Recent zbMATH articles in MSC 28https://zbmath.org/atom/cc/282021-01-08T12:24:00+00:00WerkzeugHausdorff dimension of chaotic sets caused by a continuous self-map on \(I^n\).https://zbmath.org/1449.280082021-01-08T12:24:00+00:00"Wu, Huaming"https://zbmath.org/authors/?q=ai:wu.huamingSummary: This paper extends the results of Hausdorff dimension of chaotic sets caused by continuous self-maps on \(I\) and \({I^2}\) to the \(n\)-dimensional cube. We prove that there is a residual set \(\mathscr{R}\) in \({C^0} ({I^n})\), if set \(C \subset {I^n}\) is chaotic for any given \(f \in \mathscr{R}\) in the sense of Li-Yorke, then \({\dim_H} (C) \le n-1\). Similarly, the results on high dimensional Cartesian product can be obtained. That is, there are residual sets \({\mathscr{R}_i}\) in \({C^0} ({I^{ni}}, {I^{ni}})\) such that for any \({f_i} \in {\mathscr{R}_i}, i = 1, 2\), if set \({C_i} \subset {I^{ni}}\) is chaotic in the sense of Li-Yorke, then \({\dim_H} ({C_1} \times {C_2}) \le n-1\).Integral inequalities for generalized harmonically quasi-convex functions on fractal sets.https://zbmath.org/1449.260402021-01-08T12:24:00+00:00"Sun, Wenbing"https://zbmath.org/authors/?q=ai:sun.wenbingSummary: In this paper, the author introduces the concept of generalized harmonically quasi-convex functions on fractal sets \({\mathbb{R}}^{\alpha}\) (\(0<\alpha\leq 1\)) of real line numbers and establishes generalized Hermite-Hadamard and Simpson type inequalities for generalized harmonically quasi-convex functions. Some applications for \(\alpha\)-type special means of real line numbers are given.A sufficient condition for the finite \({\mu_{M,D}}\)-orthogonal exponentials function system.https://zbmath.org/1449.280112021-01-08T12:24:00+00:00"Li, Na"https://zbmath.org/authors/?q=ai:li.na"Li, Jianlin"https://zbmath.org/authors/?q=ai:li.jianlin|li.jianlin.1Summary: Let \({\mu_{M, D}}\) be a self-affine measure uniquely determined by the iterated function system \(\{\phi_d (x) = M^{-1} (x+d)\}_{d \in D}\). The spectrality or non-spectrality of \({\mu_{M, D}}\) is directly connected with the finiteness or infiniteness of orthogonal exponentials in the Hilbert space \({L^2} (\mu_{M, D})\). In this paper, the authors provide a sufficient condition for the finite \({\mu_{M, D}}\)-orthogonal exponentials by applying the elementary matrix transformations. This sufficient condition depends only upon the determinant of the matrix \(M\), and is easy to use in the research of non-spectrality of \({\mu_{M, D}}\).Cesàro means of subsequences of double sequences.https://zbmath.org/1449.400092021-01-08T12:24:00+00:00"Taş, Emre"https://zbmath.org/authors/?q=ai:tas.emre"Orhan, Cihan"https://zbmath.org/authors/?q=ai:orhan.cihanSummary: In this paper we characterize the convergence and \((C,1,1)\) summability of a double sequence. In particular we study conditions under which the convergence or \((C,1,1)\) summability of a double sequence carry over to that of its subsequences, and conversely, whether these properties for suitable subsequences imply them for the sequence itself. We show, for instance, that a bounded double sequence is \((C,1,1)\) summable if and only if almost all of its subsequences are \((C,1,1)\) summable.Geometric representations of multivariate skewed elliptically contoured distributions.https://zbmath.org/1449.600212021-01-08T12:24:00+00:00"Richter, Wolf-Dieter"https://zbmath.org/authors/?q=ai:richter.wolf-dieter"Venz, John"https://zbmath.org/authors/?q=ai:venz.johnSummary: We derive a wide class of geometric representation formulas for multivariate skewed elliptically contoured distributions and show in a unified geometric way how some of them are related to stochastic representations known in the literature. Furthermore, we make use of the geometric measure representation to explore independence between collections of components of accordingly distributed random vectors, and to investigate contour plots of skewed normal densities from a geometric viewpoint.Some generalisations of measurable and integrable functions.https://zbmath.org/1449.280042021-01-08T12:24:00+00:00"Lipovan, Octavian"https://zbmath.org/authors/?q=ai:lipovan.octavianSummary: We define the notion of pseudosubmeasure as a generalisation of the submeasure notion, and we study some properties of the topological ring of sets defined by that. Using families of pseudosubmeasures and the associated topological rings, the pseudosubmeasurable function concept is then defined. The convergence in measure, almost everywhere convergence and almost uniform convergence are generalized to the sequence of functions with values in pseudometric space.
