Carnival of Mathematics #170

I’m hosting issue number 170 because I have a thing for the number’s largest prime factor, but it turns out there’s a reason for a Martin Gardner fan like me to appreciate the number itself: 170 is the number of steps1 needed to solve a classic mechanical puzzle called The Brain invented by computer scientist Marvin H. Allison, Jr., described by Martin Gardner in his Scientific American essay “The Binary Gray Code”, and still available from Amazon.

The Brain, aka The Brain Puzzle, aka The Brain Puzzler. Here’s what Gardner says about The Brain:

It consists of a tower of eight transparent plasticdisks that rotate horizontally around their centers. The disks are slotted, with eight upright rods going through the slots. The rods can be moved to two positions, in or out, and the task is to rotate the disks to positions that permit all the rods…

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Physicists propose a second level of quantization for quantum Shannon theory

Information theory, which was developed by Claude Shannon starting in the late 1940s, deals with questions such as how quickly information can be sent over a noisy communications channel. Both the information carriers (e.g., photons) and the channel (e.g., optical fiber cable) are assumed to be classical systems, with well-defined, perfectly distinguishable states.

In the past two decades, physicists have been developing a quantum version of information theory in which the internal state of each information carrier has quantum properties, such as superposition—the ability to occupy two or more classical states at once. But the transmission lines are generally still assumed to be classical, so that the path taken by messages in space is always well-defined.

Now in a new paper, physicists Giulio Chiribella and Hlér Kristjánsson at the University of Oxford and the University of Hong Kong have proposed a second level of quantization, in which both the information carriers and the channels can be in quantum superposition. In this new paradigm of communication, the information carriers can travel through multiple channels simultaneously.

Adapted from: Physicists propose a second level of quantization for quantum Shannon theory by Lisa Zyga , Phys.org

Sum-of-Three-Cubes Problem Solved for ‘Stubborn’ Number 33

Adapted from Quanta Magazine: https://www.quantamagazine.org/

Mathematicians long wondered whether it’s possible to express the number 33 as the sum of three cubes — that is, whether the equation 33 = x³+ y³+ z³ has a solution. They knew that 29 could be written as 3³ + 1³ + 1³, for instance, whereas 32 is not expressible as the sum of three integers each raised to the third power. But the case of 33 went unsolved for 64 years.

Read the whole article here: Sum-of-Three-Cubes Problem Solved for ‘Stubborn’ Number 33

Heisenberg’s final theory

Antoine Tilloy's research log

Last month, I was at Foundations 2018 in Utrecht. It is one of the biggest conferences on the foundations of physics, bringing together physicists, philosophers, and historians of science. A talk I found particularly interesting was that of Alexander Blum, from the Max Planck Institute for the History of Science, entitled Heisenberg’s 1958 Weltformel & the roots of post-empirical physics. Let me briefly summarize Blum’s fascinating story.

In 1958, Werner Heisenberg put forward a new theory of matter that, according to his peers (and to every physicist today) could not possibly be correct, failing to reproduce most known microscopic phenomena. Yet he firmly believed in it, worked on it restlessly (at least for a while), and presented it to the public as a major breakthrough. How was such an embarrassment possible given that Heisenberg was one of the brightest physicists of the time? One could try to find…

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Chapter 1
First Day of School

He goes to class. There is an empty seat in front. He sits in the seat. He looks around. There are different people. He says “hi” to the girl next to him. She smiles. The teacher comes in. She closes the door. Everyone is silent. The first day of school begins.

Chapter 2
Water on the Floor

She is thirsty. She gets a glass of water. She begins to walk. She drops the glass. There is water on the floor. The puddle is big. She gets a mop. She wipes the water off. The floor is clean. She gets another glass of water. She drinks it. She is happy.

Chapter 3
Babysitting

Casey wants a new car. She needs money. She decides to babysit. She takes care of the child. She feeds him lunch. She reads him a story. The story is funny. The child laughs. Casey likes him. The child’s mom comes home. The child kisses Casey. Casey leaves. She will babysit him again.

Chapter 4
Twins

Jill and Jodi are twins. They look the same. But they act differently. Jill likes sports. She is good at basketball and golf. She is also loud. She talks all day. Jodi likes reading. She can read 300 pages a day. She is also quiet. She does not like to talk. Jill and Jodi still love each other.

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