Wikipedia:Reference desk/Archives/Mathematics/2010 March 3

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

March 3

Taylor polynomials and errors (homework)

This is homework. The assignment makes me consider the fourth order taylor polynomial p(x) of the function $f(x)={\sqrt[{3}]{x+1}}$, around x=0, and then (quoting literally): "with error $E(x)=f(x)-p(x)$, show that $|E(x)|\leq {\frac {1}{3}}\cdot 10^{-6}$ for $|x|\leq 10^{-1}$

I'm quite sure I didn't make any mistakes in the polynomial, derivatives or anything of that kind. I end up with the following equation using Taylor's theorem:

$|Error|\leq {\frac {|x|^{5}}{5!}}\cdot \max |{\frac {880}{243\cdot (x+1)^{14/3}}}|$

The maximization on the right hand side being the fifth derivative. I'm quite sure about the above, too. Now comes the question: if I take $x\in [-0.1,0.1]$, the right hand side of the above equation results in approximately ${\frac {1}{2}}\cdot 10^{-6}$ at x=-0.1.

Assuming my professor isn't fallible and doesn't ask me to prove things he cannot, what did I do wrong here? Don't give the answer away please, though :) User:Krator (t c) 01:37, 3 March 2010 (UTC)

Not sure what your professor meant to have you do, but how about this: expand the series to five terms. The total error must be less than the sum of the fifth term plus the result of the error formula applied with 5 terms. For most functions seen in practice the actual error will be less than that given by the error formula.--RDBury (talk) 04:22, 3 March 2010 (UTC)

That's what I would suggest too. Note that the integral form of the remainder, evaluated with maple, gives R<0.255*10^-6. PS: sorry, I thought to 0≤x≤1/10, I didn't notice you wrote |x|≤1/10. --pma 08:09, 3 March 2010 (UTC)

According to my calculations, your calculations are incorrect, since the true error is >0.327*10^-6 at -0.1.

Also, five terms is not strong enough for the required estimate. Six terms work, though. -- Meni Rosenfeld (talk) 08:30, 3 March 2010 (UTC)

Thanks. So a possible explanation is that the original problem had 0≤x≤1/10 and not |x|≤1/10, in which case the professor's bound should be correct. --pma 10:04, 3 March 2010 (UTC)

Sorry, I wasn't clear. The professor's bound for the error of the fourth order expansion is correct for |x|≤1/10. To prove this you need to consider the sixth order expansion.

Anyway, your bound for R isn't correct for 0.1 either. -- Meni Rosenfeld (talk) 10:13, 3 March 2010 (UTC)

The assignment specifically tells me to consider the fourth order expansion, so I calculated the error using the fifth order expansion, for which the error bound is incorrect as I wrote earlier. Why do I need to consider the sixth? I'm afraid I don't understand that. User:Krator (t c) 11:42, 3 March 2010 (UTC)

The assignment asked you to calculate the error of the fourth order expansion. What you tried to do is bound the error using Taylor's theorem for the fourth order expansion, which involves the fifth derivative. As you have noticed, this doesn't give you the bound you want.

RDBury suggested using Taylor's theorem for the fifth order expansion (which involves the sixth derivative) to tighten the bound, and I corrected that fifth isn't enough - you need the theorem for the sixth order expansion, which uses the seventh derivative.

The proof involves the triangle inequality - if $p_{n}(x)$ is the nth order expansion of $f(x)$, then $|f(x)-p_{4}(x)|\leq |f(x)-p_{6}(x)|+|p_{6}(x)-p_{4}(x)|$. You need to bound each of these summands, and then the bound you want will follow.

