arXiv subjects
Publications and source records attributed to Vilmos Komornik.
We give a nonlinear representation of the duals for a class of Banach spaces. This leads to classroom-friendly proofs of the classical representation theorems $H'=H$ and $(L^p)'=L^q$. Our proofs extend to a family of Orlicz spaces, and yield as an unexpected byproduct a version of the Helly-Hahn-Banach theorem.
We investigate the internal observability of the wave equation with Dirichlet boundary conditions in a triangular domain. More precisely, the domain taken into exam is the half of the equilateral triangle. Our approach is based on Fourier analysis and on tessellation theory: by means of a suitable tiling of the rectangle, we extend earlier observability results in the rectangle to the case of a triangular domain. The paper includes a general result relating problems in general domains to their tiles, and a discussion of the triangular case. As an application, we provide an estimation of the observation time when the observed domain is composed by three strips with a common side to the edges of the triangle.
In this paper we answer several questions raised by Sidorov on the set $\mathcal B_2$ of bases in which there exist numbers with exactly two expansions. In particular, we prove that the set $\mathcal B_2$ is closed, and it contains both infinitely many isolated and accumulation points in $(1, q_{KL})$, where $q_{KL}\approx 1.78723$ is the Komornik-Loreti constant. Consequently we show that the second smallest element of $\mathcal B_2$ is the smallest accumulation point of $\mathcal B_2$. We also investigate the higher order derived sets of $\mathcal B_2$. Finally, we prove that there exists a $δ>0$ such that \begin{equation*} \dim_H(\mathcal B_2\cap(q_{KL}, q_{KL}+δ))<1, \end{equation*} where $\dim_H$ denotes the Hausdorff dimension.
Given $x\in(0, 1]$, let $\mathcal U(x)$ be the set of bases $q\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^\infty d_i/q^i$. Lü, Tan and Wu (2014) proved that $\mathcal U(x)$ is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum $\mathcal U(x)+λ\mathcal U(x)$ and product $\mathcal U(x)\cdot\mathcal U(x)^λ$ contain an interval for all $x\in(0, 1]$ and $λ\ne 0$. As an application we show that the same phenomenon occurs for the set of non-matching parameters studied by the first author and Kalle (2017).
We present a modification of Riesz's construction of the Lebesgue integral, leading directly to finite or infinite integrals, at the same time simplifying the proofs.
Glendinning and Sidorov discovered an important feature of the Komornik-Loreti constant $q'\approx1.78723$ in non-integer base expansions on two-letter alphabets: in bases $1<q<q'$ only countably numbers have unique expansions, while for $q\ge q'$ there is a continuum of such numbers. We investigate the analogous question for ternary alphabets.
Cantor's ternary function is generalized to arbitrary base-change functions in non-integer bases. Some of them share the curious properties of Cantor's function, while others behave quite differently.
A classical theorem of Ingham extended Parseval's formula of the trigonometrical system to arbitrary families of exponentials satisfying a uniform gap condition. Later his result was extended to several dimensions, but the optimal integration domains have only been determined in very few cases. The purpose of this paper is to determine the optimal connected integration domains for all regular two-dimensional lattices.
We fix a positive integer $M$, and we consider expansions in arbitrary real bases $q>1$ over the alphabet $\{0,1,...,M\}$. We denote by $U_q$ the set of real numbers having a unique expansion. Completing many former investigations, we give a formula for the Hausdorff dimension $D(q)$ of $U_q$ for each $q\in (1,\infty)$. Furthermore, we prove that the dimension function $D:(1,\infty)\to[0,1]$ is continuous, and has a bounded variation. Moreover, it has a Devil's staircase behavior in $(q',\infty)$, where $q'$ denotes the Komornik--Loreti constant: although $D(q)>D(q')$ for all $q>q'$, we have $D'<0$ a.e. in $(q',\infty)$. During the proofs we improve and generalize a theorem of Erdős et al. on the existence of large blocks of zeros in $β$-expansions, and we determine for all $M$ the Lebesgue measure and the Hausdorff dimension of the set of bases in which $x=1$ has a unique expansion.
