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The Mathematics of Duelling

Duelling with pistols. If you were the one issuing the challenge, your dilemma was that custom dictated that your adversary be allowed to shoot first. Only then, if you were still able to shoot, would you be permitted to seek “satisfaction”.

How much of an advantage does the first shooter really have? In this article, we build a simple probability model, and implement a numerical model in a few lines of R code.

Two gentleman face off in the snow.  Convention dictates the challenged shoots first.

Two gentleman face off in the snow. Convention dictates the challenged shoots first.

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A Force-Multiplier for R&D

Your computer, kitted out with specialised software (mostly free) and configured for efficiency of use, is perhaps the single greatest ‘force multiplier’ that you can have as a Scientist, R&D Engineer, or Mathematician — unless of course you have a large budget and a team of bright minds at your disposal.

With good tools, a bit of creative joinery, and a good dose of persistence, you can become an army of one…

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A Radar Tracking Approach to Data Mining

(Statistics and Data Mining II)

Automated decision problems are frequently encountered in statistical data processing and data mining. An heuristic filter or heuristic classifier typically has a limited set of input data from which to arrive at a set of conclusions and make a decision: REJECT, ACCEPT, or UNDETERMINED. In such cases, pre-processing the input data before applying the heuristic classifier can substantially enhance the performance of the decision system.

In this article, I’ll motivate the use of a radar-tracking algorithm to improve the performance of automated decision making and statistical estimation in data processing. I will illustrate using the website visitation statistics problem.

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Disaster Management for Web-Hosted Data

(Technology Infrastructure Series)

In case you’re taking seriously guarantees about uptime, reliability, or backups advertised by website hosting companies, you should know that most guarantees of service are an idealized concept, especially if you use a low-cost web hosting service. Now, this doesn’t mean you should avoid low-cost web hosts. What you should do is give a little thought to the “what if’s” that may arise, and what you can do before they arise to minimize the pain when they do.

In this article, I’ll go through a few situations you might want to consider, and some options you can use to reduce your risk.

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Analysis of Visitor Statistics: Data Mining in-the-Small

(Statistics and Data Mining I)

For a variety of reasons, meaningful website visitation and visitor behavior statistics are an elusive data set to generate. This article introduces the visitor statistics problem, and describes seven challenges that must be overcome by statistical and data analysis techniques aiming for accurate estimates. Along the way, we’ll encounter the “Good News Cheap, Bad News Expensive” Paradox of Data Mining — or, why information is often used “as-is”.

This article is the first in a series on algorithms, statistics and data analysis techniques (using free and open source tools) using the visitor statistics problem as a vehicle for illustration.

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Mathematical Finance and The Rise of the Modern Financial Marketplace

Abstract

One of the fascinating areas to arise recently in applied mathematics has been Mathematical Finance. From a technical point of view, Mathematical Finance uses a broad range of sophisticated mathematics for its financial models, and relies on state-of-the-art software engineering and computer hardware to implement these financial models, often in real-time.

Whether one is interested in technology or not, there is a kernel of core financial ideas at the heart of the global free market capitalist system that every literate citizen should understand. Whether we agree with their principles or with the inequities that are, arguably, the result, these ideas are in place across most of the world today. A closer look at mathematical finance will offer a better understanding of the mechanics of the modern financial world.

In this article, I’ll motivate the need for financial mathematics through a simplified account of the rise of the modern financial marketplace.

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Professional Grade Typesetting with LaTeX / TeX

(Mathematical Toolset Series: TeX & LaTeX, Part 1 of 3)

If symbols, formulas, and equations comprise a large portion of your professional communication, then becoming familiar with the LaTeX / TeX platform should be high on your to-do list. With the right tools and a little practice, the relative ease of creating beautiful documents with TeX may mean that you soon leave your favorite Office suite in favor of TeX for your technical writing.

This article introduces the LaTeX / TeX platform, illustrates its capabilities, and highlights the key differences between using TeX for document preparation and more commonly used word processing systems.

For those that like to know the human side of the tools they use, a little history of TeX, the philosophy motivating its development, and something about its legendary creator, is included.

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An Open Source LaTeX / TeX Platform for Windows


(Mathematical Toolset Series: TeX & LaTeX, Part 2 of 3)

LaTeX / TeX on Windows has had the rather unfortunate reputation of being difficult to install and use. But with the quality of today’s open source tools and references, this is no longer the case, and should not be a reason to deter you from trying it out.