Think of it this way - the more terms you use, the more accurate you are. Usually what we want to be accurate about is $f(x)$, but here we want to be accurate about $f(x)-p_{4}(x)$. -- Meni Rosenfeld (talk) 13:11, 3 March 2010 (UTC)

$.321419721*10^{-7}$ ? User:Krator (t c) 14:13, 3 March 2010 (UTC)

This doesn't look right. Can you post how you got it? -- Meni Rosenfeld (talk) 14:51, 3 March 2010 (UTC)

Oh, to the power -6 of course. But besides that:

${\frac {|-0.1|^{7}}{7!}}|{\frac {209440}{2187(-0.1+1)^{20/3}}}|+|{\frac {880(-0.1)^{5}}{5!\cdot 243}}-{\frac {12320(-0.1)^{6}}{6!\cdot 729}}|$

With x=-0.1 as that is where the second fraction maximizes within the domain given. User:Krator (t c) 15:01, 3 March 2010 (UTC)

The expression is correct, but the result isn't. Perhaps you have an arithmetical error? -- Meni Rosenfeld (talk) 15:30, 3 March 2010 (UTC)

You're right. Ye olde excel gives $0.32909*10^{-6}$ User:Krator (t c) 15:52, 3 March 2010 (UTC)

Precisely. -- Meni Rosenfeld (talk) 16:15, 3 March 2010 (UTC)

Maximum angle

Hello. Consider the equation tanθ=sinφ/(r+cosφ). If r<1, tanθ can equal any value, and therefore there is no limit to the value of θ. However, if r>1, then there should be a maximum value for θ. How can you find this value? I tried finding when the derivative equals 0, but I just ended up with cosφ=-r, which clearly isn't possible if r>1. —Preceding unsigned comment added by 173.179.59.66 (talk) 11:49, 3 March 2010 (UTC)

Check your calculations, you should have gotten $\cos \varphi =-1/r$ which means $|\tan \theta |\leq {\frac {1}{\sqrt {r^{2}-1}}}$. -- Meni Rosenfeld (talk) 13:18, 3 March 2010 (UTC)

Right, thanks. 173.179.59.66 (talk) 16:37, 3 March 2010 (UTC)

Topology: Is there a name for a function differentiable in its connected open domain?

Regardless of whether its domain is a sub-domain of the real line, or of the complex plane, or of any other metric space (being a group). HOOTmag (talk) 13:56, 3 March 2010 (UTC)

If you want to use the standard mathematical definition of a function, the name is "differentiable function". If you want to use something different, you'll have to be specific. By "standard mathematical definition" I mean that a "function f from A to B" is a subset of the cartesian product of A and B with certain properties. With this definition of function, information about the domain and codomain is *built into the definition of f*, and to say "f is differentiable" means that f is diffable at every point of A. 129.67.37.143 (talk) 16:13, 4 March 2010 (UTC)

I don't want to use the standard mathematical definition of a function, but rather to find a universal topological term (which could be the term: "holomorphic" - if the domain were a sub-domain of the complex plane), for a function differentiable in its connected open domain (whereas the very domain is a sub-domain of a given metric space, unnecessarily of the complex plane). HOOTmag (talk) 19:58, 4 March 2010 (UTC)

I don't think there's a standard term for such functions. How are you defining differentiability of functions on arbitrary metric spaces, anyway? Algebraist 20:06, 4 March 2010 (UTC)

Differentiability is defined as a limit involving three operations only: adding, substracion, and the absolute value, which is defined wherever a distance is defined. Hence, Differentiability is easily defined in every topological metric space, since such a space is equipped with a well defined distance. HOOTmag (talk) 21:30, 4 March 2010 (UTC)

? not every topological space has a distance function, e.g. non-Hausdorff spaces can't be metric, and not every metric space is a vector space so addition and subtraction may not make sense. If your function is from A to B you will need to be able to add and subtract elements of B, and to multiply by real scalars, thus some extra structure beyond that of a metric space is needed. In answer to your original question, if you're looking for a name for a function which is differentiable and whose domain is connected an open, I don't think there is one. 129.67.37.143 (talk) 23:03, 4 March 2010 (UTC)

And you can define multiple compatible metrics on the same topological space. A domain is, under the usual definition of a function, just a set. There is no additional structure imposed on it. It is not considered to have a topology (which is necessary for either connectedness or openness to make sense) or to be a subset of another set (which is necessary for openness to much sense). If we're talking about functions on topological spaces (in which case, we're probably talking about continuous functions) then the domain would usually be considered to be always open since any topological space is (by definition) open in itself. --Tango (talk) 05:01, 5 March 2010 (UTC)

