Weighted Hausdorff content and fractional maximal operators on self-similar sets

Based on panel data from China's 13 major grain - producing provinces between 2014 and 2023, this study constructs a multidimensional grain security evaluation index system, incorporating the dimensions of quantity, quality, ecology, and capacity. Using the entropy method, kernel density estimation, Dagum Gini coefficient, and convergence model, we analyze the regional differences, dynamic evolution, and spatial convergence of grain security levels. The results show that: (1) grain security level is generally increasing; there are obvious spatial differences in grain security level among; (2) There are obvious spatial differences in grain security levels among regions, showing the trend of "Northeast> central> eastern> western"; (3) α Convergence analysis shows that the central region converges, while the northeast, east and west regions diverge; both absolute β convergence and conditional β convergence exist in the whole region and each region. Compared with previous studies, this paper provides new insights into the dynamic evolution, spatial distribution and regional convergence of multidimensional grain security in China's major grain producing areas by integrating a multi-dimensional framework including quantity, quality, ecology and capacity. The research has enriched the evaluation system of grain security and laid a solid foundation for policy optimization.

For a fixed non-integral α>1, let PS(α)={⌊nα⌋:n=1,2,…}. We show that x+y=z has only finitely many solutions (x,y,z)∈PS(α)3 for almost every α>3. Furthermore, we show that PS(α) contains only finitely many arithmetic progressions of length 3 for almost every α>10. In addition, we give upper bounds for the Hausdorff dimension of the set of α∈[s,t] such that y=a1x1+⋯+anxn has infinitely many solutions in PS(α).

Abstract The distribution of the magnetic induction in a superconducting (SC) disk under electromagnetic influence was studied using the magneto-optical (MO) method. Particular attention was paid to the behavior of the trapped flux in a disk, being a SC permanent magnet, used in technical devices (motors, generators, etc). These magnets endure repeated electromagnetic ‘shocks’ during operation, which may influence their performance. Our results reveal a direct correlation between the electromagnetic exposure and the trapped flux response, with each increase or decrease in the field inducing a corresponding 40%–50% change in trapped flux for a 600 G step at 5 K. The magnitude of these changes depends on various external conditions, which could lead to additional energy dissipation and potential heating, which can affect the reliability of SC magnets in applications. MO imaging of the penetration field reveals the formation of a highly irregular flux front structure, which may reflect the distribution of pinning centers. The transformation of this induction structure demonstrates an enhancement in shielding properties as a result of extruded alloy annealing. A scaling analysis of the induction flux profiles reveals roughness exponents in the range of 0.435–0.475 and Hausdorff dimensions in the range of 1.525–1.565, both of them influenced by magnetic field strength, temperature, and annealing.

Intermediate dimensions of measures: Interpolating between Hausdorff and Minkowski dimensions

Urban flooding increasingly challenges rapidly expanding cities in developing countries. Migration, weak urban planning, and unregulated land use collectively intensify flood risk. Effective flood mitigation requires understanding the dynamic interactions between physical and social processes that shape urban vulnerability. This study examines how migrant households in flood-prone areas adapt over time to enhance resilience. The study applies a dynamic flood risk framework using settlement-duration cohorts from 560 peri-urban households in Phnom Penh. Findings show that rapid in-migration into flood-prone zones has increased physical exposure to flood hazards. Migrants’ adaptation and resilience, however, develop gradually, reducing vulnerability only over time. Newer migrants remain highly vulnerable due to insecure housing, limited renovation, and restricted access to flood information. Long-term migrants face structural and economic challenges, including low income, limited access to credit, and deteriorating housing conditions. Mid-term migrants demonstrate the strongest adaptive capacity, supported by stable income, housing investment, and access to flood information. Overall, the study contributes to more dynamic urban risk frameworks that incorporate demographic and socioeconomic transitions. These insights are relevant for other rapidly growing cities, particularly those in Southeast Asia.

Abstract We study infinite graph-directed iterated function systems (GIFS) whose underlying graph is not strongly connected and has countably many vertices and edges. In addition to a summability condition for the physical potential, we provide lower and upper estimates of the Hausdorff dimension of the limit set of such GIFS. Bowen type formula is also given under conformal condition and suitable separation conditions. We also introduce perturbed GIFS in which the images of arbitrarily chosen contraction mappings shrink to a single point. In other words, the graph of the perturbed GIFS differs from that of unperturbed GIFS. Assuming suitable continuity condition on contraction mappings, we prove that the Hausdorff dimension of the limit set of the perturbed GIFS converges to that of the unperturbed GIFS. Our main theorems extends the finite-graph results in Tanaka (2019 Nonlinearity 32 728–67; 2016 J. Fractal Geom. 3 119–61) to the setting of infinite graphs. As applications, we consider a perturbed nonconformal mapping, as well as convergence and non-convergence in the Hausdorff dimension for perturbed complex continued fractions with degeneration.

