## Archive for the ‘Books’ category

### Concepts of Modern Physics

6 September 2009

Read through Concepts of Modern Physics by Beiser in July, August and September.  It provided a great review and overview, and filled in several gaps in my knowledge.   I took tons of notes, but haven’t worked them up into latex (not sure how useful this would be – I have other areas that need my focus).

For me this book was perfect – high level of detail and consistent expectations of the reader.

### Outliers: The story of success

1 September 2009

Finished the book Outliers this week, a very quick read.  The chapter “10,000 hours” was re-affirming of my belief that practice to achieve skill is so overlooked – that popular view defensively jumps on talent as innate.  Tangentially related if you can find it is “The Woodcarver”: A Model for Right Action from The Active Life, which was the first to put words to this concept for me:

“We want to believe that we are so ordinary compared to “the experts,” that our action cannot possibly be as free and graceful as theirs, that we cannot be held to those standards. …. The question is: Why do so many people want to be mystified by expertise?”

### Project Euler problem 54

29 August 2009

I’m starting on Project Euler to help myself learn Python better.  Project Euler offers a great range of problems that sit in the realm between math and programming.  I finally solved problem 54 tonight, which involves judging poker hands.  A couple of observations that helped me finish the problem were:

1. Breaking ties between hands of the same value only sometimes requires high card evaluation.   E.g. when comparing four of a kind, three of a kind, or full house, there is no need to check any other cards besides the melded cards.
2. Royal flush and Straight flush require the same evaluation.

After looking at a lot of Python books and wishing for something up to date, I found Python: Essential Reference fourth edition by Beazley.  It has good coverage of both 2.6 and 3.0, and the author comes from a scientific background as well.

### The White Tiger

24 August 2009

I love that I can reserve books online from our library.  It makes getting a hold of new and popular books very easy – they even robo-call you when the book is waiting for you.   Read The White Tiger almost without stopping – felt like I was rising and falling at the same time.

### Advanced Physics Demystified

18 July 2009

I spent June and July working through Advanced Physics Demystified and finished the final exam (a lengthy 100 multiple choice question test).  The advice the author gives is good, have a friend grade the quiz and exam, and tell you your score, but not which you got right or wrong.  This really allows you to work through the process of self-assessment.

I’d say this book is a very curious one, not an overview, yet not focused well.  It offers some topics clearly, and yet I was put off by the fact that some relationships were re-arranged and offered as “different” equations (F=ma, a=F/m, etc.)

I did find it useful and put together a summary though I’m too pressed right now to re-latex it for the blog.

Equations of motion arise from simple integration on $t$, including $x = \frac{1}{2} at^2 + v_0t + x_0$.   Impulse is $\textbf{J}=\textbf{F}t$

### Math Proofs Demystified

18 July 2009

I borrowed Math Proofs Demystified from the library and worked through about half of it in May and in July.  As with the other demystified books it provides an opening to self teaching, and does a good job.  Working through the examples rather than just reading through them is the key with this book, however, since all the chapter tests are multiple choice.

### Prime Obsession

5 May 2009

Read Prime Obsession over the last five days (I got a hold of it just in time for vacation).  It is a really nicely written book on the history and math behind the Riemann hypothesis.  The areas of math involved here are number theory (prime numbers), and complex analysis.  The Euler product (shown on the cover) is the beginning of the bridge between the two fields.

Mathematicians are interested in the number of primes less than a given integer $N$ and call this $\pi(N)$.   While the function $\pi(N)$ can be calculated for any given $N$, there is no closed form (simple) solution for it.  Approximations have been proposed, the first being

$\pi(N) \sim\frac{N}{\log(N)}$

and the second being

$\pi(x) \sim Li(x) = \int \limits_0^x \frac{1}{\log(t)} dt$

and the closest approximation coming in the last chapter.

Meanwhile, also of interest are the convergent infinite series of fractions, given by the Riemann zeta function $\zeta(s)$.  This function is defined as

$\zeta(s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + ...$

The first solution to this I encountered in college for $s=2$ which is amazingly

$1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... = \frac{\pi^2}{6}$

Euler solved this generally for all $N$ even, (and they are all powers of $\pi$ to the $N$).  However, all $N$ odd are unsolved, and it was not until 1978 that $N=3$ was even proved to be irrational!  This summed series can be converted with regular algebra (and some insight using something like the sieve of Eratosthenes) to a product series.  This is the famous Euler product

$\sum \limits_n n^{-s} = \prod \limits_p (1-p^{-s})^{-1}$

where $n$ are all the integers and $p$ stands for all prime numbers.  (The use of $s$ is a convention started by Riemann, and not disturbed by mathematicians over the centuries, it could just as well be $x$).   The Riemann zeta function is continued analytically into the complex plane and gives rise to the Riemann hypothesis,

The Riemann hypothesis – all non-trivial zeros of the zeta function have real part one-half.

(now the zeta function in the complex plane is linked with primes!)