Given a homomorphism of commutative noetherian rings R → S and an S–module N , it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals sup {m ∈ Z | Torm(E,N) 6= 0}, where E is the injective hull of the residue field of R. This re...

Although Kirby and Siebenmann [13] showed that there are manifolds which do not admit PL structures, the possibility remained that all manifolds could be triangulated. In the late seventies Galewski and Stern [10] constructed a closed 5–manifold M 5 so that every n–manifold, with n 5, can be triangulated if and only if M 5 can be triangulated. Moreover, M 5 admits a triangulation if and only if...

Recently, the theory of dense graph limits has received attention from multiple disciplines including graph theory, computer science, statistical physics, probability, statistics, and group theory. In this paper we initiate the study of the general structure of differentiable graphon parameters F . We derive consistency conditions among the higher Gâteaux derivatives of F when restricted to the...

Solution: For the action mentioned in the hint, the stabilizer of the coset eH (e is the identity) is H since gH = H implies that g ∈ H. The kernel K of the homomorphism of G into the permutation group of the cosets is a normal subgroup of G which is the intersection of stabilizers of all cosets. In particular the kernel K of the above homomorphism is a subgroup of H. Its index in G is equal to...

Motivated by the Alday-Gaiotto-Tachikawa (AGT) conjecture, we construct a homomorphism from affine super Yangian [Formula: see text] to universal enveloping algebra of rectangular W-superalgebra for all m ≠ n, m, n ≥ 2 or 3, = 0. Furthermore, show that image this is dense, provided k + ( − n)( l 1)

(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

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...