Wikipedia:Reference desk/Archives/Mathematics/2006 July 3

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Semi-factorial functions

Working on the problem about pairing up people, I figured out that the number of possible sets of pairs of 2n people is f(1) = 1, f(n) = (2n - 1) * f(n - 1), or alternately for n people, f(2) = 1, f(n) = (n - 1) * f (n - 2). This is similar to the factoral function, but I haven't found a way to refactor it in terms of factorial. Is there a name for this?--Prosfilaes 06:25, 3 July 2006 (UTC)

Your function is basically f (n) = (n + 1)!!, where "!!" is a double factorial - see the factorial article. It can also be given using factorials:

$f(2n)=(2n+1)!!={\frac {(2n+1)!}{2^{n}n!}}$

-- Meni Rosenfeld (talk) 06:33, 3 July 2006 (UTC)

graph coloring and maximal clique

Is there an undirected graph for which the chromatic number exceeds the maximal clique size by more than one?

--Henning

Yes. Mycielski proved in 1955 that for every $k\geq 1$ there is a graph with chromatic number k that contains no triangle subgraphs, that is, whose maximal clique size is just 2. I'll make a drawing of such a graph with chromatic number 4 in a minute. —Bkell (talk) 10:24, 3 July 2006 (UTC)

—Bkell (talk) 10:34, 3 July 2006 (UTC)

Mycielski's proof is actually a constructive proof, so you can use it to make a graph with as large a chromatic number as you like with a maximal clique size of only 2. —Bkell (talk) 10:43, 3 July 2006 (UTC)

J. Mycielski. Sur le coloriage des graphes. Colloq. Math., 3:161–162, 1955. —Bkell (talk) 10:51, 3 July 2006 (UTC)

Thanks a bunch! :)

Base

how to calculate the value of 678 to the base of 7?

First make a list of powers of 7 until you get to a number larger than your starting value (678). 7⁰ = 1, 7¹ = 7, 7² = 49, 7³ = 343, 7⁴ = 2401. Take the largest power of 7 less than 678, 7³, and subtract it repeatedly until you get a value less than 7³: 678 - 343 = 335, so 678 = 1*7³ + something else. Continue with each smaller power of 7 until you get to 0. 335 - 6*7² = 41. 41 - 5*7¹ = 6. 6 - 6*7⁰ = 0. So in base 7, 678 = 1656 = 1*7³ + 6*7² + 5*7¹ + 6*7⁰. 128.197.81.223 17:05, 3 July 2006 (UTC)

We're assuming 678 is written in base 10, yes? The above method is fine; here's another.

(Let's work with different numbers so we don't do your homework for you. We'll convert 2097₁₀ to base 5.)

• Divide 2097 by 5, obtaining quotient 419 with remainder 2.
• Divide 419 by 5, obtaining quotient 83 with remainder 4.
• Divide 83 by 5, obtaining quotient 16 with remainder 3.
• Divide 16 by 5, obtaining quotient 3 with remainder 1.
• Divide 3 by 5, obtaining quotient 0 with remainder 3.
• The quotient is zero; stop dividing.
• Assemble the remainders left-to-right from last to first:

  2097₁₀ = 31342₅.

For computer implementation with very large numbers, more sophisticated algorithms are available.

When fractional components are involved, scaling and termination become issues; see Burger and Dybvig's paper, "Printing Floating-Point Numbers Quickly and Accurately." [1] --KSmrq^T 18:49, 3 July 2006 (UTC)

Investor Help

I am doing a project on the different types of investments. I was hoping that you could tell me the average growth and decay rate of stocks over 20-50 years. I also need the equation of this, as well as the future projected growth rate for 10-50 years. Please help me!

Thank You, HS

PS If it's not too much trouble, I would also like the same information on gold, silver, bonds, treasury bills, money market funds, mutual funds and antiques and commodities.

Since there are a lot of stock this is a difficult question to answer: there are stock that haven't seen much movement over the last 20 years while others have fluctuated greatly. But say you're interested in the major stock dealt on the New York Stock Exchange, it might be worth looking at the Dow Jones Industrial Average (DJIA). A chart of it shows that it has mainly grown over the last 20-50 years: Chart of DJIA performance (1928-present). Of course this isn't completely fair and for a real analysis of this you probably have to take into account the inflation that has occured over the same period.

The same applies in more or lesser degree to gold, silver, bonds, treasury bills, money market funds, and mutual funds. With antiques it is even more difficult to say something objective, as the kinds of antiques differ greatly and determining what is antique is a quite subjective business. -- Koffieyahoo 02:46, 4 July 2006 (UTC)

To get the average growth rate of Dow Jones, you can fit an exponential curve to its graph. That is, you can use linear regression to estimate slope of graph of log(DJ price) versus time. (Igny 04:11, 4 July 2006 (UTC))