35 research outputs found
Continuous images of Cantor's ternary set
The Hausdorff-Alexandroff Theorem states that any compact metric space is the
continuous image of Cantor's ternary set . It is well known that there are
compact Hausdorff spaces of cardinality equal to that of that are not
continuous images of Cantor's ternary set. On the other hand, every compact
countably infinite Hausdorff space is a continuous image of . Here we
present a compact countably infinite non-Hausdorff space which is not the
continuous image of Cantor's ternary set
An Introduction to Set Theory and Topology
These notes are an introduction to set theory and topology. They are the result of teaching a two-semester course sequence on these topics for many years at Washington University in St. Louis. Typically the students were advanced undergraduate mathematics majors, a few beginning graduate students in mathematics, and some graduate students from other areas that included economics and engineering. The usual background for the material is an introductory undergraduate analysis course, mostly because it provides a solid introduction to Euclidean space Rn and practice with rigorous arguments — in particular, about continuity. Strictly speaking, however, the material is mostly self-contained. Examples are taken now and then from analysis, but they are not logically necessary for the development of the material. The only real prerequisite is the level of mathematical interest, maturity and patience needed to handle abstract ideas and to read and write careful proofs. A few very capable students have taken this course before introductory analysis (even, rarely, outstanding university freshmen) and invariably they have commented later on how material eased their way into analysis.https://openscholarship.wustl.edu/books/1020/thumbnail.jp
Categoricity
After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those involving the distinction between characterizing a system and axiomatizing the truths of a syste
Follow the Flow: sets, relations, and categories as special cases of functions with no domain
We introduce, develop, and apply a new approach for dealing with the intuitive notion of function, called Flow Theory. Within our framework all functions are monadic and none of them has any domain. Sets, proper classes, categories, functors, and even relations are special cases of functions. In this sense, functions in Flow are not equivalent to functions in ZFC. Nevertheless, we prove both ZFC and Category Theory are naturally immersed within Flow. Besides, our framework provides major advantages as a language for axiomatization of standard mathematical and physical theories. Russell's paradox is avoided without any equivalent to the Separation Scheme. Hierarchies of sets are obtained without any equivalent to the Power Set Axiom. And a clear principle of duality emerges from Flow, in a way which was not anticipated neither by Category Theory nor by standard set theories. Besides, there seems to be within Flow an identification not only with the common practice of doing mathematics (which is usually quite different from the ways proposed by logicians), but even with the common practice of teaching this formal science
NASA Microgravity Materials Science Conference
The Microgravity Materials Science Conference was held June 10-11, 1996 at the Von Braun Civic Center in Huntsville, AL. It was organized by the Microgravity Materials Science Discipline Working Group, sponsored by the Microgravity Science and Applications Division at NASA Headquarters, and hosted by the NASA Marshall Space Flight Center and the Alliance for Microgravity Materials Science and Applications (AMMSA). It was the second NASA conference of this type in the microgravity materials science discipline. The microgravity science program sponsored approximately 80 investigations and 69 principal investigators in FY96, all of whom made oral or poster presentations at this conference. The conference's purpose was to inform the materials science community of research opportunities in reduced gravity in preparation for a NASA Research Announcement (NRA) scheduled for release in late 1996 by the Microgravity Science and Applications Division at NASA Headquarters. The conference was aimed at materials science researchers from academia, industry, and government. A tour of the MSFC microgravity research facilities was held on June 12, 1996. This volume is comprised of the research reports submitted by the principal investigators after the conference and presentations made by various NASA microgravity science managers
Spatio-Temporal Analyses of Cenozoic Normal Faulting, Graben Basin Sedimentation, and Volcanism around the Snake River Plain, SE Idaho and SW Montana
This dissertation analyzes the spatial distribution and kinematics of the Late Cenozoic Basin and Range (BR) and cross normal fault (CF) systems and their related graben basins around the Snake River Plain (SRP), and investigates the spatio-temporal patterns of lavas that were erupted by the migrating Yellowstone hotspot along the SRP, applying a diverse set of GIS-based spatial statistical techniques. The spatial distribution patterns of the normal fault systems, revealed by the Ripley\u27s K-function, display clustered patterns that correlate with a high linear density, maximum azimuthal variation, and high box-counting fractal dimensions of the fault traces. The extension direction for normal faulting is determined along the major axis of the fractal dimension anisotropy ellipse measured by the modified Cantor dust method and the minor axis of the autocorrelation anisotropy ellipse measured by Ordinary Kriging, and across the linear directional mean (LDM) of the fault