New Solutions of Heun’s Equation

The Laplace transform and its inverse are used to obtain new solutions of Heuns equation (HE) in the form of readily computable five-fold series. These solutions are a considerable improvement on the extremely complicated solutions of perturbation type deduced by Exton [2].AMS Subject Classification (2000), 33E10, 33E10, 33E20, 33E30, 34AO53     

A note on Wilson’s functional equation

Let S be a semigroup, and \(\mathbb {F}\) a field of characteristic \(\ne 2\). If the pair \(f,g:S \rightarrow \mathbb {F}\) is a solution of Wilson’s \(\mu \)-functional equation such that \(f \ne 0\), then g satisfies d’Alembert’s \(\mu \)-functional equation.     

Knuth’s Coherent Presentations of Plactic Monoids of Type A

We construct finite coherent presentations of plactic monoids of type A. Such coherent presentations express a system of generators and relations for the monoid extended in a coherent way to give a family of generators of the relations amongst the relations. Such extended presentations are used for representations of monoids, in particular, it is a way to describe actions of monoids on categories. Moreover, a coherent presentation provides the first step in the computation of a categorical cofibrant replacement of a monoid. Our construction is based on a rewriting method introduced by Squier that computes a coherent presentation from a convergent one. We compute a finite coherent presentation of a plactic monoid from its column presentation and we reduce it to a Tietze equivalent one having Knuth’s generators.     

A Generalization of Strassen’s Functional LIL

Let X ₁,X ₂,… be a sequence of i.i.d. mean zero random variables and let S _n denote the sum of the first n random variables. We show that whenever we have with probability one, lim?sup? _n→∞|S _n |/c _n =α ₀<∞ for a regular normalizing sequence {c _n }, the corresponding normalized partial sum process sequence is relatively compact in C[0,1] with canonical cluster set. Combining this result with some LIL type results in the infinite variance case, we obtain Strassen type results in this setting.     

An Approximate Functional Equation for the Primitive of Hardy’s Function

A formula of Atkinson type for the primitive of Hardy’s function is generalized to the case where the lengths of the two sums involved in that formula vary in wide ranges.     

A Generalized Cubic Functional Equation

In this paper, we determine the general solution of the functional equation f1 （2x ＋ y） ＋ f2（2x - y） = f3（x ＋ y） ＋ f4（x - y） ＋ f5（x） without assuming any regularity condition on the unknown functions f1,f2,f3, f4, f5 ： R→R. The general solution of this equation is obtained by finding the general solution of the functional equations f（2x ＋ y） ＋ f（2x - y） = g（x ＋ y） ＋ g（x - y） ＋ h（x） and f（2x ＋ y） - f（2x - y） = g（x ＋ y） - g（x - y）. The method used for solving these functional equations is elementary but exploits an important result due to Hosszfi. The solution of this functional equation can also be determined in certain type of groups using two important results due to Szekelyhidi.     

On a Functional Equation Associated with Simpson’s Rule

In this paper, we determine the general solution of the functional equation $$f(x)-g(y)=(x-y)\lbrack h(x+y)+\psi (x)+\phi (y)\rbrack$$ for all real numbers x and y. This equation arises in connection with Simpson’s Rule for the numerical evaluation of definite integrals. The solution of this functional equation is achieved through the functional equation $$g(x)-g(y)=(x-y)f(x+y)+(x+y)f(x-y).$$      

A New Proof of Diophantine Equation ■

By using algebraic number theory and p-adic analysis method,we give a new and simple proof of Diophantine equation ■.     

A generalization of Knopp’s Observation on Ramanujan’s tau-function

The Ramanujan Journal - We establish a vast generalization of an observation made by Marvin Knopp half a century ago concerning the nonvanishing of Ramanujan’s tau-function.     

A New Proof of a TheoremAbout Wave Equation

In book [1], Professor C. D. Sogge proved the Theorem 2.2 (page 15) by the method presented by F. John. But I think we can also prove it by another method which is simpler and more direct than the original approach.     

A link between Wilson’s and d’Alembert’s functional equations

If \({f, g : G \to \mathbb{C}}\), f ≠ 0, is a solution of Wilson’s functional equation on a group G, then g is a d’Alembert function.     

A Different Version of Flett＇s Mean Value Theorem and an Associated Functional Equation

In this paper we present a mean value theorem derived from Flett‘s mean value theorem. It turns out that cubic polynomials have the midpoint of the interval as their mean value point. To answer what class of functions have this property, we consider a functional equation associated with this mean value theorem. This equation is then solved in a general setting on abelian groups.     

Finding Sum of Powers on Arithmetic Progressions with Application of Cauchy’s Equation

In this paper we introduce a method to find the sum of powers on arithmetic progressions by using Cauchy’s equation and obtain a general formula. Then we apply our results to show how to determine some other sums of powers and sums of products. Our results are more general than those in [9]. Finally we discuss the sum of powers on arithmetic progressions in commmutative rings with characteristic 2 and find ‘full polynomials’.     

A New Discrete Analogue of Pontryagin’s Maximum Principle

By introducing the concept of a γ-convex set, a new discrete analogue of Pontryagin’s maximum principle is obtained. By generalizing the concept of the relative interior of a set, an equality-type optimality condition is proved, which is called by the authors the Pontryagin equation.     

A New Lower Bound Based on Gromov’s Method of Selecting Heavily Covered Points

Boros and Füredi (for d=2) and Bárány (for arbitrary d) proved that there exists a positive real number c _d such that for every set P of n points in R ^d in general position, there exists a point of R ^d contained in at least $c_{d}\binom{n}{d+1}$ d-simplices with vertices at the points of P. Gromov improved the known lower bound on c _d by topological means. Using methods from extremal combinatorics, we improve one of the quantities appearing in Gromov??s approach and thereby provide a new stronger lower bound on c _d for arbitrary d. In particular, we improve the lower bound on c ₃ from 0.06332 to more than 0.07480; the best upper bound known on c ₃ being 0.09375.