Wikipedia:Reference desk/Archives/Mathematics/2009 December 26

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December 26

dual spaces of Sobolev spaces

The Rellich-Kondrachev theorem gives compact embeddings of $W^{1,p}$ into $L^{p^{*}}$, but what can we say about, say, $L^{q}$ and $W^{-1,q}$? I remember it was straightforward, but I'm having trouble finding it in the references, and am rather embarassed that it's not working out easily. Many thanks. 96.235.177.218 (talk) 03:47, 26 December 2009 (UTC)

(To be precise, there is compactness only when q is strictly below the critical exponent: q<p*). I'm not sure of what's exactly your question though. One thing is that dualizing the RK embedding you still get a dense, injective, compact map of the dual of $L^{q}$ into the dual of $W^{1,p}.$ What you possibly had in mind is that if a bounded linear operator $\scriptstyle J:X\to Y$ between Banach spaces is compact/injective/dense, then the transpose operator $\scriptstyle J^{*}:Y^{*}\to X^{*}$ is respectively compact/w*-dense/injective. If 1<p<n the space $W^{1,p}(\Omega )$ is reflexive, so that "w*-dense" above is the same as just "dense". Was this your question?--pma (talk) 16:55, 26 December 2009 (UTC)

Is the word "induce" used technically or non-technically?

Suppose A and B are groups, and N is a normal subgroup of A. Suppose we have an isomorphism $f:A\to B$; then we would say that f naturally induces an isomorphism $A/N\to B/f(N)$.

I would like to know the limits of the word induce. I see two alternatives:

(1) Is the phrase "the map induced by f" rigorously defined to refer to that map which results from passing to the quotient spaces, as in my example? In this case, the phrase "induced map" would have a formal, unambiguous meaning, just as "the pullback of f" has a formal, unambiguous meaning.

(2) Is the phrase "the map induced by f" used informally to refer to any map that results from some kind of a canonical process? For example, would it be correct to say the restriction map $N\to f(N)$ is induced by f? Could we also say the pullback of f (by some other map) is "induced" by f? Maybe the lift of f is also "induced" by f? In this case, the phrase "induced map" would have a subjective meaning, depending on context to establish what particular process we mean.

Of course the actual usage of the word "induce" could differ from both of the two above descriptions, and can vary from one mathematician to another; but I am most interested in the distinction between a formal, technical meaning for "induce" and an informal, non-technical meaning. Thanks. Eric. 131.215.159.171 (talk) 08:43, 26 December 2009 (UTC)

Let A, B, C and D be objects in a category, and let $f:A\to B$ be an element of $Hom(A,B)$. Formally, I would say that f induces an element $g:C\to D$ of $Hom(C,D)$, if there exist morphisms $h_{1}:A\to C$ and $h_{2}:B\to D$ such that the following diagram commutes:

${\begin{array}{ccc}A&{\stackrel {f}{\rightarrow }}&B\\\downarrow \scriptstyle {h_{1}}&&\downarrow \scriptstyle {h_{2}}\\C&{\stackrel {g}{\rightarrow }}&D\end{array}}$

We shall now restrict our attention to abelian categories. The above diagram includes the two cases you mentioned, as is demonstrated by the following special commutative diagrams (let $h_{1}:A\to A/N$ and $h_{2}:B\to B/f(N)$ be the respective canonical homomorphisms):

${\begin{array}{ccc}A&{\stackrel {f}{\rightarrow }}&B\\\downarrow \scriptstyle {h_{1}}&&\downarrow \scriptstyle {h_{2}}\\A/N&{\stackrel {g}{\rightarrow }}&B/f(N)\end{array}}$

The above diagram commutes, as you can check. Similarly, consider the following commutative diagram (in this case, let $h_{1}:N\to A$ and $h_{2}:f(N)\to B$ be the respective inclusion maps):

${\begin{array}{ccc}A&{\stackrel {f}{\rightarrow }}&B\\\uparrow \scriptstyle {h_{1}}&&\uparrow \scriptstyle {h_{2}}\\N&{\stackrel {g}{\rightarrow }}&f(N)\end{array}}$

