analytic torsion and holomorphic determinant bundles i bott chern forms and analytic torsion we attach secondary invariants to any acyclic complex of holomorphic hermitian vector bundles on a complex manifold these were first introduced by bott and chern bot c our new definition uses quillen s superconnections we also give an axiomatic characterization of these classes these results will be used in bgs2 and bgs3 to study the determinant of the cohomology of a holomorphic vector bundle

realizations of conformal current type lie algebras in this paper we obtain the realizations of some infinite dimensional lie algebras named conformal current type lie algebras in terms of a two dimensional novikov algebra and its deformations furthermore ovsienko and roger s loop cotangent virasoro algebra which can be regarded as a nice generalization of the virasoro algebra with two space variables is naturally realized as an affinization of the tensor product of a deformation of the two dimensional novikov algebra and the laurent polynomial algebra these realizations shed new light on various aspects of the structure and representation theory of the corresponding infinite dimensional lie algebras

battle outcome prediction for an extended system of lanchester type differential equations abstract battle outcome prediction conditions are given for an extended system of lanchester type differential equations for two different types of battle termination conditions a fixed force level breakpoint battles and b fixed force ratio breakpoint battles necessary and sufficient conditions for predicting battle outcome are given in the former case for a fight to the finish while sufficient conditions are given in the latter case the former results are equivalent to those for the problem of classical analysis of determining explicitly as a function of the initial conditions the occurrence of a zero point for the solution to this extended system although such results as given here have not appeared previously for nonoscillatory in the strict sense solutions

quantum field theory grassmannians and algebraic curves this paper is devoted in part to clarifying some aspects of the relation between quantum field theory and infinite grassmannians and in part to pointing out the existence of a close analogy between conformal field theory on riemann surfaces and the modern theory of automorphic representations along the way we develop a multiplicative analog of the usual additive ward identities of current algebra we also reformulate the additive ward identities in a way which may be useful in terms of the residues of operator valued differential forms a concluding section is devoted to some remarks on string field theory in an appendix we attempt to clarify the recent construction by beilinson manin and schechtman of what might be called global virasoro algebras

on the domain of hyperbolicity of the cumulant equations in this article we consider the influence of non equilibirum values of classical variables on the eigenvalues of the advection part of the cumulant equations real and finite eigenvalues are a neccessary condition for the cumulant equations to be hyperbolic which can be used to obtain estimates on admissible deviations from equilibrium for a model of particular order still to be valid we find that this condition puts no constraints on velocity and shear stress values but specific energy must be positive normal stress must be bounded by specific energy and heat flux not be too large

the dirichlet problem for a quasilinear singularly perturbed second order system we study an equivalent form of the dirichlet problem for a quasilinear singularly perturbed second order system which is a singular singularly perturbed boundary value problem in this way we have not only eliminated the usual assumption of the existence of a vector potential function but also proved the existence and uniqueness of the exact solution of the studied problem meanwhile an asymptotic analysis and a remainder estimate of the solution are also presented

analytic fields on riemann surfaces ii the properties of analytic fields on a riemann surface represented by a branch covering of 1 are investigated in detail branch points are shown to correspond to the vertex operators with simple conformal properties as applications we compute determinants of bar partial j operators forzn symmetric surfaces and obtain various representations for the two loop measure in the bosonic string theory together with various identities for theta functions of hyperelliptic surfaces we also present an integral representation for the quantum part of the twist field correlation functions which describe propagation of the string on the orbifold background we also calculate the quantum part of the structure constants of the twist field operator algebra generalizing the results of dixon friedan martinec and shenker

liouville green approximations for a class of linear oscillatory difference equations of the second order abstract an asymptotic approximation theorem is proved for the solutions of linear oscillatory three term recurrence equations in a certain class this result represents a discrete analogue of the well known liouville green or wkbj theorem rigorously proved by olver for second order linear differential equations as in olver s theorem precise estimates are obtained for the error terms and the double asymptotic nature of the approximation is made clear applications to certain orthogonal polynomials are also given

the boltzmann equation for weakly inhomogeneous data we solve the initial value problem associated to the nonlinear boltzmann equation in the case in which the initial distribution has sufficiently small spatial gradients

higher adeles and non abelian riemann roch abstract we show a riemann roch theorem for group ring bundles over an arithmetic surface this is expressed using the higher adeles of beilinson parshin and the tame symbol via a theory of adelic equivariant chow groups and chern classes the theorem is obtained by combining a group ring coefficient version of the local riemann roch formula as in kapranov vasserot with results on k groups of group rings and an explicit description of group ring bundles over p 1 our set up provides an extension of several aspects of the classical frohlich theory of the galois module structure of rings of integers of number fields to arithmetic surfaces

