Compactness In Spaces Research Articles

On Covering Properties in Intuitionistic Fuzzy Topological Spaces: A survey

We show here some results on covering properties in intuitionistic fuzzy topological spaces. In 1983, K.T. Atanassov proposed a generalization of the notion of fuzzy set: the concept of intuitionistic fuzzy set. D. Çoker constructed the fundamental theory on intuitionistic fuzzy topological spaces, and D. Çoker and other mathematicians studied compactness, connectedness, continuity, separation, convergence and paracompactness in intuitionistic fuzzy topological spaces. Finally, G.-J Wang and Y.Y. He showed that every intuitionistic fuzzy set may be regarded as an L-fuzzy set for some appropriate lattice L. Nevertheless, the results obtained by above authors are not redundant with other for ordinary fuzzy sense.

A-priori and a-posteriori error estimates for discontinuous Galerkin method of the Maxwell eigenvalue problem

This paper is devoted to a-priori and a-posteriori error analysis of discontinuous Galerkin (DG) method for the Maxwell eigenvalue problem. The discrete compactness of DG space is proved so that the Babuška and Osborn spectral approximation theory can be applicable in the a-priori error analysis. Then we prove the optimal error estimates for DG eigenfunctions in mesh-dependent norm and DG eigenvalues. A special contribution of this work is to prove that the error in L2-norm for smooth eigenfunctions is of higher order than that in mesh-dependent norm, so that the DG eigenvalues can approximate the true eigenvalues from upper. Another contribution of this work is to provide a-posteriori error analysis for the DG method. A reliable a-posteriori error estimator is analyzed. The upper bound property of DG eigenvalues and the robustness of adaptive methods are verified through numerical experiments.

Dynamic Calculation Approach of the Collision Risk in Complex Navigable Water

It is vital to analyze ship collision risk for preventing collisions and improving safety at sea. This paper takes Ningbo-Zhoushan Port, a typical complex navigable water, as the research object. Firstly, a probabilistic conflict detection method based on an AIS data-driven dynamic ship domain model is proposed to achieve effective ship conflict detection under uncertain environments. Then, a ship group identification method is proposed, which can extract the ship groups with conflict correlation and space compactness. Finally, according to the characteristics of ship traffic in complex navigable waters, the dynamic calculation of ship collision risk is carried out from individual, regional, and local multi-scale perspectives. The experimental results show that the proposed method can detect the collision risk in a timely, reliable, and effective manner under complex dynamic conditions. As such, they provide valuable insights into ship collision risk prediction and the development of risk mitigation measures.

Novel Results on Nigh Lindelöfness in Topological Spaces

The main objective of this research paper is to introduce the concept of nigh Lindelöfness in topological spaces and nigh topological spaces. This has led us to establish several generalizations and properties of nigh Lindelöf space that are related to the nearly nigh Lindelöf space, the nigh compactness in topological spaces, and their relations with other spaces. Several examples are discussed, and many well-known theorems are generalized concerning the nigh Lindelöf spaces.

Nigh-Locally Compactness in Topological Spaces

The focus of this paper is the introduction of the notion of a nigh-locally compact topological space. To do this, concepts of "nigh-topological space" and "nigh-compact space" would be defined, and various conclusions and theorems would be derived. This would lead to a well-defined notion of a nigh-locally compact topological space, from which we would obtain a number of theorems and instances concerning this innovative idea.

On Some Aspects of Compactness in Metric Spaces

In this paper, we investigate the generalizations of the concepts from Heine-Borel Theorem and the Bolzano-Weierstrass Theorem to metric spaces. We show that the metric space X is compact if every open covering has a finite subcovering. This abstracts the Heine-Borel property. Indeed, the Heine-Borel Theorem states that closed bounded subsets of the real line R are compact. In this study, we rephrase compactness in terms of closed bounded subsets of the real line R, that is, the Bolzano-Weierstrass theorem. Let X be any closed bounded subset of the real line. Then any sequence (xn) of the points of X has a subsequence converging to a point of X. We have used these interesting theorems to characterize compactness in metric spaces.

Advancements in the Analysis of Sobolev Spaces and Function Spaces on Manifolds: Theoretical Framework and Applications

This research paper delves into Sobolev spaces and function spaces on smooth manifolds, revealing fundamental theorems such as existence, embeddings, and compactness properties. Noteworthy results include the Poincare inequality elucidating function behavior on compact manifolds and compactness properties of Sobolev spaces on Riemannian manifolds. The study establishes trace theorems for functions on the boundary and interpolation results between Sobolev spaces. Isoperimetric inequalities and stability under weak convergence contribute to a holistic understanding of geometric and analytical aspects of Sobolev spaces. The research concludes by exploring invariance under diffeomorphisms and compactness in dual spaces, providing a unified framework for analyzing function spaces on manifolds.

