Wikipedia:Reference desk/Archives/Mathematics/2007 April 22

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.

April 22

Math Songs

I wasn't sure where to post this, and I decided the Math desk would be marginally the better choice. I know of, at the moment, three really good sources of math-based music: [1], [2], and [3]. Can anyone suggest anything else? —The preceding unsigned comment was added by Black Carrot (talk • contribs) 08:23, 22 April 2007 (UTC).

Sorry I didn't sign that. Here's one other I didn't have a link to. [4] Black Carrot 09:23, 22 April 2007 (UTC)

Tom Lehrer wrote a couple of math songs, but his math-related ones tend to be more rare than the rest. One of his most famous ones has its own article, New Math (song). Maelin (Talk | Contribs) 12:27, 23 April 2007 (UTC)

Cubic Functions

Is there anything analoguous to conics for higher degree polynomials (e.g. conics are to second degree polynomials as ... are to third degree polynomials)? --AMorris (talk)●(contribs) 09:07, 22 April 2007 (UTC)

Degree 4 = quartic; degree 5 = quintic; degree 6 = sextic. Gandalf61 09:33, 22 April 2007 (UTC)

It depends on what you expect. A degree 2 polynomial in one (or more) variables is called a "quadratic", as in the famous quadratic formula. A degree 3 polynomial is a cubic. But conics, or conic sections, are special. Every degree 2 polynomial in two variables, x and y, implicitly defines a curve in the projective plane that is a conic section. With one variable the graph of the function is always a parabola, and with three variables we speak of quadrics (or quadric surfaces).

Elliptic curves for j=1728

Curves of higher degree acquire complications. The typical cubic curve, which is nonsingular, has no parametric form, and is called an elliptic curve for historical reasons. (These curves are not ellipses!) Every real-valued elliptic curve is equivalent to either of

${\begin{aligned}y^{2}&=x(x-1)(x-\lambda );\qquad &0&<\lambda <1\\y^{2}&=x(x^{2}-2\lambda x+1);\qquad &-1&<\lambda <1\end{aligned}}$

by a linear transformation. Among these, the pair of curves

${\begin{aligned}y^{2}&=x^{3}-x\\y^{2}&=x^{3}+x\end{aligned}}$

have a unique status, having to do with the number 1728 (which is 12³) and some very deep mathematics. (The final route to the proof of Fermat's Last Theorem involved elliptic curves in a subtle and surprisingly beautiful way.)

The cubic Bézier curves which are so popular in computer graphics do not fall into one of these two families of elliptic curves. In fact, simply knowing that they are parametric curves tells us that they must be singular as algebraic curves. Thus somewhere on the curve a magnified view does not look like a line.

Consider the curve with Bézier control points (−3,−9), (1,−3), (−1,−3), (3,−9). As an implicit curve, it is

$0=81x^{2}+9y^{2}+2y^{3}.\,\!$

The singular point does not appear on the parametric curve; it is at the origin, (0,0). Using lines through this acnode, we can find every other curve point's parameter

$t={\frac {y-3x}{2y}}.$

Thus we see a profound difference between degree 2 and everything higher. Bézout's theorem tells us a line will intersect a curve as many times as the degree, if we count multiplicity. For degree 2, we can fix any point on the curve and use the family of lines through it, each of which gives a second intersection point, and thus a parameterization. For degree three, we must have a "double point" — a singularity — to accomplish the same thing. And the nonsingular cubics, the elliptic curves, have extraordinary properties not shared by curves of higher degree. --KSmrq^T 16:30, 22 April 2007 (UTC)

Polynomial sequence

The generating function for the n^th powers of the positive integers has the form P_n(x)/(1-x)ⁿ⁺¹, where P_n(x) is a polynomial of degree n. In other words,

${\frac {P_{n}(x)}{(1-x)^{n+1}}}=\Sigma _{k=1}^{\infty }k^{n}x^{k}$

The first few polynomials in the sequence P_n are:

$P_{1}=x$

$P_{2}=x^{2}+x$

$P_{3}=x^{3}+4x^{2}+x$

$P_{4}=x^{4}+11x^{3}+11x^{2}+x$

$P_{5}=x^{5}+26x^{4}+66x^{3}+26x^{2}+x$

and the P_n polynomials satisfy the recurrence relation:

$P_{n}=x(1-x){\frac {dP_{n-1}}{dx}}+nxP_{n-1}$

Does anyone know if there is a name for this polynomial sequence ? I have searched the polynomial sequence articles here and at Mathworld, but I haven't found this particular sequence. Gandalf61 09:29, 22 April 2007 (UTC)

This [5] comes up on OEIS. Just skimmed it. Eulerian polynomials? --87.194.21.177 10:52, 22 April 2007 (UTC)

Ah, yes, thank you - the co-efficients of these polynomials are the Eulerian numbers as decribed here and here ... but not, apparently, currently in Wikipedia, because "Eulerian number" redirects to Euler number, which is not the same thing. I feel an article coming on ... Gandalf61 14:24, 22 April 2007 (UTC)

