Absolute Algebra III - The saturated spectrum

Let B1 denote the set {0,1} with the usual operations except that $1+1=1$, in other words, the smallest characteristic 1 semifield . We compare two possible analogues of the notion of prime ideal for B1--algebras. We then consider the relations between these notions and Deitmar's theory of F1--schemes

On affine interest rate models

Bernstein processes are Brownian diffusions that appear in Euclidean Quantum Mechanics. Knowledge of the symmetries of the Hamilton-Jacobi-Bellman equation associated with these processes allows one to obtain relations between stochastic processes (Lescot-Zambrini, Progress in Probability, vols 58 and 59). More recently it has appeared that each one--factor affine interest rate model (in the sense of Leblanc-Scaillet) could be described using such a Bernstein process

Solving stochastic differential equations with Cartan's exterior differential systems

The aim of this work is to use systematically the symmetries of the (one dimensional) bacward heat equation with potentiel in order to solve certain one dimensional It\^o's stochastic differential equations. The special form of the drift (suggested by quantum mechanical considerations) gives, indeed, access to an algebrico-geometric method due, in essence, to E.Cartan, and called the Method of Isovectors. A V singular at the origin, as well as a one-factor affine model relevant to stochastic finance, are considered as illustrations of the method

On the commuting probability and supersolvability of finite groups

For a finite group $G$, let $d(G)$ denote the probability that a randomly chosen pair of elements of $G$ commute. We prove that if $d(G)>1/s$ for some integer $s>1$ and $G$ splits over an abelian normal nontrivial subgroup $N$, then $G$ has a nontrivial conjugacy class inside $N$ of size at most $s-1$. We also extend two results of Barry, MacHale, and N\'{\i} Sh\'{e} on the commuting probability in connection with supersolvability of finite groups. In particular, we prove that if $d(G)>5/16$ then either $G$ is supersolvable, or $G$ isoclinic to $A_4$, or G/\Center(G) is isoclinic to $A_4$

PRIME AND PRIMARY IDEALS IN SEMIRINGS

International audienceWe study zero divisors and minimal prime ideals in semirings of characteristic one. Thereafter we find a counterexample to the most obvious version of primary decomposition, but are able to establish a weaker version. Lastly, we study Evans'condition in this context

Reduction theorems for characteristic functors on finite pp-groups and applications to pp-nilpotence criteria

International audienceWe formalize various properties of characteristic functors on $p$-groups, and discuss relationships between them. Applications to the Thompson subgroup and certain of its analogues are then given