This paper is meant as a (short and partial) introduction to the study of the geometry of Carnot groups and, more generally, of Carnot-Carathéodory spaces associated with a family of Lipschitz continuous vector fields. My personal interest in this field goes back to a series of joint papers with E. Lanconelli, where this notion was exploited for the study of pointwise regularity of weak solutions to degenerate elliptic partial differential equations. As stated in the title, here we are mainly concerned...

The aim of this manuscript is to determine the relative size of several functions (copulas, quasi– copulas) that are commonly used in stochastic modeling. It is shown that the class of all quasi–copulas that are (locally) associated to a doubly stochastic signed measure is a set of first category in the class of all quasi– copulas. Moreover, it is proved that copulas are nowhere dense in the class of quasi-copulas. The results are obtained via a checkerboard approximation of quasi–copulas.

It is shown that a Banach-valued Henstock-Kurzweil integrable function on an $m$-dimensional compact interval is McShane integrable on a portion of the interval. As a consequence, there exist a non-Perron integrable function $f{[0,1]}^2⟶ℝ$ and a continuous function $F{[0,1]}^2⟶ℝ$ such that $$\left(\P \right){\int }_0^x\left\{\left(\P \right){\int }_0^{y}f(u,v)\mathrm{d}v\right\}\mathrm{d}u=\left(\P \right){\int }_0^y\left\{\left(\P \right){\int }_0^{x}f(u,v)\mathrm{d}u\right\}\mathrm{d}v=F(x,y)$$ for all $(x,y)\in {[0,1]}^2$.

A topological space $X$ is called base-base paracompact (John E. Porter) if it has an open base $ℬ$ such that every base ${ℬ}^{\text{'}}\subseteq ℬ$ has a locally finite subcover $𝒞\subseteq {ℬ}^{\text{'}}$. It is not known if every paracompact space is base-base paracompact. We study subspaces of the Sorgenfrey line (e.g. the irrationals, a Bernstein set) as a possible counterexample.

In this paper, we propose a new method to generate a continuous belief functions from a multimodal probability distribution function defined over a continuous domain. We generalize Smets' approach in the sense that focal elements of the resulting continuous belief function can be disjoint sets of the extended real space of dimension n. We then derive the continuous belief function from multimodal probability density functions using the least commitment principle. We illustrate the approach on two...

In this paper, we propose a new method to generate a continuous belief functions from a multimodal probability distribution function defined over a continuous domain. We generalize Smets' approach in the sense that focal elements of the resulting continuous belief function can be disjoint sets of the extended real space of dimension n. We then derive the continuous belief function from multimodal probability density functions using the least commitment principle. We illustrate the approach on two...

Obsahuje tyto části: 1. Benoit Mandelbrot vyznamenán za velký vědecký čin. 2. J. W. Cannon: recenze knihy B. B. Mandelbrota „Fraktální geometrie přírody‟. 3. David Preiss: Něco málo matematiky k fraktálúm.

The necessary and sufficient condition for a function $f:(0,1)\to [0,1]$ to be Borel measurable (given by Theorem stated below) provides a technique to prove (in Corollary 2) the existence of a Borel measurable map $H:{\{0,1\}}^{\mathbb{N}}\to {\{0,1\}}^{\mathbb{N}}$ such that $ℒ\left(H\left({\mathbf{\text{X}}}^p\right)\right)=ℒ\left({\mathbf{\text{X}}}^{1/2}\right)$ holds for each $p\in (0,1)$, where ${\mathbf{\text{X}}}^p=(X_1^p,X_2^p,...)$ denotes Bernoulli sequence of random variables with $P[X_i^p=1]=p$.

We construct Bernstein sets in ℝ having some additional algebraic properties. In particular, solving a problem of Kraszewski, Rałowski, Szczepaniak and Żeberski, we construct a Bernstein set which is a < c-covering and improve some other results of Rałowski, Szczepaniak and Żeberski on nonmeasurable sets.

Let $E$ be a self-similar set with similarities ratio $r_j(0\le j\le m-1)$ and Hausdorff dimension $s$, let $\overrightarrow{p}(p_0,p_1)...p_{m-1}$ be a probability vector. The Besicovitch-type subset of $E$ is defined as$$E\left(\overrightarrow{p}\right)=\left\{x\in E:\underset{n\to \infty }{lim}\frac{1}{n}\sum _{k=1}^n{\chi }_j\left(x_k\right)=p_j,0\le j\le m-1\right\},$$where ${\chi }_j$ is the indicator function of the set $\left\{j\right\}$. Let $\alpha ={dim}_H\left(E\left(\overrightarrow{p}\right)\right)={dim}_P\left(E\left(\overrightarrow{p}\right)\right)=\frac{{\sum }_{j=0}^{m-1}{p}_{j}log{p}_j}{{\sum }_{j=0}^{m-1}{p}_{i}log{r}_j}$ and $g$ be a gauge function, then we prove in this paper:(i) If $\overrightarrow{p}=(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, then$${\mathscr{H}}^s\left(E\left(\overrightarrow{p}\right)\right)={\mathscr{H}}^s\left(E\right),{𝒫}^s\left(E\left(\overrightarrow{p}\right)\right)={𝒫}^s\left(E\right),$$moreover both of ${\mathscr{H}}^s\left(E\right)$ and ${𝒫}^s\left(E\right)$ are finite positive;(ii) If $\overrightarrow{p}$ is a positive probability vector other than $(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, then the gauge functions can be partitioned as follows$${\mathscr{H}}^g\left(E\left(\overrightarrow{p}\right)\right)=+\infty \iff \underset{t\to 0}{\overline{\lim }}\frac{logg\left(t\right)}{logt}\le \alpha ;{\mathscr{H}}^g\left(E\left(\overrightarrow{p}\right)\right)=0⟺\underset{t\to 0}{\overline{\lim }}\frac{logg\left(t\right)}{logt}\&gt;\alpha ,$$$$...$$