a map well defined up to scalars

zero pole interpolation for matrix meromorphic functions on a compact riemann surface and a matrix fay trisecant identity this paper presents a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact riemann surface having a prescribed weil divisor i e having prescribed zeros and poles with directional as well as multiplicity information included in the vector case this new formalism unifies the earlier approach of ball clancey in the setting of trivial bundles over an abstract riemann surface with an earlier approach of the authors where the riemann surface was assumed to be the normalizing riemann surface for an algebraic curve embedded in c 2 with determinantal representation and the vector bundles were assumed to be presented as the kernels of linear matrix pencils the main tool is a version of the cauchy kernel appropriate for flat vector bundles over the riemann surface our formula for the interpolating bundle map in the special case of a single zero and a single pole can be viewed as a generalization of the fay trisecant identity from the usual line bundle case to the vector bundle case in terms of cauchy kernels in particular we obtain a new proof of the fay trisecant identity

on the characteristic polynomials of multiparameter pencils abstract in this paper we study the characteristic polynomial of multiparameter pencil z 1 a 1 z 2 a 2 z s a s the main theorem states that a unitary representation of a finitely generated group contains a one dimensional representation if and only if the characteristic polynomial of its generators contains a linear factor it follows that a two or three dimensional unitary representation of a finitely generated group is irreducible if and only if the characteristic polynomial of the pencil of its generators is irreducible the result is of kin to the dedekind and frobenius theorem on finite group determinant

gv sheaves fourier mukai transform and generic vanishing we prove a formal criterion for generic vanishing in the sense originated by green and lazarsfeld and pursued further by hacon but in the context of an arbitrary fourier mukai correspondence for smooth projective varieties we apply this to deduce a kodaira type generic vanishing theorem for adjoint bundles associated to nef line bundles and in fact a more general generic nadel type vanishing theorem for multiplier ideal sheaves still in the context of the picard variety the same method gives various other generic vanishing results by reduction to standard vanishing theorems we further use our criterion in order to address some examples related to generic vanishing on higher rank moduli spaces

on the quadratic two parameter eigenvalue problem and its linearization we introduce the quadratic two parameter eigenvalue problem and linearize it as a singular two parameter eigenvalue problem this together with an example from model updating shows the need for numerical methods for singular two parameter eigenvalue problems and for a better understanding of such problems there are various numerical methods for two parameter eigenvalue problems but only few for nonsingular ones we present a method that can be applied to singular two parameter eigenvalue problems including the linearization of the quadratic two parameter eigenvalue problem it is based on the staircase algorithm for the extraction of the common regular part of two singular matrix pencils

descent of coherent sheaves and complexes to geometric invariant theory quotients abstract fix a scheme x over a field of characteristic zero that is equipped with an action of a reductive algebraic group g we give necessary and sufficient conditions for a g equivariant coherent sheaf on x or a bounded above complex of g equivariant coherent sheaves on x to descend to a good quotient x g this gives a description of the coherent derived category of x g as an admissible subcategory of the equivariant derived category of x

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

interpolation problems for holomorphic functions abstract in this note we discuss the nevanlinna pick problem corona problem and caratheodory fejer problem for bounded holomorphic functions

moduli of vector bundles frobenius splitting and invariant theory let x be an irreducible projective curve of genus g over an algebraically closed field of positive characteristic 2 3 in part i we prove adapting the degeneration arguments of n tr that moduli spaces of rank two parabolic bundles on x are frobenius split m r for generic smooth x a similar result holds for x nodal a consequence is the verlinde formula in positive characteristic in part ii we give a direct proof of the fact that the local rings of the moduli spaces are f split and further that they are cohen macaulay this involves showing that the ring of invariants of g copies of 2 x 2 matrices under the adjoint action of sl 2 is f split and cohen macaulay

elementary transformations of determinantal representations of algebraic curves abstract we study elementary transformations first introduced by livsic and kravitsky in an operator theoretic context of determinantal representations of algebraic curves we consider determinantal representations of a smooth irreducible curve f over an algebraically closed field when regarded as acting on the corresponding vector bundle the elementary transformations are a matrix generalization of scalar linear fractional transformations they add to the class of divisors of the vector bundle a single zero and a single pole we show that given a determinantal representation of f we can build all the nonequivalent determinantal representations of f by applying to the given representation finite sequences of at most g elementary transformations where g is the genus of f

geometric aspects of multiparameter spectral theory the paper contains a geometric description of the dimension of the total root subspace of a regular multiparameter system in terms of the intersection multiplicities of its determinantal hypersurfaces the new definition of regularity used is proved to restrict to the familiar definition in the linear case a decomposability problem is also considered it is shown that the joint root subspace of a multiparameter system may be decomposable even when the root subspace of each member is indecomposable

strange duality of verlinde spaces for g_2 and f_4 we prove that the pull back of the canonical theta divisor for e 8 bundles at level one induces a strange duality between verlinde spaces for g 2 and f 4 at level one on smooth curves of genus g we also prove a parabolic generalization in terms of conformal blocks and write down identities between conformal blocks divisors in text pic overline text m g n mathbb q

de branges spaces on compact riemann surfaces and a beurling lax type theorem abstract using the notion of commutative operator vessels this work investigates de branges rovnyak spaces whose elements are sections of a line bundle of multiplicative half order differentials on a compact real riemann surface as a special case we obtain a beurling lax type theorem in the setting of the corresponding hardy space on a finite bordered riemann surface

determinantal representations of invariant hyperbolic plane curves abstract we study hyperbolic polynomials with nice symmetry and express them as the determinant of a hermitian matrix with special structure the goal of this paper is to answer a question posed by chien and nakazato in 2015 3 by properly modifying a determinantal representation construction of dixon 1902 5 we show for every hyperbolic polynomial of degree n invariant under the cyclic group of order n there exists a determinantal representation admitted via some cyclic weighted shift matrix moreover if the polynomial is invariant under the action of the dihedral group of order n the associated cyclic weighted shift matrix is unitarily equivalent to one with real entries

hecke correspondence stable maps and the kirwan desingularization we prove that the moduli space m0 0 n 2 of stable maps of degree 2 to the moduli space n of rank 2 stable bundles of flxed odd determinant ox ix over a smooth projective curve x of genus g 3 has two irreducible components which intersect transversely one of them is kirwan s partial desingularization e mx of the moduli space mx of rank 2 semistable bundles with determinant isomorphic to ox yix for some y 2 x the other component is the partial desingularization of the git quotient phom sl 2 w pgl 2 for a vector bundle w r 1 l i2 ix of rank g over the jacobian of x we also show that the hilbert scheme h the chow scheme c of conics in n and m0 0 n 2 are related by explicit contractions

symmetric determinantal representation of polynomials abstract we give an elementary proof only using linear algebra of a result due to helton mccullough and vinnikov which says that any polynomial over the reals can be written as the determinant of a symmetric affine linear pencil we give explicit determinantal representation formulas and extend our results to polynomials with coefficients in a ring of characteristic different from 2