algebraic and differential operator equations abstract explicit expressions for solutions of boundary value problems and cauchy problems related to the operator differential equation x n a n 1 x n 1 a 0 x 0 are given in terms of solutions of the algebraic operator equation x n a n 1 x n 1 a 0 0 a method for solving this algebraic equation is studied

on a conjecture of voisin on the gonality of very general abelian varieties abstract we study orbits for rational equivalence of zero cycles on very general abelian varieties by adapting a method of voisin to powers of abelian varieties we deduce that for k at least 3 a very general abelian variety of dimension at least 2 k 2 has covering gonality greater than k this settles a conjecture of voisin we also discuss how upper bounds for the dimension of orbits for rational equivalence can be used to provide new lower bounds on other measures of irrationality in particular we obtain a strengthening of the alzati pirola bound on the degree of irrationality of abelian varieties

a simple proof that rational curves on k3 are nodal the picard group of s is generated by two effective divisors c and f with c 2 f 2 0 and c f 1 it can be realized as an elliptic fibration over p with a unique section c fibers f and 2 here c1 is the first chern class of the hodge bundle of the fibration s p see h m it is a standard result that the number of nodal fibers of an elliptic fibration are given by 12 h m p 158 so there are exactly 24 rational nodal curves in the linear series f for s general this is the same special k3 surface used by bryan and leung in their counting of curves on k3 surfaces b l it is actually the attempt to understand their method that leads us to our proof we will call a k3 surface with picard lattice 1 1 a bl k3 surface a bl k3 surface s lies on the boundary of the moduli space of k3 surfaces of genus g with c gf as the corresponding primitive divisor every curve in the linear series os c gf is totally reducible i e it consists of the 2 curve c and g elliptic tails attached to c a curve d os c gf is the image of a stable rational map only if d c m1f1 m2f2 m24f24 where f1 f2 f24 are 24

on the chow group of zero cycles of a generalized kummer variety abstract for a generalized kummer variety x of dimension 2n we will construct for each 0 i n some co isotropic subvarieties in x foliated by i dimensional constant cycle subvarieties these subvarieties serve to prove that the rational orbit filtration introduced by voisin on the chow group of zero cycles of a generalized kummer variety coincides with the induced beauville decomposition from the chow ring of abelian varieties as a consequence the rational orbit filtration is opposite to the conjectural bloch beilinson filtration for generalized kummer varieties

galois closure and lagrangian varieties abstract let x be a complex projective variety and consider the morphism k k h 0 x x 1 h 0 x x k we use galois closures of finite rational maps to introduce a new method for producing varieties such that k has non trivial kernel we then apply our result to the two dimensional case and we construct a new family of surfaces which are lagrangian in their albanese variety moreover we analyze these surfaces computing their chern invariants and proving that they are not fibred over curves of genus g 2 the topological index of these surfaces is negative and this provides a counterexample to a conjecture on lagrangian surfaces formulated in barja et al 2007 3

initializing generalized feedback shift register pseudorandom number generators the generalized feedback shift register pseudorandom number generators proposed by lewis and payne provide a very attractive method of random number generation unfortunately the published initialization procedure can be extremely time consuming this paper considers an alternative method of initialization based on a natural polynomial representation for the terms of a feedback shift register sequence that results in substantial improvements in the initialization process

some comments on computer implementation of random number generators abstract several workers in simulation studies are concerned with the correct and portable computer implementation of random number generators since approximate calculations may lead to accumulated errors in various machines all implementations must be exact in a previous work gentle this journal 1990 presented several implementations for multiplicative congruential generators the aim of this paper is to describe some more methods especially the implementation using the decomposition of the modulus is very efficient

random number generation with the recursion x t x t 3p x t 3q abstract a generalized feedback shift register gfsr algorithm proposed by lewis and payne 1973 uses a primitive trinomial to generate a sequence of pseudorandom numbers we propose a similar algorithm which uses a primitive polynomial with many nonzero terms but generates a number as fast as the original gfsr algorithm our sequence is guaranteed to be equidistributed in higher dimensions and to have a good autocorrelation property extensive statistical tests have been performed on the sequences generated by our algorithm and the results were quite satisfactory

