Related papers: Arithmetic, Infinite Trees, and Second-order Subsy…
A process of extending sets which can be used as foundation for an alternative organization for Differential and Integral Calculus is presented.
We suggest a new approach to Artin's constant that leads to its representation as an infinite sum divided by another infinite sum. The same approach works well for Stephens' constant and higher rank Artin's constants. The main results are…
This is a short exposition--mostly by way of the toy models ``double logarithm'' and ``triple logarithm''--which should serve as an introduction to a forthcoming article in which we establish a connection between multiple polylogarithms,…
I shall explore various senses in which ultrafinitism can be fruitfully understood as engaging with a potentialist perspective in mathematics. First, I explain that every model $M$ of the theory of finite arithmetic -- arithmetic with a…
Sumterms are introduced as syntactic entities, and sumtuples are introduced as semantic entities. Equipped with these concepts a new description is obtained of the notion of a sum as (the name for) a role which can be played by a number.…
In this survey we discuss the results on the finitistic dimension of various stratified algebras. We describe what is already known, present some recent estimates, and list some open problems.
I consider the expansion of transcendental functions in a small parameter around rational numbers. This includes in particular the expansion around half-integer values. I present algorithms which are suitable for an implementation within a…
Fiore and Hur recently introduced a conservative extension of universal algebra and equational logic from first to second order. Second-order universal algebra and second-order equational logic respectively provide a model theory and a…
Furstenberg-Weiss have extended Szemer\'edi's theorem on arithmetic progressions to trees by showing that a large subset of the tree contains arbitrarily long arithmetic subtrees. We study higher dimensional versions that analogously extend…
These notes, connected to a "potpourri" topics class currently underway, discuss some basic topics in analysis and connections with other areas of mathematics.
When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in…
An alternative mathematics based on qualitative plurality of finiteness is developed to make non-standard mathematics independent of infinite set theory. The vague concept "accessibility" is used coherently within finite set theory whose…
We summarize four different versions of our course notes on the limits of mathematics.
One can find lists of whole numbers having equal sum and product. We call such a creature a bioperational multiset. No one seems to have seriously studied them in areas outside whole numbers such as the rationals, Gaussian integers, or…
A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…
Extensions of the $Stirling$ numbers of the second kind and $Dobinski$ -like formulas are proposed in a series of exercises for graduates. Some of these new formulas recently discovered by me are to be found in the source paper $ [1]$.…
This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…
This paper presents a novel possible worlds semantics, designed to elucidate the underpinnings of ultrafinitism. By constructing a careful modification of the well-known Kripke models for inuitionistic logic, we seek to extend our…
This paper presents a novel approach to automatically solving arithmetic word problems. This is the first algorithmic approach that can handle arithmetic problems with multiple steps and operations, without depending on additional…
This paper proposes an algebraic view of trees which opens the doors to an alternative computational scheme with respect to classic algorithms. In particular, it is shown that this view is very well-suited for machine learning and…