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16

Euler apparently had some trouble deriving the Jacobian used in change of variables for double integrals. He began by considering congruent transformations consisting of (affine) linear functions, and got something like $$\mathrm{d}x\,\mathrm{d}y=m\sqrt{1-m^2}\,\mathrm{d}t^2+(1-2m^2)\,\mathrm{d}t\,\mathrm{d}v-m\sqrt{1-m^2}\,\mathrm{d}v^2$$ which he ...


8

Euler conjectured that for $n=2\pmod 4$ there are no mutually orthogonal Latin squares of size $n\times n$. Bose and Shrikande disproved it by construction and earned the name Euler's Spoilers. See http://en.wikipedia.org/wiki/Graeco-Latin_square


6

It was a problem for me too when I started learning conformal differential geometry in 2008. As I prefer to learn from openly available sources even though our University has an excellent library within a minute of walk, some of my references will be links to such online resources. My first encounter with the proof of the conformal transformation of the ...


4

Euler liked to play fast and loose with divergent series. Mathematicians of that era did not seem to be concerned with convergence issues. For a more concrete example, Euler made a large mistake in trying to prove Fermat's Last Theorem for $n=3$. For details, check out http://www-history.mcs.st-and.ac.uk/HistTopics/Fermat%27s_last_theorem.html


4

This actually has nothing to do with zeta values, it is pure symmetry - a special case of $$e_n=Z(P_n)(p_1,p_2,\cdots,p_n) \tag{1}$$ where $e_k$ and $p_k$ are the elementary and power-sum symmetric polynomials, in infinitely many variables $x_1,x_2,\cdots$, respectively. For zeta values we simply evaluate at $x_k:=k^{-s}$. The recursive identity you wrote ...


4

Here's another one to see how you understand calculus. $f(x)\geqslant 0, \forall x \geqslant 0$ $f(x)\leqslant c\int_0^xf(t)dt, \forall x\geqslant 0 ,\exists c>0$ Prove that $f(x)$ is identically zero. Solution : Let $F(x)=\int_0^xf(t)dt $ $$f(x)-cF(x)\leqslant 0$$ $$f(x)e^{-cx}-ce^{-cx}F(x)\leqslant0$$ $$(F(x)e^{-cx})'\leqslant 0$$ ...


4

This isn't a bona fide mistake but it's certainly a pitfall. Hopefully someone can verify the following. In Euler's original proof of the Basel Problem $(\zeta(2)=\pi^2/6$), he used the fact that $$\sin(z)=z\prod_{n\geq 0}\left(1-\frac{z^2}{n^2\pi^2}\right).$$ This was well before Weierstrass's factorization theorem, which allows for a prefactor of ...


4

Here is a proof that, for a finite group $G$, $|G/\Phi(G)|$ is divisible by all primes $p$ dividing $|G|$. It uses the Schur-Zassenhaus Theorem. I don't know whether that can be avoided. First note that each Sylow $p$-subgroup $P$ of $\Phi(G)$ is normal in $G$. (So, in particular, $\Phi(G)$ is nilpotent.) To see that, we have $G = \Phi(G)N_G(P)$ by the ...


4

Because scientists are not famous, with the exception of perhaps 10 top scientists of all times. Mathematicians and scientists are resented most people for the same reason high school kids dislike nerds: the living proof that someone is smarter than you is discomforting. The announcement of 7 Millennium Prizes made news that made the news for the same ...


3

To find a finite group with non-abelian Frattini subgroup, you'll need a group with a non-abelian maximal subgroup, but that does not suffice. The dihedral group of order 16 has a maximal subgroup isomorphic to the dihedral group of order 8, but the Frattini subgroup of a a non-cyclic group of order 16 has order at most 4, so is abelian (it is Klein 4 in ...


3

Find the (real) value of $a$ such that the curves \begin{eqnarray} y &=& a^x \\ y &=& \log_a (x) \end{eqnarray} intersect exactly once. Find also the $x$ and $y$ values where they intersect. Note that's the logarithm of base $a$ in the second curve. I think this is a pretty tough problem. It doesn't involve advanced calculus, but you need to ...


3

Let $\zeta$ be a primitive $n$th root of unity. The cyclotomic field $\Bbb Q(\zeta)$ is the splitting field of $x^n-1$. The minimal polynomial of $\zeta$ is the cyclotomic polynomial $\Phi_n(x)$ which has degree $\varphi(n)$, so the field extension $\Bbb Q(\zeta)/\Bbb Q$ has degree $\varphi(n)$. If $\zeta$ is also a quadratic integer then $\Bbb Q(\zeta)/\Bbb ...


3

I consider Folland's “Real Analysis: Modern Techniques and Their Applications” as the best textbook ever written, on any subject. I warn you, though, it is extremely dense: If you want to be thorough and check everything for yourself (as many really easy results are just mentioned in passing without proof and there are a lot of exercises, some quite easy, ...


3

My favorite analysis book is that by Pugh. It's similar to Rudin, but readable and with tons of fantastic problems. So far, I really like Runde for topology. It's rigorous, and short enough that you'd want to read it front to back. It has what you need if you want to continue on to more advanced topics, and is enough even if you don't! Another nice book to ...


