the frobenius map rank 2 vector bundles and kummer s quartic surface in characteristic 2 and 3 abstract let x be a smooth projective curve of genus g 2 defined over an algebraically closed field k of characteristic p 0 let mx r be the moduli space of semi stable rank r vector bundles with fixed trivial determinant the relative frobenius map f x x 1 induces by pull back a rational map v m x 1 r m x r we determine the equations of v in the following two cases 1 g r p 2 2 2 and x nonordinary with hasse witt invariant equal to 1 see math ag 0005044 for the case x ordinary and 2 g r p 2 2 3 we also show the existence of base points of v i e semi stable bundles e such that f e is not semi stable for any triple g r p

a smoothing property of a hyperbolic system and boundary controllability abstract we investigate a method with which one can deduce controllability results from smoothing properties previous applications of the method were for partial differential equations like the euler bernoulli beam equation petrowski hyperbolic in this paper we study the method s applicability to a strictly hyperbolic system by considering the boundary controllability of a vibrating timoshenko beam with physical characteristics that may vary along the length of the beam two cases are considered a beam which is clamped at one end the other end being controlled by a torque and transverse force and a beam which is hinged at one end where a control torque is applied and free at the other end where a control force is applied

energy decay for timoshenko systems of memory type abstract linear systems of timoshenko type equations for beams including a memory term are studied the exponential decay is proved for exponential kernels while polynomial kernels are shown to lead to a polynomial decay the optimality of the results is also investigated

stability of frobenius direct images over surfaces let x be a smooth projective surface over an algebraically closed field k of characteristic p 0 with omega x 1 semistable and mu omega x 1 0 for any semistable resp stable bundle w of rank r we prove that f w is semistable resp stable when p ge r r 1 2 1

frobenius unstable bundles and p curvature we use the theory of p curvature of connections to analyze stable vector bundles of rank 2 on curves of genus 2 which pull back to unstable bundles under the frobenius morphism we take two approaches first using explicit formulas for p curvature to analyze low characteristic cases and then using degeneration techniques to obtain an answer for a general curve by degenerating to an irreducible rational nodal curve and applying the results of additional works by the author we also apply our explicit formulas to give a new description of the strata of curves of genus 2 of different p rank

direct images of bundles under frobenius morphism let x be a smooth projective variety of dimension n over an algebraically closed field k with char k p 0 and f x x1 be the relative frobenius morphism for any vector bundle w on x we prove that instability of f w is bounded by instability of w tl 1x 0 l n p 1 corollary 4 9 when x is a smooth projective curve of genus g 2 it implies f w being stable whenever w is stable

boundary control of the timoshenko beam it is shown that the timoshenko beam can be uniformly stabilized by means of a boundary control a numerical study on the spectrum is also presented

a general decay result for a memory type timoshenko thermoelasticity system with second sound abstract in this paper we consider a system of timoshenko thermoelasticity with second sound in the presence of an infinite history term we prove under suitable conditions on the coefficients and the relaxation function a general decay result from which the exponential and polynomial decay estimates are only special cases we also establish a lack of exponential decay result and discuss the optimality for some situation

on the control of a viscoelastic damped timoshenko type system abstract in this paper we consider the following timoshenko system tt x x 0 0 1 0 tt xx 0 t g t xx d x 0 0 1 0 with dirichlet boundary conditions where g is a positive nonincreasing function we establish an exponential and polynomial decay results with weaker conditions on g than those required in f ammar khodja a benabdallah j e munoz rivera r racke energy decay for timoshenko systems of memory type j differ equations 194 2003 82 115

interlacing polynomials and the veronese construction for rational formal power series fixing a positive integer r and define for every formal power series f as jochemko recently showed that the polynomial has only non positive zeros for any and any positive integer n as a consequence jochemko confirmed a conjecture of beck and stapledon on the ehrhart polynomial of a lattice polytope of dimension n which states that has only negative real zeros whenever in this paper we provide an alternative approach to beck and stapledon s conjecture by proving the following general result if the polynomial sequence is interlacing so is our result has many other interesting applications in particular this enables us to give a new proof of savage and visontai s result on the interlacing property of some refinements of the descent generating functions for coloured permutations besides we derive a carlitz identity for refined coloured permutations

