The Painter’s Wife and Other Stories is a novella and a collection of short fiction focusing on the lives of women and men in the contemporary Western United States. In their exploration of Western life, the novella and short stories subvert the popular mythology of the West. The novella and stories are set in one of three Colorado settings: the city of Denver, the rural Western slope, and the mountain communities of the Western Slope. Beyond being linked by region, the stories are also loosely linked by characters. Characters from one story are mentioned or make brief appearances in others, reinforcing the idea of a people connected by community and landscape.

The use of stars as a radiation source for ground-based ozone remote-sensing instruments is explored and an automated prototype instrument that measures absorption due to atmospheric ozone in stellar spectra has been designed, implemented and tested. <p> This work represents the proof-of-concept development of a low-cost, low dispersion slitless imaging spectrometer that measures Chappuis-band absorption in stellar spectra. The work presented here progresses from the initial concept to a functional calibrated prototype that is capable of nightly automated observations of visible-band spectra from mid-magnitude stars. The design and calibration of the prototype and subsequent data collection and analysis are presented. <p>A slitless imaging spectrometer has been developed and integrated with a commercial self-pointing telescope and an astronomical imager. A relative intensity calibration and the development of a dynamic wavelength calibration scheme, necessitated by the slitless nature of the instrument, is presented. The calibrated prototype has been used to collect several data sets of stellar spectra, and it is shown that the instrument can detect Chappuis absorption in stellar spectra. Several issues with both the concept and design that must be addressed in further development of the prototype are identified.

Multi-core systems have become commonplace, however, theorem provers often do not take advantage of the additional computing resources in an interactive setting. This research explores automatically using these additional resources to lessen the delay between when users submit conjectures to the theorem prover and when they receive feedback from the prover that is useful in discovering how to successfully complete the proof of a particular theorem. This research contributes mechanisms that permit applicative programs to execute in parallel while simultaneously preparing these programs for verification by a semi-automatic reasoning system. It also contributes a parallel version of an automated theorem prover, with management of user interaction issues, such as output and how inherently single-threaded, user-level proof features can be configured for use with parallel computation. Finally, this dissertation investigates the types of proofs that are amenable to parallel execution. This investigation yields the result that almost all proof attempts that require a non-trivial amount of time can benefit from parallel execution. Proof attempts executed in parallel almost always provide the aforementioned feedback sooner than if they executed serially, and their execution time is often significantly reduced. / text

How can a client extract useful work from a server without trusting it to compute correctly? A modern motivation for this classic question is third party computing models in which customers outsource their computations to service providers (as in cloud computing). In principle, deep results in complexity theory and cryptography imply that it is possible to verify that an untrusted entity executed a computation correctly. For instance, the server can employ probabilistically checkable proofs (PCPs) in conjunction with cryptographic commitments to generate a succinct proof of correct execution, which the client can efficiently check. However, these theoretical solutions are impractical: they require thousands of CPU years to verifiably execute even simple computations. This dissertation describes the design, implementation, and experimental evaluation viiiof a system, called Pepper, that brings this theory into the realm of plausibility. Pepper incorporates a series of algorithmic improvements and systems engineering techniques to improve performance by over 20 orders of magnitude, relative to an implementation of the theory without our refinements. These include a new probabilistically checkable proof encoding with nearly optimal asymptotics, a concise representation for computations, a more efficient cryptographic commitment primitive, and a distributed implementation of the server with GPU acceleration to reduce latency. Additionally, Pepper extends the verification machinery to handle realistic applications of third party computing: those that interact with remote storage or state (e.g., MapReduce jobs, database queries). To do so, Pepper composes techniques from untrusted storage with the aforementioned technical machinery to verifiably offload both computations and state. Furthermore, to make it easy to use this technology, Pepper includes a compiler to automatically transform programs in a subset of C into executables that run verifiably. One of the chief limitations of Pepper is that verifiable execution is still orders of magnitude slower than an unverifiable native execution. Nonetheless, Pepper takes powerful results from complexity theory and verifiable computation a few steps closer to practicality / text

This study of student-to-student discussions focuses on a single inquiry-oriented transition to proof course. Mathematical proof is essential to a strong mathematics education but very often students complete their mathematics studies with limited abilities to construct and validate mathematical proofs (c.f. Harel & Sowder, 1998; Knuth, 2002; Almeida, 2000). The role of mathematical proof in education is to provide explanation and understanding. Both the research on mathematical discourse and the standards of the NCTM claim that participation in mathematical discourse provides opportunities for understanding. Although this link has been established, there is very little research on the role of students and the instructor during discussions on student generated proofs at the undergraduate level -- particularly in inquiry-oriented classes. This research analyzes the types of discussions that occurred in an inquiry-oriented undergraduate mathematics course in which proof was the main content. The discussions of interest involved at least two student participants and at least three separate utterances. These discussions fell along a continuum based on the level of student interaction. As a result of this research, the four main discussion types that were present in this course have been described in detail with a focus on the roles of the instructor and the students. The methodology for this research is qualitative in nature and is an exploratory case study. The data used for this research was video tapes of two to three class sessions per week of an Introduction to Number Theory course taught in the fall of 2005. / text

