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We study the properties of \begin{document}$ \Phi $\end{document} -irregular sets (sets of points for which the Birkhoff average diverges) in dynamical systems with the shadowing property. We estimate the topological entropy of \begin{document}$ \Phi $\end{document} -irregular set in terms of entropy on chain recurrent classes and prove that \begin{document}$ \Phi $\end{document} -irregular sets of full entropy are typical. We also consider \begin{document}$ \Phi $\end{document} -level sets (sets of points whose Birkhoff average is in a specified interval), relating entropy they carry with the entropy of some ergodic measures. Finally, we study the problem of large deviations considering the level sets with respect to reference measures.

The area of the Nízký Jeseník Mts. is, among other things, well known for its shale roofing tiles since the 18th century. In places where shale was intensively or extensively exploited until 1945, abandoned areas after mining works remained. In general, every mining is perceived as a activity of landscape degradation by the public. However, these indelible traces of shale mining in the form of various mining-related objects (e.g. abandoned quarries, quarry ponds, shafts, drains etc.) are also gradually becoming places that are colonised by unique plant and animal communities. There are very interesting species bond to specific environmental conditions of post-mining landscape, with frequent rare and endangered species. People have also become ‘new’ colonisers in the case of the shale landscape.

Recently Bernardes and Darji provided a very nice characterization of a residual set of maps of Cantor set in terms of covers of special type. Using their characterization, we provide a more direct description of this class. This way we are able to provide a further characterization of dynamical properties (e.g. shadowing properties, nullness) of maps in the class and further study of what features (e.g. prescribed minimal sets or the values of topological entropy) we can get by small perturbations of a given homeomorphism.

We show that every (invertible, or noninvertible) minimal Cantor system embeds in $\mathbb{R}$ with vanishing derivative everywhere. We also study relations between local shrinking and periodic points.

In this contribution, we propose a generalization of a four-fold table known from classical statistics to the case when fuzzy sets represent attributes in a data set. We define fuzzy quantifiers on the generalized four-fold table and investigate various classes of such quantifiers. We also study properties of fuzzy four-fold tables and several quality measures based on them and relations between the choice of used t-norms and chosen quality measures. We do not only present a new theoretical approach to fuzzy association analysis but, based on our results, we conclude this paper with a short discussion on the best choice of t-norm used for mining fuzzy associations.

The article is focused on the application of Energy dispersive micro X-ray fuorescence spectroscopy as a specifc method to determine the contents of potentially toxic elements and its spread in plant tissues. As a model species, Quercus spp. were selected. In order to compare the obtained results with previous research, four well-described abandoned Cu-deposits were selected for sampling: Ľubietová (Slovakia), Libiola and Caporciano (Italy), and São Domingos (Portugal). The results of micro X-ray fuorescence spectrometry confrm the irregular contamination of Quercus spp. by potentially toxic elements. The level of contamination is the highest predominantly in the root cortex, where is also the highest Ca contents (with exception of São Domingos). At Ľubietová and Caporciano, high Ni content was described in branches cortex, in branches mesoderm also Fe, Cu and Zn. At the same time, the inhibition infuence of Ca was also confrmed regarding the input of these elements into plants. info:eu-repo/semantics/publishedVersion

We are interested in dendrites for which all invariant measures of zero-entropy mappings have discrete spectrum, and we prove that this holds when the closure of the endpoint set of the dendrites is countable. This solves an open question which has been around for awhile, and almost completes the characterization of dendrites with this property.

To increase their resilience to climate change, cities are looking to apply elements of urban environmental acupuncture. The essence of such measures is many smaller sites that is functioning as mitigation measures. Many of these small places then create a large overall effect. The advantage of these small-scale measures is that they can be in densely populated areas The assessment tool described in this paper is designed for city representatives and is an aid to assess the suitability of applying a particular measure based on the parameters described. The evaluation itself then helps to decide whether the solution is suitable for a particular site or whether any of the parameters need to be adjusted to make it suitable, or whether it would be appropriate to change the proposed solution. The intention of the evaluation is not to assess the technical solution but relies primarily on the location, long-term (especially financial) sustainability and acceptance by the citizens of the city. The paper presents an example of the application of the evaluation to four sites in city Liptovský Mikuláš, describing the results and identifying parameters that can be improved to ensure the urban environmental acupuncture is accepted by citizens and thus future-proofed.

