On simultaneous approximation to (α,α) with α+kα−1=0

We show that the modified Jacobi-Perron algorithm gives the best simultaneous approximation to (α,α²) with α³+kα−1=0. We claim the following facts:

(1)

the limit set of become an ellipse, where (p_n,q_n,r_n) is the nth convergent (p_n/q_n,r_n/q_n) of (α,α²) by the modified Jacobi-Perron algorithm,

(2)

the limit set of belongs to outside of the ellipse in (1).

     

On α-times integrated C-cosine functions and abstract Cauchy problem I

In this paper we first investigate some basic properties concerning nondegenerate α-times integrated C-cosine functions on a Banach space X, and then characterize their generator A in terms of the unique existence of strong solutions of the following abstract Cauchy problem: for t>0, u(0)=x, u^′(0)=y.     

On a condition for α-starlikeness

In this paper we give a sufficient condition for function to be α-starlike function and some of its applications. We use the techniques of convolution and differential subordinations.     

On the α-invariants of cubic surfaces with Eckardt points

In this paper, we show that the αm,2-invariant (introduced by Tian (1991) [27] and (1997) [29]) of a smooth cubic surface with Eckardt points is strictly bigger than . This can be used to simplify Tian's original proof of the existence of Kähler-Einstein metrics on such manifolds. We also sketch the computations on cubic surfaces with one ordinary double points, and outline the analytic difficulties to prove the existence of orbifold Kähler-Einstein metrics.     

Points with large α-depth

We show that for every ?>0 there exists an angle α=α(?) between 0 and π, depending only on ?, with the following two properties: (1) For any continuous probability measure in the plane one can find two lines ?₁ and ?₂, crossing at an angle of (at least) α, such that the measure of each of the two opposite quadrants of angle π−α, determined by ?₁ and ?₂, is at least . (2) For any set P of n points in general position in the plane one can find two lines ?₁ and ?₂, crossing at an angle of (at least) α and moreover at a point of P, such that in each of the two opposite quadrants of angle π−α, determined by ?₁ and ?₂, there are at least points of P.     

On condition numbers in hp-FEM with Gauss–Lobatto-based shape functions

Sharp bounds on the condition number of stiffness matrices arising in hp/spectral discretizations for two-dimensional problems elliptic problems are given. Two types of shape functions that are based on Lagrange interpolation polynomials in the Gauss–Lobatto points are considered. These shape functions result in condition numbers O(p) and $\text{O(p}\text{ln}\text{p)}$ for the condensed stiffness matrices, where p is the polynomial degree employed. Locally refined meshes are analyzed. For the discretization of Dirichlet problems on meshes that are refined geometrically toward singularities, the conditioning of the stiffness matrix is shown to be independent of the number of layers of geometric refinement.     

On the Legendre constant of α-continued fractions

We prove the existence of the Legendre constant of α-continued fractions for . Moreover, we give its upper and lower bounds. The proof is based on the coding between the α-continued fraction expansions and the regular continued fraction expansions.     

On S-asymptotically ω-periodic functions on Banach spaces and applications

This paper is devoted to the study of the class of continuous and bounded functions for which exists ω>0 such that limt→∞(f(t+ω)−f(t))=0 (in the sequel called S-asymptotically ω-periodic functions). We discuss qualitative properties and establish some relationships between this type of functions and the class of asymptotically ω-periodic functions. We also study the existence of S-asymptotically ω-periodic mild solutions of the first-order abstract Cauchy problem in Banach spaces.     

Composition-closed ?-groups of almost-piecewise-linear functions

In the present work, we shall construct some non-essential H-closed epireflections of W that are not comparable with any other known H-closed epireflections of W other than the divisible hull and the epicompletion. We show first that the free objects in any H-closed epireflective subcategory must be closed under composition (see Section 2 for a precise definition), and that any epic extension of a free W-object on n generators that is closed under composition is actually the free object on n generators in some H-closed epireflective subcategory of W. We then apply these results to certain ?-groups of almost-piecewise-linear Baire functions on R. By definition, a function f:R→R is almost-piecewise-linear if there is a finite point set S⊂R such that f is piecewise-linear on the complement of any neighborhood of S.     

Two results on φ-normal functions

In this paper, we obtain two results on φ-normal functions, which extend some related results due to Lappan, and Aulaskari–Rättyä.     

On relative ?0-stability of impulsive hybrid systems via perturbing Lyapunov functions

This paper obtains some sufficient conditions for the relative ?₀-stability of the two impulsive hybrid systems by using the method of perturbing Lyapunov functions.     

