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We propose a new definition of preimage entropy dimension for continuous maps on compact metric spaces, investigate fundamental properties of the preimage entropy dimension, and compare the preimage entropy dimension with the topological entropy dimension. The defined preimage entropy dimension satisfies various basic properties of topological entropy dimension, for example, the preimage entropy dimension of a subsystem is bounded by that of the original system and topologically conjugated systems have the same preimage entropy dimension. Also, we discuss the relation between the preimage entropy dimension and the preimage entropy.

In this paper we study topological entropy dimension on an arbitrary subset and give the definitions of Bowen topological entropy dimension and packing topological entropy dimension for amenable group actions. Moreover, we introduce the α -local entropy for any Borel probability measure on a compact metric space, and investigate the relation between α -local entropy of Borel probability measures and Bowen α -topological entropy. Also, we establish a variational principle for packing topological entropy dimension on compact subsets in the context of amenable group actions.

The notion of topological entropy dimension for a Z -action has been introduced to measure the subexponential complexity of zero entropy systems. Given a Z 2 -action, along with a Z 2 -entropy dimension, we also consider a finer notion of directional entropy dimension arising from its subactions. The entropy dimension of a Z 2 -action and the directional entropy dimensions of its subactions satisfy certain inequalities. We present several constructions of strictly ergodic Z 2 -subshifts of positive entropy dimension with diverse properties of their subgroup actions. In particular, we show that there is a Z 2 -subshift of full dimension in which every direction has entropy 0.

The notion of metric entropy dimension is introduced to measure the complexity of entropy zero dynamical systems. For measure preserving systems, we define entropy dimension via the dimension of entropy generating sequences. This combinatorial approach provides us with a new insight for analyzing entropy zero systems. We also define the dimension set of a system to investigate the structure of the randomness of the factors of a system. The notion of a uniform dimension in the class of entropy zero systems is introduced as a generalization of a K-system in the case of positive entropy. We investigate joinings among entropy zero systems and prove the disjointness property among some classes of entropy zero systems using dimension sets. Given a topological system, we compare topological entropy dimension with metric entropy dimension.

For each integer k>0 , let n_k and m_k be integers such that n_k 2, m_k 2 , and let D_k be a subset of \\0, n_k-1\\ \\0, m_k-1\\ . For each w=(i,j)ın D_k , we define an affine transformation on  ^2 by _w(x)=T_k(x+w), wınD_k, where T_k=diag(n_k^-1,m_k^-1) . The non-empty compact set E=_k=1^ınfty_(w_1w_2 w_k)ın _i=1^kD_i_w_1 _w_2 _w_k is called a nonautonomous carpet .In the paper, we provide the lower, packing, box-counting and Assouad dimensions of the nonautonomous carpets E . We also explore the dimension properties of nonautonomous measures supported on E , and we provide Hausdorff, packing and entropy dimension formulas of  .

Entropy dimension is an entropy-type quantity which takes values in $[0,1]$ and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy dimension by choosing suitable rotations for the base system.

We study the topological complexities of relative entropy zero extensions acted upon by countable-infinite amenable groups. First, for a given Følner sequence { F n } n = 0 + ∞ , we define the relative entropy dimensions and the dimensions of the relative entropy generating sets to characterize the sub-exponential growth of the relative topological complexity. we also investigate the relations among these. Second, we introduce the notion of a relative dimension set. Moreover, using the method, we discuss the disjointness between the relative entropy zero extensions via the relative dimension sets of two extensions, which says that if the relative dimension sets of two extensions are different, then the extensions are disjoint.