Motivated by a recent result of Ciesielski and Jasinski we study periodic point free Cantor systems that are conjugate to systems with vanishing derivative everywhere, and more generally locally radially shrinking maps. Our study uncovers a whole spectrum of dynamical behaviors attainable for such systems, providing new counterexamples to the Conjecture of Edrei from 1952, first disproved by Williams in 1954.

Given a compact and connected metric space (continuum) X, we study topological and dynamical properties of the hyperspace of normal fuzzy sets F1(X) equipped with the Hausdorff, endograph or sendograph metric. Among the many results we show that it is contractible, path connected, locally contractible, locally path connected, locally simply connected and locally connected. For the endograph metric the hyperspace F1(X) is a continuum, and then for a topological graph X we show how, using the inverse limit approach of Barge and Martin, the inverse limit of a fuzzy dynamical system on X can be realized as an attractor of a fuzzy dynamical system on a manifold.

We prove that there exists a topologically mixing homeomorphism which is completely scrambled. We also prove that, for any integer$n\geq 1$, there is a continuum of topological dimension$n$supporting a transitive completely scrambled homeomorphism and an$n$-dimensional compactum supporting a weakly mixing completely scrambled homeomorphism. This solves a 15-year-old open problem.

The goal of this paper is to establish connections between two entropy concepts, namely the fuzzy entropy of a fuzzy set, which appears in fuzzy set theory, and the topological entropy of Zadeh's extensions of continuous maps defined on metric compact spaces, which is a topological dynamics concept. We develop several concepts that allow us to simultaneously address the two entropy concepts. Except for stating several basic properties, we are partially capable of specifying in which classes of fuzzy sets dynamically complex behavior can appear by using these instruments. Such instruments are necessary for a proper understanding of the behavior of fuzzy systems generated from crisp, discrete dynamical systems.

Zadeh’s extension is a powerful principle in fuzzy set theory which allows to extend a real-valued continuous map to a map having fuzzy sets as its arguments. In our previous work we introduced an algorithm which can compute Zadeh’s extension of given continuous piecewise linear functions and then to simulate fuzzy dynamical systems given by them. The purpose of this work is to present results which generalize our previous approach to a more complex class of maps. For that purpose we present an adaptation on optimization algorithm called particle swarm optimization and demonstrate its use for simulation of fuzzy dynamical systems.

New model of linguistic variables based on fuzzy partition and its subpartitions.System allowing to choose something in between interpretability and precision.Rules extracted with various fuzzy partitions and fuzzy association measures.Prediction by fuzzy IF-THEN rules in implicative fuzzy inference engine. We propose a new generalized model of linguistic variables based on fuzzy partition and its subpartitions. We use this new model for mining relationships between linguistic variables (linguistic associations) from a data set. These relationships can be interpreted as fuzzy IF-THEN rules in the implicative fuzzy inference engine, which is an extended version of the implicative inference called Perception-based Logical Deduction. We show that our extension leads to statistically significant improvements with respect to the previous model used with the help of original and successful Perception-based Logical Deduction. We perform the comparison with different measures of rule quality and five datasets. We can obtain improvements in prediction precision while retaining the interpretability of the models. We also compare our method with the classical machine learning methods and obtain a similar quality of precision, which is very encouraging because interpretability usually leads to worse precision.