Using the notion of control submeasure, there are introduced a criterion of functions pseudosubmeasurability. Finally, we develop an integration theory for these functions, with respect to a semigroup valued measure.An elementary proof that the Borel class of the reals has cardinality continuum.https://zbmath.org/1449.280032021-01-08T12:24:00+00:00"Kánnai, Z."https://zbmath.org/authors/?q=ai:kannai.zoltanSummary: We give a recursion-like theorem which enables us to encode the elements of the real Borel class by infinite sequences of integers. This fact implies that the cardinality of the Borel class is not above continuum, without depending on cumbrous tools like transfinite induction and Suslin operation.Conditional uncertain set and conditional membership function.https://zbmath.org/1449.280172021-01-08T12:24:00+00:00"Yao, Kai"https://zbmath.org/authors/?q=ai:yao.kaiSummary: The uncertain set, as a generation of uncertain variable, is a set-valued function on an uncertainty space. The conditional uncertain set, derived from an uncertain set restricted to a conditional uncertainty space given an uncertain event, plays a crucial role in uncertain inference systems. This paper studies conditional uncertain sets and their membership functions, and gives a sufficient condition for an uncertain set having a conditional membership function. In addition, when the uncertain set is conditioned on an independent event, this paper finds the analytic expression of the conditional membership function based on the original membership function.Pseudodifferential operators in infinite dimensional spaces: a survey of recent results.https://zbmath.org/1449.354682021-01-08T12:24:00+00:00"Jager, Lisette"https://zbmath.org/authors/?q=ai:jager.lisetteThe author gathers previously published results concerning the quantization of pseudodifferential analysis in abstract Wiener spaces, in Fock spaces and in Gaussian Hilbert spaces. She starts briefly recalling the Weyl calculus in finite dimensions. Moving to the infinite dimensional cases, she first defines abstract Wiener spaces. Taking a real, separable and infinite dimensional Hilbert space \(\mathcal{H}\), a finite dimensional subspace \(E\) of \(\mathcal{H}\) and the orthogonal projection \(\pi_{E}\) on \(E\), she introduces the pseudomeasure \(\mu_{\mathcal{H},s}(C)\) on a cylinder \(C=\{x\in H:\pi_{E}(x)\in A\}\) where \(A\) is a Borel set of \(E\) through \(\mu_{\mathcal{H},s}(C)=\int_{A}\exp (-\frac{\left\vert y\right\vert ^{2}}{2s}) (2\pi s)^{-\dim (E)/2}d\lambda_{E}(y)\), where \(\lambda_{E}\) is the Lebesgue measure on \(E\). If \(\mathcal{H}\) can be endowed with a norm \(\left\Vert \cdot \right\Vert\) which satisfies a measurability property, let \(B\) be the completion of \(\mathcal{H}\) with respect to this norm and \(i\) the injection from \(\mathcal{H}\) to \(B\). Then \((i,\mathcal{H},B)\) is an abstract Wiener space. She presents the conditions for a continuous function on \(\mathcal{H}\) to admit a stochastic extension on \(L^{2}(B,\mu_{B,h})\) and proves that measures \(\mu_{B,s}(x,\cdot)\) and \(\mu_{B,t}(y,\cdot)\), with \(\mu_{B,s}(x,A)=\mu_{B,s}(A-x)\) for every Borel subset \(A\) of \(B\), are absolutely continuous with respect to one another if and only if \(s=t\) and \(x-y\in \mathcal{H}\).
