a new algebraization of the lameequation we develop a new way of writing the lamehamiltonian in lie algebraic form this yields in a natural way an explicit formula for both the lamepolynomials and the classical non meromorphic lame functions in terms of chebyshev polynomials and of a certain family of weakly orthogonal polynomials

the quantum anharmonic oscillator and quasi exactly solvable bose systems we extend the notion of quasi exactly solvable qes models from potential ones and differential equations to bose systems we obtain conditions under which algebraization of part of the spectrum occurs in some particular cases simple exact expressions for several energy levels of an anharmonic bose oscillator are obtained explicitly the corresponding results do not exploit perturbation theory and include the strong coupling regime a generic hamiltonian under discussion cannot in contrast to qes potential models be expressed as a polynomial in generators of sl2 algebra the suggested approach is extendable to many particle bose systems with interaction

cubic structures equivariant euler characteristics and lattices of modular forms we use the theory of cubic structures to give a fixed point riemann roch formula for the equivariant euler characteristics of coherent sheaves on projective at schemes over z with a tame action of a finite abelian group this formula supports a conjecture concerning the extent to which such equivariant euler characteristics may be determined from the restriction of the sheaf to an infinitesimal neighborhood of the fixed point locus our results are applied to study the module structure of modular forms having fourier coeficients in a ring of algebraic integers as well as the action of diamond hecke operators on the mordell weil groups and tate shafarevich groups of jacobians of modular curves

the index of an algebraic variety let k be the field of fractions of a henselian discrete valuation ring mathcal o k let xk k be a smooth proper geometrically connected scheme admitting a regular model x mathcal o k we show that the index xk k of xk k can be explicitly computed using data pertaining only to the special fiber xk k of the model x

linear operators with invariant polynomial space and graded algebra the irreducible finite dimensional representations of the graded algebras osp j 2 j 1 2 3 are expressed in terms of differential operators some quantum deformations of these algebras are shown to admit similar kinds of representations these are formulated in terms of finite difference operators the results are discussed in the framework of the quasi exactly solvable equations

on the sutherland spin model of b n type and its associated spin chain the b n hyperbolic sutherland spin model is expressed in terms of a suitable set of commuting dunkl operators this fact is exploited to derive a complete family of commuting integrals of motion of the model thus establishing its integrability the dunkl operators are shown to possess a common flag of invariant finite dimensional linear spaces of smooth scalar functions this implies that the hamiltonian of the model preserves a corresponding flag of smooth spin functions the discrete spectrum of the restriction of the hamiltonian to this spin flag is explicitly computed by triangularization the integrability of the hyperbolic sutherland spin chain of b n type associated with the dynamical model is proved using polychronakos s freezing trick

third order equation for harmonic generation complex canonical transformation and jwkb solution the quantum description of third harmonic generation can be formulated as an eigenvalue problem for a third order linear differential equation we perform a semiclassical study of this third order equation generalizing the familiar jwkb theory for the second order schrodinger equation and deriving explicit albeit approximate formulas for the eigenvalues within this semiclassical context a central role in this analysis is played by a nonlinear complex canonical transformation which permits a complete description of the classical motion generated by a complex polynomial hamiltonian function in the complexified position and momentum planes

on the families of orthogonal polynomials associated to the razavy potential we show that there are two different families of weakly orthogonal polynomials associated to the quasi exactly solvable razavy potential v x cos 2x m 2 0 m one of these families encompasses the four sets of orthogonal polynomials recently found by khare and mandal while the other one is new these results are extended to the related periodic potential u x cos 2x m 2 for which we also construct two different families of weakly orthogonal polynomials we prove that either of these two families yields the ground state when m is odd and the lowest lying gaps in the energy spectrum of the latter periodic potential up to and including the m 1 th gap and having the same parity as m 1 moreover we show that the algebraic eigenfunctions obtained in this way are the well known finite solutions of the whittaker hill or hill s three term periodic differential equation thus the foregoing results provide a lie algebraic justification of the fact that the whittaker hill equation unlike for instance mathieu s equation admits finite solutions

