THOUGHTS of the week: June 2013

Thursday, 27 June 2013

OEIS Sequence of the Week [July 1, 2013]

A000554: Number of labeled trees of diameter 3 with $n$ nodes:

12, 60, 210, 630, 1736, 4536, 11430, 28050, 67452, 159588, 372554, 859950, 1965840, 4456176, 10026702, 22412970, 49806980, 110100060, 242220594, 520578950, 1157627352, 2516581800, 5452594550, 11777604930, 25367149836, 54492396756

Interesting Fact of the Week [July 4, 2013]

The name "R2-D2" from Star Wars was said to have come from when George Lucas was making an earlier film, American Graffiti. Sound editor Walter Murch says he was responsible for the utterance which sparked the name of the droid. Murch asked for Reel 2, Dialog Track 2, abbreviated "R-2-D-2". Lucas heard this request and stated it was a great name after some clarification.

Sunday, 23 June 2013

Equation of the Week [July 1, 2013]

The Navier-Stokes equations describe the motion of fluid substances. The derivation begins with the conservation of momentum, an application of Newton's second law of motion.In an inertial frame of reference, the general form of the equation of fluid motion is

$\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \nabla \cdot\boldsymbol{\mathsf{T}} + \mathbf{f}$

where $\mathbf{v}$ is the flow velocity, $\rho$ is the fluid density, $p$ is the pressure, $\boldsymbol{\mathsf{T}}$ is the deviatoric component of the total stress tensor, $\mathbf{f}$ represents the body forces acting on the fluid, and $\nabla$ is the del operator.

Assuming constant viscosity and assuming incompressible flow, the Navier-Stokes equations will read, in vector form

$\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = -\nabla p + \mu \nabla^2 \mathbf{v} + \mathbf{f}$

where $\mu$ is the (constant) dynamic viscosity.

Visual Art Piece of the Week [July 1, 2013]

Claude Monet's Impression, soleil levant (Impression, Sunrise):

Classical Music Selection of the Week [July 1, 2013]

Ludwig van Beethoven's Piano Sonata no. 17 in D minor op. 31 no.2 3rd movement "Tempest":

https://www.youtube.com/watch?v=ZYdv5jk09XQ

Quote of the Week [July 1, 2013]

The downside of my celebrity is that I cannot go anywhere in the world without being recognized. It is not enough for me to wear dark sunglasses and a wig. The wheelchair gives me away.
-Stephen Hawking

Saturday, 22 June 2013

OEIS Sequence of the Week [June 24, 2013]

A001220: Wieferich primes - primes $p$ such that $p^2$ divides $2^{p-1}-1$ :

1093, 3511

Problem of the Week [June 24, 2013]

A positive integer $n$ is called "savage" if the integers $\{1, 2,... ~n\}$ can be partitioned into three sets $A$ , $B$ , and $C$ such that:

i) The sum of the elements in each of $A$ , $B$ , and $C$ is the same.
ii) $A$ contains only odd numbers.
iii) $B$ contains only even numbers.
iv) $C$ contains every multiple of 3 (and possibly other numbers).

Determine all even savage integers less than 100.

Correct answers as well as solutions will be acknowledged.

Friday, 21 June 2013

Interesting Fact of the Week [June 24, 2013]

Naked mole rats can't get cancer, their noses and mouths are unaffected by acids, they can survive long periods of oxygen deprivation, and they can live in areas with severe pollution.

Equation of the Week [June 24, 2013]

Newton's laws of motion are three physical laws that laid the basis for classical mechanics:

1. An object remains in uniform velocity, or rest if it is in rest, with respect to an inertial reference frame, unless acted on by an outside force. Mathematically $\sum \bold F =0 \Rightarrow \frac{d\bold v}{dt}=0$ .

2. The acceleration of a body is directly proportional to, and in the same direction as, the net force acting on the body, and inversely proportional to its mass. Mathematically $\bold F=m\frac{d\bold v}{dt}=m\bold a$ .

3. Every force exerted on a body is simultaneously being exerted on a force equal in magnitude and opposite in direction. Mathematically $\sum \bold F_{ab}=-\sum \bold F_{ba}$ .

Visual Art Piece of the Week [June 24, 2013]

Piet Mondrian's Composition II in Red, Blue, and Yellow:

File:Mondrian Composition II in Red, Blue, and Yellow.jpg

Classical Music Selection of the Week [June 24. 2013]

Ludwig van Beethoven's Symphony no. 9 4th movement - Presto:
https://www.youtube.com/watch?v=ljGMhDSSGFU

Quote of the Week [June 24, 2013]

Truth is the daughter of time, and I feel no shame in being her midwife.
-Johannes Kepler

Tuesday, 18 June 2013

OEIS Sequence of the Week [June 17, 2013]

A130597: Inverse binomial transform of decimal expansion of pi:

3, -12, 5, -11, 24, -44, 60, -39, -90, 456, -1265, 2810, -5464, 9650, -15778, 24102, -34309, 44251, -46267, 17462, 1-3896, -471971, 1449364, -3850820, 9448033, -22013005, 49484952, -108500910, 233911495, -498770996, 1056140390, -2225762829

Monday, 3 June 2013

Problem of the Week [June 17, 2013]

Suppose $n$ is a positive integer. Consider an upward-pointing equilateral triangle with side length $n$ , cut up into unit triangles. For each $n$ , let $f(n)$ represent the total number of downward-pointing equilateral triangles of all sizes. For example, $f(3)=3$ and $f(4)=7$ . Determine, with justification, all positive integers $n$ for which $f(n)$ is divisible by $n$ .

Correct answers as well as solutions will be acknowledged.