New Information Theory Post and Mathematica 9

Mathematica 9 has come out and has some amazing new capabilities. I have produced a new information theory post using some of the new functions. This is at http://www.madscitech.org/information/information.html

New Information Theory Site on MAST

I have formalized my notes on information theory and placed them on the MAST website. You can find tham at http://www.madscitech.org/information/information.html

The Nature of information III

In this posting I explore the probability of finding strings and establish the principle that the chace of finding a given string is independent from the elements of the string. This result is called the Asymptotic Equipartition Property. For details go to the The Nature of Information III

The Nature of information

The Nature of Information

George Hrabovsky

MAST

Introduction

Information can be thought of as what we do not already know. We acquire information when we learn, or perhaps it is that the act of learning is acquiring information. In either case we can consider infomration in the abstract to be a phsycial quantity and it can be measured.

Bits

How do we measure information? Like any measurement of a physical quantity we choose a fundamental unit. Then we say that something has a quantity of units, a numerical result. Something can be five meters long, or it can be 4.3 pounds of weight, or 6.1 kilograms of mass, and so on. For information we use the bit as a fundamental quantity. Thus we can say that some information is composed of a number of bits.

Characterizing Information

We will use symbols to represent bits of information. A set of symbols comprising a single bit will be represented by

two bits by

 

and so on, so that a set of n such symbols is the sequence

.

Schemes

If we also associate a probability that a given symbol will be sent,

The symbols of such a set must be mutually exclusive, so that the total of all probabilies is 1. We can combine them into a single symbol,

This is called a finite scheme.

Entropy

Entropy is, among other things, the measure of the disorder of a system of symbols. In the case of a scheme the entropy is given by,

We will discuss these things in more detail later.

Been Away Again

I have been away for a couple of months. In that time my co-author on the book, Leonard Susskind, and I have been working on the final draft of our book THEORETICAL MINIMUM: Claswsical Mechanics and hope to have it done in a few weeks.

MAST is working on developing education modules for basic science.

I have had a paper on numerical modeling of the diffusion equation accepted for publication in The Mathematica Journal.

I will be working on the show The Wonders of Physics this year.

I will be doing more in the coming days.

Some new results

MAST has begun to post some research results!

1) I have started to do some dripping handrail models of accretion disks. You can find them here: http://www.madscitech.org/csg/accretion.html

2) James Firmiss and I have begun serious work on neutron star equations of state, results will be put up shortly.

3) Chris Winfield is giving a talk on local solvability at the American Mathematical Society meeting in New Orleans, that talk is located here: http://www.madscitech.org/csg/AMS10mintalk.pdf

4) We are doing research and development of database capabilities in Mathematica, mostly due to the efforts of Rocky Wenz.

More results will be reported as they occur!

Project Updates

Hello!

I have been insanely busy the last week. Had a great Thanksgiving!

Just completed Chapter 9 in the book project, this is a survey of linear algebra. I am about to start in on chapter 10, Leonard Susskind's classical mechanics Lecture 5.

I am also developing the program for the Institute for Scientific Computing Summer School program. I am also working on developing courses for teaching Mathematica 8 online.

That's all for now.

Recent Papers of Interest 16 November 2010

Here are some ArXiv papers I found interesting of late.

1) Takami Kuroda and Hideyuki Umeda, Three Dimensional Magneto Hydrodynamical Simulations of Gravitational Collapse of a 15Msun Star

This is an important paper for those of us who study compact objects. It begins with a nice overview of the theory of core-collapse supernovae (CCSN) and the efforts to model these. The next two sections details the code used, both MHD and GR. This is followed by the adaptive mesh scheme used. Finally they discuss the tests of the code. This accounts for half of the paper and makes a very nice overview of modeling MHD in a GR environment. The results are very nice, great paper!

2) Manjit Bhatia, P. Narayana Swamy, Elementary Quantum Mechanics in a Space-time Lattice

This paper explores the particle-in-a-box problem under the assumption of a fundamental length scale forming a spacetime lattice. This is an interesting approach.

3) Oleg Korobkin, Ernazar B. Abdikamalov, et al, Stability of general-relativistic accretion disks,

This paper is a thorough treatment of a model of unstable accretion disks near a black hole.

4) T. Fischer, I. Sagert, et al, The revival of an explosion mechanism of massive stars - the quark hadron phase transition during the early post bounce phase of core collapse supernovae,

This is a long paper (30 pages) explores the bounce during a supernova core collapse caused by the so-called quark hadron phase transition. The introduction is a nicce overview of the topic and its history.  The second section, no pun intended, forms the theoretical core of the paper, describing the hybrid equation of state and other issues. The third section describes the simulation scheme. The fourth section explores the results of the model runs. The conclusion is that the quark matter description leads to early onset of deconfinement, a quark-gluon plasma. A very interesting paper!

Project Updates

I am about to finish chapter six of the book project.

We (MAST) are planning a Summer School on Computational Physics; this will tentaively be held at the University of Wisconsin, Madison, Department of Physics and will be held from 15 June 2011 to 15 August 2011. Check out the MAST website for details.

We are exploring the possibility of Mathematica training over the web. This will include beginner training all the way up to advanced applications.

Update

Well, we just turned in our NSF proposal to develop the MAST Institute for Scientific Computing. We have letters of interest from Jack Horner, a software engineer from Los Alamos, and from the Unversity of Wisconsin, Madison, Department of Physics.

This is a very cool project!

Now that the propsal is in, I can get back to work on the book!

Some Recent Papers on Theoretical Physics and Mathematical Physics

1) Gary Horowitz, Surprising Connections Between General Relativity and Condensed Matter,

My only objection to this paper is that it seems to be tied to anti-deSitter space (AdS). Proving things in AdS may be easier than proving them in general relativity (AdS is essentially general relativity with a negative cosmological constant), but it cannot gaurantee applicability to the real world (since we seem to have a positive cosmological constant). The paper seems to rely on an observation that black holes in AdS behave like thermal systems of one dimensional lower, and that the string theory/holographic principle of black holes allows you to extract features from general relativity that look a lot like superconductivity.

2) Geoffrey Lovelace, Mark. A. Scheel, Bela Szilagyi, Simulating merging binary black holes with nearly extremal spins,

This fascinating paper describes the merger of two rapidly rotating black holes.

New developments at MAST

MAST (Madison Area Science and Technology) will be seeking funding from the NSF to open a computational science center and, if funded, it will allow us to offer internships for High School Seniors, Citizen Scientists, Undergraduate Students, Graduate Students, and even Post-docs who are interested in computational science, including, but not limited to: astrophysical fluid dynamics, compact objects, accretion disks, relativity, cosmology, mathematical physics, numerical methods, fluid dynamics, atmospheric modeling, shock wave phenomena, chaotic dynamical systems, scattering, quantum information theory, solution methods for ODEs and PDEs, and bioinformatics. We will handle all necessary educational requirements to bring the students up to speed (including training in Mathematica).

We will even be accepting a number of foreign internships.

The proposal will be submitted to the NSF by 27 October of this year.