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# 156 Comments

 21h comment Is there any relatively quick way to diagonalize this matrix with an orthogonal matrix? Yeah, sorry. I calculated the eigenvalues twice in my notebook and copied the wrong ones down. 21h comment Investigation Problem to challenge mathematical reasoning I don't know how "original" you need to be but this MO post mathoverflow.net/questions/48771/… seems to be related. Some of the proofs are a little above Calc 2, but you may be able to understand the first example: Euler's (not Euclid's) proof that their are infinitely primes. 22h comment Is there any relatively quick way to diagonalize this matrix with an orthogonal matrix? I learned orthogonal as $A\cdot A^{T} = I_n$ and I wasn't really aware of the orthonormality that needed preservation. Normalizing each vector gives an identity matrix. Thanks! 2d comment Showing $A+B$ is invertible? Thanks. I guess I should have kept trying... At least I was on the right track after all. :) Apr 27 comment How to derive this identity: $\lim _{x\to \infty} (1 + f(x))^{\frac{1}{g(x)}} = \mathrm e^{\lim_{x \to \infty} \frac{f(x)}{g(x)}}$ If $f(x)=1$ and $g(x)=1$ then the LHS goes to $2$ and the RHS goes to $e$... I don't think this is true in general. Apr 27 comment Necessity of Differential Forms Not entirely an answer to your question (it doesn't answer the importance in analysis) but for a more intuitive explanation, Tao gives an introduction to differential forms here: math.ucla.edu/~tao/preprints/forms.pdf and argues the algebraic laws of the differential forms derive from a roughly speaking more intuitive, what I would describe as: "infinitesimal geometry". Apr 22 comment f(x) is a function such that $\lim_{x\to0} f(x)/x=1$ You really shouldn't assume $f(x)$ is $\sin(x)$ because you have not shown that $a$ and $b$ are unique and irrespective of $f(x)$. Mar 17 comment Terry Tao, Russells Paradox, definition of a set Although it is related to lazy evaluation, I believe (although I could be wrong), as EJP has mentioned, short circuiting is a better term to describe this. The main reason being that lazy evaluation is much larger in scope and has further applications beyond this example. It refers to not computing values until they are needed, whereas short circuiting specifically refers to not computing values because they are unneeded. Feb 23 comment Prove that zero multiplied by zero is equal to zero. @ruakh: Oh yes, that's probably it: I'm thinking of fields. Feb 23 comment Prove that zero multiplied by zero is equal to zero. Huh, usually I have had to derive this. Although maybe it depends on what you take for granted... $a \cdot 0 = 0$ but $0 + 0 = 0$ therefore, $a \cdot 0 = a \cdot ( 0 + 0 ) = a \cdot 0 + a \cdot 0$. So $a \cdot 0 = a \cdot 0 + a \cdot 0 \implies 0 = a \cdot 0$ Feb 16 comment Prove that $\int_{0}^{\infty} \sin(t^2)\,dt$ equals $\sqrt{\pi/8}$ The top one is one of the two Fresnel integrals. wikiwand.com/en/Fresnel_integral The specific problem is mentioned later in the wikipedia page. Feb 11 comment Proving trefoil group is isomorphic to a fundamental group. It think so, it can now be shown that $x_1 = b^2a^-1$ and that $x_2 = ab^{-1}$, so $a$, $b$ also generates the trefoil group, therefore we have an alternative presentation, and the two groups must be isomorphic. Oct 31 comment Confused about basics of subsequences You do realize this is a sequence and not a series? I once confused the two and I sense that maybe you are confusing the two... Oct 26 comment Mathematical texts: white background or tan I don't really have a problem with white colored background. I would guess, that it's just easier to get a hold of white colored paper since it is "more standard." Not so much math but I feel like at least some of the paper writing formats (MLA, APA, Chicago etc.) require white as the color of choice. There would be less contrast between the black font and tan which may make it hard for people with poor vision to read. Just ask the professor. (This is ALL ENTIRELY speculative and a very unsupported guess, more of something to consider). Oct 25 comment Give a regular expression that generates C. This is off topic but I have never seen a language use (in particular) the #. Usually it is something like: /* and */ Oct 12 comment Neighbors of Irrational Numbers on Real Number Line Maybe this can help convince you otherwise: wikiwand.com/en/Archimedean_property Instead of arguing about what infinitesimal neighbor 0 has, the Archimedean property indicates that over $\mathbb{R}$ there exists no infinitesimal objects in the first place. Hence, the "next neighbor" is neither rational nor irrational as it doesn't exist. Oct 9 comment Relationship between perfect squares and infinite series (zeta function) I don't believe there is any nice formula: wikiwand.com/en/Ap%C3%A9ry%27s_constant Sep 29 comment Mathematical Symbols Possible duplicate: xkcd.com/927 Sep 29 comment Help with verifying integral inequality. @mercio: Plugging in the linear transformation $t$ gives an equality for the first statement. So then looking at the comment above you can use this fact to conclude: $\int_x^1 f(t) dt \geq \int_x^1 t dt$. Therefore: $$\left|\int_0^1 f(x)dx \right|^2 \geq \int_0^1 x dx \cdot \int_0^1 x dx$$ and similarly from the Cauchy-Schwarz inequality: $$\int_0^1 |f(x)|^2 dx = \int_0^1 x f(x) dx$$ Sep 29 comment Help with verifying integral inequality. @Aravind: Sorry, but do you mind clarifying why we cannot conclude thusly?