Some classes of smooth bimodules over II1_1 factors and their associated 1-cohomology spaces

We study several classes of Banach bimodules over a II$_1$ factor $M$, endowed with topologies that make them ``smooth'' with respect to $L^p$-norms implemented by the trace on $M$. Letting M\subset \B= \B(L^2M), and $2\leq p < \infty$, we consider: $(1)$ the space \B(p), obtained as the completion of \B in the norm \vertiii{T}_p := \sup \{|\varphi(T)| \mid \varphi \in \B^*, \sup\{|\varphi(xYz)| \mid Y\in (\B)_1, x, z \in M\cap (L^pM)_1\} \leq 1 \}; $(2)$ the subspace \K(p)\subset \B(p), obtained as the closure in \B(p) of the space of compact operators \K(L^2M); $(3)$ the space \K_p\subset \B of operators that are \vertiii{ \, \cdot \, }_p-limits of bounded sequences of operators in \K(L^2M). We prove that \K_p are all equal to the {\it $\tau$-rank-completion} of \K(L^2M) in \B, defined by \begin{align} \text{\rm q}\K_M:= \{K\in \B(L^2M) \mid & \exists K_n \in \K(L^2M), p_n\in \mathcal P(M), \nonumber \\ & \lim_n \|p_n(K-K_n)p_n\|= 0, \lim_n\tau(1-p_n)=0\}. \nonumber \end{align} We show that any separable II$_1$ factor $M$ admits non-inner derivations into \text{\rm q}\K_M, but that any derivation \delta:M \rightarrow \text{\rm q}\K_M is a pointwise limit in $\tau$-rank-metric of inner derivations.Comment: 30 page

Computability Theory and Ordered Groups

Ordered abelian groups are studied from the viewpoint of computability theory. In particular, we examine the possible complexity of orders on a computable abelian group. The space of orders on such a group may be represented in a natural way as the set of infinite paths through a computable tree, but not all such sets can occur in this way. We describe the connection between the complexity of a basis for a group and an order for the group, and completely characterize the degree spectra of the set of bases for a group. We describe some restrictions on the possible degree spectra of the space of orders, including a connection to algorithmic randomness

Exterior billiards: systems with impacts outside bounded domains

The book contains an account of results obtained by the author and his collaborators on billiards in the complement of bounded domains and their applications in aerodynamics and geometrical optics. We consider several problems related to aerodynamics of bodies in highly rarefied media. It is assumed that the medium particles do not interact with each other and are elastically reflected when colliding with the body boundary; these assumptions drastically simplify the aerodynamics and allow to reduce it to a number of purely mathematical problems. First we examine problems of minimal resistance in the case of translational motion of bodies. These problems generalize the Newton problem of least resistance; the difference is that the bodies are generally nonconvex in our case and therefore the particles can make multiple reflections from the body surface. It is proved that typically the infimum of resistance equals zero; thus, there exist 'almost perfectly streamlined' bodies. Next we consider the generalization of Newton's problem on minimal resistance of convex axisymmetric bodies to the case of media with thermal motion of particles. Two kinds of solutions are found: first, Newton-like bodies and second, shapes obtained by gluing together two Newton-like bodies along their rear ends. Further, we state results on characterization of billiard scattering by nonconvex and rough bodies; next we solve some special problems of optimal mass transportation. These two groups of results are applied to problems of minimal and maximal resistance for bodies that move forward and at the same time slowly rotate. It is found, in particular, that the resistance of a three-dimensional convex body can be increased at most twice and decreased at most by 3.05% by roughening its surface. Next, we consider a rapidly rotating rough disc moving in a rarefied medium on the plane. It is shown that the force acting on the disc is not generally parallel to the direction of the disc motion, that is, has a nonzero transversal component. This phenomenon is called Magnus effect (proper or inverse, depending on the direction of the transversal component). We show that the kind of Magnus effect depends on the kind of disc roughness, and study this dependence. The problem of finding all admissible values of the force acting on the disc is formulated in terms of a vector-valued problem of optimal mass transportation. Finally, we describe bodies that have zero resistance when translating through a medium, and state results on existence or non-existence of bodies with mirror surface invisible in one or several directions. We also consider the problem of constructing retroreflectors: bodies with specular surface that reverse the direction of any incident beam of light

A Process Model for the Integrated Reasoning about Quantitative IT Infrastructure Attributes

