Wikipedia:Reference desk/Archives/Mathematics/2009 April 24

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April 24

Growth of a bacterial colony

Hi I was wondering if I could obtain some help with the following question: Starting with 1 bacterium that divides every 15 minutes, how many bacteria would there be after 6 hours? Assume none die etc. I thought ok so first of all how many sets of 15 minutes in 6 hours? There are 4 lots of 15 mins in 1 hour therefore 4 x 6 = 24 15 minute intervals in 6 hours. Nothing too difficult there. I thought then to work it out I would do e^{original number x time} making the equation e^1x24 which gives an answer of 2.649 x 10¹⁰. I had a look at the answer and it said 16,777,216 which is obtained by the equation 2²⁴ but I have no idea where this equation 2²⁴ comes from. I'm also why my calculation involving e was incorecct. If anyone could explain these two things to me it would be a great help. Thanks. —Preceding unsigned comment added by 92.22.189.144 (talk) 08:24, 24 April 2009 (UTC)

Dividing every 15 minutes means doubling, so there will be twice as many as before. Hence it is the appropriate power of 2 which is wanted.81.154.108.6 (talk) 08:52, 24 April 2009 (UTC)

Right. After 15 minutes, there will be 2 (2¹) bacteria. They both will divide after 30 minutes (15x2), and there will be 4 (2²) bacteria. After 45 minutes (15x3), these 4 bacteria will double and number of bacteria will be 8 (2³), so you see there is a pattern. After 15xn minutes, there will be 2ⁿ bacteria. In your case n is 24, so there will be 2²⁴ bacteria after 15x24 = 360 minutes = 6 hours. I have no idea why you think there should be a connection with 'e'. - DSachan (talk) 11:21, 24 April 2009 (UTC)

Many thanks for the reply. I am sure there is an equation of exponential growth involving e or would it be log e^kt? Not sure. Thanks anyway! —Preceding unsigned comment added by 92.21.233.141 (talk) 13:08, 24 April 2009 (UTC)

The article on exponential growth is quite helpful in this regard. We could write the number of bacteria as a function of the number of minutes that have passed as

$b(m)=2^{m/15}$

but as the original poster guessed, we could just as well write this formula using base e.

$b(m)=e^{m/T}$

To get the value of T, we set the two formulas equal and just take the log of both sides.

$2^{m/15}=e^{m/T}$

${\frac {\ln {2}}{15}}={\frac {1}{T}}$.

The number of bacteria at a given time can now be written as $b(m)=e^{m{\frac {\ln {2}}{15}}}$. (any suggestions to make my math look better would be appreciated). mislih 13:48, 24 April 2009 (UTC)

a suggestion to make the math look better: try \scriptstyle : $\scriptstyle {\frac {\ln {2}}{15}}={\frac {1}{T}}$. Bo Jacoby (talk) 14:13, 24 April 2009 (UTC).

sci.mathresearch newsreader question

What's an easy newsreader to use for posting there? (They say you have to use a newsreader now, and I don't know a newsreader from Adam.) thanks, Rich (talk) 09:28, 24 April 2009 (UTC)

e lon to nhat la ai nhi lu khon nan nhat oi vai lon chua kia so lon thi so bi aids nhe kinh —Preceding unsigned comment added by 123.18.213.33 (talk) 11:04, 24 April 2009 (UTC)

Just go to groups.google.com and follow instructions. McKay (talk) 12:02, 24 April 2009 (UTC)

thanksRich (talk) 06:07, 25 April 2009 (UTC)

Polynomial division

This is actually a homework sum. I tried in as many ways as possible but i didn't get an answer. The question is :

If the polynomial $x^{4}-6x^{3}+16x^{2}-25x+10$ is divided by another polynomial $g(x)=x^{2}-2x+k$, the remainder comes out to be $x+a$. Find k and a.