alternative solution of the inhomogeneous linear differential equation of order two abstract in this paper we present an alternative method for solving the general inhomogeneous linear ordinary differential equation ode of order two the solution appears in the standard form as the sum of the solution of the equivalent homogeneous problem and the particular solution of the inhomogeneous problem at hand the main advantage of the method exposed herein is that the particular solution is computable from two different integrals this allows the problem solver to choose the simplest integral with which to work with in order to get the final solution for illustrative purposes we employ the method presented to aid in the solution of some example problems including the inhomogeneous klein gordon equation

interval oscillation theorems for asecond order linear differential equation interval oscillation criteria are given for the forced second order linear differential equation ly t p t y q t y t email protected 0 where p q are locally integrable functions and p t 0 for t 0 no restriction is imposed on t to be the second derivative of an oscillatory function as assumed by kartsatos 1 our results also allow both q and f to change sign in the neighborhood at infinity in particular we show that all solutions of y c sin t y t b cos t with b 0 are oscillatory for c 1 3448 this improves an estimate given by nasr 2 for the linear equation

revaluation of the first order upwind difference scheme to solve coarse grained master equations this study addresses the initial boundary value problem of coarse grained probability measure on the state space in which a differentiable vector field v is given and as a consequence the differenced continuity equation using the first order upwind difference scheme uds based on the finite volume method appears as the physical substance on the coarse grained dynamics at first the uds is theoretically shown to be equivalent to a class of coarse grained master equations cgme brought by a principle that we cannot distinguish state points in the same partition with each other the principle is based on the formulation of non equilibrium statistical mechanics to resolve the macroscopic irreversibility moreover the entropy production evaluated by the uds is also shown to be in accord with the average volume contraction rate in the steady state this is essential for the non equilibrium statistical dynamics and was numerically confirmed under the principle of coarse graining the uds is very superior to the conventional monte carlo method in computer time and storage and is very useful to solve the cgme

xiao s conjecture for general fibred surfaces we prove that the genus g the relative irregularity qf and the clifford index cf of a non isotrivial

rational curves and points on k3 surfaces we study the distribution of algebraic points on k3 surfaces 1 introduction let k be a field and k a fixed algebraic closure of k we are interested in connections between geometric properties of algebraic varieties and their arithmetic properties over k over its finite extensions k k or over k here we study certain varieties of intermediate type namely k3 surfaces and their higher dimensional generalizations calabi yau varieties to motivate the following discussion let 5 be a k3 surface over k in positive characteristic 5 may be unirational and covered by rational curves examples are supersingular k3 surfaces over fields of characteristic two or the surface

simulated division with approximate factoring for the multiple recursive generator with both unrestricted multiplier and non mersenne prime modulus abstract this paper focuses on devising a general and efficient way of generating random numbers for the multiple recursive generator with both unrestricted multiplier and non mersenne prime modulus we propose a new algorithm that embeds the technique of approximate factoring into the simulated division method the proposed new algorithm improves the decomposition method in terms of both the suitability for various word sizes of the computers and the efficiency characteristics such as the number of arithmetic operations required and the computational time empirical simulations are conducted to compare and evaluate the computational time of this algorithm with the decomposition method for various computers

an exhaustive analysis of two term multiple recursive random number generators with efficient multipliers this paper presents results of an exhaustive computer search for the maximum spectral value in full period two term kth order multiple recursive generators mrgs with efficient multipliers for orders k 2 3 7 we provide a table of kth order mrgs with largest spectral values these mrgs not only have the best spectral values found so far but also can be implemented efficiently by applying the approximate factoring method

density of rational curves on k3 surfaces using the dynamics of self rational maps of elliptic k3 surfaces together with deformation theory we prove that the union of rational curves is dense on a very general k3 surface and that the union of elliptic curves is dense in the 1st jet space of a very general k3 surface both in the strong topology