Single-Valued Neutrosophic Roughness via Ideals

In this paper, we connect the idea of single-valued neutrosophic ideal to the concept of single-valued neutrosophic approximation space to define the concept of single-valued neutrosophic ideal approximation spaces. We present the single-valued neutrosophic ideal approximation interior operator intψΦ and the single-valued neutrosophic ideal approximation closure operator clψΦ, and we present the single-valued neutrosophic ideal approximation preinterior operator pintψΦ and the single-valued neutrosophic ideal approximation pre-closure operator pclψΦ about this concerning single-valued neutrosophic ideal defined on the single-valued neutrosophic approximation space (χ˜,ϕ) related with some single-valued neutrosophic set ψ∈ξχ˜. Also, we present single-valued neutrosophic separation axioms, single-valued neutrosophic connectedness, and single-valued neutrosophic compactness in single-valued neutrosophic approximation spaces and single-valued neutrosophic ideal approximation spaces as well, and prove the associations in between.

Compact subsets of C λ,u (X)

Abstract The famous Ascoli-Arzelà theorem served as a springboard for research into compactness in function spaces, particularly spaces of continuous functions. This paper investigates compact subsets of spaces of continuous functions endowed with topologies between the topology of pointwise convergence and the topology of uniform convergence. More precisely, this paper studies necessary and sufficient conditions for a subset to be compact in C λ,u (X) for a locally-λ space X when λ ⊇ 𝓕(X), for a hemi- λ λf -space X when λ ⊆ 𝓟 𝓢(X), and for a k-space X when λ ⊇ 𝓚(X). This paper also studies that every bounded subset of C λ,u (X) has compact closure for some classes of topological spaces X.

On Generalized Compactness in Fuzzy Bitopological Spaces

The main objective of this research is to study some types of generalized closed sets in fuzzy bitopology including $(i,j)-g\alpha-cld$, $(i,j)-gs-cld$, $(i,j)-gp-cld$, and $(i,j)-g\beta-cld$. We then present basic theorems for determining their relationships and explain their properties, such as closure and interior. In addition, there are many interesting counterexamples. The last part of the research focuses on generalized compactness in fuzzy bitopological spaces and their types and explores the relationships between these concepts, their important theories, and some relevant counterexamples. The results established in this paper are new in the domain of fuzzy bitopology.

Prototypical Bidirectional Adaptation and Learning for Cross-Domain Semantic Segmentation

Cross-domain semantic segmentation, which aims to address the distribution shift while adapting from a labeled source domain to an unlabeled target domain, has achieved great progress in recent years. However, most existing work adopts a source-to-target adaptation path, which often suffers from clear class mismatching or class imbalance issues. We design PBAL, a prototypical bidirectional adaptation and learning technique that introduces bidirectional prototype learning and prototypical self-training for optimal inter-domain alignment and adaptation. We perform bidirectional alignments in a complementary and cooperative manner which balances both dominant and tail categories as well as easy and hard samples effectively. In addition, We derive prototypes efficiently from a source-trained classifier, which enables class-aware adaptation as well as synchronous prototype updating and network optimization. Further, we re-examine self-training and introduce prototypical contrast above it which greatly improves inter-domain alignment by promoting better intra-class compactness and inter-class separability in the feature space. Extensive experiments over two widely studied benchmarks show that the proposed PBAL achieves superior domain adaptation performance as compared with the state-of-the-art.

Energy estimates of Yang–Mills functional

In this paper, we use the Uhlenbeck compactness theorem and Lojasiewicz–Simon gradient inequality of the Yang–Mills [Formula: see text]-energy functional to prove some energy properties of Yang–Mills connections. In particular, we can get the discreteness of Yang–Mills energy from the compactness of the moduli space of Yang–Mills connections.

Adversarially robust neural networks with feature uncertainty learning and label embedding

Deep neural networks (DNNs) are vulnerable to the attacks of adversarial examples, which bring serious security risks to the learning systems. In this paper, we propose a new defense method to improve the adversarial robustness of DNNs based on stochastic neural networks (SNNs), termed as Margin-SNN. The proposed Margin-SNN mainly includes two modules, i.e., feature uncertainty learning module and label embedding module. The first module introduces uncertainty to the latent feature space by giving each sample a distributional representation rather than a fixed point representation, and leverages the advantages of variational information bottleneck method in achieving good intra-class compactness in latent space. The second module develops a label embedding mechanism to take advantage of the semantic information underlying the labels, which maps the labels into the same latent space with the features, in order to capture the similarity between sample and its class centroid, where a penalty term is equipped to elegantly enlarge the margin between different classes for better inter-class separability. Since no adversarial information is introduced, the proposed model can be learned in standard training to improve adversarial robustness, which is much more efficient than adversarial training. Extensive experiments on data sets MNIST, FASHION MNIST, CIFAR10, CIFAR100 and SVHN demonstrate superior defensive ability of the proposed method. Our code is available at https://github.com/humeng24/Margin-SNN.

