CS245 regarding countability decidability

I thought this assignment is really interesting and worth posting about.

CS 245 Assignment 5 Fall 2012

```Question 1 (30pt) Let ti be a term of the form s(· · · s(0) · · ·) that represents a natural number ni written in unary (e.g., s(s(s(0))) represents the number 3).```

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```• Form a set of clauses Σ such that Σ r TIMES(t1, t2, t3) whenever n1 · n2 = n3; • Show a resolution refutation of TIMES(s(s(0), s(0), s(s(0)))) w.r.t. Σ from above; • Is the set of terms {f (t1, t2, t3) | n1 · n2 = n3} recursive? Continue reading →```

Different sizes of infinity

Did you know there is infinity, and then there's a bigger infinity?

This concept is quite bizarre - one would think that infinity has only one size - that is, infinite size. But this is not true. Georg Cantor, a German mathematician, proved this idea by way of contradiction. Take two infinite sizes - the natural numbers, and real numbers. Both systems are unbounded and infinite in size. It can be proved that one infinity is larger than the other. The opposite is assumed: suppose both sets of numbers have the same size of infinity. By constructing a mapping, we can take an arbitrary mapping and show correspondence. Suppose every natural number has a real partner. We pair them off. In the diagonal construction, it is shown that there exists numbers that can't be mapped because as your mapping get arbitrarily precise, the real numbers increase still. So there will always be number left unmapped. QED.

Holographic photography

I will be exploring into the world of holographic photography!

Otherwise known as holography. This is a technique that allows the light scattered from an object to be recorded and later reconstructed so that when an imaging system (a camera or an eye) is placed in the reconstructed beam, an image of the object will be seen even when the object is no longer present. The image changes as the position and orientation of the viewing system changes in exactly the same way as if the object were still present, thus making the image appear three-dimensional.

Wouldn't it be cool to have a 3D photograph? It'll just be like a picture on a newspaper from the world of Harry Potter. I think by far the easiest way to get started is to get a kit. You still need to setup everything in the kit... my idea is to figure out how this works and simplify it down to a point and shoot device. Then try to build another one. I want to create a device that allows people to easily take a holographic photo. First, I need to figure out how holographic photography really works.

This is the kit I will be invest it with. Thank you to Dr. Shodiev for letting me take a look at this. Check it out!

A holographic photography kit

The OPERA Experiment and Special Relativity Consequences of FTL Travel

Description

This is an analysis of the consequences from the OPERA experiment, which measured the speed of neutrinos to be travelling faster than light.

The OPERA experiment was an observation of muon neutrinos apparently travelling faster than light. The OPERA instrument, which stands for Oscillation Project with Emulsion-tRacking Apparatus, was used to measure the neutrinos. This was a paper published in September 2011, observing that muon neutrinos sent from CERN were travelling faster than light by a factor of roughly 1 in 40,000.

Application for Perimeter Institute International Summer School for Young Physicists

While reading through the stuff I had on google docs, I came to this document that I wrote for my application to Perimeter Institute International Summer School for Young Physicists.

Quantum Key Distribution Summer School at the Institute for Quantum Computing

What is Quantum Key Distribution?

The purpose of the Quantum Key Distribution summer school is twofold:

1. To provide a well-structured series of lectures explaining the level of modern know-how in quantum cryptography.
2. To provides better understanding of the topic by using the lecture material in interactive tutorials.

IQC has an excellent atmosphere for research, and just being here would innovate quantum leaps to the future. This summer school really built a solid foundation for understanding concepts of quantum information and cryptography.

Nuclear Magnetic Resonance Lab at IQC

Day 1

Basic cryptography

The first day at Quantum Key Distribution class is a considered a crash course on cryptography. Our first speaker is David Jao, he covers some basics of cryptography, such as the use of a single key for encrypting & decrypting. Basically this SKES symmetric key encryption scheme. He defining a protocol as semantically secure and infeasible to crack as 280 operations. Next major topic was hashing functions, and its application and weakness. It's possible to create two files that have the same hash, and this is called a collision. This is not desirable as collision leads to a malicious hacker to change the content of the document and still pass it off as legitimate (he explained the birthday paradox). Hashing can also be used in conjunction with SKES and/or the use of public key cryptography to increase the speed of data delivery.