Formalizing Mathematical Knowledge as a Biform Theory Graph: A Case Study

A biform theory is a combination of an axiomatic theory and an algorithmic theory that supports the integration of reasoning and computation. These are ideal for formalizing algorithms that manipulate mathematical expressions. A theory graph is a network of theories connected by meaning-preserving theory morphisms that map the formulas of one theory to the formulas of another theory. Theory graphs are in turn well suited for formalizing mathematical knowledge at the most convenient level of abstraction using the most convenient vocabulary. We are interested in the problem of whether a body of mathematical knowledge can be effectively formalized as a theory graph of biform theories. As a test case, we look at the graph of theories encoding natural number arithmetic. We used two different formalisms to do this, which we describe and compare. The first is realized in ${\rm CTT}_{\rm uqe}$, a version of Church's type theory with quotation and evaluation, and the second is realized in Agda, a dependently typed programming language.Comment: 43 pages; published without appendices in: H. Geuvers et al., eds, Intelligent Computer Mathematics (CICM 2017), Lecture Notes in Computer Science, Vol. 10383, pp. 9-24, Springer, 201

Microscopic and Macroscopic Stress with Gravitational and Rotational Forces

Many recent papers have questioned Irving and Kirkwood's atomistic expression for stress. In Irving and Kirkwood's approach both interatomic forces and atomic velocities contribute to stress. It is the velocity-dependent part that has been disputed. To help clarify this situation we investigate [1] a fluid in a gravitational field and [2] a steadily rotating solid. For both problems we choose conditions where the two stress contributions, potential and kinetic, are significant. The analytic force-balance solutions of both these problems agree very well with a smooth-particle interpretation of the atomistic Irving-Kirkwood stress tensor.Comment: Fifteen pages with seven figures, revised according to referees' suggestions at Physical Review E. See also Liu and Qiu's arXiv contribution 0810.080

The Wrath of God and the Grace of God in Lutheran Theology

Every Lutheran theologian, at hearing these terms, will admit, I think, that in discussing them we deal with the very heart of theology, and not only of theology, but of religion itself. Wherever religion has not developed into a mere caricature, but is live, spontaneous, heartfelt, real, it occupies itself, among other things, with these concepts. The statements that an individual or whole groups make about them may be entirely erroneous and objectionable or highly unsatisfactory, but, at any rate, occupying one\u27s self with them is unavoidable - the human heart simply has to come to grips with these matters. The explanation is that we are moral beings, having a sense of right and wrong, and that, moreover, we have implanted in us a certain knowledge of God which compels us to ask, How about the wrath of God and the grace of God

Production of a T cell hybridoma that expresses the T cell receptor gamma/delta heterodimer.

We have produced a T cell hybridoma line by fusion of an IL-2-dependent, long-term T cell receptor (TCR) gamma/delta+ Thy-1+, bone marrow-derived, dendritic epidermal cell line to the BW5147 tumor line. The resultant hybridoma was rapidly growing, lymphokine independent, and expressed T3 in association with the TCR gamma/delta heterodimer. Several subclones of the hybridoma line produced easily detectable levels of IL-2 after stimulation by anti-T3 or Con A. The availability of these cloned cell lines should greatly facilitate further functional, biochemical, and molecular studies of the TCR delta chain