One-dimensional infinite memory imitation models with noise

In this paper we study stochastic process indexed by $\mathbb {Z}$ constructed from certain transition kernels depending on the whole past. These kernels prescribe that, at any time, the current state is selected by looking only at a previous random instant. We characterize uniqueness in terms of simple concepts concerning families of stochastic matrices, generalizing the results previously obtained in De Santis and Piccioni (J. Stat. Phys., 150(6):1017--1029, 2013).Comment: 22 pages, 3 figure

Commuting varieties of rr-tuples over Lie algebras

Let $G$ be a simple algebraic group defined over an algebraically closed field $k$ of characteristic $p$ and let \g be the Lie algebra of $G$. It is well known that for $p$ large enough the spectrum of the cohomology ring for the $r$-th Frobenius kernel of $G$ is homeomorphic to the commuting variety of $r$-tuples of elements in the nilpotent cone of \g [Suslin-Friedlander-Bendel, J. Amer. Math. Soc, \textbf{10} (1997), 693--728]. In this paper, we study both geometric and algebraic properties including irreducibility, singularity, normality and Cohen-Macaulayness of the commuting varieties C_r(\mathfrak{gl}_2), C_r(\fraksl_2) and $C_r(\N)$ where $\N$ is the nilpotent cone of \fraksl_2. Our calculations lead us to state a conjecture on Cohen-Macaulayness for commuting varieties of $r$-tuples. Furthermore, in the case when \g=\fraksl_2, we obtain interesting results about commuting varieties when adding more restrictions into each tuple. In the case of \fraksl_3, we are able to verify the aforementioned properties for C_r(\fraku). Finally, applying our calculations on the commuting variety C_r(\overline{\calO_{\sub}}) where \overline{\calO_{\sub}} is the closure of the subregular orbit in \fraksl_3, we prove that the nilpotent commuting variety $C_r(\N)$ has singularities of codimension $\ge 2$.Comment: To appear in Journal of Pure and Applied Algebr

Reasoning on Schemata of Formulae

A logic is presented for reasoning on iterated sequences of formulae over some given base language. The considered sequences, or "schemata", are defined inductively, on some algebraic structure (for instance the natural numbers, the lists, the trees etc.). A proof procedure is proposed to relate the satisfiability problem for schemata to that of finite disjunctions of base formulae. It is shown that this procedure is sound, complete and terminating, hence the basic computational properties of the base language can be carried over to schemata

Multifraction reduction I: The 3-Ore case and Artin-Tits groups of type FC

We describe a new approach to the Word Problem for Artin-Tits groups and, more generally, for the enveloping group U(M) of a monoid M in which any two elements admit a greatest common divisor. The method relies on a rewrite system R(M) that extends free reduction for free groups. Here we show that, if M satisfies what we call the 3-Ore condition about common multiples, what corresponds to type FC in the case of Artin-Tits monoids, then the system R(M) is convergent. Under this assumption, we obtain a unique representation result for the elements of U(M), extending Ore's theorem for groups of fractions and leading to a solution of the Word Problem of a new type. We also show that there exist universal shapes for the van Kampen diagrams of the words representing 1.Comment: 29 pages ; v2 : cross-references updated ; v3 : typos corrected; final version due to appear in Journal of Combinatorial Algebr

Sequences of irreducible polynomials without prescribed coefficients over odd prime fields

In this paper we construct infinite sequences of monic irreducible polynomials with coefficients in odd prime fields by means of a transformation introduced by Cohen in 1992. We make no assumptions on the coefficients of the first polynomial $f_0$ of the sequence, which belongs to \F_p [x], for some odd prime $p$, and has positive degree $n$. If $p^{2n}-1 = 2^{e_1} \cdot m$ for some odd integer $m$ and non-negative integer $e_1$, then, after an initial segment $f_0, ..., f_s$ with $s \leq e_1$, the degree of the polynomial $f_{i+1}$ is twice the degree of $f_i$ for any $i \geq s$.Comment: 10 pages. Fixed a typo in the reference

Generating Schemata of Resolution Proofs

Two distinct algorithms are presented to extract (schemata of) resolution proofs from closed tableaux for propositional schemata. The first one handles the most efficient version of the tableau calculus but generates very complex derivations (denoted by rather elaborate rewrite systems). The second one has the advantage that much simpler systems can be obtained, however the considered proof procedure is less efficient