Threshold dynamics of stochastic H7N9 model with Markov switching and hybrid strategy

This article aims to study the impact of different environmental states on the transmission dynamics of avian influenza A (H7N9) for poultry by using an epidemic model with general incidence rate and Markov switching. The model couples the essential roles of quarantine and hybrid strategy of vaccination and elimination. By establishing the basic reproduction number R0 as a sharp threshold for the spread of H7N9. When R0<1, the disease almost surely dies out exponentially. We use the compactness of space to prove that when R0>1, the disease almost surely persistent, and this method can be applied to other infectious disease models considering vaccination. In addition, under a mild extra condition, the ergodicity of the Markov process is proved. Our research indicates that: (i) theoretical results and sensitivity analysis show that implementing hybrid strategy is an effective method to control H7N9 propagation; (ii) under the influence of environmental switching, numerical simulations find that the transmission state of H7N9 will neutralize multiple regimes, leading to the possibility of multi wave transmission. This can be used to explain the multiple outbreaks of H7N9 in China.

Time-frequency analysis on flat tori and Gabor frames in finite dimensions

We provide the foundations of a Hilbert space theory for the short-time Fourier transform (STFT) where the flat tori TN2=R2/(Z×NZ)=[0,1]×[0,N] act as phase spaces. We work on an N-dimensional subspace SN of distributions periodic in time and frequency in the dual S0′(R) of the Feichtinger algebra S0(R) and equip it with an inner product. To construct the Hilbert space SN we apply a suitable double periodization operator to S0(R). On SN, the STFT is applied as the usual STFT defined on S0′(R). This STFT is a continuous extension of the finite discrete Gabor transform from the lattice onto the entire flat torus. As such, sampling theorems on flat tori lead to Gabor frames in finite dimensions. For Gaussian windows, one is lead to spaces of analytic functions and the construction allows to prove a necessary and sufficient Nyquist rate type result, which is the analogue, for Gabor frames in finite dimensions, of a well known result of Lyubarskii and Seip-Wallstén for Gabor frames with Gaussian windows and which, for N odd, produces an explicit full spark Gabor frame. The compactness of the phase space, the finite dimension of the signal spaces and our sampling theorem offer practical advantages in some applications. We illustrate this by discussing a problem of current research interest: recovering signals from the zeros of their noisy spectrograms.

Spectral clustering and embedding with inter-class topology-preserving

Spectral clustering and embedding are critical in data mining and rely heavily on pre-computed graph structure quality. Some methods simultaneously learn graph structures and low-dimensional embeddings to achieve high-quality graph structures with separable categories. However, existing approaches prioritize inter-class separability while overlooking the inter-class topological structure. Consequently, the learned low-dimensional manifold representations may fail to capture inter-class semantic information, causing the learned graph structures to represent categories partitioning inaccurately. To address these issues, we come up with the inter-class topology and utilize category anchors to indicate class locations in low-dimensional space. The inter-class topology reflects the elaborate structural neighbor information of class, which makes the neighbor classes rationally separable and helps to learn better low-dimensional embedding representations. The spectral embedded clustering method and some variants are proposed to preserve inter-class topology while making categories separable. Furthermore, we ensure intra-class sample compactness in low-dimensional space by assigning sample embeddings near the anchors. We analyze the importance of inter-class topology preserving and intra-class compactness learning. Extensive experiments on publicly available datasets demonstrate that our method is competitive with popular graph clustering and embedding baselines and effectively captures the inter-class topology.

GroomGen: A High-Quality Generative Hair Model Using Hierarchical Latent Representations

Despite recent successes in hair acquisition that fits a high-dimensional hair model to a specific input subject, generative hair models, which establish general embedding spaces for encoding, editing, and sampling diverse hairstyles, are way less explored. In this paper, we present GroomGen , the first generative model designed for hair geometry composed of highly-detailed dense strands. Our approach is motivated by two key ideas. First, we construct hair latent spaces covering both individual strands and hairstyles. The latent spaces are compact, expressive, and well-constrained for high-quality and diverse sampling. Second, we adopt a hierarchical hair representation that parameterizes a complete hair model to three levels: single strands, sparse guide hairs, and complete dense hairs. This representation is critical to the compactness of latent spaces, the robustness of training, and the efficiency of inference. Based on this hierarchical latent representation, our proposed pipeline consists of a strand-VAE and a hairstyle-VAE that encode an individual strand and a set of guide hairs to their respective latent spaces, and a hybrid densification step that populates sparse guide hairs to a dense hair model. GroomGen not only enables novel hairstyle sampling and plausible hairstyle interpolation, but also supports interactive editing of complex hairstyles, or can serve as strong data-driven prior for hairstyle reconstruction from images. We demonstrate the superiority of our approach with qualitative examples of diverse sampled hairstyles and quantitative evaluation of generation quality regarding every single component and the entire pipeline.