exponential stability for a timoshenko type system with history in this paper we consider hyperbolic timoshenko type vibrating systems that are coupled to a heat equation modeling an expectedly dissipative effect through heat conduction we use the semigroup method to prove the exponential stability result with assumptions on past history relaxation function g exponentially decaying for the equal wave speed case

initial ideals veronese subrings and rates of algebras abstract let s be a polynomial ring over an infinite field and let i be a homogeneous ideal of s let t d be a polynomial ring whose variables correspond to the monomials of degree d in s we study the initial ideals of the ideals v d i t d that define the veronese subrings of s i in suitable orders they are easily deduced from the initial ideal of i we show that in v d i is generated in degree max reg i d 2 where reg i is the regularity of the ideal i in other words the d th veronese subrings of any commutative graded ring s i has a grobner basis of degree max i d 2 we also give bounds on the regularity of i in terms of the degrees of the generators of in i and some combinatorial data this implies a version of backelin s theorem that high veronese subrings of any ring are homogeneous koszul algebras in the sense of priddy trans amer math soc 152 1970 39 60 we also give a general obstruction for a homogeneous ideal i s to have an initial ideal in i that is generated by quadrics beyond the obvious requirement that i itself should be generated by quadrics and the stronger statement that s i is koszul we use the obstruction to show that in certain dimensions a generic complete intersection of quadrics cannot have an initial ideal that is generated by quadrics for the application to backelin s theorem we require a result of backelin whose proof has never appeared we give a simple proof of a sharpened version bounding the rate of growth of the degrees of generators for syzygies of any multihomogenous module over a polynomial ring modulo an ideal generated by monomials following a method of bruns and herzog

stability of sheaves of locally closed and exact forms abstract for any smooth projective variety x of dimension n over an algebraically closed field k of characteristic p 0 with x 1 0 if the truncated symmetric powers t l x 1 0 l n p 1 of x 1 are semi stable then the sheaf b x 1 of exact 1 forms is stable when x is a surface with x 1 0 and x 1 is semi stable the sheaf b x 2 of exact 2 forms is also stable moreover under the same condition the sheaf z x 1 of closed 1 forms is stable when p 3 and z x 1 is semi stable when p 3

the s eulerian polynomials have only real roots we study the roots of generalized eulerian polynomials via a novel approach we interpret eulerian polynomials as the generating polynomials of a statistic over inversion sequences inversion sequences also known as lehmer codes or subexcedant functions were recently generalized by savage and schuster to arbitrary sequences s of positive integers which they called s inversion sequences our object of study is the generating polynomial of the ascent statistic over the set of s inversion sequences of length n since this ascent statistic over inversion sequences is equidistributed with the descent statistic over permutations we call this generalized polynomial the s eulerian polynomial the main result of this paper is that for any sequence s of positive integers the s eulerian polynomial has only real roots this result is first shown to generalize several existing results about the real rootedness of various eulerian polynomials we then show that it can be used to settle a conjecture of brenti that eulerian polynomials for all finite coxeter groups have only real roots and partially settle a conjecture of dilks petersen stembridge on type b affine eulerian polynomials it is then extended to several q analogs we show that the macmahon carlitz q eulerian polynomial has only real roots whenever q is a positive real number confirming a conjecture of chow and gessel the same holds true for the hyperoctahedral group and the wreath product groups confirming further conjectures of chow and gessel and chow and mansour respectively our results have interesting geometric consequences as well

