Wikipedia:Reference desk/Archives/Mathematics/2009 September 10

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September 10

Weaker variant of Schur for sum of two = irreducibles: two eigenvalues?

Hello,

I was considering endomorphisms of \mathbb{C} G- modules, and something funny happened with the eigenvalues. I was wondering if this were true:

If V is the \mathbb{C} G-module V with V=A+B, with A and B two isomorphic irreducible modules, then every endomorphism f has at most two complex eigenvalues. Moreover, every vector v\in V is either an eigenvector of f, or is in an invariant subspace with respect to f of dimension two.

It looks like it true, but I've never seen it before so that is why I am in doubt.. I was also wondering if there were more general things like this that are known and very important.

Many thanks, Evilbu

Homomorphisms behave nicely with respect to direct sum decompositions. In your case, where V is the direct sum of two copies of A, End(V) is isomorphic to the 2x2 matrix ring with entries in End(A). Since you work over C, this is really the 2x2 matrices over C. Now at least it's obvious that there can be at most two eigenvalues. Tinfoilcat (talk) 16:46, 10 September 2009 (UTC)

maths?

what is maths —Preceding unsigned comment added by 59.96.142.2 (talk) 17:05, 10 September 2009 (UTC)

math is using numbers to get new numbers.Accdude92 (talk) (sign) 17:07, 10 September 2009 (UTC)

Mathematics is the science and study of quantity, structure, space, and change. Mathematics is both the queen and the handmaiden of all science. --LarryMac | Talk 17:42, 10 September 2009 (UTC)

I'd say she's also science's drinking buddy, confidant, and occasional tennis doubles partner. -GTBacchus^(talk) 17:22, 11 September 2009 (UTC)

Mathematics is an esoteric discipline whose higher reaches are guarded jealously by those in the know so that it is hard for new entrants to join them. Thus in some ways mathematics, like other learned professions, provides a good living for its practitioners.86.152.79.120 (talk) 22:33, 10 September 2009 (UTC)

That is the opposite of the truth. Lots of popularizations get written by mathematicians, but far fewer people are willing to read them than mathematicians might wish. Michael Hardy (talk) 21:00, 11 September 2009 (UTC)

Is it possible for "maths" not to use numbers and still be properly considered as maths? 202.147.44.84 (talk) 22:30, 10 September 2009 (UTC)

Yes. Algebraist 22:33, 10 September 2009 (UTC)

Yes. Topology and geometry don't involve a great deal of numbers (they pop up from time to time, but mostly just to index things). --Tango (talk) 22:43, 10 September 2009 (UTC)

Someone has to properly ingrain the idea in humans' minds that mathematics is not solely about numbers, equations or formulas. Although number theory is a branch of mathematics, it is not concerned with "the arithmetic of big numbers" - such operations may be done with a computer (some cannot but that is irrelevant to mathematicians). Furthermore, although mathematics does deal with equations in a vague sense (see algebraic geometry, field theory and algebraic number theory), a mathematician is not really concerned with "solving equations" but rather how the solution sets behave. In algebraic geometry, one is concerned with the "geometric" properties of solution sets; in field theory, one is sometimes concerned with algebraic extensions of fields (an algebraic extension of a field (F) is simply another field containing F whose every element is the solution of a non-zero polynomial with coefficients in F). For instance, a non-mathematician may ask which numbers can be a solution to an equation. A mathematician will then formalize this: Define a number to be algebraic over the rational numbers if it is the solution to some non-zero polynomial with rational coefficients. The mathematician will now investigate this in greater depth by analysing some non-trival (first) examples of algebraic numbers. For instance, $x^{2}-2=0$, has one solution equating to ${\sqrt {2}}$; this is "non-trivial" since it "cannot be written as a fraction" (again, this idea must be formalized). After this, a mathematician will attempt to modify this example and investigate what one may obtain - what about, $x^{2}+1=0$, he/she will ask. I urge the OP to attempt this - it is remarkable how this question leads to a totally new branch of mathematics. Lastly, "formulas" are not of interest in mathematics - if at all they serve a purpose it is how one obtains them and not what they state. In differential geometry, these so-called "formulas" exist but the way in which they are obtained leads to new insights in differential topology, for instance.

I have told you what mathematics is not but I have not told you what it is. I would love to do so but I think that the best basic examples of this may be found in Group (mathematics) and Vector space - wonderfully written articles which may be understood by even someone who cannot count (by the way, one need not know how to count to do mathematics - I would not know how to count if my reputation as a mathematician (as viewed by "society") were not at stake). Ideas relating to numbers are basic to mathematics and existed more than 2000 years ago. If I had to give a "definition" for mathematics (which numerous people have attempted to do), I would define it as: "The logical deduction of truth from a set of axioms." The more experience I have with the subject, the further this definition changes in my mind - I am sure that I will never be able to find a definition which suits me perfectly. --PST 05:06, 11 September 2009 (UTC)

Isn't your description a bit extreme in the opposite direction? I'd say that some formulas really are of interest, et cetera. Also, would a person who cannot count know the meaning of "two vectors" and "17th century"?

