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非線形偏微分方程式への有限要素法の適用

Mathematica 12 has powerful functionality for solving partial differential equations (PDEs) both symbolically and numerically. This article focuses on, among other things, the finite element method (FEM)–based solver for nonlinear PDEs that has been newly implemented in Version 12. After briefly reviewing basic syntax of the Wolfram Language for PDEs, including how to designate Dirichlet and Neumann boundary conditions, we will delineate how Mathematica 12 finds the solution of a given nonlinear problem with FEM. We then show some examples in physics and chemistry, such as the Gray–Scott model and the time-dependent Navier–Stokes equation. More information can be found in the Wolfram Language tutorial “Finite Element Programming,” on which most of this article is based.

1. はじめに

Wolfram Research社の旗艦製品であるMathematicaは,5,000 を超える組み込み関数を有するWolfram Languageを駆動する.数理モデリング,解析の基本となる常・偏微分方程式の分野においては,これらをシンボリックに,あるいは数値的に解くための強力なソルバを搭載している.最近は有限要素法(FEM) を利用した数値的求解機能が大幅に強化され,偏微分方程式(PDE)を任意の領域上で解いたり,固有値・固有関数を求めたりすることが可能となった.ここでは,最新のバージョン12における非線形偏微分方程式のFEMによる求解を中心に,現実的な問題に応用する上での流れを例とともに紹介する.なお,有限要素法を用いて非線形PDEを解くワークフローの詳細,コードはすべて公開されている.MathematicaのWolframドキュメント内で,チュートリアル“FiniteElementProgramming”を参照いただきたい.

Best of Blog

Finally We May Have a Path to the Fundamental Theory of Physics… and It’s Beautiful

I Never Expected This

It’s unexpected, surprising---and for me incredibly exciting. To be fair, at some level I've been working towards this for nearly 50 years. But it’s just in the last few months that it’s finally come together. And it’s much more wonderful, and beautiful, than I’d ever imagined.

In many ways it's the ultimate question in natural science: How does our universe work? Is there a fundamental theory? An incredible amount has been figured out about physics over the past few hundred years. But even with everything that's been done---and it's very impressive---we still, after all this time, don't have a truly fundamental theory of physics.

Back when I used do theoretical physics for a living, I must admit I didn't think much about trying to find a fundamental theory; I was more concerned about what we could figure out based on the theories we had. And somehow I think I imagined that if there was a fundamental theory, it would inevitably be very complicated.

Academics

Hitting All the Marks: Exploring New Bounds for Sparse Rulers and a Wolfram Language Proof

The sparse ruler problem has been famously worked on by Paul Erdős, Marcel J. E. Golay, John Leech, Alfréd Rényi, László Rédei and Solomon W. Golomb, among many others. The problem is this: what is the smallest subset of so that the unsigned pairwise differences of give all values from 1 to ? One way to look at this is to imagine a blank yardstick. At what positions on the yardstick would you add 10 marks, so that you can measure any number of inches up to 36?

Another simple example is of size 3, which has differences , and . The sets of size 2 have only one difference. The minimal subset is not unique; the differences of also give .

Part of what makes the sparse ruler problem so compelling is its embodiment in an object inside every schoolchild's desk—and its enduring appeal lies in its deceptive simplicity. Read on to see precisely just how complicated rulers, marks and recipes can be.

Academics

An Intriguing Identity: Connecting Distinct and Complete Integer Partitions

Number theory is a very old subject that in modern times has branched into various large areas. One of these is additive number theory, with problems like this: when is a prime the sum of two squares? Primes are part of the more classical area now called multiplicative number theory, so as this problem of Fermat’s indicates, the two areas are intimately connected. The problem I discuss in this blog is a mix of additive and multiplicative number theory, with a dash of linear algebra.

Academics

On the Polygon Front Lines: Visualizing the Amplituhedron with the Wolfram Language

It’s rare to hear polygons mentioned in a physics class, even in higher education. This may seem unexpected given the fundamental role they play in mathematics. However, over the last few years, polygons have come to the front line in many areas of theoretical physics, helping us understand the laws of nature with their astonishing beauty.

This is particularly true in the field of particle physics, where a new geometrical object has been found to be connected to particle dynamics: the amplituhedron. It represents a novelty not only in physics but also in mathematics, generalizing the concept of a convex polygon. In this blog post, I will first discuss its relation to particle physics, and then how to visualize its geometry using the Wolfram Language.

Academics

Announcing the Rule 30 Prizes

The Story of Rule 30

How can something that simple produce something that complex? It’s been nearly 40 years since I first saw rule 30---but it still amazes me. Long ago it became my personal all-time favorite science discovery, and over the years it’s changed my whole worldview and led me to all sorts of science, technology, philosophy and more.

But even after all these years, there are still many basic things we don't know about rule 30. And I've decided that it's now time to do what I can to stimulate the process of finding more of them out. So as of today, I am offering $30,000 in prizes for the answers to three basic questions about rule 30.

Academics

Building Uniform Polyhedra for Version 12

Since I started working at Wolfram, I’ve been a part of several different projects. For Version 12, my main focus was replicating models of the uniform polyhedra with the Wolfram Language to ensure that the data fulfilled certain criteria to make our models precise, including exact coordinates, consistent face orientation and a closed region in order to create a proper mesh model of each solid.

Working with visual models of polyhedra is one thing, but analyzing them mathematically proved to be much more challenging. Starting with reference models of the polyhedra, I found that the Wolfram Language made mathematical analysis of uniform polyhedra particularly efficient and easy.

But first, what really are polyhedra, and why should we care? With Version 12, we can explore what polyhedra are and how they’ve earned their continued place in our imaginations.

Academics

How I Used Last-Mover Advantage to Make Money: An Exploration of Yahtzee and Coin Flipping

This week, I won some money applying a mathematical strategy to a completely unpredictable gambling game. But before I explain how, I need to give some background on last-mover advantage.

Some time ago, I briefly considered doing some analysis of the dice game Yahtzee. But I was put off by the discovery that several papers (including this one) had already enumerated the entire game state graph to create a strategy for maximizing the expected value of the score (which is 254.59).

However, maximizing the expected value of the score only solves the solo Yahtzee game. In a competitive game, and in many other games, we are not actually trying to maximize our score—we are trying to win, and these are not always the same thing.

Current Events & History

Drawing on Autopilot: Automated Plane (Geometry) Illustrations from The American Mathematical Monthly

Version 12 of the Wolfram Language introduces the functions GeometricScene, RandomInstance and FindGeometricConjectures for representing, drawing and reasoning about problems in plane geometry. In particular, abstract scene descriptions can be automatically supplied with coordinate values to produce diagrams satisfying the conditions of the scene. Let’s apply this functionality to some of the articles and problems about geometry appearing in the issues of The American Mathematical Monthly from February and March of 2019.