On a Type of Generalized GS_β-Closed Set in Grill Topology

This work intends to propose the concept of generalized -closed set-in grill topological space. Further, some of the properties of such generalized -closed set are evaluated and provided with suitable illustrations. Many theorems based on generalized -closed set have also been proved in this work. Furthermore, the ideas of -regular space and -normal space are also studied. Also, the relationship between -regular and - normal spaces is discussed. The properties of -regular space and -normal space concepts have also been discussed.Introduction: Alo, R. A., and Shapiro, H. L defined the semi- normal spaces and considered some of their important inheritance. The authors further studied the compactness in grill topological spaces and studied some of their properties. Data granulation serves as a vital decision-making tool in numerous real-life applications. Mashhour, A. S and Abd, M. E introduced certain new topological tools employing rough set approximations for the purpose of data granulation. The introduction of -normality helps to separate closed subsets, continuous functions. It also extends the support to define the concepts of complete regularity and Tietze extension theorem. The -regular space studied in this article is regular, that is completely regular, which is a stronger condition.Objectives: I.Arockiarani and A. Karthika introduced a new concept such as generalized -closed set in grill topological space. The above paper motivates me to do the research in generalized -closed set. In this paper a new concept of “generalized -closed set in grill topological space” are introduced. In this article we are using stronger condition for proving some theorems.Methods: The present task ambitioned at formation of the extension of topological system applying the idea of grill. The importance of the theory of grill between the topological spaces have been elaborated. A set is countable collection based on elements, without integrity of form. At the same time some kinds of algebraic actions are applied on this set, indefinitely the elements of this set are correlated into a whole, so ultimately it grows into a space. So ultimately in the whole paper we are applying the concept of grill topological space.Results: In this article a new concept of generalized -closed set in a grill topological space are introduced. Using this definition we are proving many results based on operators on grill topological space. Furthermore, the implication such as “all closed set are generalized -closed sets” is proved and the converse may not be true are also proved with proper illustration. In addition the relationship between -normal space, -normal space, -regular space and -regular space are also studied. Further many equivalent conditions are also proved using -normal space and -regular space.Conclusions: The work proposes and investigates the concept of generalized -closed sets within the framework of grill topological spaces. It explores the properties of these sets, provides illustrative examples, and proves theorems related to their behavior. Additionally, the study delves into the concepts of -regular space and -normal space, discussing their properties and implications. The research contributes to the understanding of these concepts and their applications in the context of grill topological spaces.

Neighborhood contrastive representation learning for attributed graph clustering

Attributed graph clustering is a fundamental task in graph learning field. Because of the high-dimensional node features and the complex non-Euclidean graph structure, it is challenging for attributed graph clustering methods to exploit graph information. Recent studies on graph contrastive learning (GCL) have achieved promising results. However, existing GCL-based methods neither consider a clustering-friendly node representation nor a clustering-oriented loss function, resulting in inferior performance. To this end, we propose NCAGC, a neighborhood contrastive representation learning method for attributed graph clustering task. Specifically, NCAGC constrains the representation learning of similar nodes by a neighborhood contrast module to ensure the compactness in the latent space, thus facilitating the clustering task. Meanwhile, a contrastive self-expression module is present for learning a discriminative self-expression coefficient matrix, which is crucial for the subsequent subspace clustering. Moreover, the two designed modules are trained and optimized jointly, which benefits the node representation learning and clustering to achieve mutual refinement. Extensive experimental results on four attributed graph datasets demonstrate the superiority of NCAGC compared with 16 state-of-the-art methods, which surpasses the sub-optimal method on Cora dataset by 2.1%, 4.3%, and 3.7% in terms of ACC, NMI, and ARI, respectively. Our code and dataset is available at https://github.com/wangtong627/NCAGC-NeuroCom.

ON COMPACTNESS IN BI-GENERALIZED TOPOLOGICAL SPACES

In this paper, we define compactness for all open sets defined in bigeneralized topological spaces such as: μ(m,n)-semi compactness, μ(m,n)-pre compactness, μ(m,n)-regular compactness, μ(m,n)-α-compactness, μ(m,n)-β-compactness, μ(m,n)-compactness and (m, n)-compactness. For our investigation, we choose μ(m,n)- semi compactness as a base space and studies the relationships between the μ(m,n)- semi compactness and other compactness in bi-generalized topological spaces.