Updated! February 5, 2017
The value of zero raised to the zero power, 
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If you haven’t done so already, you may want to start by reading the Preface to the Computing Series: Software as a Force Multiplier, Sections 1-3. Maxima: a Computer Algebra System (CAS) for symbolic computationLast updated: Feb 19, 2023 (fixed links). Nov 11, 2022 (added omega-math’s excellent web interface, and generating function calculation of the partition of integers problem). Maxima is a computer algebra system (CAS) for symbolic computation that is free, open source, runs on multiple operating systems (Win,Mac,Linux), and covers a wide range of mathematical capabilities and graphical capabilities. These include algebraic simplification, polynomials, methods from calculus, matrix equations, differential equations, number theory, combinatorics, hypergeometric functions, tensors, gravitational physics, PDEs, nonlinear systems, plus including 2-D/3-D plotting and animation. With a large and responsive user community, there is plenty of help to get up the learning curve, and with its active developer base, Maxima and its ecosystem continue to gain capability, including a fantastic web interface by Omega-Math/Vroom-Labs (see the screenshot below, r0*0). The result is a free, versatile, powerful mathematical computing package for engineers, scientists, mathematicians, programmers, and students. This article will help you get started with Maxima and set you up with resources to flatten the learning curve.  Omega-Math’s web interface to Maxima. Used here to calculate the first 10 elements of p(n), the number of ways to partition integer n, using a generating function comprising a truncated series of polynomials up to degree n=10 Continue reading this article… The mathematician Alfred North Whitehead1 observed that “[Advancement occurs] by extending the number of important operations which we can perform without thinking of them.” (Introduction to Mathematics, 1911 2) This is certainly true in mathematics where the development of judicious notation, accompanied by good mathematical technique, extends the capability to perform chains of complex reasoning accurately and efficiently. Through proper problem formulation (tractable yet generalizable), one can sometimes pass from a single insight to the solution of a family of problems, and in some cases, to the solution to the general question itself.3 Here, mathematical computing can provide a useful benefit: helping to efficiently explore conjectures, dispatch with false directions, and save time during the development, error-checking and validation stages of obtaining general results. In industry, where specific or semi-general results are needed fast, such tools allow rapidly working up the required material and providing the necessary certainty before the fully general results or complete proof are ready. Continue reading this article…
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 of the leisured class in ancient times to the impact of computing in modern times. 2nd ed., Feb 1, 2024, 1st ed. Jan 9th, 2010 A common misconception is that assembly language programming is a relic of the past. This is certainly not the case, and assembly language remains a core knowledge area for embedded systems development, digital design, and algorithm development in the 21st century. A second misconception, especially amongst those who are only familiar with higher level languages (Python, Ruby, C#/.NET, Perl), is that assembly language is a defective programming language and therefore not worth the time to invest in. But assembly language is more than ‘just another general purpose programming language’. It is actually the control signal specification for the microprocessor or microcontroller that will be running the instructions, and whose digital design must be reasonably well understood in order to get it to work successfully. Higher level languages typically hide the underlying toolchains behind turnkey integrated development environments (IDEs). But the toolchains are valuable in their own right, comprising various software components (pre-processor, compiler, assembler, linker, loader) which take the high level code and transform it to executable machine code that can run on the target processor, optionally producing assembly code for inspection along the way. Familiarity with this toolchain can help evaluate how much overhead the high-level tools introduce on the code, which is an important part of understanding how much you’re trading off. In this article, we’ll look first take a look at the software toolchain involved in general terms, before turning to specific tools you can use on a modern Windows computer (through Windows 11) to target an x86 chip (no longer in your PC but in a DOS Emulator). Similar skills and approaches carry over to the toolchain for the Atmel 328P and ATTiny 85 with a graphics application (TinyPhoto) on the ATTiny85 here. The development of mathematics is intimately interwoven with society and culture, influencing the course of history through its applications to science and technology. But mathematics itself has changed much over its history. Even the mathematics of the early 1800s can now seem quite strange, so great have been the changes in just the past 150 years as it has been reworked in the modern abstract approach. Though advanced mathematics may now appear arcane from the outside looking in, the present state of mathematics is the result of a natural evolution of the subject. And there is much excitement promised ahead with the rise of new mathematics and application areas in subatomic and quantum physics, in the the field of statistical learning (also called artificial intelligence or machine learning), and in numerical computing and simulation. What follows is the story of mathematics, in a nutshell. What are the characteristics of mathematics, especially contemporary mathematics? I’ll consider five groups of characteristics:
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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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