Vector Space Research Articles

Invertible topological field theories

AbstractA ‐dimensional invertible topological field theory (TFT) is a functor from the symmetric monoidal ‐category of ‐bordisms (embedded into and equipped with a tangential ‐structure) that lands in the Picard subcategory of the target symmetric monoidal ‐category. We classify these field theories in terms of the cohomology of the ‐connective cover of the Madsen–Tillmann spectrum. This is accomplished by identifying the classifying space of the ‐category of bordisms with as an ‐space. This generalizes the celebrated result of Galatius–Madsen–Tillmann–Weiss (Acta Math. 202 (2009), no. 2, 195–239) in the case , and of Bökstedt–Madsen (An alpine expedition through algebraic topology, vol. 617, Contemp. Math., Amer. Math. Soc., Providence, RI, 2014, pp. 39–80) in the ‐uple case. We also obtain results for the ‐category of ‐bordisms embedding into a fixed ambient manifold , generalizing results of Randal–Williams (Int. Math. Res. Not. IMRN 2011 (2011), no. 3, 572–608) in the case . We give two applications: (1) we completely compute all extended and partially extended invertible TFTs with target a certain category of ‐vector spaces (for ), and (2) we use this to give a negative answer to a question raised by Gilmer and Masbaum (Forum Math. 25 (2013), no. 5, 1067–1106. arXiv:0912.4706).

On the running and the UV limit of Wilsonian renormalization group flows

In nonperturbative formulation of quantum field theory, the vacuum state is characterized by the Wilsonian renormalization group (RG) flow of Feynman type field correlators. Such a flow is a parametric family of ultraviolet (UV) regularized field correlators, the parameter being the strength of the UV regularization, and the instances with different strength of UV regularizations are linked by the renormalization group equation. Important RG flows are those which reach out to any UV regularization strengths. In this paper it is shown that for these flows a natural, mathematically rigorous generally covariant definition can be given, and that they form a topological vector space which is Hausdorff, locally convex, complete, nuclear, semi-Montel, Schwartz. That is, they form a generalized function space having favorable properties, similar to multivariate distributions. The other theorem proved in the paper is that for Wilsonian RG flows reaching out to all UV regularization strengths, a simple factorization formula holds in case of bosonic fields over flat (affine) spacetime: the flow always originates from a regularization-independent distributional correlator, and its running satisfies an algebraic ansatz. The conjecture is that this factorization theorem should generically hold, which is worth future investigations.

Homogeneous isosceles-free spaces

We study homogeneity aspects of metric spaces in which all triples of distinct points admit pairwise different distances; such spaces are called isosceles-free. In particular, we characterize all homogeneous isosceles-free spaces up to isometry as vector spaces over the two-element field, endowed with an injective norm. Using isosceles-free decompositions, we provide bounds on the maximal number of distances in arbitrary homogeneous finite metric spaces.

Antijamming Schemes for the Generalized MIMO Y Channel.

Signal space alignment (SSA) is a promising technique for interference management in wireless networks. However, despite the excellent work done on SSA, its robustness against jamming attacks has not been considered in the literature. In this paper, we propose two antijamming strategies for the SSA scheme applied in the multiple-input-multiple-output (MIMO) Y channel. The first scheme involves projecting the jamming signal into the null space of each source's precoding vectors, effectively eliminating it entirely. The second scheme removes interference originating from the jammer by subtracting the disturbance estimate from the incoming signal. The estimate is derived on the basis of the criterion of minimizing the received signal energy. The block error rate (BLER) performance of the proposed strategies in various channel configurations is verified by link level simulations and is presented to show the efficiency in mitigating jamming signals within the SSA-based MIMO Y channel.

Analysis of addiction craving onset through natural language processing of the online forum Reddit.

Alcohol cravings are considered a major factor in relapse among individuals with alcohol use disorder (AUD). This study aims to investigate the frequency and triggers of cravings in the daily lives of people with alcohol-related issues. Large amounts of data are analyzed with Artificial Intelligence (AI) methods to identify possible groupings and patterns. For the analysis, posts from the online forum "stopdrinking" on the Reddit platform were used as the dataset from April 2017 to April 2022. The posts were filtered for craving content and processed using the word2vec method to map them into a multi-dimensional vector space. Statistical analyses were conducted to calculate the nature and frequency of craving contexts and triggers (location, time, social environment, and emotions) using word similarity scores. Additionally, the themes of the craving-related posts were semantically grouped using a Latent Dirichlet Allocation (LDA) topic model. The accuracy of the results was evaluated using two manually created test datasets. Approximately 16% of the forum posts discuss cravings. The number of craving-related posts decreases exponentially with the number of days since the author's last alcoholic drink. The topic model confirms that the majority of posts involve individual factors and triggers of cravings. The context analysis aligns with previous craving trigger findings related to the social environment, locations and emotions. Strong semantic craving similarities were found for the emotions boredom, stress and the location airport. The results for each method were successfully validated on test datasets. This exploratory approach is the first to analyze alcohol cravings in the daily lives of over 24,000 individuals, providing a foundation for further AI-based craving analyses. The analysis confirms commonly known craving triggers and even discovers new important craving contexts.