On (strong) α-favorability of the Wijsman hyperspace

The Banach-Mazur game as well as the strong Choquet game are investigated on the Wijsman hyperspace from the nonempty player's (i.e. α's) perspective. For the strong Choquet game we show that if X is a locally separable metrizable space, then α has a (stationary) winning strategy on X iff it has a (stationary) winning strategy on the Wijsman hyperspace for each compatible metric on X. The analogous result for the Banach-Mazur game does not hold, not even if X is separable, as we show that α may have a (stationary) winning strategy on the Wijsman hyperspace for each compatible metric on X, and not have one on X. We also show that there exists a separable 1st category metric space such that α has a (stationary) winning strategy on its Wijsman hyperspace. This answers a question of Cao and Junnila (2010) [6].     

Rings of continuous functions on σ-frames

This paper deals with the ?-rings RS of all real-valued continuous functions on a completely regular σ-frame. It shows that, in marked contrast with the situation for frames, any ?-ring homomorphism RS→RT results from a σ-frame homomorphism S→T. Further, it proves the analogue of this for integer-valued continuous functions and 0-dimensional σ-frames. In all, this demonstrates that the important classical difference between Alexandroff spaces and Tychonoff spaces with respect to the real-valued continuous functions carries over fully to the pointfree setting - indeed, it adds the integer-valued case which seems to be new in this context.     

On linear systems and τ functions associated with Lamé's equation and Painlevé's equation VI

Painlevé's transcendental differential equation PVI may be expressed as the consistency condition for a pair of linear differential equations with 2×2 matrix coefficients with rational entries. By a construction due to Tracy and Widom, this linear system is associated with certain kernels which give trace class operators on Hilbert space. This paper expresses such operators in terms of Hankel operators Γ? of linear systems which are realised in terms of the Laurent coefficients of the solutions of the differential equations. Let P(t,∞):L²(0,∞)→L²(t,∞) be the orthogonal projection; then the Fredholm determinant τ(t)=det(I−P(t,∞)Γ?) defines the τ function, which is here expressed in terms of the solution of a matrix Gelfand-Levitan equation. For suitable values of the parameters, solutions of the hypergeometric equation give a linear system with similar properties. For meromorphic transfer functions that have poles on an arithmetic progression, the corresponding Hankel operator has a simple form with respect to an exponential basis in L²(0,∞); so det(I−Γ?P(t,∞)) can be expressed as a series of finite determinants. This applies to elliptic functions of the second kind, such as satisfy Lamé's equation with ?=1.     

Interval functions related to geocze k-area †

The Geocze area theory for dimension two uses the geometrically simplest intervals on which area theory can he developed, These intervals are simple polygonal regions in the plane. For Geocze k-area with k > 2 it is shown in the present paper that the geometrically simplest interval arc not tue obvious generalizations of the- two dimensional case, that is, polyhedral regions in R^k which are topological K-cells. It is shown that the simplest interval is a polyhedral region in R^k whose complement is connected     

On the nonexistence of a relation between σ-left-porosity and σ-right-porosity

Given an arbitrarily weak notion of left-〈f〉$〈f〉$-porosity and an arbitrarily strong notion of right-〈g〉$〈g〉$-porosity, we construct an example of closed subset of R$\mathbb{R}$ which is not σ  -left-〈f〉$〈f〉$-porous and is right-〈g〉$〈g〉$-porous. We also briefly summarize the relations between three different definitions of porosity controlled by a function; we then observe that our construction gives the example for any combination of these definitions of left-porosity and right-porosity.     

On estimates of the density of Feynman-Kac semigroups of α-stable-like processes

Suppose that α∈(0,2) and that X is an α-stable-like process on Rd. Let F be a function on Rd belonging to the class Jd,α (see Introduction) and be _∑s?tF(Xs−,Xs), t>0, a discontinuous additive functional of X. With neither F nor X being symmetric, under certain conditions, we show that the Feynman-Kac semigroup defined by

     

Domb's numbers and Ramanujan-Sato type series for 1/π

In this article, we construct a general series for . We indicate that Ramanujan's -series are all special cases of this general series and we end the paper with a new class of -series. Our work is motivated by series recently discovered by Takeshi Sato.     

On a σ-ideal of compact sets

We recall from [T. Mátrai, Kenilworth, Proc. Amer. Math. Soc. 137 (3) (2009) 1115-1125] a Gδσ-ideal of compact subsets of ω² and prove that it is not Tukey reducible to the ideal . This result answers a question of S. Solecki and S. Todor?evi? in the negative.