The author then defines a Gaussian Hilbert space as a real vector \(\mathcal{M}\) of random variables \(\xi \) defined on a probability space \((\Omega,\mathcal{F},P)\), such that every random variable is centered and Gaussian. She also recalls the definition of a Fock space and the notions of coherent state and of Wick and anti-Wick symbols. She recalls the notion of symbol class in this infinite dimensional case and their properties. She then defines the heat operator on an abstract Wiener space as \(\widetilde{H}_{t}f(x)=\int_{B}f(x+y)\,d\mu_{B,t}(y)\) for every Borel bounded function \(f\) on the Wiener space \((B,\mathcal{B}(B))\) and she recalls its properties. She defines the space \(\mathcal{D}\) which replaces the Schwartz space \(\mathcal{S}(\mathbb{R}^{n})\) in finite dimension, the Wigner function \(W_{h}(f,g)\) attached to functions \(f,g\) in \(\mathcal{D}\), the quadratic form \(Q_{h}^{W}(\widetilde{F})\) associated to a bounded Borel function \(\widetilde{F}\) on \(B^{2}\), the Calderon-Vaillancourt classes and the pseudo-differential operators in this infinite dimensional case. She presents a Beals characterization, composition results and properties of the Wick and Weyl symbols. The paper ends with some applications.
Reviewer: Alain Brillard (Riedisheim)Choquet integral correlation measures of intuitionistic fuzzy sets and its application in decision making.https://zbmath.org/1449.280162021-01-08T12:24:00+00:00"Liu, Weifeng"https://zbmath.org/authors/?q=ai:liu.weifeng"He, Xia"https://zbmath.org/authors/?q=ai:he.xia"Chang, Juan"https://zbmath.org/authors/?q=ai:chang.juanSummary: The Choquet integral correlation coefficient between intuitionistic fuzzy sets and the computing formula were obtained by a previous author, but the results have contradiction with the nature of the Choquet integral correlation coefficient between intuitionistic fuzzy sets. It is found from the practical example that the range of the Choquet integral correlation coefficient between intuitionistic fuzzy sets is not appropriate, and through the study on the proof of its nature of the Choquet integral correlation coefficient, the cause of the problem existed is analyzed. Then based on the Choquet integral correlation of intuitionistic fuzzy sets, the new Choquet integral information energy of the intuitionistic fuzzy set and the Choquet integral correlation coefficient between intuitionistic fuzzy sets are defined, and their natures are also discussed. Finally, the Choquet correlation coefficient between the alternative and the positive ideal alterative is derived, and a method is developed to solve the multiple attribute decision making problem using the Choquet correlation coefficient, and an example is used to illustrate the feasibility and effectiveness of the proposed method.Entropy of infinite systems and transformations.https://zbmath.org/1449.370052021-01-08T12:24:00+00:00"Amini, Massoud"https://zbmath.org/authors/?q=ai:amini.massoudSummary: The Kolmogorov-Sinai entropy is a far reaching dynamical generalization of Shannon entropy of information systems. This entropy works perfectly for probability measure preserving (p.m.p.) transformations. However, it is not useful when there is no finite invariant measure. There are certain successful extensions of the notion of entropy to infinite measure spaces, or transformations with infinite invariant measures. The three main extensions are Parry, Krengel, and Poisson entropies. In this survey, we shortly overview the history of entropy, discuss the pioneering notions of Shannon and later contributions of Kolmogorov and Sinai, and discuss in somewhat more details the extensions to infinite systems. We compare and contrast these entropies with each