Abstract:
In the 1-parameter family of Berger spheres S^3(a), a > 0 (S^3(1) is the round 3-sphere of radius 1) we classify the stable constant mean curvature spheres, showing that in some Berger spheres (a close to 0) there are unstable constant mean curvature spheres. Also, we classify the orientable compact stable constant mean curvature surfaces in S^3(a), 1/3 <= a < 1 proving that they are spheres or the minimal Clifford torus in S^3(1/3). This allows to solve the isoperimetric problem in these Berger spheres.

Abstract:
In this work, we study the stability of Hopf vector fields on Lorentzian Berger spheres as critical points of the energy, the volume and the generalized energy. In order to do so, we construct a family of vector fields using the simultaneous eigenfunctions of the Laplacian and of the vertical Laplacian of the sphere. The Hessians of the functionals are negative when they act on these particular vector fields and then Hopf vector fields are unstable. Moreover, we use this technique to study some of the open problems in the Riemannian case.

Abstract:
We construct compact arbitrary Euler characteristic orientable and non-orientable minimal surfaces in the Berger spheres. Besides we show an interesting family of surfaces that are minimal in every Berger sphere, characterizing them by this property. Finally we construct, via the Daniel correspondence, new examples of constant mean curvature surfaces in the products S^2 x R, H^2 x R and in the Heisenberg group with many symmetries.

Abstract:
The fact that the capacitance coefficients for a set of conductors are geometrical factors is derived in most electricity and magnetism textbooks. We present an alternative derivation based on Laplace's equation that is accessible for an intermediate course on electricity and magnetism. The properties of Laplace's equation permits to prove many properties of the capacitance matrix. Some examples are given to illustrate the usefulness of such properties.

Abstract:
We introduce the new difference sequence space (Δ) . Further, it isproved that the space (Δ) is the BK-space including the space , which is the space of sequences of pboundedvariation. We also show that the spaces (Δ), and ？ are linearly isomorphic for 1≤<∞. Furthermore, the basis and the -, - and -duals of the space (Δ) are determined. We devotethe final section of the paper to examine some geometric properties of the space (Δ).

Abstract:
We study the negative gradient flow of the spinorial energy functional (introduced by Ammann, Wei{\ss}, and Witt) on 3-dimensional Berger spheres. For a certain class of spinors we show that the Berger spheres collapse to a 2-dimensional sphere. Moreover, for special cases, we prove that the volume-normalized standard 3-sphere together with a Killing spinor is a stable critical point of the volume-normalized version of the flow.

Abstract:
In the present paper we survey the most recent classification results for proper biharmonic submanifolds in unit Euclidean spheres. We also obtain some new results concerning geometric properties of proper biharmonic constant mean curvature submanifolds in spheres.

Abstract:
The main purpose of this paper is to introduce modular structure of the sequence space defined by Altay and Ba ar (2007), and to study Kadec-Klee (H) and uniform Opial properties of this sequence space on K the sequence spaces.

Abstract:
The main purpose of this paper is to introduce modular structure of the sequence space defined by Altay and Ba ar (2007), and to study Kadec-Klee ( ) and uniform Opial properties of this sequence space on K the sequence spaces.

Abstract:
In this paper, we analyze the stability of the real-valued Maxwell-Bloch equations with a control that depends on state variables quadratically. We also investigate the topological properties of the energy-Casimir map, as well as the existence of periodic orbits and explicitly construct the heteroclinic orbits.