The Collatz tree as a Hilbert hotel:a proof of the 3x + 1 conjecture

The Collatz conjecture maintains that each natural number $n$ is a node in the Collatz tree with a root path to $1$ generated by iterations of the Collatz function, which connects even $n$ to $n/2$ and odd $n$ to $3n+1$. The conjecture holds if a binary subtree $T_{\ge 0}$ exists with as nodes $\textit{all}$ branching numbers constituting the congruence classes ${\left[4,16\right]}_{18}$ with density $2/18$. For each branching number, the $\textit{upward}$ and $\textit{rightward}$ function generate arrows to two branching numbers in the directions $2n$ and $(n-1)/3$. In the automorphism graph of tree $T_{\ge 0}$. The iterates and conjugates of these two functions build outer paths from tree $T_{\ge 0}$ at its root node to two infinite sequences of nodes containing nested subtrees and disjoint cotrees. The nodes and conjugate inner arrows within each subtree and within each cotree correspond one-to-one to the nodes and arrows within tree $T_{\ge 0}$. The cumulative density of the congruence classes of numbers in disjoint cotrees of $T_{\ge 0}$ amounts to the density $2/18$ of all branching numbers, thus proving the conjecture.Comment: Collatz paths between branching classes $[4,16]_18$ and partitioning of the binary tree $T_{ge 0}$ into disjoint cotrees now validate in v5 the density proofs in previous versions (v1 31 Aug 2000

System-wide stress testing & systemic risk

The financial crisis of 2007-2009, which brought the entire system at the brink of collapse, renewed efforts to guard against financial instability. A key pillar of the post-crisis regulatory toolkit is "stress testing". Stress tests provide a forward-looking examination of firms’ potential losses during severely ad- verse conditions. And enable timely action to recapitalise those firms who experience capital shortfalls in such crisis scenarios. Today’s regulatory stress tests do not heed the key lesson of the financial crisis: amplifications in the networked financial system must be taken into account to be able to assess systemic risk. Because of this, these tests are unable to assess systemic risk and ergo to address it – defeating their raison d’ˆetre. The overarching research question in this thesis is whether new building blocks – expressing the heterogeneity of institutions, contracts, markets, constraints and behaviour in the interconnected financial system – can be supplied for system-wide stress tests to better capture the endogenous amplification of shocks in order to improve the assessment of systemic risk and the evaluation of prudential policies to address financial fragility. The cornerstone of my thesis is the development of a generic network-based method, comprised of these five building blocks (i.e. institutions, contracts, markets, constraints and behaviour), for system-wide stress testing – which has gained traction from leading central banks, including the Bank of England and the European Central Bank. Using this method, I implement two data-driven models to address some of the most salient financial stability questions of today. First, we ask how the regulatory buffer size and its usability under Basel III affect systemic risk? We find that financial resilience decreases if regulatory buffers are seen to be less usable by banks. If regulatory buffers are not treated as usable, then regulatory buffers de facto act as capital requirements. In such case, if an adverse shock threatens an institution to breach its capital buffers constraints, it is forced to delever, which tends to have a destabilising effect on the financial markets. We show that the size of usable regulatory buffers that is required to maintain stability is underestimated if the interaction between exposure loss contagion, funding contagion, overlapping portfolio contagion and margin call contagion is not taken into account. Second, we inquire what the systemic implications are of the bail-in design to resolve systemically important banks? First of all, we find that the bail-in design tremendously matters for whether bail-ins can be credibly executed in system-wide financial crises and cases of large systemically impor- tant bank failures, without significantly exacerbating financial distress. Our results demonstrate that an early bail-in, strong recapitalisation and fair distribution of equity compensation by means of debt-to-equity conversion rates makes bail-in a feasible option on the table for idiosyncratic cases of bank failure and limits – but not eliminates – contagion in cases of system-wide distress. We further show that excluding run-prone, short-term debt from the application of the bail-in tool, increasing the requirements on loss absorbing debt and providing investors with certainty about the bail-in design lowers contagion in system-wide crises to manageable levels. Our findings highlight that while well-designed bail-ins could be credibly administered in system-wide crises, it is not clear that the current bail-in design is in the regime of stability. Altogether, the methods and findings of this thesis emphasise the promise that system-wide stress tests hold for regulators to efficaciously assess systemic risk and calibrate prudential policies constituting the financial architecture

Strategies for Improving Semi-automated Topic Classification of Media and Parliamentary documents

