Sets, Sequences, and Counting

James H. Schmerl. Peano models with many generic classes. Pacific Journal of Mathematics, vol. 43 (1973), pp. 523–536. - James H. Schmerl. Correction to: “Peano models with many generic classes”. Pacific Journal of Mathematics, vol. 92 (1981), no. 1, pp. 195–198. - James H. Schmerl. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80 (Proceedings, Seminars, and Conferences in Mathematical Logic, University of Connecticut, Storrs, Connecticut, 1979/80). edited by M. Lerman, J. H. Schmerl, and R. I. Soare, Lecture Notes in Mathematics, vol. 859. Springer, Berlin, pp. 268–282. - James H. Schmerl. Recursively saturatedmodels generated by indiscernibles. Notre Dane Journal of Formal Logic, vol. 26 (1985), no. 1, pp. 99–105. - James H. Schmerl. Large resplendent models generated by indiscernibles. The Journal of Symbolic Logic, vol. 54 (1989), no. 4, pp. 1382–1388. - James H. Schmerl. Automorphism groups of models of Peano arithmetic. The Journal of Symbolic Logic, vol. 67 (2002), no. 4, pp. 1249–1264. - James H. Schmerl. Diversity in substructures. Nonstandard models of arithmetic and set theory. edited by A. Enayat and R. Kossak, Contemporary Mathematics, vol. 361, American Mathematical Societey (2004), pp. 45–161. - James H. Schmerl. Generic automorphisms and graph coloring. Discrete Mathematics, vol. 291 (2005), no. 1–3, pp. 235–242. - James H. Schmerl. Nondiversity in substructures. The Journal of Symbolic Logic, vol. 73 (2008), no. 1, pp. 193–211. - Volume 15 Issue 2

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsSolomon FefermanSOLOMON FEFERMAN is professor of Mathematics and Philosophy, and Suppes Family Professor in Humanities and Sciences at Stanford University. He is past President of the Association for Symbolic Logic and is a Fellow of the American Academy of Arts and Sciences. He received his Ph.D. at U.C. Berkeley in 1957 under the direction of Alfred Tarski. He is Editor-in-Chief of the Collected Works of Kurt Gödel, and the author of numerous articles on mathematical logic, the foundations of mathematics, and the history of modern logic.

Statistical and econometric analysis of the treatment of HIV/AIDS

This chapter treats two interrelated topics in discrete mathematics: elementary Set Theory (set notation; subsets, partitions, and power sets; and unions, intersections, set differences, and Cartesian products) and the theory of Combinatorics (sequential counting, permutations, and combinations). Strong connections to Propositional Logic (Chapter 1) are demonstrated, and applications are made to binomial expansion, discrete probability, and everyday counting. The material on Set Theory also provides the theoretical foundation for topics later in the text on infinite sets (Chapter 5), functions (Chapter 6), and relations (Chapters 6 and 7).

Constructing and estimating a model to explain the mechanism of neurotic disorders is important and significant. The model helps the rearrangement or representation of knowledge obtained from professional physicians. However, it is a very difficult problem, because the objects requiring analysis are mental activities of human beings, and they originally include a comparatively large variance between individuals. The object data were obtained from patients with neurotic disorders who were diagnosed by several doctors for 10 years in a subagricultural areas in Japan. We analyzed the data and calculated the weights attached for personal information depending upon the similarity between the information and the kinds of neurotic disorders using the theory of fuzzy sets. From the results of our analysis, we constructed and estimated a model explaining the mechanism causing neurotic disorders as several linear equations. From data processing points of view, the estimation we attempted is placed in a kind of effective data compression with respect to discrete statistical data.

The article considers approaches to activating the self-study work of cadets as one of the key factors in increasing the effectiveness of the educational process in higher mathematics in institutions of higher military education. The importance of developing in future specialists the skills of independent search, analysis and application of mathematical knowledge to solve professionally oriented tasks is emphasized. The article analyzes pedagogical conditions that contribute to the development of cognitive activity and independence of cadets, proposes methods, forms and means of organizing self-study work, in particular through the use of applied tasks, information and communication technologies, and elements of problem-based and project-based learning. Among the tasks of the professional orientation of military specialties, the use of mathematical fconcepts in the field of robotics is indicative. The work examines some key mathematical ideas that are mastered when studying linear algebra, differential and integral calculus, differential equations, probability theory, and discrete mathematics. Special attention is paid to the role of the teacher in shaping motivation for learning, developing critical thinking and self-control skills in students. The positive impact of increasing self-study work on the quality of learning mathematical material and on the formation of professional competence of future officers is highlighted. A key factor in teaching mathematical disciplines is the formation of sustainable motivation of students, which, in turn, stimulates their independent cognitive activity and contributes to deeper assimilation of knowledge. In general, the results of the research indicate that the effective organization of self-study of cadets while studying higher mathematics should be based on a combination of traditional and modern pedagogical technologies, taking into account the specifics of the military profession and the level of training of education seekers.

