analytic torsion and holomorphic determinant bundles in this paper we prove that in the case of holomorphic locally kahler fibrations the analytic and algebraic geometry constructions of determinant bundles for direct images coincide we calculate the curvature of the holomorphic hermitian connection for the quillen metric on the determinant bundle we study the behavior of the quillen metric under change of metrics in the fibers and also on the twisting vector bundles we thus generalize the conformal anomaly formula to kahler manifolds of arbitrary dimension we also study the quillen metrics on determinants associated with exact sequences of vector bundles we prove that the quillen metric is smooth on the grothendieck knudsen mumford determinant for arbitrary holomorphic fibrations

a local index theorem for families of overline partial operators on punctured riemann surfaces and a new kahler metric on their moduli spaces we prove a local index theorem for families of bar partial operators on riemann surfaces of type g n i e of genusg withn 0 punctures we calculate the first chern form of the determinant line bundle on the teichmuller spacetg n endowed with quillen s metric where the role of the determinant of the laplace operators is played by the values of the selberg zeta function at integer points the result differs from the case of compact riemann surfaces by an additional term which turns out to be the kahler form of a new kahler metric on the moduli space of punctured riemann surfaces as a corollary of this result we derive for instance an analog of mumford s isomorphism in the case of the universal curve

the second homology group of the mapping class group of an orientable surface in i 7 mumford shows that the picard group p ic is isomorphic to h2 f 2 and conjectures the latter is rank one g 3 we prove this below for g 5 another interpretation of this theorem may be obtained by identifying h2 f as bordism classes of fiber bundles f w4 t where t is a closed oriented surface sect 0 when f is closed every such bundle is bordant to f w t a bundle admitting a section s t w the theorem then says that

potentials and chern forms for weil petersson and takhtajan zograf metrics on moduli spaces abstract for the tz metric on the moduli space m 0 n of n pointed rational curves we construct a kahler potential in terms of the fourier coefficients of the klein s hauptmodul we define the space s g n as holomorphic fibration s g n s g over the schottky space s g of compact riemann surfaces of genus g where the fibers are configuration spaces of n points for the tautological line bundles l i over s g n we define hermitian metrics h i in terms of fourier coefficients of a covering map j of the schottky domain we define the regularized classical liouville action s and show that exp s is a hermitian metric in the line bundle l i 1 n l i over s g n we explicitly compute the chern forms of these hermitian line bundles c 1 l i h i 4 3 tz i c 1 l exp s 1 2 wp we prove that a smooth real valued function s s i 1 n log h i on s g n a potential for this special difference of wp and tz metrics coincides with the renormalized hyperbolic volume of a corresponding schottky 3 manifold we extend these results to the quasi fuchsian groups of type g n

on asymptotic vanishing behavior of local cohomology let r be a standard graded algebra over a field k with irrelevant maximal ideal mathfrak m and i a homogeneous r ideal we study the asymptotic vanishing behavior of the graded components of the local cohomology modules text h i mathfrak m r i n n in mathbb n for i dim r i we show that when mathrm char k 0 r i is cohen macaulay and i is a complete intersection locally on mathrm spec r setminus mathfrak m the lowest degrees of the modules text h i mathfrak m r i n n in mathbb n are bounded by a linear function whose slope is controlled by the generating degrees of the dual of i i 2 our result is a direct consequence of a related bound for symmetric powers of locally free modules if no assumptions are made on the ideal or the field k we show that the complexity of the sequence of lowest degrees is at most polynomial provided they are finite our methods also provide a result on stabilization of maps between local cohomology of consecutive powers of ideals

harmonic differentials and infinite geodesic joining two punctures on a riemann surface let m be a hyperbolic riemann surface of finite volume the harmonic dual form to an infinite geodesic joining two punctures on m is obtained in two different ways first of all using the degeneration of hyperbolic eisenstein series it is made explicit in terms of these secondly generalizing the construction of kudla and millson to the case of an infinite geodesic joining two punctures we give an automorphic realization of this harmonic form

well balanced schemes for conservation laws with source terms based on a local discontinuous flux formulation we propose and analyze a finite volume scheme of the godunov type for conservation laws with source terms that preserve discrete steady states the scheme works in the resonant regime as well as for problems with discontinuous flux moreover an additional modification of the scheme is not required to resolve transients and solutions of non linear algebraic equations are not involved our well balanced scheme is based on modifying the flux function locally to account for the source term and to use a numerical scheme especially designed for conservation laws with discontinuous flux due to the difficulty of obtaining bv estimates we use the compensated compactness method to prove that the scheme converges to the unique entropy solution as the discretization parameter tends to zero we include numerical experiments in order to show the features of the scheme and how it compares with a wellbalanced scheme from the literature

regularity and cohomology of determinantal thickenings we consider the ring s c xij of polynomial functions on the vector space cm n of complex m n matrices we let gl glm c gln c and consider its action via row and column operations on cm n and the induced action on s for every gl invariant ideal i s and every j 0 we describe the decomposition of the modules extsj s i s into irreducible gl representations for any inclusion i j of gl invariant ideals we determine the kernels and cokernels of the induced maps extsj s i s extsj s j s as a consequence of our work we give a formula for the regularity of the powers and symbolic powers of generic determinantal ideals and in particular we determine which powers have a linear minimal free resolution as another consequence we characterize the gl invariant ideals i s for which the induced maps extsj s i s hij s are injective in a different direction we verify that kodaira vanishing as described in work of bhatt blickle lyubeznik singh zhang holds for determinantal thickenings

the asymptotic behavior of the takhtajan zograf metric we obtain the asymptotic behavior of the takhtajan zograf metric on the teichmuller space of punctured riemann surfaces

