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

Generative neural networks have expanded from text and image generation to creating realistic 3D graphics, which are critical for immersive virtual environments. Physically Based Rendering (PBR)—crucial for realistic 3D graphics—depends on PBR maps, environment (env) maps for lighting, and camera viewpoints. Current research mainly generates PBR maps separately, often using fixed env maps and camera poses. This limitation reduces visual consistency and immersion in 3D spaces. Addressing this, we propose EnvMat, a diffusion-based model that simultaneously generates PBR and env maps. EnvMat uses two Variational Autoencoders (VAEs) for map reconstruction and a Latent Diffusion UNet. Experimental results show that EnvMat surpasses the existing methods in preserving visual accuracy, as validated through metrics like L-PIPS, MS-SSIM, and CIEDE2000. Full article

In this article, we will introduce a new generalized proximal $\theta$-contraction for multivalued and single-valued mappings named ${(\mathcal{f}-{\theta }_{\kappa })}_{C_P}$-proximal contraction and ${(\mathcal{f}-{\theta }_{\kappa })}_{B_P}$-proximal contraction. Using these newly constructed proximal contractions, we will establish new results for the coincidence best proximity point, best proximity point, and fixed point for multivalued mappings in the context of rectangular metric space. Also, we will reduce these contractions for single-valued mappings, named ${\left({\theta }_{\kappa }\right)}_{C_P}$-proximal contraction and ${\left({\theta }_{\kappa }\right)}_{B_P}$-proximal contraction, to establish results for the coincidence proximity point, best proximity point, and fixed point results. We will give some illustrated examples for our newly generated results with graphical representations. In the last section, we will also find the solution to the equation of motion by using our defined results. Full article

This paper investigates the existence of common attractors for generalized $\theta$-Hutchinson operators within the framework of partial metric spaces. Utilizing a finite iterated function system composed of $\theta$-contractive mappings, we establish theoretical results on common attractors, generalizing numerous existing results in the literature. Additionally, to enhance understanding, we present intuitive and easily comprehensible examples in one-, two-, and three-dimensional Euclidean spaces. These examples are accompanied by graphical representations of attractor images for various iterated function systems. As a practical application, we demonstrate how our findings contribute to solving a functional equation arising in a dynamical system, emphasizing the broader implications of the proposed approach. Full article

A model transformation method based on state refinement and semantic mapping is proposed to address the challenges of high modeling complexity and resource consumption in symbolic validation of industrial software requirements. First, a rule-based semantic mapping system is constructed through the explicit definition of element correspondence between statechart components and verification models, coupled with a composite state-level refinement strategy to structurally optimize model hierarchy. Second, an automated transformation algorithm is developed to bridge graphical modeling tools with formal verification environments, supported by quantitative evaluation metrics for mapping validity. To demonstrate its practical applicability, the methodology is systematically applied to railway infrastructure safety—specifically the railroad turnout control system—as a critical case study. The experimental implementation converts operational statecharts of turnout control logic into optimized NuSMV models. Not only did the models remain intact, but the state space was also effectively reduced through the optimization of the hierarchical structure. In the validation phase, the converted model is tested for robustness using the fault injection method, and boundary condition anomalies that are not explicitly stated in the requirement specification are successfully detected. The experimental results show that the validation model generated by this method has improved validation efficiency in the NuSMV tool, which is significantly better than the traditional conversion method. Full article

In this manuscript, we present several novel results in fixed-point theory for a complete controlled metric space. The first presented result is inspired from the Caristi contraction where we explore the existence and uniqueness of fixed points under specific conditions. Furthermore, we propose a graphical form of it by endowing the considered space with a graph and develop a new fixed-point theorem, which is illustrated by two examples. Also, we establish a theorem for the $\alpha$-admissible mapping. To demonstrate its effectiveness, the last theorem proposes an approach to solve a second-order differential equation. Full article

We introduce fuzzy enriched contraction, which extends the classical notion of fuzzy Banach contraction and encompasses specific fuzzy non-expansive mappings. Our investigation establishes both the presence and uniqueness of fixed points considering this broad category of operators using a Krasnoselskij iterative scheme for their approximation. We also show the graphical representation of fuzzy enriched contraction and analyze its graph for different values of beta. The implications of these findings extend to significant results within fuzzy fixed-point theory, enriching the understanding of iterative processes in fuzzy metric spaces. To demonstrate the versatility of our innovative concepts and the associated fixed-point theorems, we provide illustrative examples that showcase their applicability across diverse domains, including the generation of fractals. This demonstrates the relevance of fuzzy enriched contraction to iterated function systems, enabling the study of fractal structures under various contractive conditions. Additionally, we explore practical applications of fuzzy enriched contraction in dynamic market equilibrium, offering new insights into stability and convergence in economic models. Through this unified framework, we open new avenues for both theoretical advancements and real world applications in fuzzy systems. Full article

In this manuscript, we present a novel concept termed graphical ${\Theta }_c$-Kannan contraction within the context of graphically controlled metric-type spaces. Unlike traditional Kannan contraction, this novel concept presents a modified method of contraction mapping. We discuss the significance and the existence of fixed point results within the framework of this novel contraction. To strengthen the credibility of our theoretical remarks, we provide a comparison example demonstrating the efficiency of our suggested framework. Our study not only broadens the theoretical foundations inside graphically controlled metric-type spaces by introducing and examining visual ${\Theta }_c$-Kannan contraction, but it also demonstrates the practical significance of our innovations through significant examples. Furthermore, applying our findings to second-order differential equations by constructing integral equations into the domain of Fredholm sheds light on the broader implications of our research in the field of mathematical analysis and contributes to the advancement of this field. Full article

