The Math Citadel
https://themathcitadel.com
Recent articles from The Math Citadel.An Illustration of Applied Stochastic Geometry: Coverage Probability in a Simple ALOHA MANETThis article weaves high level concepts from stochastic geometry into a practical description of wireless network coverage under the Aloha protocol.
https://www.themathcitadel.com/articles/aloha-manet-coverage.html
Markov Decision Process BasicsThis article discusses the practical basics of Markov decision processes, with machine repair and a garden fertilization examples.
https://www.themathcitadel.com/articles/mdp-basics.html
An Interesting Prisoner's DilemmaThis article will explore a generalization of a problem Traylor found in S. Ross's "Applied Probability Models with Optimization Applications" involving a strange prison and some basic probability reasoning.
https://www.themathcitadel.com/articles/prisoner-dilemma.html
Concatenation as an OperationThis article explores the definition of an operation using the familiar notion of concatenation.
https://www.themathcitadel.com/articles/concatenation-operation.html
Defining an Anomaly This article details some of the challenges in anomaly detection, particularly in defining an anomaly "globally".
https://www.themathcitadel.com/articles/defining-an-anomaly.html
Taking Things for Granted: Elementary Properties of GroupsMany concepts in basic algebra are also taken for granted, such as cancellation of terms, and commutativity. This post will revisit some basic algebra (think solving for x), but with some of those things we took for granted removed.
https://www.themathcitadel.com/articles/elementary-properties-groups.html
Group Theory, XOR, and Binary Codes: Introducing Coding TheoryThis article gives an introduction to coding theory by looking at the group structure of binary words under the exclusive-or operation.
https://www.themathcitadel.com/articles/group-theory-xor.html
Interventions, not AnomaliesThis article gives a suggestion for dealing with time series anomalies.
https://www.themathcitadel.com/articles/interventions-not-anomalies.html
Maximum-Likelihood Decoding This post will now allow us to explore some more interesting topics in coding theory, courtesy of Pinter's A Book of Abstract Algebra. We'll introduce the notion of a code, informations, and parity check equations.
https://www.themathcitadel.com/articles/maximum-likelihood-decoding.html
Like Clockwork: Modulo Addition And Finite Groups Of IntegersThis article eases a reader into finite groups and modulo arithmetic.
https://www.themathcitadel.com/articles/mod-add-groups-integers.html
A Partition By Any Other NameThis article looks at cosets of a group and notes how they partition a set, then mentions an application of the result to coding theory.
https://www.themathcitadel.com/articles/partition-other-name.html
All the Same OppositesThis article examines the surprising richness in the field of two elements.
https://www.themathcitadel.com/articles/all-the-same-opposites.html
Applications of Reflections: Taking a Group to its Abelian FormThis article looks at the definition of the categorical concept of a reflection and gives a clear application with full construction.
https://www.themathcitadel.com/articles/application-reflection-abelian.html
Beyond Cookbook Mathematics, Part 1This article gives a more colloquial introduction to the importance of mathematical definitions and framing practical problems mathematically.
https://www.themathcitadel.com/articles/beyond-cookbook-math-1.html
Beyond Cookbook Mathematics, Part 2This article continues colloquially discussing how to look at and read mathematical proofs.
https://www.themathcitadel.com/articles/beyond-cookbook-math-2.html
Building a Ground Floor: Topology Basics, Part 1This article gives a short introduction to basic topological concepts, with concrete examples.
https://www.themathcitadel.com/articles/topology-basics-1.html
Cauchy Sequences: The Importance of Getting CloseThis article gives a short exposition of a Cauchy sequence of real numbers.
https://www.themathcitadel.com/articles/cauchy-sequences-close.html
Energy Levels of Molecules are Bounded BelowThis guest submission is from Nikita Lisitsa, a professional software developer and mathematician.
https://www.themathcitadel.com/articles/energy-levels-molecules-bounded.html
Equivalence v. Isomorphisms in Category TheoryThis article discusses the difference between equivalence and isomorphisms among categories, with concrete examples.