hyperelliptic jacobians and isogenies abstract in this note we mainly consider abelian varieties isogenous to hyperelliptic jacobians in the first part we prove that a very general hyperelliptic jacobian of genus g 4 is not isogenous to a non hyperelliptic jacobian as a consequence we obtain that the intermediate jacobian of a very general cubic threefold is not isogenous to a jacobian another corollary tells that the jacobian of a very general d gonal curve of genus g 4 is not isogenous to a different jacobian in the second part we consider a closed subvariety y a g of the moduli space of principally polarized varieties of dimension g 3 we show that if a very general element of y is dominated by the jacobian of a curve c and dim y 2 g then c is not hyperelliptic in particular if the general element in y is simple its kummer variety does not contain rational curves finally we show that a closed subvariety y m g of dimension 2 g 1 such that the jacobian of a very general element of y is dominated by a hyperelliptic jacobian is contained either in the hyperelliptic or in the trigonal locus

combined multiple recursive random number generators we analyze the random number generators obtained by combining two or more multiple recursive generators we study the lattice structure of such combined generators and argue that combination is a good way of obtaining robust generators based on a recurrence with many nonzero coefficients and which also possess a fast implementation

theoretical analyses of forward and backward heuristics of multiple recursive random number generators this article focuses on heuristic algorithms that are capable of finding multiple recursive generators mrgs with maximum spectral value criterion in a short period of time two promising algorithms provided by kao and tang are the forward and backward heuristics here we provide a deep understanding of the forward and backward heuristics for an mrg the forward heuristic finds a sequence of partial sets of multipliers that lead toward a better spectral value for the second order mrgs the performance of the backward heuristic theoretically outperforms that of the forward heuristic the effectiveness of the backward heuristic is numerically confirmed by the experiments we present

effective and efficient restriction on producing the multipliers for the multiple recursive random number generator abstract the most efficient and easiest way for implementing a multiple recursive random number generator is the approximate factoring af method in the literature the af method can be applied in the three types of restriction on multiplier this paper considers the effectiveness and efficiency of these restrictions from the theoretical and empirical points of view the most effective and efficient restriction is presented the number of arithmetic operations required by the af method is also discussed

boundary value problems and cauchy problems for the second order euler operator differential equation abstract explicit solutions of boundary value problems and cauchy problems for the euler operator differential equation are given in terms of solutions of algebraic operator equations

on the nonhomogeneous second order euler operator differential equation explicit solutions abstract a variation of parameters method for solving the operator differential equation t 2 x 2 ta 1 x 1 a 0 x f t is presented in terms of an appropriate pair of solutions of the algebraic operator equation z 2 a 1 i z a 0 0 under this hypothesis existence and uniqueness conditions for two point boundary value problems as well as an explicit expression for the solutions in terms of data are given

good parameters and implementations for combined multiple recursive random number generators combining parallel multiple recursive sequences provides an efficient way of implementing random number generators with long periods and good structural properties such generators are statistically more robust than simple linear congruential generators that fit into a computer word we made extensive computer searches for good parameter sets with respect to the spectral test for combined multiple recursive generators of different sizes we also compare different implementations and give a specific code in c that is faster than previous implementations of similar generators

boundary value problems for second order differential operator equations abstract it is proved that the resolution problem of an operator boundary value problem for a second order differential operator equation with constant coefficients is solved in terms of solutions of certain algebraic operator equations explicit expressions of solutions are given

constructing rational curves on k3 surfaces we develop a mixed characteristic version of the morimukai technique for producing rational curves on k3 surfaces we reduce modulo p produce rational curves on the resulting k3 surface over a finite field and lift to characteristic zero as an application we prove that all complex k3 surfaces with picard group generated by a class of degree two have an infinite number of rational curves

thoughts on pseudorandom number generators abstract much of the informal discussion at the workshop concerned the merits of different pseudorandom number generators here we record some comments based on comparing generators across a wide range of machines

complete second order differential equations in banach spaces with dynamic boundary conditions abstract in this paper we deal with the mixed initial boundary value problem for complete second order in time linear differential equations in banach spaces in which time derivatives occur in the boundary conditions general wellposedness theorems are obtained for the first time which are used to solve the corresponding inhomogeneous problems examples of applications to initial boundary value problems for partial differential equations are also presented

