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

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February 24

Limit of a maximum of functions

How would one show rigorously that the limit of the maximum of a finite set of continuous functions is the maximum of the limit, i.e. $\lim _{x\to a}max\{f_{1}(x),f_{2}(x),...,f_{n}(x)\}=max\{f_{1}(a),f_{2}(a),...,f_{n}(a)\}$? I've tried fiddling around with the inequalities of the limits a little bit but can't see a nice way to get this result out - thankyou! Mathmos6 (talk) 01:09, 24 February 2009 (UTC)Mathmos6

Hint 1 - Try the epsilon-delta definition of continuity. The epsilon you'll need for the limit will turn out to be the same (this doesn't always happen, but max is a fairly special function). Hint 2 (more complicated) - This is a special case of a more general result that the limit of a continuous function is the value of the function with the limit plugged in. You'll have to prove this (or quote it from your textbook) and show that max is continuous. 98.216.49.28 (talk) 04:29, 24 February 2009 (UTC)

Special Unitary group over a finite field

I am slightly confused by something. In the work I am doing, I have encountered the notion of $PSU(3,2^{n})$. The PSU obviously refers to the projective special unitary group... but I am only familiar with this over the complex numbers. How does this work over a finite field? Is there just one, or are there many, equivalents of conjugation here? SetaLyas (talk) 16:04, 24 February 2009 (UTC) 15:51, 24 February 2009 (UTC)

There's some kind of description in Unitary group#Finite fields. — Emil J. 16:01, 24 February 2009 (UTC)

Yeah I know, I've read it. It seems kinda badly written, and just confused me even more when I first saw it. It doesn't really explain what the special unitary group is... that's why I asked the specific questions that I did. thanks though! SetaLyas (talk) 16:10, 24 February 2009 (UTC)

If I understand it correctly, the role of conjugation is played by the automorphism $\alpha (x)=x^{q}$ (which is the unique nontrivial automorphism of $\mathbf {F} _{q^{2}}$ fixing $\mathbf {F} _{q}$; notice that $\mathbf {F} _{q^{2}}/\mathbf {F} _{q}$ is a Galois extension of degree 2). So, yes, there is just one of them. — Emil J. 16:40, 24 February 2009 (UTC)

thanks Emil J! so is there no description of SU over a finite field that doesn't involve these field extensions and Galois theory? SetaLyas (talk) 16:45, 24 February 2009 (UTC)

Well, I don't think you really need Galois theory, as everything can be seen directly. The automorphism $\alpha$ of $\mathbf {F} _{q^{2}}$ is given by an explicit formula, we have $\alpha (\alpha (x))=x$ and $\mathbf {F} _{q}=\{x\in \mathbf {F} _{q^{2}}\mid \alpha (x)=x\}$ by simple algebra, and that should be all you need to know. — Emil J. 17:01, 24 February 2009 (UTC)

I think I've found where my confusion arises. So $U(3,2^{n})$ actually denotes the group of 3x3 matrices over the field $F_{q^{2}}$ (not over the field $F_{q}$) such that $\alpha (A)=A^{-1}$? SetaLyas (talk) 17:11, 24 February 2009 (UTC)

Yes. And as mentioned in the article, here either $2^{n}=q$ or $2^{n}=q^{2}$, the notation varies in the literature. — Emil J. 17:42, 24 February 2009 (UTC)

To be clear, this is true when α(A) is the matrix whose (i,j)th is the qth power of the (j,i)th entry of A; that is, it is the conjugate transpose. The field with q elements plays the role of the real numbers, and the field with q² elements plays the role of the complex numbers, and the qth power map plays the role of complex conjugation. JackSchmidt (talk) 20:39, 24 February 2009 (UTC)

The sum of all numbers between 0 and n (including n)

I'm trying to come up with an equation for this using basic algebra. Can anyone guide me in this? First step I figured out I have to simplify the expression n+(n-1)+(n-2)+...+(n-n). By just looking at the previous expression, it turns out to equal n*n+n-[sum of all numbers between 0 and n including n]...which goes back full circle because I have to prove the term in the square brackets. I'm lost and I feel like I'm making this too complicated. 128.163.165.39 (talk) 17:44, 24 February 2009 (UTC)

You'll find it simpler if you ignore the algebra for a bit, and start at the beginning using numbers. What you're looking at are the Triangular Numbers, and the first few terms are 1, 3, 6, 10, 15, 21, 28, 36. Try using the difference between successive terms to calculate the equation that governs it. I'll give you a hint - it is a quadratic. -mattbuck (Talk) 18:11, 24 February 2009 (UTC)