other and with the entropy on finite systems.Interiors of continuous images of self-similar sets with overlaps.https://zbmath.org/1449.280122021-01-08T12:24:00+00:00"Xi, Lifeng"https://zbmath.org/authors/?q=ai:xi.lifeng"Jiang, Kan"https://zbmath.org/authors/?q=ai:jiang.kan"Zhu, Jiali"https://zbmath.org/authors/?q=ai:zhu.jiali"Pei, Qiyang"https://zbmath.org/authors/?q=ai:pei.qiyangSummary: Let \(K\) be the attractor of the following iterated function system \[\{S_1(x)=\lambda x,S_2(x)=\lambda x+c-\lambda, S_3(x)=\lambda x+1-\lambda\},\] where \(S_1(I)\cap S_2(I)\ne \emptyset, (S_1(I)\cup S_2(I))\cap S_3(I)=\emptyset\) and \(I=[0,1]\) is the convex hull of \(K\). Let \(d_1=\frac{1-c-\lambda}{\lambda}<\frac{1}{1-c-\lambda}=d_2\). Suppose that \(f\) is a continuous function defined on an open set \(U\subset \mathbb{R}^2\). Denote the image \[f_U(K,K)=\{f(x,y):(x,y)\in (K\times K)\cap U\}.\] If \(\partial_xf,\partial_yf\) are continous on \(U\), and there is a point \((x_0,y_0)\in (K\times K)\cap U\) such that \[\left|\frac{\partial_yf|_{(x_0,y_0)}}{\partial_xf|_{(x_0,y_0)}}\right|\in (d_1,d_2)\;\;\text{or} \;\; \left|\frac{\partial_xf|_{(x_0,y_0)}}{\partial_yf|_{(x_0,y_0)}}\right|\in (d_1,d_2),\] then \(f_U(K,K)\) contains an interval. As a result we let \(c=\lambda=1/3\), and if \[f(x,y)=x^{\alpha}y^{\beta}\; (\alpha\beta\ne 0),\;\; x^{\alpha}\pm y^{\alpha}\; (\alpha\ne 0),\;\; \sin(x)\cos(x),\;\; \text{or}\;\; x\sin(xy)\] then \(f_U(C,C)\) contains an interval, where \(C\) is the middle-third Cantor set.On the quasisymmetric minimality of homogeneous perfect sets.https://zbmath.org/1449.280092021-01-08T12:24:00+00:00"Xiao, Yingqing"https://zbmath.org/authors/?q=ai:xiao.yingqing"Zhang, Zhanqi"https://zbmath.org/authors/?q=ai:zhang.zhanqiSummary: In literature, the notion of homogeneous perfect sets was introduced as a generalization of Cantor type sets and their exact Hausdorff dimensions based on the length of their basic intervals and the gaps between them were determined. In this paper, we considered the quasisymmetric minimality of the homogeneous perfect sets, proved that the homogeneous perfect sets with Hausdorff dimension 1 are 1-dimensional quasisymmetrically minimal under some conditions.Axiomatic system defining an order-embedding between infinite \(\sigma\)-algebras.https://zbmath.org/1449.280012021-01-08T12:24:00+00:00"Agbeko, Nutefe Kwami"https://zbmath.org/authors/?q=ai:agbeko.nutefe-kwamiSummary: The purpose of the present paper is twofold: on the one hand to set up an axiomatic system defining an order-embedding between infinite \(\sigma\)-algebras to generalize the powering mapping and investigate some additional necessary and sufficient condition for the postulate of powering to hold in the system, and on the other hand to provide some theoretical applications.Spectral sets and tiles in Cartesian products over the \(p\)-adic fields.https://zbmath.org/1449.280022021-01-08T12:24:00+00:00"Kadir, Mamateli"https://zbmath.org/authors/?q=ai:kadir.mamateliSummary: In this paper, we prove that the Cartesian product \(\Omega_1\times \Omega_2\) of two measurable sets \(\Omega_1 \subset \mathbb{Q}_p^{d_1}\) and \(\Omega_2 \subset \mathbb{Q}_p^{d_2}\) tile the product space \(\mathbb{Q}_p^{d_1}\times \mathbb{Q}_p^{d_2}\) by translations if and only if they tile the corresponding space by translations. We also consider the similar problem for spectral sets.Space of configurations and the special measures on it.https://zbmath.org/1449.280132021-01-08T12:24:00+00:00"Berezansky, Yu. M."https://zbmath.org/authors/?q=ai:berezanskii.yurii-makarovichThe author gives a novel exposition of the basic notions of analysis on configurations, including various topologizations and classes of measures.
