(Discrete Mathematics Techniques II)
1st ed. Feb 8th, 2010
Abstract
We continue the 3-part paper exploring how one might solve for themselves the general case of the sum-of-integer-powers problem 
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(Discrete Mathematics Techniques III) 1st ed. Apr 2nd, 2010 Abstract A lovely paper by Bearden (March 1996, American Mathematical Monthly), which was shared with me by a reader, tells the mathematical story nicely, with much of the history filled in. *New!* (29 Aug 2020) – Turtle Logo v1.8 (portable) is available! Developer kit with source code included. Suitable from ages 3 years to adult. (970 lines of Forth code).
A challenge facing parents and teachers is how to help children develop ‘builder’ relationships with technology rather than being limited to the passive consumption of content created by others. The consensus on what’s important for older kids and adults is clear: coding. This enables children to participate in the creation of their own technological “micro-worlds” — environments rich in educational potential.[14] This autumn, spurred by having our own young children (one aged 4 years, the other 16 months), we began an experiment, the result of which is a Turtle Logo program for Windows computers (freely downloadable) that is simple enough to be accessible for children from 3 years and older, while providing an extensible platform that can grow with the child. The long-term goal is to enable children to express their creativity, artistry, and natural ‘builder’ impulses using coding, computer graphics, and robotics as readily as the previous generation could using paints, brushes, and building blocks.  Turtle Logo – Inspiring the next generation of technology builders. Updated: 2014-11-23 Pencil & Notebook are hard to beat when diagrams, equations, and words are desired in roughly equal measure, which is common when working as an engineer, mathematician, or product designer. To be sure, there are good digital tools for subsets of these (example LaTeX), but not when all three are combined. But now there’s a new suite of digital freehand tools that are almost good enough to set aside the notebook and pencil permanently. The core of this toolkit is a vector-based graphics canvas (software) and an ultra-precise stylus (hardware).  Example of Freehand Digital Pencil, using Adobe Ideas. If you use Apple iOS, you may already be using Adobe Ideas, which does the job brilliantly and is free (see this excellent article by Simon Raper). [1] But Adobe has no plans to release this to either Windows RT or Android, so for the rest of us, that’s a show-stopper, until now, with MetaMoji… 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.
Your computer, kitted out with best in class software tools (mostly open source), can be perhaps the single greatest force multiplier that you have as a scientist, engineer, or mathematician. With the willingness to work hard and persist, and some skill at ‘software joinery’, you can accomplish more than you realize solely through your own capabilities. This article is about achieving those visions which you can do single-handedly, and how this enables having and testing the kind of bigger visions for which you need that next level of force multiplication that comes through larger budgets, high-performance teams, and management. (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. (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.
Introducing the LaTeX typsetting platform If symbols, formulas, and equations comprise a large portion of your professional communication, then you will gain significantly by becoming proficient with the LaTeX (pronounced “lay-tech”) document preparation platform. With the right tools and a little practice, the relative ease of creating beautiful mathematical documents with LaTeX will likely mean that you leave Office in favor of LaTeX for your technical writing. This article introduces the LaTeX platform (short for Lamport-TeX, after the mathematician Leslie Lamport), illustrates its capabilities, and highlights the key differences between using LaTeX or WYSIWYG “what you see is what you get” word processing systems such as Office. For those that like to know the human side of the tools they use, we provide a brief history of the legendary TeX (pronounced “tech”) platform, which underpins all variations of which LaTeX is one, looks at the philosophy motivating the development of TeX, and something about its legendary creator Donald Knuth.  Don Knuth, Leslie Lamport, and an illustration of of why writing mathematics in LaTeX is easier than in Word.
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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© 2009-2024 Assad Ebrahim To contact me on email, use: assadebrahim2000 at gmail dot com (do the obvious) |
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independent monomials 
(
), where the 
are unknown coefficients. Using the recurrence relation 
, we obtain a linear combination of the monomials, which reduces to an easily solvable 
-by-