decoupling of recessive and nonrecessive solutions for a second order system abstract a riccati transformation is used to obtain liouville green approximations along with error bounds for a vector system u f t u on a real interval a subject to suitable assumptions on the n n matrix valued function f f t

decoupling and order reduction via the riccati transformation a program that dates back to riccati 1724 and that seeks to replace a given differential equation with an equivalent collection of effectively uncoupled equations of lower orders through the use

liouville green approximations via the riccati transformation the riccati transformation is used to obtain liouville green approximations along with error bounds possessing a particularly simple and explicit form for the equation u f t u the resulting error bounds involve the function f and its first derivative whereas the earlier bounds of olver 1961 1974 and taylor 1978 1982 involve also the second derivative of f the theorems given here resemble corresponding results of olver 1974 chapter 6 and indeed the present work owes a great debt to olver the riccati transformation used here is based on work of y sibuya k w chang and w a harris and can be used in a more general study of linear ordinary differential equations and systems of such equations cf smith 1984 1985

stability and absolute stability of a three point difference scheme consider a three point difference scheme h 2 2 yn qn h yn fn h n z 0 1 2 where h 0 h0 h0 is a given positive number 2 yn yn 1 yn 1 f h fn h n z l h l h f h f h l h f h l h supn z fn h we assume a unique a priori requirement 0 qn h for any n z and h 0 h0 the main results are a criterion of stability and absolute stability of the difference scheme 1 in the space l h

periodic manifolds with spectral gaps abstract we investigate spectral properties of the laplace operator on a class of non compact riemannian manifolds for a given number n we construct periodic manifolds such that the essential spectrum of the corresponding laplacian has at least n open gaps we use two different methods first we construct a periodic manifold starting from an infinite number of copies of a compact manifold connected by small cylinders in the second construction we begin with a periodic manifold which will be conformally deformed in both constructions a decoupling of the different period cells is responsible for the gaps

determinant bundles and virasoro algebras we consider the interplay of infinite dimensional lie algebras of virasoro type and moduli spaces of curves suggested by string theory we will see that the infinitesimal geometry of determinant bundles is governed by virasoro symmetries the mumford forms are just invariants of these symmetries the representations of virasoro algebra define twisted d modules on moduli spaces thesed modules are equations on correlators in conformal field theory

hitchin s connection and differential operators with values in the determinant bundle for a family of smooth curves we have the associated family of moduli spaces of stable bundles with fixed determinant on the curves there exists a so called theta line bundle on the family of moduli spaces when the kodaira spencer map of the family of curves is an isomorphism we prove in this paper an identification theorem between sheaves of differential operators on the theta line bundle and higher direct images of vector bundles on curves as an application the so called hitchin connection on the direct image of powers of the theta line bundle is derived naturally from the identification theorem a logarithmic extension to certain singular stable curves is also presented in this paper

a new version of the chang transformation a new decoupling transformation is proposed for singularly perturbed linear systems this transformation has the advantage over the previous one in that in order to find such a transformation only one algorithm is required and the computation can be done in parallel because the transformation matrices are obtained from two independent equations that are identical in form

regularity of the inversion problem for the sturm liouville difference equation i representation of the davies harrell type for the green difference function abstract this is the first part of a study of the inversion for a sturm liouville difference equation the main results of the paper are a special representation for the green difference function and a condition for regularity of the inversion problem which is expressed in terms of that function

regularity of the inversion problem for the sturm liouville difference equation ii two sided estimates for the diagonal value of the green function abstract this is the second part of a study of the inversion for a sturm liouville difference equation our main result consists in getting two sided sharp by order estimates for the diagonal value of the green difference function

more on asymptotic approximation for matrix differential equations an asymptotic approximation is obtained for solutions of a matrix differential equation with symmetric matrix coefficient analytic on the real line our contribution consists of an asymptotic approximation that is valid even for various singularities near infinity

singularly perturbed linear boundary value problems abstract in this paper an alternative approach to the method of asymptotic expansions for the study of a singularly perturbed linear system with multiparameters and multiple time scales is developed the method consists of developing a non singular linear transformation that transforms an arbitrary n time scale system into diagonal form this fast and slow mode decomposition provides a modern technique to find an approximate solution of the original system in terms of the solution of an auxiliary system corresponding to the decoupled system furthermore the decoupled system provides a useful mechanism to relate the asymptotic behavior of the solution of the original system and the solution of the degenerate system relative to the original system