frictional versus viscoelastic damping in a semilinear wave equation in this article we show exponential and polynomial decay rates for the partially viscoelastic nonlinear wave equation subject to a nonlinear and localized frictional damping the equation that models this problem is given by begin eqnarray u tt kappa 0 delta u int 0 t mbox div a x g t s nabla u s ds f u b x h u t 0 mbox in omega times bbb r quad end eqnarray where a b are nonnegative functions a in c 1 overline omega b in l infty omega satisfying the assumption begin eqnarray a x b x geq delta 0 quad forall x in omega end eqnarray and f and h are power like functions we observe that the assumption 0 2 gives us a wide assortment of possibilities from which to choose the functions a x and b x and the most interesting case occurs when one has simultaneous and complementary damping mechanisms taking this point of view into account a distinctive feature of our paper is exactly to consider different and localized damping mechanisms acting in the domain but not necessarily strategically localized dissipations as considered in the prior literature

instability of truncated symmetric powers of sheaves abstract let x be a smooth projective variety of dimension n over an algebraically closed field k of characteristic p 0 let f x x x be the absolute frobenius morphism and e a torsion free sheaf on x we give an upper bound of instability of truncated symmetric powers t l e 0 l rk e p 1 in terms of l max x 1 i x 1 and i e theorem 3 5 as an application we obtain an upper bound of the instability of frobenius direct image f x e and some sufficient conditions of f x e being slope semi stable in addition we study the slope semi stability of sheaves of locally exact resp closed forms b x i resp z x i

the ffrt property of two dimensional normal graded rings and orbifold curves abstract we study the finite f representation type abbr ffrt property of a two dimensional normal graded ring r in characteristic p 0 using notions from the theory of algebraic stacks given a graded ring r we consider an orbifold curve c which is a root stack over the smooth curve c proj r such that r is the section ring associated with a line bundle l on c the ffrt property of r is then rephrased with respect to the frobenius push forwards f e l i on the orbifold curve c as a result we see that if the singularity of r is not log terminal then r has ffrt only in exceptional cases where the characteristic p divides a weight of c

uniform decay in a timoshenko type system with past history abstract fernandez sare and rivera h d fernandez sare j e munoz rivera stability of timoshenko systems with past history j math anal appl 339 1 2008 482 502 considered the following timoshenko type system 1 t t k x x 0 2 t t b x x 0 g t x x t s d s k x 0 where g is a positive differentiable exponentially decaying function they established an exponential decay result in the case of equal wave speed propagation and a polynomial decay result in the case of nonequal wave speed propagation in this paper we study the same system for g decaying polynomially and prove polynomial stability results for the equal and nonequal wave speed propagation our results are established under conditions on the relaxation function weaker than those in h d fernandez sare j e munoz rivera stability of timoshenko systems with past history j math anal appl 339 1 2008 482 502

mirror symmetry for moduli spaces of higgs bundles via p adic integration we prove the topological mirror symmetry conjecture by hausel thaddeus for smooth moduli spaces of higgs bundles of type sl n and pgl n more precisely we establish an equality of stringy hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties our proof utilises p adic integration relative to the fibres and interprets canonical gerbes present on these moduli spaces as characters on the hitchin fibres using tate duality furthermore we prove for d prime to n that the number of rank n higgs bundles of degree d over a fixed curve defined over a finite field is independent of d this proves a conjecture by mozgovoy schiffmann in the coprime case

the generalized verschiebung map for curves of genus 2 let c be a smooth curve and m r c the coarse moduli space of vector bundles of rank r and trivial determinant on c we examine the generalized verschiebung map v r m r c p dashrightarrow m r c induced by pulling back under frobenius our main result is a computation of the degree of v 2 for a general c of genus 2 in characteristic p 2 we also give several general background results on the verschiebung in an appendix

existence and decay of solutions of a viscoelastic equation with a nonlinear source abstract in a bounded domain we consider u tt u 0 t g t u d u u where 0 and g is a nonnegative and decaying function we prove a local existence theorem and show for certain initial data and suitable conditions on g and that this solution is global with energy which decays exponentially or polynomially depending on the rate of the decay of the relaxation function g