Older adults with disabilities face compounded vulnerabilities due to both functional limitations and socioeconomic disadvantage. In South Korea, where public welfare systems remain fragmented and cultural values emphasize independence and productivity, understanding the mechanisms linking socioeconomic status (SES) to health outcomes is critical. This study investigates whether reserve capacity mediates the relationship between SES and self-rated health (SRH) in older adults with disabilities. Data were drawn from the supplementary survey on people with disabilities in the 18th wave (2023) of the Korea Welfare Panel Study (KWePS). The analytic sample included older adults aged 65 and above with registered disabilities. A multiple mediation analysis was conducted using Model 4 of the PROCESS macro in SPSS to examine whether three dimensions of reserve capacity—intrapsychic resources (self-esteem), interpersonal resources (social support satisfaction), and tangible resources (use of public disability services)—mediated the relationship between SES and SRH. Demographic and health-related covariates were statistically controlled. The results are as follows: The direct effect of SES on SRH was not significant; however, significant indirect effects were found through all three mediators. Higher SES was positively associated with intrapsychic and interpersonal resources and negatively associated with tangible resource use. Among the mediators, interpersonal resources had the strongest positive effect on SRH, while tangible resources showed a negative association—possibly due to compensatory activation or increased disease awareness among service users. The findings highlight the importance of psychosocial and relational resources in shaping perceived health among disabled older adults in Korea. Policy interventions should move beyond material assistance and focus on strengthening social networks and psychological resilience to reduce health disparities in this population.

We show that all Cantor sets in \mathbb{R}^{d} can be accompanied by another Cantor set in \mathbb{R}^{d} so that their product has a pinned tree distance set with nonempty interior. As a corollary, we construct Cantor sets of Hausdorff dimension d/2 in \mathbb{R}^{d} for even d that have a pinned tree distance set with nonempty interior. Our results generalize to the setting in which the Euclidean distance, |x-y| , is replaced by a general function, \phi(x,y) , satisfying a mild derivative condition.

In this paper, we introduce a new concept of dimension on fractal networks, the so-called Hausdorff Steiner [Formula: see text]-dimension. It is defined by a natural generalization of the Hausdorff dimension in Zeng’s work. In particular, this new dimension is a non-trivial higher estimate for the Hausdorff dimension. We explore its basic properties and determine the exact values of some networks. Meanwhile, we discuss the changes under different graph operations, including single node sum and the Cartesian product of graphs. Within a specific class of networks, the Hausdorff Steiner [Formula: see text]-dimension and the Hausdorff dimension exhibit an exact quantitative relationship.

Notwithstanding that digital transformation has been mandated at local government levels in developing nations, too little is understood about how policy capacity shapes implementation outcomes in rural contexts, undermining policy effectiveness. This qualitative study examines how policy capacity – encompassing institutional, operational, and social dimensions – shapes the implementation of mandated e-governance in rural Indonesian village administration. Drawing on 23 semi-structured interviews with government officials and citizens across five high-performing villages in Banyuwangi Regency, East Java, this paper contextualises current debates with the lived experiences of those mediating digital transformation. Our analysis brings out how policy capacity manifests through three interconnected dimensions: institutional frameworks, implementation approaches, and social integration mechanisms. The findings make clear a multidimensional understanding where institutional resilience, implementation adaptability, and social embeddedness reciprocally determine villages’ ability to sustainably transform public service delivery. We challenge prevailing orthodoxies that privilege technocratic solutions, arguing instead how sustainable e-governance grows from blending institutional stability, adaptive implementation, and authentic community participation.

Let H \mathbb {H} denote the three-dimensional Heisenberg group. In this paper, we study vertical curves in H \mathbb {H} and fibers of maps H → R 2 \mathbb {H}\to \mathbb {R}^2 from a metric perspective. We say that a set in H \mathbb {H} is a vertical curve if it satisfies a cone condition with respect to a homogeneous cone with axis ⟨ Z ⟩ \langle Z \rangle , the center of H \mathbb {H} . This is analogous to the cone condition used to define intrinsic Lipschitz graphs. In the first part of the paper, we prove that connected vertical curves are locally bi-Hölder equivalent to intervals. We also show that the class of vertical curves coincides with the class of intersections of intrinsic Lipschitz graphs satisfying a transversality condition. Unlike intrinsic Lipschitz graphs, the Hausdorff dimension of a vertical curve can vary; we construct vertical curves with Hausdorff dimension either strictly larger or strictly smaller than 2. Consequently, there are intersections of intrinsic Lipschitz graphs with Hausdorff dimension either strictly larger or strictly smaller than 2. In the second part of the paper, we consider smooth functions β \beta from the unit ball B B in H \mathbb {H} to R 2 \mathbb {R}^2 . We show that, in contrast to the situation in Euclidean space, there are maps such that β \beta is arbitrarily close to the projection π \pi from H \mathbb {H} to the horizontal plane, but the average H 2 \mathcal {H}^2 measure of a fiber of β \beta in B B is arbitrarily small.

