Real Computational Universality: The Word Problem for a class of groups with infinite presentation

The word problem for discrete groups is well-known to be undecidable by a Turing Machine; more precisely, it is reducible both to and from and thus equivalent to the discrete Halting Problem. The present work introduces and studies a real extension of the word problem for a certain class of groups which are presented as quotient groups of a free group and a normal subgroup. Most important, the free group will be generated by an uncountable set of generators with index running over certain sets of real numbers. This allows to include many mathematically important groups which are not captured in the framework of the classical word problem. Our contribution extends computational group theory from the discrete to the Blum-Shub-Smale (BSS) model of real number computation. We believe this to be an interesting step towards applying BSS theory, in addition to semi-algebraic geometry, also to further areas of mathematics. The main result establishes the word problem for such groups to be not only semi-decidable (and thus reducible FROM) but also reducible TO the Halting Problem for such machines. It thus provides the first non-trivial example of a problem COMPLETE, that is, computationally universal for this model.Comment: corrected Section 4.

An Abstraction of Whitney's Broken Circuit Theorem

We establish a broad generalization of Whitney's broken circuit theorem on the chromatic polynomial of a graph to sums of type $\sum_{A\subseteq S} f(A)$ where $S$ is a finite set and $f$ is a mapping from the power set of $S$ into an abelian group. We give applications to the domination polynomial and the subgraph component polynomial of a graph, the chromatic polynomial of a hypergraph, the characteristic polynomial and Crapo's beta invariant of a matroid, and the principle of inclusion-exclusion. Thus, we discover several known and new results in a concise and unified way. As further applications of our main result, we derive a new generalization of the maximums-minimums identity and of a theorem due to Blass and Sagan on the M\"obius function of a finite lattice, which generalizes Rota's crosscut theorem. For the classical M\"obius function, both Euler's totient function and its Dirichlet inverse, and the reciprocal of the Riemann zeta function we obtain new expansions involving the greatest common divisor resp. least common multiple. We finally establish an even broader generalization of Whitney's broken circuit theorem in the context of convex geometries (antimatroids).Comment: 18 page

Interaction of market and credit risk: an analysis of inter-risk correlation and risk aggregation

In this paper we investigate the interaction between a credit portfolio and another risk type, which can be thought of as market risk. Combining Merton-like factor models for credit risk with linear factor models for market risk, we analytically calculate their interrisk correlation and show how inter-risk correlation bounds can be derived. Moreover, we elaborate how our model naturally leads to a Gaussian copula approach for describing dependence between both risk types. In particular, we suggest estimators for the correlation parameter of the Gaussian copula that can be used for general credit portfolios. Finally, we use our findings to calculate aggregated risk capital of a sample portfolio both by numerical and analytical techniques. -- Die Berechnung einer bankweit aggregierten Risikokennzahl (normalerweise ausgedrückt durch das ökonomische Kapital) ist ein äußerst wichtiger Bestandteil eines modernen Risikocontrollings and als solches von besonderer Bedeutung für bankinterne als auch regulatorische Zwecke. Eine wichtige Frage dabei betrifft die Behandlung von risikoreduzierenden Diversifikationseffekten, die als Folge der Geschäftsstrategie einer Bank (z.B. durch Produktdiversifikation oder geografische Diversifikation) auftreten können. Solche Diversifikationseffekte stellen einen Wettbewerbsvorteil dar, den Banken deshalb bei der Bestimmung ihrer Kapitaladäquanz mit einbeziehen wollen. Auch die Bankenaufsicht erkennt in ihren Ausführungen über die bankinternen Kapitalbeurteilungsverfahren nach den Grundsätzen der zweiten Säule von Basel II die Existenz von Diversifikationseffekten an. Bei der praktischen Berechnung des Diversifikationseffektes unterscheidet man oft zwischen Intrarisiko- und Interrisikodiversifikation. Letztere behandelt die Diversifikation innerhalb einer Risikoart (z.B. Markt- oder Kreditrisiko), wohingegen Interrisiko-Diversifikation die Diversifikation zwischen verschiedenen Risikoarten beschreibt und meist durch eine Interrisiko-Korrelationsmatrix erfasst wird.Risk aggregation,Inter-risk correlation,economic capital,ICAAP,diversification

