Wikipedia:Reference desk/Archives/Mathematics/2011 October 23

= October 23 =

Zero integral theorem
Give a completely rigorous proof that $$\int_a^b (f(x))^2dx = 0 \Rightarrow f(x) = 0$$ for continuous $$f$$ and $$b > a$$.

I guess it follows quite intuitively from the fact that $$(f(x))^2$$ is nonnegative. I need a rigorous proof though. Widener (talk) 01:13, 23 October 2011 (UTC)


 * Here is a hint: Suppose there is some point c with f(c) ≠ 0.  Show that because of continuity, |f(x)| is greater than some ε in a neighborhood around c. Rckrone (talk) 01:42, 23 October 2011 (UTC)


 * Let g=f^2. For each positive epsilon, the open set $$g^{-1}(\epsilon,\infty)$$ contains no interval, and hence is empty. Thus $$g^{-1}(0,\infty)$$ is also empty.  So g=0.  Sławomir Biały  (talk) 01:49, 23 October 2011 (UTC)
 * I'm afraid I'm not familiar with that notation. Widener (talk) 02:13, 23 October 2011 (UTC)
 * See preimage. Sławomir Biały  (talk) 13:01, 23 October 2011 (UTC)
 * Suppose there exists a point c such that $$f(c) \ne 0$$ and hence $$(f(c))^2 > 0$$. f continuous implies f^2 continuous so $$\forall \epsilon > 0 \exists \delta > 0 : |x-c| < \delta \Rightarrow |f(x)^2 - f(c)^2| < \epsilon \Rightarrow f(c)^2 -\epsilon< f(x)^2$$. If we choose $$\epsilon$$ such that $$\epsilon < f(c)^2$$ Then this shows that $$f(x)^2$$ is greater than some positive number ($$=f(c)^2-\epsilon$$) in the $$\delta$$ neighborhood around c. I don't think this is sufficient to infer that $$\int_a^b (f(x))^2dx$$ is positive though, if it were, we could just infer $$\int_a^b (f(x))^2dx = 0 \Rightarrow f(x) = 0$$ right away. Widener (talk) 02:13, 23 October 2011 (UTC)
 * Oh wait, I suppose we have $$\int_a^b (f(x))^2dx \ge 2\delta(f(c)^2-\epsilon)$$ where $$2\delta(f(c)^2-\epsilon)$$ is the area of the rectangle with side lengths $$f(c)^2-\epsilon$$ and $$2\delta$$. Is this rigorous enough? Widener (talk) 02:19, 23 October 2011 (UTC)
 * Almost. The interval $$[c-\delta,c+\delta]$$ is not necessarily included in the interval $$[a,b]$$. 98.248.42.252 (talk) 02:33, 23 October 2011 (UTC)
 * Ok, of course you can replace $$\delta$$ with any smaller $$\delta'$$ until we have $$c - \delta' > a$$ and $$c + \delta' < b$$ Widener (talk) 02:43, 23 October 2011 (UTC)
 * Not if the problem asked you to prove that $$f(x) = 0$$ for $$x \in [a,b]$$. 98.248.42.252 (talk) 02:52, 23 October 2011 (UTC)
 * Uh, why not? The rectangle with area $$2\min({c-a,b-c})(f(c)^2-\epsilon)$$ is still inside the integral.Widener (talk) 03:02, 23 October 2011 (UTC)
 * Basically what I've shown is that if that if the function is nonzero at any point then there exists a rectangle with nonzero area enclosed entirely within the integral from a to b.Widener (talk) 03:08, 23 October 2011 (UTC)
 * The rigorous first step is "Suppose there exists a point $$c \in [a,b]$$ such that $$f(c) \ne 0$$", so you can end up with c = a or c = b. 98.248.42.252 (talk) 03:16, 23 October 2011 (UTC)
 * Oh yeah. nnnnnnngggggggg. Widener (talk) 03:30, 23 October 2011 (UTC)

Affine transformation equivalency
$$a \in \mathbb{R}; \mathbf{x},\mathbf{y} \in \mathbb{R}^n$$ Apparently $$T(a\mathbf{x} + (1-a) \mathbf{y}) = aT(\mathbf{x}) + (1-a)T(\mathbf{y})$$ implies $$T$$ is affine. Why is this? One can show easily that $$T(a\mathbf{x}) - T(\mathbf{0}) = a(T(\mathbf{x})-T(\mathbf{0}))$$ using the following argument: $$T(a\mathbf{x})-T(\mathbf{0})=T(a\mathbf{x}+(1-a)\mathbf{0})-T(\mathbf{0})=aT(\mathbf{x})+(1-a)T(\mathbf{0})-T(\mathbf{0})=a(T(\mathbf{x})-T(\mathbf{0}))$$.

