Hyers–Ulam stability of Sahoo–Riedel’s point

In this paper, we construct a counter example to show that “Theorem” of Hyers–Ulam Stability of Flett’s Point in [M. Das, T. Riedel, P.K. Sahoo, Hyers-Ulam stability of Flett’s points, Applied Mathematics Letters. 16 (3) (2003), 269–271] is incorrect. At the same time, we give the correct theorem and generalize it.     

Stationary points of O’Hara’s knot energies

In this article we study the regularity of stationary points of the knot energies E ^(α) introduced by O’Hara (Topology 30(2):241–247, 1991; Topol Appl 48(2):147–161, 1992; Topol Appl 56(1):45–61, 1994) in the range ${\alpha\in(2,3)}$ . In a first step we prove that E ^(α) is C ¹ on the set of all regular embedded curves belonging to ${{H^{(\alpha+1)/2,2}(\mathbb {R}{/}\mathbb {Z}, \mathbb {R}^n)}}$ and calculate its derivative. After that we use the structure of the Euler-Lagrange equation to study the regularity of stationary points of E ^(α) plus a positive multiple of the length. We show that stationary points of finite energy are of class C ^∞—so especially all local minimizers of E ^(α) among curves with fixed length are smooth.     

The number of integer points on Vinogradov’s quadric

An asymptotic formula is obtained for the number of integer solutions of bounded height on Vinogradov’s quadric. Two leading terms are determined, and a strong estimate for the error term is given.     

Complex Euler’s groups and values of Euler’s function at complex integer Gauss points

The complex Euler group is defined associating to an integer complex number z the multiplicative group of the complex integers residues modulo z, relatively prime to z. This group is calculated for z=(3+0i) ⁿ : it is isomorphic to the product of three cyclic group or orders (8, 3 ⁿ⁻¹ and 3 ⁿ⁻¹).     

The stability of set of generalized Ky Fan’s points

This paper is concerned with a generalized Ky Fan’s inequality. We first give an existence result of generalized Ky Fan’s (weak) efficient points, and then establish a complete metric space. Based on these results, we obtain the sufficient and necessary conditions of upper semicontinuity of efficient solution mapping to a generalized Ky Fan’s inequality. We also obtain the sufficient conditions of lower semicontinuity and continuity of efficient solution mapping to a generalized Ky Fan’s inequality. Our results are new and different from the corresponding ones in the literature.     

On the convergence of continued fractions at Runckel’s points

We consider the limit periodic continued fractions of Stieltjes type

Extraneous fixed points of Euler iteration and corresponding Sullivan’s basin

The phenomenon of “numerical extraneous roots” of Euler’s iteration has been found. By systematic searching, some polynomials and the corresponding initial values are given, which make the fixed points of Euler’s iteration not the roots of the polynomials. For those repelling extraneous fixed points, the adjoint dynamical types of Sullivan’s basins are also studied. Finally, the fractal pictures are produced.     

Chaplygin’s ball with a rotor: Non-degeneracy of singular points

A problem of a dynamically balanced asymmetric ball with a rotor rolling over a rough horizontal plane is considered. Earlier, A. Y. Moskvin constructed bifurcation diagrams of the momentum map and bifurcation complexes in order to study the dynamics of the system and to obtain singular solutions. A natural development of this research is a fine Liouville analysis of the system. The first step in this direction is presented in the paper, namely, we verify the non-degeneracy of singularities and describe the Liouville foliation in a neighborhood of singular points of the momentum map.     

Periodic points of algebraic functions and Deuring’s class number formula

The Ramanujan Journal - In this paper, transformation formulas for the function $$\begin{aligned} A_{1}\left( z,s:\chi \right) =\sum \limits _{n=1}^{\infty }\sum \limits _{m=1} ^{\infty }\chi...     

Surfaces à points doubles isolés

We give a characterization of the generic projection on P ² of an algebraic surface of P ³ with a finite number of nodes. The construction of an algebraic surface of P ³ with a given number of nodes is thus equivalent to the construction of a plane curve with nodes and cusps in some special position. Received: November 9, 1996     

Algebraic independence of the values ofE-functions at singular points and Siegel’s conjecture

We prove a generalization of Shidlovskii’s theorem on the algebraic independence of the values ofE-functions satisfying a system of linear differential equations that is well known in the theory of transcendental numbers. We consider the case in which the values ofE-functions are taken at singular points of these systems. Using the obtained results, we prove Siegel’s conjecture that, for the case of first-order differential equations, anyE-function satisfying a linear differential equation is representable as a polynomial in hypergeometricE-functions. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 174–190, February, 2000.     

Tight points of Pixley–Roy hyperspaces

In this paper, continuing with the study of tight spaces in ,  and , we investigate tight points of Pixley–Roy hyperspaces. We introduce the notion of a supertight space, and show that PR(X)$\mathit{PR}(X)$ is tight if and only if X is supertight.     

Estimating upper bounds on the limit points of majorizing sequences for Newton’s method

We present new sufficient conditions for the semilocal convergence of Newton’s method to a locally unique solution of an equation in a Banach space setting. Upper bounds on the limit points of majorizing sequences are also given. Numerical examples are provided, where our new results compare favorably to earlier ones such as Argyros (J Math Anal Appl 298:374–397, 2004), Argyros and Hilout (J Comput Appl Math 234:2993-3006, 2010, 2011), Ortega and Rheinboldt (1970) and Potra and Pták (1984).     

Kadec–Klee property and fixed points

Let X be a reflexive Banach space which does not have the Kadec–Klee property. Then there exists a compact mapping f   from the unit ball _BX$B_X$ of X   to the dual space X^?$X^{?}$ such that _infx∈BX‖f(x)‖>0${\inf }_{x\in B_X}\Vert f(x)\Vert >0$ and 〈f(x),x〉<‖f(x)‖$〈f(x),x〉<\Vert f(x)\Vert$ for every x∈_BX$x\in B_X$.     

Transseries and Todorov–Vernaeve’s asymptotic fields

We study the relationship between fields of transseries and residue fields of convex subrings of non-standard extensions of the real numbers. This was motivated by a question of Todorov and Vernaeve, answered in this paper.