Class Notes for Math 901/902: Abstract Algebra, Instructor Tom Marley

Topics include: Free groups and presentations; Automorphism groups; Semidirect products; Classification of groups of small order; Normal series: composition, derived, and solvable series; Algebraic field extensions, splitting fields, algebraic closures; Separable algebraic extensions, the Primitive Element Theorem; Inseparability, purely inseparable extensions; Finite fields; Cyclotomic field extensions; Galois theory; Norm and trace maps of an algebraic field extension; Solvability by radicals, Galois' theorem; Transcendence degree; Rings and modules: Examples and basic properties; Exact sequences, split short exact sequences; Free modules, projective modules; Localization of (commutative) rings and modules; The prime spectrum of a ring; Nakayama's lemma; Basic category theory; The Hom functors; Tensor products, adjointness; Left/right Noetherian and Artinian modules; Composition series, the Jordan-Holder Theorem; Semisimple rings; The Artin-Wedderburn Theorem; The Density Theorem; The Jacobson radical; Artinian rings; von Neumann regular rings; Wedderburn's theorem on finite division rings; Group representations, character theory; Integral ring extensions; Burnside's paqb Theorem; Injective modules.

Tratamiento N-categorial de la lógica de la programación

En el contexto de las técnicas de validación de programas, de las estructuras de axiomatización de la programación planteadas por Hoare y del desarrollo de la Lógica de la Programación que Dijkstra realiza con el operador wp, hemos dotado a los elementos esenciales que intervienen en dicho desarrollo de una estructura que permite su estudio en el ámbito de las categorias. En este trabajo se demuestra que el conjunto de precondiciones de un fragmento de programa anotado tiene estructura de N-categoría y que sucede lo mismo con el conjunto de postcondiciones. Más aún, se ha puesto de manifiesto (y probado) que el operador wp actúa como un funtor entre estos pares de N-categorías. Además, los conjuntos de guardas surgen de un modo natural en la lógica de la programación, han sido tratados desde el punto de vista de la semántica denotacional (Wlrth para el PASCAL y Scott y otros, después, en un enfoque más general) y presentados por Manes y Arbib en su semántica parcialmente aditiva. Pues bien, en este trabajo se demuestra que los conjuntos de guardas tienen también estructura de N-categoría y gue cualguier N-categoría dotada de una suma definida adecuadamente tiene estructura de conjunto de guardas, de tal modo, además, gue el preorden inducido por la suma en el conjunto de guardas coincide con la flecha de la N-categoría. Esta suma es, en concreto, la disyunción exclusiva, lo que adicionalmente supone una sorprendentemente sencilla definición alternativa a la suma de las categorías parcialmente aditivas definida por Manes y Arbib. Con todo esto, se aportan herramientas conceptuales para entender mejor y resolver más eficientemente los problemas que tiene planteados la lógica de la programación, pues se dispone de un punto de vista distinto y nuevo y de toda una familia de instrumentos adicionales.---ABSTRACT---In the context of program validation techniques, Hoare's systems for programming and Dijkstra's development of the logic of programming, based on the operator wp, we have endowed the essential features of this development with a structure that permits to study them in the frame of category theory. In this thesis we show that the set of preconditions of an annotated program segment is an N-category, and the same happens for the set of postconditions. Even more, it is shown that the operator wp acts as a functor between those pairs of Ncategories. Furthermore, guard sets come out in a natural way in the logic of programming, they have been considered from a denotational semantics point of view (Wirth for Pascal and afterwards Scott and al. in a more general setting) and they have been embodied by Manes and Arbib in their partially additive semantics. Then, it is shown in this thesis that the above mentioned guard sets also have the structure of an N-category and that any N-category with an appropiately defined sum has the structure of a guard set in such a way that, besides, the preorder defined in the guard set by the sum operation coincides with the arrows of the N-category. This sum is just the exclusive or of Boolean Logic and this fact adds a surprisingly simple alternative definition for the sum operation in Manes and Arbib partially additive categories. The present work, in summary, makes a contribution of conceptual tools for a better understanding and a more efficient solution of the problems posed to the logic of programming and it does so by providing a new different point of view and a whole family of additional techniques.

