Topological Modules Research Articles

Graded pseudo weakly prime spectrum of graded topological modules

In this study, we introduce graded pseudo weakly prime submodules of G-graded R-modules, which are an extension of graded weakly prime ideals over G-graded rings. On the graded spectrum of graded pseudo weakly prime submodules, we investigate the Zariski topology. Different aspects of this topological space are investigated, and they are linked to the algebraic properties of the G-graded R-modules under study.

Network Subgraph-based Method: Alignment-free Technique for Molecular Network Analysis

Background: Comparing directed networks using the alignment-free technique offers the advantage of detecting topologically similar regions that are independent of the network size or node identity. Objective: We propose a novel method to compare directed networks by decomposing the network into small modules, the so-called network subgraph approach, which is distinct from the network motif approach because it does not depend on null model assumptions. Methods: We developed an alignment-free algorithm called the Subgraph Identification Algorithm (SIA), which could generate all subgraphs that have five connected nodes (5-node subgraph). There were 9,364 such modules. Then, we applied the SIA method to examine 17 cancer networks and measured the similarity between the two networks by gauging the similarity level using Jensen- Shannon entropy (HJS). Results: We identified and examined the biological meaning of 5-node regulatory modules and pairs of cancer networks with the smallest HJS values. The two pairs of networks that show similar patterns are (i) endometrial cancer and hepatocellular carcinoma and (ii) breast cancer and pathways in cancer. Some studies have provided experimental data supporting the 5-node regulatory modules. Conclusion: Our method is an alignment-free approach that measures the topological similarity of 5-node regulatory modules and aligns two directed networks based on their topology. These modules capture complex interactions among multiple genes that cannot be detected using existing methods that only consider single-gene relations. We analyzed the biological relevance of the regulatory modules and used the subgraph method to identify the modules that shared the same topology across 2 cancer networks out of 17 cancer networks. We validated our findings using evidence from the literature.

Atom-ProteinQA: Atom-level protein model quality assessment through fine-grained joint learning

MotivationProtein model quality assessment (ProteinQA) is a fundamental task that is essential for biologically relevant applications, i.e., protein structure refinement, protein design, etc. Previous works aimed to conduct ProteinQA only on the global structure or per-residue level, ignoring potentially usable and precise cues from a fine-grained per-atom perspective. In this study, we propose an atom-level ProteinQA model, named Atom-ProteinQA, in which two innovative modules are designed to extract geometric and topological atom-level relationships respectively. Specifically, on the one hand, a geometric perception module exploits 3D sparse convolution to capture the geometric features of the input protein, generating fine-grained atom-level predictions. On the other hand, natural chemical bonds are utilized to construct an atom-level graph, then message passing from a topological perception module is applied to output residue-level predictions in parallel. Eventually, through a cross-model aggregation module, features from different modules mutually interact, enhancing performance on both the atom and residue levels. ResultsExtensive experiments show that our proposed Atom-ProteinQA outperforms previous methods by a large margin, regardless of residue-level or atom-level assessment. Concretely, we achieved state-of-the-art performance on CATH-2084, Decoy-8000, public benchmarks CASP13 & CASP14, and the CAMEO. AvailabilityThe repository of this project is released on: https://github.com/luyfcandy/Atom_ProteinQA.

TOPOLOGICAL MODULES GEOMETRY

The extremal structure of zero-neighbourhoods of a topological module is analyzed reaching unexpected conclusions when the module topology is not Hausdorff. These results motivate us to introduce the notion of metric modules, which are modules endowed with a translation-invariant metric, turning them into an (additive) topological group. We study the central and diametral points of additively symmetric subsets and find examples of convex sets which are not symmetric translates (translates of addivitely symmetric subsets). As a consequence of all of these, it seems natural to transport the well-known Bishop-Phelps property from the category of real topological vector spaces to general topological modules over topological rings. Then we stick to particular topological rings, the unital C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras, showing that the subset of positive elements lying below the unity is an effect algebra. We also prove that every continuous linear operator on a Hausdorff locally convex topological vector space that commutes with all continuous linear projections of one-dimensional range is a multiple of the identity. Finally, we discuss how to transport the previous result to C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebras.

Pareto vectors of continuous linear operators

The intersection of all zero-neighborhoods in a topological module over a topological ring is a bounded and closed submodule whose inherited topology is the trivial topology. In this manuscript, we prove that this is the smallest closed submodule and thus replaces the null submodule in the Hausdorff setting. This fact motivates to introduce a new notion in operator theory called topological kernel. Another new concept is also defined that of Pareto optimal element for a family of continuous linear operators between topological modules. It is then proved that topological kernels have a strong influence on the existence of Pareto optimal elements. This work is strongly motivated by the ongoing search for a consistent operator theory on topological modules over general topological rings.

