Uniaxial extension of linear flexible-chain polymers in an extremely broad range of stresses and strain rates

Summary This work summarizes the basic patterns of strainstrength behaviour of linear flexible-chain polymers at uniaxial extension. The boundaries of transition of non-cross-linked polymers under isothermal conditions from the fluid to the high-elastic and leathery states when an axial load is applied are defined. A relationship is established between the parameters of longitudinal and low-amplitude strains. A study is made into the effect of molecular parameters and liquid media on the deformation and rupture of non-cross-linked linear polymers.With 12 figures     

Rank One Perturbations in a Pontryagin Space with One Negative Square

Let N₁ denote the class of generalized Nevanlinna functions with one negative square and let N_1, 0 be the subclass of functions Q(z)∈N₁ with the additional properties lim_y→∞ Q(iy)/y=0 and lim sup_y→∞ y |Im Q(iy)|<∞. These classes form an analytic framework for studying (generalized) rank one perturbations A(τ)=A+τ[·, ω] ω in a Pontryagin space setting. Many functions appearing in quantum mechanical models of point interactions either belong to the subclass N_1, 0 or can be associated with the corresponding generalized Friedrichs extension. In this paper a spectral theoretical analysis of the perturbations A(τ) and the associated Friedrichs extension is carried out. Many results, such as the explicit characterizations for the critical eigenvalues of the perturbations A(τ), are based on a recent factorization result for generalized Nevanlinna functions.     

On classes of maps which preserve finitisticness

We shall prove the following: Let be a refinable map between paracompact spaces. Then is finitistic if and only if is finitistic. Let be a hereditary shape equivalence between metric spaces. Then if is finitistic, is finitistic.

     

Convexity numbers of closed sets in

For 2$"> let be the -ideal in generated by all sets which do not contain equidistant points in the usual metric on . For each 2$"> a set is constructed in so that the -ideal which is generated by the convex subsets of restricted to the convexity radical is isomorphic to . Thus is equal to the least number of convex subsets required to cover -- the convexity number of .

For every non-increasing function \aleph_0\}$"> we construct a model of set theory in which for each . When is strictly decreasing up to , uncountable cardinals are simultaneously realized as convexity numbers of closed subsets of . It is conjectured that , but never more than , different uncountable cardinals can occur simultaneously as convexity numbers of closed subsets of . This conjecture is true for and .     

Universal Central Extensions of Lie Groups

We call a central Z-extension of a group G weakly universal for an Abelian group A if the correspondence assigning to a homomorphism ZA the corresponding A-extension yields a bijection of extension classes. The main problem discussed in this paper is the existence of central Lie group extensions of a connected Lie group G which is weakly universal for all Abelian Lie groups whose identity components are quotients of vector spaces by discrete subgroups. We call these Abelian groups regular. In the first part of the paper we deal with the corresponding question in the context of topological, Fréchet, and Banach–Lie algebras, and in the second part we turn to the groups. Here we start with a discussion of the weak universality for discrete Abelian groups and then turn to regular Lie groups A. The main results are a Recognition and a Characterization Theorem for weakly universal central extensions.     

The Method of Approximative Extension of Mappings in the Theory of Extensors

We develop the method of approximative extension of mappings which enables us not only to simplify the proofs of many available theorems in the theory of extensors but also to obtain a series of new results. Combined with Ancel' theory of fiberwise trivial correspondences, this method leads to considerable progress in the characterization of absolute extensors in terms of local contractivity. We prove the following assertions: Suppose that a space X is represented as the union of countably many closed ANEM-subspaces X _i and a countably dimensional subspace D: 1. If each X _i is a strict deformation neighborhood retract of X and XLC, then XANE. 2. If XLEC then XANE.     

Fibers in Ore Extensions

Let R be a finitely generated commutative algebra over an algebraically closed field k and let A=R[t;,] be the Ore extension with respect to an automorphism and a -derivation . We view A as the coordinate ring of an affine noncommutative space X. The inclusion RA gives an affine map : XSpecR, and X is a noncommutative analogue of A ¹×SpecR. We define the fiber X _p of over a closed point pSpecR as a certain full subcategory ModX _p of ModA. The category ModX _p has the following structure. If p has infinite -orbit, then ModX _p is equivalent to the category of graded modules over the polynomial ring k[x] with degx=1. If p is not fixed by , but has finite -orbit, say of size n, then ModX _p is equivalent to the representations of the quiver à _n–1 with the arrows all going in the same direction. If p is fixed by , then ModX _p is equivalent to either Modk or Modk[x]. It is also shown that X is the disjoint union of the fibers X _p in a certain sense.     

Stability Zones in Whitney's Extension Problem for Ultradifferentiable Functions

We establish a connection between the growth rate of weight functions generating nonquasianalytic classes of ultradifferentiable functions of Beurling and Roumieu type and the validity of an analog of Whitney's extension theorem for these classes.     

Subsets of the Plane with Small Linear Sections and Invariant Extensions of the Two-Dimensional Lebesgue Measure

We consider some subsets of the Euclidean plane R ², having small linear sections (in all directions), and investigate those sets from the point of view of measurability with respect to certain invariant extensions of the classical Lebesgue measure on R ².     

A generalization of the Auslander-Nagata purity theorem

Let be a module finite extension of normal domains. We show that if is unramified in codimension one and if has finite projective dimension over , then is étale over . Our proof makes use of P. Roberts' New Intersection Theorem.