After surveying several earlier definitions of "chaos" , this paper is devoted to presenting the recently introduced notion of distributional chaosto a non-specialist audience. It is shown that the theory of distributional chaos avoids various shortcomings of the earlier theories and that it allows one not only to distinguish between chaotic and non-chaotic behavior, but also to measure the actual extent of any existing chaotic behavior. The whole is illustrated with numerous examples.

Erdős posed the following problem. "Let $E$ be an infinite set of real numbers. Prove that there is a set of real numbers $S$ of positive measure which does not contain a set $E^\prime$ similar (in the sense of elementary geometry) to $E$.'' The proof is known for only a few special cases; and not included among these is the geometric sequence $\{2^{-n}\}_{n=1}^\infty$. In this paper we examine the known literature, present some new results, and ask a few related questions.

In this note we present a simple proof of the inequality $\Phi \left( E^{\mathcal{A}}\xi \right) \leq E^{\mathcal{A}}\Phi (\xi )$ a.s. for separable random elements $\xi \mathcal{I}n L_{1}(\Omega ,\mathcal{F},P;X)$ in a Banach space $X,$ where $E^{\mathcal{A}}\left(\cdot\right) $ denotes conditional expectation with respect to the $\sigma $-field $\mathcal{A} \subset \mathcal{F}$, and $\Phi :X\rightarrow \mathbb{R}$ is a convex functional satisfying certain additional assumptions which are less restrictive than known till now. Some consequences of the above result are also discussed; e.g., it is shown that if $\xi $ is a Gaussian random element in $X$, then there exists a constant $0<c< \infty $ such that for each $\sigma $-field $\mathcal{A}_{0}\subset \mathcal{F}$ the family $\left\{ \exp \{c\left\| E^{\mathcal{A}}\xi \right\| ^{2}\}\mathcal{A}_{0}\subseteq \mathcal{A} \subseteq \mathcal{F}\right\} $ is uniformly integrable.

We define weak local dimension at each point in a deranged Cantor set. Using this, we classify the deranged Cantor set into a multifractal structure which is useful for computing the Hausdorff dimension and the packing dimension of the deranged Cantor set. Finally, we give examples in which the weak local dimensions are used for the calculation of the dimensions in a deterministic case and a probabilistic case.

In the paper the general connectivity property is given for multivalued maps and the Darboux property, the intermediate value property, functional connectivity property, connectivity property etc. are considered as subcases of this property. This general property is characterized locally, so as corollaries we obtain local characterization of the Darboux property, the intermediate value property etc. for multivalued maps and for real functions those classical results given by Bruckner, Ceder [2] and Garret, Nelms and Kellum [5]. Characterization of the sets of Darboux points, the intermediate value property points etc. for multivalued maps and for real functions are straightforward corollaries from one general theorem (Theorem 11).

We show that the Generalized Continuum Hypothesis GCH (its appropriate part) implies that many natural algebras on $\mathbb{R}$, including the algebra $\mathcal{B}$ of Borel sets and the interval algebra $\Sigma$, are outer Marczewski-Burstin representable by families of non-Borel sets. Also we construct, assuming again an appropriate part of GCH, that there are algebras on $\mathbb{R}$ which are not MB-representable. We prove that some algebras (including $\mathcal{B}$ and $\Sigma$) are not inner MB-representable. We give examples of algebras which are inner and outer MB-representable, or are inner but not outer MB-representable.

Sets of approximative asymmetry in the sense of category are introduced. The following theorem is proved. If $f : \mathbb{R}\to \mathbb{R}$ is a function, then the set of $\mathcal{I}$-asymmetry points of $f$ is of the type $F_{\sigma \delta \sigma}$ and is $\sigma$-well-porous. This illustrates the difference between measure and category. We give an example of a function with the set of $\mathcal{I}-$asymmetry points of the cardinality of the continuum.

It is shown that $Z$ does not contain a copy of $c_0$ if and only if for every bilinear map $B:X\times Y\to Z$, every function $f:[0,1]\to X$ of bounded $({\cal B})$-variation is $({\cal B})$-regulated.

A function is quasicontinuous if inverse images of open sets are semi-open. We generalize this definition: a collection of functions is setwise quasicontinuous if finite intersections of inverse images of open sets by functions in the collection are semi-open (so a function is quasicontinuous if and only if its singleton is a setwise quasicontinuous set). Two topologies on the same space are $\Pi$-related if each nonempty open set (in each) has non-empty interior with respect to the other. This paper demonstrates that a dynamical system is setwise quasicontinuous if and only if the original topology can be strengthened to one which is $\Pi$-related to it, and with respect to which each of the functions is continuous to the range space. Further, the set of iterates $\{1_X,f,f\circ f,\dots\}$ of a self-map $f:X\to X$, is setwise quasicontinuous if and only if the topology can be extended to a $\Pi$-related one, so that each iterate is continuous from the new space to the new space. We present a quasicontinuous function on the unit interval which is discontinuous on a dense subset of the interval; and show that conjugacies of dynamical systems via quasicontinuous bijections preserve much of the desired structure of the systems