This research empirically investigates the Environmental, Social, and Governance (ESG) sustainable determinants affecting the 24-hr around-the-year operation model of Contemporary Taiwan's convenience stores. Given the operational tensions between continuous service provision and sustainability imperatives, this research addresses the critical question of how customers, convenience stores, and the public perceive the existential viability of 24-hr operations through cross-analyzing three core dimensions: customers' need for 24-hr convenience store goods and services (CN24CSGS), convenience stores’ 24-hr goods and services supply (CS24GSS), and public expectations of 24-hr convenience store goods and services (PE24CSGS). The triadic reciprocal determinism of Social Learning Theory (SLT) was applied to integrate a comprehensive analytical framework. A mixed-methods approach employing factor analysis (FA), triple-dimension entropy analysis (TEA), and analytical network process (ANP) was utilized to identify and validate core determinants across quantitative and qualitative dimensions. Results reveal that staff health care and management service level (SHC-MSL) emerges as the most critical determinant (SCEW = 0.1659), followed by public welfare programs (PWP = 0.0694) and employee occupational training level (EOTL = 0.0596), all within the governance accountability framework. Findings indicate that carbon emission reduction (CER = 0.0503) represents the primary environmental challenge, while community public education development (CPED = 0.0308) constitutes the key social responsibility factor. Eventually, 24-hr convenience store operations remain sustainable when supported by robust ESG governance mechanisms, particularly emphasizing employee welfare and environmental stewardship. These findings provide actionable insights for managers and policymakers of convenience store in Taiwan seeking to balance operational efficiency with sustainable development goals. An Empirical Investigation on ESG Sustainable Determinants for 24-hour Around Year Operation Model of Contemporary Taiwan’s Convenience Stores This research empirically investigates the Environmental, Social, and Governance (ESG) sustainable determinants affecting the 24-hour around-the-year operation model of Contemporary Taiwan’s convenience stores. Given the operational tensions between continuous service provision and sustainability imperatives, this research addresses the critical question of how customers, convenience stores, and the public perceive the existential viability of 24-hour operations through cross-analyzing three core dimensions: customers' need for 24-hour convenience store goods and services (CN24CSGS), convenience stores’ 24-hour goods and services supply (CS24GSS), and public expectations of 24 hour convenience store goods and services (PE24CSGS). The triadic reciprocal determinism of Social Learning Theory (SLT) was applied to integrate a comprehensive analytical framework. A mixed-methods approach employing factor analysis (FA), triple-dimension entropy analysis (TEA), and analytical network process (ANP) was utilized to identify and validate core determinants across quantitative and qualitative dimensions. Results reveal that staff healthcare and management service level (SHC-MSL) emerges as the most critical determinant (0.1659 of SCEW), followed by public welfare programs (0.0694 of PWP) and employee occupational training level (EOTL = 0.0596), all within the governance accountability framework. Findings indicate that carbon emission reduction (0.0503 of CER) represents the primary environmental challenge, while community public education development (0.0308 of CPED) constitutes the key social responsibility factor. Eventually, 24-hour convenience store operations remain sustainable when supported by robust ESG governance mechanisms, particularly emphasizing employee welfare and environmental stewardship. These findings provide actionable insights for managers and policymakers of convenience store in Taiwan seeking to balance operational efficiency.

The swarm intelligence algorithm has become an important method to solve optimization problems because of its excellent self-organization, self-adaptation, and self-learning characteristics. However, when a traditional swarm intelligence algorithm faces high and complex multi-peak problems, population diversity is quickly lost, which leads to the premature convergence of the algorithm. In order to solve this problem, dimension entropy is proposed as a measure of population diversity, and a diversity control mechanism is proposed to guide the updating of the swarm intelligence algorithm. It maintains the diversity of the algorithm in the early stage and ensures the convergence of the algorithm in the later stage. Experimental results show that the performance of the improved algorithm is better than that of the original algorithm.

Multifractal analysis has been used in the exploration of soil grain size distributions (GSDs) in environmental and agricultural research. However, multifractal studies regarding the GSDs of sediments in braided delta-front are currently scarce. Open-source software designed for the realization of this technique has not yet been programmed. In this paper, the multifractal parameters of 61 GSDs from braided delta-front in the Paleogene Enping Formation in Huilu Low Uplift, Pearl River Mouth basin, are calculated and compared with traditional parameters. Multifractal generalized dimension spectrum curves are sigmoidal and decrease monotonically. Multifractal singularity spectrum curves are asymmetric, convex, and right-hook unimodal. The entropy dimension and singularity spectrum width ranges of silt-mudstones and gravelly sandstones are wider than those of fine and medium-coarse sandstones. The symmetry degree scopes from different lithologies are concentrated in distinguishing intervals. With the increase of grain sizes, the symmetry degree decreases overall. Both the symmetry degree and mean of GSDs are effective to distinguish the different lithologies from various depositional environments. A flexible and easy-to-use MATLAB (2021b)® GUI (graphic user interface) package, MfGSD (Multifractal of GSD, V1.0), is provided to perform multifractal analysis on sediment GSDs. After raw GSDs imported into MfGSD, multifractal parameters are batch calculated and graphed in the interface. Then, all multifractal parameters can be exported to an Excel file, including entropy dimension, singularity spectrum, correlation dimension, symmetry degree of multifractal spectrum, etc. MfGSD is effective, and the multifractal parameters outputted from MfGSD are helpful to distinguish depositional environments of GSDs. MfGSD is open-source software that can be used to explore GSDs from various kinds of depositional environments, including water or wind deposits.