Primes Represented by Binary Cubic Forms

Let f(x, y) be a binary cubic form with integral rational coefficients,and suppose that the polynomial f(x, y) is irreducible in Q[x,y] and no prime divides all the coefficients of f. We provethat the set f Z⁽²⁾ contains infinitely many primes unless f(a,b) is even for each (a,b) in Z², in which case the set contains infinitely many primes. 2000Mathematical Subject Classification: primary 11N32; secondary11N36, 11R44.     

INDECOMPOSABILITY OF POLYNOMIALS AND RELATED DIOPHANTINE EQUATIONS

We present a new criterion for indecomposability of polynomialsover . Using the criterion, we obtain general finiteness resulton the polynomial Diophantine equation f(x) = g(y).     

High-Order Embedded Runge-Kutta-Nystrom Formulae

New efficient embedded Runge-Kutta-Nystrom processes of orders8(6) and 12(10) are presented for the numerical solution ofthe special second-order differential equation y'(x) = f[x,y(x)]. Test results indicate their improved efficiency relativeto other RKN formulae in current use.     

A Counterexample to Uniqueness in the Riemann Mapping Theorem for Univalent Harmonic Mappings

Let f be an orientation-preserving univalent harmonic mappingof the unit disk U. Then , where h and g are analytic in U. Furthermore, f satisfies theequation in U, where a(z)=g'(z)/h'(z), and |a(z)| < 1 in U. The functiona(z) is the analytic dilatation of f. In [2], Hengartner and Schober proved the following versionof the Riemann mapping theorem for univalent harmonic mappings.1991 Mathematics Subject Classification 31A05, 31A20.     

Increased Accuracy Cubic Spline Solutions to Two-Point Boundary Value Problems

The relation between finite difference approximation and cubicspline solutions of a two-point boundary value problem for thedifferential equation y' +f(x)y^'+g(x)y = r(x) has been consideredin a previous paper. The present paper extends the analysisto the integral equation formulation of the problem. It is shownthat an improvement in accuracy (local truncation error O(h⁶)rather than O(h⁴)) now results from a cubic spline approximationand that for the particular case f(x) 0 the resulting recurrencerelations have a form and accuracy similar to the well-knownNumerov formula. For this case also a formula with local truncationerror O(h⁸) is derived.     

A finite-difference algorithm for an inverse Sturm-Liouville problem

We study a method for approximating a potential q(x) in y(0)=y()=0 from finite spectral data. When the potential is symmetric,the data are the first M Dirichlet eigenvalues. In the generalcase, the first M terminal velocities are also specified. Acentred finite-difference scheme reduces the inverse Sturm-Liouvilleproblem to a matrix inverse eigenvalue problem. Our approachis motivated by the work of Paine, de Hoog and Anderssen, whoinvestigated the discrepancy between continuous and matrix eigenvaluesunder finite differences. Our modified Newton scheme is basedon choosing the number of interior mesh points in the discretizationto be 2M. The modified Newton scheme is shown to be convergentfor both the case of a symmetric and general potential. Somenumerical experiments are given. Supported in part by Institute for Scientific Computation,Texas A&M University.     

High-accuracy P-stable Methods for y' = f(t, y)

We obtain a one-parameter family of sixth-order P-stable methodsfor the numerical integration of periodic or near-periodic differentialequations that are defined by initial-value problems of theform: y^" = f(t, y), y(t₀)= y₀, y^'(t₀)= y^’₀. Our P-stablemethods are symmetric and involve three function evaluationsper step (periteration, in case f(t, y) is non-linear in y).For non-linear problems, starting values for the solution ofthe implicit equations by modified Newton's method are suggestedand illustrated by an example.     

Complex Dynamics of Convergence Acceleration

Generalized Steffensen methods are nonderivative algorithmsfor the computation of fixed points of a function f. They replacethe functional iteration Z_m+1=f(Z_m) with Z_m+1=F_n(Z_m, where F_nis explicitly provided for every n 1 as a quotient of two Hankeldeterminants. In this paper we derive rules pertaining to thelocal behaviour of these methods. Specifically, and subjectto analyticity, given that is a bounded fixed point of f, thenit is also a fixed point of F_n. Moreover, unless f'() vanishesor is a root of unity, becomes a superattractive fixed pointof F_n of degree n; if f'() is a root of unity of minimal degreeq2, then is (as a fixed point of F_n) superattractive of degreemin {q-1, n}; if f'()=1, then is attractive for F_n; and, finally,if is superattractive of degree s (as a fixed point of f),then it becomes superattractive of degree (s + 1)^n–1(ns+ s + 1)–1. Attractivity rules change at infinity (providedthat f()=). Broadly speaking, infinity becomes less attractivefor F_n, Since one is interested in convergence to finite fixedpoints, this further enhances the appeal of generalized Steffensenmethods.     

A comparative study of functional equations characterising sine and cosine

A comparative study of the functional equationsf(x+y)f(x–y)=f ²(x)–f ²(y),f(y){f(x+y)+f(x–y)}=f(x)f(2y) andf(x+y)+f(x–y)=2f(x){1–2f ²(y/2)} which characterise the sine function has been carried out. The zeros of the functionf satisfying any one of the above equations play a vital role in the investigations. The relation of the equationf(x+y)+f(x–y)=2f(x){1–2f ²(y/2)} with D'Alembert's equation,f(x+y)+f(x–y)=2f(x)f(y) and the sine-cosine equationg(x–y)=g(x)g(y) +f(x)f(y) has also been investigated.     