In $\mathbb{R}^m$, we define the generalized Riemann integral over normal $m$-dimensional currents with compact support and bounded multiplicities, and prove the Stokes theorem for continuous $(m-1)$-forms that are pointwise Lipschitz outside sets of $\sigma$-finite $(m-1)$-dimensional Hausdorff measure. In addition, we show that the usual transformation formula holds for local lipeomorphisms, which need not be injective

We show the difference property and the double difference property for some classes of real-valued functions.

We give a condition which is shown to be equivalent to the $\Delta_2 $ condition and use it to prove a well-known result of Musielak and Orlicz.

It is known that smoothness-type conditions have several implications on continuity and differentiability properties of real functions. When these conditions hold uniformly on an interval the implications become even stronger. The aim of this paper is to extend to higher orders the relations between uniform smoothness-type conditions and differentiability, taking into account higher order divided differences.

In this paper we introduce and study the concept of ${\cal I}$-convergence of sequences in metric spaces, where ${\cal I}$ is an ideal of subsets of the set $\N$ of positive integers. We extend this concept to ${\cal I}$-convergence of sequence of real functions defined on a metric space and prove some basic properties of these concepts.

We extend the results of [2], [6] in the case of topological spaces. It is shown that given an upper semicontinuous (USC) function $f:X\to [0,\infty)$ where $X$ is a massive first countable $T_1$-space satisfying some "neighborhood conditions", there exists $F:X\to [0, \infty)$ whose oscillation equals $f$ everywhere on $X$ (Theorem 2.1). The analogous result holds for USC functions $f:X\to [0, \infty]$ if, in addition, $X$ is a normal space (Theorem 2.4). A special metrizability criterion is established (Theorem 1.1). This is to show, by exhibiting corresponding examples, that the neighborhood conditions and massiveness do not imply that $X$ is Baire or metrizable. Among other related topics, sequences of $\omega$-primitives are discussed.

For an $X\neq\emptyset$ and a given family $\mathcal{F}\subset \mathcal {P}(X)\setminus \{ \emptyset \} $, we consider the \mf \sodf which consists of sets $A\subset X$ such that each set $U\in \mathcal{F}$ contains a set $V\in \mathcal{F}$ with $V\subset A$ or $V\cap A=\emptyset$. We also study the respective ideal $S^0(\mathcal{F})$. We show general properties of $S^0(\mathcal{F})$ and certain representation theorems. For instance we prove that the interval algebra in $[0,1)$ is a Marczewski field. We are also interested in situations where $S(\mathcal{F}=S(\tau\setminus \{ \emptyset \} )$ for a topology $\tau $ on $X$. We propose a general method which establishes $S(\mathcal{F})$ and $S^0(\mathcal{F})$ provided that $\mathcal{F}$ is the family of perfect sets with respect to $\tau $, and $\tau$ is a certain ideal topology on $\mathbb{R}$ connected with measure or category.

In terms of Baire category, a typical real-valued Lipschitz function on a finite dimensional space has a local minimum at every point of a dense subset of the domain, and a Dini subdifferential that is either singleton or empty at all points. Moreover, its Dini subdifferential is empty outside a set of first category. Hence a typical Lipschitz function has no points of subdifferential regularity.

In this article we show that a function $f$, such that the complement of the set of points at which $f$ has the Darboux property and is bilaterally quasicontinuous is nowhere dense, must be the discrete limit of a sequence of bilaterally quasicontinuous Darboux functions. Moreover, there is given a construction of a function that is the discrete limit of a sequence of bilaterally quasicontinuous Darboux functions and which does not have a local Darboux property on a dense set.

The purpose of this paper is to show that given any non-zero cardinal number $n \leq {\aleph }_{0}$, the set of differentiable paths of class $C^{2}$ and of unit length in the plane having their arc length as the parameter in $[0,1]$ and tracing curves which have at least $n$ vertices is analytic non-Borel, while for any $r \in ({\N } \cup \{ \infty \} ) \setminus \{ 0,1,2 \} $, the set of differentiable paths of class $C^{r}$ and of unit length in the plane having their arc length as the parameter in $[0,1]$ and tracing curves which have at least $n$ vertices is $F_{\sigma }$ if $n<{\aleph }_{0}$ and $F_{\sigma \delta }$ if $n={\aleph }_{0}$.

We show that if \(f, g\colon [0,1]\to [0,1]\) are both Darboux Baire-1 functions, then their composition, \(f\circ g\), possesses a fixed point.

We establish a sharp integrability condition on the partial derivatives of a weakly monotone Sobolev function to guarantee differentiability almost everywhere.

