July 26, 2018 — Itai Seggev, Senior Kernel Developer, Algorithms R&D

One of the many beautiful aspects of mathematics is that often, things that look radically different are in fact the same—or at least share a common core. On their faces, algorithm analysis, function approximation and number theory seem radically different. After all, the first is about computer programs, the second is about smooth functions and the third is about whole numbers. However, they share a common toolset: asymptotic relations and the important concept of asymptotic scale.

By comparing the “important parts” of two functions—a common trick in mathematics—asymptotic analysis classifies functions based on the relative size of their absolute values near a particular point. Depending on the application, this comparison provides quantitative answers to questions such as “Which of these algorithms is fastest?” or “Is function a good approximation to function *g*?”. Version 11.3 of the Wolfram Language introduces six of these relations, summarized in the following table.

July 19, 2018 — Devendra Kapadia, Kernel Developer, Algorithms R&D

Asymptotic expansions have played a key role in the development of fields such as aerodynamics, quantum physics and mathematical analysis, as they allow us to bridge the gap between intricate theories and practical calculations. Indeed, the leading term in such an expansion often gives more insight into the solution of a problem than a long and complicated exact solution. Version 11.3 of the Wolfram Language introduces two new functions, `AsymptoticDSolveValue` and `AsymptoticIntegrate`, which compute asymptotic expansions for differential equations and integrals, respectively. Here, I would like to give you an introduction to asymptotic expansions using these new functions.

May 10, 2018 — Michael Trott, Chief Scientist, Wolfram|Alpha Scientific Content

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, Wolfram|Alpha Scientific Content

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.

November 9, 2017 — Devendra Kapadia, Kernel Developer, Algorithms R&D

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.

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

## July 17, 2019 Update

The Wolfram|Alpha 2.0 app is now available! Learn more.

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, Wolfram|Alpha Scientific Content

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.