We prove in ZFC that there is a set $A\subseteq 2^{\omega }$ and a surjective function H: A → ⟨0,1⟩ such that for every null additive set X ⊆ ⟨0,1), $H^{-1}\left(X\right)$ is null additive in $2^{\omega }$. This settles in the affirmative a question of T. Bartoszyński.

Let $R$ be a prime ring of characteristic different from 2 and 3, $Q_r$ its right Martindale quotient ring, $C$ its extended centroid, $L$ a non-central Lie ideal of $R$ and $n\ge 1$ a fixed positive integer. Let $\alpha$ be an automorphism of the ring $R$. An additive map $D:R\to R$ is called an $\alpha$-derivation (or a skew derivation) on $R$ if $D\left(xy\right)=D\left(x\right)y+\alpha \left(x\right)D\left(y\right)$ for all $x,y\in R$. An additive mapping $F:R\to R$ is called a generalized $\alpha$-derivation (or a generalized skew derivation) on $R$ if there exists a skew derivation $D$ on $R$ such that $F\left(xy\right)=F\left(x\right)y+\alpha \left(x\right)D\left(y\right)$ for all $x,y\in R$. We prove...

For each ordinal 1 ≤ α < ω₁ we present separable metrizable spaces $X_{\alpha }$, $Y_{\alpha }$ and $Z_{\alpha }$ such that (i) $f{X}_{\alpha },f{Y}_{\alpha },f{Z}_{\alpha }=\omega ₀$, where f is either trdef or ₀-trsur, (ii) $A\left(\alpha \right)-trind{X}_{\alpha }=\infty$ and $M\left(\alpha \right)-trind{X}_{\alpha }=-1$, (iii) $A\left(\alpha \right)-trind{Y}_{\alpha }=-1$ and $M\left(\alpha \right)-trind{Y}_{\alpha }=\infty$, and (iv) $A\left(\alpha \right)-trind{Z}_{\alpha }=M\left(\alpha \right)-trind{Z}_{\alpha }=\infty$ and $A(\alpha +1)\cap M(\alpha +1)-trind{Z}_{\alpha }=-1$. We also show that there exists no separable metrizable space $W_{\alpha }$ with $A\left(\alpha \right)-trind{W}_{\alpha }\ne \infty$, $M\left(\alpha \right)-trind{W}_{\alpha }\ne \infty$ and $A\left(\alpha \right)\cap M\left(\alpha \right)-trind{W}_{\alpha }=\infty$, where A(α) (resp. M(α)) is the absolutely additive (resp. multiplicative) Borel class.

CONTENTSPART IIntroduction............................................................................................... 31. General solution.................................................................................. 42. Preliminaries and notation................................................................ 53. $C^p$ solutions in $ℭ$*................................................ 74. Change of variables..............................................................................

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

We show that any ${\Sigma }_s$-product of at most $𝔠$-many $L\Sigma (\le \omega )$-spaces has the $L\Sigma (\le \omega )$-property. This result generalizes some known results about $L\Sigma (\le \omega )$-spaces. On the other hand, we prove that every ${\Sigma }_s$-product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every ${\Sigma }_s$-product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

Given a finite additive abelian group $G$ and an integer $k$, with $3\le k\le \left|G\right|$, denote by ${𝒟}_k\left(G\right)$ the simple incidence structure whose point-set is $G$ and whose blocks are the $k$-subsets $C=\{c_1,c_2,\cdots ,c_k\}$ of $G$ such that $c_1+c_2+\cdots +c_k=0$. It is known (see [Caggegi, A., Di Bartolo, A., Falcone, G.: Boolean 2-designs and the embedding of a 2-design in a group arxiv 0806.3433v2, (2008), 1–8.]) that ${𝒟}_k\left(G\right)$ is a 2-design, if $G$ is an elementary abelian $p$-group with $p$ a prime divisor of $k$. From [Caggegi, A., Falcone, G., Pavone, M.: On the additivity...

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...