where k is the first Betti number b^M), T is a flat riemannian λ -torus, M~ is a compact connected Ricci-flat (n — λ;)-manifold, and Ψ is a finite group of fixed point free isometries of T x M' of a certain sort (Theorem 4.1). This extends Calabi's result on the structure of compact euclidean space forms ([7] see [20, p. 125]) from flat manifolds to Ricci-flat manifolds. We use it to essentiall...

1.1. We denote by V a connected ^-dimensional complete Riemannian manifold, by d = d(V) the diameter of V, and by c = c(V) and c~ = c~(V), respectively, the upper and lower bounds of the sectional curvature of V. We set c = c(V) = max (| c1, | c~ |). We say that F i s ε-flat, ε > 0, if cd < ε. 1.2. Examples. a. Every compact flat manifold is ε-flat for any ε > 0. b. Every compact nil-manifold p...

It is proved that if $\varphi\colon A\to B$ a local homomorphism of commutative noetherian rings, nonzero finitely generated $B$-module $N$ whose flat dimension over $A$ at most $\mathrm{edim}\, A - \mathrm{edim}\, B$, free $B$, and $\varphi$ special type complete intersection. This result motivated by "patching method" developed Taylor Wiles, conjecture de Smit, the first author, dealing with ...

(1.1) |f(a)− f(b)| ≤ L |a− b| for every pair of points a, b ∈ A. We also say that a function is Lipschitz if it is L-Lipschitz for some L. The Lipschitz condition as given in (1.1) is a purely metric condition; it makes sense for functions from one metric space to another. In these lectures, we concentrate on the theory of Lipschitz functions in Euclidean spaces. In Section 2, we study extensio...

We point out the similarities and differences between cylindrical and disk photonic shells separating regions of flat spacetime. PACS numbers: 04.20.-q , 04.20.Jb, 04.40.-b

Given a presentation of a finitely presented group, there is a natural way to represent the group as the fundamental group of a 2-complex. The first part of this paper demonstrates one possible way to represent a finitely presented algebra S in a similarly compact form. From a presentation of the algebra, we construct a quiver with relations whose path algebra is finite dimensional. When we adj...

A rectangular rod is placed in a flow field flowing parallel to a flat plate. Effect of chord-thickness ratio of rectangular rod on developing vortex shedding downstream to the rod is studied. Then, for each one of the aspect ratios, the distance of the rod from the neighboring flat plate is reduced until the rod sticks to the flat plate. In each case, the effect of the flat plate boundary laye...

We study solutions to the static vacuum Einstein equations on domains of the form M ' R \B with prescribed Bartnik data (γ,H) on the inner boundary ∂M . It is proved that for any smooth boundary data (γ,H) close to standard round data on the unit sphere (γ+1, 2), there exists a unique asymptotically flat solution of the static vacuum Einstein equations realizing the boundary data (γ,H) which is...

Objectives: The aim of this study was to determine the prevalence of flat foot in a population of 7-12 year old students and to investigate the relationship between flat foot and age and sex. Methods: In this cross-sectional study, a total of 945 students (460 girls and 485 boys) were examined. The presence of flatfoot and the degree of its severity according to Tachdjian's system of grading...