How does competition affect efficiency and soundness in banking? New empirical evidence

A growing body of literature indicates that competition increases bank soundness. Applying an industrial organization based approach to large data sets for European and U.S. banks, we offer new empirical evidence that efficiency plays a key role in the transmission from competition to soundness. We use a two-pronged approach. First, we employ Granger causality tests to establish the link between competition and measures of profit efficiency in banking, and find that competition indeed increases bank efficiency. Second, building on these results, we examine the relation between the Boone indicator [Boone, J. (2001) Intensity of competition and the incentive to innovate. IJIO, Vol. 19, pp. 705-726], an innovative measure of competition that focuses on the impact of competition on performance of efficient banks, and relate this measure to bank soundness. We find evidence that competition robustly increases bank soundness, via the efficiency channel. JEL Classification: G21, G28, L11Bank competition, Efficiency, market structure, regulation, soundness

Dynamic Occupancy Grid Prediction for Urban Autonomous Driving: A Deep Learning Approach with Fully Automatic Labeling

Long-term situation prediction plays a crucial role in the development of intelligent vehicles. A major challenge still to overcome is the prediction of complex downtown scenarios with multiple road users, e.g., pedestrians, bikes, and motor vehicles, interacting with each other. This contribution tackles this challenge by combining a Bayesian filtering technique for environment representation, and machine learning as long-term predictor. More specifically, a dynamic occupancy grid map is utilized as input to a deep convolutional neural network. This yields the advantage of using spatially distributed velocity estimates from a single time step for prediction, rather than a raw data sequence, alleviating common problems dealing with input time series of multiple sensors. Furthermore, convolutional neural networks have the inherent characteristic of using context information, enabling the implicit modeling of road user interaction. Pixel-wise balancing is applied in the loss function counteracting the extreme imbalance between static and dynamic cells. One of the major advantages is the unsupervised learning character due to fully automatic label generation. The presented algorithm is trained and evaluated on multiple hours of recorded sensor data and compared to Monte-Carlo simulation

International Migration, Ethnicity and Economic Inequality

While the allocative efficiency of mobility is typically considered to be positive but small in the long run, the induced changes in equality may be considerable in size. In practice, however, migrants typically improve their income position in comparison to those at home, stimulate the economic situation of the sending countries through remittances and rise the economic performance of natives and of capital in the host country through complementarities. The chapter suggests that at least skilled immigration promotes economic equality in the host country under standard conditions. The context is empirically documented und theoretically explained in a core model. Also, immigrant assimilation and selection is discussed, as is the role of ethnicity and ethnic identity for relative economic performance.ethnosizing, inequality, income distribution, migration, ethnicity, minority, assimilation, integration, Gini-coefficient

Theory and Test on the Corporate Governance of Financial Cooperative Systems: Merger vs. Networks

This paper presents a study of the economic organization of systems of financial cooperatives (FC). The first part presents a theoretical framework rooted in principles of transaction cost economics (TCE) that seeks to explain empirical regularities observable in systems of FC worldwide. The second part is an empirical study that compares X-efficiency between members of the Quebec Desjardins movement (DM) and the United States Credit Union system (USCU), the first organized as a tight network of institutions and the second composed largely by independent institutions with few ties. The fundamental proposition is that networks, are a superior form of governance mechanism (over markets and mergers) for relatively wide and relevant ranges of contractual hazard and size of the institutions. Further, that networks provide substitute, hierarchy based, control mechanisms when size of the institution dilutes internal governance mechanisms, discouraging subgoal pursuits and expense preferences by agents, both occurring in large FC. The theory allows us to generate a set of testable hypothesis of which we highlight three: i) For small FC, differences in efficiency will be relatively small, if any. ii) Large institutions should display systematically lower efficiency than similar sized FC members of strategic networks. iii) Networks should display lower variance in the size as well as in performance indicators. Throughout, empirical results are consistent with our central theoretical proposition.Transaction cost economics, financial cooperatives, credit unions, networks, corporate governance, technical efficiency, X-efficiency