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Abstract: In this paper, we consider the existence and multiplicity of nodal solutions of semilinear elliptic equations. We prove that a semilinear elliptic equation in large domains does not admit any least energy nodal (sign-changing) solution and in an upper half strip with -holes has at least 2-nodal solutions. [Copyright &y& Elsevier]

We give necessary and sufficient conditions that a polynomial f ( x, y) gives an ovoid of PG(3, q). Furthermore by considering Rédei polynomials associated to ovoids of PG(3, q) we obtain sets of equations relating the coefficients of f ( x, y). One such set of equations implies that if xiyj occurs as a term in f ( x, y) + xy then i and j must have disjoint binary expansions. [ABSTRACT FROM AUTHOR]

We calculate the Hochschild cohomology of the algebra of smooth functions on a finite-dimensional real torus with coefficients in the adjoint representation, generalizing the previously developed technique to the discrete case for this. [ABSTRACT FROM AUTHOR]

We study a new class of infinite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates the name automorphic Lie algebras. For automorphic Lie algebras we present bases in which they are quasigraded and all structure constants can be written out explicitly. These algebras have useful factorisations on two subalgebras similar to the factorisation of the current algebra on the positive and negative parts. [ABSTRACT FROM AUTHOR]