Let Fn be a free group of rank n and F n the quotient group of Fn by the subgroup [Γn(3),Γn(3)][[Γn(2),Γn(2)],Γn(2)] where Γn(k) denotes the k-th subgroup of the lower central series of the free group Fn. In this paper, we determine the group structure of the graded quotients of the lower central series of the group F n by using a generalized Chen’s integration in free groups. Then we apply it ...

We describe hypergeometric functions of Deligne-Mostow type for open subsets of the configuration space of six points on P, induced from those for seven points on P. The seven point ball quotient example DM(2, 1) does not appear on Mostow’s original list [9], but does appear on Thurston’s corrected version [12]. We show that DM(2, 1) is a finite cover of the moduli space of cubic surfaces MC en...

We study the notion of a strongly normal sequence in the dual E∗ of a Banach space E. In particular, we prove that the following three conditions are equivalent: (1) E∗ has a strongly normal sequence, (2) (E∗, σ(E∗, E)) has a Schauder basic sequence, (3) E has an infinite-dimensional separable quotient.

Let (M,ω) be a connected, compact symplectic manifold equipped with a Hamiltonian S action. We prove that, as fundamental groups of topological spaces, π1(M) = π1(minimum) = π1(maximum) = π1(Mred), where Mred is the symplectic quotient at any value in the image of the moment map φ. Let (M,ω) be a connected, compact symplectic manifold equipped with a circle action. If the action is Hamiltonian,...

Proof. Define a map U → μn by u 7→ u/ū; this gives a well-defined map by exercise 2 of homework 5. Compose with the quotient μn → μn/μ 2 n, and denote the resulting map by φ : U → μn/μ 2 n. It is enough to show that ker φ = μnU , since μn/μ 2 n has order 2. Certainly μnU + ⊂ kerφ, since φ(ζu) = [ζ] = 0 in μn/μ 2 n for u + ∈ U and ζ ∈ μn. Conversely, suppose φ(u) = 0 for some u ∈ U : then u/ū = ...

The Autism Spectrum Quotient (AQ) has been used to define the 'broader' (BAP), 'medium' (MAP) and 'narrow' autism phenotypes (NAP). We used a new Italian version of the AQ to test if difference on AQ scores and the distribution of BAP, MAP and NAP in autism parents (n = 245) versus control parents (n = 300) were replicated in a Sicilian sample. Parents of children with autism spectrum condition...

In this paper, we first define an equivalence relation on the sequence space $\Sigma _{2}$. Then equip quotient set _{2}/_{\sim}$ with a metric $d_1$. We also determine isometry map between spaces $(\Sigma _{2}/_{\sim},d_1)$ and $([0,1],d_{eucl})$. Finally, investigate symmetry conditions respect to some points compare truncation errors for computations which is obtained by metrics $d_{eucl}$

Let Q be a finite acyclic valued quiver. We define bialgebra structure and an integration map on the Hall algebra associated to morphism category of projective representations Q. As application, we recover surjective homomorphism defined in [12], which realizes principal coefficient quantum cluster $$\mathcal {A}\,_q(Q)$$ as sub-quotient morphisms. Moreover, also Caldero–Chapoton formula, well ...

For a closed topological manifold M with dim(M) ≥ 5 the topological structure set S(M) admits an abelian group structure which may be identified with the algebraic structure group of M defined by Ranicki. If dim(M) = 2d−1, and the manifoldM is oriented and equipped with a map to the classifying space of a finite group G, then the reduced ρ-invariant defines a function ρ̃ : S(M)→ QR d Ĝ to a cert...

Let (M,ω,Φ) be a Hamiltonian T -space and let H ⊆ T be a closed Lie subtorus. Under some technical hypotheses on the moment map Φ, we prove that there is no additive torsion in the integral full orbifoldK-theory of the orbifold symplectic quotient [M//H ]. Our main technical tool is an extension to the case of moment map level sets the well-known result that components of the moment map of a Ha...