A Lower Bound for {a+b:aA, bB, and P(a,b)≠0}

Let A and B be two finite subsets of a field . In this paper, we provide a non-trivial lower bound for {a+b:aA, bB, and P(a,b)≠0} where P(x,y) [x,y].     

A note on parameter free Π1‐induction and restricted exponentiation

We characterize the sets of all Π₂ and all $\mathcal {B}(\Sigma _{1})We characterize the sets of all Π₂ and all $\mathcal {B}(\Sigma _{1})$ (= Boolean combinations of Σ₁) theorems of IΠ^?₁ in terms of restricted exponentiation, and use these characterizations to prove that both sets are not deductively equivalent. We also discuss how these results generalize to n > 0. As an application, we prove that a conservation theorem of Beklemishev stating that IΠ^?_{n + 1} is conservative over IΣ^?_n with respect to $\mathcal {B}(\Sigma _{n+1})$ sentences cannot be extended to Π_{n + 2} sentences. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

A note on a result of Bochner and Conrey-Ghosh about the Selberg class

We provide new proofs of a theorem of Bochner and Conrey-Ghosh about the structure of the Selberg class S_d^\sharp {\cal S}_d^\sharp for 0 ￡ d0\leq d < 1.     

A note on ℝ‐linear GMRES for solving a class of ℝ‐linear systems

We prove that ?‐linear GMRES for solving a class of ?‐linear systems is faster than GMRES applied to the related ?‐linear systems in terms of matrix–vector products. Numerical examples are given to demonstrate the theoretical result. Copyright © 2011 John Wiley & Sons, Ltd.     

A note on a result of Skiba

A subgroup H of a group G is called weakly s-permutable in G if there is a subnormal subgroup T of G such that G = HT and H ∩ T ≤ H _sG , where H _sG is the maximal s-permutable subgroup of G contained in H. We improve a nice result of Skiba to get the following

Theorem. Let ? be a saturated formation containing the class of all supersoluble groups

Open image in new window

and let G be a group with E a normal subgroup of G such that G/E ∈ ?. Suppose that each noncyclic Sylow p-subgroup P of F*(E) has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| are weakly s-permutable in G for all p ∈ π(F*(E)); moreover, we suppose that every cyclic subgroup of P of order 4 is weakly s-permutable in G if P is a nonabelian 2-group and |D| = 2. Then G ∈ ?.

     

A note on unit and class number of real quadratic fields

Letp be an odd prime withp1 (mod 4) andQ(p) be the real quadratic field. Also let andh denote the fundamental unit and the class number of Q(p), respectively. The main purpose of this paper is to study the explicit expressions of ^h and ^2h , and to discuss the problems related to the conjecture of Ankeny-Artin-Chowla.     

A note on a result of Zolotarev and Bernstein

In this note we obtain error bounds to x²ⁿ⁺¹–x²ⁿ on [–1, 1] by polynomials of degree at most (2n–1). The result proved here improves and extends some of the known results of Zolotarev and Bernstein. The proof presented here is different (and simple) from the one adopted by Zolotarev and Bernstein.     

A note on a symmetry result for traveling waves in cylinders

We prove in this note that all bounded traveling waves, in cylinders, of some -dimensional viscous conservation laws are symmetric.

     

A note on a result of Kong about p-nilpotency

Monatshefte für Mathematik - We point out an error in the paper ‘On an extension of the Frobenius’ theorem about p-nilpotency of a finite group’ (Monatsh Math...     

A note on a class of extensions

An extensionR?T of commutative integral domains is called a Δ₀-extension, provided each intermediateR-module is actually an intermediate ring, and an extensionR?T is called quadratic if eacht∈T satisfies a monic quadratic polynomial overR. Our purpose is to investigate these extensions in the context of Prüfer domains.     

A note on a result by W. Hoeffding

The Hoeffding assertion about weak convergence of the distributions of the expected values of the order statistics based on i.i.d. observations is extended to the case of dependent sample data.     

A decomposition result for a class of polling models

Boxma and Groenendijk have shown that the workload in polling models decomposes into two independent variables. This paper demonstrates a different type of decomposition that has an explicit multi-dimensional form. This decomposition does not apply to all polling models, but does, for example, apply to models with constant switch-over times and either exhaustive or gated service disciplines. For such models, we show that the population of customers present in the system (represented by a vector indicating the number of customers at each queue) at key time points breaks into two independent subpopulations: (1) the population of customers present in the related model with zero switch-over times; (2) another population, which is particularly easy to analyze. This result has a number of theoretical and applied implications.     

A note on “Convergence of a Halpern-type iteration algorithm for a class of pseudo-contractive mappings”

In [C.O. Chidume, G. De Souza, Convergence of a Halpern-type iteration algorithm for a class of pseudocontractive mappings, Nonlinear Analysis (2007), doi:10.1016/j.na.2007.08.008], the authors proved a strong convergence result for strictly pseudo-contractive mappings using a Halpern-type iteration algorithm. However, the main result is not correct. In this note, we provide a counter-example to the theorem.     

A note on a class of stronglyA-stable methods

Formulae for a class ofA-stable quadrature methods, or equivalently a certain implicit Runge-Kutta scheme, are given. A short proof of the strongA-stability is presented.On leave of absence from Chalmers University of Technology, Göteborg, Sweden.