Search Results (75)

This paper explores coherent upper conditional previsions, a class of nonlinear functionals that generalize expectations while preserving consistency properties. The study focuses on their integral representation using the countably additive Möbius transform, which is possible if coherent upper previsions are defined with respect to a monotone set function of bounded variation. In this work, we prove that an integral representation with respect to a countably additive measure is also possible, on the Borel $\sigma$-algebra, even when the coherent upper prevision is defined by the Choquet integral with respect to a Hausdorff measure, which is not of bounded variation. It occurs since Hausdorff outer measures are metric measures, and therefore every Borel set is measurable with respect to them. Furthermore, when the conditioning event has a Hausdorff measure in its own Hausdorff dimension equal to zero or infinity, coherent conditional probability is defined via the countably additive Möbius transform of a monotone set function of bounded variation. The paper demonstrates the continuity of coherent conditional previsions induced by Hausdorff measures. Full article

Let $\beta >1$ be a real number and $T_{\beta }x=\beta x\left(mod1\right)$. This paper is concerned with the quantitative recurrence properties of the system $([0,1],T_{\beta })$ in some (refined) irregular sets. Specifically, let ${\alpha }_1,{\alpha }_2>0$ and $\psi :\mathbb{N}\to (0,1)$ be a positive function; we define the set $E_{{\alpha }_1,{\alpha }_2}^{\beta }=\left\{x\in [0,1):\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{1}{n}}{S}_n(x,\beta )={\alpha }_1,\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{1}{n}}{S}_n(x,\beta )={\alpha }_2\right\},$ and calculate the Hausdorff dimension of the set $E_{{\alpha }_1,{\alpha }_2}^{\beta }\left(\psi \right):=\left\{x\in E_{{\alpha }_1,{\alpha }_2}^{\beta }:|T_{\beta }^{n}x-x|<\psi \left(n\right)i.m.n\in \mathbb{N}\right\},$ where $i.m.$ stands for infinitely many. Consequently, the Hausdorff dimension of the set ${\widehat{E}}^{\beta }\left(\psi \right)=\left\{x\in [0,1):\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}{S}_n(x,\beta )\mathrm{does}\mathrm{not}\mathrm{exist}, |T_{\beta }^{n}x-x|<\psi \left(n\right)i.m.n\in \mathbb{N}\right\}$ is also determined. Full article

The purpose of this paper is twofold. On a technical side, we propose an extension of the Hausdorff distance from metric spaces to spaces equipped with asymmetric distance measures. Specifically, we focus on extending it to the family of Bregman divergences, which includes the popular Kullback–Leibler divergence (also known as relative entropy). The resulting dissimilarity measure is called a Bregman–Hausdorff divergence and compares two collections of vectors—without assuming any pairing or alignment between their elements. We propose new algorithms for computing Bregman–Hausdorff divergences based on a recently developed Kd-tree data structure for nearest neighbor search with respect to Bregman divergences. The algorithms are surprisingly efficient even for large inputs with hundreds of dimensions. As a benchmark, we use the new divergence to compare two collections of probabilistic predictions produced by different machine learning models trained using the relative entropy loss. In addition to the introduction of this technical concept, we provide a survey. It outlines the basics of Bregman geometry, and motivated the Kullback–Leibler divergence using concepts from information theory. We also describe computational geometric algorithms that have been extended to this geometry, focusing on algorithms relevant for machine learning. Full article

Brain tumours (BTs) are among the most dangerous and life-threatening cancers in humans of all ages, and the early detection of BTs can make a huge difference to their treatment. However, grade recognition is a challenging issue for radiologists involved in automated diagnosis and healthcare monitoring. Recent research has been motivated by the search for deep learning-based mechanisms for segmentation and grading to assist radiologists in diagnostic analysis. Segmentation refers to the identification and delineation of tumour regions in medical images, while classification classifies based on tumour characteristics, such as the size, location and enhancement pattern. The main aim of this research is to design and develop an intelligent model that can detect and grade tumours more effectively. This research develops an aggregated architecture called LGCNet, which combines a local context attention network and a global context attention network. LGCNet makes use of information extracted through the task, dimension and scale. Specifically, a global context attention network is developed for capturing multiple-scale features, and a local context attention network is designed for specific tasks. Thereafter, both networks are aggregated, and the learning network is designed to balance all the tasks by combining the loss functions of the classification and segmentation. The main advantage of LGCNet is its dedicated network for a specific task. The proposed model is evaluated by considering the BraTS2019 dataset with different metrics, such as the Dice score, sensitivity, specificity and Hausdorff score. Comparative analysis with the existing model shows marginal improvement and provides scope for further research into BT segmentation and classification. The scope of this study focuses on the BraTS2019 dataset, with future work aiming to extend the applicability of the model to different clinical and imaging environments. Full article

