Recent content by TheBigBadBen

Ah, I knew something was off about my proof. Thank you for picking that up and wrapping it up neatly, and for imparting some of your own triangle-inequality magic. At any rate, I prefer to think of compactness in this sense rather than in terms of convergent subsequence, and I think this proof...

Right. Here's a sketch of the proof I have in mind: Given a continuous $f:X\to Y$, we want to show that for any $\epsilon>0$, there is a $\delta>0$ so that $d_Y(f(x),f(y))<\epsilon$ whenever $d_X(x,y)<\delta$ Consider any $\epsilon>0$. By continuity, we may state that for each $x\in X$...

We could also prove this using the topological definition of compactness (i.e. that every open cover has a finite subcover) rather than sequential compactness (i.e. that every sequence has a convergent subsequence). To me, this proof is nicer, though I can't guarantee it will be any easier on...

First of all, for convenience, I will write $p(x) \equiv a_1 x + a_2 x^2 + a_3 x^3 + \cdots $ Now, if you try substituting in, you get $$ (y + \alpha y^2)' = p(y + \alpha y^2) $$ First of all, note that $(y + \alpha y^2)' = (2\alpha y + 1)y' $. As for the other side, we can note that $$ p(y...

Re: analysis suggetion I still have this textbook from when I took complex analysis a few years ago, and it has definitely served me well. (Smile)

#2 seems to be a case of sneaky advertising. It's 10 calories! ...per half the bottle, that is, per "8 fluid oz". I don't think the others have such a justification though. I guess the quadratic formula holds for quadratics of the form $ax^2 + b(x+1)=0$.

I am sure that there's some nice little trick to all this, but here I go anyway... u-substition: $$ u = \ln(x);\; du = dx/x \rightarrow\\ \int^{\infty}_{0} \frac{\ln(x) }{x^2+a^2}\,dx= \int^{\infty}_{-\infty} \frac{u}{e^{2u}+a^2}\,e^u du\\ =\int^{\infty}_{-\infty} \frac{u}{e^{u}+a^2e^{-u}}\...

Just to elaborate on Opalg's tip: after substituting $t = \ln(x)$, you correctly ended up with $$ \int \cos (t)\, e^{t}dt $$ Now to do integration by parts, begin by proceeding as usual. Take: $$ u = \cos(t)\\ du = -\sin(t)\,dt\\ dv = e^t dt\\ v = e^t $$ And you end up with $$ \int \cos (t)\...

Here's one approach to framing the problem: Let $x$ be the height of the ball at its break-off point Let We note that an object will leave its circular orbit when the radial component of the force of gravity becomes greater than the centripetal force required to keep the ball in the circle...

No, but I can certainly exhibit an element of $\mathbb{Z}\times\mathbb{Z}$ that cannot be reached by $(x+y,x-y)$: one example is $(0,1)$. In general, $x+y$ is even iff $x-y$ is even for $(x,y)\in\mathbb{Z\times Z}$.

So I originally found this on the math stackoverflow page, and I hadn't looked back at it for a while but it's apparently become a popular problem. It started here abstract algebra - Can we ascertain that there exists an epimorphism $G\rightarrow H$? - Mathematics Stack Exchange and has since...

Your statement assumes that $\phi$ must act on the components separately. I thought this might be at first, but I don't believe this is necessarily the case. A possible counter-example: $\phi:\mathbb {Z\times Z} \to \mathbb {Z}\times \mathbb Z$ given by $\phi(x,y) = (x+y,x-y)$ is an...

Interesting question I've happened upon: If there is an epimorphism (i.e. onto homomorphism) $\phi:G\times G \to H\times H$, is there necessarily an epimorphism $\psi:G\to H$? If not, under what conditions can we ascertain such an epimorphism given the existence of $\phi$? I would think that...

The Easier Proof (via set theory) Consider the following definitions: $$ F_{X,Y}(x,y)=P(X< x \text{ and } Y < y)\\ F_{X}(x)=P(X< x)\\ F_{Y}(y)=P(Y< y) $$ We note that the set of $(x,y)$ such that $X< x \text{ and } Y < y$ is the intersection of the set of $(x,y)$ such that $X< x$ and the set...

So the proof of the first inequality via integrals would go something like this: First of all, definition. We state that there is some (joint) probability density function of the form $f_{X,Y}(x,y)$. We can then supply the following definitions of $F_X,F_Y,$ and $F_{X,Y}$ in terms of...