opening up and control of spectral gaps of the laplacian in periodic domains the main result of this work is as follows for arbitrary pairwise disjoint finite intervals j j 0 j 1 m and for arbitrary n 2 we construct a family of periodic non compact domains e n e 0 such that the spectrum of the neumann laplacian in e has at least m gaps when e is small enough moreover the first m gaps tend to the intervals j j as e 0 the constructed domain e is obtained by removing from n a system of periodically distributed trap like surfaces the parameter e characterizes the period of the domain e also it is involved in a geometry of the removed surfaces

chiral de rham complex we define natural sheaves of vertex algebras over smooth manifolds which may be regarded as semi infinite de rham complexes of certain d modules over the loop spaces for calabi yau manifolds they admit n 2 supersymmetry connection with wakimoto modules is discussed

unified canonical forms for matrices over a differential ring abstract linear differential equations with variable coefficients of the vector form i x a t x and the scalar form ii y n n t y n 1 2 t y 1 t y 0 can be studied as operators on a differential module over a differential ring using this differential algebraic structure and a classical result on differential operator factorizations developed by cauchy and floquet a new variable eigenvalue theory and an associated set of matrix canonical forms are developed in this paper for matrices over a differential ring in particular eight new spectral and modal matrix canonical forms are developed that include as special cases the classical companion jordan diagonal and generalized vandermonde canonical forms as traditionally used for matrices over a number ring field therefore these new matrix canonical forms can be viewed as unifications of these classical ones two additional canonical matrices that perform order reductions of a cauchy floquet factorization of a linear differential operator are also introduced general and explicit formulas for obtaining the new canonical forms are derived in terms of the new variable eigenvalues the results obtained here have immediate applications in diverse engineering and scientific fields such as physics control systems communications and electrical networks in which linear differential equations i and ii are used as mathematical models

a unified approach to string scattering amplitudes 1 physics in the calculation of g loop string tachyon amplitudes withn scattering points the distinguished polyakov measure d g n on the moduli spacem g n of riemann surfaces of genus g withn punctures arises we give an interpretation of this measure as the modulus squared of a holomorphic section g n the mumford form of a certain holomorphic line bundle i e we prove an analog of the belavin knizhnik theorem d g n g n 2 in the amplitudic case we give an expression for this measure through the determinants of the laplace operators over ghosts and over multivalued fields with monodromy prescribed by momenta at the scattering points we show also that the form g n n 0 n 0 for the partition function andn point amplitudes can be obtained from a unified over alln universal mumford form 2 mathematics the following new concepts from the theory of complex algebraic curves are investigated divisors with complex coefficients complex powers of holomorphic line bundles determinants of laplace operators over multivalued functions etc the corresponding generalizations of the determinant line bundles the weil deligne pairings the quillen and the arakelov deligne metrics are constructed a suggested by string amplitude considerations analog of the mumford theorem on holomorphic triviality of the bundle 2 1 13 over the moduli space is given this analog asserts the existence of a canonical flat metric on a certain line bundle lambda 2 otimes lambda 1 13 otimes left mathop otimes limits v 1 13 left langle mathcal o d v mathcal o d v right rangle 1 right see the main body of the text there exist two differences the latter bundle is not holomorphically trivial but has a canonical flat metric and being defined on the teichmuller spacet g n this bundle can be pulled down only on an infinite sheeted covering of the moduli spacem g n the universal isometries and the relative curvatures from the second part of the paper may be interesting too

on holomorphic principal bundles over a compact riemann surface admitting a flat connection let g be a connected reductive linear algebraic group over mathbb c and x a compact connected riemann surface let l subset g be a levi factor of some parabolic subgroup of g with l 0 l l l its maximal abelian quotient we prove that a holomorphic g bundle e g over x admits a flat connection if and only if for every such l and every reduction e l subset e g of the structure group of e g to l the l 0 bundle obtained by extending the structure group of e l is topologically trivial for g rm gl n mathbb c this is a well known result of a weil

liouville green asymptotic approximation for a class of matrix differential equations and semi discretized partial differential equations abstract a liouville green or wkb asymptotic approximation theory is developed for the class of linear second order matrix differential equations y f t a g t y on a where a and g t are matrices and f t is scalar this includes the case of an asymptotically constant not necessarily diagonalizable coefficient a when f t 1 an explicit representation for a basis of the right module of solutions is given and precise computable bounds for the error terms are provided the double asymptotic nature with respect to both t and some parameter entering the matrix coefficient is also shown several examples some concerning semi discretized wave and convection diffusion equations are given

