The spectrum of the QCD Dirac operator and chiral random matrix theory: the threefold way

We argue that the spectrum of the QCD Dirac operator near zero virtuality can be described by random matrix theory. As in the case of classical random matrix ensembles of Dyson we have three distinct classes: the chiral orthogonal ensemble (chGOE), the chiral unitary ensemble (chGUE) and the chiral symplectic ensemble (chGSE). They correspond to gauge groups $SU(2)$ in the fundamental representation, $SU(N_c), N_c \ge 3$ in the fundamental representation, and gauge groups for all $N_c$ in the adjoint representation, respectively. The joint probability density reproduces Leutwyler-Smilga sum rules.Comment: 7 pages, SUNY-NTG-94/

Ben Marais (1909-1999): The influences on and heritage of a South African Prophet during two periods of transformation

University of Pretoria / Dissertation / Department of Church History and Church Policy / Advised by Prof J W HofmeyrThis thesis in Church History presents a biographic study on the life of Ben Marais against the political and ecclesiastic background of South Africa of the 20th century. The significance of Ben Marais’ life is approached through his correspondence with the secretaries of the World Council of Churches during the 1960s and 1970s. The letters, pertaining to the World Council of Churches financial and moral support for the organisations fighting against Apartheid, reflect on Ben Marais’ involvement with the World Council and his particular concerns. Through a study on the life of Ben Marais insight can be gained into the thinking of the leadership of the NG Kerk. The study presents Ben Marais as a prophet who challenged the then popular tendencies in the NG Kerk theology on policy justification and on the relation between religion and nationalism. The central question in this study asks, what led an ordinary man, of humble background, to the insights he reflected, and guided him through times of transparent opposition to maintain his belief in what was right and just? What was the essence of his theology and understanding of the South African problem? To what extent could the church leaders of the present, and the future learn from his example and life, in terms of the tribulations faced, different schools of thought, and sentiments, both nationalistic and spiritual? The study then wishes to test the following hypothesis: Ben Marais can be considered as one of the steadfast and humble prophets of the church in Southern Africa during the 20th century, who serves as an example of Christian Brotherhood, regardless of the perplexities, for present and future generations on relations between the affairs of faith, state and society. The thesis presents a broader introduction on Church Historiography. Ben Marais’ own historiographical reflection is considered. The approaches to history are summarised as background to the periodisation model adopted by the study. The study wishes to work with a thematic model set against a chronological framework. Sensitivity to geographical concerns is also expressed. Afrikaner Nationalism is not seen in isolation, but in relation to African, English and Indian Nationalism

Conditioning in tropical probability theory

We define a natural operation of conditioning of tropical diagrams of probability spaces and show that it is Lipschitz continuous with respect to the asymptotic entropy distance.Comment: 12 pages, V2 - updated reference

Eno-Osher schemes for Euler equations

The combination of the Osher approximate Riemann solver for the Euler equations and various ENO schemes is discussed for one-dimensional flow. The three basic approaches, viz. the ENO scheme using primitive variable reconstruction, either with Cauchy-Kowalewski procedure for time integration or the TVD Runge-Kutta scheme, and the flux-ENO method are tested on different shock tube cases. The shock tube cases were chosen to present a serious challenge to the ENO schemes in order to test their ability to capture flow discontinuities, such as shocks. Also the effect of the ordering of the eigen values, viz. natural or reversed ordering, in the Osher scheme is investigated. The ENO schemes are tested up to fifth order accuracy in space and time. The ENO-Osher scheme using the Cauchy-Kowalewski procedure for time integration is found to be the most accurate and robust compared with the other methods and is also computationally efficient. The tests showed that the ENO schemes perform reasonably well, but have problems in cases where two discontinuities are close together. In that case there are not enough points in the smooth part of the flow to create a non-oscillatory interpolation

Tropical probability theory and an application to the entropic cone

In a series of articles, we have been developing a theory of tropical diagrams of probability spaces, expecting it to be useful for information optimization problems in information theory and artificial intelligence. In this article, we give a summary of our work so far and apply the theory to derive a dimension-reduction statement about the shape of the entropic cone.Comment: 18 pages, 1 figure, V2 - updated reference

Tropical Limits of Probability Spaces, Part I: The Intrinsic Kolmogorov-Sinai Distance and the Asymptotic Equipartition Property for Configurations

The entropy of a finite probability space $X$ measures the observable cardinality of large independent products $X^{\otimes n}$ of the probability space. If two probability spaces $X$ and $Y$ have the same entropy, there is an almost measure-preserving bijection between large parts of $X^{\otimes n}$ and $Y^{\otimes n}$. In this way, $X$ and $Y$ are asymptotically equivalent. It turns out to be challenging to generalize this notion of asymptotic equivalence to configurations of probability spaces, which are collections of probability spaces with measure-preserving maps between some of them. In this article we introduce the intrinsic Kolmogorov-Sinai distance on the space of configurations of probability spaces. Concentrating on the large-scale geometry we pass to the asymptotic Kolmogorov-Sinai distance. It induces an asymptotic equivalence relation on sequences of configurations of probability spaces. We will call the equivalence classes \emph{tropical probability spaces}. In this context we prove an Asymptotic Equipartition Property for configurations. It states that tropical configurations can always be approximated by homogeneous configurations. In addition, we show that the solutions to certain Information-Optimization problems are Lipschitz-con\-tinuous with respect to the asymptotic Kolmogorov-Sinai distance. It follows from these two statements that in order to solve an Information-Optimization problem, it suffices to consider homogeneous configurations. Finally, we show that spaces of trajectories of length $n$ of certain stochastic processes, in particular stationary Markov chains, have a tropical limit.Comment: Comment to version 2: Fixed typos, a calculation mistake in Lemma 5.1 and its consequences in Proposition 5.2 and Theorem 6.