Units of measurement

It is shown that the symbols for the fundamental units of mechanics, namely length, time and mass, are capable of a meaningful interpretation as positive real parameters. Then a suitable parameter domain allows one to take the derived units into account. The formal manipulations usually carried out with symbols of physical quantities, involving real coefficients and units of different kinds affected by rational exponents, are consistent with that interpretation. The note is expected to serve as a well-motivated introduction to a deeper study of dimensional analysis.     

The modulus of matrix semigroups

We treat a result concerning the generator of the modulus semigroup of a strongly continuous semigroup acting on the product of Banach lattices with order continuous norm. Received: 28 March 2003     

The modulus of Rudin-Carleson extensions

We prove the followingTheorem. LetF be a closed subset of the unit circleT which has Lebesgue measure zero. Suppose thatp is a smooth positive function onT. GivenfC(F) which satisfies|f(s)|p(s) (sF) and a neighbourhoodU ofF there is an extension off in the disc algebra such that and .     

On the modulus of -convexity

In this paper, we prove that the moduli of W^*-convexity, introduced by Ji Gao [J. Gao, The W^*-convexity and normal structure in Banach spaces, Appl. Math. Lett. 17 (2004) 1381–1386], of a Banach space X and of the ultrapower of X itself coincide whenever X is super-reflexive. Moreover, we improve a sufficient condition for uniform normal structure of the space and its dual. This generalizes and strengthens the main results of [J. Gao, The W^*-convexity and normal structure in Banach spaces, Appl. Math. Lett. 17 (2004) 1381–1386].     

Extremal metrics and modulus

We give a new proof of Beurling's result related to the equality of the extremal length and the Dirichlet integral of solution of a mixed Dirichlet-Neuman problem.Our approach is influenced by Gehring's work in space. Also, some generalizations of Gehring's result are presented.     

On the generalized modulus

The generalized modulus has numerous applications in geometric function theory and analytic number theory. In this paper, we study the monotonicity and convexity of the generalized modulus and obtain sharp functional inequalities for this function. Our results are the extensions of some well-known inequalities of the modulus of the Grötzsch ring.     

Minimum of modulus of dirichlet multisequence

Conditions are established under which the following relation is satisfied:

On Sums of Units

It is shown that if R is a finitely generated integral domain of zero characteristic, then for every n there exist elements of R which are not sums of at most n units. This applies in particular to rings of integers in finite extensions of the rationals. On the other hand there are many infinite algebraic extensions of the rationals in which every integer is a sum of two units.     

Traces of Torsion Units

A conjecture due to Zassenhaus asserts that if G is a finite group then any torsion unit in ?G is conjugate in ?G to an element of G. Here, a weaker form of this conjecture is proved for some infinite groups.     

Combinatorial modulus and type of graphs

Let A be the 1-skeleton of a triangulated topological annulus. We establish bounds on the combinatorial modulus of a refinement A^′, formed by attaching new vertices and edges to A, that depend only on the refinement and not on the structure of A itself. This immediately applies to showing that a disk triangulation graph may be refined without changing its combinatorial type, provided the refinement is not too wild. We also explore the type problem in terms of disk growth, proving a parabolicity condition based on a superlinear growth rate, which we also prove optimal. We prove our results with no degree restrictions in both the EEL and VEL settings and examine type problems for more general complexes and dual graphs.     

Elastic modulus of a nonpolar polymer

A theoretical calculation of the elastic modulus of a nonpolar polymer (polyethylene) is carried out after deriving the appropriate relationships. Intramolecular forces (retarded rotation) play a far more important part than intermolecular forces. A simple method is proposed for finding the entropy and energy forces from the corresponding elastic moduli. The intra- and intermolecular energy effects are separated for one particular case.K. D. Ushinskii Yaroslav State Pedagogical Institute. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 450–454, May–June, 1973.     

最大模原理的推广

本文把最大模原理及 Phragmén—— Lindelof定理推广到具有保域性的一类连续函数上 ,得到若干结果 .     

Complex Pisot numbers of small modulus

We use an algorithm of Chamfy to determine all complex Pisot numbers of modulus less than 1.17. To cite this article: D. Garth, C. R. Acad. Sci. Paris, Ser. I 336 (2003).