I'm talking universally about any metric space which is also a group. Is there a universal term describing the fact that a given function is differentiable in its connected open domain - given that the topolgical space is metric and constitutes a group? HOOTmag (talk) 11:58, 5 March 2010 (UTC)

How is being a group going to help you? The definition of a derivative, $f'(x)=\lim _{y\to x}{\frac {f(y)-f(x)}{y-x}},$ calls for at least subtraction, division and topology (T₂, if we want the limit to be unique) so you need something like a topological division ring. Better make it a topological field so that we don't have to worry whether to use left or right division in the definition.—Emil J. 14:10, 5 March 2010 (UTC)

You've forgotten to add the absolute value.

So, yes, I've meant "a metric field".

HOOTmag (talk) 18:46, 6 March 2010 (UTC)

Where is absolute value used? Perhaps you were referring to the fact that the topology of the reals can be defined in terms of absolute value, and so the classical definition of limit uses it. But there is no need for absolute value\distance on top of Emil's requirements. -- Meni Rosenfeld (talk) 19:38, 6 March 2010 (UTC)

Yes, so is there a universal term describing the fact that a given function is differentiable in its connected open domain - given that the topolgical space is a field? HOOTmag (talk) 20:11, 6 March 2010 (UTC)

Ring Z[sqrt(2)]: complete set of units

Hi all.

I'm wondering why it is that $\pm (1\pm {\sqrt {(}}2))^{n}$ is the complete set of units in the ring $\mathbb {Z} [{\sqrt {2}}]$ - with the norm |a+bsqrt(2)|, we must have $a^{2}-2b^{2}=\pm 1$, and I can see easily why the above values are solutions to this equation - but why are they necessarily all the values? I did some fiddling around with modular values, but I haven't really managed to get anywhere with it successfully. I thought perhaps there would be a nice argument looking at the absolute value of $a+b{\sqrt {2}}$ - for example, something about the fact that if (a+bsqrt(2)) was a unit, then (a+bsqrt(2))(1±sqrt(2)) is also a unit, because it's a product of 2 units, and then since |1+sqrt(2)|>1,|1-sqrt(2)|<1, we can just multiply arbitrarily many times by these until we eventually get a modulus of something less than 1: in this case, we must have reached a point in (1,3), and then since there's only one unit in [1,3], and we must at some point have jumped from >1 to <1 in modulus, before the jump we must have had modulus at most 1+sqrt(2) (since dividing by that made the value <1) and clearly >1, thus ${\frac {a+b{\sqrt {2}}}{(1+{\sqrt {2}})^{n}}}$must have been equal to 1+sqrt(2), and hence result: the same argument works if our initial modulus of a+bsqrt(2) <1 too - but I couldn't see how to prove the crux of the argument, i.e. that the only x>1, <3 which solves $a^{2}-2b^{2}=\pm 1$ is 1+sqrt(2).

Could anyone help, or suggest a better argument? Spalton232 (talk) 14:58, 3 March 2010 (UTC)

Use the value of b (or a) instead of the absolute value of $a+b{\sqrt {2}}$. If $a^{2}-2b^{2}=\pm 1$ and $a,b>0$ (the cases with a or b negative can be reduced to this one), then it is easy to see that $b\leq a<2b$. Putting $a'+b'{\sqrt {2}}=(a+b{\sqrt {2}})/(1+{\sqrt {2}})=(2b-a)+(a-b){\sqrt {2}}$, this means that $0\leq b'<b$. Iterating this process eventually brings the solution down to b = 0, which gives a = 1, i.e., $a+b{\sqrt {2}}=(1+{\sqrt {2}})^{0}$.—Emil J. 15:25, 3 March 2010 (UTC)

There may be more info in the Pell's equation article.—Emil J. 15:31, 3 March 2010 (UTC)

