Monthly Archives: May 2011

All models are wrong!

The actual quote, attributed to George Box, according to here is all models are wrong but some are useful How true.

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Number Theory by

Pommersheim, Marks and Flapan. The full subtitle of the book is “A Lively Introduction with Proofs, Applications, and Stories.” I have to admit I only browsed through a couple of pages of the book but it already lives up to … Continue reading

Posted in Books, Number Theory | 1 Comment

A dicey past year exam question

A die consists of six faces with each face representing precisely one of the numbers 1, 2, 3, 4, 5, 6. Suppose that n such dice are rolled for some positive integer n. The number of the upper face of … Continue reading

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Modelling with exponential generating functions

Let [tex] a_n[/tex] be the number of ways to distribute n distinct objects to four distinct boxes, such that the total objects is even. Find the exponential generating function. Either all four boxes have even number of objects, or exactly … Continue reading

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A nonlinear first order recurrence relation

Mr Ding! asked me the following question from Bona’s textbook. Question 6 of supplementary exercises in chapter 3. [tex] a_n = (n+1) a_{n-1} +3^n, a_0 = 1[/tex] My solution as follows. Let [tex]b_{n+1} = a_n [/tex]then [tex] b_{n+1} = (n+1) … Continue reading

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Bell Numbers

defined as [tex] \displaystyle B(n) = \sum_{k=0}^n \left\{ {n \atop k}\right\}[/tex] counts the total number of ways to partition n distinct objects into disjoint subsets (or blocks) where [tex] \displaystyle \left\{ {n \atop k}\right\} [/tex] is the well known Stirling … Continue reading

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Binomial identity and probability

The identity [tex]\displaystyle \sum k \binom{n}{k} = n 2^{n-1} [/tex] is pretty standard, and one can prove it algebraically by cancelling the k in the sum with the binomial coefficient and then using the binomial theorem summation or a combinatorial … Continue reading

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Double Factorial

Using the double factorial notation to denote the following [tex] \displaystyle n!! = \prod_{i=0}^{\lfloor \frac{n-1}{2} \rfloor} (n-2i) [/tex] seems pretty standard. (See Wolfram and Wiki.) So [tex] 4!! = 4 \times 2 = 8[/tex] but [tex] (4!)! = 24! [/tex]. … Continue reading

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Pascal’s triangle

Perhaps the most famous triangle of all. Take your calculator, and compute [tex] 11, 11^2, 11^3, 11^4[/tex] … cute! Can you explain why? It’s so famous that there’s lots of information on the web about it. Named after Pascal but … Continue reading

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Zeta(5) is irrational ?

The answer is that it probably is but mathematicians do not yet know how to prove it. A paper has been put up in arXiv (dated 4 May) that claims to have used very elementary methods to prove that [tex]\zeta(5)[/tex] … Continue reading

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pendulum waves

Great video illustrating waves with 15 pendulums. Watch to the end to see the pendulums make one complete cycle. First saw this via john d cook.

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Soap bubbles and current research

That’s the title of Frank Morgan’s talk today. The main takeaway is how, (motivated?) students can be exposed to interesting problems and work on current research. For example, it has been known for almost 2000 years that the circle in … Continue reading

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