Wikipedia:Reference desk/Archives/Mathematics/2008 November 17

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November 17

Polynomials

Are things like the mod function or the floor function allowed in polynomials? --Melab^±1 01:37, 17 November 2008 (UTC)

Well, a polynomial is just a function of the form $\sum \limits _{n}a_{n}x^{n}$ - you get a polynomial mod p quite often in various branches of mathematics, but such functions aren't really polynomials themselves, so not really. -mattbuck (Talk) 02:19, 17 November 2008 (UTC)

The answer is definitely not (for polynomials as in algebra). See the page polynomial for the formal definition.

Topology Expert (talk) 05:48, 17 November 2008 (UTC)

Polynomials over the integers mod p are polynomials, you can have polynomials over any ring. I don't think that's what the OP is talking about, though. --Tango (talk) 11:32, 17 November 2008 (UTC)

I have seen this term being used so often. What does 'OP' mean?

(My answer assumed that the person who asked the question was referring to calculus).

Topology Expert (talk) 11:43, 17 November 2008 (UTC)

OP here stands for original poster. Algebraist 11:51, 17 November 2008 (UTC)

It's hard to determine what the question really means, but if the OP means whether it's possible to define a mod function on a polynomial ring analogous to one of the possible mod functions in the ring of integers, then sure, why not? Polynomial division is even nicer than integer division as the remainders are made unique more naturally. Similarly, if we want to define floor as a function from rational expressions to polynomials, analogous to the floor mapping Q to Z, then why not? -- Jao (talk) 12:03, 17 November 2008 (UTC)

How would you define floor on rational expressions? Algebraist 13:27, 17 November 2008 (UTC)

Expand as a Laurent polynomial and truncate the negative order terms? That seems analogous to taking a decimal expansion and truncating it at the decimal point. It's going to depend on what point you expand around, of course. --Tango (talk) 15:56, 17 November 2008 (UTC)

Hang on, that could give a power series rather than a polynomial, couldn't it? I guess it's more analogous to taking the fractional part of a rational number. There are lots of analogies between (Laurent) polynomials and integers (rational numbers), but you have to reverse the order in some sense (positive powers of X in a (Laurent) polynomial correspond to negative powers of 10 in the decimal expansion of the rational number). --Tango (talk) 16:01, 17 November 2008 (UTC)

I had simply thought to define $\scriptstyle {\big \lfloor }{\frac {f(x)}{g(x)}}{\big \rfloor }$ as the polynomial $\scriptstyle q(x)$ such that $\scriptstyle f(x)=q(x)g(x)+r(x)$ with $\scriptstyle \deg r(x)<\deg g(x)$. I don't know how much sense it would make to call it floor, or how useful it would be, but there's at least a clear analogy with the Q → Z floor function. -- Jao (talk) 17:21, 17 November 2008 (UTC)

No not in Calculus or rings an 8th grade definition of a polynomial. —Preceding unsigned comment added by Melab-1 (talk • contribs) 21:56, 17 November 2008 (UTC)

Could you give an example of what you mean, then? I suspect the answer is "no". A polynomial is just something of the form a+bx+cx²+...+zxⁿ, where all the a,b,c,...,z are numbers (usually rational numbers if it isn't specified), anything else, and it isn't a polynomial. --Tango (talk) 00:50, 18 November 2008 (UTC)

I like Jao's idea. Say we are in the real coefficients case. His q is also the only polynomial such that $\scriptstyle {\frac {f(x)}{g(x)}}=q(x)+o(1)$ as $|x|\to \infty$, thus it is a kind of closest integer approximation in this context. One could ask why q should be named the floor and not the ceiling part of $\scriptstyle {\frac {f(x)}{g(x)}}$: in fact I would call it just the "closest polynomial at infinity"; if we really want a floor, that is, the greatest integer thing below $\scriptstyle {\frac {f(x)}{g(x)}}$, then I would vote for $\scriptstyle {\Big \lfloor }{\frac {f(x)}{g(x)}}{\Big \rfloor }$ to be defined as $\scriptstyle q(x)+\inf _{x\in \mathbb {R} }{\frac {r(x)}{g(x)}}$. --PMajer (talk) 18:48, 18 November 2008 (UTC)

Matrix

What are matrices really for? The article on wikipedia doesn't really explain that. ----The Successor of Physics 13:22, 17 November 2008 (UTC) —Preceding unsigned comment added by Superwj5 (talk • contribs)

