Invariant densities for random β\beta-expansions

Let $\beta >1$ be a non-integer. We consider expansions of the form $\sum_{i=1}^{\infty} d_i \beta^{-i}$, where the digits $(d_i)_{i \geq 1}$ are generated by means of a Borel map $K_{\beta}$ defined on $\{0,1\}^{\N}\times [ 0, \lfloor \beta \rfloor /(\beta -1)]$. We show existence and uniqueness of an absolutely continuous $K_{\beta}$-invariant probability measure w.r.t. $m_p \otimes \lambda$, where $m_p$ is the Bernoulli measure on $\{0,1\}^{\N}$ with parameter $p$ $(0 < p < 1)$ and $\lambda$ is the normalized Lebesgue measure on $[0 ,\lfloor \beta \rfloor /(\beta -1)]$. Furthermore, this measure is of the form $m_p \otimes \mu_{\beta,p}$, where $\mu_{\beta,p}$ is equivalent with $\lambda$. We establish the fact that the measure of maximal entropy and $m_p \otimes \lambda$ are mutually singular. In case the number 1 has a finite greedy expansion with positive coefficients, the measure $m_p \otimes \mu_{\beta,p}$ is Markov. In the last section we answer a question concerning the number of universal expansions, a notion introduced in [EK]

A two-dimensional univoque set

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 $q$.Comment: 12 pages, 0 figures. Comments of the referee are incorporated. Accepted for publication in Fundamenta Mathematica

On the number of unique expansions in non-integer bases

Let $q > 1$ be a real number and let $m=m(q)$ be the largest integer smaller than $q$. It is well known that each number $x \in J_q:=[0, \sum_{i=1}^{\infty} m q^{-i}]$ can be written as $x=\sum_{i=1}^{\infty}{c_i}q^{-i}$ with integer coefficients $0 \le c_i < q$. If $q$ is a non-integer, then almost every $x \in J_q$ has continuum many expansions of this form. In this note we consider some properties of the set $\mathcal{U}_q$ consisting of numbers $x \in J_q$ having a unique representation of this form. More specifically, we compare the size of the sets $\mathcal{U}_q$ and $\mathcal{U}_r$ for values $q$ and $r$ satisfying $1< q < r$ and $m(q)=m(r)$.Comment: typo corrected in Theorem 1.

Theoretical asset pricing under behavioral decision making

This dissertation is a collection of chapters that analyze the role of non-standard beliefs and preferences in investor behavior and thereby for financial markets. The first chapter sheds light on the role of non-standard preferences for an ongoing debate on the slope of the term structure of equity risk premia. More specifically, the finding that individuals may act in a risk seeing way, and may not always be risk averse, helps to explain the empirical findings in the literature. The second chapter gives a general theoretical result about learning the distribution of a random variable under limited attention. The main result is that limited attention gives rise to some familiar biases in behavioral finance or implies similar behavior as: a preference for positive skewness, overestimation of volatility, overextrapolative beliefs and probability weighting. The third chapter provides a new empirical finding, namely that the trading volume in the first three years after an IPO is u-shaped. Subsequently, the analysis shows that this new finding can be explained by a model with speculative investors and ambiguity averse investors and thereby also illustrates the importance of ambiguity in finance

Optimal expansions in non-integer bases

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 $x$.Comment: 11 pages, 0 figures, to appear in Proc. Amer. Math. So

Simulations of the merging cluster of galaxies Cygnus A

The archetype FR-II galaxy Cygnus A lies in a moderately rich cluster about to undergo a major merger. We study the pre-merger Cygnus cluster environment using smoothed particle hydrodynamics simulations constrained by 2Ms of Chandra observations of the hot intracluster medium. The observations constrain the total gravitating mass and concentration parameter, and the simulations provide the quiescent and merger-enhanced temperature profiles of the pre- and post merger of the cluster excluding the central active galactic nucleus. We present the first detailed model of the sub cluster north west of Cygnus A, named CygNW. We find a lower baryon fraction and higher concentration parameter for CygA than expected from known scaling relations in the literature. The model suggests the Cygnus cluster hosts a pre-merger with a progenitor mass ratio of about 1.5:1 at the virial radius. We notice that the intra cluster medium is heated as a result of the merger, but we find no evidence for a (pre-)merger shock in the interstitial region between both cluster haloes. We attribute the merger-induced heating to compression of the cluster outskirts. The smooth model obtained from our simulations is subtracted from the observed cluster state and shows residual temperature structure that is neither hydrostatic nor merger-heated cluster gas. We speculate that this residual heating may result from previous AGN activity over the last ~100 Myr

Adapting Monolingual Models:Data can be Scarce when Language Similarity is High

For many (minority) languages, the resources needed to train large models are not available. We investigate the performance of zero-shot transfer learning with as little data as possible, and the influence of language similarity in this process. We retrain the lexical layers of four BERT-based models using data from two low-resource target language varieties, while the Transformer layers are independently fine-tuned on a POS-tagging task in the model's source language. By combining the new lexical layers and fine-tuned Transformer layers, we achieve high task performance for both target languages. With high language similarity, 10MB of data appears sufficient to achieve substantial monolingual transfer performance. Monolingual BERT-based models generally achieve higher downstream task performance after retraining the lexical layer than multilingual BERT, even when the target language is included in the multilingual model