3

I would recommend practicing bounding things with inequalities, so when you encounter a problem that requires the use of the triangle inequality (it comes up fairly frequently), you can find a bound fast. Many of the definitions used in analysis deal with inequalities as well, such as convergence of a sequence, limit of a function, continuity, etc... Also ...


2

This will test all of your analytical and mathematical skills. $f(x)$ is a differentiable function and $g(x)$ is a double differentiable function such that $|f(x)|\leqslant 1$ and $f'(x)=g(x)$. If $$f(0)^2+g(0)^2=9$$ then prove that there exists some $c\in(-3,3)$ such that$ \ \ g(c) \cdot g''(c)<0$.


2

Here is a proof of equivalence of 1, 3 and 4. Below, $S$ is a compact connected surface without boundary. Definition. A maximal cut in $S$ is a 1-dimensional submanifold $L\subset S$ such that $S\setminus L$ is connected and contains no nonseparating simple loops. The latter condition just means that $S\setminus L$ is the 2-dimensional sphere with ...


2

Concerning https://de.wikipedia.org/wiki/Elementzeichen#Geschichte it was first used 1889 in the work Arithmetices principia nova methodo exposita (page X): „Signum ϵ significat est. Ita a ϵ b legitur a est quoddam b“ This means something like (my translation which isn't the best!): The symbol ϵ means is. So a ϵ b has to be read as a is a b Here ...


2

Game Theory does not make any restrictions on what the set of actions $\mathcal{A}$ can be. The set $\mathcal{A}$ can be a set of functions, a set of sets, etc... Players may even have different "actions" available to them, $\mathcal{A}=\mathcal{A}_1\times...\times\mathcal{A}_I$, where $\mathcal{A}_i$ is the set of strategies of player $i$. We define ...


2

There is an innovative course Coding the Matrix offered by Philip Klein which consists of a book and a course offered on Coursera and other places. It even has a Twitter account for keeping updated. The reviews are controversial, see also here and here, but it looks as an interesting challenge to try. It is designed, according to the author's website, as a ...


2

As OP stated he was a physics student in a comment, there are a lot of books called (something similar) to "Mathematical Methods for Physicists", which have relevant references, details and techniques for physicists. I'd probably start with the first two. Arfken, "Mathematical Methods for Physicists" Boas, "Mathematical Methods in the Physical Sciences " ...


2

The diagrams are a way of describing a group generated by reflections. Any collection of reflections (in Euclidean space, say) will generate a group. To know what this group is like, you need to know more than just how many generators there are: you need to know the relationships between the generators. The Coxeter diagram tells you that information. There ...


2

As Adam mentions, my paper with Benedikt Löwe on the The modal logic of forcing is concerned with the modality $\square\varphi$, which means that $\varphi$ holds in all forcing extensions, and the corresponding modality $\Diamond\varphi$, which means that $\varphi$ holds in some forcing extension. It is remarkable, in my opinion, that these set-theoretic ...


2

Here's one way of viewing it. First $\mathbb{Q}\otimes \mathbb{Z}/p^i\mathbb{Z}=0$ for all $i$, so the right hand side is zero. There is an injection $$0\to\mathbb{Z}_p\to\prod_i \mathbb{Z}/p^i\mathbb{Z}$$ where $\mathbb{Z}_p$ is the $p$-adic integers. Since $\mathbb{Q}$ is flat we obtain an injection ...


2

You should think of $\text{Ext}^{\bullet}_A(R, R)$ as a shadow of a more fundamental operation, namely taking derived endomorphisms $\text{RHom}_A(R, R)$ of $R$, regarded as an object in the derived category of $A$-modules. Here the ring structure is obvious; the multiplication comes from composition. More generally, there is a composition map ...


2

A main point I would make is that connected unions of non connected spaces are commonplace, and an example is given on groupoids page which you are welcome to use, of course. See also arxiv:1404.0556 for a correction to a proof in my book about the Phragmen-Brouwer Property, with a useful new result on pushouts of groupoids. The virtue of the use of ...


1

I really liked "Riemannian Geometry" by Manfredo do Carmo - http://www.amazon.com/dp/0817634908/ref=rdr_ext_sb_ti_sims_1 (a PDF copy can be found by googling "Riemannian geometry do Carmo" and looking at the first few search results) Chapter 0 discusses the preliminaries from smooth manifold theory and subsequent chapters immediately begin in Riemannian ...


1

First a disclaimer: I have never read Petersen's book, so I'll answer based on having skimmed through the contents and first chapters and assuming it is not much different from other books of the same level. Therefore take this with a grain of salt. I really like Lee's Smooth Manifolds, but since you want a brief introduction so you can study Riemmanian ...


1

Petersen's book is challenging, but very clear and thorough. If you want to learn the prerequisites quickly-as I'm sure all graduate students who want to begin research do-then John Lee's books aren't really the best option for you. They're wonderfully lucid and comprehensive texts,but thier sheer length means they're really best for self study when you have ...


1

Mathematical physicists see much of their work as an generalization of exterior algebra, so try: The geometry of physics by Theodore Frenkel. A lot of this stuff was invented before good computer graphics, but Wikipedia has some neat figures as well:



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