How does one show that the other property of affine transformations $$T(\mathbf{x}+\mathbf{y}) - T(\mathbf{0}) = T(\mathbf{x}) + T(\mathbf{y}) - 2 T(\mathbf{0})$$ holds under this condtion?Widener (talk) 02:59, 23 October 2011 (UTC)
 * $1/2$T(x+y)+$1/2$T(0)=T((x+y)/2)=$1/2$T(x)+$1/2$T(y).--RDBury (talk) 03:47, 23 October 2011 (UTC)

When do you raise the rate on a certificate of deposit if there are unpredictable increases in interest rates?
A certain online bank has a 4-Year CD that allows you to increase your interest rate if rates go up. Over a CD's term of 4 years, this bank (if the interest rates go up) allows you to raise the rate up to the current rate TWICE over the four year period. For instance, if the rate is locked in at 2% for three years and the bank's 4-year CD rates rise to 2.5% at the end of the third year you can raise your CD's rate to 2.5% for the remainder of the term. You aren't required to raise your rate.

Let's say I get a CD at the interest rate right now. I'm not really expecting interest rates to rise anytime soon, but if they started to rise, what would be a good mathematical way of deciding when was the best time to use those TWO instances that I'm allowed?

I know there's probably not a simple solution, as it depends on many variables (much like the stock market), but is there some sort of mathematical reasoning that would take some of this into account and give me a nonrandom strategy of when to raise the rate? Perhaps a formula that assumes that rates will continue to rise? How about a formula that assumes nothing?

I'm just kind of confused as to the best time to raise the rate if rates start to rise. Sooner rather than later is my thought, but I don't know. Thanks for any help you can give.

173.247.7.89 (talk) 03:41, 23 October 2011 (UTC)
 * You need to have some sort of assumptions (there's work on how to minimize the so-called "weak regret" in multi-armed bandit problems with essentially no assumptions at all, but I don't think that applies here). Once you figure out the assumptions you want to make, describing the solution mathematically is straightforward, though actually finding the optimal solution could be very difficult.
 * Let's get warmed up with a simple case. The problem is continuous; you can change the rate at any time; the bank's interest rate is deterministic, and is known to increase linearly from $$r_0$$ at the start to $$r_e$$ after n years. I will use the continuous interest rate; the annual rate is $$(\exp(r)-1)\cdot100\%$$. If you switch at $$n_1$$ and $$n_2$$, then at the end you'll multiply your investment by $$\exp\left(n_1r_0+(n_2-n_1)(r_0+(r_e-r_0)n_1/n)+(n_e-n_2)(r_0+(r_e-r_0)n_2/n)\right)$$. The exponentiation is monotonic and can be ignored when optimizing (it will be important in the stochastic case); differentiating and equating to 0 gives the solution $$n_1=n_e/3,\ n_2=2n_3/3$$, so you'll want to update after 1y4m and 2y8m.
 * Now let's assume there's a discrete number n of terms, and the rate changes only after each term. I'll use the per-term interest; if the term is a month (or anything else) but you're given the annual rate you'll need to translate it. Even if the rate is really updated more frequently, you may want to use a small n to simplify the solution.
 * This time we will not assume the interest rate is known, rather that it follows some stochastic process. This still requires specification of the process; we'll assume Gaussian random walk which is very reasonable, which means that the interest rate each term changes by a normal variable with mean 0 and known variance $$\sigma^2$$ (which can be estimated from historic data). We want to maximize our expected savings at the end of n terms. We'll denote by $$f(n,o,r,u)$$ our expected eventual savings, as a multiple of the current savings, given that we have n terms left, our current rate is o, the bank's current rate is r, and we have u switches left. Then f is given by some recurrence relations:
 * $$f(0,o,r,u)=1\,\!$$
 * $$f(n,o,r,0)=(1+o)^n\,\!$$
 * $$f(n,o,r,u)=\max\left((1+o)\int_{-\infty}^{\infty}\frac{\exp(-x^2/(2\sigma^2))}{\sqrt{2\pi}{\sigma}}f(n-1,o,r+x,u),(1+r)\int_{-\infty}^{\infty}\frac{\exp(-x^2/(2\sigma^2))}{\sqrt{2\pi}{\sigma}}f(n-1,r,r+x,u-1)\right)$$
 * Solving this exactly is probably impossible, but with a variety of techniques an approximation can be found for any given parameters.
 * I've run a simulation for $$n=4,\ r_0=0.025,\ \sigma=0.002$$. The approximate optimal strategy is:
 * After 1 year: Switch if $$r>0.0251668$$.
 * After 2 years: If you haven't switched yet, switch if $$r>0.025$$. If you have, switch if $$r>o+0.0005329$$.
 * After 3 years: If you have a switch left, switch if $$r>o$$.
 * We can also consider the continuous case where $$n\to\infty$$. Then the stochastic process becomes Brownian motion, and the recurrence relations become differential equations. I don't think it would be realistic to solve these, but we can approximate with a large but finite n. -- Meni Rosenfeld (talk) 12:02, 23 October 2011 (UTC)

