On the definition of the cross-product

In this note it is shown how, starting with the geometric definition of cross-product, one can prove the distributivity of cross-multiplication over vector addition, thereby recovering the usual algebraic definition of cross-product. This approach gives direct access to geometric problems and enhances geometric and conceptual understanding.     

Limit distribution of a roundoff error

Let [X] and {X} be the integer and the fractional parts of a random variable X. The conditional distribution function F_n(x)=P({X}≤x|[X]=n) for an integer n is investigated. F_n for a large n is regarded as the distribution of a roundoff error in an extremal event. For most well-known continuous distributions, it is shown that F_n converges as n→∞ and three types of limit distributions appear as the limit distribution according to the tail behavior of F.     

On the definition of bolic spaces

We discuss notions of bolic metric spaces introduced by Kasparov and Skandalis. We make some observations which clarify the involved metric axioms and explain all the known basic examples.     

On the definition of Hilbert space

The projection theorem expresses a central feature of classical Hilbert space. Do other infinite dimensional sesquilinear spaces share this property? We show here that this is not the case for several prominent candidates; in particular Kalish's p-adic Hilbert spaces, Springers non archimedean normed spaces, the positive definite spaces over ordered fields. This yields interesting characterizations of classical Hilbert space.     

On the definition of Bourgain points

This paper is devoted to study of the so-called Bourgain points (B-points) for functions from L^∞(ℝ). In 1993, Bourgain showed that for a real-valued bounded function f, the set E_f of B-points is everywhere dense and has the maximal Hausdorff dimension, dim _H(E_f ) = 1; in addition, the vertical variation of the harmonic extension of f to the upper half-plane is finite at B-points. An essentially simpler definition of B-points is given compared to that in the original works by Bourgain. A geometric characterization of B-points for Cantor-like sets is obtained. Bibliography: 7 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 219–236.     

On the definition of purity of modules

In the paper the coincidence is established of the concepts of purity and bitriangular purity in module theory, introduced in the book of A. P. Mishina and L. A. Skornyakov, Abelian Groups and Modules.Translated from Matematieheskie Zametki, Vol. 11, No. 4, pp. 375–380, April, 1972.In conclusion, the author wishes to thank D. K. Faddeev for interest in this work.     

On the definition of a Lachlan semilattice

We study the algorithmic properties of the semilattices introduced in 1972 by Lachlan in his work on recursively enumerable m-degrees, the so-called Lachlan semilattices. We show that in Lachlan’s definition the effectivity condition on the meet can be omitted in the sequence determining such a semilattice.     

On the definition of word hyperbolic groups

Formal languages based on multiplication tables of finitely generated groups are investigated and used to give a linguistic characterization of word hyperbolic groups. Received: 19 October 2000 / in final version: 6 June 2001 / Published online: 1 February 2002     

On fuzzy tree definition

The connectedness and acyclicity levels are presented for a fuzzy graph. Several fuzzy tree definitions are given too, and their relations with previous studies on the matter are analyzed.     

On the definition of isomorphisms of linear spaces

Let (P, ) and (P, ) be linear spaces satisfying the exchange axiom with dim P=dim P . Then a bijection :PP which maps collinear points onto collinear points is an isomorphism. Also a surjection :PP which maps any three non-collinear points to non-collinear points is an isomorphism. This assertion is not true if dim P is not finite.