On solutions of a common generalization of the Go?a?b-Schinzel equation and of the addition formulae

Under some additional assumptions we determine solutions of the equation

f(x+M(f(x))y)=f(x)○f(y),     

The stability of Cauchy's gamma-beta functional equation

We obtain the super stability of Cauchy's gamma-beta functional equation

B(x,y)f(x+y)=f(x)f(y),     

On Second Order Bifurcations of Limit Cycles

The paper derives a formula for the second variation of thedisplacement function for polynomial perturbations of Hamiltoniansystems with elliptic or hyperelliptic Hamiltonians H(x, y)=y²–U(x)in terms of the coefficients of the perturbation. As an application,the conjecture stated by C. Chicone that a specific cubic systemappearing in a deformation of singularity with two zero eigenvalueshas at most two limit cycles is proved.     

Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay

In this note we propose a method for the integration of y'(t) = f(t, y(t), y(rt)), 0 t t_f y(0) = y₀, where 0 < r < 1, by a superconvengent s-stage continuousRK method of discrete global order p and continuous uniformorder q < p – 1 for the approximation of the delayedterm y(rt). We prove that, although the maximum attainable orderof the method on an arbitrary mesh is q' = min{p, q + 1}, byusing a quasi-geometric mesh, introduced by Bellen et al. (1997,Appl. Numer. Math. 24, 1997, 279–293), the optimal accuracyorder p is preserved.     

Finite difference methods for two-point boundary value problems involving high order differential equations

We discuss the construction of finite difference schemes for the two-point nonlinear boundary value problem:y ⁽²ⁿ⁾+f(x,y)=0,y ^(2j)(a)=A _2j ,y ^(2j)(b)=B _2j ,j=0(1)n–1,n2. In the case of linear differential equations, these finite difference schemes lead to (2n+1)-diagonal linear systems. We consider in detail methods of orders two, four and six for two-point boundary value problems involving a fourth order differential equation; convergence of these methods is established and illustrated by numerical examples.     

A sixth order tridiagonal finite difference method for non-linear two-point boundary value problems

We present a new sixth order finite difference method for the second order differential equationy=f(x,y) subject to the boundary conditionsy(a)=A,y(b)=B. An interesting feature of our method is that each discretization of the differential equation at an interior grid point is based onfive evaluations off; the classical second order method is based on one and the well-known fourth order method of Noumerov is based on three evaluations off. In case of linear differential equations, our finite difference scheme leads to tridiagonal linear systems. We establish, under appropriate conditions, the sixth order convergence of the finite difference method. Numerical examples are considered to demonstrate computationally the sixth order of the method.     

Solving two-point boundary value problems with spline functions

We consider a collocation method for the approximation of thesolution of the nonlinear two-point boundary value problem y'(x)=f(x,y(x)), y(a)=A, y(b)=B, using splines of degree m3. The methodwhich we shall use leads to a system of recurrence relationswhich can be solved by Newton's method. By obtaining asymptotic error bounds we verify a conjectureof Khalifa & Eilbeck, i.e. splines of even degree can giveeven better solutions than splines of odd degree in certaincases.     

A Fourth-order Tridiagonal Finite Difference Method for General Non-linear Two-point Boundary Value Problems with Mixed Boundary Conditions

We present a new fourth-order finite difference method for thegeneral second-order non-linear differential equation y^N = f(x,y, y') subject to mixed two-point boundary conditions. An interestingfeature of our method is that each discretization of the differentialequation at an interior grid point is based on just three evaluationsof f. We establish, under appropriate conditions, O(h⁴)-convergenceof the finite difference scheme. In the case of linear differentialequations, our finite difference scheme leads to tridiagonallinear systems. Numerical examples are considered to demonstratecomputationally the fourth order of the method.     

Sommes D'Exponentielles et Entiers Sans Grand Facteur Premier

Let S(x,y) be the set S(x,y)= 1 n x : P(n) y, where P(n) denotesthe largest prime factor of n. We study , where f is a multiplicative function. When f=1and when f=µ, we widen the domain of uniform approximationusing the method of Fouvry and Tenenbaum and making explicitthe contribution of the Siegel zero. Soit S(x,y) l'ensemble S(x,y)= 1 n x : P(n) y, désigne le plus grand facteur premier den. Nous étudions , lorsque f est une fonction multiplicative. Quand f=1 et quand f=µ,nous élargissons le domaine d'approximation uniformeenutilisant la méthode développée par Fouvryet Tenenbaum et en explicitant la contribution du zérode Siegel. 1991 Mathematics Subject Classification: 11N25, 11N99.     

A Fourth-order Tridiagonal Finite Difference Method for General Two-point Boundary Value Problems with Non-linear Boundary Conditions

We present a fourth-order finite difference method for the generalsecond-order nonlinear differential equation y^" = f(x, y, y‘)subject to non-linear two-point boundary conditions g₁(y(a), — y^’()) = 0, g₂(y(b), y'(b)) = 0. When both the differential equation and the boundary conditionsare linear, the method leads to a tridiagonal linear system.We show that the finite difference method is O(h⁴)-convergent.Numerical examples are given to illustrate the method and itsfourth-order convergence. The present paper extends the methodgiven in Chawla (1978) to the case of non-linear boundary conditions.     

Decomposer and associative functional equations

In the previous researches [2,3] b-integer and b-decimal parts of real numbers were introduced and studied by M.H. Hooshmand. The b-parts real functions have many interesting number theoretic explanations, analytic and algebraic properties, and satisfy the functional equation f (f(x) + y - f(y)) = f(x). These functions have led him to a more general topic in semigroups and groups (even in an arbitrary set with a binary operation [4] and the following functional equations have been introduced: Associative equations:

f(xf(yz))=f(f(xy)z),f(xf(yz))=f(f(xy)z)=f(xyz)