I tried by the following method:

$x^{4}-6x^{3}+16x^{2}-25x+10=(x^{2}-2x+k)*q(x)+x+a$

putting remainder on L.H.S, $x^{4}-6x^{3}+16x^{2}-26x+10-a=(x^{2}-2x+k)*q(x)$

If i divide L.H.S by g(x), i get $q(x)=x^{2}-4x-8+k$ and $r(x)=(10+8k)x+10-a-8k-k^{2}$. I equate r(x) to zero coz g(x) is a factor of L.H.S. I get $(10+8k)x+10-a-8k-k^{2}=0$. I am unable to continue. Please help me--harish (talk) 13:51, 24 April 2009 (UTC)

The remainder must be the zero polynomial, meaning that both coefficients are zero. $10+8k=10-a-8k-k^{2}=0$. Bo Jacoby (talk) 14:07, 24 April 2009 (UTC).

(e/c) If $(10+8k)x+10-a-8k-k^{2}$ is the zero polynomial, this means that all its coefficients are zero, i.e., $10+8k=0$ and $10-a-8k-k^{2}=0$. You can extract k from the first equation, and then you get a from the second equation. However, I think that you made a numerical error, I got a different result for q and r. — Emil J. 14:09, 24 April 2009 (UTC)

solve(identity(x^4-6*x^3+16*x^2-25*x+10 = (x^2-2*x+k)*(x^2+b*x+c) + (x+a),x),{a,b,c,k}); McKay (talk) 00:40, 25 April 2009 (UTC)

Holomorphic mapping

Hello, I am trying to prove that any one-to-one and onto holomorphic mapping of $\mathbb {C} \setminus \{0\}$ with a removable singularity at 0 satisfies $\lim _{z\to 0}f(z)=0$ and I am stuck.

I assumed it was something else, c. Then, there must exist some other point that also maps to c, say p. I assume I need to do something with an integral to count the number of zeros of the function f(z) - c, which has two (if I just abuse notation and let this f now represent f with the removable singularity removed). But, I am not sure what I can do. Form a circle big enough to contain 0 and p, call it D. Then, consider the function

$\int _{\partial D}{\frac {f'(z)}{f(z)-q}}\,dz$

which is a function of q. At c, the value is 2. I am guessing that I want to prove that for some small neighborhood around c, all numbers in that disc must also have value 2 for this integral, which will contradict the fact that the original map was one-to-one.

So, is this integral function a continuous function of q at c? I am pretty sure I know it is continuous in z but that's not what I want here. I am not sure exactly how this works.

Thanks StatisticsMan (talk) 21:52, 24 April 2009 (UTC)

Your argument is correct, but just observe that the extended function on $\scriptstyle \mathbb {C}$ is a nonconstant holomorphic function, hence it is an open map. Terefore if the pre-image of c has at least 2 points, the same is true in a nbd of c, loosing the injectivty on $\scriptstyle \mathbb {C} \setminus \{0\}$. --pma (talk) 06:56, 25 April 2009 (UTC)

Let me see if I understand this. If I take a disc around 0, maybe with radius |p|/2 to ensure p is not in it. Then I map this with the holomorphic function, it must be an open set and it contains c. Then, by continuity, the inverse image of that is an open set. But, it now contains p and also at least a small disc around p. But, that means every point in that disc is an inverse image of some point (I didn't name these discs so hard to talk about discs now), which means every non-p point in that disc around p goes to the same value as some nonzero point in the disc around 0, contradiction. Okay, this makes sense and it's much simpler.

Just for my knowledge, can someone please help me understand why that integral is continuous in q? (Nevermind on that, I actually found this in my book finally... and it's in the proof of the open mapping theorem, which makes sense.) Thanks for the help! StatisticsMan (talk) 14:08, 25 April 2009 (UTC)

How to compute statistical significance of correlation

I have a list correlations R[] and the associated number of samples for each correlation N[]. And I am not sure how to calculate a weight for each correlation/amount which is proportional to the chance that the correlation is non-spurious. Right now I have weight set to sqrt(N)*R^2 but this does a lousy job at discriminating one pair of R,N from another.

I believe you mean "statistically insignificant" instead of "non spurious." Spurious (at least in econometrics) refers to a correlation that is statistically significant due to random chance. Determining whether a correlation is spurious is, at best, non-trivial and, depending on your particular circumstances, quite possibly impossible. Wikiant (talk) 23:22, 24 April 2009 (UTC)

If you have an assumption of joint normality, and if you mean what I suspect you might mean (but I can't be sure, given what you've said and what you haven't said), an F-test should do it. Michael Hardy (talk) 03:21, 26 April 2009 (UTC)