So one way to define a group presentation is to say, well, let’s generate the free group with some number of generators, and then quotient by saying certain elements (relators) cancel just as $aa^{-1}$ does (and other things, you have to take the normal subgroup generated by the elements, but the basic idea works, I […]

I have read some of the books suggested in What are the prerequisites to Jech's Set theory text?, so I have some beginning experience with transfinite recursion, ordinals, cardinals, order types, and the axioms in ZFC. My question is this: How much formal logic should I know before reading Jech’s Set Theory (In particular, how […]

Cartan Theorem: Let $M$ be a compact riemannian manifold. Let $\pi_1(M)$ be the set of all the classes of free homotopy of $M.$ Then in each non trival class there is a closed geodesic. (i.e a closed curve which is geodesic in all of its points.) My question: Why free classes? Why the theorem does […]

What is the name of the following summation formula? $$\sum_{k = 1}^n f(k)) = \int_1^{n + 1} f – \frac{f(n + 1) + f(0)}2 + \int_1^{n + 1} f’w,$$ where $w$ is the “sawtooth” function, defined by $w(x) = (x – (k + 1/2))$, for $k < x <= k + 1$, if $k$ is […]

I always wanted to ask this question since when I joined MSE, but because I was afraid of asking too many soft questions I never asked it. I’ve seen some pretty complicated integrals and infinite products and infinite series and other math equations that have been reduced to simple closed form expressions using special functions […]

The weak topologies of a Banach Space are constructed by taking a family $\tilde{B}$ consisting on all finite intersections of $\mathbb{U}$ and then taking arbitrary unions of sets of $\tilde{B}$. I want to show that this is indeed a topology, for which I need to show that the family constructed is stable under finite intersections. […]

I was wandering if anybody knew any books with difficult exercises on Infinite Series, Mean Value Theorem, perhaps some limits and DE’s. Thanks!

I’m looking for a source (or sources) which develop a complete theory of the trigonometric functions with no reference to circle geometry. That is, it is purely analytic. The starting point could be (for example) $$\arcsin x := \int_0^x \frac{dt}{\sqrt{1-t^2}} $$ or alternatively defining $\sin$ and $\cos$ as solutions to differential equations.

What are the common methods and tools to tackle optimization problemsinvolving integrals. To be precise lets consider the following optimization problem that I came across with: $$\text{maximize}\,\,F(a,b)=\int_0^a\int_0^bf(x,y)dxdy,$$ Subject to $ab=1$. Any reference will be highly appreciated.

Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $\gcd(q,n)=1$ and $q \equiv k \equiv 1 \pmod 4$). (That is, $2N=\sigma(N)$ where $\sigma$ is the classical sum-of-divisors function.) Since $\gcd(q^k,\sigma(q^k))=1$, it follows that $q \mid \sigma(n^2)$. My question is this: Is the Euler prime $q$ of […]

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