Let $L$ be a second order linear elliptic differential operator on an open bounded subset $U\subset \mathbb R^n$, with smooth uniformly bounded coefficients. Suppose the boundary of $U$ is $C^\infty$. Suppose $f\in C_c^\infty(U)$ ($f$ is smooth and has compact support in $U$). Must there exist a solution $u$ to the PDE $Lu = f$, $u|_{\partial […]

I am looking for either a reference, a proof, or a suitable proof sketch that can explain Ulam’s original argument about measure theory and measurable cardinals. Here is the result I am looking for: The smallest cardinal $\kappa$ that admits a non-trivial countably-additive two-valued measure must be inaccessible. The original paper can be found below […]

Would you advise me on the references of Pringsheim Convergence about interchanging the order of limits? Where can I find the most general statement?

In Bitopological spaces, Proc. London Math. Soc. (3) 13 (1963) 71–89 MR0143169, J.C. Kelly introduced the idea of bitopological spaces. Is there any paper concerning the generalization of this concept, i.e. a space with any number of topologies?

Can anyone recommend a good for MCMC? I have worked with HMMs, Markov Chains in the past but nothing on simulation. So something in the intermediate level would be great. Also, if you know of any introductory books on stochastic simulation, that would be great.

It seems that the cumulative mean of the Mertens function is very similar in behaviour to $x$ raised to the power of the first zeta zero. I tentatively notate it as: $$\frac{1}{x}\left(\sum_{k=1}^xM(k)\right)+2\sim\Im((\text{K}x) ^{\rho_{1}}/\text{c})$$ where $\text{c}\approx64,\ \rm{K}$ is Catalan’s constant, $M(k)$ is the Mertens function of $k$, and $\rho_n$ is the nth zeta zero, and the […]

While attending a math puzzle contest, my friend (a math student) asked me to prove that $$\sum_{k=1}^n \frac{1}{k} \notin \mathbb{Z} \quad \forall n \geq 2$$ Being the first time seeing this problem, I came up with a proof that required the following conjecture: (1) Given a composite number $n \geq 4$, $\exists p$ prime such […]

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. […]

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