@YCor made a suggestion in an answer to another post (What is the intended use of the (proofs) tag?) a bit over a year ago. It received several upvotes, but does not appear to have been acted upon. I thought it might receive more attention here: It's not exactly about the original question, bu...

I love the book Proofs Without Words by Roger B. Nelsen. One of the proofs I liked the most was this: Area under one arch of a cycloid is 3 times the area of the wheel that traces it. You break the cycloid in to three parts and show that each part has an area equal to that of the wheel. I have a...

In Upper Limit on the Central Binomial Coefficient, Noam Elkies and David Speyer have given a nice proof that the central binomial coefficient $\binom{2n}{n} \sim \frac{4^n}{\sqrt{\pi n}}$. This can be used to derive Stirling's formula $$ a_n = \frac{n!e^n}{n^n \sqrt{n}} \sim \sqrt{2\pi} $$ by sh...

While my question topic is that of mathematical writing of papers, which is a broad subject, the particular question is specific. I am writing a paper, in which we have a section called "Outline of Proof". (It's Section 2.) The outline is fairly informal, and we omit some technical details, makin...

I'm teaching an honors differential equations class and have been using linear algebra heavily. I thought it would be interesting to include a proof of the method of undetermined coefficients along the lines described below, which I figured is already known. Nonetheless, Google turns up no proo...

I recently heard a talk about these topics and found them very interesting. The talk was centered on the formal structure and didn't really focus on examples. So my question is: what is your favorite application of topological chiral homology? (or its other variants and specialisations)

I am going to compute some intersection numbers on certain RZ spaces and therefore need to fully understand the deformation of $p$-divisible groups. This can be understood as deformation of displays, where displays are generalized notions of Dieudonné modules. The question is about the paper Tho...

Let $K$ be a finite extension of $\mathbb{Q}_p$. If $A_K$ is a semistable Abelian variety over $K$, then we have a Frobenius endomorphism on $H_{dR}^1(A_K)$, whose definition depends on a choice of a branch of the $p$-adic logarithm on $K^\times$. The unit root subspace $W$ of $H_{dR}^1(A_K)$ is ...

I'm studying relationships between trace entropy functionals and combinatorics and I'm faced with the following problem. Lets $\mathcal {D}$ be the following differential operator $1 -x\cdot \cfrac{d}{dx}$ i.e. $\mathcal {D} g = g - x\cdot g'$. For $m\ge 0$ integer, if $\Phi_m(x) := x\cdot \log(x...

I am having trouble proving this modified mean-value inequality. Suppose that $\Delta u+cu\ge 0$ for $u:\mathbb{R}^n\to [0,\infty).$ Prove that there exists constants $r_0,C>0$ depending only on $c$ so that $$u(0)\le \frac{C}{r^n}\int_{B(r)}u\,\mathrm{dVol},$$ for all $r\le r_0$. This was mention...

I came across Villani's paper titled "Hypocoercive diffusion operators" and couldn't figure out a computation that is skipped in that paper. Specifically, consider the following transformed Fokker-Planck equation, where $h(t,x,v)$ is the unknown, $(x,v) \in \mathbb{R}^n \times \mathbb{R}^n$, $V(x...

Here are two common ways of obtaining chain complexes with vanishing homology: Chain complexes that compute the reduced homology of a contractible space Chain complexes that arise as a "long exact sequence in homology" induced by a short exact sequence of chain complexes These two examples seem...

I'm looking for a nontrivial, but not super difficult question concerning Fibonacci numbers. It should be at a level suitable for an undergraduate course. Here is a (not so good) example of the sort of thing I am looking for. a) Prove that every positive integer can be represented in binary o...

Can you prove that 8 is the largest cube in the Fibonacci sequence?

I wonder whether all (composites of) trigonometric and inverse trigonometric functions with periodic functional iterations can be found. In order to specify what I mean by that, let's introduce some notation first. Let $f^{[n]} (x)$ denote the function $f$ that has been iterated $n$ times with it...