the cassels tate pairing on polarized abelian varieties let a e be a principally polarized abelian variety defined over a global field k and let a be its shafarevich tate group let a nd denote the quotient of a by its maximal divisible subgroup cassels and tate constructed a nondegenerate pairing a nd a nd q z if a is an elliptic curve then by a result of cassels the pairing is alternating but in general it is only antisymmetric using some new but equivalent definitions of the pairing we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on a nd these criteria are expressed in terms of an element c a nd that is canonically associated to the polarization e in the case that a is the jacobian of some curve a down to earth version of the result allows us to determine effectively whether a if finite is a square or twice a square we then apply this to prove that a positive proportion in some precise sense of all hyperelliptic curves of even genus g 2 over q have a jacobian with nonsquare if finite for example it appears that this density is about 13 for curves of genus 2 the proof makes use of a general result relating global and local densities this result can be applied in other situations

uniruled surfaces of general type we give a systematic construction of uniruled surfaces in positive characteristic using this construction we find surfaces of general type with non trivial global vector fields surfaces with arbitrarily non reduced picard schemes as well as surfaces with inseparable canonical maps in particular we show that some previously known pathologies are not sporadic but exist in abundance

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

modeles semi factoriels et modeles de neron let s be the spectrum of a discrete valuation ring with function field k let x be a scheme over s we will say that x is semi factorial over s if any invertible sheaf on the generic fiber xk can be extended to an invertible sheaf on x here we show that any proper geometrically normal scheme over k admits a proper flat normal and semi factorial model over s we also construct some semi factorial compactifications of regular s schemes such as neron models of abelian varieties the semi factoriality property for a scheme x s corresponds to the neron property of its picard functor in particular one can recover the neron model of the picard variety rm pic x k k rm red 0 of xk from the picard functor picx s as in the known case of curves this provides some information on the relative algebraic equivalence on the s scheme x

neron models lie algebras and reduction of curves of genus one let k be a discrete valuation field let ok denote the ring of integers of k and let k be the residue field of ok of characteristic p 0 let s specok let x k be a smooth geometrically connected projective curve of genus 1 over k denote by ek the jacobian of x k let x s and e s be the minimal regular models of x k and ek respectively in this article we investigate the possible relationships between the special fibers xk and ek in doing so we are led to study the geometry of the picard functor pic x s when x s is not necessarily cohomologically flat as an application of this study we are able to prove in full generality a theorem of gordon on the equivalence between the artin tate and birch swinnertondyer conjectures recall that when k is algebraically closed the special fibers of elliptic curves are classified according to their kodaira type which is denoted by a symbol t in i n n z 0 ii ii iii iii iv iv given a type t and a positive integer m we denote by mt the new type obtained from t by multiplying all the multiplicities of t by m when k is algebraically closed the relationships between the type of a curve of genus 1 and the type of its jacobian can be summarized as follows

failure of the hasse principle for enriques surfaces abstract we construct an enriques surface x over q with empty etale brauer set and hence no rational points for which there is no algebraic brauer manin obstruction to the hasse principle in addition if there is a transcendental obstruction on x then we obtain a k3 surface that has a transcendental obstruction to the hasse principle

the structure of quasi exactly solvable systems the linear differential operators that preserve the n dimensional vector space whose entries are polynomials of fixed but arbitrary degrees in one variable are considered herein they are relevant for the classification of quasi exactly solvable systems of equations generating elements are explicitly constructed and rules for ordering their products are derived some examples of equations that can be reduced to such systems are discussed

the symmetric form of the koekoeks laguerre type differential equation abstract koekoek and koekoek 1991 proved that the laguerre type polynomials ln n x n 0 which are orthogonal on 0 with respect to the measure d n defined on 0 by d n x x e x 1 n x dx satisfy a differential equation l2 4 n y x ny x of order 2 4 when is a nonnegative integer in this paper we announce that the differential expression l2 4 n is symmetrizable on 0 with symmetry factor x e x and explicitly give this formally symmetric expression x e xl2 4 n y

linearized factorization techniques for multidimensional reaction diffusion equations an iterative predictor corrector technique for the elimination of the approximate factorization errors which result from the factorization of linearized q methods in multidimensional reaction diffusion equations is proposed and its convergence and linear stability are analyzed four approximate factorization techniques which do not account for the approximate factorization errors are developed the first technique uses the full jacobian matrix of the reaction terms requires the inversion of in general dense matrices and its approximate factorization errors are second order accurate in time the second and third methods approximate the jacobian matrix by diagonal or triangular ones which are easily inverted but their approximate factorization errors are however first order accurate in time the fourth approximately factorized method has approximate factorization errors which are second order accurate in time and requires the inversion of lower and upper triangular matrices the techniques are applied to a nonlinear two species two dimensional system of reaction diffusion equations in order to determine the approximate factorization errors and those resulting from the approximations to the jacobian matrix as functions of the allocation of the reaction terms space and time