Circle question
A, B, and C are three different points on a circle O. The tangents at A and B meet at point P, while the line AB meets the tangent at C at the point Q. Prove that $$ |PQ|^2 = |PA|^2 + |QC|^2 $$. — Preceding unsigned comment added by OsamaMahmoodKhawar (talk • contribs) 08:00, 23 October 2011 (UTC) AndrewWTaylor (talk) 08:18, 23 October 2011 (UTC)


 * Let O be the center of the circle and let AB and OP meet at R. Then
 * PQ2=PR2+RQ2=AP2-AR2+RQ2=AP2+AQ.QB=AP2+QC2. You still need to justify the four equals signs though.--RDBury (talk) 12:23, 23 October 2011 (UTC)

Inflection
What is the difference between a point of inflection and a midpoint of inflection. "Estimate the force at the midpoint of this inflection." Plasmic Physics (talk) 11:00, 23 October 2011 (UTC)


 * Could you be more specific about the context please? An inflection point of a plane curve is a point where the curve has at least three point contact with its tangent line, e.g. the point (0,0) on the curve y = x3. However, it doesn't look like that's the kind of inflection point you mean. — Fly by Night  ( talk )  11:10, 23 October 2011 (UTC)

A one variable function plot, modelled by a smoothly increasing curve interupted by a feature called a "shoulder". Plasmic Physics (talk) 13:20, 23 October 2011 (UTC)


 * I believe the midpoint of inflection is the point halfway between a local maximum and local minimum. This is the inflection point of the cubic polynomial that interpolates between the two local extrema, and should be close to the actual inflection point.  (Precise estimates on the error in this approximation can probably be derived using Taylor's formula, if some information about the size of the fourth derivative of the function is known a priori.)  The main advantage of a midpoint of inflection over an actual point of inflection is that it is much easier to estimate from numerical data.   Sławomir Biały  (talk) 14:13, 23 October 2011 (UTC)

Thank you, that is most helpful. Plasmic Physics (talk) 23:00, 23 October 2011 (UTC)

p-adic integers and isomorphisms
Hello everyone,

I have a problem I'm trying to work on, and I was hoping for some help:

-Show that the inclusion $$\mathbb{Z} \subset \mathbb{Z}_p$$ induces an isomorphism $$\mathbb{Z}/p^n \mathbb{Z} \cong \mathbb{Z}_p /p^n \mathbb{Z}_p$$ for all n ≥ 1, where $$\mathbb{Z}_p$$ denotes the p-adic integers. Show also $$|(\mathbb{Z}_p /m \mathbb{Z}_p )| = \|m\|_p^{-1} \, \forall \, m \in \mathbb{Z}_p$$, where $$\| \cdot \|_p$$ denotes the p-adic absolute value.

I'm having some trouble however, because I feel like I don't have a very good intuition of what $$\mathbb{Z}_p$$ actually is, and therefore what $$\mathbb{Z}_p /p^n \mathbb{Z}_p$$ is. As far as I can understand, the p-adic integers consist of elements of the p-adic numbers $$\mathbb{Q}_p$$ with p-adic absolute value at most 1. However, although I know that $$\mathbb{Q}_p$$ is the completion of $$\mathbb{Q}$$ under the absolute value, I can't say I have a full understanding of what elements are actually added in the completion (I believe the square root of -1 is, for example, but what else? Is it somehow related to infinite series expansions in powers of p?), and so I can't really figure out how to get this isomorphism to work as required (I would think it just falls out nicely from the inclusion when you know what you're constructing the isomorphism into, but haven't managed to get that far) - similarly, working out the number of elements in $$(\mathbb{Z}_p /m \mathbb{Z}_p )$$ is also quite confusing when I don't fully understand what I'm working with.

Could anyone help me, generally with my confusion and more specifically with making some progress on these problems? Once I get going I will do my best to work out the rest of the problem on my own, don't want you to feel like you're doing my work for me! Thanks all - Typeships17 (talk) 15:15, 23 October 2011 (UTC)


 * I would take as a definition that $$Z_p$$ is the inverse limit of $$Z/p^nZ$$ (i.e., formal power series in p). The first claim should be almost immediate.  For the second claim, think about the first $$Z/p^nZ$$ in which m is nonzero. (Sorry, I'd explain more, but I'm in a rush at the moment.)   Sławomir Biały  (talk) 15:48, 23 October 2011 (UTC)


 * What definition of the p-adic numbers are you working with? Rckrone (talk) 16:13, 23 October 2011 (UTC)


 * The definition I am working with is that $$\mathbb{Q}_p$$ is the completion of $$\mathbb{Q}$$ under the p-adic metric, and $$\mathbb{Z}_p$$ likewise the completion of $$\mathbb{Z}$$: equivalently I believe this is $$\{x \in \mathbb{Q}_p: \|x\|_p \leq 1\}$$. Sławomir, no problem: I'll have a think about that in the meantime and see if I can get anywhere. Thanks. Typeships17 (talk) 22:11, 23 October 2011 (UTC)