In situ thermal desorption is one of the most promising remediation techniques for soils contaminated with non-aqueous phase liquids (NAPLs), but its remediation efficiency is limited by the thermal conductivity (k) of NAPL-contaminated soils. The fractal dimension is an important factor affecting k. To systematically study the influence of the fractal dimension on k, firstly, this research establishes a three-dimensional numerical model of NAPL-contaminated soils and calculates its k. Subsequently, the reliability of the numerical simulation results is verified through experiments. Combining the numerical simulation method with Hausdorff fractal theory, we explored the relationship between the fractal dimension and k. This research shows that k decreases with increasing porosity and increases with increasing saturation. The liquid phase can form a “liquid bridge” between solid phases, greatly shortening the path of heat flux and increasing k. k is more affected by porosity. With the increase in porosity, the pore fractal dimension and liquid phase fractal dimension of NAPL-contaminated soils increase, while the solid phase fractal dimension and pore curvature fractal dimension decrease. The fractal dimension of the liquid phase increases with the increase in NAPL content. k increases with the increase in the solid phase fractal dimension, liquid phase fractal dimension, and pore curvature fractal dimension and decreases with the increase in the pore fractal dimension. This study provides a basis for the investigation of the thermal conductivity of NAPL-contaminated soils and the development of in situ thermal desorption technology. Full article

This paper conducts an in-depth investigation into the fundamental relationship between the fractal dimensions and convergence properties of mathematical sequences. By concentrating on three representative classes of sequences, namely, the factorial-decay, logarithmic-decay, and factorial–exponential types, a comprehensive framework is established to link their geometric characteristics with asymptotic behavior. This study makes two significant contributions to the field of fractal analysis. Firstly, a unified methodology is developed for the calculation of multiple fractal dimensions, including the Box, Hausdorff, Packing, and Assouad dimensions, of discrete sequences. This methodology reveals how these dimensional quantities jointly describe the structures of sequences, providing a more comprehensive understanding of their geometric properties. Secondly, it is demonstrated that different fractal dimensions play distinct yet complementary roles in regulating convergence rates. Specifically, the Box dimension determines the global convergence properties of sequences, while the Assouad dimension characterizes the local constraints on the speed of convergence. The theoretical results presented herein offer novel insights into the inherent connection between geometric complexity and analytical behavior within sequence spaces. These findings have immediate and far-reaching implications for various applications that demand precise control over convergence properties, such as numerical algorithm design and signal processing. Notably, the identification of dimension-based convergence criteria provides practical and effective tools for the analysis of sequence behavior in both pure mathematical research and applied fields. Full article

The Weierstrass function $W\left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{a}^{n}cos\left(2\pi b^{n}x\right)$ is a function that is continuous everywhere and differentiable nowhere. There are many investigations on fractal dimensions of the Weierstrass function, and the investigation of its Hausdorff dimension is still ongoing. In this paper, we summarize past researchers’ investigations on fractal dimensions of the Weierstrass function graph. Full article

In this work, the fractal continuum Maxwell law for the creep phenomenon is introduced. By mapping standard integer space-time into fractal continuum space-time using the well-known Balankin’s approach to variable-order fractal calculus, the fractal version of Maxwell model is developed. This methodology employs local fractional differential operators on discontinuous properties of fractal sets embedded in the integer space-time so that they behave as analytic envelopes of non-analytic functions in the fractal continuum space-time. Then, creep strain $\epsilon \left(t\right)$, creep modulus $J\left(t\right)$, and relaxation compliance $G\left(t\right)$ in materials with fractal linear viscoelasticity can be described by their generalized forms, ${\epsilon }^{\beta }\left(t\right),J^{\beta }\left(t\right) \mathrm{and} G^{\beta }\left(t\right)$, where $\beta =dimS/dimH$ represents the time fractal dimension, and it implies the variable-order of fractality of the self-similar domain under study, which are $dimS$ and $dimH$ for their spectral and Hausdorff dimensions, respectively. The creep behavior depends on beta, which is characterized by its geometry and fractal topology: as beta approaches one, the fractal creep behavior approaches its standard behavior. To illustrate some physical implications of the suggested fractal Maxwell creep model, graphs that showcase the specific details and outcomes of our results are included in this study. Full article

Two decades ago, Ngai and Wang introduced a well-known finite type condition (FTC) on the self-similar iterated function system (IFS) with overlaps and used it to calculate the Hausdorff dimension of self-similar sets. In this paper, inspired by Ngai and Wang’s idea, we define a new FTC on self-affine IFS and obtain an analogous formula on the generalized dimensions of self-affine sets. The generalized dimensions raised by He and Lau are used to estimate the Hausdorff dimension of self-affine sets. Full article