a trace formula for varieties over a discretely valued field we study the motivic serre invariant of a smoothly bounded alge braic or rigid variety x over a complete discretely valued field k with perfect residue field k if k has characteristic zero we extend the definition to arbi trary k varieties using bittner s presentation of the grothendieck ring and a process of neron smoothening of pairs of varieties the motivic serre invariant can be considered as a measure for the set of unramified points on x under certain tameness conditions it admits a cohomological interpretation by means of a trace formula in the curve case we use t saito s geometric criterion for cohomological tameness to obtain more detailed results we discuss some applications to weil chatelet groups chow motives and the structure of the grothendieck ring of varieties msc 2000 11s15 14g05 14g22

grothendieck s pairing on component groups of jacobians let r be a discrete valuation ring with field of fractions k let ak be an abelian variety over k with dual ak denote by a and a the corresponding neron models and by a and a their component groups in gr exp vii ix grothendieck used the notion of biextension invented by mumford to investigate how the duality between ak and ak is reflected on the level of neron models in fact the essence of the relationship between a and a is captured by a bilinear pairing a a q z

on some polynomial potentials in d dimensions the d dimensional schrodinger s equation is analyzed with regard to the existence of exact solutions for polynomial potentials under certain conditions on the interaction parameters we show that the polynomial potentials v8 r k 18 krk 8 0 and v10 r k 110 krk 10 0 are exactly solvable by examining the polynomial solutions of certain linear differential equations with polynomial coefficients the necessary and sufficient conditions for the existence of these exact solutions are discussed finding accurate solutions for arbitrary values of the potential parameters using the asymptotic iteration method is also presented

on orthogonal polynomials with respect to a class of differential operators we consider orthogonal polynomials with respect to a linear differential operatorl m k 0m r k z d kdz k where r k k 0 m are complex polynomials such that deg r k

on one dimensional schrodinger problems allowing polynomial solutions we discuss the explicit construction of the schrodinger equations admitting a representation through some family of general polynomials almost all solvable quantum potentials are shown to be generated by this approach some generalization has also been performed in higher dimensional problems

invariant modules and the reduction of nonlinear partial differential equations to dynamical systems abstract we completely characterize all nonlinear partial differential equations leaving a given finite dimensional vector space of analytic functions invariant existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinary differential equations and provides a nonlinear counterpart to quasi exactly solvable quantum hamiltonians these results rely on a useful extension of the classical wronskian determinant condition for linear independence of functions in addition new approaches to the characterization of the annihilating differential operators for spaces of analytic functions are presented

period and index of genus one curves over global fields the period of a curve is the smallest positive degree of galois invariant divisor classes the index is the smallest positive degree of rational divisors we construct examples of genus one curves with prescribed period and index over a given global field as long as the characteristic of the field does not divide the period

criterion for polynomial solutions to a class of linear differential equations of second order we consider the differential equations y 0 x y s0 x y where 0 x s0 x are c functions we prove i if the differential equation has a polynomial solution of degree n 0 then n nsn 1 n 1sn 0 where n n 1 sn 1 0 n 1andsn s n 1 s0 k 1 n 1 2 conversely ii if n n 1 0 and n 0 then the differential equation has a polynomial solution of degree at most n we show that the classical differential equations of laguerre hermite legendre jacobi chebyshev first and second kinds gegenbauer and the hypergeometric type etc obey this criterion further we find the polynomial solutions for the generalized hermite laguerre legendre and chebyshev differential equations

on algebraic classification of hermitian quasi exactly solvable matrix schrodinger operators on line we construct six multi parameter families of hermitian quasi exactly solvable matrix schrodinger operators in one variable the method for finding these operators relies heavily upon a special representation of the lie algebra o 2 2 sl 2 sl 2 whose representation space contains an invariant finite dimensional subspace furthermore we select those quasi exactly solvable matrix models that have square integrable eigenfunctions on these models are in direct analogy with the quasi exactly solvable scalar schrodinger operators obtained by turbiner and ushveridze

