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# Mathematics

## New in 13: Cryptography, Blockchains & NFTs

Two years ago we released Version 12.0 of the Wolfram Language. Here are the updates in cryptography, blockchains and NFTs since then, including the latest features in 13.0. The contents of this post are compiled from Stephen Wolfram's Release Announcements for 12.1, 12.2, 12.3 and 13.0.

### Cryptography & Security (December 2020)

One of the things we want to do with Wolfram Language is to make it as easy as possible to connect with pretty much any external system. And in modern times an important part of that is being able to conveniently handle cryptographic protocols. And ever since we started introducing cryptography directly into the Wolfram Language five years ago, I’ve been surprised at just how much the symbolic character of the Wolfram Language has allowed us to clarify and streamline things to do with cryptography.

## The Physicalization of Metamathematics and Its Implications for the Foundations of Mathematics

One of the many surprising (and to me, unexpected) implications of our Physics Project is its suggestion of a very deep correspondence between the foundations of physics and mathematics. We might have imagined that physics would have certain laws, and mathematics would have certain theories, and that while they might be historically related, there wouldn’t be any fundamental formal correspondence between them.

But what our Physics Project suggests is that underneath everything we physically experience there is a single very general abstract structure—that we call the ruliad—and that our physical laws arise in an inexorable way from the particular samples we take of this structure. We can think of the ruliad as the entangled limit of all possible computations—or in effect a representation of all possible formal processes. And this then leads us to the idea that perhaps the ruliad might underlie not only physics but also mathematics—and that everything in mathematics, like everything in physics, might just be the result of sampling the ruliad.

## Learning Differential Equations in 10 Hours or Fewer with the Wolfram Language

Differential equations are a cornerstone of modern mathematics. From quantum mechanics to population dynamics and stock market predictions, they play a crucial role in understanding the world around us. For this reason, courses on differential equations are core for many undergraduate degrees in the natural sciences, engineering and other fields.
Current Events & History

## The Singular Euler–Maclaurin Expansion A New Twist to a Centuries-Old Problem

Of all mathematical operations, addition is the most basic: It’s what we learn first in school. Historically, it is the most ancient. While the simple task of getting the sum of two numbers is simple, sums of many numbers can easily turn into a challenging numerical problem if the number of summands is very large.

Current Events & History

## Is Your Function Continuous? Squaring Away the New Function Properties in the Wolfram Language

The Wolfram Language has several hundred built-in functions, ranging from sine to Heun. As a user, you can extend this collection in infinitely many ways by applying arithmetic operations and function composition. This could lead you to defining expressions of bewildering complexity, such as the following:

 ✕ f = SinhIntegral[ LogisticSigmoid[ ScorerHi[Tanh[AiryAi[HermiteH[-(1/2), x] - x + 1]]]]];
You may then ask, “Is continuous?” or “Can be written as a composition of an increasing function with another function?” The powerful new tools for studying function properties in Version 12.2 provide quick answers to such questions—opening the doors for applying a network of theorems and ideas that have been developed by mathematicians during the last few centuries.

## 3D-Printed Jewelry Made with the Wolfram Language Showcases the Beauty of Mathematics

I enjoy turning mathematical concepts into wearable pieces of art. That’s the idea behind my business, Hanusa Design. I make unique products that feature striking designs inspired by the beauty and precision of mathematics. These pieces are created using the range of functionality in the Wolfram Language. Just in time for Valentine’s Day, we recently launched Spikey earrings in the Wolfram Store, which are available in rose gold–plated brass and red nylon. In this blog, I’ll give a look under the hood and discuss how an idea becomes a product through the Wolfram Language.

Math is one of the main things that deters students from wanting to learn more about chemistry. Being a chemical engineering student, I understand this, especially for students who just have to get chemistry out of the way as a general education requirement. Essentially, step-by-step solutions are like your own on-demand math tutor: in addition to calculating the answer, Wolfram|Alpha shows you how it got there. Here are six important math skills that you will definitely use on a regular basis in your chemistry class and how they relate to different chemistry concepts.