My personal definition for mathematics used to be along the lines of "The logical deduction of truth from a set of axioms", but lately I'm of the opinion that "mathematics" is a name for two distinct things - one is a method of obtaining truths (using axioms, proofs and the like), and one is a body of knowledge in subject matters which are traditionally explored with this method (quantity, structure, ...). Theoretical computer science is explored with the mathematical method, but is not usually considered to be included in the mathematical body of knowledge. -- Meni Rosenfeld (talk) 06:05, 11 September 2009 (UTC)

By my comment I implied that "counting" is simply an assignment of symbols and sounds to commonsense. In particular, it is not essential that mathematics be based on an agreement upon universal terminology; each and every person may define his own truth and do mathematics. It is just a question of mathematical interest - can a deep theory emerge from this, for instance. Thus, if a student finished high school without knowing how to count, I would not form any opinion of his/her mathematical ability but I would ask him/her for a substitute that he/she has developed.

With regards to formulas, I do not know whether they are really of interest except for the method that obtained them. For instance, one can give a formula for the area of a circle but that does not mean much except for the applications that it may produce. The general ways in which one may obtain that formula (which I found interesting the first time I saw them) are of actual interest. Although a formula can sometimes reflect the method which produced it, the method will have deep insights which the formula cannot encapsulate. --PST 02:19, 12 September 2009 (UTC)

Factor in a load of things, then factor out what it's all about. That's maths by my WP:OR, though it doesn't say what it's all for. Dmcq (talk) 10:55, 11 September 2009 (UTC)

I'd say mathematics is the study of abstract structures. And two abstract structures are the same precisely if they're isomorphic. Hence mathematics is the study of truths that are preserved by isomorphism. Michael Hardy (talk) 21:01, 11 September 2009 (UTC)

I think that Michael's definition above concurs with what Cantor once famously quoted - "Mathematics is not the study of objects but the relations between them." I agree that mathemaics is the study of truth preserved by isomorphism, as well. Nevertheless one runs into the point that Meni Rosenfield stated above - "Theoretical computer science is explored with the mathematical method, but is not usually considered to be included in the mathematical body of knowledge." If one thinks about it, all fields require the logical deduction of truth but the nature of the "axioms" change. For instance, scientists will logically deduce truth from experimental evidence to build a theory. Although my next comment will be debatable, I think that it is the case - all bodies of knowledge are in some sense mathematical in nature. This is perhaps explained by the fact that there are many mathematicians in history who have explored other fields (to name a few, Emanuel Lasker and John Nash, but there are far more). In effect, every field is mathematics once the "truth" has been decided. Deciding the truth is what is unique to the field. --PST 02:19, 12 September 2009 (UTC)

Then where does Philosophy come into play then? Because isn't that like the general case of all the sciences? After all, all science originated from philosophy... so would it make more sense to say that math is philosophical in nature, or that philosophy is mathematical in nature?--12.48.220.130 (talk) 20:56, 16 September 2009 (UTC)

You can choose the one you like more among these: Definitions of mathematics --78.13.142.34 (talk) 20:52, 13 September 2009 (UTC)

Paper towels

If you know the radius of the roll and the radius of the inner cylinder, what is the formula for the length? Thickness is harder to measure, so I guess what I really want is a formula for how much is left when the radius is half what it started out to be.Vchimpanzee · talk · contributions · 22:23, 10 September 2009 (UTC)

The length of the paper towel on the roll will depend upon the thickness of an individual sheet of paper; there is no way around that. If you find it difficult to measure the thickness of one sheet, you could try measuring the thickness of ten or so sheets (say, wrapped around the roll to make it easier) and divide by ten.

Once you know the thickness of an individual sheet, there are a few ways to estimate the length. One way might be to measure the volume of the paper towels (by measuring the volume of a cylinder with the larger radius and subtracting the volume of the inner cylinder), and use that and the thickness to get the length. If you are careful, you will notice that this method does not depend upon the height of the roll (if you leave the height h as a variable when computing the volume, you will notice the h cancels out at the end when computing the length of the paper towel).

As for your second question, "how much is left when the radius is half...", I don't understand what you are asking. Do you mean, what proportion of the paper towel has not been used when the outer radius has been halved? Eric. 216.27.191.178 (talk) 23:02, 10 September 2009 (UTC)

Usually they tell you on the package how many sheets are in a full roll, and the size of each sheet. That lets you figure out the length of a full roll (call this L). The length will also be (approximately) proportional to the area of the annulus that you see when you look at the roll from the end. Let R be the radius of the full roll and r be the radius of the inner cylinder. Then the area of the annulus is $A_{1}=\pi (R^{2}-r^{2})$. The thickness of the annulus is of course R-r. When enough towels are used up that the outer radius of the roll is half what it started at (i.e. it is R/2), the area of the annulus is of course $A_{2}=\pi \left(({R/2})^{2}-r^{2}\right)$. The length of that partial roll will be ${A_{2} \over A_{1}}L$. If what you are really asking is the length once the thickness is half what it started as, then just find the new outer radius $R_{2}=r+{{R-r} \over 2}$ and similarly calculate the new area and new length. —Preceding unsigned comment added by 70.90.174.101 (talk) 23:21, 10 September 2009 (UTC)

A similar question was posed here. Clearly, a common mathematical thought induced by the closet --"When I'll get out, I'll post on the RD/M" ;-) --pma (talk) 06:32, 11 September 2009 (UTC)