So… can we expect cheaper and better lenses?
Better? Yes. Truly sharper from corner to corner.
Source: Goodbye Aberration: Physicist Solves 2,000-Year-Old Optical Problem
Tag Archives: mathematics
The geometry of Islamic art becomes a treasure of a game
But in Engare’s case, every massive, crisscrossing slew of curves and lines and patterns has already been proven out by the puzzles you’ve solved. Your reward for doing well in Engare isn’t unlocking more pattern-generation options; it’s the ability to understand the incredible combination of rotations and line patterns that went into each one and how they’re all geometrically solvable thanks to their adherence to X and Y axes.
Source: Engare review: The geometry of Islamic art becomes a treasure of a game | Ars Technica
Verdict: Buy two copies; donate the second one to a school computer lab.
Main site: http://www.engare.design/
Only $6 on Steam. I Will be trying it out on my Steam account.
A Solution of the P versus NP Problem
Berg and Ulfberg and Amano and Maruoka have used CNF-DNF-approximators to prove exponential lower bounds for the monotone network complexity of the clique function and of Andreev’s function. We show that these approximators can be used to prove the same lower bound for their non-monotone network complexity. This implies P not equal NP.
Source: [1708.03486] A Solution of the P versus NP Problem
More background at: The P-versus-NP page
This page collects links around papers that try to settle the “P versus NP” question (in either way). Here are some links that explain/discuss this question:
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A clear formulation of the “P versus NP” question by Stephen Cook.
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Math and the Best Life
If mathematics is a medium for human flourishing, it stands to reason that everyone should have a chance to participate in it. But in his talk Su identified what he views as structural barriers in the mathematical community that dictate who gets the opportunity to succeed in the field — from the requirements attached to graduate school admissions to implicit assumptions about who looks the part of a budding mathematician.
Source: Math and the Best Life — an Interview With Francis Su | Quanta Magazine
Image Kernels explained visually
An image kernel is a small matrix used to apply effects like the ones you might find in Photoshop or Gimp, such as blurring, sharpening, outlining or embossing. They’re also used in machine learning for ‘feature extraction’, a technique for determining the most important portions of an image. In this context the process is referred to more generally as “convolution” (see: convolutional neural networks.)
Source: Image Kernels explained visually
Two-hundred-terabyte maths proof is largest ever
The puzzle that required the 200-terabyte proof, called the Boolean Pythagorean triples problem, has eluded mathematicians for decades. In the 1980s, Graham offered a prize of US$100 for anyone who could solve it. (He duly presented the cheque to one of the three computer scientists, Marijn Heule of the University of Texas at Austin, earlier this month.) The problem asks whether it is possible to colour each positive integer either red or blue, so that no trio of integers a, b and c that satisfy Pythagoras’ famous equation a2 + b2 = c2 are all the same colour. For example, for the Pythagorean triple 3, 4 and 5, if 3 and 5 were coloured blue, 4 would have to be red.
Source: Two-hundred-terabyte maths proof is largest ever
There are more than 102,300 ways to colour the integers up to 7,825, but the researchers took advantage of symmetries and several techniques from number theory to reduce the total number of possibilities that the computer had to check to just under 1 trillion. It took the team about 2 days running 800 processors in parallel on the University of Texas’s Stampede supercomputer to zip through all the possibilities. The researchers then verified the proof using another computer program.
Math whizzes of ancient Babylon figured out forerunner of calculus
During that interval, Jupiter’s motion across the sky appears to slow. (Such erratic apparent motion stems from the complex combination of Earth’s own orbit around the sun with that of Jupiter.) A graph of Jupiter’s apparent velocity against time slopes downward, so that the area under the curve forms a trapezoid. The area of the trapezoid in turn gives the distance that Jupiter has moved along the ecliptic during the 60 days. Calculating the area under a curve to determine a numerical value is a basic operation, known as the integral between two points, in calculus. Discovering that the Babylonians understood this “was the real ‘aha!’ moment,” Ossendrijver says.
Source: Math whizzes of ancient Babylon figured out forerunner of calculus | Science | AAAS
After cuneiform died out around 100 C.E., Babylonian astronomy was thought to have been virtually forgotten, he notes. It was left to French and English philosophers and mathematicians in the late Middle Ages to reinvent what the Babylonians had developed.
Shtetl-Optimized
According to the University of Chicago’s theory seminar calendar, on Tuesday of next week (November 10), the legendary Laszlo Babai will be giving a talk about a new algorithm that solves the graph isomorphism problem in quasipolynomial time.
Edit Distance Reveals Hard Computational Problems
As far as computer scientists know, the only general-purpose method to find the correct answer to a SAT problem is to try all possible settings of the variables one by one. The amount of time that this exhaustive or “brute-force” approach takes depends on how many variables there are in the formula. As the number of variables increases, the time it takes to search through all the possibilities increases exponentially. To complexity theorists and algorithm designers, this is bad. (Or, technically speaking, hard.)
SETH takes this situation from bad to worse. It implies that finding a better general-purpose algorithm for SAT — even one that only improves on brute-force searching by a small amount — is impossible.
Source: Edit Distance Reveals Hard Computational Problems | Quanta Magazine
The Jocks of Computer Code Do It for the Job Offers
The Hacker Cup goes much the same way as other sport-coding contests: five puzzles to finish in any order over three hours. Keep the programming as efficient as possible. The cleanest, most accurate code in the fastest time takes first place. A common type of problem might ask for the shortest route between San Francisco and Los Angeles given a number of constraints. Or perhaps the problem is about how to tile a floor in a specific pattern. The questions typically revolve around a well-known algorithm or mathematical structure with a fresh twist. Elite sport coders must figure out the underlying logic quickly and then trust their abilities.