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Education & Academic

Creating Mathematical Gems in the Wolfram Language

The Wolfram Community group dedicated to visual arts is abound with technically and aesthetically stunning contributions. Many of these posts come from prolific contributor Clayton Shonkwiler, who has racked up over 75 "staff pick" accolades. Recently I got the chance to interview him and learn more about the role of the Wolfram Language in his art and creative process. But first, I asked Wolfram Community's staff lead, Vitaliy Kaurov, what makes Shonkwiler a standout among mathematical artists.
Michael Gammon, Blog Administrator, Document & Media Systems
Education & Academic

Limits without Limits in Version 11.2

Here are 10 terms in a sequence: And here's what their numerical values are: But what is the limit of the sequence? What would one get if one continued the sequence forever? In Mathematica and the Wolfram Language, there's a function to compute that: Limits are a central concept in many areas, including number theory, geometry and computational complexity. They're also at the heart of calculus, not least since they're used to define the very notions of derivatives and integrals. Mathematica and the Wolfram Language have always had capabilities for computing limits; in Version 11.2, they've been dramatically expanded. We've leveraged many areas of the Wolfram Language to achieve this, and we've invented some completely new algorithms too. And to make sure we've covered what people want, we've sampled over a million limits from Wolfram|Alpha.
Devendra Kapadia, Manager of Calculus and Algebra, Algorithms R&D
Education & Academic

How Laplace Would Hide a Goat: The New Science of Magic Windows

Last week, I read Michael Berry’s paper, “Laplacian Magic Windows.” Over the years, I have read many interesting papers by this longtime Mathematica user, but this one stood out for its maximizing of the product of simplicity and unexpectedness. Michael discusses what he calls the magic window. For 70+ years, we have known about holograms, and now we know about magic windows. So what exactly is a magic window? Here is a sketch of the optics of one:

Michael Trott, Chief Scientist, Wolfram|Alpha Scientific Content
Education & Academic

New in the Wolfram Language: Enhanced Derivatives

Derivatives of functions play a fundamental role in calculus and its applications. In particular, they can be used to study the geometry of curves, solve optimization problems and formulate differential equations that provide mathematical models in areas such as physics, chemistry, biology and finance. The function D computes derivatives of various types in the Wolfram Language and is one of the most-used functions in the system. My aim in writing this post is to introduce you to the exciting new features for D in Version 11.1, starting with a brief history of derivatives.
Devendra Kapadia, Manager of Calculus and Algebra, Algorithms R&D
Best of Blog

Hidden Figures: Modern Approaches to Orbit and Reentry Calculations

The movie Hidden Figures was released in theaters recently and has been getting good reviews. It also deals with an important time in US history, touching on a number of topics, including civil rights and the Space Race. The movie details the hidden story of Katherine Johnson and her coworkers (Dorothy Vaughan and Mary Jackson) at NASA during the Mercury missions and the United States' early explorations into manned space flight. The movie focuses heavily on the dramatic civil rights struggle of African American women in NASA at the time, and these struggles are set against the number-crunching ability of Johnson and her coworkers. Computers were in their early days at this time, so Johnson and her team's ability to perform complicated navigational orbital mechanics problems without the use of a computer provided an important sanity check against the early computer results.
Jeff Bryant, et al.
Education & Academic

Gardening à la Gardner

When looking through the posts on Wolfram Community, the last thing I expected was to find exciting gardening ideas. The general idea of Ed Pegg's tribute post honoring Martin Gardner, "Extreme Orchards for Gardner," is to find patterns for planting trees in configurations with constraints like "25 trees to get 18 lines, each having 5 trees." Most of the configurations look like ridiculous ideas of how to plant actual trees. For example:
Kathryn Cramer, Technical Communications and Strategy Group
Education & Academic

The Semantic Representation of Pure Mathematics

Introduction

Building on thirty years of research, development and use throughout the world, Mathematica and the Wolfram Language continue to be both designed for the long term and extremely successful in doing computational mathematics. The nearly 6,000 symbols built into the Wolfram Language as of 2016 allow a huge variety of computational objects to be represented and manipulated---from special functions to graphics to geometric regions. In addition, the Wolfram Knowledgebase and its associated entity framework allow hundreds of concrete "things" (e.g. people, cities, foods and planets) to be expressed, manipulated and computed with. Despite a rapidly and ever-increasing number of domains known to the Wolfram Language, many knowledge domains still await computational representation. In his blog "Computational Knowledge and the Future of Pure Mathematics," Stephen Wolfram presented a grand vision for the representation of abstract mathematics, known variously as the Computable Archive of Mathematics or Mathematics Heritage Project (MHP). The eventual goal of this project is no less than to render all of the approximately 100 million pages of peer-reviewed research mathematics published over the last several centuries into a computer-readable form. In today's blog, we give a glimpse into the future of that vision based on two projects involving the semantic representation of abstract mathematics. By way of further background and motivation for this work, we first briefly discuss an international workshop dedicated to the semantic representation of mathematical knowledge, which took place earlier this year. Next, we present our work on representing the abstract mathematical concepts of function spaces and topological spaces. Finally, we showcase some experimental work on representing the concepts and theorems of general topology in the Wolfram Language.
Eric Weisstein, Senior Researcher, Wolfram|Alpha Scientific Content Ian Ford, Pure Math Developer, Wolfram|Alpha Scientific Content
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