1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) $x=1/d₁\left(x\right)+1/\left(d₁\left(x\right)d₂\left(x\right)\right)+...+1/(d₁\left(x\right)d₂\left(x\right)...d_n\left(x\right))+...$, where $d_j\left(x\right),j\ge 1$ is a sequence of positive integers satisfying d₁(x) ≥ 2 and $d_{j+1}\left(x\right)\ge d_j\left(x\right)$ for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) $li{m}_{n\to \infty }{d}_n^{1/n}\left(x\right)=e.$He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. $di{m}_{H}x\in (0,1]:\left(2\right)fails=1$. We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and $di{m}_H$ to denote...

We prove the interior Hölder continuity of weak solutions to parabolic systems $$\frac{\partial u^j}{\partial t}-D_{\alpha }{a}_j^{\alpha }(x,t,u,\nabla u)=0\text{in}Q(j=1,...,N)$$ ($Q=\Omega \times (0,T),\Omega \subset {ℝ}^2$), where the coefficients $a_j^{\alpha }(x,t,u,\xi )$ are measurable in $x$, Hölder continuous in $t$ and Lipschitz continuous in $u$ and $\xi$.

Let X be a compactum and let $A=(A_i,B_i):i=1,2,...$ be a countable family of pairs of disjoint subsets of X. Then A is said to be essential on Y ⊂ X if for every closed $F_i$ separating $A_i$ and $B_i$ the intersection $\left(\cap F_i\right)\cap Y$ is not empty. So A is inessential on Y if there exist closed $F_i$ separating $A_i$ and $B_i$ such that $\cap F_i$ does not intersect Y. Properties of inessentiality are studied and applied to prove:  Theorem. For every countable family of pairs of disjoint open subsets of a compactum X there exists an open set G ∩ X on...

A partition of a positive integer n is a nonincreasing sequence of positive integers with sum $n.$ Here we define a special class of partitions. 1. Let $t\ge 1$ be a positive integer. Any partition of n whose Ferrers graph have no hook numbers divisible by t is known as a t- core partitionof $n.$ The hooks are important in the representation theory of finite symmetric groups and the theory of cranks associated with Ramanujan’s congruences for the ordinary partition function [3,$$4,$$6]. If $t\ge 1$ and $n\ge 0$,...

We study ergodicity of cylinder flows of the form   $T_f:T\times ℝ\to T\times ℝ$, $T_f(x,y)=(x+\alpha ,y+f\left(x\right))$, where $f:T\to ℝ$ is a measurable cocycle with zero integral. We show a new class of smooth ergodic cocycles. Let k be a natural number and let f be a function such that $D^{k}f$ is piecewise absolutely continuous (but not continuous) with zero sum of jumps. We show that if the points of discontinuity of $D^{k}f$ have some good properties, then $T_f$ is ergodic. Moreover, there exists ${\epsilon }_f>0$ such that if $v:T\to ℝ$ is a function with zero integral such that $D^{k}v$ is of bounded...

Introduction. An old conjecture of P. Erdős repeated many times with a prize offer states that the counting function A(n) of a $B_r$-sequence A satisfies $limin{f}_{n\to \infty }(A\left(n\right)/\left(n^{1/r}\right))=0$. The conjecture was proved for r=2 by P. Erdős himself (see [5]) and in the cases r=4 and r=6 by J. C. M. Nash in [4] and by Xing-De Jia in [2] respectively. A very interesting proof of the conjecture in the case of all even r=2k by Xing-De Jia is to appear in the Journal of Number Theory [3]. Here we present a different, very short proof...

We parametrize Cichoń’s diagram and show how cardinals from Cichoń’s diagram yield classes of small sets of reals. For instance, we show that there exist subsets N and M of $w^w\times 2^w$ and continuous functions $e,f:w^w\to w^w$ such that  • N is $G_{\delta }$ and $N_x:x\in w^w$, the collection of all vertical sections of N, is a basis for the ideal of measure zero subsets of $2^w$;  • M is $F_{\sigma }$ and $M_x:x\in w^w$ is a basis for the ideal of meager subsets of $2^w$;  •$\forall x,y{N}_{e\left(x\right)}\subseteq N_y\Rightarrow M_x\subseteq M_{f\left(y\right)}$. From this we derive that for a separable metric space X,  •if for all Borel (resp. $G_{\delta }$) sets...

We give sufficient and necessary conditions to be fulfilled by a filter $\Psi$ and an ideal $\Lambda$ in order that the $\Psi$-quotient space of the $\Lambda$-ideal product space preserves $T_k$-properties ($k=0,1,2,3,3\frac{1}{2}$) (“in the sense of the Łos theorem”). Tychonoff products, box products and ultraproducts appear as special cases of the general construction.