in this paper, we have dened and studied a generalized form of topological vectorspaces called s-topological vector spaces. s-topological vector spaces are dened by using semi-open sets and semi-continuity in the sense of levine. along with other results, it is provedthat every s-topological vector space is generalized homogeneous space. every open subspaceof an s-topological vector space is ...

Let G be a locally compact abelian group with Haar measure, and Y measure space. Suppose that H is reproducing kernel Hilbert space of functions on $$G\times Y$$ , such naturally embedded into $$L^2(G\times Y)$$ invariant under the translations associated elements G. Under some additional technical assumptions, we study W*-algebra $${\mathcal {V}}$$ translation-invariant bounded linear operator...

We consider random translation-invariant frustration-free quantum spin Hamiltonians on $\mathbb Z^D$ in which the nearest-neighbor interaction every direction is randomly sampled and then distributed across lattice. Our main result that, under a small rank constraint, are automatically they gapped with positive probability. This extends previous results 1D chains to all dimensions. The argument...