In studies on fractal geometry, it is important to determine whether the classification by means of computational complexity is independent of the classification by means of fractal dimension. In this paper, we show that each self-similar set defined by polynomial time computable functions is polynomial time computable, if the self-similar set satisfies a polynomial time open set condition. This fact provides us examples of sets whose computational complexity are polynomial time computable, and which have non integer Hausdorff dimension. We also construct a set with computational complexity NP-complete and with an integer Hausdorff dimension. These two examples establish the independence of computational complexity and Hausdorff dimension.

The formulae for determining the quantization dimensions of self--similar probabilities satisfying the open set condition are proved by a new method. In addition, this method gives the exact order of convergence for the quantization errors.

Let $f_{(n)}$ and ${\underline f}{_{(n)}}$ denote the $n^{\rm th}$ Peano derivative and the $n^{\rm th}$ lower Peano derivative of the function $f:[a,b]\to $\mathbb{R} .$ We investigate the validity of the following statements.

$(M_n)$ If the set $H=\{ x\in [a,b]: {\underline f}{_{(n)}} (x)>0\}$ is of positive outer measure, then $f$ is $n$-convex on a subset of $H$ having positive outer measure.

$(Z_n)$ The set $E_n (f)=\{ x\in [a,b] : f_{(n)} (x)=\infty \}$ is of measure zero for every $f:[a,b]\to \mathbb{R} .$

We prove that ($M_n$) and ($Z_n$) are true for $n=1$ and $n=2,$ but false for $n\ge 3.$ More precisely we show that for every $n\ge 3$ there is an $(n-1)$ times continuously differentiable function $f$ on $[a,b]$ such that $f_{(n)} (x)=\infty$ a.e. on $[a,b] ,$ and that such a function cannot be $n$-convex on any set of positive outer measure. We also show that the category analogue of ($Z_n$) is false for every $n.$ Moreover, the set $E_n (f)$ can be residual. On the other hand, the category analogue of ($M_n$) is true for every $n.$ More precisely, if $\{ x\in [a,b] : {\underline f}{_{(n)}} (x)>0\}$ is of second category, then $f$ is $n$-convex on a subinterval of $[a,b].$ As a corollary we find that $E_n (f)$ cannot be residual and of full measure simultaneously.

Let $f$ be a real function on ${\mathbb R}$, let $\{I_v\}$ be a family of intervals covering a set $E$ such that $m(E \cap I_v) \ge m\bigl (f(E \cap I_v)\bigr )$ for each $I_v$. We prove that $m\bigl (f(E)\bigr ) \le 2 \cdot m(E)$. No coefficient smaller than $2$ will suffice here in general.

Packing measures are counterparts to Hausdorff measures, used in measuring fractal dimension of sets. C. Tricot defined them for subsets of finite-dimensional Euclidean space. We consider here the proper way to phrase the definitions for use in general metric spaces, and for Hausdorff functions other than the simple powers, in particular non-blanketed Hausdorff functions. The question of the Vitali property arises in this context. An example of a metric space due to R. O. Davies illustrates the concepts.

The present paper concerns with the introduction of a new type of generalized Stieltjes integral with an integrator function which depends on multiple points in a division and cannot be induced by a measure. Some properties of this integral were studied.

In response to a question posed by O.E. Lanford III, it is shown that for each \(\mu>0\) there is a differentiable and non-linearizable interval map $g$ with non-vanishing derivative defined on a neighborhood of a fixed point $0$ with \(g^\prime(0)=\mu\) such that $g$ has infinitely many differentiable and non-linearizable orientation-reversing composition square roots with non-vanishing first derivatives on a neighborhood of $0$.

In this paper, we give a characterization of the Kurzweil-Henstock integral in n-dimensional space.

We show that some sets in the spaces $c_0\times c_0$, $L^1[0,1]\times L^1[0,1]$ and $C[0,1]$, that appear in analysis, are meager.

K. Ciesielski asked whether the composition of two derivatives from the unit interval to itself always has a fixed point. The question is equivalent to asking if the composition of two Darboux, Baire one maps of $[0,1]$ to $[0,1]$ has a fixed point. The question is answered affirmatively for three subclasses of the Darboux, Baire one maps of $[0,1]$ to $[0,1]$

A subset $E$ of $\Bbb{R}^p$ is s-straight if $E$ has finite Hausdorff s-dimensional outer measure which equals its Method I s-outer measure. The graph of a continuously differentiable function is shown to be the countable union of closed 1-straight sets together with a set of Hausdorff 1-measure zero. This result is extended to the graphs of absolutely continuous functions and to regular sets.

For a finite set $\Lambda$ on the circle we consider a family of the multiplier operators $T_m$ in $l_2 ({\mathbb Z})$ generated by the $2^{-m}$-neighborhoods of $\Lambda$. We show that the norm of the corresponding maximal operator $T$ can not be estimated by an absolute constant.

