On the number of generators of the torsion module of differentials

In this paper we study the (minimum) global number of generators of the torsion module of differentials of affine hypersurfaces with only isolated singularities. We show that for reduced plane curves the torsion module of differentials can be generated by at most two elements, whereas for higher codimensions there is no universal upper bound. We then proceed to give explicit examples. In particular (when ) , we give examples of a reduced hypersurface with a single isolated singularity at the origin in that require

generators for the torsion module, Torsion .

     

On the number of generators of the module of derivations and multiplicity of certain rings

The present paper studies the module of derivations of certain rings and the multiplicity of certain rational surface singularities. We also consider some relation between these two for 2-dimensional quotient singularities. We conjecture a relation between the multiplicity of a rational surface singularity and the order of the divisor class group of the singularity and verify the same for several cases.     

Minimum number of generators of the lattice of submodules of a semisimple module

It is proved that the minimum number of generators of the lattice of all subspaces of a finite-dimensional vector space over a field is finite if and only if the field is finitely generated over the prime field. An upper bound is given for this number, which does not depend on the dimension of the space and is linearly dependent on the number of elements generating the field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 132, pp. 110–113, 1983.     

On the expected number of generators of a submodule of a free module over a finite principal ideal ring

Résumé  On généralise un résultat de Dobbs-Lancaster sur les espaces vectorielles au cadre des modules sur un anneau principal finiR. SoitV une module libre de type fini. On trouve une formule explicite et des valeurs dans la limite du nombre anticipé de générateurs d'un sousmodule arbitraire deV. De plus, on calcule ces valeurs anticipés quand tout sousmodule a un poids naturel (par exemple, quant a sa cardinalité). Enfin, on trouve une formule explicite et des valeurs dans la limite du rang anticipé du sousmodule libre le plus grand contenu dan un sousmodule arbitraire deV.      

On the expected number of generators of a submodule of a free module over a finite special principal ideal ring

Résumé  On généralise un résultat de Dobbs-Lancaster sur les espaces vectorielles au cadre des modules sur un anneau principal spécial finiR. SoitV une module libre de type fini. On trouve une formule explicite et des valeurs dans la limite du nombre anticipé de générateurs d’un sousmodule arbitraire deV. De plus, on calcule ces valeurs anticipés quand tout sousmodule a un poids naturel (par exemple, quant a sa cardinalité)      

On the number of generators of a finite group

Partially supported by NSF Grant DMS 8700961.     

On generators of ideals defining projective toric varieties

 For any ample line bundle L on a projective toric variety of dimension n, it is proved that the line bundle L ^⊗i is normally generated if i is greater than or equal to n−1, and examples showing that this estimate is best possible are given. Moreover we prove an estimate for the degree of the generators of the ideals defining projective toric varieties. In particular, when L is normally generated, the defining ideal of the variety embedded by the global sections of L has generators of degree at most n+1. When the variety is embedded by the global sections of L ^⊗(n−1) , then the defining ideal has generators of degree at most three. Received: 11 July 2001 / Revised version: 17 December 2001     

On a new class of multiplicative pseudo-random number generators

In the computing literature, there are few detailed analytical studies of the global statistical characteristics of a class of multiplicative pseudo-random number generators.We comment briefly on normal numbers and study analytically the approximately uniform discrete distribution or (j,)-normality in the sense of Besicovitch for complete periods of fractional parts {x _{0 1} ⁱ /p} on [0, 1] fori=0, 1,..., (p–1)p^–1–1, i.e. in current terminology, generators given byx _{n+1 1} x _n mod p wheren=0, 1,..., (p–1)p ^–1–1,p is any odd prime, (x ₀,p)=1, ₁ is a primitive root modp ², and 1 is any positive integer.We derive the expectationsE(X, ),E(X ², ),E(X _nX_n+k); the varianceV(X, ), and the serial correlation coefficient k. By means of Dedekind sums and some results of H. Rademacher, we investigate the asymptotic properties of k for various lagsk and integers 1 and give numerical illustrations. For the frequently used case =1, we find comparable results to estimates of Coveyou and Jansson as well as a mathematical demonstration of a so-called rule of thumb related to the choice of ₁ for small k.Due to the number of parameters in this class of generators, it may be possible to obtain increased control over the statistical behavior of these pseudo-random sequences both analytically as well as computationally.     

On the number of generators of Cohen-Macaulay ideals

Several bounds on the number of generators of Cohen-Macaulay ideals known in the literature follow from a simple inequality which bounds the number of generators of such ideals in terms of mixed multiplicities. Results of Cohen and Akizuki, Abhyankar, Sally, Rees and Boratynski-Eisenbud-Rees are deduced very easily from this inequality.

     

A polynomial bound on the number of comaximal localizations needed in order to make free a projective module

Let A be a commutative ring and M be a projective module of rank k with n generators. Let h=n-k. Standard computations show that M becomes free after localizations in comaximal elements (see Theorem 5). When the base ring A contains a field with at least hk+1 non-zero distinct elements we construct a comaximal family G with at most (hk+1)(nk+1) elements such that for each g∈G, the module M_g is free over A[1/g].