On the definite integral of two confluent hypergeometric functions related to the Kampé de Fériet double series

We study the Kampé de Fériet double series $ F_{1:1;1}^{1:1;1 } $ through the solution to the associated first-order nonhomogeneous differential equation. We show that the integral of t ^β+l M(?; β; λt)M(?; β;?λt) over t ∈ [0, T], T ? 0, $ l=0,1,\ldots,\Re \beta +l>-1 $ , is a linear combination of functions $ F_{1:1;1}^{1:1;1 } $ . The integral is a generalization of a class of the so-called Coulomb integrals involving regular Coulomb wave functions.     

A reduction formula for the Kampé de Fériet function

A generalization is provided for a reduction formula for the Kampé de Fériet function due to Cvijovi?.     

Some reduction formulae for double power series and Kampé de Fériet functions

Application of suitable ₄F₃[1] summation theorems leads to a number of formulae expressing certain double power series with equal variables as single power series. These formulae yield as particular cases reduction formulae expressing Kampé de Fériet functions F_q:1:1^p:2:2[X,X] with certain parameter conditions in terms of either _p+3F_q+2[X] or _p+2F_q+1[X].     

A note on a generalization of Diliberto’s Theorem for certain differential equations of higher dimension

In the theory of autonomous perturbations of periodic solutions of ordinary differential equations the method of the Poincaré mapping has been widely used. For the analysis of properties of this mapping in the case of two-dimensional systems, a result first obtained probably by Diliberto in 1950 is sometimes used. In the paper, this result is (partially) extended to a certain class of autonomous ordinary differential equations of higher dimension.This research was supported by Grant No. 201/99/0295 of the Grant Agency of the Czech Republic.This revised version was published online in April 2005 with a corrected missing date string.     

A note on the probabilistic approach to Turán's problem

The Turán number T(n, l, k) is the smallest possible number of edges in a k-graph on n vertices such that every l-set of vertices contains an edge. Given a k-graph H = (V(H), E(H)), we let X_s(S) equal the number of edges contained in S, for any s-set S?V(H). Turán's problem is equivalent to estimating the expectation E(X_l), given that min(X_l) ≥ 1. The following lower bound on the variance of X_s is proved:

$\text{Var(X}{}_{\mathrm{s}}\text{)?mm}\text{n?2k}\text{s?k}\text{n}\text{s}{}^{\mathrm{?1}}\text{n}\text{k}{}^1$

, where m = |E(H)| and $\text{m}\text{= (}{}_{\mathrm{k}}{}^{\mathrm{n}}\text{) ? m}$. This implies the following: putting t(k, l) = lim_n→∞T(n, l, k)(_kⁿ)^?1 then t(k, l) ≥ T(s, l, k)((_k^s) ? 1)^?1, whenever s ≥ l > k ≥ 2. A connection of these results with the existence of certain t-designs is mentioned.     

A note on the Frank—Tardos bi-truncation algorithm for crossing-submodular functions

Recently, Frank and Tardos (1988) presented an algorithm, called the bi-truncation algorithm, for discerning whether a polyhedron defined by $$Q(b) = \{ x|x \in ^s ,\forall X \subseteq S:x(X) \le b(X),X(S) = b(S)\}$$ is nonempty, whereb″ is a crossing-submodular function on 2 ^s . We point out an error in their algorithm and show how it is corrected.     

A remark on the discriminant of Hill’s equation and Herglotz functions

We establish a link between the basic properties of the discriminant of periodic second-order differential equations and an elementary analysis of Herglotz functions. Some generalizations are presented using the language of self-adjoint extensions.     

A note on Lavi’s paper “A Ganzstellensatz for semi-algebraic sets and a boundedness criterion for rational functions”

This article is intended to indicate and discuss some errors in N. Lavi’s paper “A Ganzstellensatz for semi-algebraic sets and a boundedness criterion for rational functions”.     

A note on the equilibria of the unbounded traveler’s dilemma

When there is no upward limit on admissible claims, the traveler’s dilemma admits a continuum of symmetric mixed strategy equilibria in addition to the pure strategy equilibrium in which both players ask and obtain the minimum. The payoff of any of these equilibria exceeds the payoff of the pure strategy one and any claim represents an attainable payoff. If the distinction between a large and an unbounded action set is fuzzy, this result can explain some puzzling stylized facts on the behavior of experimental subjects in the game.     

A note on solutions of the Korteweg–de Vries hierarchy

olutions of the Korteweg–de Vries hierarchy are discussed. It is shown that results by Wazwaz [Wazwaz AM. Multiple-soliton solutions of the perturbed KdV equation. Commun Nonlinear Sci Simul 2010;15911:3270–73] are the well-known consequences of the full integrability for the Korteweg–de Vries hierarchy.     

A note on Fröberg's conjecture for forms of equal degrees

In this note we study ideals generated by generic forms in polynomial rings over any algebraicly closed field of characteristic zero. We prove for many cases that the $(d+k)$-th graded component of an ideal generated by generic forms of degree d has the expected dimension (given by dimension count). And as a consequence of our result, we obtain that ideals generated by several generic forms of degrees d usually have the expected Hilbert series. The precise form of this expected Hilbert series, in general, is known as Fröberg's conjecture.     

A note on pépin’s counter examples to the hasse principle for curves of genus 1

In a series of articles published in the C.R. Paris more than a century ago, T. PéPIN announced a list of “theorems” concerning the solvability of diophantine equations of the type ax⁴ +by ⁴ = z². In this article, we show how to prove these claims using the structure of 2-class groups of imaginary quadratic number fields. We will also look at a few related results from a modern point of view.     

A note on asymptotic behaviour of Mittag–Leffler functions

In this paper, we give a short note on the asymptotic behaviour of the two parameter Mittag–Leffler function. Useful results are collected for the reader and also explicit estimation formulas for this function are obtained which will play a role in existence and stability theory of fractional differential equations.