An n log(n) algorithm for multiplication

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Credit: MathIsFun.com

The discovery of decimal arithmetic

The discovery of decimal arithmetic in ancient India, together with the well-known schemes for long multiplication and long division, surely must rank as one of the most important discoveries in the history of science. The date of this discovery, by an unknown Indian mathematician or group of mathematicians, was recently pushed back to the third century CE, based on the recent dating of the Bakhshali manuscript, but it probably happened earlier, perhaps around 0 CE.

Arithmetic on modern computers

Computers, of course, do not use

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LENR: A skeptical perspective

LENR: Science or pseudoscience?

As chronicled in the earlier Math Scholar blog, a community of researchers has been pursuing a new green energy source, known as “low energy nuclear reaction” (LENR) physics, or, variously, “lattice-assisted nuclear reaction” (LANR) physics or “condensed matter nuclear reaction” (CMNR) physics. This work harkens back to 1989, when University of Utah researchers Martin Fleischmann and Stanley Pons that they had achieved desktop “cold fusion,” in a hastily called news conference. After several other laboratories failed to reproduce these findings, the scientific community quickly concluded that the Utah researchers were mistaken, to put it mildly.

But

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Karen Uhlenbeck wins the Abel Prize

The Norwegian Academy of Science and Letters has awarded the Abel Prize, a mathematical award often regarded as on a level with the Nobel Prize, to Karen Uhlenbeck of the University of Texas, USA.

The award cited her work in geometric analysis, gauge theory and global analysis, which has application across a broad range of modern mathematics and mathematical physics, including models for particle physics, string theory and general relativity.

Her career began in the mid-1960s, under the advisor Richard Palais. Palais had been exploring some connections between analysis (generalizations of calculus) and topology and geometry (the mathematical theory of

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LENR energy: Science or pseudoscience?

Credit: IPCC

The threat of climate change

The threat of climate change is emerging as the premier global issue of our time. As a recent report by the Intergovernmental Panel on Climate Change (IPCC) grimly warns, even a 1.5 degree C (2.7 degree F) rise in global temperatures would have “substantial” consequences, in terms of extreme weather, damage to ecosystems and calamitous impact on human communities. But limiting the increase to 1.5 degree C will still require a wrenching change away from fossil fuels and an equally wrenching realignment of global economies, all over the next decade or as

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New result for Mordell’s cube sum problem

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Mordell’s cube sum problem

In 1957, British-American mathematician Louis Mordell asked whether, given some integer $k$, there are integers $x, y, z$ such that $x^3 + y^3 + z^3 = k$. Like Fermat’s last theorem, this problem is very easily stated but very difficult to explore, much less solve definitively.

Some solutions are easy. When $k = 3$, for instance, there are two simple solutions: $1^3 + 1^3 + 1^3 = 3$ and $4^3 + 4^3 + (-5)^3 = 3$. It is also known that there are no solutions in other cases, including

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A Pi Day crossword puzzle

Once again Pi Day (March 14, or 3/14 in North American notation) is here, a day when both professional mathematicians and students alike celebrate this most famous of mathematical numbers.

San Francisco’s Exploratorium is featuring several events, culminating with a “Pi Procession” at 1:59pm Pacific Time (corresponding to 3.14159) and pie served at 2:15pm. The website teachpi.org lists 50 ideas to make Pi Day “entertaining, educational, tasty and fun.”

For this year’s Pi Day festivities, the Math Scholar blog presents a Pi Day crossword puzzle (see below), created by the present author. We will announce the first correct solver! Send

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Simple proofs: Archimedes’ calculation of pi

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Credit: Ancient Origins

Introduction

Archimedes is widely regarded as the greatest mathematician of antiquity. He was a pioneer of applied mathematics, for instance with his discovery of the principle of buoyancy, and a master of engineering designs, for instance with his “screw” to raise water from one level to another. But his most far-reaching discovery was the “method of exhaustion,” which he used to deduce the area of a circle, the surface area and volume of a sphere and the area under a parabola. Indeed, with this method Archimedes anticipated, by nearly

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Simple proofs: The fundamental theorem of calculus

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Isaac Newton, Credit: sjisblog.com

Introduction: The fundamental theorem of calculus, namely the fact that integration is the inverse of differentiation, is indisputably one of the most important results of all mathematics, with applications across the whole of modern science and engineering. It is not an exaggeration to say that our entire modern world hinges on the fundamental theorem of calculus. It has applications in astronomy, astrophysics, quantum theory, relativity, geology, biology, economics, just to name a few fields of science, as well as countless applications in all types of engineering — civil,

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Is scientific progress real?

Top 500 supercomputer performance (orange = #1, blue = #500, green = sum)

Postmodern denials of scientific progress

The fact that scientific research has made immense progress over the past years, decades and centuries is taken for granted among professional scientists and most of the lay public as well. But there are others, from both the left wing and the right wing of society, who question, dismiss or even reject the notion that science progresses. One group, which is mostly rooted in the right wing of society, rejects the scientific consensus on evolution, as with the creationism and intelligent

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2018: The year that artificial intelligence came of age

AI’s tortuous history

The field of artificial intelligence (AI) is actually rather old. Ancient Greek, Chinese and Indian philosophers developed principles of formal reasoning several centuries before Christ. In 1651, British philosopher Thomas Hobbes wrote in Leviathan that “reason … is nothing but reckoning (that is, adding and subtracting).” In 1843 century Ada Lovelace, widely considered to be the first computer programmer, ventured that machines such as Babbage’s analytical engine “might compose elaborate and scientific pieces of music of any degree of complexity or extent.”

In 1950 Alan Turing’s landmark paper Computing machinery and intelligence outlined the principles of AI

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