methods for reducing approximate factorization errors in two and three factored schemes three methods are presented for reducing approximate factorization af errors that exist in two and three factored schemes such as the popular adi lu and lu ssor methods for problems in which af errors are larger than both time discretization and time linearization errors these methods can be used to lower the af errors so that larger time step sizes can be used when transient solutions are of interest when only steady state solutions are of interest these methods can be used to accelerate the convergence rate the methods presented also can be used to stabilize schemes that are unstable because of af errors e g the three factored adi scheme applied to the linear advection equation with central difference approximation of the spatial derivatives the three methods presented can be added easily to existing codes using any two or three factored schemes

quommutator deformations of spl n 1 we analyze the witten woronowicz s type deformations of the lie superalgebras spl n 1 and obtain deformations parametrized by n n 1 2 independent parameters for n 2 and by three parameters for n 2 for some of these algebras finite dimensional representations can be constructed in terms of finite difference operators providing operators that are relevant for the classification of quasiexactly solvable finite difference equations

fisher information of special functions and second order differential equations we investigate a basic question of analytic information theory namely the evaluation of the fisher information and the relative fisher information with respect to a non negative function for the probability distributions obtained by squaring the special functions of mathematical physics which are solutions of second order differential equations we obtain explicit expressions for these information theoretic properties via the expectation values of the coefficients of the differential equation we illustrate our approach for various nonrelativistic d dimensional wavefunctions and some special functions of physicomathematical interest emphasis is made in the nikiforov uvarov hypergeometric type functions which include and generalize the hermite functions and the gauss and kummer hypergeometric functions among others

umbral calculus discretization and quantum mechanics on a lattice umbral calculus deals with representations of the canonical commutation relations we present a short exposition of it and discuss how this calculus can be used to discretize continuum models and to construct representations of lie algebras on a lattice related ideas appeared in recent publications and we show that the examples treated there are special cases of umbral calculus this observation then suggests various generalizations of these examples a special umbral representation of the canonical commutation relations given in terms of the position and momentum operator on a lattice is investigated in detail

the schrodinger equation and canonical perturbation theory lett0 ev be the schrodinger operator corresponding to the classical hamiltonianh0 ev whereh0 is thed dimensional harmonic oscillator with non resonant frequencies 1 d and the potentialv q1 qd is an entire function of order d 1 1 we prove that the algorithm of classical canonical perturbation theory can be applied to the schrodinger equation in the bargmann representation as a consequence each term of the rayleigh schrodinger series near any eigenvalue oft0 admits a convergent expansion in powers of of initial point the corresponding term of the classical birkhoff expansion moreover ifv is an even polynomial the above result and the kam theorem show that all eigenvalues n e oft0 ev such thatn coincides with a kam torus are given up to order e by a quantization formula which reduces to the bohr sommerfeld one up to first order terms in

normalizability of one dimensional quasi exactly solvable schrodinger operators we completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi exactly solvable schrodinger operator on the line methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi exactly solvable hamiltonians and explicit normalizability conditions in general coordinate systems

neron models and compactified picard schemes over the moduli stack of stable curves we construct modular deligne mumford stacks representable over parametrizing neron models of jacobians as follows let b be a smooth curve and k its function field let be a smooth genus g curve over k admitting stable minimal model over b the neron model b is then the base change of via the moduli map b of f i e b moreover is compactified by a deligne mumford stack over giving a completion of neron models naturally stratified in terms of neron models

logarithmic transformations of rigid analytic elliptic surfaces we give new examples of algebraic elliptic surfaces and non algebraic rigid analytic elliptic surfaces by means of logarithmic transformations in the complex analytic case it is known that all multiple fibers of elliptic surfaces are obtained by logarithmic transformations using rigid analytic geometry we construct similar transformations of elliptic surfaces over complete non archimedean valuation base fields these operations yield rigid analytic elliptic fibrations with multiple fibers when the resulting surface admits an ample line bundle we may algebraize the surface in the positive characteristic case we obtain new types of algebraic elliptic surfaces we also obtain a non algebraic rigid analytic surface the combination of whose invariants appears neither in the algebraic case nor in the complex analytic case

motivic zeta functions of abelian varieties and the monodromy conjecture we prove for abelian varieties a global form of denef and loeser s motivic monodromy conjecture in arbitrary characteristic more precisely we prove that for every tamely ramified abelian variety a over a complete discretely valued field with algebraically closed residue field its motivic zeta function has a unique pole at chai s base change conductor c a of a and that the order of this pole equals one plus the potential toric rank of a moreover we show that for every embedding of ql in c the value exp 2 ic a is an l adic tame monodromy eigenvalue of a the main tool in the paper is edixhoven s filtration on the special fiber of the neron model of a which measures the behavior of the neron model under tame base change

