minimal rational curves on moduli spaces of stable bundles let c be a smooth projective curve of genus g 2 and l be a line bundle on c of degree d let m uc r l be the moduli space of stable vector bundles on c of rank r and with the fixed determinant l assume that r d 1 then m is a smooth projective fano variety with picard number 1 for any projective curve on m we can define its degree with respect to the ample anti canonical line bundle km the first result of this paper determines all rational curves of minimal degree passing through a generic point of m which answers a question of jun muk hwang see question 1 in hw

very weak solutions of degenerate parabolic systems with non standard p x t growth abstract we study higher integrability of very weak solutions to certain degenerate parabolic systems whose model is the parabolic p x t laplacian system t u div d u p x t 2 d u div f p x t 2 f under natural assumptions on the exponent function p 0 t 2 we prove that any very weak solution u 0 t r n with d u p 1 e l 1 belongs to the natural energy space i e d u p l loc 1 provided e 0 is small enough

gradient enhanced damage for quasi brittle materials summary conventional continuum damage descriptions of material degeneration suffer from loss of well posedness beyond a certain level of accumulated damage as a consequence numerical solutions are obtained which are unacceptable from a physical point of view the introduction of higher order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models a consistent numerical solution procedure of the governing partial differential equations is presented which is shown to be capable of properly simulating localization phenomena

strain gradient theory in orthogonal curvilinear coordinates in this short note general formulations of the toupin mindlin strain gradient theory in orthogonal curvilinear coordinate systems are derived and are then specified for the cases of cylindrical coordinates and spherical coordinates expressions convenient for practical use are presented for the corresponding equilibrium equations boundary conditions and the physical components for strains and strain gradients in the two coordinate systems the results obtained in this paper are general and complete and can be useful for a wide range of applications such as asymptotic crack tip field analysis cylindrical and spherical cavity expansion problems and the interpretation of micro nano indentation tests and bending twisting tests on small scales

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

plastic continuum with microstructure local second gradient theories for geomaterials localization studies many plastic second gradient models have been developed in the last 10 years however some plastic second gradient models are nonlocal ones this paper is an attempt to give a general framework to deal with local second gradient theories within theories with microstructure keeping in mind future applications for geomaterials it is advocated that particular elasto plastic local models with microstructure namely local second gradient and cosserat second gradient models which are the least developed in the literature have some advantages which are somewhat promising one main objective of this paper is to present these two families of models the first one cosserat second gradient model is shown to be well adapted to granular materials the second model family is likely to be a good model for cohesive geomaterials another aim of this work is to give a method to obtain basic solutions which can be seen as localization analysis in one and two dimensions cases the key point of this method is the use of patch conditions between loading and unloading parts

very weak solutions of subquadratic parabolic systems with non standard p x t growth abstract the aim of this paper is to establish a higher integrability result for very weak solutions of certain parabolic systems whose model is the parabolic p x t laplacian system under assumptions on the exponent function p t 0 t 2 n n 2 2 it is shown that any very weak solution u t r n with d u p 1 e l 1 t belongs to the natural energy spaces i e d u p l loc 1 t provided e 0 is small enough this extends the main result of bogelein and li 2014 to the subquadratic case

partial desingularisations of quotients of nonsingular varieties and their betti numbers when a reductive group g acts linearly on a nonsingular complex projective variety x one can define a projective quotient variety x g using mumford s geometric invariant theory if the condition that every semistable point be properly stable is satisfied this quotient is the ordinary topological quotient of an open subset xss of the variety by the group in k a formula is obtained for the rational cohomology of x g under this condition the formula involves the rational cohomology of x and various linear sections of x together with the rational cohomology of the classifying spaces of g and certain reductive subgroups of g in many interesting examples the condition required in k is not satisfied thus the question arises as to what information we can obtain in general the quotient x g can now have serious singularities in contrast to the good case considered in k where the only singularities are those caused by finite isotropy groups it will be shown in this paper that there is a systematic way of blowing up x along a sequence of nonsingular subvarieties to obtain a variety x with a linear action of g such that every semistable point of x is stable the only assumption that we have to make is that there exists at least one stable point of x then x g is almost a resolution of singularities of x1 g in the sense that the most serious singularities have been resolved moreover there is a formula for the rational cohomology of x g again involving the rational cohomology of x and certain linear sections of x together with the rational cohomology of the classifying spaces of g and some reductive subgroups of g see theorem 8 14 for convenience we shall assume throughout that g is connected however the construction of x works in general and it is straightforward to modify the cohomological formulas to apply to the general case the construction of x g can also be modified to apply in some cases when x has no stable points the layout of the paper is as follows section 1 is a review of the basic facts of geometric invariant theory which will be needed and section 2 describes the relationship of geometric invariant theory with symplectic geometry and the moment map in section 3 semistability and stability in a blow up of x along a nonsingular c invariant subvariety are related to semistability and stability in x

