We study the ground-state properties of hard-core bosons trapped by arbitrary confining potentials on one-dimensional optical lattices. A recently developed exact approach based on the Jordan-Wigner transformation is used. We analyze the large distance behavior of the one-particle density matrix, the momentum distribution function, and the lowest natural orbitals. In addition, the low-density l...

(a) Prove that [G,G]CG. (b) Prove that the quotient Gab := G/[G,G] (called the abelianization of G) is abelian. (c) If N CG is any normal subgroup such that G/N is abelian, prove that [G,G] ≤ N . (d) Put everything together to prove the universal property of abelianization: Given a homomorphism φ : G→ A to an abelian group A, there exists a unique homomorphism φ̄ := Gab → A such that φ = φ̄ ◦ π, ...

A geometric graph G is a simple graph G together with a straight line drawing of G in the plane with the vertices in general position. Two geometric realizations of a simple graph are geo-isomorphic if there is a vertex bijection between them that preserves vertex adjacencies and non-adjacencies, as well as edge crossings and non-crossings. A natural extension of graph homomorphisms, geo-homomo...

Graph homomorphism has been studied intensively. Given an m ×m symmetric matrix A, the graph homomorphism function is defined as

In this paper, we give a characterization of strongly Jordan zero-product preserving maps on normed algebras as a generalization of  Jordan zero-product preserving maps. In this direction, we give some illustrative examples to show that the notions of strongly zero-product preserving maps and strongly Jordan zero-product preserving maps are completely different. Also, we prove that the direct p...

We consider the question of the existence of homomorphisms between Gn,p and odd cycles when p = c/n, 1 < c ≤ 4. We show that for any positive integer l, there exists ε = ε(l) such that if c = 1 + ε then w.h.p. Gn,p has a homomorphism from Gn,p to C2l+1 so long as its odd-girth is at least 2l+1. On the other hand, we show that if c = 4 then w.h.p. there is no homomorphism from Gn,p to C5. Note t...

Let (M,ω) be a closed symplectic manifold and Ham(M,ω) the group of Hamiltonian diffeomorphisms of (M,ω). Then the Seidel homomorphism is a map from the fundamental group of Ham(M,ω) to the quantum homology ring QH∗(M ; Λ). Using this homomorphism we give a sufficient condition for when a nontrivial loop ψ in Ham(M,ω) determines a nontrivial loop ψ × idN in Ham(M ×N, ω ⊕ η), where (N, η) is a c...

Let R be an oriented compact surface without boundary of genus exceeding one, and let θ : π1(R;O) → Γ ⊂ PSL(2,C) be a homomorphism of its fundamental group onto a nonelementary group Γ of Möbius transformations. We present a complete, self-contained proof of the following facts: (i) θ is induced by a complex projective structure for some complex structure on R if and only if θ lifts to a homomo...