Mathematical Science & Technologies » Mathematics

A Radar Tracking Approach to Data Mining

Assad Ebrahim — Mon, 15 Nov 2010 06:18:06 +0000

(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 [...]

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

Assad Ebrahim — Fri, 03 Sep 2010 06:23:53 +0000

(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 [...]

Mathematical Finance and The Rise of the Modern Financial Marketplace

Assad Ebrahim — Mon, 05 Jul 2010 18:12:24 +0000

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 [...]

Professional Grade Typesetting with LaTeX / TeX

Assad Ebrahim — Thu, 20 May 2010 19:38:12 +0000

(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 [...]

An Open Source LaTeX / TeX Platform for Windows

Assad Ebrahim — Thu, 20 May 2010 18:44:19 +0000

(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 [...]

Writing Modular TeX Documents

Assad Ebrahim — Thu, 20 May 2010 17:14:55 +0000

(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 [...]

Meta-Questions and Educational Success

Assad Ebrahim — Thu, 15 Apr 2010 16:53:29 +0000

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 will explore this conjecture, and suggest a [...]

A Course in the Philosophy and Foundations of Mathematics

Assad Ebrahim — Thu, 15 Apr 2010 16:36:30 +0000

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 [...]

Finite Summation of Integer Powers (Part 3)

Assad Ebrahim — Fri, 02 Apr 2010 10:41:39 +0000

(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 . Having established in Part 2 that the closed-form solution is a polynomial, the summation is here rewritten as the sum of the independent monomials (), where the [...]

Why Zero Raised to the Zero Power IS One

Assad Ebrahim — Thu, 25 Feb 2010 13:56:16 +0000

The question of what value 0^0 should evaluate to has been discussed since the time of Euler (1700s). There are three candidate choices: 1,0, or “indeterminate” (i.e., throw an error).

In this article, I argue that the only reasonable choice (for discrete mathematics) is 0^0=1 (), and I’ll give a tangible, feel-the-grit-in-your-palms reason [...]

Maxima for Symbolic Computation

Assad Ebrahim — Mon, 22 Feb 2010 06:33:13 +0000

…By virtue of its unbeatable low cost (free), ready availability for all three major operating systems, and its raw power in all areas of mathematics and analytic engineering, Maxima is a mathematical computing package that ought to be in the toolbelt of every programmer, engineering, scientist, and mathematician.

From garden-variety algebraic simplification, polynomials, calculus, [...]

Finite Summation of Integer Powers (Part 2)

Assad Ebrahim — Mon, 08 Feb 2010 19:57:14 +0000

(Discrete Mathematics Techniques II)

Abstract We solve the general case of the finite-summation-of-integer-powers problem for arbitrary , and obtain a -th order recurrence relation that can be used to iteratively obtain the closed form polynomial for for any given . Source code is given for computing these polynomials using Maxima, an open-source (free) symbolic [...]

Finite Summation of Integer Powers (Part 1)

Assad Ebrahim — Mon, 08 Feb 2010 08:50:47 +0000

(Discrete Mathematics Techniques I)

Abstract We motivate an approach that uses recurrence relations to find closed form solutions to the finite-summation-of-integer-powers problem for any individual . The approach is illustrated for small : . Maxima, an open-source (free) software package, is used to demonstrate how a symbolic computation platform can speed up the accurate [...]

Mathematics Toolset

Assad Ebrahim — Wed, 03 Feb 2010 11:52:59 +0000

…For industry or research.

Over the coming months, I’ll be posting articles as part of a series on setting up a toolset for Mathematics work in industry or research.

I’ll be emphasizing open source software. Though the primary target is the Windows PC platform (dominant in industry), I will list alternatives for Linux/Unix.

The current plan is [...]

Good mathematical technique and the case for mathematical insight

Assad Ebrahim — Sun, 31 Jan 2010 19:16:14 +0000

Good mathematical technique can bring the solution to certain mathematical questions within reach. By a proper formulation (one that is both tractable and that generalizes readily) and the use of mechanical techniques, one can often pass from a single insight to the solution of a family of problems, and in some cases, to the [...]

Teaching Mathematics “in Tunic”

Assad Ebrahim — Tue, 19 Jan 2010 16:04:11 +0000

…Thoughts on the Teaching of Exploratory, Topical Mathematics

Mathematics is a richly spun tapestry threaded with interconnections from a multiplicity of endeavors, perspectives, and disciplines, both theoretical and applied. Contrary to its typical presentation, mathematics is not a linear subject.

For an instructor, this presents a number of challenges: how best to address the non-linear, inter-woven [...]

Catalysts in the Development of Mathematics

Assad Ebrahim — Fri, 15 Jan 2010 13:57:04 +0000

The development of mathematics has had many encouraging forces: societal, technological, cultural. These have served to accelerate mathematics and have been accelerated in turn, in many cases the pair becoming locked into a mutually beneficial resonance that has dramatically energized both.

In this article, I look at some of the significant catalysts, from the rise [...]

The Development of Mathematics

Assad Ebrahim — Sun, 03 Jan 2010 17:23:48 +0000

… in a nutshell

The development of mathematics is intimately interwoven with the progress of civilization, influencing the course of history through its application to science and technology.

But mathematics has changed. Even the mathematics of the 1800s can seem quite strange now, so greatly has mathematics evolved in the past 100 years and [...]

Characteristics of Modern Mathematics

Assad Ebrahim — Sat, 02 Jan 2010 14:17:25 +0000

What are the characteristics of mathematics, especially contemporary mathematics?

I’ll consider five groups of characteristics:

Applicability and Effectiveness, Abstraction and Generality, Simplicity, Logical Derivation, Axiomatic Arrangement, Precision, Correctness, Evolution through Dialectic…

Characteristics of Modern Mathematics

In the article What is Mathematics?, I have posited that Mathematics arises from Man’s attempt to summarize the variety of empirical phenomena that [...]

What is Mathematics?

Assad Ebrahim — Wed, 30 Dec 2009 14:34:25 +0000

Can a definition be given that captures the meaning of Mathematics across the millennia of its recorded history? What unites the practice of mathematics throughout its history and into the present time?

In this article, I will try for a short answer by proceeding iteratively — convergence will be reached in two iterations….

What is Mathematics?

Though [...]

Mathematics in Pre-History

Assad Ebrahim — Tue, 29 Dec 2009 14:28:02 +0000

…Taking into account early human evolution, and in particular, the development of speech, it is likely that the understanding of counting and keeping time far pre-dates the discovered fossil evidence of Homo Sapiens from 30,000 years ago, and reaches back at least as far as 250,000 years ago, coincident with speech, fire, and cooperative hunting, [...]