Les références principales pour l’interprétation du produit de mélange de deux séries en termes d’automates et les applications aux identités syntaxiques sont [Lo 2002] 1.3 et [J 1980]. Dans le cours du 14 mars les exemples suivant sont traités: 1. Mélange de deux éléments de K < X > 2. L’identité x∗0x(x1 · · ·xn) = x∗0x1x∗0x2 · · ·x∗0xnx∗0 3. L’identité (x0 + x1)∗ = x∗0xx∗1 4. L’identité (1 + ...

If X0 = i then we say that i is the initial state of process X. If Xn = j then process X is in state j in step n, for n > 0. If Xn−1 = k and Xn = ` then process X enters state ` from state k in step n, for n > 0. Let us consider two examples of a process X = (X0, X1, X2, . . .) where Xi ∈ {0, 1}. One is when the random variables X0, X1, X2, . . . are independent and determined by tosses of a co...

the main goal of this note is to introduce another second-order differenceequation where every nontrivial solution is of minimal period 5, namelythe difference equation:xn+1 =1 + xn−1xnxn−1 − 1, n = 1, 2, 3, . . .with initial conditions x0 and x1 any real numbers such that x0x1 6= 1.

Under a map T , a point x recurs at rate given by a sequence {rn} near a point x0 if d(T(x), x0) < rn infinitely often. Let us fix x0, and consider the set of those x’s. In this paper, we study the size of this set for expanding maps and obtain its measure and sharp lower bounds on its dimension involving the entropy of T , the local dimension near x0 and the upper limit of 1 n log 1 rn . We ap...

In this work we consider a simple system of piecewise linear discontinuous 1D map with two discontinuity points: X0 = aX if jXj < z, X0 = bX if jXj > z, where a and b can take any real value, and may have several applications. We show that its dynamic behaviors are those of a linear rotation: either periodic or quasiperiodic, and always structurally unstable. A generalization to piecewise monot...

Suppose we wish to recover a vector x0 ∈ R (e.g. a digital signal or image) from incomplete and contaminated observations y = Ax0 + e; A is a n by m matrix with far fewer rows than columns (n m) and e is an error term. Is it possible to recover x0 accurately based on the data y? To recover x0, we consider the solution x to the `1-regularization problem min ‖x‖`1 subject to ‖Ax− y‖`2 ≤ , where i...

In many practical situations, we would like to maximize (or minimize) several different criteria, and it is not clear how much weight to assign to each of these criteria. Such situations are ubiquitous and thus, it is important to be able to solve the corresponding multi-objective optimization problems. There exist many heuristic methods for solving such problems. In this paper, we reformulate ...

We consider a Lévy flyer that starts from a point x0 on an interval [O,L] with absorbing boundaries. Using fractional differential equations, we show that the average number of flights the flyer takes before it is absorbed is 2(πα)−1 sin(πα/2) [ x0(L− x0)l −2 0 ]α/2 , where 0 < α < 2 and l0 > 0 are the parameters of the distribution of the flight length l: P (|l| > r) = (l0/r) α, r > l0, and l0...

In 1995 Magnus [15] posed a conjecture about the asymptotics of the recurrence coefficients of orthogonal polynomials with respect to the weights on [−1, 1] of the form (1− x) (1 + x) |x0 − x| × { B, for x ∈ [−1, x0) , A, for x ∈ [x0, 1] , with A,B > 0, α, β, γ > −1, and x0 ∈ (−1, 1). We show rigorously that Magnus’ conjecture is correct even in a more general situation, when the weight above h...

Let κ = Fq be a finite field of characteristic p > 0, and k be a fixed algebraic closure of κ. We fix a prime ` 6= p, and an isomorphism τ : Q` → C. Whenever we want to denote something (e.g. scheme, sheaf, morphism, etc.) defined over κ, we will put a subscript 0 (e.g. X0 is a scheme over κ, F0 is a Weil sheaf defined over X0, etc.), and when the subscript is dropped, this means the correspond...