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.     

A Simpler Proof of Mirzakhani’s Simple Curve Asymptotics

Maryam Mirzakhani (in her doctoral dissertation) has proved the author’s conjecture that the number of simple closed curves of length bounded by L on a hyperbolic surface S is asymptotic to a constant times L^d, where d is the dimension of the Teichmüller space of S. In this note we clarify and simplify Mirzakhani’s argument.     

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.     

Asymptotics of the solution of Laplace’s equation with mixed boundary conditions

A uniform asymptotic expansion of the solution of a two-dimensional elliptic problem with mixed boundary conditions is found. A physical application of the result is discussed.     

An Improvement of Ostrowski’s and King’s Techniques with Optimal Convergence Order Eight

In this paper, we first establish a new class of three-point methods based on the two-point optimal method of Ostrowski. Analysis of convergence shows that any method of our class arrives at eighth order of convergence by using three evaluations of the function and one evaluation of the first derivative per iteration. Thus, this order agrees with the conjecture of Kung and Traub (J. ACM 643–651, 1974) for constructing multipoint optimal iterations without memory. We second present another optimal eighth-order class based on the King’s fourth-order family and the first attained class. To support the underlying theory developed in this work, we examine some methods of the proposed classes by comparison with some of the existing optimal eighth-order methods in literature. Numerical experience suggests that the new classes would be valuable alternatives for solving nonlinear equations.     

Convergence of Rump’s method for computing the Moore-Penrose inverse

We extend Rump’s verified method (S.Oishi, K.Tanabe, T.Ogita, S.M.Rump (2007)) for computing the inverse of extremely ill-conditioned square matrices to computing the Moore-Penrose inverse of extremely ill-conditioned rectangular matrices with full column (row) rank. We establish the convergence of our numerical verified method for computing the Moore-Penrose inverse. We also discuss the rank-deficient case and test some ill-conditioned examples. We provide our Matlab codes for computing the Moore-Penrose inverse.     

The Global Bifurcation of a Cubic System

In this paper,we study the perturbation of certain of cubic system.By using the method of multi-parameter perturbation theory and qualitative analysis,we infer that the system under consideration can havefive limit cycles.     

The automorphisms of Petit’s algebras

Let σ be an automorphism of a field K with fixed field F. We study the automorphisms of nonassociative unital algebras which are canonical generalizations of the associative quotient algebras K[t;σ]∕fK[t;σ] obtained when the twisted polynomial f∈K[t;σ] is invariant, and were first defined by Petit. We compute all their automorphisms if σ commutes with all automorphisms in Aut_F(K) and n≥m?1, where n is the order of σ and m the degree of f, and obtain partial results for n<m?1. In the case where K∕F is a finite Galois field extension, we obtain more detailed information on the structure of the automorphism groups of these nonassociative unital algebras over F. We also briefly investigate when two such algebras are isomorphic.     

The converse of Schur’s theorem

Schur’s theorem states that for a group G finiteness of G/Z(G) implies the finiteness of G′. In this paper, we show the converse is true provided that G/Z(G) is finitely generated and in such case, we have |G/Z(G)| ≤ |G′| ^d(G/Z(G)). In the special case of G being nilpotent, we prove |G/Z(G)| divides |G′| ^d(G/Z(G)).     

The Topological Structure of Subseries Convergence

Let E be an abstract set, G be an Abelian topological group, FEG and be afamily of subsets of F. Let the sequence {xj}E be subseries aF convergent. In this paper,we present a sufficient and necessary condition for {xj} to be also subseries convergent in thetopology of uniform convergence on the sets in ,     

Global Asymptotics of the Second Painlevé Equation in Okamoto’s Space

We study the solutions of the second Painlevé equation (P II) in the space of initial conditions first constructed by Okamoto, in the limit as the independent variable, x, goes to infinity. Simultaneously, we study solutions of the related equation known as the thirty-fourth Painlevé equation (P 34). By considering degenerate cases of the autonomous flow, we recover the known special solutions, which are either rational functions or expressible in terms of Airy functions. We show that the solutions that do not vanish at infinity possess an infinite number of poles. An essential element of our construction is the proof that the union of exceptional lines is a repeller for the dynamics in Okamoto’s space. Moreover, we show that the limit set of the solutions exists and is compact and connected.     

Kantorovich-Like Convergence Theorems for Newton’s Method Using Restricted Convergence Domains

The convergence set for Newton’s method is small in general using Lipschitz-type conditions. A center-Lipschitz-type condition is used to determine a subset of the convergence set containing the Newton iterates. The rest of the Lipschitz parameters and functions are then defined based on this subset instead of the usual convergence set. This way the resulting parameters and functions are more accurate than in earlier works leading to weaker sufficient semi-local convergence criteria. The novelty of the paper lies in the observation that the new Lipschitz-type functions are special cases of the ones given in earlier works. Therefore, no additional computational effort is required to obtain the new results. The results are applied to solve Hammerstein nonlinear integral equations of Chandrasekhar type in cases not covered by earlier works.     

On the Convergence of Chebyshev’s Method for Multiple Polynomial Zeros

In this paper we investigate the local convergence of Chebyshev’s iterative method for the computation of a multiple polynomial zero. We establish two convergence theorems for polynomials over an arbitrary normed field. A priori and a posteriori error estimates are also provided. All of the results are new even in the case of simple zero.     

Convergence of Newton’s Method for Sections on Riemannian Manifolds

The present paper is concerned with the convergence problems of Newton’s method and the uniqueness problems of singular points for sections on Riemannian manifolds. Suppose that the covariant derivative of the sections satisfies the generalized Lipschitz condition. The convergence balls of Newton’s method and the uniqueness balls of singular points are estimated. Some applications to special cases, which include the Kantorovich’s condition and the γ-condition, as well as the Smale’s γ-theory for sections on Riemannian manifolds, are given. In particular, the estimates here are completely independent of the sectional curvature of the underlying Riemannian manifold and improve significantly the corresponding ones due to Dedieu, Priouret and Malajovich (IMA J. Numer. Anal. 23:395–419, 2003), as well as the ones in Li and Wang (Sci. China Ser. A. 48(11):1465–1478, 2005).     

The mystery of ten wooden blocks: Hjelmslev’s geometry of reality

Mathematische Semesterberichte - I&nbsp;shall argue that a&nbsp;collection of wood models at the University of Copenhagen was made by Johannes Hjelmslev as an aid in his elementary...