On a generalization of Kronecker’s limit formula

In this paper, we first generalize the Kronecker limit formula for a class of Epstein zeta functions using new approximation formulas. This enables us to derive some applications to the class number of quadratic imaginary number fields K and the period ratios of elliptic curves with complex multiplication.     

Some improvements of Ostrowski’s method

In this paper, we present some new variants of Ostrowski’s method with order of convergence eight. For each iteration the new methods require three evaluations of the function and one evaluation of its first derivative and therefore they have the efficiency index equal to 1.682. Numerical tests verifying the theory are also given.     

An adaptive finite element method in reconstruction of coefficients in Maxwell’s equations from limited observations

We propose an adaptive finite element method for the solution of a coefficient inverse problem of simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions in the Maxwell’s system using limited boundary observations of the electric field in 3D.     

Kronecker’s limit formula and the height of Euclidean lattices

The height \({h(\Lambda)}\) of a Euclidean lattice \({\Lambda}\) is defined as the derivative at the point 0 of the spectral zeta function of the associated flat torus \({{\bf R}^n /\Lambda}\). The quest of the lattices which realise a minimum of h is an open problem in dimension higher than 3. In this paper, we show that a lattice which realises a local minimum of the height is irreducible, i.e. it cannot be decomposed into the orthogonal sum of two non-zero sublattices. The proof is based on a formula found by Terras, which generalises Kronecker’s limit formula for the Epstein zeta function.     

Convergence field of Abel’s summation method

Let F(A) denote the set of all bounded sequences summable by Abel’s method. It is known, that F(A) is a linear subspace of the linear metric space (S, ρ) of all bounded sequences endowed with the sup metric. It is shown in [KOSTYRKO, P.: Convergence fields of regular matrix transformations 2, Tatra Mt. Math. Publ. 40 (2008), 143–147] that the convergence field of a regular matrix transformation is a σ-porous set. We show that F(A) is very porous in S.     

An hybrid method that improves the accessibility of Steffensen’s method

Steffensen’s method is known for its fast speed of convergence and its difficulty in applying it in Banach spaces. From the analysis of the accessibility of this method, we see that we can improve it by using the simplified secant method for predicting the initial approximation of Steffensen’s method. So, from both methods, we construct an hybrid iterative method which guarantees the convergence of Steffensen’s method from approximations given by the simplified secant method. We also emphasize that the study presented in this work is valid for equations with differentiable operators and non-differentiable operators.     

Rate of convergence of Euler’s approximations for SDEs with non-Lipschitz coefficients

We prove that Euler＇s approximations for stochastic differential equations driven by infinite many Brownian motions and with non-Lipschitz coefficients converge almost surely. Moreover, the rate of convergence is obtained.     

Extension of Euler’s method to parabolic equations

Euler generalized d’Alembert’s solution to a wide class of linear hyperbolic equations with two independent variables. He introduced in 1769 the quantities that were rediscovered by Laplace in 1773 and became known as the Laplace invariants. The present paper is devoted to an extension of Euler’s method to linear parabolic equations with two independent variables. The new method allows one to derive an explicit formula for the general solution of a wide class of parabolic equations. In particular, the general solution of the Black–Scholes equation is obtained.     

Fourier coefficients of half-integral weight cusp forms and Waring’s problem

Extending the approach of Iwaniec and Duke, we present strong uniform bounds for Fourier coefficients of half-integral weight cusp forms of level N. As an application, we consider a Waring-type problem with sums of mixed powers.     

Kantorovich’s majorants principle for Newton’s method

We prove Kantorovich’s theorem on Newton’s method using a convergence analysis which makes clear, with respect to Newton’s method, the relationship of the majorant function and the non-linear operator under consideration. This approach enables us to drop out the assumption of existence of a second root for the majorant function, still guaranteeing Q-quadratic convergence rate and to obtain a new estimate of this rate based on a directional derivative of the derivative of the majorant function. Moreover, the majorant function does not have to be defined beyond its first root for obtaining convergence rate results. The research of O.P. Ferreira was supported in part by FUNAPE/UFG, CNPq Grant 475647/2006-8, CNPq Grant 302618/2005-8, PRONEX–Optimization(FAPERJ/CNPq) and IMPA. The research of B.F. Svaiter was supported in part by CNPq Grant 301200/93-9(RN) and by PRONEX–Optimization(FAPERJ/CNPq).     

Lord Kelvin’s method of images in semigroup theory

We show that Lord Kelvin’s method of images is a way to prove generation theorems for semigroups of operators. To this end we exhibit three examples: a more direct semigroup-theoretic treatment of abstract delay differential equations, a new derivation of the form of the McKendrick semigroup, and a generation theorem for a semigroup describing kinase activity in the recent model of Kaźmierczak and Lipniacki (J. Theor. Biol. 259:291–296, 2009).     

An effective modification of He’s variational iteration method

It is well known that one of the advantages of He’s variational iteration method is the free choice of initial approximation. Therefore, in this paper, we use this advantage to propose a reliable modification of He’s variational iteration method. Indeed, this constructs an initial trial-function without unknown parameters, which is called the modified variational iteration method. Some of the nonlinear and linear equations are examined by the modified method to illustrate the effectiveness and convenience of this method, and in all cases, the modified technique performed excellently. The results reveal that the proposed method is very effective and simple and gives exact solutions. The modification could lead to a promising approach for many applications in applied sciences.     

Convergence ball and error analysis of Ostrowski-Traub’s method

Under the hypotheses that the second-order and third-order derivatives of a function are bounded, an estimate of the radius of the convergence ball of Ostrowski-Traub’s method is obtained. An error analysis is given which matches the convergence order of the method. Finally, two examples are provided to show applications of our theorem.     

New variants of Jarratt’s method with sixth-order convergence

In this paper, by using the two-variable Taylor expansion formula, we introduce some new variants of Jarratt’s method with sixth-order convergence for solving univariate nonlinear equations. The proposed methods contain some recent improvements of Jarratt’s method. Furthermore, a new variant of Jarratt’s method with sixth-order convergence for solving systems of nonlinear equations is proposed only with an additional evaluation for the involved function, and not requiring the computation of new inverse. Numerical comparisons are made to show the performance of the presented methods.     

Global convergence of Newton’s method on an interval

The solution of an equation f(x)= given by an increasing function f on an interval I and right-hand side , can be approximated by a sequence calculated according to Newtons method. In this article, global convergence of the method is considered in the strong sense of convergence for any initial value in I and any feasible right-hand side. The class of functions for which the method converges globally is characterized. This class contains all increasing convex and increasing concave functions as well as sums of such functions on the given interval. The characterization is applied to Keplers equation and to calculation of the internal rate of return of an investment project.An earlier version was presented at the Joint National Meeting of TIMS and ORSA, Las Vegas, May 7–9, 1990. Financial support from Økonomisk Forskningsfond, Bodø, Norway, is gratefully acknowledged. The author thanks an anonymous referee for helpful comments and suggestions.