Mathematics

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Best Book For Differential Equations?

I know this is a subjective question, but I need some opinions on a very good book for learning differential equations. Ideally it should have a variety of problems with worked solutions and be easy to read. Thanks
Finance
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Are all mathematicians human calculators?

I asked my dad why he did not major in math he said "because he is not good at math". I think I like math, and I think I'm ok at it, but I'm not gifted or anything like that, I just like math. I think I'd like to major in math, but I see all these…
Kat
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Compute $ \lim\limits_{n \to \infty }\sin \sin \dots\sin n$

I need your help with evaluating this limit: $$ \lim_{n \to \infty }\underbrace{\sin \sin \dots\sin}_{\text{$n$ compositions}}\,n,$$ i.e. we apply the $\sin$ function $n$ times. Thank you.
user6163
81
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7 answers

Is there a slowest rate of divergence of a series?

$$f(n)=\sum_{i=1}^n\frac{1}{i}$$ diverges slower than $$g(n)=\sum_{i=1}^n\frac{1}{\sqrt{i}}$$ , by which I mean $\lim_{n\rightarrow \infty}(g(n)-f(n))=\infty$. Similarly, $\ln(n)$ diverges as fast as $f(n)$, as $\lim_{n \rightarrow…
Meow
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3 answers

String Theory: What to do?

This is going to be a relatively broad/open-ended question, so I apologize before hand if it is the wrong place to ask this. Anyways, I'm currently a 3rd year undergraduate starting to more seriously research possible grad schools. I find myself in…
Jonathan Gleason
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81
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6 answers

Can someone explain the Yoneda Lemma to an applied mathematician?

I have trouble following the category-theoretic statement and proof of the Yoneda Lemma. Indeed, I followed a category theory course for 4-5 lectures (several years ago now) and felt like I understood everything until we covered the Yoneda Lemma,…
Chris Taylor
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4 answers

A simple explanation of eigenvectors and eigenvalues with 'big picture' ideas of why on earth they matter

A number of areas I'm studying in my degree (not a maths degree) involve eigenvalues and eigvenvectors, which have never been properly explained to me. I find it very difficult to understand the explanations given in textbooks and lectures. Does…
robintw
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81
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8 answers

How to prove this identity $\pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;$?

How to prove this identity? $$\pi=\sum_{k=-\infty}^{\infty}\left(\dfrac{\sin(k)}{k}\right)^{2}\;$$ I found the above interesting identity in the book $\bf \pi$ Unleashed . Does anyone knows how to prove it? Thanks.
Neves
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81
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4 answers

What is the difference between the Frobenius norm and the 2-norm of a matrix?

Given a matrix, is the Frobenius norm of that matrix always equal to the 2-norm of it, or are there certain matrices where these two norm methods would produce different results? If they are identical, then I suppose the only difference between them…
Ricket
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9 answers

What is mathematical research like?

I'm planning on applying for a math research program over the summer, but I'm slightly nervous about it just because the name math research sounds strange to me. What does math research entail exactly? For other research like in economics, or…
TheHopefulActuary
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81
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What is lost when we move from reals to complex numbers?

As I know when you move to "bigger" number systems (such as from complex to quaternions) you lose some properties (e.g. moving from complex to quaternions requires loss of commutativity), but does it hold when you move for example from naturals to…
Юрій Ярош
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4 answers

Two curious "identities" on $x^x$, $e$, and $\pi$

A numerical calculation on Mathematica shows that $$I_1=\int_0^1 x^x(1-x)^{1-x}\sin\pi x\,\mathrm dx\approx0.355822$$ and $$I_2=\int_0^1 x^{-x}(1-x)^{x-1}\sin\pi x\,\mathrm dx\approx1.15573$$ A furthur investigation on OEIS ( A019632 and A061382 )…
zy_
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$\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this?

I came across a question where I needed to find the sum of the factorials of the first $n$ numbers. So I was wondering if there is any generic formula for this? Like there is a generic formula for the series: $$ 1 + 2 + 3 + 4 + \cdots + n =…
vikiiii
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What's a good place to learn Lie groups?

Ok so I read the following article the other day: http://www.aimath.org/E8/ and I wanted to learn more about lie groups. Using my exceptional deduction skills I thought "oh it must have something to do with groups" So I picked up a copy of Dummit…
Asinomás
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81
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3 answers

Mathematicians shocked(?) to find pattern in prime numbers

There is an interesting recent article " Mathematicians shocked to find pattern in "random" prime numbers " in New Scientist . (Don't you love math titles in the popular press? Compare to the source paper's Unexpected Biases in the Distribution of…
Tito Piezas III
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