Univoque graphs for non-integer base expansions

Science China Mathematics - Unique expansions in non-integer bases have been investigated in many papers during the last thirty years. They are often conveniently generated by labeled directed...     

ON THE NONLINEAR BOUNDARY STABILIZATION OF THE WAVE EQUATION

The author studies the energy decay rate for the wave equation in a bounded domain under-weak growth assumptions on the feedback function and, applying an integral inequality well adapted to this type of problems, improves some earlier results of Zuazua and of Rao ans Conrad.     

Subexpansions,superexpansions and uniqueness properties in non-integer bases

The β-expansions, i.e., greedy expansions with respect to non-integer bases q>1, were introduced by Réenyi and then investigated by many authors. Some years ago, Erdős, Horváth and Joó found the surprising fact that there exist infinitely many numbers 1<q<2 for which the β-expansion of 1 is the unique possible expansion with coefficients 0 or 1. Subsequently, the unique expansions were characterized in [9] and this characterization led to the determination (in [17]) of the smallest number q having this curious property. It is intimately related to the classical Thue-Morse sequence. Allouche and Cosnard recently proved that this q is transcendental. The purpose of this paper is to extend the previous results for expansions in arbitrary non-integer bases q>1. We also determine the smallest q having the corresponding uniqueness property in each case, and we prove that all of them are transcendental. We will also obtain some probably new properties of the Thue-Morse sequence. In the last section we answer a question concerning the existence of universal expansions, a notion introduced in [12]. This revised version was published online in June 2006 with corrections to the Cover Date.     

Partial Observability of Coupled Linear Systems

In [25] Lions proved several partial observability theorems for coupled linear distributed systems with small enough coupling parameters by a perturbation argument. He asked whether analogous results hold for large values of the parameters. Adapting an approach of Graham and Russell [7] we obtain positive results in particular cases.     

A simplified multidimensional integral

We present a simplified integral of functions of several variables. Although less general than the Riemann integral, most functions of practical interest are still integrable. On the other hand, the basic integral theorems can be obtained more quickly. We also give a characterization of the integrable functions and their primitives.