Try Triangular numbers. --Tango (talk) 18:31, 24 February 2009 (UTC)

Add the first and last numbers together, then add the 2nd and penultimate numbers, then the 3rd and 3rd to last and so on. Do you spot a pattern (it's a really easy pattern!)? Use that pattern to work out a formula. --Tango (talk) 18:31, 24 February 2009 (UTC)

Although what you have shown seems to have gotten you back to where you started, you can use it to find the formula. If we let $S_{n}=0+1+2+...+n$, then you have shown that $S_{n}=n^{2}+n-S_{n}$. Hence $2S_{n}=n^{2}+n$ and so $S_{n}={\frac {n^{2}+n}{2}}$. --Matthew Auger (talk) 20:16, 24 February 2009 (UTC)

Read Knuth's Concrete Mathematics, it's a really good book and explains this well. (It also has some advanced material in the later chapters, feel free to skip that part, don't let it scare you away.) – b_jonas 09:43, 25 February 2009 (UTC)

Arithmetic progression —Preceding unsigned comment added by 129.67.37.225 (talk) 15:40, 25 February 2009 (UTC)

Another valuable technique to learn when looking for patterns in sequences and series is proof by mathematical induction. Zunaid 12:02, 27 February 2009 (UTC)

length across an ellipse derivation

I am trying to figure out the length across an ellipse at any given angle theta (Θ). The ellipse is given as 2a across the major axis, and 2b across the minor axis. I am basically trying to figure out how to make a plot of the lengths as I go from the major axis and vertical axis being at the same direction, to the major and vertical axis that are 90 degrees apart.

This question is related to non destructive evaluation. I need to find the lengths of an ellipsoidal crack when the crack is in different orientations. The lengths will then help me find the intensities of the x-rays that are going through the part and the crack at different orientations.

Thanks —Preceding unsigned comment added by 12.216.236.227 (talk) 19:47, 24 February 2009 (UTC)

I think you want the equation for this ellipse in polar coordinates, that is

$\rho ={\sqrt {a^{2}\cos ^{2}(\theta )+b^{2}\sin ^{2}(\theta )}}$

Then if you mean the length of the chord at angle $\textstyle \theta$, it's of course $\textstyle 2\rho$.--pma (talk) 22:56, 24 February 2009 (UTC)

Investment question

I have an investment question regarding rental property.

In an online game I have the option to invest in two different properties, with the following considerations:

Property 1 will pay for itself in 6 years, and will require about 3 years of saving to purchase

Property 2 will pay for itself in 4 years, but will require about 11 years of saving to purchase

The catch is that property 1 costs $200,000 and property 2 costs $750,000

Assume that there is no inflation, and that income and costs remain constant for the duration, and that property values will not change.

Which is more efficient for me to purchase?

I am leaning toward the cheeper one, but am not sure since I can only get 1 property at that price. 12.216.168.198 (talk) 23:16, 24 February 2009 (UTC)

Assuming no inflation and 0% interest rates (pretty realistic at the moment, actually!), and assuming that by "pay for itself in X years" you mean it generates a profit of price/X each year, then the answer is going to depend on the planned length of the investment. Obviously, if you are going to invest for less than 11 years, option 2 makes you nothing, so option 1 would be better. Over a very long period of time (100's of years, say), the amount of time it takes to save up and then pay off the purchase costs are irrelevant and you want the one with the greater return, which is option 2. There will be some point inbetween where the best choice switches from one to the other. I'll do some maths and see if I can work out what it is... --Tango (talk) 23:32, 24 February 2009 (UTC)

The switch is around 16 or 17 years. If you're investing for less than 16 years, take option 1, more than 17 years, take option 2. (Between 16 and 17 years would require me to do the arithmetic more carefully!) --Tango (talk) 23:42, 24 February 2009 (UTC)

I take the estimates of time to save up the money as implying that money is saved up at a constant rate and one investment or the other is made as soon as enough is saved; after which it gives a constant return of $33,333/year or $187,500/year. The estimates imply slightly different rates of saving, so I can only say that I get turnover at around 13 years. —Tamfang (talk) 07:32, 2 March 2009 (UTC)

Thanks for the help. —Preceding unsigned comment added by 12.216.168.198 (talk) 23:54, 24 February 2009 (UTC)

Types of functions

How many types of functions are there? I have polynomial, exponential, and trigonometric so far but I was wondering if there are any more.