Network Biomarker Detection From Gene Co-Expression Network Using Gaussian Mixture Model Clustering.

Finding network biomarkers from gene co-expression networks (GCNs) has attracted a lot of research interest. A network biomarker is a topological module, i.e., a group of densely connected nodes in a GCN, in which the gene expression values correlate with sample labels. Compared with biomarkers based on single genes, network biomarkers are not only more robust in separating samples from different categories, but are also able to better interpret the molecular mechanism of the disease. The previous network biomarker detection methods either employ distance based clustering methods or search for cliques in a GCN to detect topological modules. The first strategy assumes that the topological modules should be spherical in shape, and the second strategy requires all nodes to be fully connected. However, the relations between genes are complex, as a result, genes in the same biological process may not be directly, strongly connected. Therefore, the shapes of those modules could be oval or long strips. Hence, the shapes of gene functional modules and gene disease modules may not meet the aforementioned constraints in the previous methods. Thus, previous methods may break up the genes belonging to the same biological process into different topological modules due to those constraints. To address this issue, we propose a novel network biomarker detection method by using Gaussian mixture model clustering which allows more flexibility in the shapes of the topological modules. We have evaluated the performance of our method on a set of eight TCGA cancer datasets. The results show that our method can detect network modules that possess better discriminate power, and provide biological insights.

FUZZY TOPOLOGICAL MODULES INDUCED BY FUZZY PSEUDO NORMED MODULE

In this article, we give defines about fuzzy pseudo norm module space and fuzzy metric module space which induces by this space. Complete metric space was proved. Also we are defining fuzzy topological module space which induces by fuzzy metric module space.

FUZZY TOPOLOGICAL MODULES INDUCED BY FUZZY PSEUDO NORMED MODULE

In this article, we give defines about fuzzy pseudo norm module space and fuzzy metric module space which induces by this space. Complete metric space was proved. Also we are defining fuzzy topological module space which induces by fuzzy metric module space.

On M∗-Irresolute Topological Rings

The main aim of this paper is to introduce and study the new notions namely M∗-irresolute topological rings and M∗-irresolute topological R-modules by virtue of M∗-open sets. Examples of an M∗-irresolute topological ring and module have been put forth. Further, we provide several fundamental properties and characterizations of M∗-irresolute topological rings and M∗-irresolute topological R-modules. In addition, we shall define boundedness in these two structures and present several results on them.

Tensorized topological graph learning for generalized incomplete multi-view clustering

The success of the current multi-view clustering lies in the default assumption of completeness on each view, while it is hardly satisfied for real-world applications. Incomplete multi-view clustering (IMC) studies clustering multi-view instances that are partially observed due to data corruption or sensor failure. Although making some progress, the current research suffers from the following obstacles: (1) Existing works are always built on pairwise similarity measurement, which cannot fully capture the topological structure of incomplete multi-view samples for precise clustering; (2) Due to the absence of partially-observed samples across multiple views, how could we recover and enhance the high-order similarities across different views becomes one of the main bottlenecks of IMC; (3) Current research mainly aims to handle specific dual-view IMC, which greatly limits the deployment in real-world applications. In this paper, we propose a novel Tensorized Topological Graph Learning (TTGL) for the generalized incomplete multi-view clustering, which jointly considers the topological graph construction, missing feature completion, and high-order uncertain correlation enhancement. Specifically, instead of using pairwise similarity measurement, we formulate a consensus topological graph construction module, which can coalesce affinity matrices to analyze the topological structure of incomplete multi-view data. Moreover, we build an iterative missing similarity imputation scheme for feature completion, which ensures the intrinsic similarity preservation on observed instances as well as an imputation on unobserved ones. Furthermore, a low-rank tensor structure is imposed to capture the high-order similarities via both feature and similarity enhancement and completion across different views. Our method can cluster incomplete data with an arbitrary number of views and any missing statuses. Extensive experiments on several benchmark datasets validate the effectiveness of our method when compared with a series of state-of-the-art IMC clustering algorithms. The source code of our TTGL is available at: https://github.com/DarrenZZhang/TTGL.

Fixed Point Theorems in Cone Metric Spaces via c-Distance Over Topological Module

In 2011, Wang and Guo introduced c-distance in cone metric spaces. The idea of cone metric spaces over topological modules was presented by Branga and Olaru in 2020. Combining these two ideas, we introduce cone metric spaces with c-distance over topological module and establish a fixed point theorem.

