Normal crossings in local analytic geometry

Das Hauptziel dieser Dissertation ist eine effektive algebraische Charakterisierung von Divisoren (= Hyperflächen) mit normalen Kreuzungen in komplexen Mannigfaltigkeiten anzugeben. Um eine derartige Charakterisierung zu finden, studieren wir sowohl logarithmische Vektorfelder entlang eines Divisors, d.h., Vektorfelder des umgebenden Raumes, die in allen glatten Punkten des Divisors tangential an ihn sind, als auch logarithmische Differentialformen. Mit Hilfe der zugehörigen Theorie, entwickelt von K. Saito, wird eine Charakterisierung von Divisoren mit normalen Kreuzungen durch logarithmische Differentialformen (Vektorfelder) gezeigt. Des weiteren wird eine Charakterisierung durch das logarithmische Residuum vorgestellt (diese beruht auf Ergebnissen von Granger und Schulze). Damit kann eine Frage von K. Saito beantwortet werden. Im zweiten Kapitel werden Singularitäten eines Divisors mit normalen Kreuzungen untersucht, insbesondere betrachten wir das Jacobi Ideal, das den singulären Ort des Divisors definiert. Unser Hauptsatz besagt, dass ein Divisor genau dann normale Kreuzungen in einem Punkt besitzt, wenn er frei in diesem Punkt, sein Jacobi Ideal radikal und seine Normalisierung Gorenstein ist. Freie Divisoren werden durch logarithmische Differentialformen definiert und bilden eine Klasse von Divisoren, die insbesondere Divisoren mit normalen Kreuzungen enthält. Da eine algebraische Charakterisierung von freien Divisoren durch deren Jacobi Ideale existiert (nach A. G. Aleksandrov), ergibt sich aus unserem Resultat eine rein algebraische Charakterisierung der normalen Kreuzungsbedingung. Im Laufe des Beweises des Hauptsatzes werden gespreizte Divisoren eingeführt, die eine leichte Verallgemeinerung von Divisoren mit normalen Kreuzungen darstellen. Im letzten Teil der Arbeit werden weiterreichende Probleme betrachtet: Zuerst fragen wir, welche radikalen Ideale Jacobi Ideale von Divisoren sein können. Dann werden gespreizte Divisoren genauer untersucht, insbesondere zeigen wir, dass ihre Hilbert-Samuel Polynome eine gewisse Additivitätsbedingung erfüllen. Schließlich wird eine weitere Verallgemeinerung von Divisoren mit normalen Kreuzungen betrachtet, sogenannte Mikado Divisoren. Hier charakterisieren wir ebene Mikado Kurven durch ihr Jacobi Ideal.The main objective of this thesis is to give an effective algebraic characterization of normal crossing divisors (= hypersurfaces) in complex manifolds. In order to obtain such a characterization we study logarithmic vector fields along a divisor, i.e., vector fields defined on the ambient space, which are tangent to the divisor at its smooth points, as well as logarithmic differential forms. Using the corresponding theory, which was developed by K. Saito, a characterization of a normal crossing divisor in terms of logarithmic differential forms (vector fields) is shown. Also a characterization of a normal crossing divisor in terms of the logarithmic residue is given (which is essentially due to Granger and Schulze). With this a question posed by K. Saito in 1980 can be answered. In the second chapter we study singularities of normal crossing divisors, in particular we consider Jacobian ideals, which define the singular locus of a divisor. The main theorem is that a divisor has normal crossings at point if and only if it is free at the point, its Jacobian ideal is radical and its normalization is Gorenstein. Free divisors are defined via logarithmic vector fields and form a class of divisors containing normal crossing divisors. Since there exists an algebraic characterization of free divisors by their Jacobian ideals, our result yields a purely algebraic characterization of the normal crossings property. During the proof of the main theorem splayed divisors are introduced, which are a slight generalization of normal crossing divisors. In the last part we consider further-reaching questions: first we ask, which radical ideals can be Jacobian ideals of divisors. Then splayed divisors are studied in more detail, in particular, we show that their Hilbert-Samuel polynomials satisfy a certain additivity property. Finally, we consider another generalization of normal crossing divisors, so-called mikado divisors. Here the plane curve case is studied and we characterize mikado curves by their Jacobian ideal

Culture, worldview and transformative philosophy of mathematics education in Nepal: a cultural-philosophical inquiry

