Universal expansions in negative and complex bases.
Book review: "Control and Estimation of Distributed Parameter Systems" by W. Desch, F. Kappel and K. Kunisch,
A new method of exact controllability in short time and applications
Oscillations of anharmonic Fourier series and the wave equation.
In this paper we have collected some partial results on the sign of u(t,x) where u is a (sufficiently regular) solution of ⎧ utt + (-1)m Δmu = 0 (t,x) ∈ R x Ω ⎨ ⎩ u|Γ = ... = Δm-1 u|Γ = 0 t ∈ R. These results rely on the study of a sign of almost periodic functions of a special form restricted...
Well posedness and control of semilinear wave equations with iterated logarithms
On the sequence of numbers of the form ,
Ingham type theorems and applications to control theory
Ingham [6] ha migliorato un risultato precedente di Wiener [23] sulle serie di Fourier non armoniche. Modificando la sua funzione di peso noi otteniamo risultati ottimali, migliorando precedenti teoremi di Kahane [9], Castro e Zuazua [3], Jaffard, Tucsnak e Zuazua [7] e di Ullrich [21]. Applichiamo poi questi risultati a problemi di osservabilità simultanea.
A two-dimensional univoque set
Let J ⊂ ℝ² be the set of couples (x,q) with q > 1 such that x has at least one representation of the form with integer coefficients satisfying , i ≥ 1. In this case we say that is an expansion of x in base q. Let U be the set of couples (x,q) ∈ J such that x has exactly one expansion in base q. In this paper we deduce some topological and combinatorial properties of the set U. We characterize the closure of U, and we determine its Hausdorff dimension. For (x,q) ∈ J, we also prove new properties...
Well posedness and control of semilinear wave equations with iterated logarithms
Motivated by a classical work of Erdős we give rather precise necessary and sufficient growth conditions on the nonlinearity in a semilinear wave equation in order to have global existence for all initial data. Then we improve some former exact controllability theorems of Imanuvilov and Zuazua.
Characterization of the unique expansions and related problems
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.
Generalized golden ratios of ternary alphabets
Expansions in noninteger bases often appear in number theory and probability theory, and they are closely connected to ergodic theory, measure theory and topology. For two-letter alphabets the golden ratio plays a special role: in smaller bases only trivial expansions are unique, whereas in greater bases there exist nontrivial unique expansions. In this paper we determine the corresponding critical bases for all three-letter alphabets and we establish the fractal nature of these bases in dependence...
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