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Equivalence v. Isomorphisms in Category Theory

Equivalence v. Isomorphisms in Category Theory

Introduction Editor's Note: The article is co-written by Rachel Traylor (The Math Citadel/Marquette University) and Valentin Fadeev (The Open University, UK). Substantial additional review, contributions, and discussions were provided by Matt Kukla and Jason Hathcock. A pdf is available for download at the end of this post. The biggest challenge we have found in studying…

The Cartesian Product of Two Graphs

The Cartesian Product of Two Graphs

(Ed. Note: A pdf version of this article is attached at the end of the post for offline reading.) Introduction and Preliminaries Graphs are objects like any other, mathematically speaking. We can define operations on two graphs to make a new graph. We'll focus in particular on a type of graph product- the Cartesian product,…

Exploiting Chemistry for Better Packet Flow Management 5: Chemical Congestion Control, Design Motifs, and Conclusion

Exploiting Chemistry for Better Packet Flow Management 5: Chemical Congestion Control, Design Motifs, and Conclusion

This represents the final installment of the series reviewing the 2011 technical report by Meyer and Tschudin. Part 1 gave an overview of the report and the problems it aimed to solve, as well as the chemistry basics necessary for further understanding. Part 2 discussed artificial chemistries and the extension to an artificial packet chemistry,…

Exploiting Chemistry for Better Packet Flow Management 4: Scheduler Implementation

Exploiting Chemistry for Better Packet Flow Management 4: Scheduler Implementation

This article is the fourth part in a series based on a report reviewing the technical report of Meyer and Tschudin[11] who have extended the notion of an artificial chemistry to an artificial packet chemistry with the intention of exploiting the natural behavior of chemical reactions to design better flow management policies for computer networks.…

Exploiting Chemistry for Better Packet Flow Management 3: Formal Analysis

Exploiting Chemistry for Better Packet Flow Management 3: Formal Analysis

The previous two posts introduced the ideas of Meyer and Tschudin [11] involving the application and exploitation of chemical kinetic theory to flow management in computer networking. The first part introduced the ideas and gave an overview of the entire work, and the second part took a deeper look into the formal model of a…

Exploiting Chemistry for Better Packet Flow Management 2: Formal Model

Exploiting Chemistry for Better Packet Flow Management 2: Formal Model

This post is the second breaking down a report/review of a technical report by Meyer and Tschudin [11] that modifies the formal notion of an artificial chemistry and creates an artificial packets chemistry with the goal of designing better flow management by exploiting the natural behavior of chemical reactions.    Note: for those more interested…

Exploiting Chemistry for Better Packet Flow Management 1: Introduction

Exploiting Chemistry for Better Packet Flow Management 1: Introduction

Perhaps the phrase "don't reinvent the wheel" is overused. However, many newer disciplines, particularly in the technology sector, seem to insist on it. One thing physical engineers learned long ago was to study the world around them, work with it, and emulate it in their designs. Network engineering should be no different. In a technical…

Expecting the Unexpected: Borel’s Paradox

Expecting the Unexpected: Borel’s Paradox

One of the best ways to shorten a proof in statistics or probability is to use conditioning arguments. I myself have used the Law of Total Probability extensively in my work, as well as other conditioning arguments in my PhD dissertation. Like many things in mathematics, there are subtleties that, if ignored, can cause quite…

Using Boolean Algebra to Find all Maximal Independent Sets in a Graph

Using Boolean Algebra to Find all Maximal Independent Sets in a Graph

Graph theory may be one of the most widely applicable topics I've seen in mathematics. It's used in chemistry, coding theory, operations research, electrical and network engineering, and so many other places. The subject is mainly credited to have begun with the famous  Seven Bridges of Königsberg problem posed by Leonard Euler in 1736. Frank Harary…

Beyond Cookbook Mathematics, Part 2

Beyond Cookbook Mathematics, Part 2

The previous article discussed the importance of definitions to mathematical thought. We looked at a definition (of an end-vertex in a graph), and picked it apart by finding multiple ways to look at it. We also directly used the definition in a practical manner to find “weak links” in a network. This time, we’ll look…