Search Results (44)

This study introduces a new metric structure called the Graphical Fuzzy Cone Metric Space (GFCMS) and explores its essential properties in detail. We examine its topological aspects in detail and introduce the notion of Hausdorff distance within this setting—an advancement not previously explored in any graphical structure. Furthermore, a fixed-point result is proven within the framework of GFCMS, accompanied by examples that demonstrate the applicability of the theoretical results. As a significant application, we construct fractals within GFCMS, marking the first instance of fractal generation in a graphical structure. This pioneering work opens new avenues for research in graph theory, fuzzy metric spaces, topology, and fractal geometry, with promising implications for diverse scientific and computational domains. 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

Purpose: This study aims to create a deep learning (DL) model capable of accurately delineating the ventricles, and by extension, the periventricular space (PVS), following the 2021 EPTN Neuro-Oncology Atlas guidelines on T1-weighted contrast-enhanced MRI scans (T1CE). The performance of this DL model was quantitatively and qualitatively compared with an off-the-shelf model. Materials and Methods: An nnU-Net was trained for ventricle segmentation using both CT and T1CE MRI images from 78 patients. Its performance was compared to that of a publicly available pretrained segmentation model, SynthSeg. The evaluation was conducted on both internal (N = 18) and external (n = 18) test sets, with each consisting of paired CT and T1CE MRI images and expert-delineated ground truths (GTs). Segmentation accuracy was assessed using the volumetric Dice Similarity Coefficient (DSC), 95th percentile Hausdorff distance (HD95), surface DSC, and added path length (APL). Additionally, a local evaluation of ventricle segmentations quantified differences between manual and automatic segmentations across both test sets. All segmentations were scored by radiotherapy technicians for clinical acceptability using a 4-point Likert scale. Results: The nnU-Net significantly outperformed the SynthSeg model on the internal test dataset in terms of median [range] DSC, 0.93 [0.86–0.95] vs. 0.85 [0.67–0.91], HD95, 0.9 [0.7–2.5] mm vs. 2.2 [1.7–4.8] mm, surface DSC, 0.97 [0.90–0.98] vs. 0.84 [0.70–0.89], and APL, 876 [407–1298] mm vs. 2809 [2311–3622] mm, all with p < 0.001. No significant differences in these metrics were found in the external test set. However clinical ratings favored nnU-Net segmentations on the internal and external test sets. In addition, the nnU-Net had higher clinical ratings than the GT delineation on the internal and external test set. Conclusions: The nnU-Net model outperformed the SynthSeg model on the internal dataset in both segmentation metrics and clinician ratings. While segmentation metrics showed no significant differences between the models on the external set, clinician ratings favored nnU-Net, suggesting enhanced clinical acceptability. This suggests that nnU-Net could contribute to more time-efficient and streamlined radiotherapy planning workflows. Full article

In this manuscript, we analyze fuzzy-fixed-point results for fuzzy-mappings under some fuzzy contraction conditions in the setting of a complete fuzzy metric space. Fuzzy-fixed-point techniques are used in mathematical modeling to solve problems where traditional methods fail due to imprecise or uncertain data. To obtain fuzzy-fixed-points, different contraction conditions are implemented in a fuzzy context. To emphasize the impact of our research, we have furnished several intriguing examples. Applications are also incorporated to furnish the results. Previous results are given as corollaries from the relevant research. Our results extend and combine many results that exist in a significant area of related research. Full article

In this manuscript, we present the classical Hutchinson–Barnsley theory on the product neutrosophic fractal spaces by utilizing an iterated function system, which is enclosed by neutrosophic Edelstein contractions and a finite number of neutrosophic b-contractions. Further, we provide a sequence of sets that, under appropriate conditions and in terms of the Hausdorff neutrosophic metric, converge to the attractor set of specific neutrosophic iterated function systems. Furthermore, we present a fuzzy variant of α-dense curves that can accurately approximate the attractor set of certain iterated function systems with barely noticeable and controlled errors. In the end, we make a connection between the above-discussed concepts of neutrosophic theory and α-density theory. Full article

The family $\mathcal{F}\left(U\right)$ of all fuzzy sets in a normed space cannot be a vector space. This deficiency affects its application to practical problems. The topics of functional analysis and nonlinear analysis in mathematics are based on vector spaces. Since these two topics have been well developed such that their tools can be used to solve practical economics and engineering problems, lacking a vector structure for the family $\mathcal{F}\left(U\right)$ diminishes its applications to these kinds of practical problems when fuzzy uncertainty has been detected in a real environment. Embedding the whole family $\mathcal{F}\left(U\right)$ into a Banach space is still not possible. However, it is possible to embed some interesting and important subfamilies of $\mathcal{F}\left(U\right)$ into some suitable Banach spaces. This paper presents the concrete and detailed structures of these kinds of Banach spaces such that their mathematical structures can penetrate the core of practical economics and engineering problems in fuzzy environments. The important issue of uniqueness for these Banach spaces is also addressed via the concept of isometry. 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