Reviewer: Anatoly N. Kochubei (Kyïv)Fractal balls.https://zbmath.org/1449.280102021-01-08T12:24:00+00:00"Bevilacqua, Luiz"https://zbmath.org/authors/?q=ai:bevilacqua.luizSummary: To the best of our knowledge, the analysis of densely folded media has not deserved special attention. The stress and strain analysis of this type of structures involves considerable difficulties concerning very strong non-linear effects. This paper presents a theory that could be classified as a geometric theory of folded media, in the sense that it ultimately leads to a kind of geometric constitutive law, or, in other words, a law that establishes the relationship between the geometry of the folded media and other variables such as the confinement capacity and the plastic strain energy. The discussion presented here is restricted to the particular case of compact balls produced by crushing together very thin plates or sheets. It is shown that both the geometry of the folded sheet and the plastic work density can be used as self-similarity tests. These criteria are equivalent for the case of thin plates or sheets made of the same material and with the same thickness. For the general case, the geometry of the folded sheet is not valid anymore as similarity criterion but there are strong arguments in favor of the plastic work density as a general criterion. If self-similarity is obtained for a ball set resulting from crumpling thin plates or sheets, it is possible to define two variables, the packing capacity and the slenderness ratio, that are related according to a power law. That is, the balls have a fractal representation. The power law scaling is derived from the mass conservation principle. The theory is claimed to be valid provided that certain assumptions referring to the geometry and material properties are satisfied. The results have shown that the theory is coherent and worthwhile of experimental validation. Some applications are suggested. A possible challenging investigation is related to the optimal geometry of biological membranes.Poincáre recurrence theorem in regular uncertain dynamic system.https://zbmath.org/1449.370032021-01-08T12:24:00+00:00"Yao, Xiao"https://zbmath.org/authors/?q=ai:yao.xiao"Ke, Hua"https://zbmath.org/authors/?q=ai:ke.huaSummary: Poincáre recurrence theorem in an uncertain dynamic system is proved in the framework of uncertainty theory, which claims that almost every point of an uncertain event with positive uncertain measure will iterate back to the event for infinite times. This recurrence behaviour can be used to develop new results of uncertain variable in an uncertain dynamic system.Hausdorff dimension for range and graph of multi-parameter operator stable Lévy processes.https://zbmath.org/1449.280062021-01-08T12:24:00+00:00"Chen, Xiaoping"https://zbmath.org/authors/?q=ai:chen.xiaoping"Lin, Huonan"https://zbmath.org/authors/?q=ai:lin.huonanSummary: The lower Hausdorff dimension results for the range and the graph of multi-parameter operator stable Lévy processes are established. The consequences are completely determined by the eigenvalues of its exponent matrix.Characterization of a b-metric space completeness via the existence of a fixed point of Ciric-Suzuki type quasi-contractive multivalued operators and applications.https://zbmath.org/1449.540482021-01-08T12:24:00+00:00"Alolaiyan, Hanan"https://zbmath.org/authors/?q=ai:alolaiyan.hanan-abdulaziz"Ali, Basit"https://zbmath.org/authors/?q=ai:ali.basit"Abbas, Mujahid"https://zbmath.org/authors/?q=ai:abbas.mujahidSummary: The aim of this paper is to introduce Ćirić-Suzuki type quasi-contractive multivalued operators and to obtain the existence of fixed points of such mappings in the framework of b-metric spaces. Some examples are presented to support the results proved herein. We establish a characterization of strong b-metric and b-metric spaces completeness. An asymptotic estimate of a Hausdorff distance between the fixed point sets of two Ćirić-Suzuki type quasi-contractive multivalued operators is obtained. As an application of our results, existence and uniqueness of multivalued fractals in the framework of b-metric spaces is proved.Signed topological measures on locally compact spaces.https://zbmath.org/1449.280142021-01-08T12:24:00+00:00"Butler, S. V."https://zbmath.org/authors/?q=ai:butler.svetlana-vIn this paper a (signed) topological measure on a locally compact space \(X\) is a function \(\mu\) defined on the union of the families of open sets, \(\mathcal{O}(X)\), and closed sets, \(\mathcal{C}(X)\), with values in \([0,\infty]\) (in \([-\infty,\infty]\)) that is finitely additive on \(\mathcal{O}(X)\cup\mathcal{K}(X)\), where \(\mathcal{K}(X)\) is the family of compact sets. It is also required to satisfy two regularity conditions: if \(U\) is open then \(\mu(U)=\lim\{\mu(K):K\in\mathcal{K}(X), K\subseteq U\}\) and if \(F\) is closed then \(\mu(F)=\lim\{\mu(O):O\in\mathcal{O}(X), F\subseteq O\}\), where the limit is taken along the family on the right hand side, directed by (reverse) inclusion.