the pfaffian line bundle we analyze the holomorphic pfaffian line bundle defined over an infinite dimensional isotropic grassmannian manifold using the infinite dimensional relative pfaffian we produce a fock space structure on the space of holomorphic sections of the dual of this bundle on this fock space an explicit and rigorous construction of the spin representations of the loop groupslo n is given we also discuss and prove some facts about the connection between the pfaffian line bundle over the grassmannian and the pfaffian line bundle of a dirac operator

complete manifolds with bounded curvature and spectral gaps abstract we study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius we give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps in particular for any noncompact covering of a compact manifold there is a metric on the base so that the lifted metric has an arbitrarily large finite number of gaps in its essential spectrum also for any complete noncompact manifold with bounded curvature and positive injectivity radius we construct a metric uniformly equivalent to the given one also of bounded curvature and positive injectivity radius with an arbitrarily large finite number of gaps in its essential spectrum

curvature of determinant bundles for degenerate families we calculate the 1 1 curvature of the beilinson schechtman connection for the determinant bundle associated to a family of riemann surfaces with ordinary singularities as consequences we obtain generalizations of theorems of bismut and bost

maximal and minimal eigenvalues and their associated nonlinear equations the spectral theory of uniformly elliptic operators a under perturbations v giving rise to operators of the form hv a v x on a bounded or unbounded region such as schrodinger operators are considered suppose that v p is constrained but v is otherwise unspecified the theory of the potentials v that maximize or minimize the eigenvalues of hv is presented the optimizing potentials are typically determined by equations of the form u w x u cu u the optimization of eigenvalues also turns out to be related to the determination of the best constants in sobolev s inequality and in its one dimensional simplification to a classical oscillator problem with instanton properties

new bosonization and conformal field theory over z new formulation of bosonization is given so that it is defined over the ring z of integers the charge zero sector of the new boson fock space is the completion of the coordinate ring of the universal witt scheme by using new bosonization conformal field theory of free fermions over z is given

comportement semi classique pour l operateur de schrodinger a potentiel periodique abstract in this paper the schrodinger operator with a periodic potential is considered let v a smooth periodic function we study the semi classical behavior for a continuum spectrum of h2 v h 0 we are interested by localization and width of bands we give the interaction matrix up to an exponentially small error measured by agmon s distance between the wells a detailed investigation of the spectrum is made for the case where v has one nondegenerate minima per unit cell we also investigate the spectral properties of h2 v v where v is a smooth positive perturbation with compact support

gerbes of chiral differential operators ii vertex algebroids introduction 605 0 recollections and notation 612 1 vertex algebroids 615 2 from vertex algebras to vertex algebroids 622 3 category of vertex algebroids 628 4 cofibered structure 633 5 computation of a cocycle 637 6 characteristic classes of vector bundles 643 7 torseurs and gerbes 647 8 gerbes of vertex algebroids 653 9 chiral differential operators on curves and sheaves of virasoro algebras 657 10 enveloping algebra of a vertex algebroid 664 11 calabi yau structures 668 12 virasoro and calabi yau structures 671 references 680

twisted wess zumino witten models on elliptic curves investigated is a variant of the wess zumino witten model called a twisted wzw model which is associated to a certain lie group bundle on a family of elliptic curves the lie group bundle is a non trivial bundle with flat connection and related to the classical elliptic r matrix the usual non twisted wzw model is associated to a trivial group bundle with trivial connection on a family of compact riemann surfaces and a family of its principal bundles the twisted wzw model on a fixed elliptic curve at the critical level describes the xyz gaudin model the elliptic knizhnik zamolodchikov equations associated to the classical elliptic r matrix appear as flat connections on the sheaves of conformal blocks in the twisted wzw model

periodic riemannian manifold with preassigned gaps in spectrum of laplace beltrami operator it is known e l green 1997 o post 2003 that for an arbitrary m n one can construct a periodic non compact riemannian manifold m with at least m gaps in the spectrum of the corresponding laplace beltrami operator m in this work we want not only to produce a new type of periodic manifolds with spectral gaps but also to control the edges of these gaps the main result of the paper is as follows for arbitrary pairwise disjoint intervals j j 0 j 1 m m n for an arbitrarily small 0 and for an arbitrarily large l 0 we construct a periodic non compact riemannian manifold m with at least m gaps in the spectrum of the operator m moreover the edges of the first m gaps belong to neighbourhoods of the edges of the intervals j j while the remaining gaps if any are located outside the interval 0 l