conformal limits and the bialynicki birula stratification of the space of λ connections abstract the bialynicki birula decomposition of the space of connections restricts to the morse stratification on the moduli space of higgs bundles and to the partial oper stratification on the de rham moduli space of holomorphic connections for both the morse and partial oper stratifications every stratum is a holomorphic lagrangian fibration over a component of the space of complex variations of hodge structure in this paper we generalize known results for the hitchin section and the space of opers to arbitrary strata these include the following a biholomorphic identification of the fibers of the two strata over a stable variation of hodge structure via the conformal limit of gaiotto a proof that the fibers of the morse and partial oper stratifications are transverse at the base point and an explicit parametrization of the fibers as half dimensional affine spaces

a stability result of a timoshenko system with a delay term in the internal feedback abstract in this paper we consider a timoshenko system with a delay term in the feedback and prove a stability result the beam is clamped at the endpoints and has in addition to an internal damping a feedback with a delay under an appropriate assumption on the weights of the two feedbacks we prove the well posedness of the system and establish an exponential decay result for the case of equal speed wave propagation

ends of the moduli space of higgs bundles we associate to each stable higgs pair a 0 phi 0 on a compact riemann surface x a singular limiting configuration a infinity phi infinity assuming that det phi has only simple zeroes we then prove a desingularization theorem by constructing a family of solutions a t t phi t to hitchin s equations which converge to this limiting configuration as t infinity this provides a new proof via gluing methods for elements in the ends of the higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space

decay rate estimates for the quasi linear timoshenko system with nonlinear control and damping terms in this work we consider a class of quasi linear timoshenko systems with nonlinear control and damping terms we investigate the stabilization of weak solutions to the initial boundary value problem and obtain explicit decay rate estimation depending on control and damping terms

general energy decay in a timoshenko type system of thermoelasticity of type iii with a viscoelastic damping in this paper we consider a one dimensional linear thermoelastic system of timoshenko type where the heat conduction is given by green and naghdi theories we prove a general decay result from which the exponential and polynomial decays are only special cases

on the log concavity of hilbert series of veronese subrings and ehrhart series for every positive integer n consider the linear operator un on polynomials of degree at most d with integer coefficients defined as follows if we write frac h t 1 t d 1 sum m geq 0 g m t m for some polynomial g m with rational coefficients then frac rm u n h t 1 t d 1 sum m geq 0 g nm t m we show that there exists a positive integer nd depending only on d such that if h t is a polynomial of degree at most d with nonnegative integer coefficients and h 0 1 then for n nd unh t has simple real negative roots and positive strictly log concave and strictly unimodal coefficients applications are given to ehrhart polynomials and unimodular triangulations of dilations of lattice polytopes as well as hilbert series of veronese subrings of cohen macauley graded rings

surface group representations and u p q higgs bundles using the l2 norm of the higgs field as a morse function we study the moduli spaces of u p q higgs bundles over a riemann surface we require that the genus of the surface be at least two but place no constraints on p q a key step is the identification of the function s local minima as moduli spaces of holomorphic triples in a companion paper 7 we prove that these moduli spaces of triples are nonempty and irreducible because of the relation between flat bundles and fundamental group representations we can interpret our conclusions as results about the number of connected components in the moduli space of semisimple pu p q representations the topological invariants of the flat bundles are used to label subspaces these invariants are bounded by a milnor wood type inequality for each allowed value of the invariants satisfying a certain coprimality condition we prove that the corresponding subspace is nonempty and connected if the coprimality condition does not hold our results apply to the closure of the moduli space of irreducible representations 1 members of vbac vector bundles on algebraic curves which is partially supported by eager ec fp5 contract no hprn ct 2000 00099 and by edge ec fp5 contract no hprn ct 2000 00101 2 partially supported by the national science foundation under grant dms0072073 3 partially supported by the ministerio de ciencia y tecnolo ia spain under grant bfm2000 0024 4 partially supported by the fundacao para a ciencia e a tecnologia portugal through the centro de matematica da universidade do porto and through grant no sfrh bpd 1606 2000 5 partially supported by the portugal spain bilateral programme acciones integradas grant nos hp2000 0015 and ai 01 24 6 partially supported by a british epsrc grant october december 2001 received 11 27 2002