Disclaimer: this question is a "big picture" one aiming at shedding light on something I've thought of for some time and seems to be also a matter of investigation by number theorists. I've just dowloaded a recent (May 2020) pdf: http://davidlowryduda.com/wp-content/uploads/2020/06/darmouth2020....

Let $(R,m)$ be a regular local ring of dimension $d$ and char $p>0.$ Let $F^e:R\longrightarrow R$ defined by $r\longrightarrow r^{p^e}$be the Frobenius map. How to compute $l(R/m^{[p^e]})?.$ I know the answer is $p^{ed}$ but I do not know how to prove it.

In case this is too general, here is a more specific question. Is there a hyperbolic threefold which admits a Hessian metric (hyperbolic or otherwise)? Background A Hessian manifold is a Riemannian manifold which admits an atlas of coordinate charts whose transition maps are affine (i.e. $x \ma...

This was asked and bountied at MSE with no response: My question is the following: Is there a nonstandard model $\mathcal{M}\models\mathsf{PA}$ such that $\mathcal{M}$ has no $\Delta^1_1$-with-parameters-definable nonempty proper successor-closed initial segments? Here "$\Delta^1_1$" is meant i...

This is a modified version of a question which was asked and bountied at MSE without success. Below, "$\mathsf{PA}$" refers to first-order Peano arithmetic. There are various "schematic" theories out there, like $\mathsf{PA}$ and $\mathsf{ZFC}$, which basically consist of three components: a "base"

(For simplicity, the background theory for this post is NBG, a set theory directly treating proper classes which is a conservative extension of ZFC.) Vopenka's Principle ($VP$) states that, given any proper class $\mathcal{C}$ of structures in the same (set-sized, relational) signature $\Sigma$,...

Is there an algebrization of second-order logic, analogous to Boolean algebras for propositional logic and cylindric and polyadic algebras for first-order logic?

In their paper "On Löwenheim-Skolem-Tarski numbers for extensions of first order logic", Magidor and Väänänen make the following statement: "For second order logic, $LS(L^{2})$ [the Löwenheim-Skolem number for second order logic--my comment] is the supremum of $\Pi_{2}$-definable ordinals..., wh...

Thinking about the recent threads on structural consequences of the Axiom of Foundation (AF) over ZF-AF, I've been trying to find some conservativity result which explains why AF doesn't seem to have consequences in "ordinary mathematics." Here's a candidate that came to mind. Let's call a sente...

Hi, It's shown by an easy cardinality argument that there are complete second-order theories that are not categorical (have more than one model up to isomorphism). Anyone knows of a concrete example of such a theory? Thanks in advance

So a common method used to construct non-zero $\omega$-REA arithmetic degrees with various properties is to build an $\omega$-REA operator $J$ satisfying the constraints that (for all $X$) $$\tag{1} J(X') \equiv_T J(X) \oplus X'$$ $$\tag{2} J(X) >_T X$$ Inductively, 1 implies that $J(X^n) \equiv_...

Suppose that $A$ is an $\omega$-REA set (so $A^{[n]}$ is r.e. in the prior columns). It is a well-known result that if each column of $A$ is computable then $A \leq_T \emptyset^2$ ($\emptyset^2$ can recover the computable index for each column from indices of prior columns and thus compute $A$)....

Remember a degree $\mathbb{d}$ is the $n$-lub of $\mathbb{c}_j$ in the Turing degrees if it is the least element (not merely a minimal element) set of $\mathbb{c}^{(n)}$ such that $\mathbb{c}$ computes every $\mathbb{c}_j$. It is non-trivial if it's not the $n$-th jump of a finite join of the d...

So, we take $\frac{\text{sgn}(x-1)}{x}$ and apply $\mathcal{L}_t[t f(t)](x)$ four times. The transform is known to keep area under the curve. These integrals, I think, are equal to minus Euler-Mascheroni constant. Since they all have infinite parts that cancel each other, their values are finite....