This paper generalizes some results from another one, namely [3]. We have studied the issues of expressing natural numbers as a sum of powers of natural numbers in paper [3]. It means we have studied sets of type $A=\{n_1^{k_1}+n_2^{k_2}+\cdots +n_m^{k_m}\mid n_i\in \mathbb{N}\cup \left\{0\right\},i=1,2\cdots ,m,(n_1,n_2,\cdots ,n_m)\ne (0,0,\cdots ,0)\},$ where $k_1,k_2,\cdots ,k_m\in \mathbb{N}$ were given natural numbers. Now we are going to study a more general case, i.e. sets of natural numbers that are expressed as sum of integral parts of functional values of some special functions. It means that we are interested in sets of natural numbers in the form...

In this paper we study integral operators of the form $$Tf\left(x\right)=\int |x-a_1{y|}^{-{\alpha }_1}\cdots {|x-a_{m}y|}^{-{\alpha }_m}f\left(y\right)\mathrm{d}y,$$ ${\alpha }_1+\cdots +{\alpha }_m=n$. We obtain the $L^p\left(w\right)$ boundedness for them, and a weighted $(1,1)$ inequality for weights $w$ in $A_p$ satisfying that there exists $c\ge 1$ such that $w\left(a_{i}x\right)\le cw\left(x\right)$ for a.e. $x\in {ℝ}^n$, $1\le i\le m$. Moreover, we prove ${\parallel Tf\parallel }_{\mathrm{B}MO}\le c{\parallel f\parallel }_{\infty }$ for a wide family of functions $f\in L^{\infty }\left({ℝ}^n\right)$.

Shelah’s pcf theory describes a certain structure which must exist if ${\aleph }_{\omega }$ is strong limit and $2^{{\aleph }_{\omega }}>{\aleph }_{{\omega }_1}$ holds. Jech and Shelah proved the surprising result that this structure exists in ZFC. They first give a forcing extension in which the structure exists then argue that by some absoluteness results it must exist anyway. We reformulate the statement to the existence of a certain partially ordered set, and then we show by a straightforward, elementary (i.e., non-metamathematical) argument that...

In this note we investigate the relationship between the convergence of the sequence $\left\{S_n\right\}$ of sums of independent random elements of the form $S_n={\sum }_{i=1}^n{\epsilon }_i{x}_i$ (where ${\epsilon }_i$ takes the values $\pm 1$ with the same probability and $x_i$ belongs to a real Banach space $X$ for each $i\in \mathbb{N}$) and the existence of certain weakly unconditionally Cauchy subseries of ${\sum }_{n=1}^{\infty }{x}_n$.