The Rubik’s Cube, a widely recognized combinational puzzle which provides a rich mathematical structure that has become an important object of study in discrete mathematics. Algorithms developed for solving it reveals how systematic sequences of moves can efficiently guide an immense state space of over possible configurations. Within the mathematical field, these algorithms describe core concepts such as permutation, symmetry, computation, optimization, and the study of algorithmic complexity. Thus, the Rubik’s Cube serves as a unique and comprehensible model for exploring how mathematical reasoning can be applied to solve high-dimensional, contrived problems. Regardless of significant progress in deriving optimal solutions such as God’s Number, heuristic algorithms, and group-theoretic approaches gaps remain in connecting theoretical optimality with practical algorithmic performance. Existing research basically focuses either on pure mathematical analysis of the cube’s structure or on speed-solving techniques, leaving limited work that combines these perspectives. Furthermore, there remains a need for deeper exploration of how algorithms help with variations of the cube, including higher-order cubes and non-standard configurations. The aim of this research is to analyze and evaluate the mathematical foundations that support algorithms for solving the Rubik’s Cube, and to investigate improved algorithmic strategies that justify theoretical optimality with practical usability. This study will adopt a mixed approach of mathematical and computational methods, involving group-theoretic modelling, analysis of permutation structures, algorithmic model, and comparative assessment of existing solving methods. Through this, the research seeks to contribute to a deeper understanding of the mathematical principles commanding Rubik’s Cube algorithms.

A class-specific feature selection and classification approach using neighborhood rough set and K-nearest neighbor theories

Based on double-helicoid swept-volume display system,this paper proposes and implements a 3D datasets volume visualization scheme in real stereoscopic space.Homogenous image space is created by the screen's rotational scanning.In rendering process,3D regular discrete data is subdivided into a series of helicoid slices evenly.Then each data slice is rendered into an image frame separately.The rendered images are projected into the imaging space in rapid succession through a high-speed projector.By using multi-thread technique,synchronization between projected images sequence and screen's rotation is achieved.Finally the whole spatial image is scanned out layer by layer.The experimental results show that elements of the presented spatial images are uniformly distributed and well defined.All three dimensionalities of the images are reserved inherently;this approach offers omni-directional viewing freedom around the display.

We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity. Like the inconsistent “ideal calculus” for set theory, it is essentially based on just two set-theoretical principles: extensionality and comprehension (to which we add ∈-induction and optionally the axiom of choice). Comprehension is formulated as: \(x\in\{x\mid\varphi\}\leftrightarrow\varphi\), where {x|ϕ} is a legal set term of the theory. In order for {x|ϕ} to be legal, ϕ should be safe with respect to {x}, where safety is a relation between formulas and finite sets of variables. The various systems we consider differ from each other mainly with respect to the safety relations they employ. These relations are all defined purely syntactically (using an induction on the logical structure of formulas). The basic one is based on the safety relation which implicitly underlies commercial query languages for relational database systems (like SQL).Our framework makes it possible to reduce all extensions by definitions to abbreviations. Hence it is very convenient for mechanical manipulations and for interactive theorem proving. It also provides a unified treatment of comprehension axioms and of absoluteness properties of formulas.KeywordsFunction SymbolAtomic FormulaPredicate SymbolSafety RelationTransitive ModelThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

What is your dissertation about? And what contribution do you aim to achieve? What will be new or different about your work? How would you justify the time and resources that you will devote to it? These fundamental questions will seem very pressing in the beginning stages of your research, as Yeats’ intangible process of locking you into a long-run project begins. But they do not go away later on. You can often push such issues into the background in the central stages of the thesis, during field visits, case studies or the hard slog of library or archive work or data collection and analysis. But they tend to return during the ‘mid-term slump’ in morale that often afflicts dissertation authors. And they invariably crop up again when you have a first draft of your complete thesis, and have to fashion it into a polished and defensible final version. This chapter is about the importance of thinking through some reasonable answers before you invest too heavily in a particular research topic and approach. I consider first how to define one or several questions that will inform your project as a whole. The second section looks at the demands of doing ‘original’ and interesting research.KeywordsCreative PeopleCentral Research QuestionThesis TopicProfessional ResearcherArchive WorkThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