Another way to describe the g.f. of n^m is

$\sum _{n=0}^{\infty }n^{m}x^{n}=(xD)^{m}{\frac {1}{(1-x)}}.$

where the xD operator is the application of differentiation followed by multiplication with x. (xD)^m means m-times application of it. A generalization would be

$\sum _{n=0}^{\infty }c^{n}n^{m}x^{n}=(xD)^{m}{\frac {1}{(1-cx)}}.$

--85.179.17.199 16:10, 22 April 2007 (UTC)

As promised, I have now created an article on Eulerian numbers. Gandalf61 16:05, 23 April 2007 (UTC)

USB symbol

Is the USB (Universial Serial Bus) symbol a mathematical symbol? Or at least resembles one? Thanks very much for your response. --Mayfare 15:27, 22 April 2007 (UTC)

It resembles loosely both lowercase epsilon (ε) and capital sigma (Σ), used respectively to denote an infinitesimal quantity and show the process of summation. But more or less any shape can be matched to a symbol in some alphabet.…217.43.211.203 15:40, 22 April 2007 (UTC)

Besides for looking like an epsilon and sigma, I don't think the symbol means anything other than the device can connect with triangles, squares, and circles, meaning it's made to be universal. --Wirbelwindヴィルヴェルヴィント (talk) 19:08, 22 April 2007 (UTC)

In a certain (pointing upward) orientation, it also resembles the letter psi, which may be used for various mathematical things. It also resembles the Unicode Mathematical Operators #22D4 (points downward), #22F2 (points to the right), and #22FA (points to the left). --Spoon! 19:29, 22 April 2007 (UTC)

Aha! It's a metro map! —Bromskloss 22:56, 22 April 2007 (UTC)

its obviously the symbol of.... a data bus. Not anything to do with a math symbol.--Dacium 04:30, 27 April 2007 (UTC)

Quadratic Equations

If I want to find the roots of a quadratic equation with the quadratic formula, does C always need to be positive? I've noticed that if C is negative, the discriminant will be negative but we will not cover complex numbers (i) in this school unit.

For example, 2x^2 + 14x - 240 would have to be multiplied by -1 to -2x^2 - 14x + 240 = 0 to work in the quadratic formula.

Thank you very much!

Vertciel 18:48, 22 April 2007 (UTC)

The discriminant is given by $b^{2}-4ac$ - this will always be greater than zero provided b² > 4ac, which can be achieved for any variety of numbers. In the case of a negative c, a positive a will ensure a positive discriminant. Try for a variety of signs and magnitudes of a, b and c, and you'll see what I mean. Icthyos 19:01, 22 April 2007 (UTC)

Indeed, there seems to be some confusion. For the given polynomial,

$2x^{2}+14x-240,\,\!$

we have

$a=2,\quad b=14,\quad c=-240.\,\!$

Thus the discriminant is

${\begin{aligned}b^{2}-4ac&=(14)^{2}-4(2)(-240)\\&=196+1920\\&=2116,\end{aligned}}$

which is positive; no complex numbers are needed.

If you think better with pictures than with equations, consider this: The graph of a quadratic equation is a parabola. Since x² is always positive, the arms rise up if a is positive, and they drop down if it is negative. The effect of adding the constant c is to shift the curve higher or lower. So what does b do? It shifts the curve diagonally (both horizontally and vertically). We are trying to find where the curve crosses zero (the height of the x axis), and we have three possibilities.

• Perhaps the apex of the parabola just touches the x axis. (This is unlikely.)
• The x axis slices through both arms.
• The curve lies entirely above or entirely below the x axis.

The discriminant is zero, positive, or negative, depending on which case we have. In an example like x²+1, the arms rise up, the curve is shifted up, and there is no diagonal shift; thus the parabola lies entirely above the x axis, consistent with a discriminant of −4.

Teaser: A zero of x²+1 would have to be a quantity, i, that squares to −1. Since both positive and negative real numbers have positive squares, i must be something new and different. Thus we meet complex numbers, one of the most fruitful discoveries in the history of mathematics. --KSmrq^T 21:20, 22 April 2007 (UTC)

uploading of math diagrams

How to upload simple mathematical diagrams of lines and curves, etc. on the wiki page from word documents with such diagrams?--Profvk 23:22, 22 April 2007 (UTC)

You need to export these as images (PNG or SVG, only pictures should be saved as JPEG) and Upload them before adding to the article. For any further help, just ask the guys at the Help Desk. — Kieff | Talk 23:46, 22 April 2007 (UTC)

See Help:Images and other uploaded files for more details. And to be more precise, JPEG is a lossy compression format best reserved for natural photographs. Sharp line drawings and other graphic art are best presented in SVG format, which is not only highly efficient, but also scales cleanly to any size. The function article contains a typical example, Image:Graph of example function.svg. But as Kieff suggests, questions like this about Wikipedia itself are more appropriate to the Help Desk. --KSmrq^T 02:47, 23 April 2007 (UTC)