moment closure hierarchies for kinetic theories this paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory the first member of the hierarchy is the euler system which is based on maxwellian velocity distributions while the second member is based on nonisotropic gaussian velocity distributions the closure proceeds in two steps the first ensures that every member of the hierarchy is hyperbolic has an entropy and formally recovers the euler limit the second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct navier stokes behavior this is achieved through the introduction of a generalization of the bgk collision operator the simplest such system in three spatial dimensions is a 14 moment closure which also recovers the behavior of the grad 13 moment system when the velocity distributions lie near local maxwellians the closure procedure can be applied to a general class of kinetic theories

automorphism group of k t applications to the bosonic string this paper is concerned with the formulation of a non pertubative theory of the bosonic string we introduce a formal group g which we propose as the universal moduli space for such a formulation this is motivated because g establishes a natural link between representations of the virasoro algebra and the moduli space of curves

local conformal field algebras the local conformal field algebras with the multiplication corresponding to the regularized pointwise product of fields in the operator algebras of the quantum conformal field theory are investigated

geometric realization of conformal field theory on riemann surfaces conformal field theory on a family of riemann surfaces is formulated we derive equations of motion of vacua which are parametrized by moduli of riemann surfaces and show that these vacua are characterized uniquely by these equations our theory has a deep connection with sato s theory of kp equations

diagonals of green s functions and applications abstract we describe the diagonal of the green s function for some sturm liouville operators as it was introduced and studied in a number of papers already and we discuss a number of applications that depend on its properties semilinear elliptic pde s with rotationally invariant and possibly singular coefficients with power type nonlinearities provide a particularly well suited class of examples where the results of this article apply

exact slow fast decomposition of the singularly perturbed matrix differential riccati equation in this paper the hamiltonian matrix formulation of the riccati equation is used to derive the reduced order pure slow and pure fast matrix differential riccati equations of singularly perturbed systems these pure slow and pure fast matrix differential riccati equations are obtained by decoupling the singularly perturbed matrix differential riccati equation of dimension n 1 n 2 into the pure slow regular matrix differential riccati equation of dimension n 1 and the pure fast stiff matrix differential riccati equation of dimension n 2 a formula is derived that produces the solution of the original singularly perturbed matrix differential riccati equation in terms of solutions of the pure slow and pure fast reduced order matrix differential riccati equations and solutions of two reduced order initial value problems in addition to its theoretical importance the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear time invariant systems independently in pure slow and pure fast time scales

weight summability of solutions of the sturm liouville equation we consider an equation y x q x y x f x x r 1 wheref x lp r p 1 l r c r 1 q x lloc1 r we study requirements for a weight functionr x llocp r and forq x under which for a givenp 1 regardless off x lp r the solutiony x lp r of eq 1 satisfies the inequalities r x y x p c f x p y x p q x y x p c f x p c const

an identification of the connections of quillen and beilinson schechtman given a family of riemann surfaces and a holomorphic vector bundle beilinson and schechtman construct a canonical connection on the associated determinant bundle we prove the conjecture which states that their connection coincides with the quillen connection this is done by reducing to the case where bar partial along fibers are invertible both connection forms become more accessible in this case

opers and theta functions abstract we construct natural maps the klein and wirtinger maps from moduli spaces of semistable vector bundles over an algebraic curve x to affine spaces as quotients of the nonabelian theta linear series we prove a finiteness result for these maps over generalized kummer varieties moduli space of torus bundles leading us to conjecture that the maps are finite in general the conjecture provides canonical explicit coordinates on the moduli space the finiteness results give low dimensional parametrizations of jacobians in p 3g 3 for generic curves described by 2 functions or second logarithmic derivatives of theta we interpret the klein and wirtinger maps in terms of opers on x opers are generalizations of projective structures and can be considered as differential operators kernel functions or special bundles with connection the matrix opers analogues of opers for matrix differential operators combine the structures of flat vector bundle and projective connection and map to opers via generalized hitchin maps for vector bundles off the theta divisor the szego kernel gives a natural construction of matrix oper the wirtinger map from bundles off the theta divisor to the affine space of opers is then defined as the determinant of the szego kernel this generalizes the wirtinger projective connections associated to theta characteristics and the associated klein bidifferentials