Here, a function is a different "type" if it can not be expressed in terms of functions of other types in a finite number of terms. Inverse functions do not count as different types. So hyperbolic functions are not of a different type because they can be expressed in terms of exponential functions and polynomial functions. $\sinh(x)={\frac {e^{x}-e^{-x}}{2}}$ can be expressed as $f(x)=e^{x}$ and $g(x)={\frac {x}{2}}$ and $h(x)=-x$: $\sinh(x)=g(f(x)-f(h(x)))$

Trigonometric functions are a different type from polynomial functions even though $\sin(x)=x-{\frac {x^{3}}{6}}+{\frac {x^{5}}{120}}\cdot \cdot \cdot$ because it requires an infinite amount of terms.

Are there any other types?--Yanwen (talk) 23:52, 24 February 2009 (UTC)

The question "how many types of functions are there?" is quite ambiguous, just as the questions "how many languages are there?" or "how many colors are there?". Also, mathematicians are constantly making new functions as the need arises. Putting those issues aside, there are some other things you want to consider: like, what do you mean when you say "function"? All of your examples are functions from reals to reals (or complex numbers to complex numbers), but perphaps you're also interested in functions from integers to integers (like the factorial function, Euler's totient function, divisor function, and many number theoretical functions, as well as functions from computer science like Busy beaver function and the Ackermann function), integers to reals, or more abstract mathematical structures (sets, groups, rings, fields, topological spaces, metric spaces, etc. etc.). Also, what restrictions do you want to impose on these functions? Your example functions are all infinitely differentiable. Are you interested in continuous functions, uniformly continuous functions (though note that polynomials of degree above 1 are not uniformly continuous on the real line), differentiable functions, holomorphic functions, homomorphisms, etc.? My goal is not to overwhelm you (depending on your mathematical background), but to emphasize that there are many, many different kinds of functions used in mathematics.

Maybe if you're looking for more "types" of functions, maybe you will want to think about what type of function each of the following is: square-root, Riemann Zeta function, Gamma function, sawtooth function, Dirichlet function, question mark function, Weierstrass function. I hope you find something interesting/new in there somewhere.

Here's an example of one type of function that you might want to add to your list: Rational functions. Combining rational functions with trigonometric ones gives you, for example, the sinc function.

You may find the Stone-Weierstrass theorem interesting. Eric. 131.215.158.184 (talk) 05:09, 25 February 2009 (UTC)

There are also difficulties in deciding which functions are of the same type. For example, I would consider exponential and trigonometric functions to be of the same type, since they are readily interdefinable, but that only works over the complex numbers. Algebraist 09:30, 25 February 2009 (UTC)

But it seems to me that the original question about "how many types of functions are there" is quite precise and not ambiguous (although the term "type of function" is unhappy, because it does not refer to an equivalence relation; rather, he has in mind a functional dependence). Anyay, the OP says that a function is of "different type" wrto a set of functions $\scriptstyle {\mathcal {F}}$, if it can not be obtained by $\scriptstyle {\mathcal {F}}$ with finitely many operations (he thinks to sum, product, composition, inverse; and we can decide whether to allow derivatives, and antiderivatives too; one could even allow solutions of algebraic or differential equations with coefficients in $\scriptstyle {\mathcal {F}}$). Now, to decide if a certain function f is or is not in the enlarged class $\textstyle {\mathcal {F}}$ generated by a certain class of functions $\scriptstyle {\mathcal {F}}$, by means of a certain set of operations, may be an extremely difficult algebraic problem; I do not know much about the spectacular results that have been obtained in this direction. Nevertheless, usually you can quite easily show that starting with a countable $\scriptstyle {\mathcal {F}}$ there is always a "new" analytic entire function f which is not in the enlarged class $\textstyle {\mathcal {F}}$. The standard argument, without going into the details, is from their order at infinity: the functions in $\textstyle {\mathcal {F}}$, are all o(f), what implies that f is actually "new". In other words you can produce new functions for an uncountable ordinal of times, so to speak. On the other hand, if you allow limits, the scenario completely changes; you enter in the realm of the approximation theory of functions, where density theorems like the one quoted by Eric allow to generate all functions in suitable classes, by means of very simple base systems. At least, I hope you liked the notation for enlarged classes. --pma (talk) 10:15, 25 February 2009 (UTC)