In “Proof-Theoretic Justification of Logic”, building on work by Dummett and Prawitz, I show how to construct use-based meaning-theories for the logical constants. The assertability-conditional meaning-theory takes the meaning of the logical constants to be given by their introduction rules; the consequence-conditional meaning-theory takes the meaning of the logical constants to be given by their elimination rules. I then consider the question: given a set of introduction (elimination) rules \(\mathcal{R}\), what are the strongest elimination (introduction) rules that are validated by an assertability (consequence) conditional meaning-theory based on \(\mathcal{R}\)? I prove that the intuitionistic introduction (elimination) rules are the strongest rules that are validated by the intuitionistic elimination (introduction) rules. I then prove that intuitionistic logic is the strongest logic that can be given either an assertability-conditional or consequence-conditional meaning-theory. In “Grounding Grounding” I discuss the notion of grounding. My discussion revolves around the problem of iterated grounding-claims. Suppose that \(\Delta\) grounds \(\phi\); what grounds that \(\Delta\) grounds that \(\phi\)? I argue that unless we can get a satisfactory answer to this question the notion of grounding will be useless. I discuss and reject some proposed accounts of iterated grounding claims. I then develop a new way of expressing grounding, propose an account of iterated grounding-claims and show how we can develop logics for grounding. In “Is the Vagueness Argument Valid?” I argue that the Vagueness Argument in favor of unrestricted composition isn’t valid. However, if the premisses of the argument are true and the conclusion false, mereological facts fail to supervene on non-mereological facts. I argue that this failure of supervenience is an artifact of the interplay between the necessity and determinacy operators and that it does not mean that mereological facts fail to depend on non-mereological facts. I sketch a deflationary view of ontology to establish this. / Philosophy

This report discusses the history, use, and future of elliptic curves. Uses of elliptic curves in various number theory settings are presented. Fermat’s Last Proof is shown to be proven with elliptic curves. Finally, the future of elliptic curves with respect to cryptography and primality is shown. / text

Η παρούσα εργασία έχει ως θέμα τη μαθηματική απόδειξη και την διαδικασία επίλυσης προβλήματος. Στόχος της είναι αρχικά, να παρουσιάσει το θεωρητικό υπόβαθρο που διέπει αυτά τα δύο θέματα και να κάνει μια σύγκριση ώστε να αναδειχθούν οι διαφορές τους και οι ομοιότητες τους. Στην συνέχεια, γίνεται μια σύντομη παρουσίαση των Αναλυτικών Προγραμμάτων και των διδακτικών εγχειριδίων των Μαθηματικών του Λυκείου για το χρονικό διάστημα από τα τέλη της δεκαετίας του 1980 έως σήμερα έχοντας ως κύριο άξονα, την απόδειξη και την επίλυση προβλήματος. Κατόπιν, με την βοήθεια μιας δραστηριότητας κατάλληλα διαμορφωμένης εξετάζετε ο ρόλος των παραπάνω στους μαθητές και τέλος, γίνετε μια σύντομη ανάλυση της Γραμμικής και Δομικής μορφής της απόδειξης, όπως αυτή είχε προταθεί από τον Uri Leron και μια συγκριτική παρουσίαση των αποδείξεων κάποιων θεωρημάτων του σχολικού βιβλίου της Γεωμετρίας της Α΄ Λυκείου (Αργυρόπουλος Η.) και με τις δύο μορφές. / The objective of this Master Thesis is the presentation of the Mathematical Proof and Problem Solving. Its aim is initially to present the theoretical background behind these two issues and a comparison between the Mathematical Proof and Problem Solving with respect to their similarities and differences takes place. Then a brief presentation of the curriculum programs as well as the school books of mathematics is given. This presentation is about the time period from the late decade of 1980 up to date, mostly concerning the Mathematical Proof and Problem Solving. Moreover, using a suitably formulated activity, the role of the above over the students is studied. Finally, a concise analysis of Linear and Structural style of proof as it suggested by Uri Leron is given. The thesis is completed with the presentation of three theorems along with their proofs (Linear style) as they are stated in the section of "Inequality Relationships" of Geometry school book of A Lyceum class (Αργυρόπουλος Η. 2008), while for each proof its Structural style is also given.

The Painter’s Wife and Other Stories is a novella and a collection of short fiction focusing on the lives of women and men in the contemporary Western United States. In their exploration of Western life, the novella and short stories subvert the popular mythology of the West. The novella and stories are set in one of three Colorado settings: the city of Denver, the rural Western slope, and the mountain communities of the Western Slope. Beyond being linked by region, the stories are also loosely linked by characters. Characters from one story are mentioned or make brief appearances in others, reinforcing the idea of a people connected by community and landscape.

In this dissertation it is shown that abstract data types (ADTs) can be specified by the Classified Model (CM) specification language - a first-order Horn language with equality and sort "classification" assertations. It is shown how these sort assertations generalize the traditional syntactic signatures of ADT specifications, resulting in all of the specification capability of traditional equational specifications, but with the improved expressibility of the Horn-with-equality language and additional theorem proving applications such as program synthesis. This work extends corresponding results from Many Sorted Algebra (MSA), Order Sorted Algebra (OSA) and Order Sorted Model (OSM) specification techniques by promoting their syntactic signatures to assertions in the Classified Model Specification language, yet retaining sorted quantification. It is shown how this solves MSA problems such as error values, polymorphism and subtypes in a way different from the OSA and OSM solutions. However, the CM technique retains the MSA and order sorted approach to parameterization. The CS generalization also suggests the use of CM specifications to axiomatize modules as a generalization of variables within Hoare Logic, with application to a restricted, but safe, use of procedures as state changing operations and functions as value returning operations of a module. CM proof theory and semantics are developed, including theorems for soundness, completeness and the existence of a free model.