frictional versus viscoelastic damping in timoshenko systems with different speeds of wave propagation abstract we consider the decay rate of timoshenko type equations for beams with frictional damping and memory effect in an interval when the speeds of wave propagation are different we show that the solution energy of timoshenko systems decays at the rate t 1 under only basic conditions on the memory kernels without any decaying conditions the proof is based on some appropriate functionals and it improves the previously related results in the literature

some well posedness and general stability results in timoshenko systems with infinite memory and distributed time delay in this paper we consider a timoshenko system in one dimensional bounded domain with infinite memory and distributed time delay both acting on the equation of the rotation angle without any restriction on the speeds of wave propagation and under appropriate assumptions on the infinite memory and distributed time delay convolution kernels we prove first the well posedness and second the stability of the system where we present some decay estimates depending on the equal speed propagation case and the opposite one the obtained decay rates depend on the growths of the memory and delay kernels at infinity in the nonequal speed case the decay rate depends also on the regularity of initial data our stability results show that the only dissipation resulting from the infinite memory guarantees the asymptotic stability of the system regardless to the speeds of wave propagation and in spite of the presence of a distributed time delay applications of our approach to specific coupled timoshenko heat and timoshenko wave systems as well as the discrete time delay case are also presented

some combinatorial and algebraic properties of coxeter complexes and tits buildings a des groupes de coxeter et des groupes a bn paire sont associes certaines structures combinatoires les complexes de coxeter et les constructions de tits on etudie ces structures du point de vue des complexes de cohen macaulay et des complexes decorticables

hitchin mochizuki morphism opers and frobenius destabilized vector bundles over curves abstract let x be a smooth projective curve of genus g 2 defined over an algebraically closed field k of characteristic p 0 for p r r 1 r 2 g 1 we construct an atlas for the locus of all frobenius destabilized bundles of rank r i e we construct all frobenius destabilized bundles of rank r and degree zero up to isomorphism this is done by exhibiting a surjective morphism from a certain quot scheme onto the locus of stable frobenius destabilized bundles further we show that there is a bijective correspondence between the set of stable vector bundles e over x such that the pull back f e under the frobenius morphism of x has maximal harder narasimhan polygon and the set of opers having zero p curvature we also show that after fixing the determinant these sets are finite which enables us to derive the dimension of certain quot schemes and certain loci of stable frobenius destabilized vector bundles over x the finiteness is proved by studying the properties of the hitchin mochizuki morphism an alternative approach to finiteness has been realized in 3 in particular we prove a generalization of a result of mochizuki to higher ranks

on the stabilization of timoshenko systems with memory and different speeds of wave propagation in this work we consider a one dimensional timoshenko system with different speeds of wave propagation and with only one control given by a viscoelastic term on the angular rotation equation for a wide class of relaxation functions and for sufficiently regular initial data we establish a general decay result for the energy of solution unlike the past history and internal feedback cases the second energy is not necessarily decreasing to overcome this difficulty a precise estimate of the second energy in terms of the initial data and the relaxation function is proved

a characterization of ordinary abelian varieties by the frobenius push forward of the structure sheaf for an ordinary abelian variety x f e mathcal o x is decomposed into line bundles for every positive integer e conversely if a smooth projective variety x satisfies this property and the kodaira dimension of x is non negative then x is an ordinary abelian variety

f vectors of barycentric subdivisions for a simplicial complex or more generally boolean cell complex we study the behavior of the f and h vector under barycentric subdivision we show that if has a non negative h vector then the h polynomial of its barycentric subdivision has only simple and real zeros as a consequence this implies a strong version of the charney davis conjecture for spheres that are the subdivision of a boolean cell complex or the subdivision of the boundary complex of a simple polytope for a general d 1 dimensional simplicial complex the h polynomial of its n th iterated subdivision shows convergent behavior more precisely we show that among the zeros of this h polynomial there is one converging to infinity and the other d 1 converge to a set of d 1 real numbers which only depends on d