In this article, we further develop the thermodynamic formalism of affine iterated function systems with countably many transformations by showing the existence and extending earlier characterisations of the equilibrium states of finite affine iterated function systems to the countably infinite case. As an application, under mild conditions, we prove that the affinity dimension of a countable affine iterated function system is equal to the supremum of the affinity dimensions of its finite subsystems. We deduce corollaries concerning the Hausdorff dimension of countably generated self-affine sets in dimensions 1 1 , 2 2 , and 3 3 satisfying mild deterministic assumptions and in arbitrary dimension with generic translations.

Objective: To explore the application effect of combined exercise intervention based on the hospital-community-family model on intrinsic capacity in elderly patients with diabetes mellitus complicated by chronic kidney disease. Methods: Using convenience sampling, 100 elderly patients with diabetes mellitus complicated by chronic kidney disease who received treatment in the endocrinology department of a tertiary A-level hospital from May 2024 to May 2025 were selected as the study subjects. They were randomly divided into an experimental group (50 cases) and a control group (50 cases) using a random number table method. The control group received routine health education and telephone follow-up, while the experimental group, in addition to the control group’s interventions, underwent combined exercise intervention based on the hospital-community-family model. Remote medical guidance was utilized to monitor and study the application effect of exercise intervention on intrinsic capacity in elderly patients with diabetes mellitus complicated by chronic kidney disease. Fasting blood glucose, 2-hour postprandial blood glucose, glomerular filtration rate, 6-minute walk distance, and scores in five dimensions of intrinsic capacity (exercise, cognition, psychology, vitality, and sensation) were measured before the intervention, at 4 weeks of intervention, and at 12 weeks of intervention for both groups. Results: Before the exercise intervention, there were no statistically significant differences (p > 0.05) between the two groups in terms of fasting blood glucose, 2-hour postprandial blood glucose, glomerular filtration rate, 6-minute walk distance, and scores across five dimensions of intrinsic capacity: mobility, cognition, psychology, vitality, and sensation. After 12 weeks of intervention, the experimental group demonstrated significantly higher scores than the control group in glomerular filtration rate, 6-minute walk distance, and the dimensions of mobility, cognition, and vitality within intrinsic capacity, with all differences being statistically significant (p < 0.05). Conversely, the experimental group showed significantly lower scores than the control group in fasting blood glucose, 2-hour postprandial blood glucose, and the psychological dimension of intrinsic capacity, with these differences also being statistically significant (p < 0.05). Conclusion: Continuous nursing care utilizing telemedicine based on a hospital-community-family model combined with exercise intervention can effectively enhance exercise tolerance and intrinsic capacity in elderly patients with diabetes mellitus complicated by chronic kidney disease, thereby improving their quality of life. The effectiveness of the intervention is positively correlated with the duration of the intervention.

More quantitative evidence is necessary on the link between livelihood resilience and livelihood adaptive capacity (LAC) in disaster resettlement. This study used 459 field research data collected from Ankang Prefecture, southern Shaanxi, China, examining how livelihood resilience influences adaptive capacity in the context of disaster-induced relocation. The resilience of rural household livelihood systems is described in terms of two components, general resilience, and specific resilience, which are quantified using the space vector method from systems engineering. The awareness, ability, and action framework is used to measure the LAC of rural households, and quantile regression is applied to explore the impact of livelihood resilience on LAC. Guided by the Sustainable Livelihoods Approach (SLA) and awareness, ability, action framework, we differentiate between cognitive, resource-based, and behavioral dimensions of adaptive capacity. The space vector method further reveals that individual adaptive capacity is reinforced by community-level resilience. It is found that: 1. livelihood resilience has a significant positive effect on high LAC levels, with the strongest effects observed at lower quantiles; as livelihood resilience increases, LAC also increases significantly. For rural households with low LAC levels, the impact is not significant; 2. general resilience and education have significant positive effects on all levels of LAC, with high levels being the most affected; 3. specific resilience has a significant negative effect on the lowest level of LAC only, and no significant effect on other levels. This study deepens our understanding of the relationship between livelihood resilience and LAC in the context of disaster resettlement, while testing the relationship between the two provides a methodological contribution to the study of disaster resettlement and community development.

