In order to achieve gauge unification at the string scale M X ∼ 5 × 10 17 GeV in the minimal supersymmetric standard model (MSSM) it is necessary to add extra gauge non-singlet representations at an intermediate scale M I < M X , leading to a class of models which we refer to as MSSM+X models. We perform a detailed analysis of a large class of MSSM+X models and find that the number of (3,1) rep...

We consider classes of second order boundary value problems with a nonlinearity f (t, x) in the equations and subject to a multi-point boundary condition. Criteria are established for the existence of nontrivial solutions, positive solutions, and negative solutions of the problems under consideration. The symmetry of solutions is also studied. Conditions are determined by the relationship betwe...

Abstract. In this paper, we investigate the existence and controllability of mild solutions for a damped second order impulsive functional differential equation with state-dependent delay in Banach spaces. The results are obtained by using Sadovskii’s fixed point theorem combined with the theories of a strongly continuous cosine family of bounded linear operators. Finally, an example is provide...

Let X be a compact 2-manifold with nonempty boundary ∂X and let f : (X, ∂X) → (X, ∂X) be a boundary-preserving map. Denote by MF∂ [f ] the minimum number of fixed point among all boundary-preserving maps that are homotopic through boundary-preserving maps to f . The relative Nielsen number N∂(f) is the sum of the number of essential fixed point classes of the restriction f̄ : ∂X → ∂X and the num...

We compute the fixed point action of a properly defined renormaliza-tion group transformation for the Schwinger model through an expansion in the gauge field. It is local, with couplings exponentially suppressed with the distance. We check its perfection by computing the 1-loop mass gap at finite spatial volume, finding only exponentially vanishing cut off effects, in contrast with the standard...