This paper is devoted to the study of a nonlinear coupled system of two Korteweg-de Vries equations in a periodic domain under the effect of an internal damping term. The system was introduced Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. Designing a time-varying feedback law and using a Lyapunov approach we establish the exponential stability of the solutions in Sobolev spaces of any positive integral order.
Since the works of Haraux and Jaffard we know that rectangular plates may be observed by subregions not satisfying the geometrical control condition. We improve these results by observing only on an arbitrarily short segment inside the domain. The estimates may be strengthened by observing on several well-chosen segments. In the second part of the paper we establish various observability theorems for rectangular membranes by applying Mehrenberger's recent generalization of Ingham's theorem.
For a given positive integer $m$, let $A=\set{0,1,...,m}$ and $q \in (m,m+1)$. A sequence $(c_i)=c_1c_2 ...$ consisting of elements in $A$ is called an expansion of $x$ if $\sum_{i=1}^{\infty} c_i q^{-i}=x$. It is known that almost every $x$ belonging to the interval $[0,m/(q-1)]$ has uncountably many expansions. In this paper we study the existence of expansions $(d_i)$ of $x$ satisfying the inequalities $\sum_{i=1}^n d_iq^{-i} \geq \sum_{i=1}^n c_i q^{-i}$, $n=1,2,...$ for each expansion $(c_i)$ of $x$.
By the m-spectrum of a real number q>1 we mean the set Y^m(q) of values p(q) where p runs over the height m polynomials with integer coefficients. These sets have been extensively investigated during the last fifty years because of their intimate connections with infinite Bernoulli convolutions, spectral properties of substitutive point sets and expansions in noninteger bases. We prove that Y^m(q) has an accumulation point if and only if q<m+1 and q is not a Pisot number. Consequently a number of related results on the distribution of points of this form are improved.
Let $\mathbf{J} \subset \mathbb{R}^2$ be the set of couples $(x,q)$ with $q>1$ such that $x$ has at least one representation of the form $x=\sum_{i=1}^{\infty} c_i q^{-i}$ with integer coefficients $c_i$ satisfying $0 \leq c_i < q$, $i \ge 1$. In this case we say that $(c_i)=c_1c_2...$ is an expansion of $x$ in base $q$. Let $\mathbf{U}$ be the set of couples $(x,q) \in \mathbf{J}$ such that $x$ has exactly one expansion in base $q$. In this paper we deduce some topological and combinatorial properties of the set $\mathbf{U}$. We characterize the closure of $\mathbf{U}$, and we determine its Hausdorff dimension. For $(x,q) \in \mathbf{J}$, we also prove new properties of the lexicographically largest expansion of $x$ in base $q$.
We generalize several schedule matching theorems of Baiou-Balinski (Math. Oper. Res., 27 (2002), 485) and Alkan-Gale (J. Econ. Th. 112 (2003), 289) by applying a fixed point method of Fleiner (Math. Oper. Res., 28 (2003), 103). Thanks to a more general construction of revealing choice maps we develop an algorithm to solve rather complex matching problems. The flexibility and efficiency of our approach is illustrated by various examples. We also revisit the mathematical structure of the matching theory by comparing various definitions of stable sets and various classes of choice maps. We demonstrate, by several examples, that the revealing property of the choice maps is the most suitable one to ensure the existence of stable matchings; both from the theoretical and the practical point of view.
We formulate and discuss a conjecture which would extend a classical inequality of Bernstein.
Expansions in noninteger positive bases have been intensively investigated since the pioneering works of Rényi (1957) and Parry (1960). The discovery of surprising unique expansions in certain noninteger bases by Erd\H os, Horváth and Joó (1991) was followed by many studies aiming to clarify the topological and combinatorial nature of the sets of these bases. In the present work we extend some of these studies to more general, negative or complex bases.
Baiocchi et al. generalized a few years ago a classical theorem of Ingham and Beurling by means of divided differences. The optimality of their assumption has been proven by the third author of this note. The purpose of this note to extend these results to vector coefficient sums.