Call a sum of exponentials of the form $f(x)=\exp\left( 2\pi iN_{1}x\right) +\exp\left( 2\pi iN_{2}x\right) +\cdot\cdot\cdot+\exp\left( 2\pi iN_{m}x\right) $, where the $N_{k}$ are distinct integers, an idempotent. We have $L^{p}$ interval concentration if there is a positive constant $a$, depending only on $p$, such that for each interval $I\subset\left[ 0,1\right] $ there is an idempotent $f$ so that $\int _{I}\left\vert f\left( x\right) \right\vert ^{p}dx\diagup\int_{0} ^{1}\left\vert f\left( x\right) \right\vert ^{p}dx>a$. We will explain how to produce such concentration for each $p>0$. The origin of this question and the history of the development of its solution will be surveyed.

Let $X$ be a nonempty, topologically complete metric space with no isolated points. We show that there exists a closed upper porous set (in~a~strong sense) $F\subset X$ which is not $\s$-lower porous (in a weak sense). More precisely, we show that there exists a closed $(g_1)$-shell porous set $F\subset X$ which is not $\s$-$(g_2)$-lower porous, where $g_1$ and~$g_2$ are arbitrary admissible functions.

The concept of composite differentiation was introduced by O'Malley and Weil to generalize approximate differentiation. The concept of continuous path systems was introduced by us. This paper combines these concepts to introduce the notion of composite continuous path systems into differentiation theory. It is shown that a number of results that hold for composite differentiation and for continuous path differentiation also hold for composite continuous path differentiation. In particular, a composite continuous path derivative of a continuous function is a Baire class one function on some dense open set, and extreme composite continuous path derivatives of a continuous function are Baire class two functions. It is also shown that extreme composite continuous path derivatives of a Borel measurable function are Lebesgue measurable. Finally, for each composite continuous path system $E$, continuous functions typically do not have $E-$derived numbers with $E-$index less than one.

Let $f:[a, b]\to \R$ be a continuous function. Dividing the interval $[a, b]$ into subintervals of equal length, we obtain partitions of $[a, b]$ for which the upper and lower Darboux sums of $f$ constitute two sequences, which converge to the definite integral of $f$ in $[a, b]$ from above and below respectively. We study the monotonicity properties of these sequence and we prove that their non-monotonicity is a generic (quasi-sure) property in the space $C([a, b])$.

In this paper, new classes of non-continuous functions and multifunctions stronger than lower (upper) $\beta$-continuous functions and multifunctions due to Popa and Noiri [28] and [30] are introduced and investigated for characterizing the covering property $\beta$-closedness [7] from different angles.

In this paper we introduce the notion of $\Phi$-bounded variation for metric space valued mappings defined on a subset of the real line. Such a notion generalizes the one for real functions introduced by M. Schramm, and many previous generalized variations. We prove a structural theorem for mappings of $\Phi$-bounded variation. As an application we show that each mapping of $\Phi$-bounded variation defined on a subset of $\mathbb{R}$ possesses a $\Phi$-variation preserving extension to the whole real line.

We define a perturbed iterated function system (pIFS) in $\R^d$ as, loosely speaking, a sequence of iterated function systems (IFSs) whose constituent transformations converge towards some limiting IFS. We define the attractor of such a system in a similar style to that of an IFS, and prove that such a set exists uniquely. We define a partially perturbed IFS (ppIFS) to be a perturbed IFS with a constant tail. In a setup with similitudes and the strong separation condition we show that a pIFS attractor can be approximated by a sequence of ppIFS attractors in such a way that the Hausdorff measure is preserved in the limit. We use this result to calculate the exact Hausdorff measure of the pIFS attractor from that of the limiting IFS.

In this paper, we characterize scaling sets consisting of three intervals. In addition, we provide a procedure to obtain scaling sets possessing finitely many intervals.

A function $f : \mathbb{R}^n \to \mathbb{R}$ is called vertically rigid if $graph(cf)$ is isometric to $graph (f)$ for all $c \neq 0$. In [1] settled Janković's conjecture by showing that a continuous function $f : \mathbb{R}\to \mathbb{R}$ is vertically rigid if and only if it is of the form $a+bx$ or $a+be^{kx}$ ($a,b,k \in )$ prove that a continuous function $f :\mathbb{R}^2 \to \mathbb{R}$ is vertically rigid if and only if, after a suitable rotation around the $z$-axis, $f(x,y)$ is of the form $a + bx + dy$, $a + s(y)e^{kx}$ or $a + be^{kx} + dy$ ($a,b,d,k \in \mathbb{R}$, $k \neq 0$, $s : \mathbb{R} \to \mathbb{R}$ continuous). The problem remains open in higher dimensions.

For any $0<\epsilon <1,\ p\geq 1$ and each function $f\in L^{p}[0,1]$ one can find a function $g\in L^{p}[0,1],\ mes\{x\in \lbrack 0,1] ;\ g\neq f\}<\epsilon $, such that the sequence $\{|c_{k}(g)|,\ k\in spec(g)\}$ is monotonically decreasing, where $\{c_{k}(g)\}$ is\ the sequence of Fourier-Walsh coefficients of the function $g(x)$.

Let $f : [a,b] \subseteq \bar{\mathbb{R}}\to \mathcal{E}$ and $\phi : [a,b]\to \mathcal{F}$, where $(\mathcal{E},\mathcal{F},\mathcal{G})$ is a Banach space triple. a) We prove that if $\phi$ is continuous $[c,d ]\to [a,b]$ and $f\circ \psi \cdot d\phi\circ \psi$ is Kurzweil or Henstock variationally integrable, then so is $f\cdot d\phi$ and fulfills the well known change of variable formula. It follows that if $\psi$ is an indefinite Henstock integral and if $f\circ\psi\, \psi' dx $ is K-H integrable, then so is $f dx$ and the change of variable formula applies. b) We produce several versions of the converse of a), that is, we give necessary and sufficient conditions in order that with $\psi$ as above, the integrability of $f\cdot d\phi$ implies that of $f\circ \psi \cdot d\phi\circ\psi$ and the change of variable formula.

We give a new simpler proof of a theorem of Jayne and Rogers.

We consider an approach to differentiation that involves least squares lines of best fit rather than the traditional secant lines and use elementary techniques to show how this leads to the Lanczos derivative. A number of examples are presented to illustrate this concept and to show that the product, quotient, and chain rules fail for the Lanczos derivative. Several results giving conditions for which these rules do hold are discussed and proved. A brief introduction to higher order Lanczos derivatives is included.

The aim of this paper is to give some equivalent ways of defining McShane integral for vector valued functions.

Let $C(0,1)$ be the space of all continuous real-valued functions defined on an open interval $(0,1)$. We shall show that the multiplication is a weakly open operation in $C(0,1)$.

We characterize the families of quotients of functionally connected, connected and almost continuous functions. We prove also theorems concerning common divisor for the families of the quotients of functionally connected (resp. connected, almost continuous) functions with respect to functional connectivity (resp. connectivity, almost continuity).