semistable bundles over an elliptic curve abstract we identify the moduli space m n d of semistable bundles of rank n and degree d on an elliptic curve c as a symmetric product of the curve its dimension being the greatest common divisor of n and d under this identification the determinant map det m n d corresponds to the abel jacobi map shc jh so that the fixed determinant moduli space is identified with a projective space we then describe the brill noether loci theta divisors and picard groups and compute the space of theta functions for both the full moduli space and the fixed determinant moduli space of any rank and any degree we extend to elliptic curves a theorem of drezet and narasimhan on the picard group of moduli spaces as well as two formuulas of beauville narasimhan and ramanan on theta functions of level one our results verify in particular for an elliptic curve a formula of e verlinde and an equality first noticed by bott and szenes relating theta functions over the full moduli space and over the fixed determinant moduli space

analysis and computation of a discrete costly observation model for growth estimation and management of biological resources abstract early estimation of biological growth of organisms is an indispensable task in ecology and related research areas the biological growth is always time continuous while our observations of the phenomenon are time discrete in practice the formalism of the discrete costly observation dco enables us to mathematically bridge the two qualitatively different processes this formalism is still germinating and its practical applications have not been carried out this paper presents a first application of the dco formalism to a cost effective early estimation problem of the biological growth and its mathematical and numerical analysis growth dynamics of organisms which are fishery resources in this paper is governed by a stochastic differential equation whose solution is observed discretely the optimality equation to be solved for finding the most cost effective observation policy is derived as a fixed point problem based on degenerate parabolic partial differential equations the fixed point problem turns out to be uniquely solvable a recursive approximation of the fixed point problem is presented and its solvability in a viscosity sense is discussed a finite different scheme is then employed to fully discretize the recursive equations the present model is finally applied to a problem of japanese smelt plecoglossus altivelis altivelis ayu an important inland fishery resource in japan

third order maximum principle satisfying dg schemes for convection diffusion problems with anisotropic diffusivity abstract for a class of convection diffusion equations with variable diffusivity we construct third order accurate discontinuous galerkin dg schemes on both one and two dimensional rectangular meshes the dg method with an explicit time stepping can well be applied to nonlinear convection diffusion equations it is shown that under suitable time step restrictions the scaling limiter proposed in liu and yu 2014 23 when coupled with the present dg schemes preserves the solution bounds indicated by the initial data i e the maximum principle while maintaining uniform third order accuracy these schemes can be extended to rectangular meshes in three dimension the crucial for all model scenarios is that an effective test set can be identified to verify the desired bounds of numerical solutions this is achieved mainly by taking advantage of the flexible form of the diffusive flux and the adaptable decomposition of weighted cell averages numerical results are presented to validate the numerical methods and demonstrate their effectiveness

on the stack of semistable g bundles over an elliptic curve in a recent paper ben zvi and nadler proved that the induction map from b bundles of degree 0 to semistable g bundles of degree 0 over an elliptic curve is a small map with galois group isomorphic to the weyl group of g we generalize their result to all connected components of text bun g for an arbitrary reductive group g we prove that for every degree i e topological type there exists a unique parabolic subgroup such that any semistable g bundle of this degree has a reduction to it and moreover the induction map is small with galois group the relative weyl group of the levi this provides new examples of simple automorphic sheaves which are constituents of eisenstein sheaves for the trivial local system

a finite volume difference scheme for a model of settling particle dispersion from an elevated source in an open channel flow abstract a finite volume fitted difference scheme is constructed to solve the unsteady convective diffusion equation transformed on a finite domain modeling longitudinal dispersion of suspended particles with settling velocity in a turbulent shear flow over a rough bed surface first we discuss the well posedness of the differential problem and the non negativity of its solution then to overcome the degeneracy at the part of the boundary starting from the divergent form of the equation we perform a local fitted space discretization this approximation is determined by a set of two point boundary value problems non negativity of the numerical concentration of suspended fine particles is proved some results from computational experiments are presented to illustrate the properties of the constructed scheme

