Wikipedia:Reference desk/Archives/Mathematics/2011 March 30

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 30

Wing (insect)

If someone can help me and check the math on the page, Wing (insect), under the the subsections Hovering and Elasticity. In the previous discussion, there was a mention of a few mistakes, but I'm not a mathematician (note all the equautions came from a book, could they make mistakes, or don't they have someone check these things?). Another thing is that this pdf gives other equations for what seems to be the same thing, which one is correct? What exactly is one explaining and the other not? Bugboy52.4 ¦ =-= 02:40, 30 March 2011 (UTC)

Definition of a polynomial

What is the exact formal definition of a polynomial? 95.112.199.61 (talk) 13:26, 30 March 2011 (UTC)

There are several more or less equivalent ones, depending on who you ask, and what they are doing when you ask them. Our article Polynomial discusses various options. –Henning Makholm (talk) 13:28, 30 March 2011 (UTC)

If I had found one in the article I wouldn't have asked here. 95.112.199.61 (talk) 13:31, 30 March 2011 (UTC)

The very first statement is a definition and the second sentence gives an example and a non-example. What is wrong with that for you? Dmcq (talk) 13:38, 30 March 2011 (UTC)

It is not clear from the context (to me, at least) that this is a formal definition and not only informal description. If this really is the definition then my followup question is: when are 2 polynomials equal? 95.112.199.61 (talk) 13:54, 30 March 2011 (UTC)

If you think the definition in the first sentence of the article is too informal, there is a more formal definition outlined in the Abstract algebra section. Basically, you can define a polynomial over a ring $\mathbf {R}$ as a member of $\mathbf {R} ^{\infty }$ in which only a finite number of coefficients are non-zero. You then define addition and subtraction of polynomials using pointwise addition and subtraction, and the product of two polynomials using discrete convolution. Two polynomials P and Q are then equal if P - Q = (0,0,0,...) Gandalf61 (talk) 14:06, 30 March 2011 (UTC)

It is not only too informal, it even looks wrong to me, as the two expressions 1x+3x and 2x+2x are clearly different when seen as expressions. The definition in the abstract algebra section is much better, but only defines polynomials in one variable. 95.112.199.61 (talk) 14:18, 30 March 2011 (UTC)

Well, the correct place to raise issues about the content of the polynomial article is at Talk:Polynomial. There has been considerable discussion there in the past about the lead section. Overall consensus was that the lead is not the right place for a formal definition, as it reduces the accessibility of the article for the general reader. But by all means kick off a new debate over there if you want to. Gandalf61 (talk) 14:36, 30 March 2011 (UTC)

I had a look there, and you would be right if not for Talk:Polynomial. But at least I can see the problem more clearly now. 95.112.199.61 (talk) 14:51, 30 March 2011 (UTC)

What's wrong with saying x+x is a polynomial? I would have considered the R^∞ definition as giving all the different polynomial values with real coefficients rather than all the possible forms of expression. Dmcq (talk) 15:22, 30 March 2011 (UTC)

x+x is an expression that represents a polynomial. 3x-x is a different expression representing the same polynomial. In colloquial language it is perfectly OK to say x+x is a polynomial but as a formal definition this gives raise to misunderstanding. You can see the difference when you try to decide (by proof, not by consensus) if these "polynomials" are equal or not. And $\mathbf {R}$ does by no means stand for "real numbers" but for an arbitrary ring. 95.112.199.61 (talk) 15:44, 30 March 2011 (UTC)

I think there is nothing wrong at all with saying x+x is a polynomial - as Henning Makholm says below, there is no single "right" definition of a polynomial. If you want to define x + x and 2x as being distinct but equal polynomials then that is fine. The abstract algebraic approach is somewhat cleaner and simpler - for example, it sidesteps the problem of explaining why 2x and 2y are equal as polynomials - but that doesn't make it automatically right. Gandalf61 (talk) 15:48, 30 March 2011 (UTC)

Thank you for that "outdent", I didn't know that and it is very useful. If the many possible definitions of polynomials were not equivalent then the polynomials from the different definitions would have different properties and if so, they should be clearly distinguished one from the other so people can see what the other one is talking about. And 2x and 2y are not equal because the variable is different. $\mathbf {R} [x]$ and $\mathbf {R} [y]$ are isomorphic (and I should say here by what isomorphism) but not equal, otherwise x-y as element of $\mathbf {R} [x,y]$ would be zero. 95.112.199.61 (talk) 16:14, 30 March 2011 (UTC)

