Thursday, March 31, 2011

A Proof by Induction

This is mostly a test to see how well the latex script stuff from watchmath.com works in blogger. This came up on the Physics Forums and the problem was stated as:

Prove for $$n \ge 2$$ that $$\sum_{k=1}^n \frac{1}{k^2} < 1 - \frac{1}{n}$$

The above is false because $$\sum_{k=1}^n \frac{1}{k^2}$$ will always be greater than 1. What I believe was meant was:

Prove for $$n \ge 2$$ that $$\sum_{k=2}^n \frac{1}{k^2} < 1 - \frac{1}{n}$$

First, we prove it for the most simple case, $$n=2$$.

\begin{aligned}\sum_{k=2}^2 \frac{1}{k^2} &< 1 - \frac{1}{2}\\ \frac{1}{2^2} &< \frac{1}{2}\\ \frac{1}{4} &< \frac{1}{2}\end{aligned}

Now let's assume that $$\sum_{k=2}^m \frac{1}{k^2}$$ is true. Then $$\sum_{k=2}^m \frac{1}{k^2} < 1 - \frac{1}{m}$$ .

Now prove that $$\sum_{k=2}^{m+1} \frac{1}{k^2} < 1 - \frac{1}{m+1}$$ is true.

First we know that $$\sum_{k=2}^{m+1} = \frac{1}{{(m+1)}^2} + \sum_{k=2}^m$$, so prove that $$\frac{1}{{(m+1)}^2} + 1 - \frac{1}{m} \le 1 - \frac{1}{(m+1)}$$.

\begin{aligned}\frac{1}{{(m+1)}^2} + 1 - \frac{1}{m} &\le 1 - \frac{1}{(m+1)}\\ \frac{m}{m(m+1)^2} + \frac{m(m+1)^2}{m(m+1)^2} - \frac{(m+1)^2}{m(m+1)^2} &\le \frac{m}{(m+1)}\\ \frac{m+m(m+1)^2-(m+1)^2}{m(m+1)^2} &\le \frac{m}{m+1}\\ \frac{m+(m-1)(m+1)^2}{m(m+1)} &\le m\\ m+(m-1)(m+1)^2 &\le m^2(m+1)\\ m+(m-1)(m^2+2m+1) &\le m^3+m^2\\ m+m^3+2m^2+m-m^2-2m-1 &\le m^3+m^2\\ m^3+m^2-1 &\le m^3+m^2 \end{aligned}

$$\therefore\sum_{k=2}^n \frac{1}{k^2} < 1 - \frac{1}{n}$$ is true.

Note: This could quite possibly be wrong since it has been about 14 years since I have done any of this stuff.

Edit: Here is a picture illustrating this, using Maxima and Gnuplot.

Wednesday, March 30, 2011

Review: Russian Silhouettes

Russian Silhouettes by Genna Sosonko

My rating: 5 of 5 stars

Very interesting collection of essays by Genna Sosonko about chess personalities from the Soviet period.

View all my reviews

Monday, March 28, 2011

Absolution Gap

I finally got around to finishing Reynold’s Revelation Space Trilogy this evening. I had started on the final book a long time ago, but had set it aside. Overall I enjoyed the series, but the final book was disappointing, with the story with the Inhibitors being wrapped up in about 4 pages at the end. Seeing as this is a hard science fiction series I imagine many people were wanting a lot more detail about what took place. The Amazon reviews would seem to support that view. Still I would say the series was worth reading.

Wednesday, March 23, 2011

Poincare’s Prize

Tonight I finished reading this book about the solution of Poincare’s Conjecture. It discussed the life of Poincare, the history of attempts to solve the conjecture, and finally discusses the solving of the conjecture by Grisha Perelman. Szpiro does a fine job of making this interesting story accessible to the layperson.

Thursday, March 17, 2011

I had this book brought to my attention by Lind's Maneuver Warfare Handbook. The book relates Capt. von Schell's experiences as a small unit commander in World War I and the lessons learned from those experiences. It is very similar to Rommel's better known title Attacks.

It is a quick and easy read at 94 pages, but contains a lot of interesting information. Lind's recommendation of the book was well deserved.

Wednesday, March 9, 2011

Maneuver Warfare Handbook

This evening I just finished reading William Lind’s primer on maneuver warfare. It was written in the 80s when there was a movement afoot to get the USMC sold on the tenets of maneuver warfare.