Well, they have many uses. Possibly the simplest use is in geometry. Any n-dimensional transformation can be written as an n by n matrix T, so if you want to find out what happens when you apply the transformation to some vector, you take T(v) and multiply out the matrices to get your new vector. -mattbuck (Talk) 13:53, 17 November 2008 (UTC)

Linear transformation. --Tango (talk) 15:52, 17 November 2008 (UTC)

Matrices can be used to model real-world data as well. For example, if you have three types of shoes at a store, and want to model this for men and women, you could have a 3x2 matrix showing how many men and women bought each type of shoe. Then, to this 3x2 matrix, matrix operations can be applied to model different things. For example, you could have a matrix showing the cost for each type of shoe (perhaps a 1x3) and then multiply this matrix by your original matrix to give the total money men and women spent. Hope this helps, Ζρς ι'β' ^¡hábleme! 00:07, 18 November 2008 (UTC)

Markov chains are pretty sweet (see the subsection about Markov chains with finite state space). 76.126.116.54 (talk) 05:17, 18 November 2008 (UTC)

Jacobian matrix (or even the related Hessian matrix)

Topology Expert (talk) 03:35, 20 November 1991 (UTC)

Matrices are the natural transformations of vector spaces, so they show up in the same places. The world is a vector spaces, so they're used in physics. A lot of algebraic systems are vector spaces over some smaller algebraic system, so they're used in algebra. Black Carrot (talk) 21:03, 20 November 2008 (UTC)

A very important point: I can name zillions (no joke) of applications of matrices in both maths and physics (the real world example given by User:Zrs_12 also illustrates how natural matrix multiplication is. Of course, they are also natural in pure maths in many ways). For example, see rank (mathematics) or even the best page to look would be matrix (of course!).

Topology Expert (talk) 05:44, 22 November 2008 (UTC)

Mathmatic

The perimeter of a rhombus is 100 cm if one of its diagonals is 14 cm then the area of the rhombus is? —Preceding unsigned comment added by Nebul (talk • contribs) 14:59, 17 November 2008 (UTC)

This looks like homework to me, so I'll suggest looking at Rhombus#Area, and leave it at that. --Tango (talk) 15:46, 17 November 2008 (UTC)

Basic mechanics

It's a long time since I studied this, so wonder if I've got it right. Does statical equilibrium correspond to the position of lowest potential energy? Obviously true for a weight on a string or a ball at the bottom of a bowl, but it seems to work for more complicated systems as well. For example, A and B are fixed points at the same height, A has a light rod freely pivoted at it, B has a light string attached to it. The string passes through a smooth eye at the free end of the rod and has a weight at the end. The system will be in equilibrium when the two portions of string make equal angles with the rod and, using a numerical case to simplify things, some trig and calculus gave exactly the same position of the rod for the weight to be at its lowest. So if I'm right, equilibrium can be determined either by resolving forces/taking moments, or by minimising potential energy. In a particular case, could one way be simpler than the other?…81.154.108.92 (talk) 15:12, 17 November 2008 (UTC)

Stable equilibria correspond to *local* minima of potential energy (unstable equilibria correspond to local maxima, or inflection points), it doesn't necessarily have to be the overall minimum. Finding the potential energy and differentiating (twice, to check stability) can often be much easier than resolving forces, etc., especially for more complicated systems (you can use conservation of energy, so you know if kinetic energy is at a maximum, potential energy must be at a minimum, for example). --Tango (talk) 15:51, 17 November 2008 (UTC)

Agreed. Now, for a real-word example: A bowl of cereal on the table is in equilibrium, even though it would have less potential energy when knocked on the floor. The reason, as stated above, is that you would need to momentarily increase the potential energy to lift the cereal out of the bowl. StuRat (talk) 18:04, 17 November 2008 (UTC)

Real Analysis: Inverse of continuous function of interior set

Hi, can anyone please help with solution for this problem from:

Considering a function f:R->R, how can we show that f is continuous iff for every subset B $f^{-1}$(int(B) ⊆ int($f^{-1}$(B). Thank You. —Preceding unsigned comment added by 130.166.160.76 (talk) 21:06, 17 November 2008 (UTC)

This characterisation of continuity (which works for arbitrary topological spaces, not just R), is fairly easily shown to be equivalent to the standard definition of continuity for functions on top spaces. It's even mentioned in our article continuous function (topology). Algebraist 21:12, 17 November 2008 (UTC)

Emacs + LaTex on Windows XP

Any LaTeX head fancy helping this person out:

http://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Computing#Setting_up_LaTex —Preceding unsigned comment added by 78.86.164.115 (talk) 21:16, 17 November 2008 (UTC)