We study the additive and fractal structure of digit-restricted subsets of the unit interval where AD=∑n=1∞anb−n:an∈D⊆{0,…,b−1},|D|≥2, defined by allowing only digits from D in base-b expansions. These sets generalize the middle-third Cantor set and include a wide range of missing-digit and structured-digit fractals. We develop a rigorous framework for base-b digit arithmetic that separates purely discrete digit-combinatorics from carry effects. We give sharp sufficient criteria for intervals in AD+AD and AD−AD via carry-free digit blocks, establish an arithmetic obstruction to interval formation for all iterated sumsets, and prove a dimension-jump dichotomy: either a gcd obstruction prevents intervals for every k, or else some iterated sumset AD(k) contains an interval, and hence has full Hausdorff dimension 1. We also discuss the similarity-dimension formula under the open set condition, include definitions and preliminaries for a broad audience, and situate the results within classical and modern literature on Cantor sets and sumsets of self-similar sets.

Industrial policies are resurging across the world in response to climate change and geopolitical challenges. Current scholarship predominantly argues that state capacity for industrial policy depends on nationally oriented features, such as bureaucratic efficiency, financial resources, and expertise. However, the increasing uncertainties in globalized industrial supply chains mean that today, industrial policies must often expand into other countries to be successful. We propose complementing the domestically oriented dimension of state capacity with an internationally oriented one, drawing from the international political economy literature on state power. To demonstrate the merit of this approach, we analyze China's and the United States’ state capacity and actions to internationalize their electric vehicle industrial policies, in order to secure critical minerals. We find that both countries use their large market shares of global trade, while China leverages its powerful state bureaucracy and the United States its dominant military and macrofinancial regime. Thus, we argue that in order for a country to internationalize its industrial policy, a country must combine market dominance with capacities that are likely to be highly specific to their own comparative advantage.

Distances in sparse sets of large Hausdorff dimension

ABSTRACT Tourist psychological capacity, a critical dimension of tourism environmental capacity, significantly influences visitor satisfaction. This study develops a methodological framework and predictive model based on crowding perception to estimate this capacity for ancient towns in metropolitan suburbs. Using field survey data from Anren Ancient Town in Chengdu, key findings reveal the following: (1) Psychological capacity is shaped by occupation, travel companions, motivation, expectations, encounter preferences and environmental factors; (2) The instantaneous psychological capacity threshold is 501–800 visitors, with satisfaction peaking at 301–500 individuals; (3) For tour groups, satisfaction is highest with 1–5 concurrent groups, while the threshold spans 6–10 groups; (4) A tiered management framework integrating flow control, spatial restructuring and long-term system development is proposed to enhance capacity. The study provides a replicable analytical tool and crucial baseline data for specific regions, offering practical pathways for the sustainable management of similar heritage destinations.

Birth preparedness and complication readiness (BPCR) is a critical strategy to improve maternal and neonatal outcomes by reducing delays in accessing skilled care. Despite its importance, knowledge levels remain inadequate, particularly among primigravid women. This study evaluated the effectiveness of a companion-integrated BPCR (CI-BPCR) awareness package in improving knowledge and childbirth experience among primigravid women in Eastern India. A quasi-experimental study was conducted at two tertiary care hospitals serving comparable demographic populations. Eighty primigravid women (40 per group) were enrolled through purposive sampling. The intervention group received a structured CI-BPCR awareness package at 28-32 weeks of gestation, incorporating education on birth preparedness, danger sign recognition, and birth companion training through interactive sessions. The control group received standard antenatal care. Knowledge was assessed at baseline and 2 weeks postintervention using a validated questionnaire adapted from the JHPIEGO tool. Childbirth experience was evaluated 24-48 h postpartum using the Childbirth Experience Questionnaire. Thirty-five participants per group completed the study. The intervention group demonstrated significant improvements in knowledge across all domains: birth preparedness, danger signs in pregnancy, labor, puerperium, and newborn ( P < 0.001). Childbirth experience was significantly better in the intervention group across dimensions of own capacity ( P < 0.001), professional support ( P < 0.001), perceived safety ( P = 0.003), participation ( P = 0.001), and sense of security ( P < 0.001). Pain levels showed no significant difference ( P = 0.607). The CI-BPCR awareness package significantly improved knowledge and childbirth experience among primigravid women, demonstrating its potential for integration into routine antenatal care in resource-limited settings.