interior and boundary higher integrability of very weak solutions for quasilinear parabolic equations with variable exponents abstract we prove boundary higher integrability for the spatial gradient of very weak solutions of quasilinear parabolic equations of the form u t div a x t u 0 on t t u 0 on t t where the non linear structure a x t u is modeled after the variable exponent p x t laplace operator given by u p x t 2 u to this end we prove that the gradients satisfy a reverse holder inequality near the boundary by constructing a suitable test function which is lipschitz continuous and preserves the boundary values in the interior case such a result was proved in verena bogelein and qifan li 2014 provided p x t p 2 holds and was then extended to the singular case 2 n n 2 p p x t p 2 in qifan li 2017 this restriction was necessary because the intrinsic scalings for quasilinear parabolic problems are different in the case p 2 and p 2 in this paper we develop a new approach using which we are able to extend the results of verena bogelein and qifan li 2014 qifan li 2017 to the full range 2 n n 2 p p x t p and also obtain analogous results up to the boundary the main novelty of this paper is that we make use of a unified intrinsic scaling using which our methods are able to handle both the singular case and degenerate case simultaneously our techniques improve and simplify many aspects of the method of parabolic lipschitz truncation even in the constant exponent case studied extensively in existing literature to simplify the exposition we will only prove the higher integrability result near the boundary provided the domain satisfies a uniform measure density condition and are non perturbative in nature hence we make no regularity assumptions for the coefficients of the nonlinear operator our techniques are also applicable to higher order equations as well as systems

quantum cohomology of the moduli space of stable bundles over a riemann surface we determine the quantum cohomology of the moduli space m of odd degree rank two stable vector bundles over a riemann surface of genus g 1 this work together with 10 complete the proof of the existence of an isomorphism qh m hf s1

shear banding analysis of geomaterials by strain gradient enhanced damage model shear band localization is investigated by a strain gradient enhanced damage model for quasi brittle geomaterials this model introduces the strain gradients and their higher order conjugate stresses into the framework of continuum damage mechanics the influence of the strain gradients on the constitutive behaviour is taken into account through a generalized damage evolutionary law a weak form variational principle is employed to address the additional boundary conditions introduced by the incorporation of the strain gradients and the conjugate higher order stresses damage localization under simple shear condition is analytically investigated by using the theory of discontinuous bifurcation and the concept of the second order characteristic surface analytical solutions for the distributions of strain rates and strain gradient rates as well as the band width of localised damage are found numerical analysis demonstrates the shear band width is proportionally related to the internal length scale through a coefficient function of poissons ratio and a parameter representing the shape of uniaxial stress strain curve it is also shown that the obtained distributions of strains and strain gradients are well in accordance with the underlying assumptions for the second order discontinuous shear band boundary and the weak discontinuous bifurcation theory 2005 elsevier ltd all rights reserved

moduli of sheaves fourier mukai transform and partial desingularization we study birational maps among 1 the moduli space of semistable sheaves of hilbert polynomial 4m 2 on a smooth quadric surface 2 the moduli space of semistable sheaves of hilbert polynomial m 2 3m 2 on mathbb p 3 3 kontsevich s moduli space of genus zero stable maps of degree 2 to the grassmannian gr 2 4 a regular birational morphism from 1 to 2 is described in terms of fourier mukai transforms the map from 3 to 2 is kirwan s partial desingularization we also investigate several geometric properties of 1 by using the variation of moduli spaces of stable pairs

hilbert scheme of rational cubic curves via stable maps the space of smooth rational cubic curves in projective space bbb p r r ge 3 is a smooth quasi projective variety which gives us an open subset of the corresponding hilbert scheme the moduli space of stable maps or the moduli space of stable sheaves by taking its closure we obtain three compactifications bf h bf m and bf s respectively in this paper we compare these compactifications first we prove that bf h is the blow up of bf s along a smooth subvariety parameterizing planar stable sheaves next we prove that bf s is obtained from bf m by three blow ups followed by three blow downs and the centers are described explicitly using this we calculate the cohomology of bf s

