(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 [...]
By Assad Ebrahim, on March 19th, 2010
Topic: Technology-Advanced
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), [...]
By Assad Ebrahim, on March 19th, 2010
Topic: Technology-Advanced
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 [...]
… Integrating Sensors into the Ubiquitous Computing Stack
“Smart dust”, tiny leaf sensors, wearable computing — these and a host of other sensors that make measurements and communicate without requiring human intervention can now be readily integrated into dispersed systems to provide ambient intelligence, situational awareness, and the capability for adaptive behaviors or intelligent process automation. [...]
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 [...]
…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, [...]
By Assad Ebrahim, on February 12th, 2010
Topic: Software-Programming…for Embedded and Low-Level Systems Development
C provides the convenience of learning one language (a significant efficiency over assembly language), while retaining the ability to target a variety of platforms — from modern operating systems (Linux, Windows, Mac) to real-time operating systems, systems-on-a-chip, and a host of microcontrollers for embedded development. And whenever you have [...]
By Assad Ebrahim, on February 8th, 2010
Topic: Mathematics-Technical(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 [...]
By Assad Ebrahim, on February 8th, 2010
Topic: Mathematics-Technical(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 [...]
…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.
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Articles by Topic Professional Tools
Mathematics-Applications
Mathematics-Technical
Engineering-Software
Engineering-Systems
Mathematical Education
Mathematics-Phenomenology
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