IT infrastructures can be quantitatively described by attributes, like performance or energy efficiency. Ever-changing user demands and economic attempts require varying short-term and long-term decisions regarding the alignment of an IT infrastructure and particularly its attributes to this dynamic surrounding. Potentially conflicting attribute goals and the central role of IT infrastructures presuppose decision making based upon reasoning, the process of forming inferences from facts or premises. The focus on specific IT infrastructure parts or a fixed (small) attribute set disqualify existing reasoning approaches for this intent, as they neither cover the (complex) interplay of all IT infrastructure components simultaneously, nor do they address inter- and intra-attribute correlations sufficiently. This thesis presents a process model for the integrated reasoning about quantitative IT infrastructure attributes. The process model’s main idea is to formalize the compilation of an individual reasoning function, a mathematical mapping of parametric influencing factors and modifications on an attribute vector. Compilation bases upon model integration to benefit from the multitude of existing specialized, elaborated, and well-established attribute models. The achieved reasoning function consumes an individual tuple of IT infrastructure components, attributes, and external influencing factors to expose a broad applicability. The process model formalizes a reasoning intent in three phases. First, reasoning goals and parameters are collected in a reasoning suite, and formalized in a reasoning function skeleton. Second, the skeleton is iteratively refined, guided by the reasoning suite. Third, the achieved reasoning function is employed for What-if analyses, optimization, or descriptive statistics to conduct the concrete reasoning. The process model provides five template classes that collectively formalize all phases in order to foster reproducibility and to reduce error-proneness. Process model validation is threefold. A controlled experiment reasons about a Raspberry Pi cluster’s performance and energy efficiency to illustrate feasibility. Besides, a requirements analysis on a world-class supercomputer and on the European-wide execution of hydro meteorology simulations as well as a related work examination disclose the process model’s level of innovation. Potential future work employs prepared automation capabilities, integrates human factors, and uses reasoning results for the automatic generation of modification recommendations.IT-Infrastrukturen können mit Attributen, wie Leistung und Energieeffizienz, quantitativ beschrieben werden. Nutzungsbedarfsänderungen und ökonomische Bestrebungen erfordern Kurz- und Langfristentscheidungen zur Anpassung einer IT-Infrastruktur und insbesondere ihre Attribute an dieses dynamische Umfeld. Potentielle Attribut-Zielkonflikte sowie die zentrale Rolle von IT-Infrastrukturen erfordern eine Entscheidungsfindung mittels Reasoning, einem Prozess, der Rückschlüsse (rein) aus Fakten und Prämissen zieht. Die Fokussierung auf spezifische Teile einer IT-Infrastruktur sowie die Beschränkung auf (sehr) wenige Attribute disqualifizieren bestehende Reasoning-Ansätze für dieses Vorhaben, da sie weder das komplexe Zusammenspiel von IT-Infrastruktur-Komponenten, noch Abhängigkeiten zwischen und innerhalb einzelner Attribute ausreichend berücksichtigen können. Diese Arbeit präsentiert ein Prozessmodell für das integrierte Reasoning über quantitative IT-Infrastruktur-Attribute. Die grundlegende Idee des Prozessmodells ist die Herleitung einer individuellen Reasoning-Funktion, einer mathematischen Abbildung von Einfluss- und Modifikationsparametern auf einen Attributvektor. Die Herleitung basiert auf der Integration bestehender (Attribut-)Modelle, um von deren Spezialisierung, Reife und Verbreitung profitieren zu können. Die erzielte Reasoning-Funktion verarbeitet ein individuelles Tupel aus IT-Infrastruktur-Komponenten, Attributen und externen Einflussfaktoren, um eine breite Anwendbarkeit zu gewährleisten. Das Prozessmodell formalisiert ein Reasoning-Vorhaben in drei Phasen. Zunächst werden die Reasoning-Ziele und -Parameter in einer Reasoning-Suite gesammelt und in einem Reasoning-Funktions-Gerüst formalisiert. Anschließend wird das Gerüst entsprechend den Vorgaben der Reasoning-Suite iterativ verfeinert. Abschließend wird die hergeleitete Reasoning-Funktion verwendet, um mittels “What-if”–Analysen, Optimierungsverfahren oder deskriptiver Statistik das Reasoning durchzuführen. Das Prozessmodell enthält fünf Template-Klassen, die den Prozess formalisieren, um Reproduzierbarkeit zu gewährleisten und Fehleranfälligkeit zu reduzieren. Das Prozessmodell wird auf drei Arten validiert. Ein kontrolliertes Experiment zeigt die Durchführbarkeit des Prozessmodells anhand des Reasonings zur Leistung und Energieeffizienz eines Raspberry Pi Clusters. Eine Anforderungsanalyse an einem Superrechner und an der europaweiten Ausführung von Hydro-Meteorologie-Modellen erläutert gemeinsam mit der Betrachtung verwandter Arbeiten den Innovationsgrad des Prozessmodells. Potentielle Erweiterungen nutzen die vorbereiteten Automatisierungsansätze, integrieren menschliche Faktoren, und generieren Modifikationsempfehlungen basierend auf Reasoning-Ergebnissen

Essays on Quantitative Risk Management

The costly lessons from global crisis in the past decade reinforce the importance as well as challenges of risk management. This thesis explores several core concepts of quantitative risk management and provides further insight. We start with rating migration risk and propose a Mixture of Markov Chains (MMC) model to account for stochastic business cycle effects in credit rating migration risk. The model shows superior in-sample estimation and out-of-sample predication than its rivals. Compared with the naive approach the economic application suggests banks with MMC estimator will increase capital requirement in economic expansion and free up capital during recession hence it is aligned with Basel III macroprudential imitative by reducing the recession-vs-expansion gap in capital buffers. Subsequently we move to the key concept of dependence by investigating the importance of dynamic linkages between credit and equity markets. We propose a flexible regime-switching copula model to explore the dynamics of dependence and possible structure breaks with special consideration on tail dependence. The study reveals a high-dependence regime that coincides with the recent financial crisis. The backtesting results acknowledge the new model's superiority on out-of-sample VaR forecasting over purely dynamic or static copula. It can serve to emphasise the relevance for risk management of appropriately modeling complex dependence structures. Finally we discuss the risk measures and how they affect the portfolio optimisation. We contend that more successful portfolio management can be achieved by combining extreme value analysis to describe downside tail risk and dynamic copulas to model nonlinear dependence structures. Conditional Value-at-Risk is adopted as pertinent measure of downside tail risk for portfolio optimisation. Using both realised portfolio returns and a set of out-of-sample Monte Carlo experiments, our novel portfolio strategy is confronted with the de facto mean-variance approach. The results suggest that the MV approach produces suboptimal portfolios or a less desirable risk-return tradeoff

Pioneering in Psychology, 1942

A portrayal and history of psychological frontiers in which Iowa has pioneered under the leadership of Dean Seashell. This covers a time period of forty years from 1897-1937