traces. Trajectories of the LDMs for the cross faults around each caldera define asymmetric sub-parabolic patterns similar to the reported parabolic distribution of the epicenters, and indicate sub-elliptical extension about each caldera that may mark the shape of hotspot’s thermal doming that formed each generation of cross faults. The decrease in the spatial density of the CFs as a function of distance from the axis of the track of the hotspot (SRP) also suggests the role of the hotspot for the formation of the cross faults. The parallelism of the trend of the exposures of the graben filling Sixmile Creek Formation with the LDM of their bounding cross faults indicates that the grabens were filled during or after the CF event. The global and local Moran’s I analyses of Neogene lava in each caldera along the SRP reveal a higher spatial autocorrelation and clustering of rhyolitic lava than the coeval basaltic lava in the same caldera. The alignment of the major axis of the standard deviational ellipses of lavas with the trend of the eastern SRP, and the successive spatial overlap of older lavas by progressively younger mafic lava, indicate the migration of the centers of eruption as the hotspot moved to the northeast
Spatio-Temporal Analyses of Cenozoic Normal Faulting, Graben Basin Sedimentation, and Volcanism around the Snake River Plain, SE Idaho and SW Montana
This dissertation analyzes the spatial distribution and kinematics of the Late Cenozoic Basin and Range (BR) and cross normal fault (CF) systems and their related graben basins around the Snake River Plain (SRP), and investigates the spatio-temporal patterns of lavas that were erupted by the migrating Yellowstone hotspot along the SRP, applying a diverse set of GIS-based spatial statistical techniques. The spatial distribution patterns of the normal fault systems, revealed by the Ripley\u27s K-function, display clustered patterns that correlate with a high linear density, maximum azimuthal variation, and high box-counting fractal dimensions of the fault traces. The extension direction for normal faulting is determined along the major axis of the fractal dimension anisotropy ellipse measured by the modified Cantor dust method and the minor axis of the autocorrelation anisotropy ellipse measured by Ordinary Kriging, and across the linear directional mean (LDM) of the fault traces. Trajectories of the LDMs for the cross faults around each caldera define asymmetric sub-parabolic patterns similar to the reported parabolic distribution of the epicenters, and indicate sub-elliptical extension about each caldera that may mark the shape of hotspot’s thermal doming that formed each generation of cross faults. The decrease in the spatial density of the CFs as a function of distance from the axis of the track of the hotspot (SRP) also suggests the role of the hotspot for the formation of the cross faults. The parallelism of the trend of the exposures of the graben filling Sixmile Creek Formation with the LDM of their bounding cross faults indicates that the grabens were filled during or after the CF event. The global and local Moran’s I analyses of Neogene lava in each caldera along the SRP reveal a higher spatial autocorrelation and clustering of rhyolitic lava than the coeval basaltic lava in the same caldera. The alignment of the major axis of the standard deviational ellipses of lavas with the trend of the eastern SRP, and the successive spatial overlap of older lavas by progressively younger mafic lava, indicate the migration of the centers of eruption as the hotspot moved to the northeast
Holy Trinity and villainous multiplicity in the Christian shape of Macbeth
This thesis recapitulates the significance of number symbolism in Macbeth as outlined by A.L. Johnson and T. McAlindon and the significance of Christian imagery as outlined especially by R.M. Frye, Roy Walker and Glynne Wickham. It suggests that the three good kings in Macbeth are conceived as one of the play's trinities, with a level of allusion to the Christian Holy Trinity. Secondly, the thesis identifies linguistic and metrical similarities between allusions to biblical villains, suggesting a more coherent pattern of such allusions than is usually suggested. Allusions (several not original) are suggested to the biblical narratives of Adam, Cain and Abel, Judas Iscariot, Pontius Pilate, King Saul, Herod the Great, Herod Antipas and Lucifer. Incidentally, new sources are suggested for Lady Macbeth's sleepwalking, Macbeth's 'naked babe' soliloquy, and the death of Young Siward. Finally, the thesis attempts to bring these insights into relationship with more recent general criticism of Macbeth
Semantic Domains and Denotational Semantics
The theory of domains was established in order to have appropriate spaces on which to define semantic functions for the denotational approach to programming-language semantics. There were two needs: first, there had to be spaces of several different types available to mirror both the type distinctions in the languages and also to allow for different kinds of semantical constructs - especially in dealing with languages with side effects; and second, the theory had to account for computability properties of functions - if the theory was going to be realistic. The first need is complicated by the fact that types can be both compound (or made up from other types) and recursive (or self-referential), and that a high-level language of types and a suitable semantics of types is required to explain what is going on. The second need is complicated by these complications of the semantical definitions and the fact that it has to be checked that the level of abstraction reached still allows a precise definition of computability