cusps and d modules 1 1 v modules on singular varieties let y denote a variety over a field and let vy denote the full sheaf of differential operators on y in characteristic zero vy is the familiar sheaf of differential operators and in characteristic p the sheaf vy will contain all divided powers of operators d it is well known that on a general singular variety y the ring of differential operators is badly behaved in particular it need not be noetherian nor if y is affine a simple ring see bgg following grothendieck there is an alternative notion of a sheaf with infinitesi mal parallel transport called a co stratification see be2 be3 bdi or section 2 which agrees with the notion of left respectively right py module when y is smooth but will not agree in general when y is singular our first result presents a simple vanishing condition that guarantees that the notions of x module and co stratification on a singular variety coincide

cyclic homology of differential operators the virasoro algebra and a q analogue we show how methods from cyclic homology give easily an explicit 2 cocycle on the lie algebra of differential operators of the circle such that restricts to the cocycle defining the virasoro algebra the same methods yield also aq analogue of as well as an infinite family of linearly independent cocycles arising when the complex parameterq is a root of unity we use an algebra ofq difference operators andq analogues of koszul and the rham complexes to construct these quantum cocycles

improved error bounds for the liouville green or wkb approximation a variant of the gronwall bellman inequality is used to develop new bounds on solutions to the fundamental singular integral equations that arise in the error analysis of the liouville green or wkb approximation consequently new improved error bounds are obtained from the liouville green approximation to the solution of the differential equation d2wdt2 q t w 0 where the function q is real and twice continuously differentiable and does not vanish

brief paper eigenvector approach for order reduction of singularly perturbed linear quadratic optimal control problems in this paper we show how to decompose the singularly perturbed algebraic riccati equation and the corresponding linear quadratic optimal control problem at steady state in terms of reduced order pure slow and pure fast problems by using the eigenvector approach the eigenvector approach should be used for decomposition of singularly perturbed control systems in the cases when the singular perturbation parameter is not very small in such cases the decomposition methods based on series expansions fixed point iterations subspace iterations and newton iterations either fail to produce solutions of the corresponding algebraic equations or display very slow convergence

conformally flat riemannian metrics schrodinger operators and semiclassical approximation abstract a relationship between laplace beltrami and schrodinger operators on euclidean domains is analyzed and exploited for several purposes we use the schrodinger equation to analyze the spectra of laplace beltrami operators with periodic metrics on r v and use geometric notions and nonlinear differential equations to bound spectra and green functions of schrodinger operators in various ways we also have a new more operator theoretic analysis of the semiclassical limit and the liouville green or jwkb approximation in one dimension

moduli of curves grobner bases and the krichever map abstract we study moduli spaces of possibly non nodal curves c p 1 p n of arithmetic genus g with n smooth marked points equipped with nonzero tangent vectors such that o c p 1 p n is ample and h 1 o c a 1 p 1 a n p n 0 for given integer weights a a 1 a n such that a i 0 and a i g we show that each such moduli space u g n n s a is an affine scheme of finite type and the krichever map identifies it with the quotient of an explicit locally closed subscheme of the sato grassmannian by the free action of the group of changes of formal parameters we study the git quotients of u g n n s a by the natural torus action and show that some of the corresponding stack quotients give modular compactifications of m g n with projective coarse moduli spaces more generally using similar techniques we construct moduli spaces of curves with chains of divisors supported at marked points with prescribed number of sections which in the case n 1 corresponds to specifying the weierstrass gap sequence at the marked point

solution of lanchester equation models for combat abstract this paper provides a methodology for the study of quantitative behavior of solutions to the extended system of lanchester type differential equations for the prediction of the outcome of battles under different condition the method is sufficiently general for use with nonlinear and or stochastic adaptations as well

annihilation prediction for lanchester type models of modern warfare this paper introduces important new functions for analytic solution of lanchester type equations of modem warfare for combat between two homogeneous forces modeled by power attrition rate coefficients with no offset tabulations of these lanchester clifford schlafli or lcs functions allow one to study this particular variable coefficient model almost as easily and thoroughly as lanchester s classic constant coefficient one lcs functions allow one to obtain important information in particular force annihilation prediction without having to spend the time and effort of computing force level trajectories the choice of these particular functions is based on theoretical considerations that apply in general to lanchester type equations of modern warfare and provide guidance for developing other canonical functions moreover our new lcs functions also provide valuable information about related variable coefficient models also we introduce an important transformation of the battle s time scale that not only simplifies the force level equations but also shows that relative fire effectiveness and intensity of combat are the only two weapon system parameters determining the course of such variable coefficient lanchester type combat