Ah, well, if $\mathbf {R} [x]$ and $\mathbf {R} [y]$ are different polynomial rings then you have to explain how you can subtract a member of one ring, $\mathbf {R} [y]$, from a member of another ring, $\mathbf {R} [x]$, and get as a result a member of a third ring $\mathbf {R} [x,y]$. Yes, I know you can do this by establishing an isomorphism between elements of the first two rings and elements of the third ring. But the abstract algebraic approach is much cleaner here because it clearly distinguishes x in $\mathbf {R} [x]$ - which is (0,1,0,0,...) - from x in $\mathbf {R} [x,y]$ - which is ((0,1,0,...), (0,0,0,...), (0,0,0,...),...). And it shows that x in $\mathbf {R} [x]$ and y in $\mathbf {R} [y]$ are indeed equal because they are both (0,1,0,0,...). Gandalf61 (talk) 18:19, 30 March 2011 (UTC)

By assuming that my x and my y are both in $\mathbf {R} [x,y]$ from the start and that both, $\mathbf {R} [x]$ and $\mathbf {R} [y]$ are subrings of the former. Well I should have said this at the beginning of my statement. This is why all mathematics starts with something like "let x, y be elements of Z". And I know that the abstract approach is much cleaner, but wikipedia gives another, different and not equivalent definition at the start of the article. 95.112.199.61 (talk) 18:30, 30 March 2011 (UTC)

(ec) It is not "the" defintion (there is no such thing); it is a definition. As I said, there are several possible definitions; that first sentence is one of them. Another possible definition is to require the expression to have a particular standard form; a third possibility is to say that a polynomial is simply a finite list (or a finitely supported countable list) of coefficients drawn from some ring with certain operations defined on the set of such lists.

One can define various equality relations between polynomials, with the details of the definition of course depending on how one chooses to define polynomials. Under the definition at the beginning of Polynomial, what one usually means by equality as polynomials is that the two expressions can be proved to be equal using only the ring axioms and computations within the ring that does not involve variables. This corresponds to requiring that all coefficients are equal when the polynomial has been reduced to canonical form. –Henning Makholm (talk) 14:21, 30 March 2011 (UTC)

When I see R[x] that looks like the ring R with a symbolic element x adjoined, something like a Galois extension of a field. Polynomials would then just be elements of the extension ring, and polynomial equality is equality in the extension ring. I haven't looked at the article or it definition, but the above seems like a natural and adequate way to conceptualize polynomials (maybe I'm missing something). Perhaps there's some fancier way (using categories or something) to formalize the notion of adjoining a symbol to a ring. 75.57.242.120 (talk) 19:24, 30 March 2011 (UTC)

Abstractly it's the monoid ring of R over the monoid $(\mathbb {Z} _{\geq 0},+)$ (here the symbol "x" is invisible). When we identify the monoid with the set $\{1,x,x^{2},\dots \}$ under the usual multiplication, we obtain the more familiar "adjoining a formal symbol" interpretation of the polynomial ring. This is sometimes called the free algebra on the single element "x". See, for instance, Bourbaki's Algebra, III.2.7 and III.2.9. Sławomir Biały (talk) 19:42, 30 March 2011 (UTC)

Personally I prefer the straightforward definition in terms of the form like x+x and then go on from there. If you go by value you've got to deal with all the problems of if for instance x³=x+1 is an identity before you've even started thinking about them. Dmcq (talk) 19:47, 30 March 2011 (UTC)

75's description is a good intuition, and my understanding is that the similarity between the notation R[X] for a general polynomial ring and, say, Z[i] for the Gaussian integers is deliberate in order to reinforce this intuition. I think it works less well as a definition, because "polynomials are just elements of the extension ring" simply punts to the problem of defining what that extension ring is in the first place.

I think I have seen a categorical definition-up-to-isomorphism of polynomial rings by making the substitution homomorphisms the unique solutions to a universal property, but damned if I can remember (or reconstruct) the details. –Henning Makholm (talk) 20:55, 30 March 2011 (UTC)

Row echelon form

What are the practical uses of row echelon form (which I barely remember studying back in high school but, for some reason, was thinking about today)? Kansan (talk) 16:22, 30 March 2011 (UTC)

When a system of equations is in row echelon form, you can determine whether the system has a solution, if that solution is unique, and if not what the dimension of the space of solutions is (is it a line, plane, etc.) Moreover, it is very easy to solve the system explicitly from the row echelon form (using back substitution). When the system is put into the related reduced row echelon form, it actually is solved, and the space of solutions can be explicitly parametrized in terms of some free variables. Sławomir Biały (talk) 19:13, 30 March 2011 (UTC)

Thank you. Kansan (talk) 19:41, 30 March 2011 (UTC)

Geometry term ?

When to lines are almost parallel but never cross but infinity closer together never intersect? The Resident Anthropologist (talk)•(contribs) 17:50, 30 March 2011 (UTC)

Do you mean asymptote? 75.57.242.120 (talk) 18:03, 30 March 2011 (UTC)

Yes thank you The Resident Anthropologist (talk)•(contribs) 18:08, 30 March 2011 (UTC)