We study notions of absolute continuity for functions defined on $\mathbb{R}^n$ similar to the notion of $\alpha$-absolute continuity in the sense of Bongiorno. We confirm a conjecture of Malý that 1-absolutely continuous functions do not need to be differentiable a.e., and we show several other pathological examples of functions in this class. We establish some containment relations of the class $1\textit{-} AC_{\rm WDN}$ which consits of all functions in $1\textit{-}AC$ which are in the Sobolev space $W^{1,2}_{loc}$, are differentiable a.e. and satisfy the Luzin (N) property, with previously studied classes of absolutely continuous functions.

In this paper we study a class of random Cantor sets. We determine their almost sure Hausdorff, packing, box, and Assouad dimensions. From a topological point of view, we also compute their typical dimensions in the sense of Baire category. For the natural random measures on these random Cantor sets, we consider their almost sure lower and upper local dimensions. In the end we study the hitting probabilities of a special subclass of these random Cantor sets.

Some inequalities similar to Jensen inequalities for general Lebesgue integral are obtained. Applications for functions of self adjoint operators and functions of unitary operators on complex Hilbert spaces are provided as well.

We approximate the uniform measure on an equilateral triangle by a measure supported on $n$ points. We find the optimal sets of points ($n$-means) and corresponding approximation (quantization) error for $n\leq4$, give numerical optimization results for $n\leq 21$, and a bound on the quantization error for $n\to\infty$. The equilateral triangle has particularly efficient quantizations due to its connection with the triangular lattice. Our methods can be applied to the uniform distributions on general sets with piecewise smooth boundaries.

In this paper we introduce, for any coideal basis $\mathcal{B}$ on the set $\nat$ of natural numbers, the notions of a $\mathcal{B}$-sequence, a $\mathcal{B}$-subsequence of a $\mathcal{B}$-sequence, and a $\mathcal{B}$-convergent sequence in a metric space. The usual notions of a sequence, subsequence, and convergent sequence obtain for the coideal $\mathcal{B}$ of all the infinite subsets of $\nat$. We first prove a Bolzano-Weierstrass theorem for $\mathcal{B}$-sequences: if $\mathcal{B}$ is a Ramsey coideal basis on $\nat$, then every bounded $\mathcal{B}$-sequence of real numbers has a $\mathcal{B}$-convergent $\mathcal{B}$-subsequence; and next, with the help of this extended Bolzano-Weierstrass theorem, we establish an extension of the fundamental Rosenthal's $\ell_1$-dichotomy theorem: if $\mathcal{B} $ is a semiselective coideal basis on $\mathbb{N}$, then every bounded $\mathcal{B}$-sequence of real valued functions $(f_n)_{n\in A}$ has a $\mathcal{B}$-subsequence $(f_n)_{n\in B}$, which is either $\mathcal{B}$-convergent or equivalent to the unit vector basis of $\ell_1(B)$.

Smoothness conditions on a function $f:\mathbb{R}^{2}\rightarrow \mathbb{R}$ that are weaker than being differentiable or Lipschitz at a point are defined and studied.