Filters An Introduction
We apply filters to existing sets to express otherwise inexpressible statements.They effectively allow us to refer to infinitely small or infinitely large sets and conduct mathematical analysis to develop valuable insights.
Continuity as Convergence of Sequences—Expanding Our Definition of Continuity
I feel that continuity is best understood when we consider convergence at different levels of abstraction. While it’s fairly easy to understand the continuity of functions when they’re defined in spaces like R2, with standards like: The left hand limit must equal the right hand limit. The function should have a finite value at each […]
New Edition of the Book “Algebraic General Topology. Book 1: Basics”
I’ve published a new edition of my book Algebraic General Topology. The new edition features “unfixed morphisms” a way to turn a category into a semigroup. (Certain additional structure on the category is needed.) The book features a wide generalization of general topology done in an algebraic way. Now we can operate on general topological […]
Understanding Functional Discontinuities – The Building Blocks Of AGT
A discontinuity is any point on a function where one of the three possibilities arise: The right-side limit is unequal to left-side limit The function jumps suddenly The function goes to infinity at a certain point in the domain.
Understanding Continuity from the Perspective of AGT
Continuity and limits, as understood in traditional calculus, rely on infinitesimally small sets to arrive at limits for arbitrary functions. This idea, when translated into other definitions of continuities, relies on ideas about convergence of sequences and the like.
What Is The Axiomatic Theory Of Formulas?
The methods or syntax to express mathematical truths are, obviously, essential to the study of mathematics. Without a taxonomy that highlights the correct use of mathematical symbols, rules which dictate the validity of mathematical statements and theory about how these statements are to be construed, you can’t do math.
A Popular Introduction into the Generalized Limit (of arbitrary discontinuous function)
Abstract. A review of my book “Generalized limit (of arbitrary discontinuous function)”. A popular introduction with graphs to the following topic: I consider (a generalized) limit of an arbitrary (discontinuous) function, defined in terms of funcoids. The definition of the generalized limit makes it obvious to define such things as the derivative of an arbitrary […]
Generalized Limit Expressed through Ultralimits
In my book Limit of a Discontinuous Function I defined the generalized limit defined for every (even discontinuous) function. The definition of my generalized limit uses the concept of funcoids. Funcoids are a little advanced topic, what somehow hinder understanding of generalized limit by other mathematicians. But in the special case of a function p […]
An Introduction to Filters—What’s So Special About This Partially Ordered Set?
AGT isn’t new math that’s intended to replace existing mathematical theory, but rather a novel way to express mathematical concepts which were previously inexpressible. I indulge these ideas because I believe, as is the modus operandi for all academics, that any worthwhile, logically consistent idea must be pursued until it is falsified or verified for […]
What is Algebraic General Topology (Video)
Algebraic general topology – what is it? First, what is a general topology? General topology is the theory of topological spaces, as well as uniform spaces, proximity spaces, and metric spaces. I made a rather big discovery – a general theory that generalizes all these types of spaces in one algebra. That is, instead of […]