If \(\mu\) is only required to be additive on \(\mathcal{K}(X)\) then it is called a (signed) deficient topological measure.
The author proves some structural results on these measures: they are the difference of their positive and negative variations; the latter add up to the total variation. In special cases a signed topological measure can be written as the difference of two topological easures: if \(X\) is connected, locally connected and its one-point compactification has genus \(0\).
Reviewer: K. P. Hart (Delft)Calculus and nonlinear integral equations for functions with values in \(L\)-spaces.https://zbmath.org/1449.450092021-01-08T12:24:00+00:00"Babenko, V."https://zbmath.org/authors/?q=ai:babenko.v-t|babenko.vira|babenko.v-n|babenko.vladislav-f|babenko.vladimir-i|babenko.v-e|babenko.viktor-v|babenko.v-aSummary: In this paper, the calculus of functions with values in \(L\)-spaces is developed. We then consider nonlinear integral equations of Fredholm and Volterra types for functions with values in \(L\)-spaces. Such class of equations includes set-valued integral equations, fuzzy integral equations, and many others. We prove theorems of existence and uniqueness of the solutions of such equations and investigate data dependence of their solutions.On the laws of large numbers in possibilistic theory.https://zbmath.org/1449.600492021-01-08T12:24:00+00:00"Gal, Sorin G."https://zbmath.org/authors/?q=ai:gal.sorin-gheorgheSummary: In this paper we obtain some possibilistic variants of the probabilistic laws of large numbers, different from those obtained by other authors, but very natural extensions of the corresponding ones in probability theory. Our results are based on the possibility measure and on the ``maxitive'' definitions for possibility expectation and possibility variance. In the frame of this paper, we have only strong law of large numbers, because the weak form of the law of large numbers, will always imply the strong law of large numbers.The entropy of Cantor-like measures.https://zbmath.org/1449.280072021-01-08T12:24:00+00:00"Hare, K. E."https://zbmath.org/authors/?q=ai:hare.kathryn-e"Hare, K. G."https://zbmath.org/authors/?q=ai:hare.kevin-g"Morris, B. P. M."https://zbmath.org/authors/?q=ai:morris.b-p-m"Shen, W."https://zbmath.org/authors/?q=ai:shen.wanqiang|shen.weichang|shen.wenjun|shen.wen|shen.weiwei|shen.wenjing|shen.weixiang|shen.wenqiang|shen.wenzhong|shen.wenbin|shen.wenhuai|shen.wenji|shen.wanqiu|shen.weimin|shen.wenguo|shen.weicheng|shen.weixi|shen.wenhao|shen.wanfang|shen.weibo|shen.weiping|shen.wenhe|shen.wenyi|shen.wuqiang|shen.weixin|shen.wenqian|shen.weina|shen.wenda|shen.weixiao|shen.wensheng|shen.wenxuan|shen.weidong|shen.wenting|shen.weiming|shen.weining|shen.wenxia|shen.weilei|shen.wei|shen.wenxian|shen.weijie|shen.wenjie|shen.wanchun|shen.wang|shen.weihua|shen.wenwenSummary: By a Cantor-like measure we mean the unique self-similar probability measure \(\mu\) satisfying \(\mu=\sum_{i=0}^{m-1}p_i\mu\circ S_i^{-1}\), where \(S_i(x)=\frac{x}{d}+\frac{i}{d}\cdot \frac{d-1}{m-1}\), for integers \(2 \le d < m\le 2d - 1\) and probabilities \(p_i > 0, \sum p_i = 1\). In the uniform case (\(p_i = 1/m\) for all \(i\)) we show how one can compute the entropy and Hausdorff dimension to arbitrary precision. In the non-uniform case we find bounds on the entropy.Investigation of multifractal features of the relationship between price and volume in international carbon