Note that the vertical arrows in the above commutative diagram point up, as opposed to those in the other commutative diagrams, which point down. I have merely given you a formal definition (in my view) of "induce" in the mentioned situations. I do not quite understand what you would call an "informal definition"; could you please clarify? Hope this helps (and try not to notice the ugly commutative diagrams...). --PST 11:52, 26 December 2009 (UTC)

Thank you for your reply; it was helpful. Perhaps I should be more clear. I am not so interested in what the definition, whether technical or non-technical, of "induce" is, per se, as whether mathematicians view the word "induce" as a technical term (like the terms "pullback", "lift", "inverse limit"), or as a non-technical term (like the terms "trivial", "characterization", "equivalent", "canonical", "natural"). I gave examples of what a technical definition for "induce" (the result when passing to the quotient space) and a non-technical definition for "induce" (the canonical result of some natural process) to clarify my meaning of a technical term vs. a non-technical term, but I am not necessarily convinced that either one of those is what most mathematicians use the word to mean. Eric. 131.215.159.171 (talk) 12:55, 26 December 2009 (UTC)

I think that in specific cases, many mathematicians view "induce" as a technical term; one example being "pullback" (or "pushforward") as you mentioned. However, in general, I do not think that all mathematicians have a specific view as to what induce should mean. If I was talking about the pushforward measure in measure theory, the tangent bundle in differential topology, or even quotient spaces in algebra, I would employ specific aspects of the term "induce"; I would not use it in its full generality. I think that this is the case in most of mathematics - often we would generalize a term if we feel that the generalization sheds new light on concrete (or even abstract) cases. For instance, the snake lemma in homological algebra, amidst all this "abstract nonsense" about generalized abelian categories, actually allows one to construct long exact sequences in homology (Zig-zag lemma); a particularly basic tool in homology. Although it seems unnaturally general at first to the beginning student, it actually does shed light on basic tools in singular homology theory (as an example). To summarize, I do not think that mathematicians have found a "specific purpose" of viewing "induce" as a formal term, as people have done in many other branches of mathematics such as point-set topology or abstract algebra. Rather they have formalized the term in specific situations such as the ones I have mentioned ("pushforward", "pullback" etc) and this has been particularly useful. Does this answer your question? --PST 13:30, 26 December 2009 (UTC)

I personally use "X induces Y (via Z)" for any object or situation Y whose existence and (essential) unicity is guaranteed by X (as a consequence of the theorem or the construction Z, to be specified unless it is clear). It seems to me that this generic use is the most common. In some cases "deduce" or "produce" may be valid alternatives (although probably nobody cares about the etymology). --pma (talk) 17:26, 26 December 2009 (UTC)

Thanks, you have answered my question thoroughly. Eric. 131.215.159.171 (talk) 21:44, 27 December 2009 (UTC)

Does U(1) = SU(1)

Does the circle group equal the special unitary group of one dimension? -Craig Pemberton 08:44, 26 December 2009 (UTC)

No. SU(1) is the trivial group {1} - see special unitary group. Gandalf61 (talk) 09:12, 26 December 2009 (UTC)

An element of U(1) is of the form ${\begin{bmatrix}a+bi\\\end{bmatrix}}$ where $a+bi\in \mathbb {C}$, and ${\begin{bmatrix}a+bi\\\end{bmatrix}}\cdot {\begin{bmatrix}a-bi\\\end{bmatrix}}={\begin{bmatrix}1\\\end{bmatrix}}$ (since ${\begin{bmatrix}a-bi\\\end{bmatrix}}$ is the conjugate transpose of ${\begin{bmatrix}a+bi\\\end{bmatrix}}$). This fact allows one to conclude that which you note; U(1) is isomorphic to the circle group. Now, SU(1) is the set of all elements of U(1) having determinant 1. However, a matrix ${\begin{bmatrix}a+bi\\\end{bmatrix}}$ has determinant 1 iff $a+bi=1$; equivalently, iff ${\begin{bmatrix}a+bi\\\end{bmatrix}}={\begin{bmatrix}1\\\end{bmatrix}}$. Thus, SU(1) is the trivial group, whereas U(1) is the circle group (perhaps I have over-explained a little...). --PST 12:05, 26 December 2009 (UTC)

