Note on a versatile Liapunov functional: applicability to an elliptic equation

A novel, very effective Liapunov functional was used in previous papers to derive decay and asymptotic stability estimates (with respect to time) in a variety of thermal and thermo‐mechanical contexts. The purpose of this note is to show that the versatility of this functional extends to certain non‐linear elliptic boundary value problems in a right cylinder, the axial co‐ordinate in this context replacing the time variable in the previous one. A steady‐state temperature problem is considered with Dirichlet boundary conditions, the condition on the boundary being independent of the axial co‐ordinate. The functional is used to obtain an estimate of the error committed in approximating the temperature field by the two‐dimensional field induced by the boundary condition on the lateral surface. Copyright © 2002 John Wiley & Sons, Ltd.     

Note on a Fermat-type diophantine equation

Let p>5 be a prime number and ζ a pth root of unity. Let c be an integer divisible only by primes of the form kp−1,(k,p)=1.Let C_p⁽ⁱ⁾ be the eigenspace of the p-Sylow subgroup of ideal class group C of corresponding to ωi,ω being the Teichmuller character.In this article we extend the main theorem in Sitaraman (J. Number Theory 80 (2000) 174) and get the following: For any fixed odd positive integer n<p−4, assume:

(a)

At least one of C_p⁽³⁾,C_p⁽⁵⁾,…,C_p⁽ⁿ⁾ is non-trivial.

(b)

C_p⁽ⁱ⁾=0 for p−n−1?i?p−2.

(c)

for 1?i?n+1.

Let q be an odd prime such that , and such that there is a prime ideal Q over q in whose ideal class is of the form I^pJ where J is non-trivial, not a pth power and J∈C_p⁽³⁾⊕C_p⁽⁵⁾⊕?⊕C_p⁽ⁿ⁾.For such p and q, if x^p+y^p=pcz^p has a non-trivial solution , with (x,y,z)=1, then .Let t(n)=n²²⁴ⁿ⁴. If , then applying a result of Soulé (J. Reine Angew. Math. 517 (1999) 209), we show that the above result holds with only condition (a) because the others are automatically satisfied.We also make a remark about the effect of Soulé's result on the p-divisibility of h_p⁺ (the class number of the maximal real subgroup of ) which is relevant to the existence of integral solutions to x^p+y^p=pcz^p.     

Note on a remarkable superposition for a nonlinear equation

We give a simple proof of--and extend--a superposition principle for the equation div, discovered by Crandall and Zhang. An integral representation comes as a byproduct. It follows that a class of Riesz potentials is -superharmonic.

     

Note on primitive permutation groups and a diophantine equation

It is shown that there are no transitive rank 3 extensions of the projective linear groups H, PSL(m,q) ? H ? PFL(m,q), for any prime power q and integer m ? 3. In the course of the proof the diophantine equation 5^m + 11 = x^p2, where m, x are positive integers, arose. As such equations can now be solved completely we had the choice of using number theory or geometry to complete the proof.     

A remark on a logarithmic functional equation

We revisit the logarithmic functional equation of Heuvers and Kannappan [K.J. Heuvers, Pl. Kannappan, A third logarithmic functional equation and Pexider generalizations, Aequationes Math. 70 (2005) 117-121] and give a simple proof of the result and discuss the locally integrable solutions of the equation.     

Hyperstability of a functional equation

The aim of this paper is to prove that the parametric fundamental equation of information is hyperstable on its open as well as on its closed domain, assuming that the parameter is negative. As a corollary of the main result, it is also proved that the system of equations that defines the alpha-recursive information measures is stable.     

An alternative Cauchy functional equation on a semigroup

More than 33 years ago M. Kuczma and R. Ger posed the problem of solving the alternative Cauchy functional equation ${f(xy) - f(x) - f(y) \in \{ 0, 1\}}$ where ${f : S \to \mathbb{R}, S}$ is a group or a semigroup. In the case when the Cauchy functional equation is stable on S, a method for the construction of the solutions is known (see Forti in Abh Math Sem Univ Hamburg 57:215–226, 1987). It is well known that the Cauchy functional equation is not stable on the free semigroup generated by two elements. At the 44th ISFE in Louisville, USA, Professor G. L. Forti and R. Ger asked to solve this functional equation on a semigroup where the Cauchy functional equation is not stable. In this paper, we present the first result in this direction providing an answer to the problem of G. L. Forti and R. Ger. In particular, we determine the solutions ${f : H \to \mathbb{R}}$ of the alternative functional equation on a semigroup ${H = \langle a, b| a^2 = a, b^2 = b \rangle }$ where the Cauchy equation is not stable.