An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization

In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring $R$, i.e., a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over $R$. For a finitely generated maximal ideal $\mathfrak{m}$ in a commutative ring $R$ we show how solving (in)homogeneous linear systems over $R_{\mathfrak{m}}$ can be reduced to solving associated systems over $R$. Hence, the computability of $R$ implies that of $R_{\mathfrak{m}}$. As a corollary we obtain the computability of the category of finitely presented $R_{\mathfrak{m}}$-modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a by-product, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Mora's algorithm.Comment: Fixed a typo in the proof of Lemma 4.3 spotted by Sebastian Posu

Authentic African community development practices in a diverse society: A transdisciplinary approach

The South African people continuously engage in social actions characterised by intolerance, pointing to frustrations and disillusionment in a post-apartheid era. A need to find creative ways to engage diverse communities to work together to participate in their own development and well-being was identified. This article is based on long-term transdisciplinary discourse and work. The aim is to explore how the disciplines of social work, theology and the arts could contribute together towards the development of communities where participation, collaboration and cooperation as key principles of authentic community development are actively implemented. Within a transdisciplinary framework, the disciplines engaged in participatory research projects that resulted in findings that informed the development of a process where people at grassroots level become aware and more tolerant of each other, begin to work together and as such become involved in their own futures. It is concluded that by encouraging participation, collaboration and cooperation in social change processes, the South African people can be empowered towards working together and becoming involved in their own futures

Deterministically Computing Reduction Numbers of Polynomial Ideals

We discuss the problem of determining reduction number of a polynomial ideal I in n variables. We present two algorithms based on parametric computations. The first one determines the absolute reduction number of I and requires computation in a polynomial ring with (n-dim(I))dim(I) parameters and n-dim(I) variables. The second one computes via a Grobner system the set of all reduction numbers of the ideal I and thus in particular also its big reduction number. However,it requires computations in a ring with n.dim(I) parameters and n variables.Comment: This new version replaces the earlier version arXiv:1404.1721 and it has been accepted for publication in the proceedings of CASC 2014, Warsaw, Polna

Authentic African community development practices in a diverse society : a transdisciplinary approach

The South African people continuously engage in social actions characterised by intolerance, pointing to frustrations and disillusionment in a post-apartheid era. A need to find creative ways to engage diverse communities to work together to participate in their own development and well-being was identified. This article is based on long-term transdisciplinary discourse and work. The aim is to explore how the disciplines of social work, theology and the arts could contribute together towards the development of communities where participation, collaboration and cooperation as key principles of authentic community development are actively implemented. Within a transdisciplinary framework, the disciplines engaged in participatory research projects that resulted in findings that informed the development of a process where people at grassroots level become aware and more tolerant of each other, begin to work together and as such become involved in their own futures. It is concluded that by encouraging participation, collaboration and cooperation in social change processes, the South African people can be empowered towards working together and becoming involved in their own futures. INTRADISCIPLINARY AND/OR INTERDISCIPLINARY IMPLICATIONS: The disciplines of social work, theology and the arts entered into a transdisciplinary dialogue and work over the past years. The transdisciplinary team engaged in four participatory research projects to include input from grassroots levels to inform their understanding of how the different disciplines can better contribute towards a process of authentic community development in the diverse South African society. This resulted in the process proposed in this article.http://www.hts.org.zapm2021Practical Theolog

Generating Non-Linear Interpolants by Semidefinite Programming

Interpolation-based techniques have been widely and successfully applied in the verification of hardware and software, e.g., in bounded-model check- ing, CEGAR, SMT, etc., whose hardest part is how to synthesize interpolants. Various work for discovering interpolants for propositional logic, quantifier-free fragments of first-order theories and their combinations have been proposed. However, little work focuses on discovering polynomial interpolants in the literature. In this paper, we provide an approach for constructing non-linear interpolants based on semidefinite programming, and show how to apply such results to the verification of programs by examples.Comment: 22 pages, 4 figure

Mathematical models as research data via flexiformal theory graphs

Mathematical modeling and simulation (MMS) has now been established as an essential part of the scientific work in many disciplines. It is common to categorize the involved numerical data and to some extent the corresponding scientific software as research data. But both have their origin in mathematical models, therefore any holistic approach to research data in MMS should cover all three aspects: data, software, and models. While the problems of classifying, archiving and making accessible are largely solved for data and first frameworks and systems are emerging for software, the question of how to deal with mathematical models is completely open. In this paper we propose a solution -- to cover all aspects of mathematical models: the underlying mathematical knowledge, the equations, boundary conditions, numeric approximations, and documents in a flexi\-formal framework, which has enough structure to support the various uses of models in scientific and technology workflows. Concretely we propose to use the OMDoc/MMT framework to formalize mathematical models and show the adequacy of this approach by modeling a simple, but non-trivial model: van Roosbroeck's drift-diffusion model for one-dimensional devices. This formalization -- and future extensions -- allows us to support the modeler by e.g. flexibly composing models, visualizing Model Pathway Diagrams, and annotating model equations in documents as induced from the formalized documents by flattening. This directly solves some of the problems in treating MMS as "research data'' and opens the way towards more MKM services for models

On the Geometry of Super Yang-Mills Theories: Phases and Irreducible Polynomials

We study the algebraic and geometric structures that underly the space of vacua of N=1 super Yang-Mills theories at the non-perturbative level. Chiral operators are shown to satisfy polynomial equations over appropriate rings, and the phase structure of the theory can be elegantly described by the factorization of these polynomials into irreducible pieces. In particular, this idea yields a powerful method to analyse the possible smooth interpolations between different classical limits in the gauge theory. As an application in U(Nc) theories, we provide a simple and completely general proof of the fact that confining and Higgs vacua are in the same phase when fundamental flavors are present, by finding an irreducible polynomial equation satisfied by the glueball operator. We also derive the full phase diagram for the theory with one adjoint when Nc is less than or equal to 7 using computational algebraic geometry programs.Comment: 87 pages; v2: typos and eq. (4.44) correcte

The Degree and regularity of vanishing ideals of algebraic toric sets over finite fields

Let X* be a subset of an affine space A^s, over a finite field K, which is parameterized by the edges of a clutter. Let X and Y be the images of X* under the maps x --> [x] and x --> [(x,1)] respectively, where [x] and [(x,1)] are points in the projective spaces P^{s-1} and P^s respectively. For certain clutters and for connected graphs, we were able to relate the algebraic invariants and properties of the vanishing ideals I(X) and I(Y). In a number of interesting cases, we compute its degree and regularity. For Hamiltonian bipartite graphs, we show the Eisenbud-Goto regularity conjecture. We give optimal bounds for the regularity when the graph is bipartite. It is shown that X* is an affine torus if and only if I(Y) is a complete intersection. We present some applications to coding theory and show some bounds for the minimum distance of parameterized linear codes for connected bipartite graphs