bertini theorems over finite fields let x be a smooth quasiprojective subscheme of pn of dimension m i 0 over fq then there exist homogeneous polynomials f over fq for which the intersection of x and the hypersurface f 0 is smooth in fact the set of such f has a positive density equal to aex m 1 1 where aex s zx q s is the zeta function of x an analogue for regular quasiprojective schemes over z is proved assuming the abc conjecture and another conjecture

a quasiclassical analysis of second harmonic generation we investigate a quantum two oscillator model for second harmonic generation the total hamiltonian is the sum of two commuting hamiltonians with eigenvalues e0 and eint the exact determination of these eigenvalues is studied using tridiagonal matrices we present two general equations in the quasiclassical regime yielding the largest eint for a given e0 and a representation for both eigenvalues in terms of an additional quantum number some numerical analysis shows that both equations are fairly consistent for suitably large quantum numbers with the exact quantum results an approximate analytical expression for the wavefunction is also given

quasi exact solvability resonances and trivial monodromy in ordinary differential equations a correspondence between the sextic anharmonic oscillator and a pair of third order ordinary differential equations is used to investigate the phenomenon of quasi exact solvability for eigenvalue problems involving differential operators with order greater than 2 in particular links with bender dunne polynomials and resonances between independent solutions are observed for certain second order cases and extended to the higher order problems this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

lie algebras and polynomials in one variable a classification of linear differential and difference equations in one variable having polynomial solutions the generalized bochner problem is presented the idea of the approach is based on consideration of the eigenvalue problem for a polynomial in generators of the algebra sl2 r for differential equations or sl2 r q for finite difference equation given in the finite dimensional projectivized representation

heights and arakelov s intersection theory introduction this paper relates the neron tate height on jacobian varieties to an intersection theory on arithmetic surfaces constructed by s arakelov a1 here i extend and prove a statement of arakelov to the effect that his intersection pairing is a neron pairing a2i i use this to prove an analogue of the hodge index theorem for arithmetic surfaces a result due independently to g faltings the results of this paper have been extracted from my thesis written under the direction of michael artin it differs from the thesis in three ways firstly it employs the language of local height functions as defined by a neron and j tate and expounded by s lang li in the thesis i used neron s language of local intersection pairings n1 the second major difference is in the proof of theorem 1 3 in my thesis i proved this result by using work of raynaud and picard functors r 1 this method was suggested by m artin in this paper i provide a simpler proof observed by s lang and independently by b gross finally the development of arakalov s theory in the context of neron functions in section 2 is due to s lang who has kindly given me leave to publish it this exposition of my results is due in considerable part to s lang it is essentially an edited transcript of a series of conversations between us which he provided it is my great pleasure to thank both him and michael artin for their help and encouragement 1 the neron pairing neron ne has given a pairing between divisors and points on abelian varieties and also on arbitrary varieties under suitable conditions we shall give a complement to the neron theory on curves let c be a curve by which we mean a complete regular curve geometrically irreducible over a field k with an absolute value v let

on torsors under elliptic curves and serre s pro algebraic structures let mathcal o k be a complete discrete valuation ring with algebraically closed residue field of positive characteristic and field of fractions k let x k be a torsor under an elliptic curve a k over k x the proper minimal regular model of x k over s hbox spec mathcal o k and j the identity component of the neron model of mathrm pic x k k 0 we study the canonical morphism q mathrm pic 0 x s rightarrow j which extends the natural isomorphism on generic fibres we show that q is pro algebraic in nature with a construction that recalls serre s work on local class field theory furthermore we interpret our results in relation to shafarevich s duality theory for torsors under abelian varieties

quasi exactly solvable quartic bose hamiltonians we consider hamiltonians which are even polynomials of the fourth order with respect to bose operators we find subspaces preserved by the action of the hamiltonian these subspaces being finite dimensional include nonetheless states with an infinite number of quasi particles corresponding to the original bose operators the basis functions look rather simple in the coherent state representation and are expressed in terms of the degenerate hypergeometric function with respect to the complex variable labelling the representation in some particular degenerate cases they turn up to the power factor into trigonometric or hyperbolic functions bessel functions or combinations of the exponent and hermite polynomials we find explicitly the relationship between coefficients at different powers of bose operators that ensure quasi exact solvability of hamiltonians