desingularizations of the moduli space of rank 2 bundles over a curve let x be a smooth projective curve of genus g 3 and m0 be the moduli space of rank 2 semistable bundles over x with trivial determinant there are three desingularizations of this singular moduli space constructed by narasimhan ramanan nr78 seshadri ses77 and kirwan kir86b respectively the relationship between them has not been understood so far the purpose of this paper is to show that there is a morphism from kirwan s desingularization to seshadri s which turns out to be the composition of two blow downs in doing so we will show that the singularities of m0 are terminal and the plurigenera are all trivial as an application we compute the betti numbers of the cohomology of seshadri s desingularization in all degrees this generalizes the result of bs90 which computes the betti numbers in low degrees another application is the computation of the stringy e function see bat98 for definition of m0 for any genus g 3 which generalizes the result of kie03

gradient dependent plasticity formulation and algorithmic aspects a plasticity theory is proposed in which the yield strength not only depends on an equivalent plastic strain measure hardening parameter but also on the laplacian thereof the consistency condition now results in a differential equation instead of an algebraic equation as in conventional plasticity to properly solve the set of non linear differential equations the plastic multiplier is discretized in addition to the usual discretization of the displacements for appropriate boundary conditions this formulation can also be derived from a variational principle accordingly the theory is complete

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

limit theorems for gradient dependent elastoplastic geomaterials in this paper the limit theorems proposed by drucker and co workers are reformulated to address a class of gradientdependent elastoplastic geomaterials the gradient effects are accounted for by incorporating strain gradients and their conjugate higher order stresses into the constitutive descriptions gradient dependent equilibrium equations and higher order boundary conditions are defined for both statically admissible and kinematically admissible states and the associated lower bound and upper bound theories are recast for the gradient enhanced limit theorems a generalised drucker prager yield criterion that includes the gradient influence on the deviatoric stress is proposed and then employed to investigate the collapse loads for soil layers under conditions of generalised plane strain simple shear and uniaxial compression the corresponding lower and upper bounds are found for these problems it is demonstrated that the predicted collapse bounds are generally dependent on both the conventional and gradient properties with normalised length scale s being present in the results this feature enables us to give physically reasonable interpretations for size effects and shear banding during material collapse comparisons are also made between the gradient dependent bounds and those obtained through conventional plasticity theories influences of model parameters and sample dimensions on the predictions are also discussed it is shown that the proposed gradient dependent limit theorems can be used to provide physically meaningful predictions for general geotechnical applications 2006 elsevier ltd all rights reserved

a variational principle for gradient plasticity abstract we elaborate on a generalized plasticity model which belongs to the class of gradient models suggested earlier by aifantis and co workers the generalization of the conventional theory of plasticity has been accomplished by the inclusion of higher order spatial gradients of the equivalent plastic strain in the yield condition first it is shown how these gradients affect the critical condition for the onset of localization and allow for a wavelength selection analysis leading to estimates for the width and or spacing of shear bands due to the presence of higher order gradients additional boundary conditions for the equivalent plastic strain are required this question and also the associated problem of the formulation and solution of general boundary value problems were left open in the previous work we demonstrate here that upon assuming a certain type of additional boundary conditions the structural symmetries of the gradient dependent constitutive model are such that there exists a variational principle for the displacement rates and the rate of the equivalent plastic strain the variational principle can serve as a basis for the numerical solution of boundary value problems in the sense of the finite element method explicit expressions for the tangent stillness matrix and the generalized nodal point forces are given

properties of discontinuous bifurcation solutions in elasto plasticity explicit expressions for the spectral properties of the bifurcation problem involving discontinuities for general elastic plastic materials are presented it then follows that the classical value of the critical hardening modulus derived by rice 1976 proc 14th iutam congr delft the netherlands pp 207 220 north holland amsterdam is in fact the only possible one furthermore from the spectral analysis it follows in a very straightforward fashion that bifurcation displaying elastic unloading on one side of the singular surface can never precede bifurcation with plastic loading on both sides of this surface explicit analytical results for the critical bifurcation directions and the corresponding hardening modulus are derived for non associated volumetric flow rules while the devialoric portion is associated the considered yield and potential functions may depend on all three stress invariants and may involve mixed isotropic and kinematic hardening the result obtained by rudnicki and rice 1975 j mech phys solids 23 371 394 for a drucker pruger material appear as a special case other criteria that are investigated arc those of coulomb and rankine less

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