Posts

Featured Post

Publications

Publications Z. Janelidze, Characterization of pointed varieties of universal algebras with normal projections , Theory and Applications of Categories 11, 2003, 212-214 Z. Janelidze, Varieties of universal algebras with normal local projections , Georgian Mathematical Journal 11, 2004, 93-98 Z. Janelidze, Subtractive categories , Applied Categorical Structures 13, 2005, 343-350 Z. Janelidze, Closedness properties of internal relations I: A unified approach to Mal’tsev, unital and subtractive categories , Theory and Applications of Categories 16, 2006, 236-261 Z. Janelidze, Closedness properties of internal relations II: Bourn localization , Theory and Applications of Categories 16, 2006, 262-282 Z. Janelidze, Closedness properties of internal relations IV: Expressing additivity of a category via subtractivity , Journal of Homotopy and Related Structures 1, 2006, 219-227 Z. Janelidze, Closedness properties of internal relations III: Pointed protomodular categories

Reverse Prime Composite Numbers

The story begins with an observation made to me by my father, on 22 February 2022, that it is a special date. Afterwards, I found out he was representing this date as 20220222 (year first, month and then day). In my original interpretation, it was 22022022. I was not satisfied just with the fact that this is a palindrome (a number which, when reversed, gives back the same number). So I looked into its prime factors: 22022022 = 2 x 3 x 11 x 2 x 3 x 11 x 333667. After a while of staring at this, I checked what would happen if I reversed 333667. That number (766333) turned out to be a prime as well! I learned afterwards that prime numbers whose reverse (in decimal notation) are prime, are called reverse primes . So then 22022022 is a composite reverse prime , i.e., a natural number who all prime factors are reverse primes. The question was: how special does this make the number 22022022? To be continued.

The Transition from High School Mathematics to University Mathematics

Image
These are notes in progress for a talk given at the online user group conference of the advanced programme mathematics organized by ieb (19 February 2022) 1. Introduction In my experience, what makes transition from school mathematics to university mathematics hard is the depth of engagement with mathematics that university mathematics requires of students, compared to the depth of engagement that school mathematics requires. Do you agree or disagree with the following thesis: A school learner must understand school mathematics at the same depth that a university student is expected to understands university mathematics. If you do not agree and think that a university student should understand university mathematics more deeply than a school learner understands school mathematics, this means that you expect a learner transitioning to university not only having to learn more advanced mathematics, but to understand it more deeply than they understand its foundation, the school mathematic

Python-Based Introduction to Mathematical Proofs

Image
Scroll down for video lectures 1. What is a Mathematical Proof?   Mathematical proof is a method of discourse which allows a human being to:  discover new mathematical knowledge, analyze existing mathematical knowledge, verify truthfulness of a piece of mathematical knowledge.  The ability to construct a mathematical proof is part of human nature. It is closely related to the ability to form thoughts and reason. Mathematical knowledge is knowledge of abstract principles about our universe. As such, it requires use of symbols to represent entities that are inherently abstract. For example, the symbol 2 may represent 2 apples or 2 pears . The number 2 is an abstract entity, since it is not confined to any of these concrete representations.  Mathematics functions at different levels of abstraction too. For instance, we may write a symbol, such as n , to represent any number. In one case we could have n = 2 , and in another case we could have n = 3 . This is a second layer of abstractio

Noetherian information systems

Image
These are notes for a colloquium talk to be given at NITheCS. The Snake Lemma from this fragment of a 1980 film ("It's My Turn", starring Jill Clayburgh and Michael Douglas), along with many other similar theorems in abstract algebra, known to be true for a variety different algebraic settings, can all be established in a unified setting of noetherian forms. This post attempts to give a preliminary step towards a possibly ambitious goal of applying noetherian forms outside abstract mathematics. In this light we propose a variation of this notion, a "noetherian information system", which is intended to be more agile in terms of identifying applications. General Information Systems By a network  we mean a web of devices and directed binary channels between them ( = a graph in the sense of category theory). An information system  over a network consisting of just one device and no channels, is a collection of information clusters , where some clusters may be par

Research Interests of 2+ Year Level 2022 Mathematics Teaching Staff at Stellenbosch University

Image
The content of this post is author's adaption of photos and information sourced from open internet. Liam Baker Broad research interest: number theory Year of first publication: 2016 Bruce Bartlett Broad research interests: topology, geometry, category theory Year of first publication: 2011 Dirk Basson Broad research interests: number theory, algebra Year of first publication: 2017 Ronalda Benjamin Broad research interests: analysis, topology, algebra Year of first publication: 2016 Gareth Boxall   Broad research interests: logic, number theory, analysis Year of first publication: 2011 James Gray Broad research interests: category theory, algebra Year of first publication: 2010 Retha Heymann Broad research interest: analysis Year of first publication: 2014 Michael Hoefnagel Broad research interests: category theory, algebra Year of first publication: 2019  Karin-Therese Howell Broad research interest: algebra Year of first publication: 2010  Zurab Janelidze Broad research interests:

The poset of matrix properties

Image
Below are the notes for the talk above, given at the Algebra, Geometry, Topology & Applications seminar. 1. Bird's-Eye View of Exactness Properties  One of the active areas of research in Categorical Algebra is the study of various properties of categories expressed using limits and colimits. Such properties are usually referred to as exactness properties . This terminology comes from the fact that, historically, the first such properties emerged in the study of exact sequences in the sense of Homological Algebra. The matrix properties  in the title of this post are particular types of exactness properties, which can be encoded using integer matrices. Before explaining what they are, let us first recall the notions of limit and colimit. Given a diagram of objects and arrows (objects are certain mathematical structures and arrows are morphisms between them), a limit (of the diagram) is a way to encode the information about the diagram in a single object; it is a terminal (commu