on the brauer group of a surface artin and tate 15 5 1 have shown in 1966 the existence of a canonical skew symmetric pairing on the non p part of br x whose kernel is exactly the set of divisible elements it follows from this fact that if the non p part of br x is finite then its order is a square or twice a square a few years later manin published examples of rational surfaces that is surfaces birational over k to the projective plane with brauer groups equal to z 2z it is only in 1996 that the examples of manin were revisited by urabe who found a mistake in them for rational surfaces the brauer group is relatively easy to understand and urabe 16 remark after 1 17 showed improving on a result of milne 9 that the brauer group of a rational surface has order a square in 17 0 3 urabe then proves in full generality that when p 2 the 2 part of br x modulo its divisible subgroup has order a square thus to complete the proof of theorem 1 it remains to treat the case where p 2 and x k is not rational as we remarked above it was wrongly assumed for about 30 years that br x was not always a square our method of proof provides for any p that the 2 part of br x has order a square via the knowledge of the 2 part

quasiexactly solvable 2 2 matrix equations the conditions under which systems of two differential eigenvalue equations are quasiexactly solvable are investigated these systems reveal a rich set of algebraic structures some of them are explicitly described an example of a quasiexactly solvable system which provides a direct counterpart of the lame equation is studied

implicit compact linearized θ methods with factorization for multidimensional reaction diffusion equations an iterative predictor corrector technique for the elimination of the approximate factorization errors which result from the factorization of implicit three point compact linearized q methods in multidimensional reaction diffusion equations is proposed and its convergence and linear stability are analyzed four compact approximate factorization techniques which do not account for the approximate factorization errors and which involve three point stencils for each one dimensional operator are developed the first technique uses the full jacobian matrix of the reaction terms requires the inversion of in general dense matrices and its approximate factorization errors are second order accurate in time the second and third methods approximate the jacobian matrix by diagonal or triangular ones which are easily inverted but their approximate factorization errors are however first order accurate in time the fourth approximately factorized compact implicit method has approximate factorization errors which are second order accurate in time and requires the inversion of lower and upper triangular matrices the techniques are applied to a nonlinear two species two dimensional system of reaction diffusion equations in order to determine the approximate factorization errors and those resulting from the approximations to the jacobian matrix as functions of the allocation of the reaction terms space and time

iterative and non iterative full and approximate factorization methods for multidimensional reaction diffusion equations a variety of second order accurate in both space and time full and approximate factorization methods for the numerical solution of two dimensional reaction diffusion equations is presented these methods may use time linearization and yield linearly implicit techniques and one dimensional operators in each direction it is shown that if the factorization errors are neglected linearly implicit approximate factorization methods provide uncoupled equations whereas if these errors are considered the equations are coupled and must be solved iteratively it is also shown that the allocation of the reaction and diffusion terms to the one dimensional operators plays a paramount role in determining the accuracy of approximate factorization methods and preserving the symmetry of the original differential problem iterative full and approximate factorization methods that do require iterations are also presented and for the problem considered here these methods are shown to converge in about two iterations and provide solutions in agreement with those obtained with linearly implicit full and approximate factorization techniques

the class group pairing and p descent on elliptic curves we give explicit formulae for the logarithmic class group pairing on an elliptic curve defined over a number field then we relate it to the descent relative to a suitable cyclic isogeny this allows us to connect the resulting selmer group with the logarithmic class group of the base these constructions are explicit and suitable for computer experimentation from a conceptual point of view the questions that arise here are analogues of visibility questions in the sense of cremona and mazur

on the existence of certain families of curves introduction 25 w 1 generalities on families of curves in ip 28 w the map 0 d 33 w 3 the number of moduli of a family 40 w 4 plane curves existence of families with the expected number of moduli 43 w 5 smoothing certain reducible curves in ip r r 3 47 w 6 curves in ip r 3 existence of families with the expected number of moduli 52 w curves in ip3 a more general existence theorem 55

calculation of manin s invariant for del pezzo surfaces for r 7 and 8 we consider an action of the weyl group of type e r on a unimodular lattice of rank r 1 we give the tables of the first cohomology groups for all cyclic subgroups of the weyl group with respect to this action these are important in the arithmetic theory of del pezzo surfaces

