We treat the classical notion of convexity in the context of hard real analysis. Definitions of the concept are given in terms of defining functions and quadratic forms, and characterizations are provided of different concrete notions of convexity. This analytic notion of convexity is related to more classical geometric ideas. Applications are given both to analysis and geometry.

In this paper we introduce the notion of $y$-lower semi-continuity and point out a distinction between a point of lower semi-continuity in global sense and a point of lower semi-continuity in local sense in general topological spaces after classifying points of $y$-lower semi-continuity (resp. lower semi-continuity) and also study their interrelationships. In particular, we find a necessary and sufficient condition for a bijective open multifunction on a $T_{2}$ space to be lower semi-continuous. Finally, a sufficient condition for an open bijective multifunction on the real line to have at most countable points of lower semi-discontinuity is formulated.

We have characterized the spaces $X$ for which the smallest $z$-ideal containing $\cinfty$ is prime. It turns out that $\cinfty$ is a $z$-ideal in $C(X)$ if and only if every zero-set contained in an open locally compact $\sigma$-compact set is compact. Some interesting ideals related to $\cinfty$ are introduced and corresponding to the relations between these ideals and $\cinfty$, topological spaces $X$ are characterized. Some compactness concepts are explicitly stated in terms of ideals related to $\cinfty$. Finally we have shown that a $\sigma$-compact space $X$ is Baire \ifif every ideal containing $\cinfty$ is essential.

We have characterized the spaces $X$ for which the smallest $z$-ideal containing $c_\infty$ is prime. It turns out that $c_\infty$ is a $z$-ideal in $C(X)$ if and only if every zero-set contained in an open locally compact $\sigma$-compact set is compact. Some interesting ideals related to $c_\infty$ are introduced and corresponding to the relations between these ideals and $c_\infty$, topological spaces $X$ are characterized. Some compactness concepts are explicitly stated in terms of ideals related to $c_\infty$. Finally we have shown that a $\sigma$-compact space $X$ is Baire if and only if every ideal containing $c_\infty$ is essential.

We say that $f:\mathbb{R}\to \mathbb{R}$ is LIF if it is linearly independent over $\mathbb{Q}$ as a subset of $\mathbb{R}^2$ and that it is a Hamel function (HF) if it is a Hamel basis of $\mathbb{R}^2$. In this paper we present a discussion on the lattices generated by the classes HF and LIF. We also investigate extensions of partial LIF functions to HF and LIF functions defined on whole $\mathbb{R}

We construct a $d$ Hausdorff dimensional compact set in $\R^d$ that does not contain the vertices of any parallelogram. We also prove that for any given triangle ($3$ given points in the plane) there exists a compact set in $\R^2$ of Hausdorff dimension $2$ that does not contain any similar copy of the triangle. On the other hand, we show that the set of the $3$-point patterns of a $1$-dimensional compact set of $\R$ is dense.

For subsets of a separable metric space $X$ we introduce the notion of upper porosity with respect to a~Borel regular probabilistic measure $\mu$ on $X$ (called $\mu$-upper porosity) that generalizes the concept of upper porosity of the measure $\mu$. We explore several natural definitions and further provide a definition of even more general type of $\mu$-upper porosity given by suitable porosity functions. As the main consequence of achieved results concerning general $\mu$-upper porosities we get that every $\sigma$-$\mu$-upper porous set can be decomposed to a $\sigma$-strongly upper porous set and a $\mu$-null set.

If a force is applied to a particle undergoing Brownian motion, the resulting motion has a state function which satisfies a diffusion or Schrödinger-type equation. We consider a process in which Brownian motion is replaced by a process which has Brownian transitions at all times other than random times at which the transitions have an additional "impulsive" displacement. Using a Feynman-Kac formulation based on generalized Riemann integration, we examine the resulting equation of motion.

We introduce the notion of $B_1$-retract and investigate the connection between $B_1$- and $H_1$-retracts.

Existence of measure preserving maps has been discussed in books on real analysis where the Axiom of Choice is instrumental. In this note we introduce a method to \textit{construct} such maps. For our construction we use the distribution function and elementary differential equations.

The purpose of this article is to offer a couple of short arguments for results describing the interaction between the norm and pointwise products in certain spaces of functions of generalized bounded variation.

In this paper we introduce $\mathcal{I^K}$-convergence which is a common generalization of the $\mathcal{I}^*$-convergence of sequences, double sequences, and nets. We show that many results that were shown before for these special cases are true for the $\mathcal{I^K}$-convergence, too.

A function $g$ between directed sets $\langle \Sigma, \succeq' \rangle$ and $\langle \Lambda, \succeq \rangle$ is called \emph{ultimately increasing} if for each $\sigma_1 \in \Sigma$ there exists $\sigma_2 \succeq' \sigma_1$ such that $\sigma \succeq' \sigma_2 \Rightarrow g(\sigma)\succeq g(\sigma_1)$. A subnet of a net $a$ defined on $\langle \Lambda, \succeq \rangle$ \cite {Ke} is nothing but a composition of the form $a \circ g$ where $g$ is ultimately increasing and $g(\Sigma)$ is a cofinal subset of $\Lambda$. While even for linearly ordered sets, an increasing net defined on a cofinal subset of the domain need not have an increasing extension, in complete generality, it must have an ultimately increasing extension, and conversely when the domain is linearly ordered. Applications are given in the context of functions with values in a linearly ordered set equipped with the order topology - in particular, the extended real numbers. For example, we show that a real sequence $\langle a_n \rangle$ converges to the supremum of its set of terms if and only if $\langle a_n \rangle$ is the supremum of the ultimately increasing sequences that it majorizes.

We extend the results of Peres and Solomyak on absolute continuity and singularity of homogeneous Bernoulli convolutions to inhomogeneous ones and generalize the result to random power series given by inhomogeneous Markov chains. In addition we prove an Erd\"{o}s-Salem type theorem for inhomogeneous Bernoulli convolutions.

We say that a function $f:\mathbb{R}\to \mathbb{R}$ is a {\it Hamel function} if $f$, considered as a subset of $\mathbb{R}^2$, is a Hamel basis of $\mathbb{R}^2$. We show that there is a Marczewski measurable Hamel function. Additionally, we show that there is a Hamel function which is both Lebesgue measurable and with the Baire property.

We say that $f: \mathR\to\mathR$ is a Hamel function if $f$, considered as a subset of $\mathR^2$, is a Hamel basis of $\mathR^2$. For a Cantor set $C\subset\mathR$ we construct a quasi-continuous Hamel function such that $f\restr(\mathR\setminus C)$ is of Baire class one.

By virtue of the uniqueness theorem for the Haar measure on a topological group, a simple argument is sufficient to show that the Haar measures on a locally--compact topological field, corresponding to the additive and multiplicative structures of the field, are absolutely continuous with respect to one another.

This note corrects three typographical errors in the paper \textit{Wavelet Sets Accumulating at the Origin}, Real Anal. Exchange \textbf{35(2)}, 463-478. The corrections are listed according to the page of the original article.