An approach to ontology for institutional facts in the semantic web

Refinement in software engineering allows a specification to be developed in stages, with design decisions taken at earlier stages constraining the design at later stages. Refinement in complex data models is difficult due to lack of a way of defining constraints, which can be progressively maintained over increasingly detailed refinements. Category theory provides a way of stating wide scale constraints. These constraints lead to a set of design guidelines, which maintain the wide scale constraints under increasing detail. Previous methods of refinement are essentially local, and the proposed method does not interfere very much with these local methods. The result is particularly applicable to semantic web applications, where ontologies provide systems of more or less abstract constraints on systems, which must be implemented and therefore refined by participating systems. With the approach of this paper, the concept of committing to an ontology carries much more force. (c) 2005 Elsevier B.V. All rights reserved.

Dating paleochannel iron ore by (U-Th)/He analysis of supergene goethite, Hamersley province, Australia

(U-Th)/He dating of late-stage authigenic goethite, combined with corrections for diffusive loss of He-4 by the He-4/He-3 methodology, reveals strong correlation between a sample's age and its depth in ferruginized channel sediments from the Yandicoogina deposit, Western Australia. Corrected ages, ranging from ca. 18 Ma near the surface to ca. 5 Ma at the bottom of the profile, indicate that ferruginization of the aggraded channels becomes progressively younger with depth. This trend is consistent with goethite precipitation at the groundwater-atmosphere interface during water table drawdown driven by the aridification of Western Australia during the Neogene. The results demonstrate that the (U-Th)/He system is ideal for dating goethite if diffusive loss corrections are applied. The approach is suitable for dating weathering reactions on Earth and should also be suitable for dating Fe oxyhydroxides in the Martian regolith.

Capital 2.0 capital formation and legal risk in a new global economic order from fiat to exit: including case studies of the proposed transatlantic trade and investment partnership between the United States and the European Union and the financing relation between the United States and the People’s Republic of China