piecewise linearized and linearized θ methods for ordinary and partial differential equations lnitial and boundary value problems appear frequently in many branches of physics in this paper several numerical methods based on linearization techniques for solving these prob lems are reviewed first piecewise linearized methods and linearized 0 methods are considered for the solution of initial value problems in ordinary differential equations second piecewise linearized techniques for two point boundary value problems in ordinary differential equations are developed and used in conjunction with a shooting method in order to overcome the lack of convergence asso ciated with shooting piecewise linearized methods which provide piecewise analytical solutions and yield nonstandard finite difference schemes axe presented third methods of lines in either space or time for the solution of one dimensional convection reaction diffusion problems that transform the original problem into an initial or boundary value one are reviewed methods of lines in time that result in boundary value problems at each time step can be solved by means of the techniques described here whereas methods of lines in space that yield initial value problems and employ ei ther piecewise linearized techniques or linearized 0 methods in time are also developed finally for multidimensional problems approximate factorization methods are first used to transform the multi dimensional problem into a sequence of one dimensional ones which are then solved by means of the linearized and piecewise linearized methods presented here 2003 elsevier science ltd all rights reserved

multiple fibers on rational elliptic surfaces our main result theorem 0 1 classifies multiple fibers on rational elliptic surfaces over algebraically closed fields of arbitrary characteristic one result of this is the existence in positive characteristics of tame multiple fibers of additive type for several of the kodaira fiber types for which no examples were previously known 0 introduction we work in this paper over an algebraically closed field k of arbitrary characteristic p an elliptic surface f x b is a fibration of a complete smooth surface x over a complete smooth curve b such that the generic fiber is a smooth curve of genus one it is natural to wonder what kind of special fibers arise we answer this in the case of rational surfaces giving as a result examples of tame multiple fibers of additive type for every p o and for all such types except ib b 4 cf 5 to our knowledge the only example heretofore recognized is of multiplicity 2 and kodaira type iox due to katsura ka l1 although since our results were obtained we have seen a preprint of katsura and ueno ku giving other examples mostly for p 7 2 3 and kodaira type not ib b 0 any special fiber is an effective divisor on x and so can be written as mf menjcj where each curve cj is reduced and irreducible and the coefficients are positive integers with g c d nj 1 the positive integer m is the rnultiplicity of the fiber mf which is said to be a multiple fiber whenever m 1 the form of the divisor f enjcj is very restricted such a divisor is an indecomposable curve of canonical type mu a topological classification of all such curves has been given by kodaira ko cf 5 we refer to the classes which arise in this classification as the kodaira fiber types more generally the kodaira type of the fiber mf is mt where t is the type of f over an algebraically closed field k it is known that any multiple fiber mf of multiplicity m on a surface x is of the form mt where t is a kodaira type and m is divisible by and in characteristic 0 equal to the order of the normal bundle ox f s of e pic f of f in x but the extent to which these conditions are sufficient for there to occur on some elliptic surface a multiple fiber of type mt is not known of particular interest to us here are the tame multiple fibers of additive type these being fibers mf of multiplicity m where pic f is the additive group ga of the ground field k and m equals the order in ga f ox f s of these are possible only if p the characteristic of k is positive since ga only then has ptorsion but their occurrence is poorly understood received by the editors march 21 1987 1980 mathematzcs subject classzficatzon 1985 re szon primary 14j25 14m20 14d99

wild quotient singularities of surfaces let b mathcal m b be a noetherian regular local ring of dimension 2 with residue field b mathcal m b of characteristic p 0 assume that b is endowed with an action of a finite cyclic group h whose order is divisible by p associated with a resolution of singularities of mathrm spec b h is a resolution graph g and an intersection matrix n we prove in this article three structural properties of wild quotient singularities which suggest that in general one should expect when h mathbb z p mathbb z that the graph g is a tree that the smith group mathbb z n mathrm im n is killed by p and that the fundamental cycle z has self intersection z 2 le p we undertake a combinatorial study of intersection matrices n with a view towards the explicit determination of the invariants mathbb z n mathrm im n and z we also exhibit explicitly the resolution graphs of an infinite set of wild mathbb z 2 mathbb z singularities using some results on elliptic curves with potentially good ordinary reduction which could be of independent interest