HFDPS for suitable applications have demonstrated a compact solution for the overall power system while ensuring enhanced system efficiency and reliability. Moreover, adding a communication link for information exchange with limited overhead cost has always been of perpetual interest. However, designing a powerline communication (PLC) network for simultaneous HF power and data transfer for an HFDPS poses various challenges. Firstly, a complex high-order analog filtering circuit is required for coupling/decoupling the power and data signals. Secondly, the use of PLC based technique results in a spectral overlap between the HF power and data signals, which following Shannon’s theorem, reduces the channel capacity for data transmission. In lieu of the above discussion, in this research, a new scheme enabling discrete HF power and data transfer over an HF channel in an HFDPS is outlined and is termed as “Sequential co-transfer scheme.” The proposed scheme is in contrast to conventional PLC, where the power and data co- transmission is simultaneous. Sequential co-transfer avoids data corruption by temporally distributing HF power and data signals over an HF channel and limiting their overlap. In this research, a communication data-transfer architecture mechanism required to realize the sequential co-transfer approach has been outlined. Simple transmitter and receiver circuits, synthesized without the use of any analog-filtering circuitry, are designed, and an asynchronous serial-communication-interface (SCI) protocol is implemented. The sequential HF power and data co-transfer is experimentally achieved and validated on a single-input-single-output (SISO) and a Single-input-multi-output (SIMO) power electronics system. Various applications encompassing packetized power delivery empowered by the sequential co-transfer scheme are envisioned and experimentally verified. Additionally, an in-depth analysis of GaN-based power electronics system required for creating the HF power signals is undertaken, and specific design guidelines for reducing HF noise in the circuit are discussed.

Student-centered approach to preparing Junior Academy of Sciences (MAN) projects in mathematics is substantiated, combining sound topic selection with bringing results to a reproducible state. The paper proposes explicit topic-selection criteria (student comprehensibility, amenability to algorithmic treatment, visualizability, attainable scholarly novelty, practical relevance) and shows how to turn them into a step-by-step research trajectory – from problem formulation and a concise literature review to prototype implementations, computational checks, interpretation, manuscript preparation, and defense. Special emphasis is placed on types of novelty realistic for MAN level: constructing new catalogs and bounds for small parameters, improving algorithms, and validating known hypotheses on new parameter ranges, with all steps accompanied by transparent success metrics. Building on analyzed works, a minimal reproducibility standard is outlined for mathematical MAN projects: unambiguous definitions and notation; explicit assumptions, statements, and scope; illustrative examples/ counterexamples; tables documenting test parameters; and a brief description of verification procedures (which steps were executed, stopping criteria, and counting methods). Visualizations are treated as evidence rather than decoration: a “key figure” carries a self-contained caption and can be reproduced by the stated sequence of steps (e.g., diagrams of first/second neighborhoods in graphs; coverage maps of pairs in lottery problems; comparative plots contrasting naïve enumeration with improved constructions). A concise assessment rubric for students, supervisors, and juries keeps attention on relevance, novelty, correctness of arguments/constructions, reproducibility, and figure quality. The approach is illustrated by two representative cases in discrete mathematics. First, Seymour’s Second Neighborhood Conjecture: using properties of potential counterexamples (density, diameter ≥ 3, penalty-function filters, and the vertex-weighted equivalence) to narrow the search space, formalize pruning conditions, and build informative neighborhood diagrams. Second, combinatorial coverings for lottery problems: demonstrating algorithmic construction of coverings and the alignment of upper and lower bounds (including realistic parameters such as 6-out-of-36), enabling rigorous justification of optimal solutions and independent verification. Both cases show how to balance accessibility with scholarly novelty, ensure reproducibility, and prepare results for peer-reviewed publication and a successful defense.

In this chapter the reader is introduced to the idea of a concept. Historically, two main approaches can be discerned toward formalization of this idea. First, a holistic approach by Aristotle, in which a concept is construed as a term (category) by which any object can be either styled or not. This fact of calling an object a by a name term A is denoted by the formula a is A. With those formulas, Aristotle built a complete theory of concepts, the Calculus of Syllogisms, briefly exposed in section 1.1, below.The alternative approach emerged with Georg Cantor’s set theory in mid–19th century and its essence was a representation of a concept by its elements; the relation of being an element was expressed as with Aristotle by the predicate is written down as ε (the ’esti’ symbol): the formula aεA reads an object a is an element of the concept (set) A. We offer in section 1.2 and following, an outline of modern set theory, aimed at establishing a terminology and basic facts of set theory, which seem to be indispensable to any researcher in computer science.KeywordsEquivalence RelationBoolean AlgebraPropositional CalculusMathematical InductionTolerance RelationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Set theory is the most fundamental part of mathematics. The definition of almost any kind of mathematical object (a group, a ring, a vector space, a topological space, a Hilbert space...) begins: a <thing> consists of a set, together with some extra structure in the form of operations, relations, subsets, sets of subsets, functions to the real numbers, or whatever. Also, as we will see, these operations, relations, etc. are themselves special kinds of sets.KeywordsNatural NumberEquivalence RelationChoice FunctionInjective FunctionStrict OrderThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.