on nonobtuse simplicial partitions this paper surveys some results on acute and nonobtuse simplices and associated spatial partitions these partitions are relevant in numerical mathematics including piecewise polynomial approximation theory and the finite element method special attention is paid to a basic type of nonobtuse simplices called path simplices the generalization of right triangles to higher dimensions in addition to applications in numerical mathematics we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics

on the hall algebra of an elliptic curve i these rings admit numerous algebraic and geometric realizations but one of the historically first constructions dating to the work of steinitz in 1900 completed later by hall was given in terms of what is now called the classical hall algebra h see ma chapter ii this algebra has a basis consisting of isomorphism classes of abelian q groups where q is a fixed prime power and the structure constants are defined by counting extensions between such abelian groups in fact these structure constants are polynomials in q and we can therefore consider h as a c q 1 algebra a theorem of steinitz and hall provides an isomorphism h q c q x1 x2 under this isomorphism the natural basis of h resp the natural scalar product is mapped to the basis of hall littlewood polynomials resp the hall littlewood scalar product in addition zelevinsky z endowed q with a structure of a cocommutative hopf algebra and the whole algebra q c q 1

on the consistent two side estimates for the solutions of quasilinear convection diffusion equations and their approximations on non uniform grids abstract a new second order in space linearized difference scheme on non uniform grid that approximates the dirichlet problem for multidimensional quasilinear convection diffusion equation with unbounded nonlinearity is constructed proposed algorithm is a novel nonlinear generalization of difference schemes for linear problems developed earlier by the authors nontrivial two side pointwise estimates of the solution of the scheme fully consistent with the corresponding estimates for the differential problem are established such estimates permit to prove the nonnegativity of the exact solution important in physical problems and also to find sufficient conditions on the input data when the nonlinear problem is parabolic as a result a priori estimates of the approximate solution in the grid norm c that depend on the initial and boundary conditions and on the right hand side only are proved

weakened acute type condition for tetrahedral triangulations and the discrete maximum principle we prove that a discrete maximum principle holds for continuous piecewise linear finite element approximations for the poisson equation with the dirichlet boundary condition also under a condition of the existence of some obtuse internal angles between faces of terahedra of triangulations of a given space domain this result represents a weakened form of the acute type condition for the three dimensional case

qualitative properties of discrete nonlinear parabolic operators this paper is devoted to the qualitative properties of discretized parabolic operators such as nonnegativity and nonpositivity preservation maximum minimum principles and maximum norm contractivity in the linear case earlier papers of the authors farago and horvath in siam sci comput 28 2313 2336 2006 ima j numer anal 29 606 631 2009 have established the connections between the above qualitative properties and have given sufficient conditions for their validity the present paper extends the above results to nonlinear discretized parabolic operators also motivated by the authors recent paper farago and horvath in j math anal appl 448 473 497 2017 which has given related results on the continuous pde level a systematic study is presented ranging from general discrete mesh operators to proper finite element applications

an energy conserving second order numerical scheme for nonlinear hyperbolic equation with an exponential nonlinear term we present a second order accurate numerical scheme for a nonlinear hyperbolic equation with an exponential nonlinear term the solution to such an equation is proven to have a conservative nonlinear energy due to the special nature of the nonlinear term the positivity is proven to be preserved under a periodic boundary condition for the solution for the numerical scheme a highly nonlinear fractional term is involved for the theoretical justification of the energy stability we propose a linear iteration algorithm to solve this nonlinear numerical scheme a theoretical analysis shows a contraction mapping property of such a linear iteration under a trivial constraint for the time step we also provide a detailed convergence analysis for the second order scheme in the 0 t norm such an error estimate in the maximum norm can be obtained by performing a higher order consistency analysis using asymptotic expansions for the numerical solution as a result instead of the standard comparison between the exact and numerical solutions an error estimate between the numerical solution and the constructed approximate solution yields an o t 3 h 4 convergence in 0 t 2 norm which leads to the necessary error estimate using the inverse inequality under a standard constraint t c h a numerical accuracy check is given and some numerical simulation results are also presented