Automated perimetrium segmentation of transvaginal ultrasound images is an important process for computer-aided diagnosis of uterine diseases. However, ultrasound images often contain various structures and textures, and these structures have different shapes, sizes, and contrasts; therefore, accurately segmenting the parametrium region of the uterus in transvaginal uterine ultrasound images is a challenge. Recently, many fully supervised deep learning-based methods have been proposed for the segmentation of transvaginal ultrasound images. Nevertheless, these methods require extensive pixel-level annotation by experienced sonographers. This procedure is expensive and time-consuming. In this paper, we present a bidirectional copy–paste Mamba (BCP-Mamba) semi-supervised model for segmenting the parametrium. The proposed model is based on a bidirectional copy–paste method and incorporates a U-shaped structure model with a visual state space (VSS) module instead of the traditional sampling method. A dataset comprising 1940 transvaginal ultrasound images from Tongji Hospital, Huazhong University of Science and Technology is utilized for training and evaluation. The proposed BCP-Mamba model undergoes comparative analysis with two widely recognized semi-supervised models, BCP-Net and U-Net, across various evaluation metrics including Dice, Jaccard, average surface distance (ASD), and Hausdorff_95. The results indicate the superior performance of the BCP-Mamba semi-supervised model, achieving a Dice coefficient of 86.55%, surpassing both U-Net (80.72%) and BCP-Net (84.63%) models. The Hausdorff_95 of the proposed method is 14.56. In comparison, the counterparts of U-Net and BCP-Net are 23.10 and 21.34, respectively. The experimental findings affirm the efficacy of the proposed semi-supervised learning approach in segmenting transvaginal uterine ultrasound images. The implementation of this model may alleviate the expert workload and facilitate more precise prediction and diagnosis of uterine-related conditions. Full article

We consider a class of metrics which are equivalent to the Hausdorff metric in some sense to establish the well-posedness of fixed point problems associated with multifunctions of metric spaces, satisfying various generalized contraction conditions. Examples are provided to justify the applicability of new results. Full article

In this paper, we introduce the novel concept of generalized distance denoted as $J_{\theta }$ and call it an extended b-generalized pseudo-distance. With the help of this generalized distance, we define a generalized point to set distance $J_{\theta }(\mathfrak{u},{\mathcal{H}}^{★}),$ a generalized Hausdorff type distance and a $P^{J_{\theta }}$-property of a pair $({\mathcal{H}}^{★},{\mathcal{K}}^{★})$ of nonempty subsets of extended b-metric space $({\mathfrak{U}}^{★},{\rho }_{\theta }).$ Additionally, we establish several best proximity point theorems for multi-valued contraction mappings of Nadler type defined on b-metric spaces and extended b-metric spaces. Our findings generalize numerous existing results found in the literature. To substantiate the introduced notion and validate our main results, we provide some concrete examples. Full article

A certain Grothendieck topology assigned to a metric space gives rise to a sheaf cohomology theory which sees the coarse structure of the space. Already constant coefficients produce interesting cohomology groups. In degree 0, they see the number of ends of the space. In this paper, a resolution of the constant sheaf via cochains is developed. It serves to be a valuable tool for computing cohomology. In addition, coarse homotopy invariance of coarse cohomology with constant coefficients is established. This property can be used to compute cohomology of Riemannian manifolds. The Higson corona of a proper metric space is shown to reflect sheaves and sheaf cohomology. Thus, we can use topological tools on compact Hausdorff spaces in our computations. In particular, if the asymptotic dimension of a proper metric space is finite, then higher cohomology groups vanish. We compute a few examples. As it turns out, finite abelian groups are best suited as coefficients on finitely generated groups. Full article

We consider a general metric Steiner problem, which involves finding a set $\mathcal{S}$ with the minimal length, such that $\mathcal{S}\cup A$ is connected, where A is a given compact subset of a given complete metric space X; a solution is called the Steiner tree. Paolini, Stepanov, and Teplitskaya in 2015 provided an example of a planar Steiner tree with an infinite number of branching points connecting an uncountable set of points. We prove that such a set can have a positive Hausdorff dimension, which was an open question (the corresponding tree exhibits self-similar fractal properties). Full article

The applications of non-zero self distance function have recently been discovered in both symmetric and asymmetric spaces. With respect to invariant point results, the available literature reveals that the idea has only been examined for crisp mappings in either symmetric or asymmetric spaces. Hence, the aim of this paper is to introduce the notion of invariant points for non-crisp set-valued mappings in metric-like spaces. To this effect, the technique of $\kappa$-contraction and Feng-Liu’s approach are combined to establish new versions of intuitionistic fuzzy functional equations. One of the distinguishing ideas of this article is the study of fixed point theorems of intuitionistic fuzzy set-valued mappings without using the conventional Pompeiu–Hausdorff metric. Some of our obtained results are applied to examine their analogues in ordered metric-like spaces endowed with an order and binary relation as well as invariant point results of crisp set-valued mappings. By using a comparative example, it is observed that a few important corresponding notions in the existing literature are complemented, unified and generalized. Full article

The AP–Henstock–Kurzweil-type integral is defined on $\mathtt{X},$ where $\mathtt{X}$ is a complete measure metric space. We present some properties of the integral, continuing the study’s use of a Radon measure $\mu .$ Finally, using locally finite measures, we extend the AP–Henstock–Kurzweil integral theory to second countable Hausdorff spaces that are locally compact. A Saks–Henstock-type Lemma is proved here. Full article