The development of artificial intelligence makes it possible to rapidly segment landslides. However, there are still some challenges in landslide segmentation based on remote sensing images, such as low segmentation accuracy, caused by similar features, inhomogeneous features, and blurred boundaries. To address these issues, we propose a novel deep learning model called AST-UNet in this paper. This model is based on structure of SwinUNet, attaching a channel Attention and spatial intersection (CASI) module as a parallel branch of the encoder, and a spatial detail enhancement (SDE) module in the skip connection. Specifically, (1) the spatial intersection module expands the spatial attention range, alleviating noise in the image and enhances the continuity of landslides in segmentation results; (2) the channel attention module refines the spatial attention weights by feature modeling in the channel dimension, improving the model’s ability to differentiate targets that closely resemble landslides; and (3) the spatial detail enhancement module increases the accuracy for landslide boundaries by strengthening the attention of the decoder to detailed features. We use the landslide data from the area of Luding, Sichuan to conduct experiments. The comparative analyses with state-of-the-art (SOTA) models, including FCN, UNet, DeepLab V3+, TransFuse, TranUNet, and SwinUNet, prove the superiority of our AST-UNet for landslide segmentation. The generalization of our model is also verified in the experiments. The proposed AST-UNet obtains an F1-score of 90.14%, mIoU of 83.45%, foreground IoU of 70.81%, and Hausdorff distance of 3.73, respectively, on the experimental datasets. Full article

Let $\phi :\mathbb{N}\to (0,1]$ be a positive function. We consider the size of the set ${\mathbb{E}}_f\left(\phi \right):=\{\beta >1:|T_{\beta }^n\left(x\right)-f\left(\beta \right)|<\phi \left(n\right)i.o.n\}$, where “$i.o.n$” stands for “infinitely often”, and $f:(1,\infty )\to [0,1]$ is a Lipschitz function. For any $x\in (0,1]$, it is proved that the Hausdorff measure of ${\mathbb{E}}_f\left(\phi \right)$ fulfill a dichotomy law according to $\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{log\phi \left(n\right)}{n}}=-\infty$ or not, where $T_{\beta }$ is the $\beta$-transformation. In ergodic theory, the phenomenon of shrinking targets is crucial for understanding the long-term behavior of systems. By studying the shrinking target problem of the $\beta$ dynamical system, we can reveal the relationship between randomness and determinism, which is significant for constructing more complex mathematical models. Moreover, there is a close connection between the $\beta$ transformation and number theory. Investigating the contraction target problem helps uncover new properties and patterns in number theory, advancing the development of this field. In this work, we establish a significant relationship between the decay rate of the positive function $\phi \left(n\right)$ and the structural properties of the set $E_f\left(\phi \right)$. Specifically, we show that: The Hausdorff dimension of $E_f\left(\phi \right)$ either vanishes or is positive based on the behavior of $\phi \left(n\right)$ as n approaches infinity. The establishment of this dichotomy can help us more effectively understand the geometric characteristics and dynamical behavior of the system, thereby aiding our acceptance and comprehension of complex theories. Researching this shrinking target problem can help us uncover new properties in number theory, leading to a better understanding of the structure of numbers and promoting the development of related fields in number theory. Full article

The problems associated with the visible set have been explored by various scholars. In this paper, we investigate the Hausdorff dimension and the topological properties of the visible set in relation to the products of homogeneous Cantor sets. To address these issues and establish our results, we employ beta expansion theory, numerical calculations and several technical results from fractal geometry. Our research reveals that the case of the homogeneous Cantor set differs from those of the middle Cantor sets. Furthermore, we identify a critical number that is linked to both the Hausdorff dimension and the topological properties of the visible set. Full article

More than 50 years ago, Rufus Bowen noticed a natural relation between the ergodic theory and the dimension theory of dynamical systems. He proved a formula, known today as the Bowen’s formula, that relates the Hausdorff dimension of a conformal repeller to the zero of a pressure function defined by a single conformal map. In this paper, we extend the result of Bowen to a sequence of conformal maps. We present a dynamical solenoid, i.e., a generalized dynamical system obtained by backward compositions of a sequence of continuous surjections ${(f_n:X\to X)}_{n\in \mathbb{N}}$ defined on a compact metric space $(X,d).$ Under mild assumptions, we provide a self-contained proof that Bowen’s formula holds for dynamical conformal solenoids. As a corollary, we obtain that the Bowen’s formula holds for a conformal surjection $f:X\to X$ of a compact Full article

Following the work on non-stationary fractal interpolation (Mathematics 7, 666 (2019)), we study non-stationary or statistically self-similar fractal interpolation on the Sierpiński gasket (SG). This article provides an upper bound of box dimension of the proposed interpolants in certain spaces under suitable assumption on the corresponding Iterated Function System. Along the way, we also prove that the proposed non-stationary fractal interpolation functions have finite energy. Full article