May 10, 2018 — Michael Trott, Chief Scientist

The Shape of the Differences of the Complex Zeros of Three-Term Exponential Polynomials

In my last blog, I looked at the distribution of the distances of the real zeros of functions of the form with incommensurate , . And after analyzing the real case, I now want to have a look at the differences of the zeros of three-term exponential polynomials of the form for real , , . (While we could rescale to set and for the zero set , keeping and will make the resulting formulas look more symmetric.) Looking at the zeros in the complex plane, one does not see any obvious pattern. But by forming differences of pairs of zeros, regularities and patterns emerge, which often give some deeper insight into a problem. We do not make any special assumptions about the incommensurability of , , .

The differences of the zeros of this type of function are all located on oval-shaped curves. We will find a closed form for these ovals. Using experimental mathematics techniques, we will show that ovals are described by the solutions of the following equation:

… where:

April 24, 2018 — Michael Trott, Chief Scientist

Identifying Peaks in Distributions of Zeros and Extrema of Almost-Periodic Functions: Inspired by Answering a MathOverflow Question

One of the Holy Grails of mathematics is the Riemann zeta function, especially its zeros. One representation of is the infinite sum . In the last few years, the interest in partial sums of such infinite sums and their zeros has grown. A single cosine or sine function is periodic, and the distribution of its zeros is straightforward to describe. A sum of two cosine functions can be written as a product of two cosines, . Similarly, a sum of two sine functions can be written as a product of . This reduces the zero-finding of a sum of two cosines or sines to the case of a single one. A sum of three cosine or sine functions, , is already much more interesting.

Fifteen years ago, in the notes to chapter 4 of Stephen Wolfram’s *A New Kind of Science*, a log plot of the distribution of the zero distances…

… of the zero distribution of —showing characteristic peaks—was shown.

February 2, 2018 — Ed Pegg Jr, Editor, Wolfram Demonstrations Project

Some trees are planted in an orchard. What is the maximum possible number of distinct lines of three trees? In his 1821 book *Rational Amusement for Winter Evenings*, J. Jackson put it this way:

Fain would I plant a grove in rows

But how must I its form compose

With three trees in each row;

To have as many rows as trees;

Now tell me, artists, if you please:

’Tis all I want to know.

Those familiar with tic-tac-toe, three-in-a-row might wonder how difficult this problem could be, but it’s actually been looked at by some of the most prominent mathematicians of the past and present. This essay presents many new solutions that haven’t been seen before, shows a general method for finding more solutions and points out where current best solutions are improvable.

December 14, 2017 — Michael Gammon, Blog Administrator, Document and Media Systems

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.

September 7, 2017 — Greg Hurst, Wolfram|Alpha Math Content Manager

In our continued efforts to make it easier for students to learn and understand math and science concepts, the Wolfram|Alpha team has been hard at work this summer expanding our step-by-step solutions. Since the school year is just beginning, we’re excited to announce some new features.

August 25, 2017 — Michael Trott, Chief Scientist

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:

May 25, 2017 — Devendra Kapadia, Kernel Developer, Algorithms R&D

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.

February 24, 2017

Jeffrey Bryant, Research Programmer, Wolfram|Alpha Scientific Content

Paco Jain, Research Programmer, Wolfram|Alpha Scientific Content

Michael Trott, Chief Scientist

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.

December 28, 2016 — Kathryn Cramer, Technical Communications and Strategy Group

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:

December 22, 2016

Eric Weisstein, Senior Researcher, Wolfram|Alpha Scientific Content

Ian Ford, Pure Math Developer, Wolfram|Alpha Scientific Content

## 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.