Summary Many mining operations produce tailings that dewater very slowly under self-weight consolidation. On way of reducing the water content of such tailings is by coagulation/flocculation processes (C/F processes). This paper describes a study using aluminium chloride and pre-hydrated aluminium salts as aluminium chlorohydrate – ACH (F1.2A, F1AM, FD15, FDM17H, F1S) and polyaluminium chloride – PACl (PAX-18) in dewatering of flotation tailing slurries, resulting from coal flotation in the coal preparation at the Paskov Mine (Czech Republic). Experiments were conducted at different temperature and mixing conditions to simulate full-scale conditions. The results point to a potentially technique for reducing the water content of tailings streams, thus increasing storage space, improving economic and environmental aspects.

We show that for any map f on an arc-like continuum X, the induced map f ˆ on the hyperspace of subcontinua C ( X ) fixes a point in any f ˆ -invariant subcontinuum of C ( X ) . This extends a result of Robatian [21] , who proved it for the arc. However, as we show, the result does not extend to tree-like continua. We conclude with a list of related problems. Our proof builds on Hamilton's proof of the fixed point property of arc-like continua [12] .

Zadeh’s extension principle is one of the most classical techniques in fuzzy set theory. It is a tool which, for example, can naturally extend a real-valued continuous map to a map having fuzzy sets as its arguments. Theoretically, it is a nice mathematical tool used in many theories, e.g. in studies on fuzzy dynamical systems. However, concrete calculations or even approximations can be very difficult in general and, consequently, many approaches trying to solve this problem appeared. In this work we present a novel algorithm which can compute Zadeh’s extension of given continuous piecewise linear functions. Among other things, an advantage of this approach is that, unlike almost all former approaches, it can deal with discontinuities which naturally appear in simulations of fuzzy dynamical systems.

In this paper we provide several results concerning the approximation of Zadeh's extension of a given function. We provide several simple observations due to which we provide a general procedure allowing to approximate Zadeh's extension of any continuous map. Among other things, we demonstrated that the technique of F-transform can also be used to approximate this extension. Further, we studied how the choice of the metric on the space of fuzzy sets (resp. fuzzy numbers) can affect the quality of the approximation, and we clarified that, in comparison to recent habits, the choice of the levelwise metric need not be the most appropriate one, and that the approximation process could be simplified. Our results are general in the sense that we mostly require nothing but the continuity of the original map.

Environmental contamination caused by mercury is a serious problem worldwide. The study was conducted in order to identify Hg contamination in soil, technosoil from dumps, groundwater, and surface water in the surroundings of the abandoned Hg deposit of Malachov in Central Slovakia. Soil from the Malachovský brook valley was classified as cambi-soil (rendzina). The highest Hg concentrations (44.24 mg kg(-1)) were described in the soil from the mining area at the Vel'ká Studňa locality. In the groundwater, the maximal Hg content is 0.84 μg L(-1), and in the surface water it is 394 μg L(-1). The speciation study proved that in most samples, Hg occurs in the form of cinnabarite. The release of Hg into the environment as a consequence of weathering is limited.

In the main result of this article, we prove that the topological entropies of a given interval map and its Zadeh's extension (fuzzification) to the space of fuzzy numbers (i.e., the space of fuzzy sets with connected α-cuts) are the same. This result is in contrast with our previous result, which claimed that the topological entropy of the Zadeh's extension significantly increases for a majority of simple interval maps. In addition, we prove some properties of the limit sets of trajectories that are generated by iterating the fuzzy set valued function on connected fuzzy sets; for instance, we specify the shapes of the possible limit sets. Furthermore, the presented topics are studied for set-valued (induced) discrete dynamical systems. The main results are proved with a variational principle that describes the relations between topological entropy and measure-theoretical entropy.

Let X be a compact metric space , let φ be a continuous self-map on X, and let F ( X ) denote the space of fuzzy sets on X equipped with the levelwise topology. In this paper we study relations between a given (crisp) dynamical system ( X , φ ) and its g-fuzzification ( F ( X ) , Φ g ) . Among other things we study various (weak, strong, mild etc.) mixing properties and also several kinds of chaotic behaviors (Li–Yorke chaos, ω -chaos, distributional chaos, topological chaos etc.). We specified subspaces of fuzzy sets which makes sense to study further. Additionally, we proved that numerous local characteristics of the original dynamical system can be found in its fuzzy extension, while many global characteristics are preserved in the opposite direction. Thus, among other things, we obtained easy-to-check criteria for checking absence of complex behavior in induced fuzzy dynamical systems.