We will show that the following set theoretical assumption

$\mathfrak{c}=\omega_2$, the dominating number $\mathfrak {d}$ equals to $\omega_1$, and there exists an $\omega_1$-generated Ramsey ultrafilter on $\omega$

(which is consistent with ZFC) implies that for an arbitrary sequence $f_n\colon\mathbb{R}\to\mathbb{R}$ of uniformly bounded functions there is a set $P\subset\mathbb{R}$ of cardinality continuum and an infinite $W\subset\omega$ such that $\{f_n\restriction P\colon n\in W\}$ is a monotone uniformly convergent sequence of uniformly continuous functions. Moreover, if functions $f_n$ are measurable or have the Baire property then $P$ can be chosen as a perfect set. We will also show that cof$(\mathcal{N})=\omega_1$ implies existence of a magic set and of a function $f\colon\mathbb{R}\to\mathbb{r}$ such that $f\restriction D$ is discontinuous for every $D\notin\mathcal{N}\cap\mathcal{M}$.

In this paper we shall show that every function $f\colon\Re^2\rightarrow \Re$ having the Baire property and ${\cal I}$-approximately continuous in ${\cal I}_1$-almost everywhere direction at every point is of the first class of Baire.

In this note we will show that for every natural number $n>0$ there exists an $S\subset[0,1]$ such that its $n$-th algebraic sum $nS=S+\cdots +S$ is a nowhere dense measure zero set, but its $n+1$-st algebraic sum $nS+S$ is neither measurable nor it has the Baire property. In addition, the set $S$ will be also a Hamel base, that is, a linear base of $\mathbb{R}$ over $\mathbb{Q}$.

Using a standard complete metric $w$ on the set $F$ of continuous functions of bounded variation on the interval $[0,1]$, we find that a typical function in $F$ has an infinite derivative at continuum many points in every subinterval of $[0,1]$. Moreover, for a typical function in $F$, there are continuum many points in every subinterval of $[0,1]$ where it has no derivative, finite nor infinite. The restriction of the derivative of a typical function in $F$ to the set of points of differentiability has infinite oscillation at each point of this set.

In this article we prove that if a function $f:X \to {\cal R}$ is the pointwise (discrete) [transfinite] limit of a sequence of real functions $f_n$ with closed graphs defined on complete separable metric space $X$ then $f$ is the pointwise (discrete) [transfinite] limit of a sequence of continuous functions. Moreover we show that each Lebesgue measurable function $f:{\cal R} \to {\cal R}$ is the discrete limit of a sequence of functions with closed graphs in the product topology $T_d\times T_e$, where $T_d$ denotes the density topology and $T_e$ the Euclidean topology.

We give an example of a non-integrable function $f$ on $[0,1]\times [0,1]$ such that $$ \int_{\alpha}^{\beta} (\int_{\gamma}^{\delta} f(x,y)\,dy)\,dx =\int_{\gamma}^{\delta} (\int_{\alpha}^{\beta} f(x,y)\,dx)\,dy$$ for each subinterval $[\alpha,\beta] \times [\gamma, \delta]$ of $[0,1] \times [0,1]$.

In this article we investigate approximate semicontinuity of a function related to Grande's SAC problem.

We consider the set valued function $\Omega $ taking $f$ in $\ C(I,I)$ to its collection of $\omega $-limit sets $\Omega (f)=\{\omega (x,f):x\in I\}$, and consider how $\Omega (f)$ is affected by pertubations of $f$. Our main result characterizes those functions $f$ in $C(I,I)$ at which $\Omega $ is upper semicontinuous, so that whenever $g$ is sufficiently close to $f$, every $\omega $-limit set of $g$ is close to some $\omega $-limit set of $f$ in the Hausdorff metric space. We also develop necessary and sufficient conditions for a function $f$ in $C(I,I)$ to be a point of lower semicontinuity of the map $\Omega$.

The following two questions were submitted by F. S. Cater. Let $\mathcal {F}$ denote the family of all absolutely continuous , nondecreasing functions on $[0,1]$. Endow $\mathcal{F}$ withe the complete metric $d$ defined by $$ d(f,g) = |f(0)-g(0)| + \int_o^1 |f'-g'|$$

Let

$\mathcal{G}=$ {$g\in \mathcal{F}; g'(x) = \infty$ for uncountablely many $x$ in each subinterval of $[0,1]$}.

It is easy to prove that $\mathcal {G}$ and $\mathcal {F}\setminus \mathcal{G}$ are dense subsets of $\mathcal {F}$.

1. Is $\mathcal {G}$ is a first category subset of $\mathcal {F}$?

2. Is $\mathcal {F}\setminus \mathcal {G}$ a first category subset of $\mathcal {F}$?