For a normal variety X defined over an algebraically closed field with an action of the multiplicative group T = Gm, we consider the “hyperbolic localization” functor D(X)→ D(X ), which localizes using closed supports in the directions flowing into the fixed points, and compact supports in the directions flowing out. We show that the hyperbolic localization of the intersection cohomology sheaf ...

Nontopological spatially localized time periodic excitations coined discrete breathers are generic solutions for lattice Hamiltonians. I present recent results including the spatial decay of discrete breathers in systems with interactions decaying algebraically in space, the properties of the static deformation accompanying a breather excitation in low-symmetry lattice Hamiltonians with Goldsto...

Let d > 1, and let (X,α) and (Y, β) be two zeroentropy Z-actions on compact abelian groups by d commuting automorphisms. We show that if all lower rank subactions of α and β have completely positive entropy, then any measurable equivariant map from X to Y is an affine map. In particular, two such actions are measurably conjugate if and only if they are algebraically conjugate.

In this paper we introduce the cone bounded linear mapping and demonstrate a proof to show that the cone norm is continuous. Among other things, we prove the open mapping theorem and the closed graph theorem in TVS-cone normed spaces. We also show that under some restrictions on the cone, two cone norms are equivalent if and only if the topologies induced by them are the same. In the sequel, we...

‎Let $L$ be a Lie algebra‎, ‎$mathrm{Der}(L)$ be the set of all derivations of $L$ and $mathrm{Der}_c(L)$ denote the set of all derivations $alphainmathrm{Der}(L)$ for which $alpha(x)in [x,L]:={[x,y]vert yin L}$ for all $xin L$‎. ‎We obtain an upper bound for dimension of $mathrm{Der}_c(L)$ of the finite dimensional nilpotent Lie algebra $L$ over algebraically closed fields‎. ‎Also‎, ‎we classi...

In this paper we consider Z-actions, d ≥ 1, by automorphisms of compact connected abelian groups which contain at least one expansive automorphism (such actions are called algebraic Z-actions of expansive rank one). If α is such a Z-action on an infinite compact connected abelian group X, then every expansive element α of this action has a dense group ∆αn(X) of homoclinic points. For different ...