The book consists of 2 parts. The first part is written by Lind and discusses the ideas behind maneuver warfare. The 2nd part is a series of lectures that were given at the Amphibious Warfare School by Col. Wyly. The lectures are intended to teach the basics of maneuver warfare, and have exercises included to check whether you understood the concepts or not. In addition there is a nicely annotated bibliography to give readers a list of additional texts to read to further their knowledge and understanding. It would be nice if there were a whole book of exercises, basically tactical decision games I guess, that would give the reader more practice in applying the knowledge. I also would have liked it to cover maneuver warfare in the defense in a more thorough manner.

It is a fairly quick read, and I am sure I will go back and study it again many times to try and absorb more info from it. I would recommend the book to anyone with interest in the subject or interested in military history that would like a new lens through which to evaluate what they read.

Monday, March 7, 2011

Counter Ambush

I recently picked up the 2nd generation of Combat Mission games, both the Shock Force series, and Combat Mission: Afghanistan. I decided to fire up CM:A and play a battle.

Briefing: Enemy forces were recently spotted in the vicinity of the road. There will be an important convoy coming along this route in the near future so the area must be secured.

Mission: Insert to LZ, and then move north along the road looking for and defeating enemy forces, in particular on the hills overlooking the road. Move from PL Alpha to PL Bravo.

Time to complete mission: 1 hour

I was commanding a company of airborne infantry. 3 infantry platoons, a weapons platoon, and the HQ platoon. I didn’t capture any screenshots during the game so all I have is a zoomed out screenshot of the battlefield at the beginning of the battle, where the company has just been inserted to their LZ.

My plan was to first establish some OPs on the ridge in front of us so I could have eyes on the other 2 hills. I was going to need to clear the hill on the right before tackling the other hill otherwise I would be exposed to fire from there while attacking the northern hill. I moved 2 infantry platoons over towards the road, using the 3rd as a reserve and to establish the OPs. I moved the weapons platoon over by the OP watching the hill on the right so I could set up a base of fire once we spotted something.

The plan then was to have the 2 infantry platoons move across the road and then move in bounding overwatch up the hill and secure it. We have some mortar support to call in so I was going to call that in to try and hit just before the 2 platoons reached the top of the hill. Once they secured the 1st hill, I was then going to move my weapons platoon to set up a base of fire to support attacking the 2nd hill. Once again the 2 platoons would move in bounding overwatch attacking the 2nd hill. I also decided to swing the 3rd platoon around from the other side to hit the 2nd hill from multiple directions, but leaving me without a reserve if I truly needed one. Once the 2nd hill was secured the weapons, and HQ platoons would meet up with the infantry at an assembly area between the 2nd hill and the road and proceed on to PL Bravo. One the map the red marking denote phase 1 of the plan, blue markings phase 2, and green markings phase 3.

In the end we lost 4 soldiers in a squad to a minefield on the 1st hill, with no other casualties. We killed 22 insurgents, and wounded 19 more securing a victory for the battle. Apparently we didn’t kill enough though as it was not a major victory. Not bad for the 1st time out though.

Farnborough FC gets off to good start

I have been a fan of this game back to when it was still called Championship Manager. I was looking for a team to manage in one of the conference leagues, and noticed Farnborough being listed in Blue Square South. My knowledge of Farnborough was pretty much limited to the fact that there is a huge airshow there every year. In the late 80s a Mig-29 crashed there during a demonstration.

This was mostly started as a test run to playing with the latest edition of Football Manager. I didn’t know anything about the club, so I jumped right in. Finances are such that I couldn’t really do anything with the squad so I just tried to get familiar with the players during our friendlies. I did pick up a couple young players though to try and bolster the squad, grabbing a striker, a midfielder, and a center back.

Our pre-season friendlies went horribly with the boys only salvaging 1 point in 7 games. It was looking like the board’s decision to hire me was going to be a mistake and my stint as manager was going to be short.

We opened the season away at Braintree. Braintree jumped to an early lead, but we leveled a few minutes later. We then managed to score the winning goal early in the 2nd half and hold on for an opening day victory.

A few days later we opened at home against Ebbsfleet. In this game we were taking a lot more shots, and opened the scoring in the 20th minute. Ebbsfleet leveled in the 76th minute, but Dean McDonald saved the day for Farnborough scoring in the 90th minute to grab the 3 points.

We finished the week off with a home fixture against Dartford. It was a pretty tame affair with the only goal coming in the 76th minute from Kezie Ibe.

So we are sitting atop the table at the moment with Eastleigh and Maidenhead with a full 9 points after 3 games. Our goal is to finish mid-table so this is a pleasant beginning to the season for the club.

Next up we travel to Weston-super-Mare.