On curves in K-theory and TR

We prove that TR is corepresentable by the reduced topological Hochschild homology of the flat affine line \mathbf{S}[t] as a functor defined on the \infty -category of cyclotomic spectra with values in the \infty -category of spectra with Frobenius lifts, refining a result of Blumberg–Mandell. We define the notion of an integral topological Cartier module using Barwick’s formalism of spectral Mackey functors on orbital \infty -categories, extending the work of Antieau–Nikolaus in the p -typical setting. As an application, we show that TR evaluated on a connective \mathbf{E}_1 -ring admits a description in terms of the spectrum of curves on algebraic K-theory, generalizing the work of Hesselholt and Betley–Schlichtkrull.

On topological M-injective modules

Let R be a topological ring and M be a topological R-module. A topological R-module U is called a topological M-injective if for every continuous monomorphism f:K→M, where K is an open submodule of M, and for every continuous homomorphism g:K→U, there exists a continuous homomorphism h:M→U such that hf=g. A topological M-injective module is a generalization of a topological injective module defined by Goldman and Sah [14]. An infinite direct sum of injective modules is not necessary injective. In this paper, the properties of topological M-injective modules are investigated. We prove that an infinite direct sum of topological M-injective modules is also topological M-injective if the direct sum is an open submodule of the direct product of topological M-injective modules.

The Stereographic Projection in Topological Modules

The stereographic projection is constructed in topological modules. Let A be an additively symmetric closed subset of a topological R-module M such that 0∈int(A). If there exists a continuous functional m*:M→R in the dual module M*, an invertible s∈U(R) and an element a in the topological boundary bd(A) of A in such a way that m*−1({s})∩int(A)=⌀, a∈m*−1({s})∩bd(A), and s+m*bd(A)\{−a}⊆U(R), then the following function b↦−a+2s(m*(b)+s)−1(b+a), from bd(A)\{−a} to (m*)−1({s}), is a well-defined stereographic projection (also continuous if multiplicative inversion is continuous on R). Finally, we provide sufficient conditions for the previous stereographic projection to become a homeomorphism.

Trajectory Forecasting Using Graph Convolutional Neural Networks Based on Prior Awareness and Information Fusion

Predicting the future trajectories of multiple agents is essential for various applications in real life, such as surveillance systems, autonomous driving, and social robots. The trajectory prediction task is influenced by many factors, including individual historical trajectory, interactions between agents, and the fuzzy nature of an agent’s motion. While existing methods have made great progress on the topic of trajectory prediction, a lot of trajectory prediction methods take into account all pedestrians in the scene when simply modeling the influence of nearby pedestrians, and this inevitably brings redundant information. We propose a pedestrian trajectory prediction model based on prior awareness and information fusion. To make the input information more effective, for the different levels of importance of input trajectory information, we design a time information weighting module to weigh the observed trajectory information differently at different moments based on the original observed trajectory information. To reduce the impact of redundant information on trajectory prediction and to improve interaction between pedestrians, we present a spatial interaction module of multi-pedestrians and a topological graph fusion module. In addition, we use a temporal convolutional network module to obtain the temporal interactions between pedestrians. Compared to Social-STGCNN, the experimental results show that the model we propose reduces the average displacement error (ADE) and final displacement error (FDE) by 32% and 38% in the datasets of ETH and UCY, respectively. Moreover, based on this model, we design an autonomous driving obstacle avoidance system that can effectively ensure the safety of road pedestrians.

Transcriptomic congruence analysis for evaluating model organisms

Model organisms are instrumental substitutes for human studies to expedite basic, translational, and clinical research. Despite their indispensable role in mechanistic investigation and drug development, molecular congruence of animal models to humans has long been questioned and debated. Little effort has been made for an objective quantification and mechanistic exploration of a model organism's resemblance to humans in terms of molecular response under disease or drug treatment. We hereby propose a framework, namely Congruence Analysis for Model Organisms (CAMO), for transcriptomic response analysis by developing threshold-free differential expression analysis, quantitative concordance/discordance scores incorporating data variabilities, pathway-centric downstream investigation, knowledge retrieval by text mining, and topological gene module detection for hypothesis generation. Instead of a genome-wide vague and dichotomous answer of "poorly" or "greatly" mimicking humans, CAMO assists researchers to numerically quantify congruence, to dissect true cross-species differences from unwanted biological or cohort variabilities, and to visually identify molecular mechanisms and pathway subnetworks that are best or least mimicked by model organisms, which altogether provides foundations for hypothesis generation and subsequent translational decisions.

Технология создания цифровой карты сельскохозяйственных угодий на территории сельскохозяйственных организаций Ленинградской области с применением QGIS

The article deals with the technology of creating a digital map of agricultural lands in QGIS in order to form an information base on their state in individual farms. As a result of the conducted research, the authors identified the main stages of creating a digital map, defined key tasks at the stage of spatial reference of raster images and proposed ways to solve them. We developed an algorithm for creating an attribute table using the Python programming language, which enables avoiding errors in the process of filling in the semantic part of the generated information base. The main tools for digitizing raster images are highlighted, and in order to identify mistakes in the process of forming its graphical part, a topological editing module is to be used. The obtained research results can be helpful for the executive authorities of the Russian Federation subjects, local self-government bodies to make decisions in the field of land and property management of a separate entity.