The isomorphism problem for oligomorphic groups with weak elimination of imaginaries

AbstractIn Kechris et al. [J. Symb. Log. 83 (2018), no. 3, 1190–1203], it was asked if equality on the reals is sharp as a lower bound for the complexity of topological isomorphism between oligomorphic groups. We prove that under the assumption of weak elimination of imaginaries, this is indeed the case. Our methods are model theoretic and they also have applications on the classical problem of reconstruction of isomorphisms of permutation groups from (topological) isomorphisms of automorphisms groups. As a concrete application, we give an explicit description of for any vector space of dimension over a finite field, in affinity with the classical description for finite‐dimensional spaces due to Schreier and van der Waerden.

L vector spaces and L fields

L vector spaces and L fields

A generalization of Riesz* homomorphisms on order unit spaces

Riesz homomorphisms on vector lattices have been generalized to Riesz* homomorphisms on ordered vector spaces by van Haandel using a condition on sets of finitely many elements. Van Haandel attempted to prove that it suffices to take sets of two elements. We show that this is not true, in general. The description by two elements motivates to introduce mild Riesz* homomorphisms. We investigate their properties on order unit spaces, where the geometry of the dual cone plays a crucial role. Hereby, we mostly focus on the finite-dimensional case.

Coercive and Noncoercive Mixed Generalized Complementarity Problems

Impressed with the very recent developments of noncoercive complementarity problems and the use of recession sets in complementarity problems, here, we discuss mixed generalized complementarity problems in Hausdorff topological vector spaces. We used the Tikhonov regularization procedure, as well as arguments from the recession analysis, to establish the existence of solutions for mixed generalized complementarity problems without coercivity assumptions in Banach spaces.

Centrality measures-based sensitivity analysis and entropy of nonzero component graphs

The nonzero component graph of a finite-dimensional vector space over a finite field is a graph whose vertices are the nonzero vectors in the vector space, and any two vertices are adjacent if the corresponding linear combination contains a common basis vector. In this paper, we discuss the centrality measures and entropy of the nonzero component graph and also analyze the sensitivity of the graph using the centrality measures.

Equivariant Cohomology and Conditional Oriented Matroids

Abstract We give a cohomological interpretation of the Heaviside filtration on the Varchenko–Gelfand ring of a pair $({\mathcal{A}},{\mathcal{K}})$, where ${\mathcal{A}}$ is a real hyperplane arrangement and ${\mathcal{K}}$ is a convex open subset of the ambient vector space. This builds on work of the first author, who studied the filtration from a purely algebraic perspective, as well as work of Moseley, who gave a cohomological interpretation in the special case where ${\mathcal{K}}$ is the ambient vector space. We also define the Gelfand–Rybnikov ring of a conditional oriented matroid, which simultaneously generalizes the Gelfand–Rybnikov ring of an oriented matroid and the aforementioned Varchenko–Gelfand ring of a pair. We give purely combinatorial presentations of the ring, its associated graded, and its Rees algebra.

The RDF2vec family of knowledge graph embedding methods

Knowledge graph embeddings represent a group of machine learning techniques which project entities and relations of a knowledge graph to continuous vector spaces. RDF2vec is a scalable embedding approach rooted in the combination of random walks with a language model. It has been successfully used in various applications. Recently, multiple variants to the RDF2vec approach have been proposed, introducing variations both on the walk generation and on the language modeling side. The combination of those different approaches has lead to an increasing family of RDF2vec variants. In this paper, we evaluate a total of twelve RDF2vec variants on a comprehensive set of benchmark models, and compare them to seven existing knowledge graph embedding methods from the family of link prediction approaches. Besides the established GEval benchmark introducing various downstream machine learning tasks on the DBpedia knowledge graph, we also use the new DLCC (Description Logic Class Constructors) benchmark consisting of two gold standards, one based on DBpedia, and one based on synthetically generated graphs. The latter allows for analyzing which ontological patterns in a knowledge graph can actually be learned by different embedding. With this evaluation, we observe that certain tailored RDF2vec variants can lead to improved performance on different downstream tasks, given the nature of the underlying problem, and that they, in particular, have a different behavior in modeling similarity and relatedness. The findings can be used to provide guidance in selecting a particular RDF2vec method for a given task.