That's been a great help, very nice argument too, thankyou! :) Spalton232 (talk) 15:44, 3 March 2010 (UTC)

Look at Dirichlet's unit theorem, which characterizes the group of units of the algebraic integers of any number field as the group generated by the roots of unity and $r=r_{1}+r_{2}-1$ multiplicatively independent elements, where $r_{1}$ is the number of real embeddings and $r_{2}$ is the number of complex embeddings. In this case, we have 1 and -1 as the roots of unity, and just one generator: $1+{\sqrt {2}}$. (Note that $1-{\sqrt {2}}=-(1+{\sqrt {2}})^{-1}$) If you list the embeddings as $\left(\sigma _{1},\ldots ,\sigma _{r_{1}},\rho _{1},{\bar {\rho }}_{1},\ldots ,\rho _{r_{2}},{\bar {\rho }}_{r_{2}}\right)$ and consider the mapping from the number field to $\mathbb {R} ^{r_{1}+r_{2}}$ defined by $\left(\log(|\sigma _{1}|),\ldots ,\log(|\sigma _{r_{1}}|),\log(|\rho _{1}|),\ldots ,\log(|\rho _{r_{2}}|)\right)$, then the units map to a lattice on the hyperplane $z_{1}+z_{2}+\ldots +z_{r_{1}+r_{2}}=0$, since the units have norm plus or minus 1. Once you have a set of generators of full rank, you can search a fundamental domain of this lattice for any "smaller" units you missed, which is a finite search among the algebraic integers. I remember doing this for a take-home exam problem several years ago, and it was very messy. EmilJ's solution is very nice, but there's a "brute force" approach, too. 146.186.131.95 (talk) 17:01, 3 March 2010 (UTC)

Ah, I did spot a page online saying to use Dirichlet's unit theorem, but I fear it may be slightly above my level right now! Perhaps by June (exam time!), it might make a little more sense to me - either way, I'll certainly come take a look at it then, could be handy! Many thanks again to both of you. Spalton232 (talk) 20:38, 3 March 2010 (UTC)

Casual language usage, or is there an example?

In Axiom#Non-logical_axioms, there is a sentence that starts out "Almost every modern mathematical theory starts from a given set of non-logical axioms,..." Does the word Almost not technically belong there, or is there an example of a modern mathematical theory which does not start from a given set of non-logical axioms? 20.137.18.50 (talk) 17:57, 3 March 2010 (UTC)

I think it depends on what you mean by "theory" and "starts with". Arguably many theories don't "start with" axioms at all — they start with an intuitive conception of the objects being described, and axioms are chosen that intuitively appear to be true (or even, that are found to be true via evidence) of those objects. --Trovatore (talk) 18:47, 3 March 2010 (UTC)

In mathematical logic, the theory of "pure equality" is a first-order theory in which:

• The signature has no function symbols, no constant symbols, only the equality relation symbol
• There are no non-logical axioms at all, if as usual we view the axioms for the equality relation as axioms of logic.

— Carl (CBM · talk) 20:16, 3 March 2010 (UTC)

Well, that's not really a counterexample — the set of non-logical axioms there is just the empty set. The quoted statement is problematic for other reasons, though, as I alluded to in my response. --Trovatore (talk) 20:35, 3 March 2010 (UTC)

Obviously the quoted sentence didn't intend to include the empty set as a set of non-logical axioms. I am happy with the theory of pure equality as a formal theory that "does not start from a given set of non-logical axioms", even though one can mince words about the empty set

It's true that theories are motivated by their intended interpretation. But I think everyone agrees that a first-order theory qua first-order theory is defined by specifying both a signature and a set of axioms for the theory. — Carl (CBM · talk) 20:44, 3 March 2010 (UTC)

Well, "a first-order theory qua first-order theory" is just a set of sentences (maybe closed under logical consequence, maybe not, depending on your convention). You may or may not call out some of those sentences specially as "axioms". --Trovatore (talk) 20:50, 3 March 2010 (UTC)