Topological structure and global features enhanced graph reasoning model for non-small cell lung cancer segmentation from CT

Objective. Accurate and automated segmentation of lung tumors from computed tomography (CT) images is critical yet challenging. Lung tumors are of various sizes and locations and have indistinct boundaries adjacent to other normal tissues. Approach. We propose a new segmentation model that can integrate the topological structure and global features of image region nodes to address the challenges. Firstly, we construct a weighted graph with image region nodes. The graph topology reflects the complex spatial relationships among these nodes, and each node has its specific attributes. Secondly, we propose a node-wise topological feature learning module based on a new graph convolutional autoencoder (GCA). Meanwhile, a node information supplementation (GNIS) module is established by integrating specific features of each node extracted by a convolutional neural network (CNN) into each encoding layer of GCA. Afterwards, we construct a global feature extraction model based on multi-layer perceptron (MLP) to encode the features learnt from all the image region nodes which are crucial complementary information for tumor segmentation. Main results. Ablation study results over the public lung tumor segmentation dataset demonstrate the contributions of our major technical innovations. Compared with other segmentation methods, our new model improves the segmentation performance and has generalization ability on different 3D image segmentation backbones. Our model achieved Dice of 0.7827, IoU of 0.6981, and HD of 32.1743 mm on the public dataset 2018 Medical Segmentation Decathlon challenge, and Dice of 0.7004, IoU of 0.5704 and HD of 64.4661 mm on lung tumor dataset from Shandong Cancer Hospital. Significance. The novel model improves automated lung tumor segmentation performance especially the challenging and complex cases using topological structure and global features of image region nodes. It is of great potential to apply the model to other CT segmentation tasks.

Chaos in Topological Modules

Chaotic and pathological phenomena in topological modules are studied in this manuscript. In particular, constructions of noncontinuous linear functionals are provided for a wide variety of topological modules. In addition, constructions of balanced and absorbing sets which are not neighborhoods of zero are also given in an extensive class of topological modules. Finally, we construct a linearly open set with empty interior in a large amount of topological modules. All these constructions are related to each other. Prior to developing all these results, we provide an axiomatization of the topological concept of limit by introducing the limit operators in a similar context as hull operators or closure operators are defined.

SUBATOMIC: a SUbgraph BAsed mulTi-OMIcs clustering framework to analyze integrated multi-edge networks

BackgroundRepresenting the complex interplay between different types of biomolecules across different omics layers in multi-omics networks bears great potential to gain a deep mechanistic understanding of gene regulation and disease. However, multi-omics networks easily grow into giant hairball structures that hamper biological interpretation. Module detection methods can decompose these networks into smaller interpretable modules. However, these methods are not adapted to deal with multi-omics data nor consider topological features. When deriving very large modules or ignoring the broader network context, interpretability remains limited. To address these issues, we developed a SUbgraph BAsed mulTi-OMIcs Clustering framework (SUBATOMIC), which infers small and interpretable modules with a specific topology while keeping track of connections to other modules and regulators.ResultsSUBATOMIC groups specific molecular interactions in composite network subgraphs of two and three nodes and clusters them into topological modules. These are functionally annotated, visualized and overlaid with expression profiles to go from static to dynamic modules. To preserve the larger network context, SUBATOMIC investigates statistically the connections in between modules as well as between modules and regulators such as miRNAs and transcription factors. We applied SUBATOMIC to analyze a composite Homo sapiens network containing transcription factor-target gene, miRNA-target gene, protein–protein, homologous and co-functional interactions from different databases. We derived and annotated 5586 modules with diverse topological, functional and regulatory properties. We created novel functional hypotheses for unannotated genes. Furthermore, we integrated modules with condition specific expression data to study the influence of hypoxia in three cancer cell lines. We developed two prioritization strategies to identify the most relevant modules in specific biological contexts: one considering GO term enrichments and one calculating an activity score reflecting the degree of differential expression. Both strategies yielded modules specifically reacting to low oxygen levels.ConclusionsWe developed the SUBATOMIC framework that generates interpretable modules from integrated multi-omics networks and applied it to hypoxia in cancer. SUBATOMIC can infer and contextualize modules, explore condition or disease specific modules, identify regulators and functionally related modules, and derive novel gene functions for uncharacterized genes. The software is available at https://github.com/CBIGR/SUBATOMIC.