Linear combination of matrices

Is there a reference book or wiki article or something to direct me to research of eigenvalue problem of matrices like $A+xB$, in particular, properties of the roots of the characteristic polynomial ${\rm {det}}(A+xB-\lambda I)$ with the parameter $x\in \mathbb {C}$? (Igny (talk) 17:29, 26 December 2009 (UTC))

Have you tried this article? If not, I recommend it. If so, do you have a specific question?--Leon (talk) 17:50, 26 December 2009 (UTC)

Well I know the general theory of eigenproblem, and I know implicit differentiation well enough to figure out, for example, $\lambda '(x)$. However I thought that there was some more obscure research of the roots $\lambda _{j}(x)$ from the point of view of the Galois theory, for example. (Igny (talk) 18:22, 26 December 2009 (UTC))

Tosio Kato, Perturbation theory for linear operators. --pma (talk) 20:05, 26 December 2009 (UTC)

Thank you, I think I read it quite a while ago, I will look it up again. (Igny (talk) 02:16, 27 December 2009 (UTC))

Algebra over a ring that is a field?

Is it possible for an algebra over a ring-that is, a ring that is not also a field-to be a field? I'm aware that you can describe rational numbers as pairs of integers, but in as much as I understand the term "algebra over a ring", that does not qualify as addition needs to be defined differently to that on a vector space over the ring of integers.--Leon (talk) 17:48, 26 December 2009 (UTC)

What about $\mathbb {Q}$ as an algebra over $\mathbb {Z}$..?--pma (talk) 20:12, 26 December 2009 (UTC)

Didn't I just mention that, and further explain why I figured that it didn't count?--Leon (talk) 21:05, 26 December 2009 (UTC

I don't see why it doesn't count. An algebra over a ring is a module with a suitably-behaved multiplication, that's the only definition I know. $\mathbb {Q}$ can certainly be considered a module over $\mathbb {Z}$ and the usual multiplication is suitably-behaved. --Tango (talk) 21:50, 26 December 2009 (UTC)

Sorry: actually you did, but your explanation was (and I fear, will remain) rather obscure to me. I do not understand your doubts: the definition of algebra is very clear, unambiguous, and standard; and obviously any field is an algebra over any sub-ring of it. --pma (talk) 23:57, 26 December 2009 (UTC)

I suspect that the motivation for this question comes from the fact that a finite-dimensional algebra (over a field) which is also an integral domain, must necessarily be itself a field; the OP may wish to know whether this can be generalized to an arbitrary ring in place of the field over which the algebra is defined. In general, as pma suggested, the "field of fractions" of an integral domain is an algebra (and the integral domain need not be a field). If a ring has zero divisors, then no algebra over it can be a field. However, if a ring is not necessarily commutative, but has no zero divisors, we can replace the "field of fractions" idea by a "noncommutative division ring" idea via the Ore condition (that is, we can obtain a "noncommutative division ring of fractions" which is an algebra over a given noncommutative ring with no zero divisors satisfying the Ore condition). Hope this helps. --PST 03:47, 27 December 2009 (UTC)

If K is a field that is also an R-algebra for R some commutative ring with 1, AND such that K is free as an R-module, then I believe R is a field. R is a subring of K, K is a free, divisible module, so R is a divisible R-module, so R is a field. "Free" basically means that the elements of K are ordered pairs, triples, tuples of elements of R with addition defined coordinate-wise, like for vector spaces. If you require the field to be "free", then the coefficient ring R has to have been a field. However, if the elements of K need not have any specific form, but merely need to be able to be multiplied by elements of R, then of course every field is an R-algebra for some R that is not a field (and if you wish, also not the integers). JackSchmidt (talk) 07:53, 27 December 2009 (UTC)