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Category: Spines

Applications of Reflections: Taking a Group to its “Abelian” Form

Applications of Reflections: Taking a Group to its “Abelian” Form

In continuing the exploration of explicit applications and examples of category-theoretic concepts, we highlight the versatility of reflections and reflective subcategories. This concept can be used to perform all kinds of desired actions on a category to yield a subcategory that "nicer" in some way. This article explores how we can use reflections to make…

On the optimal storage of bulk cargoes: an exercise in calculus.

On the optimal storage of bulk cargoes: an exercise in calculus.

Editor's Note: This article is authored by Valentin Fadeev, a physics PhD candidate at The Open University, UK. You can reach him via Twitter at @valfadeev. Despite their undeniable and enormous role in applications, the methods of calculus are often seen by students present and past as a form of mathematical art removed from everyday…

Using Dirac Delta Formalism to Calculate Shear Force and Bending Moment

Using Dirac Delta Formalism to Calculate Shear Force and Bending Moment

Editor's note: This article was written by Valentin Fadeev, a physics PhD candidate in the UK. He also co-authored this article on equivalences and isomorphisms in category theory. You can find him on Twitter via the handle @ValFadeev. The delta function was introduced by P.A.M. Dirac, one of the founders of quantum electrodynamics. The delta…

Should I accept this shipment?

Should I accept this shipment?

Suppose you work for an engineering or manufacturing firm, and you receive shipments of different parts from various suppliers. It's not good business practice to just blindly accept a shipment, because some parts in your batch may be defective. Too many defective parts, and you'll slow down manufacturing (in addition to wasting valuable time and…

Energy Levels of Molecules are Bounded Below

Energy Levels of Molecules are Bounded Below

Editor's Note: This guest submission is from Nikita Lisitsa, a professional software developer and mathematician. You can follow him on Twitter. Are Molecules Perpetual Motion Machines? Short answer: no, of course not. Perpetual motion machines do not exist. There are deep theoretical reasons for that, as well as less deep but nevertheless convincing experimental data.…

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…