Given a bicubic planar graph consisting of faces with degree $4$ and $6$, so called Barnette graphs. We can show that there are exactly six squares. Kundor and I found six types of arrangements of the six squares: three pairs of squares $(2+2+2)$ two triples arranged in row $(\bar3+\bar3)$ two triples arranged like a triangle […]

Let $l_p$ be the space of all complex sequences $x=\{x_n\}$ equipped with the finite norm $\|x\|_p=\left(\sum_n |x_n|^p\right)^{1/p}$. Hilbert’s double series theorem states that $$\sum_{n,m}\frac{a_n b_m}{m+n}<\frac{\pi}{\sin(\pi/p)} \|a\|_p \|b\|_q$$ for non negative $a=\{a_n\}\in l_p$ and $b=\{b_n\}\in l_q$. In literature there are known some generalizations of this theorem, e.g. when $$\sum_{n,m}\frac{a_n b_m}{(m+n)^\lambda}$$ but, for example, I did not […]

Let $A = \{a_1,\dots,a_m\}$ be a finite set endowed with a discrete topology and let $X = A^{\Bbb N}$ be the product topological space. I wonder which bounded functions $f:X\to\Bbb R$ are continuous on $X$. For example, it is clear that if $f$ depends only on a finite number of coordinates then $f\in C(X)$, i.e. […]

I am looking for a reference (not proof) to the following theorem: If $X$ is a compact and locally connected topological space, Y is a Hausdorff topological space, $f:X\to Y$ is continuous and $f(X)=Y$, then $Y$ is locally connected. I found reference in Kuratowski, Topology II for the case where $X$ is metrizable. Any one […]

Consider a bounded region $\Omega\subset\mathbb R^n$ with a finite number of “holes” $X=\{x_1,\ldots,x_k\}$ that are isolated points in its interior. I’m pretty sure that in more than one dimension, it doesn’t make sense to solve Laplace’s equation with Dirichlet boundary conditions on $X$. That is, there does not exist any “valid”* $f$ such that $$\begin{align} […]

I am interested in whether a quantity \begin{align*} E[(X-E[X|Y])^4] \end{align*} has been studied in the literature before. I am not even sure if “least mean fourth error” is a correct name, since $g(Y)=E[X|Y]$ might not be the best estimator for the $\inf_{g(y)} E[(X-g(Y))^4]$. However, I am intersted in $E[(X-E[X|Y])^4]$ rather then $\inf_{g(y)} E[(X-g(Y))^4]$. Specifically, I […]

I’m studying differential equations (specifically Laplace Transforms) right now with my college assigned ‘Differential Equations with Application and Historical Notes’-George F Simmons. While I like the text, I’m not a big fan of the fact that there are not many solved examples and a solution manual isn’t available. Can you guys suggest a decent book […]

I am looking for good textbooks for lattice and coding theory. Lattice and coding theory are very interesting on their own, but I have application of the theory to K3 surfaces & modular forms (and vice versa) in mind. My goal is probably to go through Conway & Slone’s “Sphere Packings, Lattices and Groups” but […]

In constructive mathematics we often hear expressions such as “extracting computational content from proofs”, “the constructivity of mathematics lies in its computational content”, “realizability models allow to study the computational content”, “intuitionistic double-negation forgets the computational content of a proposition”, and so on. I was wondering if there is a formal notion of computational content […]

There is an exercise in Burris and Sankappanavar’s “A Course in Universal Algebra”: Problem: Find two algebras $\mathbf{A}_1$, $\mathbf{A}_2$ such that neither can be embedded in $\mathbf{A}_1 \times \mathbf{A}_2$. My solution: Consider $\mathbb{Z}_2$ and $\mathbb{Z}_3$ as a rings with unity (in $\langle +, \cdot, -, 0, 1 \rangle$ signature). Assume $\varphi \colon \mathbb{Z}_2 \to \mathbb{Z}_2 […]

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