Following the intrinsically linked balance sheets in his Capital Formation Life Cycle, Lukas M. Stahl explains with his Triple A Model of Accounting, Allocation and Accountability the stages of the Capital Formation process from FIAT to EXIT. Based on the theoretical foundations of legal risk laid by the International Bar Association with the help of Roger McCormick and legal scholars such as Joanna Benjamin, Matthew Whalley and Tobias Mahler, and founded on the basis of Wesley Hohfeld’s category theory of jural relations, Stahl develops his mutually exclusive Four Determinants of Legal Risk of Law, Lack of Right, Liability and Limitation. Those Four Determinants of Legal Risk allow us to apply, assess, and precisely describe the respective legal risk at all stages of the Capital Formation Life Cycle as demonstrated in case studies of nine industry verticals of the proposed and currently negotiated Transatlantic Trade and Investment Partnership between the United States of America and the European Union, TTIP, as well as in the case of the often cited financing relation between the United States and the People’s Republic of China. Having established the Four Determinants of Legal Risk and its application to the Capital Formation Life Cycle, Stahl then explores the theoretical foundations of capital formation, their historical basis in classical and neo-classical economics and its forefathers such as The Austrians around Eugen von Boehm-Bawerk, Ludwig von Mises and Friedrich von Hayek and most notably and controversial, Karl Marx, and their impact on today’s exponential expansion of capital formation. Starting off with the first pillar of his Triple A Model, Accounting, Stahl then moves on to explain the Three Factors of Capital Formation, Man, Machines and Money and shows how “value-added” is created with respect to the non-monetary capital factors of human resources and industrial production. Followed by a detailed analysis discussing the roles of the Three Actors of Monetary Capital Formation, Central Banks, Commercial Banks and Citizens Stahl readily dismisses a number of myths regarding the creation of money providing in-depth insight into the workings of monetary policy makers, their institutions and ultimate beneficiaries, the corporate and consumer citizens. In his second pillar, Allocation, Stahl continues his analysis of the balance sheets of the Capital Formation Life Cycle by discussing the role of The Five Key Accounts of Monetary Capital Formation, the Sovereign, Financial, Corporate, Private and International account of Monetary Capital Formation and the associated legal risks in the allocation of capital pursuant to his Four Determinants of Legal Risk. In his third pillar, Accountability, Stahl discusses the ever recurring Crisis-Reaction-Acceleration-Sequence-History, in short: CRASH, since the beginning of the millennium starting with the dot-com crash at the turn of the millennium, followed seven years later by the financial crisis of 2008 and the dislocations in the global economy we are facing another seven years later today in 2015 with several sordid debt restructurings under way and hundred thousands of refugees on the way caused by war and increasing inequality. Together with the regulatory reactions they have caused in the form of so-called landmark legislation such as the Sarbanes-Oxley Act of 2002, the Dodd-Frank Act of 2010, the JOBS Act of 2012 or the introduction of the Basel Accords, Basel II in 2004 and III in 2010, the European Financial Stability Facility of 2010, the European Stability Mechanism of 2012 and the European Banking Union of 2013, Stahl analyses the acceleration in size and scope of crises that appears to find often seemingly helpless bureaucratic responses, the inherent legal risks and the complete lack of accountability on part of those responsible. Stahl argues that the order of the day requires to address the root cause of the problems in the form of two fundamental design defects of our Global Economic Order, namely our monetary and judicial order. Inspired by a 1933 plan of nine University of Chicago economists abolishing the fractional reserve system, he proposes the introduction of Sovereign Money as a prerequisite to void misallocations by way of judicial order in the course of domestic and transnational insolvency proceedings including the restructuring of sovereign debt throughout the entire monetary system back to its origin without causing domino effects of banking collapses and failed financial institutions. In recognizing Austrian-American economist Schumpeter’s Concept of Creative Destruction, as a process of industrial mutation that incessantly revolutionizes the economic structure from within, incessantly destroying the old one, incessantly creating a new one, Stahl responds to Schumpeter’s economic chemotherapy with his Concept of Equitable Default mimicking an immunotherapy that strengthens the corpus economicus own immune system by providing for the judicial authority to terminate precisely those misallocations that have proven malignant causing default perusing the century old common law concept of equity that allows for the equitable reformation, rescission or restitution of contract by way of judicial order. Following a review of the proposed mechanisms of transnational dispute resolution and current court systems with transnational jurisdiction, Stahl advocates as a first step in order to complete the Capital Formation Life Cycle from FIAT, the creation of money by way of credit, to EXIT, the termination of money by way of judicial order, the institution of a Transatlantic Trade and Investment Court constituted by a panel of judges from the U.S. Court of International Trade and the European Court of Justice by following the model of the EFTA Court of the European Free Trade Association. Since the first time his proposal has been made public in June of 2014 after being discussed in academic circles since 2011, his or similar proposals have found numerous public supporters. Most notably, the former Vice President of the European Parliament, David Martin, has tabled an amendment in June 2015 in the course of the negotiations on TTIP calling for an independent judicial body and the Member of the European Commission, Cecilia Malmström, has presented her proposal of an International Investment Court on September 16, 2015. Stahl concludes, that for the first time in the history of our generation it appears that there is a real opportunity for reform of our Global Economic Order by curing the two fundamental design defects of our monetary order and judicial order with the abolition of the fractional reserve system and the introduction of Sovereign Money and the institution of a democratically elected Transatlantic Trade and Investment Court that commensurate with its jurisdiction extending to cases concerning the Transatlantic Trade and Investment Partnership may complete the Capital Formation Life Cycle resolving cases of default with the transnational judicial authority for terminal resolution of misallocations in a New Global Economic Order without the ensuing dangers of systemic collapse from FIAT to EXIT.

Automated service provisioning in programmable network infrastructures

Modern networks are undergoing a fast and drastic evolution, with software taking a more predominant role. Virtualization and cloud-like approaches are replacing physical network appliances, reducing the management burden of the operators. Furthermore, networks now expose programmable interfaces for fast and dynamic control over traffic forwarding. This evolution is backed by standard organizations such as ETSI, 3GPP, and IETF. This thesis will describe which are the main trends in this evolution. Then, it will present solutions developed during the three years of Ph.D. to exploit the capabilities these new technologies offer and to study their possible limitations to push further the state-of-the-art. Namely, it will deal with programmable network infrastructure, introducing the concept of Service Function Chaining (SFC) and presenting two possible solutions, one with Openstack and OpenFlow and the other using Segment Routing and IPv6. Then, it will continue with network service provisioning, presenting concepts from Network Function Virtualization (NFV) and Multi-access Edge Computing (MEC). These concepts will be applied to network slicing for mission-critical communications and Industrial IoT (IIoT). Finally, it will deal with network abstraction, with a focus on Intent Based Networking (IBN). To summarize, the thesis will include solutions for data plane programming with evaluation on well-known platforms, performance metrics on virtual resource allocations, novel practical application of network slicing on mission-critical communications, an architectural proposal and its implementation for edge technologies in Industrial IoT scenarios, and a formal definition of intent using a category theory approach.