This thesis portrays my multifaceted and emergent inquiry into the protracted problem of culturally decontextualised mathematics education faced by students of Nepal, a culturally diverse country of south Asia with more than 90 language groups. I generated initial research questions on the basis of my history as a student of primary, secondary and university levels of education in Nepal, my Master’s research project, and my professional experiences as a teacher educator working in a university of Nepal between 2004 and 2006. Through an autobiographical excavation of my experiences of culturally decontextualised mathematics education, I came up with several emergent research questions, leading to six key themes of this inquiry: (i) hegemony of the unidimensional nature of mathematics as a body of pure knowledge, (ii) unhelpful dualisms in mathematics education, (iii) disempowering reductionisms in curricular and pedagogical aspects, (iv) narrowly conceived ‘logics’ that do not account for meaningful lifeworld-oriented thinking in mathematics teaching and learning, (v) uncritical attitudes towards the image of curriculum as a thing or object, and (vi) narrowly conceived notions of globalisation, foundationalism and mathematical language that give rise to a decontextualised mathematics teacher education program.With these research themes at my disposal my aim in this research was twofold. Primarily, I intended to explore, explain and interpret problems, issues and dilemmas arising from and embedded in the research questions. Such an epistemic activity of articulation was followed by envisioning, an act of imagining futures together with reflexivity, perspectival language and inclusive vision logics.In order to carry out both epistemic activities – articulating and envisioning – I employed a multi-paradigmatic research design space, taking on board mainly the paradigms of criticalism, postmodernism, interpretivism and integralism. The critical paradigm offered a critical outlook needed to identify the research problem, to reflect upon my experiences as a mathematics teacher and teacher educator, and to make my lifetime’s subjectivities transparent to readers, whereas the paradigm of postmodernism enabled me to construct multiple genres for cultivating different aspects of my experiences of culturally decontextualised mathematics education. The paradigm of interpretivism enabled me to employ emergence as the hallmark of my inquiry, and the paradigm of integralism acted as an inclusive meta-theory of the multi-paradigmatic design space for portraying my vision of an inclusive mathematics education in Nepal.Within this multi-paradigmatic design space, I chose autoethnography and small p philosophical inquiry as my methodological referents. Autoethnography helped generate the research text of my cultural-professional contexts, whereas small p philosophical inquiry enabled me to generate new knowledge via a host of innovative epistemologies that have the goal of deepening understanding of normal educational practices by examining them critically, identifying underpinning assumptions, and reconstructing them through scholarly interpretations and envisioning. Visions cultivated through this research include: (i) an inclusive and multidimensional image of the nature of mathematics as an im/pure knowledge system, (ii) the metaphors of thirdspace and dissolution for conceiving an inclusive mathematics education, (iii) a multilogical perspective for morphing the hegemony of reductionism-inspired mathematics education, (iv) an inclusive image of mathematics curriculum as montage that provides a basis for incorporating different knowledge systems in mathematics education, and (v) perspectives of glocalisation, healthy scepticism and multilevel contextualisation for constructing an inclusive mathematics teacher education program

Categorical Ontology of Complex Systems, Meta-Systems and Theory of Levels: The Emergence of Life, Human Consciousness and Society

Single cell interactomics in simpler organisms, as well as somatic cell interactomics in multicellular organisms, involve biomolecular interactions in complex signalling pathways that were recently represented in modular terms by quantum automata with ‘reversible behavior’ representing normal cell cycling and division. Other implications of such quantum automata, modular modeling of signaling pathways and cell differentiation during development are in the fields of neural plasticity and brain development leading to quantum-weave dynamic patterns and specific molecular processes underlying extensive memory, learning, anticipation mechanisms and the emergence of human consciousness during the early brain development in children. Cell interactomics is here represented for the first time as a mixture of ‘classical’ states that determine molecular dynamics subject to Boltzmann statistics and ‘steady-state’, metabolic (multi-stable) manifolds, together with ‘configuration’ spaces of metastable quantum states emerging from complex quantum dynamics of interacting networks of biomolecules, such as proteins and nucleic acids that are now collectively defined as quantum interactomics. On the other hand, the time dependent evolution over several generations of cancer cells --that are generally known to undergo frequent and extensive genetic mutations and, indeed, suffer genomic transformations at the chromosome level (such as extensive chromosomal aberrations found in many colon cancers)-- cannot be correctly represented in the ‘standard’ terms of quantum automaton modules, as the normal somatic cells can. This significant difference at the cancer cell genomic level is therefore reflected in major changes in cancer cell interactomics often from one cancer cell ‘cycle’ to the next, and thus it requires substantial changes in the modeling strategies, mathematical tools and experimental designs aimed at understanding cancer mechanisms. Novel solutions to this important problem in carcinogenesis are proposed and experimental validation procedures are suggested. From a medical research and clinical standpoint, this approach has important consequences for addressing and preventing the development of cancer resistance to medical therapy in ongoing clinical trials involving stage III cancer patients, as well as improving the designs of future clinical trials for cancer treatments.\ud \ud \ud KEYWORDS: Emergence of Life and Human Consciousness;\ud Proteomics; Artificial Intelligence; Complex Systems Dynamics; Quantum Automata models and Quantum Interactomics; quantum-weave dynamic patterns underlying human consciousness; specific molecular processes underlying extensive memory, learning, anticipation mechanisms and human consciousness; emergence of human consciousness during the early brain development in children; Cancer cell ‘cycling’; interacting networks of proteins and nucleic acids; genetic mutations and chromosomal aberrations in cancers, such as colon cancer; development of cancer resistance to therapy; ongoing clinical trials involving stage III cancer patients’ possible improvements of the designs for future clinical trials and cancer treatments. \ud \u

Association of Christians in the Mathematical Sciences Proceedings 2017

The conference proceedings of the Association of Christians in the Mathematical Sciences biannual conference, May 31-June 2, 2017 at Charleson Southern University

The Significance of Evidence-based Reasoning for Mathematics, Mathematics Education, Philosophy and the Natural Sciences

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines

Dagstuhl News January - December 2011

"Dagstuhl News" is a publication edited especially for the members of the Foundation "Informatikzentrum Schloss Dagstuhl" to thank them for their support. The News give a summary of the scientific work being done in Dagstuhl. Each Dagstuhl Seminar is presented by a small abstract describing the contents and scientific highlights of the seminar as well as the perspectives or challenges of the research topic

Dagstuhl News January - December 2000

"Dagstuhl News" is a publication edited especially for the members of the Foundation "Informatikzentrum Schloss Dagstuhl" to thank them for their support. The News give a summary of the scientific work being done in Dagstuhl. Each Dagstuhl Seminar is presented by a small abstract describing the contents and scientific highlights of the seminar as well as the perspectives or challenges of the research topic