The present study investigates mercury contamination of soil, groundwater, surface water, vegetation and food commodities (fruit, vegetable, mushroom, and fish) close to the abandoned Hg deposit in Malachov, Central Slovakia. The soil was classified as cambi-soil (rendzina). Maximum concentrations of Hg (44.24 ppm) were found in the soil from the area nearby old mining activities in the Veľká Studňa locality. In the groundwater, 0.84 μg.L-1, and in the surface water, even 394 μg.L-1 of Hg content was found. Also plant tissues and mushrooms are Hg-contaminated (in the root of Salix fragilis, the Hg content was 22 mg.kg-1). The Hg concentration decreases in general in the following order: root, branch/stem, and leaf/needle. In the food commodities, the Hg content does not exceed the Slovak law limits. The Hg content in trout muscle (from the local Malachovský brook) is 252 μg.kg-1 and in liver 402 μg.kg-1. The calculation of an AMD formation potential (neutralisation potential, total acidity production, net neutralisation potential) is discussed as well. The value of the total acidity production potential (sensu Sobek et al., 1978) is low (1.562 - 3.125; with exception of a single sample, in which it increased to 9.375). On the other hand, the abundant presence of dolomite rocks causes a high neutralization potential value (up to 812.84). This result suggests that the assumption of the AMD production is excluded and the environmental risk from this viewpoint might be neglected

The development of the industry in last two centuries, and especially intensive coal mining activities definitely changed the local landscape character. It led to the destruction of indigenous biotopes with natural flora and fauna communities. Many ecological studies use a group of molluscs as indicators of changes in the landscape. The advantage of this group is a relatively small number of taxa, relatively simple determination and very good knowledge of ecological requirements and distribution of individual species. The territory of Loucké ponds represents an interesting area in the industrial landscape of Karvinsko, where based on the above aspects of malacocenosis, it is possible to use and compare historical data of their incidence as well as to compare the data with current research. The pond system affected by the declension has an important function in the landscape today. The first malacological researches in the Loucké ponds territory were conducted in 1954 and 1955 (Brabenec, 1954, Ložek, 1964). Detailed survey was conducted by Macha (1978, 1979) who found 29 aquatic species (12 Bivalvia and 17 Gastropoda) and 19 land species of molluscs. This research was followed by the authors' survey in 2006, 2007 and 2013 which confirmed the presence of 48 molluscs species altogether (42 Gastropoda and 6 Bivalvia), from which 21 species were aquatic molluscs and 27 species were land molluscs.

This paper is a contribution to theoretical background of data mining, more precisely to fuzzy association analysis. We consider three the most commonly used confirmation measures and we study relations among found and known associations given by them. Good understanding of such relationships is necessary for creating more efficient algorithms or for subsequent work with found associations as well as for cooperation with the consumer of the data mining process. Even if our motivation to this work arose from mining of linguistic associations, found properties that coincide with semantics of mined associations are valid in general. Additionally, some examples showing how to use obtained properties are also contained in this paper.

It is well known that any given discrete dynamical system uniquely induces its fuzzified counterpart, i.e. a discrete dynamical system on the space of fuzzy sets. In this paper we study relations between dynamical properties of the original and fuzzified dynamical system. Especially, we study conditions used in the definition of Devaney chaotic maps, i.e. periodic density and transitivity. Among other things we show that dynamical behavior of the set-valued and fuzzy extensions of the original system mutually inherits some global characteristics and that the space of fuzzy sets admits a transitive fuzzification. This paper contains the solution of the problem that was partially solved by Roman-Flores and Chalco-Cano [Some chaotic properties of Zadeh's extension, Chaos, Solitons and Fractals 35(3) (2008) 452-459].

