Articles of reference request

References mentioning the relationship between cumulants of uniform distribution and the Bernoulli numbers?

Is there anyone knows where is some official reference mentioning the relationship between cumulants of uniform distribution and the Bernoulli numbers (

Is this equality in a double category true?

Caveat: This is a utterly trivial question from a person who always learned to manipulate diagrams in a double category “from the ground”; I’ll be glad even if you simply address me to any source which gives precise rules of transformations for such diagrams (and this explains the “reference-request” tag). My problem is the following: […]

Essays on the real line?

Are there any essays on real numbers (in general?). Specifically I want to learn more about: The history of (the system of) numbers; their philosophical significance through history; any good essays on their use in physics and the problems of modeling a ‘physical’ line. Cheers. I left this vague as google only supplied Dedekind theory […]

Weakly convex functions are convex

Let us consider a continuous function $f \colon \mathbb{R} \to \mathbb{R}$. Let us call $f$ weakly convex if $$ \int_{-\infty}^{+\infty}f(x)[\varphi(x+h)+\varphi(x-h)-2\varphi(x)]dx\geq 0 \tag{1} $$ for all $h \in \mathbb{R}$ and all $\varphi \in C_0^\infty(\mathbb{R})$ with $\varphi \geq 0$. I was told that $f$ is weakly convex if, and only if, $f$ is convex; although I can […]

Comparing the deviation of a function from its mean on concentric balls

Let $\Omega \subset \mathbb{R}^n$ ($n\geq 2$) a domain with smooth boundary. Consider a ball $B(x_0,R) \Subset \Omega$ . Is true that $$ \int_{B(x_0,r)} | u – u_{x_0 , r}|^p \leq \int_{B(x_0,R)} | u – u_{x_0 , R}|^p,$$ for $u \in W^{1,p}(\Omega)(p >1)$ and for all $0 < r \leq R$, where $u_{x_0 , r} = […]

Every two positive integers are related by a composition of these two functions?

How would one prove/disprove this? … Conjecture: Suppose $p$, $q$ are distinct primes, and define $\ f(n) = n p, \ g(n) = \left \lfloor \frac{n}{q} \right \rfloor$ for all $n \in \mathbb{N_+}$; then for all $x,y \in \mathbb{N_+}$, there exists a composition $F = g \circ g \circ \cdots \circ g \circ f \circ […]

Building a hidden markov model with an absorbing state.

I’m working on trying to implement a hidden markov model to model the affect of a specific protein that can cut an RNA when the ribosome is translating the RNA slowly. Some brief background: The ribosome translates mRNA into protein The ribosome can occasionally pause on the mRNA due to things such as secondary structure […]

Representation of Cyclic Group over Finite Field

The post Irreducible representations of a cyclic group over a field of prime order discusses the irreducible representations of a cyclic group of order $N$ over a finite field $\mathbb{F}_p$ where $N$ does not divide $p$. Where can I find information about the irreducible representations in the case where $p$ does divide $N$? (I’m interested […]

The Strong Whitney Embedding Theorem-Any Recommended Sources?

Just about all of the standard textbooks on manifold theory give proofs of weak versions of the Whitney Embedding theorem. But other then Whitney’s original 1944 paper,are there any standard sources that contain a full proof of the strong version of the theorem? The only textbook I know that contains a full proof is Prasolov’s […]

Is every irreducible operator unitary equivalent to a banded operator?

This issue continues this question. Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : Let $(e_{n})_{n \in \mathbb{N}}$ be an orthonormal basis. $T \in B(H)$ is banded if $\exists r \in \mathbb{N}$ such that $ (Te_{n}, e_{m})\ne 0 \Rightarrow \vert n-m \vert \leq r$. Definition : […]