In this study, trapezoidal-type inequalities in fuzzy settings have been investigated. The theory of fuzzy analysis has been discussed in detail. The integration by parts formula of analysis of fuzzy mathematics has been employed to establish an equality. Trapezoidal-type inequality for functions with values in the fuzzy number-valued space is proven by applying the proven equality together with the properties of a metric defined on the set of fuzzy number-valued space and Höler’s inequality. The results proved in this research provide generalizations of the results from earlier existing results in the field of mathematical inequalities. An example is designed by defining a function that has values in fuzzy number-valued space and validated the results numerically using the software Mathematica (latest v. 14.1). The p-levels of the defined fuzzy number-valued mapping have been shown graphically for different values of $p\in \left[0,1\right]$. Full article

The aim of this paper is to introduce to a pair of fuzzy graphic rational F-contraction multivalued mappings and to study the necessary condition for the existence of common fixed points of fuzzy multivalued mappings in the setup of generalized parametric metric space endowed with a directed graph. A non-trivial example is presented to support the results presented herein. Our results improve and extend some recent results in the existing literature. Full article

This paper explores fixed points for both contractive and non-contractive mappings in traditional b-metric spaces, preordered b-metric spaces, and random b-metric spaces. Our findings provide insights into the behavior of mappings under various constraints and extend our approach to include coincidence and common fixed-point theorems in these spaces. We present new examples and graphical representations for the first time, offering novel results and enhancing several related findings in the literature, while broadening the scope of earlier works of Ran and Reurings, Nieto and Rodríguez-López, Górnicki, and others. Full article

Patients are complex and heterogeneous; clinical data sets are complicated by noise, missing data, and the presence of mixed-type data. Using such data sets requires understanding the high-dimensional “space of patients”, composed of all measurements that define all relevant phenotypes. The current state-of-the-art merely defines spatial groupings of patients using cluster analyses. Our goal is to apply topological data analysis (TDA), a new unsupervised technique, to obtain a more complete understanding of patient space. We applied TDA to a space of 266 previously untreated patients with Chronic Lymphocytic Leukemia (CLL), using the “daisy” metric to compute distances between clinical records. We found clear evidence for both loops and voids in the CLL data. To interpret these structures, we developed novel computational and graphical methods. The most persistent loop and the most persistent void can be explained using three dichotomized, prognostically important factors in CLL: IGHV somatic mutation status, beta-2 microglobulin, and Rai stage. In conclusion, patient space turns out to be richer and more complex than current models suggest. TDA could become a powerful tool in a researcher’s arsenal for interpreting high-dimensional data by providing novel insights into biological processes and improving our understanding of clinical and biological data sets. Full article

In this study, we significantly extend the concept of modular metric-like spaces to introduce the notion of b-metric-like spaces. Furthermore, by incorporating a binary relation $\mathfrak{R}$, we develop the framework of $\mathfrak{R}$-modular b-metric-like spaces. We establish a groundbreaking fixed point theorem for certain extensions of Geraghty-type contraction mappings, incorporating both $𝒵$ simulation function and $E$-type contraction within this innovative structure. Moreover, we present several novel outcomes that stem from our newly defined notations. Afterwards, we introduce an unprecedented concept, the graphical modular b-metric-like space, which is derived from the binary relation $\mathfrak{R}$. Finally, we examine the existence of solutions for a class of functional equations that are pivotal in dynamic programming and in solving initial value problems related to the electric current in an RLC parallel circuit. Full article

This study aims to establish common fixed point theorems for a pair of compatible self-mappings within the framework of extended parametric $S_b$-metric spaces. To support our assertions, we provide corollaries and examples accompanied with graphical representations. Moreover, we leverage our principal outcome to guarantee the existence of a common solution to a system of integral equations. Full article

With the rapid advancement of computer graphics and three-dimensional modeling technology, the processing and optimization of three-dimensional (3D) models have become contentious research topics. In the context of mobile devices or web applications, situations may arise where it becomes necessary to load a 3D model with a substantial memory footprint in real-time or dynamically adjust the level of detail of a model based on the scene’s proximity. In such cases, it is imperative to optimize the original model to ensure smoothness and responsiveness. Due to the simplicity of their algorithm, quadric error metrics (QEMs) can deliver excellent results in simplifying 3D models while maintaining high efficiency. Therefore, QEM is widely employed in engineering applications within the realm of computer graphics development. Moreover, in the pursuit of enhanced quality and efficiency, numerous scholars have improved it based on QEM algorithms. This study aims to provide a systematic review and summary of the principles and applications of current research on QEM algorithms. First, we conducted a bibliometric analysis of 128 studies in related fields spanning from 1998 to 2022 using CiteSpace. This allowed us to sort QEM algorithms and gain insights into their development status and emerging trends. Second, we delve into the fundamental principles and optimizations of the QEM algorithms to provide a deeper understanding of their implementation process. Following that, we explore the advantages and limitations of the QEM algorithms in practical applications and analyze their potential in various domains, including virtual reality and game development. Finally, this study outlines future research directions, which encompass the development of more efficient error metric calculation methods, the exploration of adaptive simplification strategies, and the investigation of potential synergies with deep learning technologies. Current research primarily centers on enhancing QEM algorithms by incorporating additional geometric constraints to better differentiate between flat and irregular areas. This enables a more accurate determination of the areas that should be prioritized for folding. Nevertheless, it is important to note that these improvements may come at the cost of reduced computational efficiency. Therefore, future research directions could involve exploring parallel computing techniques and utilizing GPUs to enhance computational efficiency. Full article