market.https://zbmath.org/1449.910702021-01-08T12:24:00+00:00"Zhang, Chen"https://zbmath.org/authors/?q=ai:zhang.chen"Fan, Fangfang"https://zbmath.org/authors/?q=ai:fan.fangfang"Yang, Xianzi"https://zbmath.org/authors/?q=ai:yang.xianzi"Wang, Wenjun"https://zbmath.org/authors/?q=ai:wang.wenjunSummary: This paper verifies the existence of the cross-correlation between price and volume in European Union allowance (EUA) market firstly. Then it is found that the relationship between price and volume is multifractal because of long-range correlations and fat-tail distributions with multifractal detrended cross-correlation analysis (MF-DCCA). At the same time, the differences of the relationship between the price and volume in the three periods (2005--2007, 2008--2012 and 2013--2020) of European Union carbon emissions trading system are studied. It is revealed that the multifractal characteristics of the relationship between the price and volume are different in each period of the market, and the risks of second and third periods are obviously lower than that of the first period while the market effectiveness is improved. Furthermore, the asymmetry of the relationship under different market situations is also examined with multifractal asymmetric detrended cross-correlation analysis (MF-ADCCA). It is discovered that the multifractality and the risk of market are different when the trend of price or volume changes in three periods. By analyzing the multifractal characteristics of the relationship between the price and volume in carbon market, the market effectiveness and risk status and their changes are explored. The results contribute to providing a theoretical reference for regulators and investors to understand the carbon market fully.Uniform integrability of sequence of generalized functions described by \(K\)-quasi additive Sugeno integral.https://zbmath.org/1449.260502021-01-08T12:24:00+00:00"Li, Yanhong"https://zbmath.org/authors/?q=ai:li.yanhongSummary: \(K\)-quasi additive Sugeno integral is a new non-additive integral defined by the induced operator. It plays an important role in the generalized integral theory and some practical applications. In order to overcome the inborn deficiency of \(K\)-quasi additive measure: without additivity, a new non-additive integral model ``\(K\)-quasi additive Sugeno integral'' is introduced. This provides a new way to further study the theory of non-additive integral. On the one hand, on the \(K\)-quasi additive measure space, the \(K\)-quasi additive Sugeno integral with the generalized measurable function is defined by the induced operator, and the uniform integrability and uniform boundedness of sequence of generalized functions are discussed by using the analytic representation of the integral. On the other hand, on the \(K\)-quasi additive measure space, it is proved that the uniform boundedness of a sequence of nonnegative generalized functions contains uniform integrability. Then a sufficient and necessary condition for the uniform integrability of the sequence of nonnegative generalized functions is given in the sense of \(K\)-quasi additive Sugeno integral.On the convergence of sequences of probability measures.https://zbmath.org/1449.280052021-01-08T12:24:00+00:00"Zaharopol, Radu"https://zbmath.org/authors/?q=ai:zaharopol.raduIn this paper \((X, d)\) is a Polish space, \(\mathcal{M}(X)\) all real valued Borel measures on \(X\) and \(C_b(X)\) all bounded continuous functions on \(X\). A function \(f\in C_b(X)\) is said to have bounded support if the set \(\overline{\{x\in X:f(x)\neq 0\}}\) is contained in an open ball. \(C^{(b)}_{bs}(X)\) are those functions in \(C_b(X)\) which