a numerical study for optimal portfolio regime switching model i 2d black scholes equation with an exponential non linear term a system of weakly coupled semilinear parabolic equations of optimal portfolio in a regime switching model is suggested by valdez and vargiolu 2013 to study the effects from the typical features of the system i e the boundary degeneration exponential non linearity and mixed derivative we consider a representative 2d black scholes semilinear equation model we construct and analyze sign preserving flux limited finite difference schemes for the scalar problem numerical experiments are discussed

an upwind finite difference method for a nonlinear black scholes equation governing european option valuation under transaction costs in this paper we develop a numerical method for a nonlinear parabolic partial differential equation arising from pricing european options under transaction costs the method is based on an upwind finite difference scheme for the spatial discretization and a fully implicit time stepping scheme we prove that the system matrix from this scheme is an m matrix and that the approximate solution converges unconditionally to the viscosity solution to the equation by showing that the scheme is consistent monotone and unconditionally stable a newton iterative algorithm is proposed for solving the discretized nonlinear system of which the jacobian matrix is shown to be also an m matrix numerical experiments are performed to demonstrate the accuracy and robustness of the method

some integral properties of the heat equation in this paper we consider the one dimensional heat conduction equation on the interval 0 1 we investigate the integrals of the solution u with respect to the space and time variables and the equivalents of the integrals in the numerical solution we give the properties of the functions e e t 10 u x t dx and f 0 1 f x 0 u x t dt we perform the numerical solution applying the so called method 1 we show that with the additional conditions of the nonnegativity preservation and maximum norm contractivity 2 similar statements are valid as in the continuous case

discrete maximum principle and adequate discretizations of linear parabolic problems in this paper we analyze the connections between the different qualitative properties of numerical solutions of linear parabolic problems with dirichlet type boundary condition first we formulate the qualitative properties for the differential equations and shed light on their relations then we show how the well known discretization schemes can be written in the form of a one step iterative process we give necessary and sufficient conditions of the main qualitative properties of these iterations we apply the results to the finite difference and galerkin finite element solutions of linear parabolic problems in our main result we show that the nonnegativity preservation property is equivalent to the maximum minimum principle and they imply the maximum norm contractivity in one two and three dimensions we list sufficient a priori conditions that ensure the required qualitative properties finally we demonstrate the above results on numerical examples

the cutoff method for the numerical computation of nonnegative solutions of parabolic pdes with application to anisotropic diffusion and lubrication type equations the cutoff method which cuts off the values of a function less than a given number is studied for the numerical computation of nonnegative solutions of parabolic partial differential equations a convergence analysis is given for a broad class of finite difference methods combined with cutoff for linear parabolic equations two applications are investigated linear anisotropic diffusion problems satisfying the setting of the convergence analysis and nonlinear lubrication type equations for which it is unclear if the convergence analysis applies the numerical results are shown to be consistent with the theory and in good agreement with existing results in the literature the convergence analysis and applications demonstrate that the cutoff method is an effective tool for use in the computation of nonnegative solutions cutoff can also be used with other discretization methods such as collocation finite volume finite element and spectral methods and for the computation of positive solutions

moments of energy level distributions in vibrational spectra expressions for moments of the spectral density distribution of a system described by a non symmetric n particle hamiltonian are derived in particular this kind of hamiltonian may represent n coupled oscillators then its spectrum models the vibrational spectrum of a molecule it is assumed that the hamiltonian is defined in a finite dimensional space formed by products of orthogonal eigenstates of some one body hamiltonians possible applications of the formalism are briefly discussed

the sum rule for spectroscopic factors in the seniority scheme of identical particles spectroscopic factors partitioning an identical particle state into a couple of substates with definite seniorities are summed into a simple form this sum rule is an extension of the reduction relation for the fixed seniority average of a many body operator the seniority projection operator is utilized throughout the present formulation

