In this paper we construct a paramaterized family of annular homeomorphisms with Birkhoff-like rotational attractors that vary continuously with the parameter, are all homeomorphic to a unique ...topological object, called the R.H. Bing's pseudo-circle, yet display an interesting boundary dynamics. Namely, in the constructed family of homeomorphisms the outer prime ends rotation number vary continuously with the parameter through the interval 0,1/2. This, in particular, answers a question from Boroński et al. (2020) 15. Furthermore, these attractors preserve the induced Lebesgue measure from the circle and have strong measure-theoretic and statistical properties. To show main results of the paper we first prove a result of an independent interest, that Lebesgue-measure preserving circle maps generically satisfy the crookedness condition which implies that generically the inverse limits of Lebesgue measure-preserving circle maps are the pseudo-solenoids. For degree one circle maps, this implies that the generic inverse limit in this context is the pseudo-circle.
•To study Δ-Choquets integral on time scales with respect to non-additive measure, more precisely, a distorted Lebesgue Δ-measure. That is also aspecial case of Choquet integral with respect to ...abstract fuzzy measure.•This integral unifies both continuous and discrete Choquet integral. Further, results unify and extend the results reported in the literature.•A short note on Δ-Choquet integral on time scale and Caputo-Fabrizio fractional derivative on time scale is discussed.•Stieltjes distorted type-I and II Lebesgue measures on time scales are provided to present another special Choquet integral.•Various examples are given to illustrate the outcomes.
The fundamental purpose of this work is to analyze Δ-Choquet integrals on time scales which is a special case of Choquet integral on abstract fuzzy (non-additive) measure space. We first present a Δ-Choquet integral with respect to non-additive Δ-measure or more precisely a distorted Lebesgue Δ-measure on an arbitrary time scale. Consequently, we come up with a more general integral than the standard Choquet integral of continuous and discrete calculus. Its use can be seen as convenient in economics, decision making, artificial intelligence, and many more. Particularly, in economics, most of the models are dynamic models (continuous and/or discrete), and those can be easily studied on time scales. Further, some basic essential results and properties of the general integral are studied. For instance, we discuss translation, homogeneity, linearity, and many more with respect to the functions and measures of the integral.
Then, after that, we present some theorems for computing the integral. The findings agree to unify and extend a number of well-known results reported in the literature to a broader calculus, including continuous, discrete, and quantum calculus, among others. We also evaluate the integral on an invariant under the translation of time scales. Besides, a short note on Δ-Choquet integral with the Caputo-Fabrizio fractional derivative on the time scales is given. The significance of the outcomes is also further enhanced by a variety of interesting examples.
Moreover, eventually, we stop findings after discussing an another way to calculate the Δ-Choquet integral on the time scales. To do this, we define Stieltjes distorted types-I and II Lebesgue Δ-measures on time scales which are accomplished with the help of distorted Lebesgue Δ-measure.
For incomplete data with mixed numerical and symbolic attributes, attribute reduction based on neighborhood multi-granulation rough sets (NMRS) is an important method to improve the classification ...performance. However, most classical attribute reduction methods can only handle finite sets as to produce more attributes and lower classification accuracy. This paper proposes a novel NMRS-based attribute reduction method using Lebesgue and entropy measures in incomplete neighborhood decision systems. First, some concepts of optimistic and pessimistic NMRS models in incomplete neighborhood decision systems are given, respectively. Then, a Lebesgue measure is combined with NMRS to study neighborhood tolerance class-based uncertainty measures. To analyze the uncertainty, noise and redundancy of incomplete neighborhood decision systems in detail, some neighborhood multi-granulation entropy-based uncertainty measures are developed by integrating Lebesgue and entropy measures. Inspired by both algebraic view with information view in NMRS, the pessimistic neighborhood multi-granulation dependency joint entropy is proposed. What is more, the corresponding properties are further deduced and the relationships among these measures are discussed, which can help to investigate the uncertainty of incomplete neighborhood decision systems. Finally, the Fisher linear discriminant method is used to eliminate irrelevant attributes to significantly reduce computational complexity for high-dimensional datasets, and a heuristic attribute reduction algorithm with complexity analysis is designed to improve classification performance of incomplete and mixed datasets. Experimental results under seven UCI datasets and eight gene expression datasets illustrate that the proposed method is effective to select most relevant attributes with higher classification accuracy, as compared with representative algorithms.