castelnuovo mumford regularity of ext modules and homological degree bounds for the castelnuovo mumford regularity of ext modules over a polynomial ring over a field are given in terms of the initial degrees castelnuovo mumford regularities and the number of generators of the two graded modules involved these general bounds are refined in the case where the second module is the ring other estimates for instance on the size of graded pieces of these modules are given we also derive a bound on the homological degree in terms of the castelnuovo mumford regularity this answers positively a question raised by vasconcelos

the riemann problem near a hyperbolic singularity the classification of solutions of quadratic riemann problems 1 the purpose of this paper is to classify the solutions of riemann problems near a hyperbolic singularity in a nonlinear system of conservation laws hyperbolic singularities play the role in the theory of riemann problems that rest points play in the theory of ordinary differential equations indeed generically only a finite number of structures can appear in a neighborhood of such a singularity in this the first of three papers the program of classification is discussed in general and the simplest structure that occurs is characterized

the generalized riemann problem method for the shallow water equations with bottom topography this paper extends the generalized riemann problem method grp to the system of shallow water equations with bottom topography the main contribution is that the generalized riemann problem method j comput phys 1984 55 1 1 32 is used to evaluate the midpoint values of solutions at each cell interface so that the bottom topography effect is included in numerical fluxes and at the same step the source term is discretized with an interface method in which only mid point values are plugged in this scheme is well balanced between the flux gradient and bottom topography when incorporating the surface gradient method sgm j comput phys 2001 168 1 1 25 into data reconstruction step and it is also suitable for both steady and unsteady flow simulations we illustrate the accuracy of this scheme by several 1 d and 2 d numerical experiments copyright 2005 john wiley sons ltd

global solutions to the compressible euler equations with geometrical structure we prove the existence of global solutions to the euler equations of compressible isentropic gas dynamics with geometrical structure including transonic nozzle flow and spherically symmetric flow due to the presence of the geometrical source terms the existence results themselves are new especially as they pertain to radial flow in an unbounded region vec x geqq 1 and to transonic nozzle flow arbitrary data withl bounds are allowed in these results a shock capturing numerical scheme is introduced to compute such flows and to construct approximate solutions the convergence and consistency of the approximate solutions generated from this scheme to the global solutions are proved with the aid of a compensated compactness framework

analysis and approximation of conservation laws with source terms we consider a conservation law of the form cl i ut f u x ax where a cdot is a bounded piecewise smooth source term and f an even convex function we first characterize the solution to the riemann problem through a new lax type formula then we prove that for a cdot fixed the semigroup associated with cl is an l1 contraction and we obtain an existence theorem for weak solutions to cl we conclude by constructing godunov type difference schemes and proving that these schemes are l infty stable and have stable steady solutions similar in structure to those of cl some numerical tests are reported

first and second order finite volume methods for the one dimensional nonconservative euler system gas flow in porous media with a nonconstant porosity function provides a nonconservative euler system we propose a new class of schemes for such a system for the one dimensional situations based on nonconservative fluxes preserving the steady state solutions we derive a second order scheme using a splitting of the porosity function into a discontinuous and a regular part where the regular part is treated as a source term while the discontinuous part is treated with the nonconservative fluxes we then present a classification of all the configurations for the riemann problem solutions in particularly we carefully study the resonant situations when two eigenvalues are superposed based on the classification we deal with the inverse riemann problem and present algorithms to compute the exact solutions we finally propose new sod problems to test the schemes for the resonant situations where numerical simulations are performed to compare with the theoretical solutions the schemes accuracy first and second order is evaluated comparing with a nontrivial steady state solution with the numerical approximation and convergence curves are established

riemann problem with nonlinear resonance effects and well balanced godunov scheme for shallow fluid flow past an obstacle we are interested in characterizing the resonant behavior that occurs in a shallow water system with an arbitrary bottom topography the inhomogeneity introduced by the bottom topography elevation term in the system of equations is modeled by a linearly degenerate field strict hyperbolicity fails when one of the different nonlinear waves has a zero speed a proliferation of reflected waves can occur by interactions and then resonance effects are important for a transonic flow classical nonlinear waves interact with standing waves and exhibit overcompressive undercompressive and marginal compressive waves analytical understanding is used to introduce a new kind of numerical processing of the source term the riemann problem with nonlinear resonance is solved the so called well balanced godunov scheme is constructed which yields more accurate asymptotic states and preserves the balance between the source terms and flux terms the numerical flux and source terms are simultaneously solved in both space and time to overcome resonance

non completeness of zograf takhtajan s kahler metric for teichmuller space of punctured riemann surfaces zograf and takhtajan introduced a new kahler metric on the teichmuller space tg n n 0 in calculating the first chern form of the quillen metric for families of operators the metric is described in terms of the eisenstein maass series we prove that it is incomplete and we also give an alternative proof of non completeness of the weil petersson metric for that we use the pinching family constructed by wolpert whose tangent vectors are always represented by using the relative poincare series associated with the pinched geodesic

on determinants of laplacians on riemann surfaces determinants of laplacians on tensors and spinors of arbitrary weights on compact hyperbolic riemann surfaces are computed in terms of values of selberg zeta functions at half integer points

grafting hyperbolic metrics and eisenstein series the family hyperbolic metric for the plumbing variety zw t and the non holomorphic eisenstein series e zeta 2 are combined to provide an explicit expansion for the hyperbolic metrics for degenerating families of riemann surfaces applications include an asymptotic expansion for the weil petersson metric and a local form of symplectic reduction

cohen macaulayness of powers of two dimensional squarefree monomial ideals abstract two dimensional squarefree monomial ideals can be seen as the stanley reisner ideals of graphs the main results of this paper are combinatorial characterizations for the cohen macaulayness of ordinary and symbolic powers of such an ideal in terms of the associated graph