Since 1995 the techniques and capacities to store new electronic data and to make it available to many persons have become a common good. As of then, different organizations, such as research institutes, universities, libraries, and private companies (Google) started to scan older documents and make them electronically available as well. This has generated a lot of new research opportunities for all kinds of academic disciplines. The use of software to analyze large datasets has become an important part of doing research in the social sciences. Most academics rely on human coded datasets, both in qualitative and quantitative research. However, with the increasing amount of datasets and the complexity of the questions scholars pose to the datasets, the quest for more efficient and effective methods is now on the agenda. One of the most common techniques of content analysis is the Boolean key-word search method. To find certain topics in a dataset, the researcher creates first a list of keywords, added with certain parameters (AND, OR etc.). All keys are usually grouped in families and the entire list of keys and groups is called the ontology. Then the keywords are searched in the dataset, retrieving all documents containing the specified keywords. The online newspaper dataset, LexisNexis, provides the user with such a Boolean search method. However, the Boolean key-word search is not always satisfying in terms of reliability and validity. For that reason social scientists rely on hand-coding. Two projects that do so are the congressional bills project (www.congressionalbills.org ) and the policy agenda-setting project (see www.policyagendas.org ). They developed a topic code book and coded various different sources, such as, the state of the union speeches, bills, newspaper articles etcetera. The continuous improving automated coding techniques, and the increasing number of agenda setting projects (in especially European countries), however, has made the use of automated coding software a feasible option and also a necessity

The effect of realistic geometries on the susceptibility-weighted MR signal in white matter

Purpose: To investigate the effect of realistic microstructural geometry on the susceptibility-weighted magnetic resonance (MR) signal in white matter (WM), with application to demyelination. Methods: Previous work has modeled susceptibility-weighted signals under the assumption that axons are cylindrical. In this work, we explore the implications of this assumption by considering the effect of more realistic geometries. A three-compartment WM model incorporating relevant properties based on literature was used to predict the MR signal. Myelinated axons were modeled with several cross-sectional geometries of increasing realism: nested circles, warped/elliptical circles and measured axonal geometries from electron micrographs. Signal simulations from the different microstructural geometries were compared to measured signals from a Cuprizone mouse model with varying degrees of demyelination. Results: Results from simulation suggest that axonal geometry affects the MR signal. Predictions with realistic models were significantly different compared to circular models under the same microstructural tissue properties, for simulations with and without diffusion. Conclusion: The geometry of axons affects the MR signal significantly. Literature estimates of myelin susceptibility, which are based on fitting biophysical models to the MR signal, are likely to be biased by the assumed geometry, as will any derived microstructural properties.Comment: Accepted March 4 2017, in publication at Magnetic Resonance in Medicin

The impact of the explosion of EU news on voter choice in the 2014 EU Elections

The European elections in 2014 were the first to be held after a long period in which EU-related news was prominent in the media. They were held after years of daily news about the euro crisis and after months of news about the popular uprising in the Ukraine against president Yanukovych, who had refused to sign the association agreement with the EU. This could have invited political parties to overcome the usual problem of low salience of EU issues by strongly profiling themselves on EU issues. Turnout at the 2014 EU elections, however, remained low, hinting that parties were unable to convert the attention for European issues into enthusiasm for their party at the European elections. This paper asks how vote choice was influenced by party campaigning on EU related issues. A news effects analysis based on a content analysis of Dutch newspapers and television, and on a panel survey among Dutch voters revealed that EU issues functioned as wedge issues: the more strongly parties were associated in the news with the euro crisis and the Ukraine, the less they succeeded in mobilizing voters

Politieke kennis en effecten van nieuws

Welk nieuws doet ertoe en hoeveel nieuws doet ertoe, en in welke mate hangt dit af van de politieke kennis van de ontvanger? Dit artikel beschrijft een longitudinale studie naar de verkiezingsstrijd voor het Nederlandse parlement in 2006

News, Ads, Chats, and Property Rights over Algorithms

The success of tech firms rests on their ownership of the algorithms for operating new platforms for the interactions among five groups of stakeholders in the markets of news, ads, and chats: stakeholders from the spheres of politics, journalism, the citizenry, the tech firms themselves, and other firms. Recent regulations that touch on property rights such as the German Netzwerkdurchsetzungsgesetz and the European Directive on Copyright in the Digital Market have turned ownership of algorithms into exclusive ownership. Thereby tech firms obtain also the right to censor and the exclusive right to micro-target clients for advertisers. Coase’s theorem is used to discuss alternative allocations of property rights that could improve the quality of news, ads, and chats