Recall that a metric space M is said to be complete if every Cauchy sequence in M converges to a limit in M . Not all metric spaces are complete, but it is a fact that all metric spaces can be “completed”, in a way that preserves the essential structure of the metric space. If the space in question is a normed linear space this process completes the space to a Banach space, and an inner product...

1. The notion of a "curvature structure" was introduced in §8, Chapter 1 of [ l ] . In this note we shall consider some of its applications. The details will be presented elsewhere. Let (M, g) be a Riemann manifold. Whenever convenient, we shall denote the inner product defined by g, by ( ). DEFINITION. A curvature structure on (M, g) is a (1, 3) tensor field T such that, for any vector fields ...

Let M be a complete Riemannian manifold with a free cocompact Z-action. Let k(t, m1, m2) be the heat kernel on M . We compute the asymptotics of k(t, m1, m2) in the limit in which t → ∞ and d(m1, m2) ∼ √ t. We show that in this limit, the heat diffusion is governed by an effective Euclidean metric on R coming from the Hodge inner product on H(M/Z;R).

‎Let $R$ be commutative ring with identity and $M$ be an $R$-module‎. ‎The zero divisor graph of $M$ is denoted $Gamma{(M)}$‎. ‎In this study‎, ‎we are going to generalize the zero divisor graph $Gamma(M)$ to submodule-based zero divisor graph $Gamma(M‎, ‎N)$ by replacing elements whose product is zero with elements whose product is in some submodules $N$ of $M$‎. ‎The main objective of this pa...

Abstract. We provide a detailed development of a function valued inner product known as the bracket product and used effectively by de Boor, Devore, Ron and Shen to study translation invariant systems. We develop a version of the bracket product specifically geared to Weyl-Heisenberg frames. This bracket product has all the properties of a standard inner product including Bessel’s inequality, a...

Hermitian Matrices It is simpler to begin with matrices with complex numbers. Let x = a + ib, where a, b are real numbers, and i = √ −1. Then, x∗ = a− ib is the complex conjugate of x. In the discussion below, all matrices and numbers are complex-valued unless stated otherwise. Let M be an n× n square matrix with complex entries. Then, λ is an eigenvalue of M if there is a non-zero vector ~v su...

In this paper we consider contact CR-warped product submanifolds of the type $M = N_Ttimes_f N_perp$, of a nearly Kenmotsu generalized Sasakian space form $bar M(f_1‎, ‎f_2‎, ‎f_3)$ and by use of Hopf's Lemma we show that $M$ is simply contact CR-product under certain condition‎. ‎Finally‎, ‎we establish a sharp inequality for squared norm of the second fundamental form and equality case is dis...

It is well known that for the Hilbert space H the minimum value of the functional Fμ(f) = ∫ H ‖f−g‖2dμ(g), f ∈ H, is achived at the mean of μ for any probability measure μ with strong second moment on H. We show that the validity of this property for measures on a normed space having support at three points with norm 1 and arbitrarily fixed positive weights implies the existence of an inner pro...

A Hilbert C∗-module is a generalisation of a Hilbert space for which the inner product takes its values in a C∗-algebra instead of the complex numbers. We use the bracket product to construct some Hilbert C∗-modules over a group C∗-algebra which is generated by the group of translations associated with a wavelet. We shall investigate bracket products and their Fourier transform in the space of ...