A Way to Choquet Calculus Sugeno, Michio
IEEE transactions on fuzzy systems,
2015-Oct., 2015-10-00, 20151001, Letnik:
23, Številka:
5
Journal Article
Recenzirano
In this paper, we deal with the Choquet integral and derivative with respect to fuzzy measures on the nonnegative real line and present a way to Choquet calculus as a new research paradigm. In ...Choquet calculus, a representation for calculating the continuous Choquet integral is first given by restricting the integrand to a class of nondecreasing and continuous functions and the fuzzy measure to a class of distorted Lebesgue measures. Next, the derivative of functions with respect to distorted Lebesgue measures is defined as the inverse operation of the Choquet integral. Then, elementary properties in Choquet calculus are explored. In addition, we clarify the relation of Choquet calculus with fractional calculus, where the fractional Choquet integral and derivative are newly defined. In addition, we consider differential equations with respect to distorted Lebesgue measures and give their solutions. Finally, we introduce conditional distorted Lebesgue measures and explore their properties.
In this paper we show that generic continuous Lebesgue measure-preserving circle maps have the s-limit shadowing property. In addition, we obtain that s-limit shadowing is a generic property also for ...continuous circle maps. In particular, this implies that classical shadowing, periodic shadowing and limit shadowing are generic in these two settings as well.
Given a strictly increasing differentiable function f:R→R whose derivative vanishes on a dense subset of R, it has been shown by Ciesielski that there is at least a comeager amount of t∈R such that ...f(x)−f(x−t) is nowhere-monotone. In this paper, we show that the amount of their shifted difference that are nowhere-monotone is not only comeager but also of full Lebesgue measure in R. Interestingly, we also provide a specific pair of strictly increasing differentiable functions φ,ψ:R→R such that φ(x)−ψ(x−t) is nowhere-monotone for every t∈R.
Feature selection for mixed and incomplete data in terms of numerical and categorical features with missing values has currently gained considerable attention. The development of the neighborhood ...rough sets-based feature selection method is an important step in improving classification performance, especially in incomplete data with mixed continuous numerical and categorical features. In this paper, a novel feature selection method based on the neighborhood rough sets using Lebesgue and entropy measures in incomplete neighborhood decision systems is proposed, and the method has the capacity to handle mixed and incomplete datasets; further, it can simultaneously maintain the original classification information. First, a Lebesgue measure based on the neighborhood tolerance class is developed to study the positive region and dependency degree. To thoroughly analyze the uncertainty, noise and incompleteness of incomplete neighborhood decision systems, some neighborhood tolerance entropy-based uncertainty measures are presented based on Lebesgue and entropy measures. Then, by combining an algebraic view with an information view in neighborhood rough sets, the neighborhood tolerance dependency joint entropy is defined in incomplete neighborhood decision systems. Moreover, all the corresponding properties are discussed, and the relationships among these measures are established to meaningfully convey the knowledge essence and investigate the uncertainty of incomplete neighborhood decision systems. Finally, for all high-dimensional datasets, the Fisher score method is used to preliminarily eliminate irrelevant features to significantly reduce the computational complexity, and a heuristic feature selection algorithm is designed to improve the classification performance of mixed and incomplete datasets. Experiments under an instance and fifteen public datasets demonstrate that the proposed feature selection method is effective in selecting the most relevant features, achieving great classification ability for incomplete neighborhood decision systems.
It is well known that there exist real-valued, non-measurable functions on the plane, whose restrictions to the graph of all measurable functions on the real line are measurable. In this paper, it is ...shown that there is, in fact, a continuum-dimensional vector space all of whose members, except for zero, are functions on the plane satisfying the mentioned property. Versions of this fact are proved in the Baire category setting and in the case where the graphs are generated by softer functions.
This paper mainly discusses the characteristic of the basin of attraction of two coupled chaotic systems, which provides a theoretical basis for the application of nonlinear systems in the fields of ...communication, control and artificial intelligence. Thus, in order to increase the complexity of the chaotic systems, two identical linear couplings are used to hide a system composed of attractors, and therefore the system has chaotic characteristic within a specific coupling strength range. We use an ode45 algorithm to solve the coupled chaotic systems to obtain a chaotic phase diagram, Lyaponov exponential spectrum and a bifurcation diagram of the system and prove that the attractors of the coupled systems are attractors in the sense of Milnor, and the basin of attraction of the coupled chaotic systems has a sieve-type property. A Lyaponov exponential function of the nonlinear system can be used to analyze the stability of the system, when the system operates in an initial state, the system will move in the direction that the Lyaponov exponential function decreases until it reaches a local minimum, a local minimum point of the Lyaponov exponential function represents a stable point of a phase space, and each attractor surrounds one substantial basin of attraction. Therefore, we use a Lyaponov exponent to describe and analyze the basin of attraction of the coupled chaotic systems. And meanwhile, when analyzing the system, we found that the system has rich dynamic behavior and multi-stability for hiding of the attractors.
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