https://www.themathcitadel.com/articles/equivalence-isomorphisms-category.html
Using Theoretical Chemistry to Engineer Better Packet FlowThis article reviews a technical report on the application of chemical kinetics to packet flow management in computer networks. Much more detail is given.
https://www.themathcitadel.com/articles/exploiting-chemistry-packet-flow.html
The Convenience of IsomorphismsThis article gives a simple introduction to the utility of isomorphisms, using an example on Z_4 and a group of complex numbers.
https://www.themathcitadel.com/articles/isomorphisms-convenient.html
Mailbox Answers: Calculating New Parity After an OverwriteThis article is inspired by an e-mail question regarding parity for a data stripe on a storage system. It provides motivation for an elegant mod 2-based proposition and proof.
https://www.themathcitadel.com/articles/mailbox-answers-parity.html
On the Optimal Storage of Bulk CargoesThis article gives an extremely practical application of basic (though nontrivial) calculus involving the optimization of storing bulk cargoes on ships.
https://www.themathcitadel.com/articles/optimal-storage-bulk-cargo.html
Networking Mathematics: Random Early Detection and TCP SynchronizationThis article discusses some computer networking basics and reviews an interesting algorithm for buffering and queue management.
https://www.themathcitadel.com/articles/networking-math-tcp.html
Paper Review: Active Queue Management with Non-Linear Packet Dropping FunctionThis article reviews a network engineering paper and points out some issues.
https://www.themathcitadel.com/articles/paper-review-packet-drop.html
Reduce the Problem: Permutations and Modulo ArithmeticThis article explores the elegant connections between permutations and modulo arithmetic.
https://www.themathcitadel.com/articles/permutation-mod-arithmetic.html
Should I Accept This Shipment? This article gives a manufacturing application of the hypergeometric distribution in shipment sampling and acceptance.
https://www.themathcitadel.com/articles/accept-ship.html
Sequences and Tendency: Topology Basics Part 2This article helps the reader to understand what convergence means when we strip the idea of "distance" away.
https://www.themathcitadel.com/articles/topology-basics-2.html
The Cartesian Product of Two GraphsAn exploration of what "multiplication" can mean for
two graphs.
https://www.themathcitadel.com/articles/cartesian-prod-graph.html
The Red-Headed Step-DistributionsThis article discusses the notion of singular
continuous distributions, and a lone constructive example.
https://www.themathcitadel.com/articles/red-head-step-dist.html
Regression and Time SeriesThis article discusses the similarities and
differences between ordinary linear regression and time series
models.
https://www.themathcitadel.com/articles/time-series-regression.html
Topologies v. sigma-algebrasThis article discusses the similarities and
differences between topologies and σ−algebras, aiming to clear up
the two definitions with explicit examples.
https://www.themathcitadel.com/articles/topologies-sigma-algebras.html
Using Boolean Algebra to Find all Maximal Independent Sets
in a GraphThis article details some frustration in finding
maximal independent sets of a graph.
https://www.themathcitadel.com/articles/maximal-independent-set-graph.html
Using Dirac Delta Formalism to Calculate Shear Force and
Bending MomentThis article gives a very interesting application of
the Dirac delta function to a common mechanics problem.
https://www.themathcitadel.com/articles/dirac-delta-moment.html
Welcome to GF(4)This article discusses the Galois field on 4 elements,
with an emphasis on solving linear equations with altered
definitions of addition and multiplication.
https://www.themathcitadel.com/articles/welcome-to-gf4.html
Poisson Processes and Data LossThis article discusses the Poisson process as it may
(or may not) apply to data loss in enterprise storage systems.
https://www.thecommathcitadel.com/articles/poisson-processes-data-loss.html
The Hathlor Classification SystemThis article explains our novel technical book
classification system that we employ in our library.
https://www.themathcitadel.com/articles/hathlor-system.html
Simulating Soundscapes Using ConvolutionsThis article gives a fantastic application of
convolutions as applied to music and signal processing. Audio is
included.
https://www.themathcitadel.com/articles/simulating-soundscapes.html