have bounded support and \(C^{(ucb)}_{bs}(X)\) are those in \(C^{(b)}_{bs}(X)\) which are uniformly continuous. For a non-negative \(\nu\in\mathcal{M}(X)\), a Borel set \(A\subset X\) is said to be \(\nu\)-continuous if \(\nu(\bar A) =\nu(A^\circ)\). In this paper the author gives an extension of the Portmanteau Theorem. The main result is: Let \(\{\mu_n\}\) be a sequence of probability measures in \(\mathcal{M}(X)\) and \(\mu\in \mathcal{M}(X), \mu\ge 0\). Then following statements are equivalent: (a) \(\mu_n\to\mu\) pointwise on \(C^{(b)}_{bs}(X)\); (b) \(\mu_n\to\mu\) pointwise on \(C^{(ucb)}_{bs}(X)\); (c) for every closed bounded set \(F\subset X\) we have lim sup \(\mu_n(F) \le \mu(F)\), and for every open bounded set \(G\subset X\) we have lim inf \(\mu_n(G) \ge \mu(G)\); (d) \(\mu_n(A)\to\mu(A)\) for every \(\mu\)-continuous Borel set \(A\subset X\). Many sophisticated lemmas are proved to reach this result.
Reviewer: Surjit Singh Khurana (Iowa City)Measures of maximal entropy.https://zbmath.org/1449.370042021-01-08T12:24:00+00:00"Amini, M."https://zbmath.org/authors/?q=ai:amini.mostafa|amini.morteza.1|amini.massih-reza|amini.morteza|amini.mohammad-m|amini-dehak.mohammad|amini.marzei|amini.massoud|amini.mahraz|amini.m-hadi|amini.m-rSummary: We extend the results of \textit{P. Walters} [Trans. Am. Math. Soc. 236, 127--153 (1978; Zbl 0375.28009); Proc. Lond. Math. Soc. (3) 28, 500--516 (1974; Zbl 0319.28011)] on the uniqueness of invariant measures with maximal entropy on compact groups to an arbitrary locally compact group. We show that the maximal entropy is attained at the left Haar measure and the measure of maximal entropy is unique.A fuzzy information measure based on cross entropy.https://zbmath.org/1449.940502021-01-08T12:24:00+00:00"Lu, Guoxiang"https://zbmath.org/authors/?q=ai:lu.guoxiangSummary: By borrowing some relative concepts in Shannon information theory, this paper improves the fuzzy symmetric cross entropy (FSCE) based on the fuzzy entropy measure, and presents a new distance measure symmetric improved fuzzy cross entropy (SIFCE). Next, this paper proves that the new measure is a metric which satisfies the conditions of non-negativity, symmetry and triangle inequality. It also satisfies boundedness. Then the relationship between SIFCE and fuzzy entropy is discussed. At last, this paper presents a new fuzzy nearness degree \({\sigma_{SIF}}\) based on the new measure. By using numerical examples in fuzzy pattern recognition, we illustrate that the recognition results are accordant with \({\sigma_{SIF}}\) and other most common fuzzy nearness degree. So it can provide a new research approach for fuzzy pattern recognition.An elementary proof of the general Poincaré formula for \(\lambda\)-additive measures.https://zbmath.org/1449.280152021-01-08T12:24:00+00:00"Dombi, József"https://zbmath.org/authors/?q=ai:dombi.jozsef-daniel|dombi.jozsef"Jónás, Tamás"https://zbmath.org/authors/?q=ai:jonas.tamasSummary: In [\textit{J. Dombi} and \textit{T. Jónás}, Inf. Sci. 490, 285--291 (2019; Zbl 07283994)] we presented the general formula for \(\lambda\)-additive measure of union of \(n\) sets and gave a proof of it. That proof is based on the fact that the \(\lambda\)-additive measure is representable. In this study, a novel and elementary proof of the formula for \(\lambda\)-additive measure of the union of \(n\) sets is presented. Here, it is also demonstrated that, using elementary techniques, the well-known Poincaré formula of probability theory is just a limit case of our general formula.