Uncountable sets of unit vectors that are separated by more than 1

Let $X$ be a Banach space. We study the circumstances under which there exists an uncountable set $\mathcal A\subset X$ of unit vectors such that $\|x-y\|>1$ for distinct $x,y\in \mathcal A$. We prove that such a set exists if $X$ is quasi-reflexive and non-separable; if $X$ is additionally super-reflexive then one can have $\|x-y\|\geqslant 1+\varepsilon$ for some $\varepsilon>0$ that depends only on $X$. If $K$ is a non-metrisable compact, Hausdorff space, then the unit sphere of $X=C(K)$ also contains such a subset; if moreover $K$ is perfectly normal, then one can find such a set with cardinality equal to the density of $X$; this solves a problem left open by S. K. Mercourakis and G. Vassiliadis.Comment: to appear in Studia Mat

Untapped Instrument. Sovereign Wealth Funds and Chinese policy toward the Central and Eastern European countries

Although there has been vivid academic debate as to what extent Sovereign Wealth Funds (SWFs) are motivated by political reasons, it is rather clear that countries can use state-owned investment funds as a tool of their foreign policy. Even Barack Obama, during his initial presidential campaign in 2008 commented: “I am obviously concerned if these… sovereign wealth funds are motivated by more than just market consideration and that’s obviously a possibility”. This book looks at SWF activities in Central and Eastern Europe (CEE) to determine the main motives for SWF presence in CEE. Are the potential financial gains the only reason behind their investments? Are SWF activities in the region dangerous for the stability and security of the CEE countries? The book is pioneering analyses of SWFs behaviour in the region, based on empirical data collected from the Sovereign Wealth Fund Institute Transaction Database, arguably the most comprehensive and authoritative resource tracking SWF investment behaviour globally.Rozdział pochodzi z książki: Political Players? Sovereign Wealth Funds’ Investments in Central and Eastern Europe, T. Kamiński (ed.), Wydawnictwo Uniwersytetu Łódzkiego, Łódź 2017.This chapter aims at looking at the role of Sovereign wealth funds (SWFs) in China’s policy toward Central and Eastern European (CEE) countries in the 21st century, especially since the enlargement of the European Union (EU) in 2004. During this time, we could observe an increase of Chinese interest in the region resulting in growing trade, investments and number of contacts on all levels. China has used a wide array of different instruments to achieve its goals in the region: from a big political project such as the “16+1 format” to an unprecedented frequency of contacts between Chinese provinces and their European counterparts (Kaczmarski, Jakóbowski 2015). Despite a visible growth of economic ties, Beijing presented a very limited will to use investments as a political instrument. Even if Chinese investments in CEE are booming, they are possibly less politically biased and more market-driven than those in other developing countries, like African ones.This book was published in frames of project “Political significance of the Sovereign Wealth Funds’ investments in the Central and Eastern Europe”. The project was financed by the Polish National Science Centre (Decision no. DEC-2012/07/B/HS5/03797)

The ideal of weakly compactly generated operators acting on a Banach space

We call a bounded linear operator acting between Banach spaces weakly compactly generated ($\mathsf{WCG}$ for short) if its range is contained in a weakly compactly generated subspace of its codomain. This notion simultaneously generalises being weakly compact and having separable range. In a comprehensive study of the class of $\mathsf{WCG}$ operators, we prove that it forms a closed surjective operator ideal and investigate its relations to other classical operator ideals. By considering the $p$th long James space $\mathcal{J}_p(\omega_1)$, we show how properties of the ideal of $\mathsf{WCG}$ operators (such as being the unique maximal ideal) may be used to derive results outside ideal theory. For instance, we identify the $K_0$-group of $\mathscr{B}(\mathcal{J}_p(\omega_1))$ as the additive group of integers

On ergodicity of some Markov processes

We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak-$^*$ ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution, is established. The principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity. The latter is called the e-property. The general result is applied to solutions of some stochastic evolution equations in Hilbert spaces. As an example, we consider an evolution equation whose solution describes the Lagrangian observations of the velocity field in the passive tracer model. The weak-$^*$ mean ergodicity of the corresponding invariant measure is used to derive the law of large numbers for the trajectory of a tracer.Comment: Published in at http://dx.doi.org/10.1214/09-AOP513 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Martin representation and Relative Fatou Theorem for fractional Laplacian with a gradient perturbation

Let $L=\Delta^{\alpha/2}+ b\cdot\nabla$ with $\alpha\in(1,2)$. We prove the Martin representation and the Relative Fatou Theorem for non-negative singular $L$-harmonic functions on ${\mathcal C}^{1,1}$ bounded open sets.Comment: 28 pages, editorial change