character d modules via drinfeld center of harish chandra bimodules the category of character d modules is realized as drinfeld center of the abelian monoidal category of harish chandra bimodules tensor product of harish chandra bimodules is related to convolution of d modules via the long intertwining functor radon transform by a result of beilinson and ginzburg represent theory 3 1 31 1999 exactness property of the long intertwining functor on a cell subquotient of the harish chandra bimodules category shows that the truncated convolution category of lusztig adv math 129 85 98 1997 can be realized as a subquotient of the category of harish chandra bimodules together with the description of the truncated convolution category bezrukavnikov et al in isr j math 170 207 234 2009 this allows us to derive under a mild technical assumption a classification of irreducible character sheaves over obtained by lusztig by a different method

some sufficient conditions for the non negativity preservation property in the discrete heat conduction model in the course of the numerical approximation of mathematical models there is often a need to solve a system of linear equations with a tridiagonal or a block tridiagonal matrices usually it is efficient to solve these systems using a special algorithm tridiagonal matrix algorithm or tdma which takes advantage of the structure the main result of this work is to formulate a sufficient condition for the numerical method to preserve the non negativity for the special algorithm for structured meshes we show that a different condition can be obtained for such cases where there is no way to fulfill this condition moreover as an example the numerical solution of the two dimensional heat conduction equation on a rectangular domain is investigated by applying dirichlet boundary condition and neumann boundary condition on different parts of the boundary of the domain for space discretization we apply the linear finite element method and for time discretization the well known q method the theoretical results of the paper are verified by several numerical experiments

extension of wick s theorem by means of the state operator formalism reordering of operators in the second quantization formalism is described by using state operators as basic operators where the state operator is either a cluster of creation operators or that of annihilation operators specified by a set of quantum numbers of any representation the commutation relation between state operators is simply expressed in terms of the coefficient of fractional parentage cfp the factor specific to state operator de composition wick s theorem prevalently applied to a string of creation and annihilation operators is extended to a string of state operators similar extension is done for the contraction theorem

preserving nonnegativity of an affine finite element approximation for a convection diffusion reaction problem an affine finite element scheme approximation of a time dependent linear convection diffusion reaction problem in 2d and 3d is presented for these equations which do not satisfy an underlying maximum principle sufficient conditions are given in terms of the coefficient functions the computational grid and the discretization parameters to ensure that the nonnegativity property of the true solution is also satisfied by its approximation numerical examples are given which confirm the necessity and sufficiency of the discretization conditions to guarantee the nonnegativity of the approximation we present a monotone fem scheme for a convection diffusion reaction problem in 2 3d the considered equation does not possess an underlying maximum principle sufficient conditions are given to ensure the nonnegativity of the approximations numerical examples confirm the necessity and sufficiency of the conditions

numerical analysis of one dimensional motion of magma without mass forces abstract in this paper we present numerical treatment for a compaction driven darcian flow viscoelastic rock magma model this problem is a strongly coupled system of one quasi linear parabolic equation and one integro differential equation for the density and the porosity of the flow the numerical discretization uses cell centered finite difference method combined with semi implicit and implicit time stepping implicit explicit schemes as well as implicit explicit iterative algorithms have been developed to solve the corresponding discrete problems some properties positivity boundness conservation of the numerical solutions are investigated convergence study of the iteration processes is also presented the efficiency and the accuracy of the proposed methods are illustrated numerically by test examples with near real data

a positive and bounded finite element approximation of the generalized burgers huxley equation abstract we present a finite element scheme capable of preserving the nonnegative and bounded solutions of the generalized burgers huxley equation proofs of existence and uniqueness of a solution to the continuous problem together with some results concerning the boundedness and the nonnegativity of the solution are given under appropriate conditions on the mesh and the initial and boundary data boundedness and nonnegativity of the finite element approximation are established an a priori error estimate for the approximation is also derived numerical experiments are presented which support the derived theoretical results