Central limit theorem for generalized Weierstrass functions

Global exponential periodicity and global exponential stability of a class of recurrent neural networks with various activation functions and time-varying delays

Let $$\lambda _{1},\ldots ,\lambda _{n}$$ be real numbers in $$(0,1)$$ and $$p_{1},\ldots ,p_{n}$$ be points in $$\mathbb {R}^{d}$$ . Consider the collection of maps $$f_{j}:\mathbb {R}^{d}\rightarrow \mathbb {R}^{d} $$ given by $$\begin{aligned} f_{j}(x)=\lambda _{j} x +\left( 1-\lambda _{j}\right) p_{j}. \end{aligned}$$ It is a well known result that there exists a unique nonempty compact set $$\Lambda \subset \mathbb {R}^{d}$$ satisfying $$\Lambda =\cup _{j=1}^{n} f_{j}(\Lambda ).$$ Each $$x\in \Lambda $$ has at least one coding, that is a sequence $$(\epsilon _{i})_{i=1}^{\infty }\in \{1,\ldots ,n\}^{\mathbb {N}}$$ that satisfies $$\lim _{N\rightarrow \infty }f_{\epsilon _{1}}\ldots f_{\epsilon _{N}} (0)=x.$$ We study the size and complexity of the set of codings of a generic $$x\in \Lambda $$ when $$\Lambda $$ has positive Lebesgue measure. In particular, we show that under certain natural conditions almost every $$x\in \Lambda $$ has a continuum of codings. We also show that almost every $$x\in \Lambda $$ has a universal coding. Our work makes no assumptions on the existence of holes in $$\Lambda $$ and improves upon existing results when it is assumed $$\Lambda $$ contains no holes.

It is natural to expect that the arithmetic sum of two Cantor sets should have positive Lebesgue measure if the sum of their dimensions exceeds 1, but there are many known counterexamples, e.g. when both sets are the middle-$\alpha$ Cantor set and $\alpha \in ({1 \over 3}, \frac 12)$. We show that for any compact set $K$ and for a.e. $\alpha \in (0,1)$, the arithmetic sum of $K$ and the middle-$\alpha$ Cantor set does indeed have positive Lebesgue measure when the sum of their Hausdorff dimensions exceeds 1. In this case we also determine the essential supremum, as the translation parameter $t$ varies, of the dimension of the intersection of $K+t$ with the middle-$\alpha$ Cantor set. We also establish a new property of the infinite Bernoulli convolutions $\nu _\lambda ^p$ (the distributions of random series $\sum _{n=0}^\infty \pm \lambda ^n ,$ where the signs are chosen independently with probabilities $(p,1-p)$). Let $1 \leq q_1<q_2 \leq 2$. For $p \neq \frac 12$ near $\frac 12$ and for a.e. $\lambda$ in some nonempty interval, $\nu _\lambda ^p$ is absolutely continuous and its density is in $L^{q_1}$ but not in $L^{q_2}$. We also answer a question of Kahane concerning the Fourier transform of $\nu _\lambda$.

A continuous function, chaotic in the sense of Li and Yorke, is constructed which possesses a scrambled set of positive Lebesgue measure.

Differentiability of continuous functions in terms of Haar-smallness

We show that there exists a transcendental entire function whose Julia set has positive finite Lebesgue measure.

We know from Netto’s theorem (Theorem 6.4) that plane curves that do not cross (or touch) themselves cannot be space-filling. We will show in this chapter that it is still possible for such curves to have a positive two-dimensional Lebesgue measure.

For λ ∈ [0.6439, 0.6441]∪[0.6458, 0.6466]∪[0.6470, 0.6472], the “fat” Sierpiński triangle—the unique compact invariant set (“attractor”) Λλ of the iterated function system on , where p0 = (0, 0), p1 = (1, 0), p2 = (1/2, 1)—has interior points and hence positive Lebesgue measure and Hausdorff dimension 2. For these λ, standard techniques for determining the Hausdorff dimension (or Lebesgue measure) of Λλ do not apply, and its Hausdorff dimension has not been known for any specific such λ. The novelty of our approach is that instead of extending techniques developed for small λ, we significantly extend geometric methods developed by Broomhead, Montaldi, and Sidorov for larger values of λ.

Let us denote the nth difference between consecutive primes by d_n. The Prime Number Theorem clearly implies that d_n is logn on average. Paul Erd\H{o}s conjectured about 60 years ago that the sequence d_n/logn is everywhere dense on the nonnegative part of the real line. He and independently G. Ricci proved in 1954-55 that the set J of limit points of the sequence {d_n/logn} has positive Lebesgue measure. The first and until now only concrete known element of J was proved to be the number zero in the work of Goldston, Yildirim and the present author. The author of the present note showed in 2013 (arXiv: 1305.6289) that there is a fixed interval containing 0 such that all elements of it are limit points. In 2014 it was shown by W.D. Banks, T. Freiberg and J. Maynard (arXiv: 1404.5094) that one can combine the Erd\H{o}s-Rankin method (producing large prime gaps) and the Maynard-Tao method (producing bounded prime gaps) to obtain the lower bound (1+o(1))T/8 for the Lebesgue measure of the subset of limit points not exceeding T. In the present note we improve this lower bound to (1+o(1))T/4, using a refinement of the argument of Banks, Freiberg and Maynard.

In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient algorithm for computing approximate equilibria in these games. Then we turn our attention to penalty games, which encompass biased games and games in which players take risk into account. Here we show that if the penalty function is Lipschitz continuous, then we can provide a quasi-polynomial time approximation scheme. Finally, we study distance biased games, where we present simple strongly polynomial time algorithms for finding best responses in L_1 and L_2^2 biased games, and then use these algorithms to provide strongly polynomial algorithms that find 2/3 and 5/7 approximate equilibria for these norms, respectively.

The main goal of this paper is to introduce a link between the thermodynamics and control systems theory. More precisely, the paper focuses on Lyapunov based control of process systems, specially the non isothermal Continuous Stirred Tank Reactors in a thermodynamic framework, using either the jacket temperature or the inlet molar flow rate as the only control input. As soon as the constraint on the total mass is considered and the reaction kinetics is a Lipschitz continuous function with respect to the temperature, it shows that the stabilization of thermal solicitations reciprocally entails the one of matter using La Salle’s invariance principle. As a consequence, these control problems can be solved if the closed loop Lyapunov functions are chosen to be proportional to the thermal part or material part of the so called thermodynamic availability function. Some numerical simulations for a first order chemical reaction with multiple steady states are given to validate our theoretical developments. The performance of the obtained nonlinear controllers with regard to the conversion rate is also discussed.

International audience

Nonhyperbolic persistent attractors near the Morse–Smale boundary

Bounded point evaluations and capacity

In this paper, we study the existence and concentration of positive solution of a class of coupled Schrödinger equations. We admit that the potentials may not be non-negative and suppose that the intersection of the sets has positive Lebesgue measure. By studying the modified functional of the associated functional carefully, we establish the existence of positive least energy solutions for the coupled Schrödinger system. Moreover, we prove the concentration phenomenon of the positive solution when the parameter goes to infinity.