the relation between polynomial deformations of sl 2 r and quasi exact solvability a general method based on the polynomial deformations of the lie algebra sl 2 r is proposed in order to exhibit the quasi exact solvability of specific hamiltonians implied by quantum physical models this method using the finite dimensional representations and differential realizations of such deformations is illustrated on the sextic oscillator as well as on second harmonic generation

heun equations and quasi exact solubility we consider the heun equations in the context of the quasi exactly solvable spectral problems and establish the conditions for this class of equations to admit algebraic solutions the schrodinger operators that can be associated with heun equations are constructed explicitly in some cases we present their supersymmetric partners we study a heun like system of equations which is relevant for the classical solution of the two dimensional abelian higgs model

quasi exactly solvable models in nonlinear optics we study a large class of models with an arbitrary finite number of degrees of freedom described by hamiltonians which are polynomial in bosonic creation and annihilation operators and including as particular cases nth harmonic generation and photon cascades for each model we construct a complete set of commuting integrals of motion of the hamiltonian fully characterize the common eigenspaces of the integrals of motion and show that the action of the hamiltonian on these common eigenspaces can be represented by a quasi exactly solvable reduced hamiltonian whose expression in terms of the usual generators of 2 is computed explicitly

third harmonic generation as a third order quasi exactly solvable system the two degrees of freedom in the usual third harmonic generation effective hamiltonian can be separated both classically and quantum mechanically into a harmonic oscillator and a nonlinear oscillator in turn the quantum hamiltonian of the nonlinear oscillator can be written as a cubic polynomial in the generators of a quasi exactly solvable lie algebra and the physically relevant eigenstates are precisely those that can be determined exactly although in general not explicitly since the standard jeffreys wentzel kramers brillouin methods are not easily applicable to general third order differential equations we use a bohr sommerfeld quantization of the classical orbits to obtain approximate explicit formulas for the corresponding quantum eigenvalues

maximal dimension of invariant subspaces admitted by nonlinear vector differential operators in this paper the dimension of invariant subspaces admitted by m component nonlinear vector differential operators is estimated it is shown that if the m component nonlinear vector differential operators of order k preserves the invariant subspace wn11 wnmm where wnqq is the space generated by solutions of linear ordinary differential equations of order nq q 1 m then max n1 nm 2mk 1 to illustrate the approach examples of nonlinear vector differential operators admitting invariant subspace wn11 wn22 with max n1 n2 9 and three component nonlinear vector differential operators admitting invariant subspace wn11 wn22 wn33 with max n1 n2 n3 13 are presented

semiclassical analysis of a quasi exactly solvable system second harmonic generation the usual second harmonic generation effective hamiltonian is shown to be equivalent to a one dimensional schrodinger operator with a sextic polynomial potential this operator belongs to the class of quasi exactly solvable models for which a finite number of eigenvalues and eigenvectors can be determined exactly a jeffreys wentzel kramers brillouin analysis provides accurate asymptotic expansions for these eigenvalues in the limit of large unperturbed energies

polynomial families and schrodinger equation one example for nonhypergeometric type of correspondence we discuss the explicit construction of the schrodinger equations admitting representation through some family of general nonorthogonal polynomials the specific choice of the third order polynomial coefficient functions that lead to quasi solvable families of schrodinger potentials is considered in detail

tridiagonalization and the heun equation it is shown that the tridiagonalization of the hypergeometric operator l yields the generic heun operator m the algebra generated by the operators l m and z l m is quadratic and a one parameter generalization of the racah algebra the racah heun orthogonal polynomials are introduced as overlap coefficients between the eigenfunctions of the operators l and m an interpretation in terms of the racah problem for su 1 1 and separation of variables in a superintegrable system are discussed

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

logarithmic good reduction of abelian varieties let k be a field which is complete for a discrete valuation we prove a logarithmic version of the neron ogg shafarevich criterion if a is an abelian variety over k which is cohomologically tame then a has good reduction in the logarithmic setting i e there exists a projective log smooth model of a over mathcal o k this implies in particular the existence of a projective regular model of a generalizing a result of kunnemann the proof combines a deep theorem of gabber with the theory of degenerations of abelian varieties developed by mumford faltings chai et al

exact solvability of superintegrable systems it is shown that all four superintegrable quantum systems on the euclidean plane possess the same underlying hidden algebra sl 3 the gauge rotated hamiltonians as well as their integrals of motion once rewritten in appropriate coordinates preserve a flag of polynomials this flag corresponds to highest weight finite dimensional representations of the sl 3 algebra realized by first order differential operators