This article takes you through the practical business of getting a LaTeX / TeX platform running on Windows, using free, open-source software. From downloading and installing, to providing reference materials and basic templates, this article should get you going quickly and provide a decent toolkit from which to build.

Note: All the steps in this article have been tested on Windows XP and Windows 7.

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Writing Modular TeX Documents


(Mathematical Toolset Series: TeX & LaTeX, Part 3 of 3)

If you write frequently, it is likely that you have certain stock or administrative material that is included in each of your documents. You also likely spend a substantial portion of your overall effort re-writing, editing, or re-arranging material. In this situation, one of the best ways of preserving your time and your sanity is to adopt a modular approach to document development.

In this final article of the three part series on LaTeX / TeX, I will discuss a modular approach to document preparation using TeX. I’ll also provide modular templates that should make your use of TeX more efficient.

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Assembly Language and Embedded Systems Development

On the Value of Assembly Language, and Resources to get you started in Digital Logic, Computer Architecture, and Assembly Language programming.

Despite advances in programming technologies since the 1970s, there are still reasons to understand and learn assembly language programming. Contrary to what one may imagine, Assembly Language is not a relic of the past, even though a typical applications programmer will almost never need to drop into assembly.

In this article, we’ll look at practical situations in embedded systems development in which assembly language programming is still used, pedagogical reasons to learn assembly language, and provide resources and projects for gaining a working knowledge of digital logic, computer architecture, and assembly language programming.

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Meta-Questions and Educational Success

Abstract

“It seems that students who are hard-working and otherwise successful, but whose milieu (peers, mentors, and home environment) are at once non-technical and disengaged from the ideas behind science and technology, are at a higher risk of disorientation, dissatisfaction, or disillusionment with mathematics and science.”

In this article, I’ll explore this conjecture, and suggest that a ‘liberal arts’ approach that addresses the humanistic & philosophical elements of technical subjects may help reverse an otherwise tough pedagogical situation.

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A Course in the Philosophy and Foundations of Mathematics

An examination of mathematical methods and the search for mathematical meaning.

During your studies of mathematics, physics and engineering, you may find yourself distracted or troubled by meta questions about mathematics — questions that fall outside the syllabi of most of the coursework that you’ll take.

For those for whom this itch is persistent, what follows is an outline and reading list for a Course in the Philosophy and Foundations of Mathematics. Among the topics included are the relation of mathematics to science, the examination of mathematical method, and the search for mathematical meaning.

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Finite Summation of Integer Powers (Part 3)

(Discrete Mathematics Techniques III)

Abstract
We find a direct closed-form solution, i.e. one that does not require iteration, for the general case of the finite-summation-of-integer-powers problem S_p(N) = \sum_{k=1}^{N} k^p. Having established in Part 2 that the closed-form solution is a polynomial, the summation is here rewritten as the sum of the p+1 independent monomials a_j N^j (1 \leq j \leq p+1), where the a_j are unknown coefficients. Using the recurrence relation S_p(N+1) = S_p(N) + (N+1)^p, we obtain a linear combination of the monomials, which reduces to an easily solvable (p+1)-by-(p+1) triangular linear system in the unknown coefficients a_j of the closed-form polynomial solution. Maxima and Octave/Matlab codes for directly computing the closed-form solutions are included in the Appendices.

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The Advance of Sensor Networks and Autonomous Systems

The past five years have seen the emergence of a growing array of autonomous swimming, flying, and rolling vehicles, each highly sensored and capable of real-time communication with processors external to themselves. Practical designs are now commercially available for each of the four primary areas of our environment: terrestrial, marine (subsea, surface, and amphibian), atmospheric (gravity constrained), and space (orbital and planetary).

A look at a selection of these achievements in networked sensor systems will set the stage to discuss the communications layer of the ubiquitous computing stack.

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The Advance of Marine Micro-ROVs

A Versatile Tool for Marine Operations, and a Portable Undersea Platform for Small Sensors

Micro-ROVs (Remotely Operated Vehicles) are becoming increasingly capable even as their size and cost drop, opening up new possibilities for the application of undersea inspection, imaging, and measurement.

In this article, I’ll discuss four reasons why Micro-ROVs should be a routinely used part of a marine and water-side operations toolkit, and review some stand-out choices in the Micro-ROV category.

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