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Apr
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answered Is learning (theoretical) physics useful/important for a mathematician?
Apr
28
comment A question about non-atomic measures on $S^1$
What may have confused the OP is that this doesn't depend on $1/3$ being $1/3$; you could do it for any $\epsilon > 0$ and get $\delta$ so that whenever an interval $I$ satisfies $|I| < \delta$, $\mu(I) < \epsilon$. If you knew the same for all measurable $I$, this would be the statement that $\mu$ is absolutely continuous with respect to Lebesgue measure. I think there's nothing wrong: I don't see how to get it for all measurable sets from knowing it for intervals. (Who knew: it's not enough to check the condition for absolute continuity on a generating set for the algebra.) Am I right?
Apr
27
comment Proof that 1-1 analytic functions have nonzero derivative
I'd argue that although the idea is simple, any proof will have to be at least a little "awkward" or "technical" for the reason that the statement is analogous to other statements that are false. For example $f: \mathbb{R} \to \mathbb{R}$ given by $x \mapsto x^3$ is one-to-one and in many senses it's as nice a function from $\mathbb{R}$ to $\mathbb{R}$ as you might want (e.g. it is real analytic) but $f'(0) = 0$. Of course the $'$ in $f'$ means something different here, but it's certainly analogous. So any proof of your statement will really have to do something. Just my two cents.
Apr
27
comment Is learning (theoretical) physics useful/important for a mathematician?
I don't have the book, but I'd guess that the intent of the quote is not that pure mathematics requires or uses physics or applied considerations, as much as that if you want a "proper appreciation" of pure math, in the sense of where it comes from, it helps to understand a little of physics and applied math. Because so many areas of pure math have their origins in applied problems, even if they are their own thing now. e.g. anything using the ideas of calculus owes a debt to Newton's mechanics. You don't need to know this, but shouldn't you, in order to appreciate calculus?
Apr
26
comment How is it shown that a Hermitian matrix will be positive definite?
A related generalization: if $M$ is the block matrix $\begin{pmatrix} A & B \\ -B & A \end{pmatrix}$, since the block matrix $U = \frac{1}{\sqrt{2}} \begin{pmatrix} I & -iI \\ I & iI \end{pmatrix}$ is unitary and $U M U^* = \begin{pmatrix} A + iB & 0 \\ 0 & A - iB \end{pmatrix}$, you see that $M$ is positive definite if and only if $A+iB$ and $A-iB$ both are. (When $A = m$ and $B = i$ are $1 \times 1$ scalar matrices the condition is that $m \pm 1$ must both be positive, ie, that $m > 1$. This is nonelementary enough that I'm not seriously trying to help the OP here; just having some fun.)
Mar
28
comment Geometric interpretation of $\frac {\partial^2} {\partial x \partial y} f(x,y)$
Seems you're asking a subquestion of what you want--- not how to interpret mixed partials, but why the sign of $D(a,b)$ can give the nature of a saddle point. For this, do elementary analytic geometry on the graph of a function $Ax^2 + By^2 + Cxy$ at $(0,0)$ (add $Ex + Fy + G$, and at $(a,b)$, if you don't see how to reduce to this case). What conditions on $A,B,C$ do you get bowl that opens up, bowl that opens down, or saddle? Once you "get" this, you "get" all $f$, by the Taylor theorem. (Personally, I understand this via the algebra, not "geometric understanding" of $f_{xy}$.)
Mar
21
answered Why are gauge integrals not more popular?
Mar
12
comment property of vector-space
Assume $av = 0$. If $a = 0$ you're done; otherwise, by the field axioms there is $b \in F$ with $ba = 1$; then the vector space axioms imply that $v = 1v = (ba)v = b(av) = b(0) = 0$ and you're done. [The fact that $b0 = 0$ may itself require proof.]
Mar
12
answered Proving ellipsoid A is a subset of ellipsoid B iff (B-A) is positive semi-definite
Mar
12
answered Irreducibility and Splitting Fields
Mar
9
comment Meaning of the terms operation, function and map
As Arturo says, it is a matter of definition. There is no one rulebook people follow on this kind of thing (making it difficult to catalog mathematical terminology in encyclopedias; what you see on Wikipedia is often just one possible choice out of many). The best any writer can do is make his or her definitions explicit before using them. For what it's worth, in a lot of usages, "function" and "map" are exact synonyms; in others, "map" may carry more information (e.g. for some topologists "map" may mean "continuous function"). One can't tell from the words; one must be told by the writer.
Mar
6
comment What is the significance of “classes”?
There is almost an overlap: in many categories, the notion of isomorphism would be an equivalence relation, but for the fact that it doesn't fit in a set. The informal mathematical practice of ignoring this, e.g. talking about isomorphism as an equivalence relation and referring to the "isomorphism class" of a group or "homeomorphism class" of a topological space without specifying a reference set on which the isomorphism is a set-theoretic equivalence relation, just (coincidentally) happens to be completely in line with formal use. (These things are proper classes).
Mar
6
comment What is the significance of “classes”?
This is more an informal remark than an answer: for many if not most professional users of mathematics, the term "class" is indeed only used as a way to speak of collections of objects that are too big to be sets (if you like, just a near-synonym for the informal term "collection" that has the value of signaling that the collection referred to might not be a set). It does have a precise role in specific formalizations of mathematics (see JBeardz' answer), but many people using the term do not have the details of this or any other formalization in mind when they use it.