The orthogonal subcategory problem in homotopy theory

It is known that, in a locally presentable category, localization exists with respect to every set of morphisms, while the statement that localization with respect to every (possibly proper) class of morphisms exists in locally presentable categories is equivalent to a large-cardinal axiom from set theory. One proves similarly, on one hand, that homotopy localization exists with respect to sets of maps in every cofibrantly generated, left proper, simplicial model category M whose underlying category is locally presentable. On the other hand, as we show in this article, the existence of localization with respect to possibly proper classes of maps in a model category M satisfying the above assumptions is implied by a large-cardinal axiom called Vopënka's principle, although we do not know if the reverse implication holds. We also show that, under the same assumptions on M, every endofunctor of M that is idempotent up to homotopy is equivalent to localization with respect to some class S of maps, and if Vopënka's principle holds then S can be chosen to be a set. There are examples showing that the latter need not be true if M is not cofibrantly generated. The above assumptions on M are satisfied by simplicial sets and symmetric spectra over simplicial sets, among many other model categories.

On the trace of an endofunctor of a small category

The trace of a square matrix can be defined by a universal property which, appropriately generalized yields the concept of "trace of an endofunctor of a small category". We review the basic definitions of this general concept and give a new construction, the "pretrace category", which allows us to obtain the trace of an endofunctor of a small category as the set of connected components of its pretrace. We show that this pretrace construction determines a finite-product preserving endofunctor of the category of small categories, and we deduce from this that the trace inherits any finite-product algebraic structure that the original category may have. We apply our results to several examples from Representation Theory obtaining a new (indirect) proof of the fact that two finite dimensional linear representations of a finite group are isomorphic if and only if they have the same character.

Development of premature children: caregivers' understanding according to the Bioecological Theory

Abstract OBJECTIVE Understanding the conceptions of premature children caregivers on child development and associated factors. METHOD An exploratory-descriptive qualitative study of 12 families with children under three years of age. Interviews were submitted to thematic content analysis, systematized into the categories of Bioecological Theory of Human Development: Process, Person, Context and Time, and in the Functional Development category. RESULTS There are concerns about impairment in the current and future development of a Person/child defined as fragile as a result of premature birth (Time dimension), minimized by the scope of observable competencies such as motor skills. The Context, especially family and health services, and Proximal Processes, described as one-way caregiver interactions, are considered determinants of development. Functional Development is considered a natural consequence and result of education. The support network is crucial, supporting or limiting care. CONCLUSION Concerns about the development mobilize caregivers to stimulate the premature child/person and requests family and healthcare assistance.

Towards conceptually comprehensive and integrated theory of entrepreneurial opportunity

Insufficient understanding of entrepreneurial opportunities characterizes entrepreneurship research (Companys & McMullen 2006, 302). Subsequently, the purpose of this study was to analyze the concept of entrepreneurial opportunity and to explore and synthetize the integrated theory of entrepreneurial opportunity. A theoretical and concept analytical approach was adapted. Findings of this study was that entrepreneurial opportunity concept was used for variety of different phenomena. No commonly accepted definition existed. Altogether 24 attributes that described the concept were found. The most frequently attached attributes were agent and action, new goods and services, market, value, new means ends or both, and future. Further, the results implied that opportunity could be best understood as a part of a process. Opportunity emerges out of intervened factors. Changes in the environment together with factors related to knowledge, cognition and social ties are the most important drivers of opportunity. Preventing factors that impeded the emergence of opportunity were typically related to cognitive and organizational factors. This study found a tendency towards more integrated theory of entrepreneurial opportunity. The integrated theory acknowledged the usefulness of both discovery and creation theories of opportunity in explaining opportunity. Yet three argument types of integrating two different opportunity theories were identified. These were process category, contextual category and complementing category. Opportunity is at the same time cognitive, social and linguistic construct, although it is shaped by the objective environment. Opportunity requires linguistic endeavors to become explicit. Materialization of opportunity occurs in a social context. Moreover, it is always characterized by some extent of subjectivity, as opportunities cannot appear without the agent and their action. Due to these the concept remains always to some extent ambiguous. Tolerating and harnessing change and investing in human and social capital create the preeminent environment for the entrepreneurial opportunity to be identified.