matrices and the structure of random number sequences abstract this note provides a short self contained treatment using linear algebra and matrix theory for establishing maximal periods underlying structure and choice of starting values for shift register and lagged fibonacci random number generators

explicit solutions for second order operator differential equations with two boundary value conditions ii abstract boundary value problems for second order operator differential equations with two boundary value conditions are studied explicit expressions of the solutions in terms of data problems are given by means of the application of algebraic techniques analogous expressions to the ones known for the scalar case are obtained

several extensively tested multiple recursive random number generators abstract the multiple recursive generator mrg has been considered by many scholars as a very good random number rn generator this paper applies a sequential search to identify the mrgs of orders one two and three which are able to produce rns with good lattice structure in terms of the spectral value and beyer quotient to detect departures from local randomness and homogeneity extensive statistical tests including runs auto correlation chi square serial and the sparse occupancy tests have been conducted in approximately 19 3 billion candidates only four mrgs namely 1280550 45991 0 45991 1758790 885300443 0 1552858447 and 885300443 1546795921 598295599 have passed all the theoretical and empirical tests among which 0 45991 1758790 can be implemented efficiently by applying the approximate factoring method and is therefore most recommended

counting rational curves on text k3 surfaces the aim of these notes is to explain the remarkable formula found by yau and zaslow y z to express the number of rational curves on a k3 surface projective k3 surfaces fall into countably many families fg g 1 a surface in fg admits a gdimensional linear system of curves of genus g a naive count of constants suggests that such a system will contain a positive number say n g of rational highly singular curves the formula is

solving algebraic and differential riccati operator equations abstract by application of the riesz dunford functional calculus a method for solving generalized algebraic riccati operator equations in hilbert space is given results are applied to the problem of finding explicit solutions for cauchy problems and two point boundary value problems for generalized riccati operator differential equations

rational orbits on three symmetric products of abelian varieties let a be an n dimensional abelian variety n 2 let ch 0 a be the group of zero cycles of a modulo rational equivalence by regarding an effective degree k zero cycle as a point on s k a the k symmetric product of a and by considering the associated rational equivalence class we get a map s k a ch 0 a whose fibres are called orbits for any n 2 in this paper we determine the maximal dimension of the orbits when k 2 or 3 it is respectively 1 and 2 and the maximal dimension of families of orbits moreover for generic a we get some refinements and in particular we show that if dim a 4 s 3 a does not contain any orbit note that it implies that a generic abelian four fold does not contain any trigonal curve

asymptotic iteration technique for second order dynamic equations on time scales an asymptotic iteration method for solving second order homogeneous linear dynamic equation of the form y a0 t y b0 t y is introduced where a0 t and b0 t are crdm t functions and the positive integer m depends on the differentiability of a0 t b0 t and iteration steps

multiplicity of solutions of second order linear differential equations on networks abstract an upper optimal bound is obtained for the multiplicity of solutions of second order linear differential equations on networks with general boundary and node conditions this bound depends only on the abstract topological associated graph in particular the result holds for the multiplicity of eigenvalues of the corresponding operators

computer implementation of random number generators abstract good theoretical work on algorithms is often defeated by a poor computer implementation this is particularly true of random number generators for two reasons the overwhelming concern for speed and the inherent ill conditioning of the problem the emphasis on speed has often resulted in shortcuts see carta 1990 these shortcuts often accept approximations which can be disastrous in random number generation other shortcuts are very machine specific and then often get used in the wrong environment the problem of random number generation using a deterministic recursion is ill conditioned in the sense that small deviations relative errors of the order of 10 10 can cause complete loss of accuracy this is why any approximation is disastrous in this paper we will concentrate on the coding considerations for a special case of a good algorithm namely a multiplicative congruential generator with properly chosen multiplier and modulus the discussion will use fortran constructs but it is applicable to any programming language we will also address implementation of recursive generators on vector processors