Allan L. Edelson

All 2-manifolds have finitely many minimal triangulations

A triangulation of a 2-manifoldM is said to be minimal provided one cannot produce a triangulation ofM with fewer vertices by shrinking an edge. In this paper we prove that all 2-manifolds have finitely many minimal triangulations. It follows that all triangulations of a given 2-manifold can be generated from the minimal triangulations by a process called vertex splitting.

Linear and semilikear eigenvalue problems in Rn

We consider the problem for , with respect to the existence and asymptotic behavior of solutions as For the linear, radially symmetric problem with , the number of zeros and asymptotic behavior of the solutions is known as λ> 0 varies. In this paper we study the linear equation as well as bounded and sublinear perturbations of the linear equation. For the nonsyrnmetric, linear and semilinear equations we investigate the structure of the set of X for which the solutions are positive and decay to zero. Unlike the symmetric linear case, these results cannot be derived directly from the fundamental structure theorem for linear ordinary differential equations. We work with topological methods and comparison theorems.

ALL ORIENTABLE 2-MANIFOLDS HAVE FINITELY MANY MINIMAL TRIANGULATIONS

We show that for every orientable 2-manifold there is a finite set of triangulations from which all other triangulations can be generated by sequences of vertex splittings.

Asymptotic behaviour of higher order semilinear equations

Equation (1.1) is said to be sublinear if 0 c y < 1, and singular if 1 < y < 0. Because of the lack of a maximum principle for higher order equations, methods which have been successfully applied to equations of order 2 are not available for equation (1 .l). This includes, most importantly, the method of sub and super solutions, or barriers. As a consequence, ordinary differential equation methods have been used, and the results have generally been applicable only to radially symmetric equations. In their study of the semilinear biharmonic equation

THE ASYMPTOTIC FORM OF NONOSCILLATORY SOLUTIONS TO FOURTH ORDER EQUATIONS

Publisher Summary This chapter presents the asymptotic form of the nonoscillatory solutions of the nonlinear fourth-order equation. It presents a study strongly motivated by Kusano and Naito, who obtained the asymptotic form of the maximal nonoscillatory solutions of the equation (r(t)y”)“ + yF(t, y) = 0, and by Edelson and Kreith, who found qualitative relationships between oscillation and the asymptotic behavior of nonoscillatory solutions. The chapter presents a few results that determine the possible forms of a nonoscillatory solution of a nonlinear fourth-order equation. The chapter focuses on only eventually positive solutions, the case of eventually negative solutions being analogous, and presents an assumption where y(t) > 0 for t > 0.

Asymptotically linear elliptic equations

SuntoIn questo lavoro si studiano le proprietà di biforcazione delle soluzioni minimali di un’equazione ellittica semilineare. In particolare si presenta una rassegna di risultati noti quando il secondo membro dell’equazione Δu=f(λ,x, u) ha un comportamento sublineare o asintoticamente lineare nella variabileu. Si dimostra poi un teorema di esistenza, unicità e biforcazione quando la crescita dif(., .,u) è sublineare inu=0 e asintoticamente lineare inu=∞.

VARIATIONAL SYSTEMS DISCONJUGACY CRITERIA FOR WNSELFADJOINT DIFFERENTIAL EQUATIONS

Abstract A systems representation for fourth order differential equations is used to develop variational criteria for the existence and nonexistence of systems conjugate points. The novelty of the results is due to the fact that techniques usually restricted to selfadjoint equations are extended to certain non-self adjoint problems.