A substantive theory of the realization of stroke patients’ dignity in hospital care

Dignity is seen important in health care context but considered as a controversial and complex concept. In health care context, it is described as being influenced by for example autonomy, respect, communication, privacy and hospital environment. Patient dignity is related to satisfaction with care, reduced stress, better confidence in health services, enhanced patient outcomes and shorter stay in a hospital. Stroke patients may struggle for dignity as being dependent on other people has impact on the patients’ self-image. In all, stroke patients are very specific patient group and considered vulnerable from emotional aspect. Therefore study findings from other patient groups in the area of ethical problems cannot be transferred to the stroke patients. This master’s thesis consists of two parts. The first part is the literature review of patients’ dignity in hospital care. The literature defined dignity and described factors promoting and reducing it. The results were ambiguous and thus a clear understanding was not able to create. That was the basis for the second part of the master’s thesis, the empirical study. This part aimed to develop theoretical construction to explore the realization of stroke patients’ dignity in hospital care. The data of the second part was collected by interviewing 16 stroke patients and analyzed using the constant comparison of Grounded Theory. The result was ‘The Theory of Realization of Stroke Patients’ Dignity in Hospital Care’ which is described not only in this master’s thesis but also as a scientific article. The theory consists of the core category, four generic elements and five specific types on realization. The core category emerged as ‘dignity in a new situation’. After a stroke, dignity is defined in a new way which is influenced by the generic elements: life history, health history, individuality and a stroke. Stroke patient’s dignity is realized through five specific types on realization: person related dignity type, control related dignity type, independence related dignity type, social related dignity type and care related dignity type. The theory points out possible special characteristics of stroke patients’ dignity in control related dignity type and independence related dignity type. Before implementing the theory, the relation between the core category, generic elements and specific types on realization needs to be studied further.

An Abstract Algebraic Theory of L-Fuzzy Relations for Relational Databases

Classical relational databases lack proper ways to manage certain real-world situations including imprecise or uncertain data. Fuzzy databases overcome this limitation by allowing each entry in the table to be a fuzzy set where each element of the corresponding domain is assigned a membership degree from the real interval [0…1]. But this fuzzy mechanism becomes inappropriate in modelling scenarios where data might be incomparable. Therefore, we become interested in further generalization of fuzzy database into L-fuzzy database. In such a database, the characteristic function for a fuzzy set maps to an arbitrary complete Brouwerian lattice L. From the query language perspectives, the language of fuzzy database, FSQL extends the regular Structured Query Language (SQL) by adding fuzzy specific constructions. In addition to that, L-fuzzy query language LFSQL introduces appropriate linguistic operations to define and manipulate inexact data in an L-fuzzy database. This research mainly focuses on defining the semantics of LFSQL. However, it requires an abstract algebraic theory which can be used to prove all the properties of, and operations on, L-fuzzy relations. In our study, we show that the theory of arrow categories forms a suitable framework for that. Therefore, we define the semantics of LFSQL in the abstract notion of an arrow category. In addition, we implement the operations of L-fuzzy relations in Haskell and develop a parser that translates algebraic expressions into our implementation.

A Character Formula for the Category (O)over-tilde of Modules for Affine sl(2)

One may construct, for any function on the integers, an irreducible module of level zero for affine sl(2) using the values of the function as structure constants. The modules constructed using exponential-polynomial functions realize the irreducible modules with finite-dimensional weight spaces in the category (O) over tilde of Chari. In this work, an expression for the formal character of such a module is derived using the highest weight theory of truncations of the loop algebra.

Categorical Structures in Non-Commutative Measure Theory.

In this paper, we investigate effect algebras and base normed spaces from the categorical point of view. We prove that the category of effect algebras is complete and cocomplete as well as the category of base normed spaces is complete, and discuss the contravariant functor from the category of effect algebras to the category of base normed spaces.

Adaptive Pointing Theory (APT) Artificial Neural Network

The choice value and the testing process against the vigilance parameter, characteristic of ART Neural Network, are merged. Only, a single unique test is required to determine if a committed category node can represent the current input or not. Advantages of APT over ART are: 1-Avoid testing every committed category node before deciding to train a committed category node or a new node must be committed, 2-The vigilance parameter is fixed during training, and 3-The choice value parameter is eliminated.