Natural Density Distribution of Hermite Normal Forms of Integer Matrices

The Hermite Normal Form (HNF) is a canonical representation of matrices over any principal ideal domain. Over the integers, the distribution of the HNFs of randomly looking matrices is far from uniform. The aim of this article is to present an explicit computation of this distribution together with some applications. More precisely, for integer matrices whose entries are upper bounded in absolute value by a large bound, we compute the asymptotic number of such matrices whose HNF has a prescribed diagonal structure. We apply these results to the analysis of some procedures and algorithms whose dynamics depend on the HNF of randomly looking integer matrices

Some Inequalities Related to the Seysen Measure of a Lattice

Given a lattice $L$, a basis $B$ of $L$ together with its dual $B^*$, the orthogonality measure $S(B)=\sum_i ||b_i||^2 ||b_i^*||^2$ of $B$ was introduced by M. Seysen in 1993. This measure is at the heart of the Seysen lattice reduction algorithm and is linked with different geometrical properties of the basis. In this paper, we explicit different expressions for this measure as well as new inequalities.Comment: Typos correcte

Learning Process and Contract Adaptation with Quality Uncertainty: Some Paradoxes in Retailer-Producer Relationships

The optimal level of formalisation of contracts and their dynamic is at stake in the economic literature on the optimum design of ex post renegotiation with third party enforcement. Another theoretical interpretation is that contract adaptations may also reflect mutual learning process between contractors. Why transactors write explicit contract that they know cannot be court enforced? The central idea is that explicit contract terms makes it clearer to the transactors what has been agreed upon, thus are decreasing the cost of private enforcement sanctions (Klein, 1996). Empirical evidences are provided by the diachronic analysis of the full set of tri-partite contracts between one of the top-ten french large retailer and its beef suppliers before and after the BSE crisis (period 1993-1999). The analysis emphasizes also the role of this increased codification of supply contracts with a progressive change in the internal retailer's organization, i.e. increased centralization of decision and supervision mechanisms. Contract design and organizational choices are then strongly interrelated.Contract design, Hold-up problems, beef sector, Self-Enforcement, Agribusiness,

Neuroprotective Properties of Xenon.

Xenon is a rare noble gas that was introduced into clinical practice more than 70 years ago. Xenon's clinical properties are predicated by its ability to fit into preformed cavities of macromolecules thereby altering their biological functions. One such action targets the NMDA-subtype of the glutamate receptors thereby inhibiting its excitatory action. As the glutamate receptors are pivotal for both anesthesia and acute neurological injury, its clinical use has included both general anesthesia as well as neuroprotection. In this manuscript, the efficacy and safety of xenon in clinical trials that address both the anesthetic and neuroprotective applications are discussed. Because of the clinical safety of this chemically inert monatomic gas, the lack of an alternative for neuroprotection, and encouraging phase 2 trial data, a multinational pivotal randomized clinical trial (XePOHCAS) has been launched to assess the utility of xenon for patients that have been successfully resuscitated following an out of hospital cardiac arrest but still remain comatose, indicating ongoing neurological ischemic-perfusion injury. If successful, the trial will herald a new era of treatments for previously intractable conditions such as traumatic brain injury, ischemic and hemorrhagic strokes, and anesthetic-induced developmental neurotoxicity

Natural Density of Rectangular Unimodular Integer Matrices

In this paper, we compute the natural density of the set of k x n integer matrices that can be extended to an invertible n x n matrix over the integers. As a corollary, we find the density of rectangular matrices with Hermite normal form [O Id]. Connections with Cesaro's Theorem on the density of coprime integers and Quillen-Suslin's Theorem are also presented.Comment: 8 page

Public Key Cryptography based on Semigroup Actions

A generalization of the original Diffie-Hellman key exchange in $(\Z/p\Z)^*$ found a new depth when Miller and Koblitz suggested that such a protocol could be used with the group over an elliptic curve. In this paper, we propose a further vast generalization where abelian semigroups act on finite sets. We define a Diffie-Hellman key exchange in this setting and we illustrate how to build interesting semigroup actions using finite (simple) semirings. The practicality of the proposed extensions rely on the orbit sizes of the semigroup actions and at this point it is an open question how to compute the sizes of these orbits in general and also if there exists a square root attack in general. In Section 2 a concrete practical semigroup action built from simple semirings is presented. It will require further research to analyse this system.Comment: 20 pages. To appear in Advances in Mathematics of Communication