Let X denote a locally compact metric space and @f : X->X be a continuous map. In the 1970s Zadeh presented an extension principle helping us to fuzzify the dynamical system (X,@f), i.e., to obtain a map @F for the space of fuzzy sets on X. We extend an idea mentioned in [P. Diamond, A. Pokrovskii, Chaos, entropy and a generalized extension principle, Fuzzy Sets Syst. 61 (1994) 277-283] to generalize Zadeh's original extension principle. In this paper we study basic properties of so-called g-fuzzifications, such as their continuity properties. We also show that, for any g-fuzzification: (i) a uniformly convergent sequence of uniformly continuous maps on X induces a uniformly convergent sequence of fuzzifications on the space of fuzzy sets and (ii) a conjugacy (resp., a semi-conjugacy) between two discrete dynamical systems can be extended to a conjugacy (resp., a semi-conjugacy) between fuzzified dynamical systems. Throughout this paper we consider different topological structures in the space of fuzzy sets, namely, the sendograph, endograph and levelwise topologies.

A discrete dynamical system is given by a compact metric space X and any continuous self-map defined on X. This discrete dynamical system can be naturally extended to the space of fuzzy sets on X. In this paper we study relations between the sizes of the topological entropies of the original dynamical system and of its fuzzy counterpart. Among other things, we present a constructive proof of the fact that even very weak assumptions on the crisp discrete dynamical system ensure infinite topological entropy of the fuzzy system. However, we also show that there are subsystems of the fuzzy dynamical system with topological entropy equal to that of the crisp dynamical system.

In a recent paper we provided a characterization of triangular maps of the square, i.e., maps given by F ( x , y ) = ( f ( x ) , g x ( y ) ) , satisfying condition (P1) that any chain recurrent point is periodic. For continuous maps of the interval, there is a list of 18 other conditions equivalent to (P1), including (P2) that there is no infinite ω-limit set, (P3) that the set of periodic points is closed and (P4) that any regularly recurrent point is periodic, for instance. We provide an almost complete classification among these conditions for triangular maps, improve a result given by C. Arteaga [C. Arteaga, Smooth triangular maps of the square with closed set of periodic points, J. Math. Anal. Appl. 196 (1995) 987–997] and state an open problem concerning minimal sets of the triangular maps. The paper solves partially a problem formulated by A.N. Sharkovsky in the eighties. The mentioned open problem, the validity of (P4) ⇒ (P3), is related to the question whether some regularly recurrent point lies in the fibres over an f-minimal set possessing a regularly recurrent point. We answered this question in the positive for triangular maps with nondecreasing fiber maps. Consequently, the classification is completed for monotone triangular maps.

Fuzzy association analysis extracts relationships from data. The result of fuzzy association analysis depends on a chosen t-norm that is used for calculating confidence and support measures of mined association rules. We show that the set of mined association rules might change depending on the t-norm. We measure the distances of sets of mined rules with different t-norms and also with set of rules mined by crisp association analysis. We experiment with various datasets and partitioning methods to examine relationships of mined rules by different t-norms. Our experiments shed new light on application of fuzzy association mining and confirm that fuzzy association analysis usually brings signifficantly different results when compared to results given by crisp (non-fuzzy) association analysis.

Within this contribution we establish a theoretical background necessary for studying inverse limits of fuzzy dynamical systems induced by crisp (non-fuzzy) ones. First, we elaborate topological properties of the space of fuzzy sets, such as connectedness and compactness. Second, we focus our attention on dynamical conditions sufficient for the existence of indecomposable continua in the inverse limit space.

The Mazák National Natural Reserve provides suitable habitat conditions for many rare and endangered animals which are limited to low-disturbed or near-natural ecosystems. The malacofauna of the Mazák NNR was studied in 2012. Altogether, 37 species (35 land and 2 freshwater snails) were recorded. Two endangered species are interesting from ecological and faunal point of view: Vitrea transsylvanica and Eucobresia nivalis.