Incompleteness and jump hierarchies

This paper is an investigation of the relationship between G\"odel's second incompleteness theorem and the well-foundedness of jump hierarchies. It follows from a classic theorem of Spector's that the relation $\{(A,B) \in \mathbb{R}^2 : \mathcal{O}^A \leq_H B\}$ is well-founded. We provide an alternative proof of this fact that uses G\"odel's second incompleteness theorem instead of the theory of admissible ordinals. We then derive a semantic version of the second incompleteness theorem, originally due to Mummert and Simpson, from this result. Finally, we turn to the calculation of the ranks of reals in this well-founded relation. We prove that, for any $A\in\mathbb{R}$, if the rank of $A$ is $\alpha$, then $\omega_1^A$ is the $(1 + \alpha)^{\text{th}}$ admissible ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a cone, the rank of $X$ is $\omega_1^X$.Comment: 11 pages. Corrects a mistake in the statements of two result

Are Antidumping Duties an Antidote for Predation?

Since price discrimination and selling below cost arise in the normal course of business and are usually legal for home firms, countering these practices by foreign firms provides a very weak rationale for antidumping duties. If antidumping duties were to provide a systematic defense against predation by foreign firms, however, a strong ''fair-trade'' justification would remain. This paper adapts the classic entry-deterrence analysis of Dixit (1979) and Brander and Spencer (1981) to provide a simple treatment of predation, which is applicable with price leadership as well as quantity leadership. Although situations of cross-border predation appear to be quite rare, foreign firms may sometimes find themselves in leadership positions if they have to make shipments and/or set prices before their home rivals. This paper shows that, in the context of such an international leadership game, predation ma y occur without dumping and vice versa. Further, when dumping and predation do coexist, a sophisticated form of antidumping duty would prevent predation, but the simple antidumping duties that are generally observed in practice will often be insufficient. Consequently, the paper challenges the ''fair-trade'' view of antidumping policy as an antidote for predation and strengthens the foundation of the counter-argument that antidumping constitutes a new insidious form of protectionism and trade harassment, which is of particularly serious concerns for small countries.trade, duopoly, Stackelberg, Cournot, antidumping, predation

The Aesthetic and Technical Problems in Designing and Mounting the Tingalary Bird: A Production Design Thesis

This thesis documents the designing and mounting of the children\u27s play, The Tingalary Bird. Designing included the set, properties and furniture, costume, make-up, lighting, and sound. A stylized concept was followed throughout the entire production. A bird cage set, furniture, and properties were built with function, mobility, color, and shape as the primary design elements. Costumes, built from day-glow colored fabric and make-up characterizing doll-like features were coordinated for an overall stylized effect. Lighting, including ultraviolet light, and tape-recorded sound, ranging from pipe organ to acid rock which periodically added a definite mood and emphasis to the plot, accentuated the make-believe and magic of children\u27s theatre. The play ran for seven performances. The children\u27s audience reacted to the fifty-minute productions by sending the cast and crew letters and drawings. The stylized design concept enabled the children\u27s audience to combine their imagination with the plot of the play. The use of ultraviolet light proved to be a spectacular technical element in the overall stylized design concept. As indicated by their drawings, the footlight shields on the forestage were distracting visual barriers to the children. The designer felt a need for the addition of more stylized trees to the forest setting which would have given the feeling of greater dimension. Designing for mobility and action, light and sound with aesthetic sensitivity as the foundation for the stylized design concept challenged the writer as designer and technical director for The Tingalary Bird

Concurrent system design: Applied mathematics & modeling in software engineering education

A hallmark of engineering design is the use of models to explore the consequences of design decisions. Sometimes these models are physical prototypes or informal drawings, but the sine qua non of contemporary practice is the use of formal, mathematical models of system structure and behavior. Whether circuit models in electrical engineering, heat-transfer models in mechanical engineering, or queuing theory models in industrial engineering, mathematics makes it possible to perform rigorous analysis that is the cornerstone of modern engineering. Until recently, such modeling was impractical for software systems. Informal models abounded, such as those created in UML1, but rigorous models from which one could derive significant properties were either so rudimentary or so tedious to use that it was difficult to justify the incremental benefit in other than the most critical of systems. In part this is a reflection of the relative immaturity of software engineering, but it also reflects a key distinction between software and traditional engineering: whereas the latter builds on numerical computation and continuous functions, software is more appropriately modeled using logic, set theory, and other aspects of discrete mathematics. Most of the models stress relationships between software components, and numerical computation is the exception rather than the norm. Recent advances in both theory and application have made it possible to model significant aspects of software behavior precisely, and to use tools to help analyze the resulting properties2,3,4. In this paper, we focus on a course developed by James Vallino and since taught and modified by Michael Lutz, to present formal modeling to our software engineering students at RIT. Our overall goals were three-fold: To acquaint our students with modern modeling tools, to connect the courses they take in discrete mathematics to real applications, and to persuade them that mathematics has much to offer to the engineering of quality software

CC187 Agriculture in Nebraska\u27s Economy

Campaign circular 187 is about agriculture in Nebraska’s Economy

Comparative Genomics of Natural Killer Cell Receptor Gene Clusters

Many receptors on natural killer (NK) cells recognize major histocompatibility complex class I molecules in order to monitor unhealthy tissues, such as cells infected with viruses, and some tumors. Genes encoding families of NK receptors and related sequences are organized into two main clusters in humans: the natural killer complex on Chromosome 12p13.1, which encodes C-type lectin molecules, and the leukocyte receptor complex on Chromosome 19q13.4, which encodes immunoglobulin superfamily molecules. The composition of these gene clusters differs markedly between closely related species, providing evidence for rapid, lineage-specific expansions or contractions of sets of loci. The choice of NK receptor genes is polarized in the two species most studied, mouse and human. In mouse, the C-type lectin-related Ly49 gene family predominates. Conversely, the single Ly49 sequence is a pseudogene in humans, and the immunoglobulin superfamily KIR gene family is extensive. These different gene sets encode proteins that are comparable in function and genetic diversity, even though they have undergone species-specific expansions. Understanding the biological significance of this curious situation may be aided by studying which NK receptor genes are used in other vertebrates, especially in relation to species-specific differences in genes for major histocompatibility complex class I molecules