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Abstract: This paper focuses on a nonlinear equation from thin plate theory of the form . We obtain maximum principles for certain functions defined on the solution of this equation using P-functions or auxiliary functions of the types used by Payne [L.E. Payne, Some remarks on maximum principles, J. Anal. Math. 30 (1976) 421–433] and Schaefer [P.W. Schaefer, Solution, gradient, and laplacian bounds in some nonlinear fourth order elliptic equations, SIAM J. Math. Anal. 18 (1987) 430–434]. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence of curved front solution of a partial differential equation coming from a mathematical model of stroke. The equation is of reaction–diffusion type in a cylinder of radius R and of diffusion and absorption type outside of the cylinder. We prove the nonexistence of a travelling front when R is small enough and the existence if R is large enough using a recent energy method. We construct the travelling front as the limit in time of a solution with a well-chosen initial condition, in a travelling referential. [Copyright &y& Elsevier]

Abstract: It is conjectured by Erdős, Graham and Spencer that if with , then this sum can be decomposed into n parts so that all partial sums are ⩽1. This is not true for as shown by , , . In 1997, Sándor proved that Erdős–Graham–Spencer conjecture is true for . In this paper, we reduce Erdős–Graham–Spencer conjecture to finite calculations and prove that Erdős–Graham–Spencer conjecture is true for . Furthermore, it is proved that Erdős–Graham–Spencer conjecture is true if and no partial sum (certainly not a single term) is the inverse of an positive integer. [Copyright &y& Elsevier]

Abstract: The arithmetic Kodaira–Spencer class of the universal elliptic curve was introduced in [A. Buium, Differential modular forms, J. Reine Angew. Math. 520 (2000) 95–167]; its reduction mod p was explicitly computed by Hurlburt [C. Hurlburt, Isogeny covariant differential modular forms modulo p, Compos. Math. 128 (1) (2001) 17–34]. In this paper the complicated expression of Hurlburt is shown to be the unique solution of a simple partial differential equation subject to a certain initial condition and weight condition. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence of multiple positive solutions for the one-dimensional p-Laplacian subject to one of the following boundary conditions: or where , . By means of a fixed point theorem due to Avery and Peterson, sufficient conditions are obtained that guarantee the existence of at least three positive solutions. The interesting point is the nonlinear term f is involved with the first-order derivative explicitly. [Copyright &y& Elsevier]