Image recovery method for underwater targets with complex polarization characteristics

Polarization imaging techniques have been effective in improving the clarity of turbid underwater images affected by water scattering. These techniques offer valuable additional information compared to traditional methods. However, previous descattering methods have mostly been designed for targets with uniform distribution of polarimetric characteristics. Therefore, targets with complex polarization characteristics have non-uniform distribution of polarization characteristics when dealing with such problems, additional parameter estimation errors can limit the results of image recovery. This paper proposes what we believe is a novel approach to address this issue. The method involves obtaining a new set of images using the polarization images vector space transformation method. The angle of polarization (AOP) of the target reflected light is estimated using the additivity law of the Stokes vector. This information is then combined with the physical model of underwater imaging to recover the underwater images affected by scattering. Experimental results conducted under varying levels of water turbidity validate the effectiveness of the proposed method. Moreover, the proposed method significantly broadens the range of application scenarios.

Guided Diverse Concept Miner (GDCM): Uncovering Relevant Constructs for Managerial Insights from Text

The Guided Diverse Concept Miner (GDCM) is an innovative deep learning algorithm tailored for the extraction of managerially relevant concepts from textual data, emphasizing the autonomy in discovering insights without predefined labels or guidance. This tool stands out by embedding words, documents, and concepts within the same vector space, which simplifies the interpretation of unearthed concepts and ensures their alignment with managerial outcomes. Central to GDCM’s methodology is its capacity to focus on concepts that are highly correlated with user-specified managerial outcomes, termed guiding variables, thereby enhancing the relevance and application of extracted insights in decision-making processes. The algorithm’s design inherently promotes the diversity of the recovered concepts, ensuring a broad spectrum of insights. Through practical application in analyzing customer reviews related to online purchases, GDCM not only identified key concepts influencing conversion rates but also validated its findings against established theories and prior causal research. This validation underscores GDCM’s utility in generating actionable, diverse insights tailored to specific managerial contexts, marking a significant advancement in how businesses leverage textual data for strategic decisions.

On some pre-orders and partial orders of linear operators on infinite dimensional vector spaces

AbstractThis article is devoted to the generalization of the Drazin pre-order and the G-Drazin partial order to Core-Nilpotent endomorphisms over arbitrary k-vector spaces, namely, infinite dimensional ones. The main properties of this orders are described, such as their respective characterizations and the relations between these orders and other existing ones, generalizing the existing theory for finite matrices. In order to do so, G-Drazin inverses are also studied in this framework. Also, it includes a generalization of the space pre-order to linear operators over arbitrary k-vector spaces.

Enhancing Knowledge Graph Embedding with Hierarchical Self-Attention and Graph Neural Network Techniques for Drug-Drug Interaction Prediction in Virtual Reality Environments

In biomedicine, the critical task is to decode Drug–Drug Interactions (DDIs) from complex biomedical texts. The scientific community employs Knowledge Graph Embedding (KGE) methods, enhanced with advanced neural network technologies, including capsule networks. However, existing methodologies primarily focus on the structural details of individual entities or relations within Biomedical Knowledge Graphs (BioKGs), overlooking the overall structural context of BioKGs, molecular structures, positional features of drug pairs, and their critical Relational Mapping Properties. To tackle the challenges identified, this study presents HSTrHouse an innovative hierarchical self-attention BioKGs embedding framework. This architecture integrates self-attention mechanisms with advanced neural network technologies, including Convolutional Neural Network (CNN) and Graph Neural Network (GNN), for enhanced computational modeling in biomedical contexts. The model bifurcates the BioKGs into entity and relation layers for structural analysis. It employs self-attention across these layers, utilizing PubMedBERT and CNN for position feature extraction, and a GNN for drug pair molecular structure analysis. Then, we connect the position and molecular structure features to integrate them into the self-attention calculation of entity and relation. After that, the output of the self-attention layer is combined with the connected vectors of the position feature and molecular structure feature to obtain the final representation vector, and finally, to model the Relational Mapping Properties (RMPs), the representation vector is embedded into the complex vector space using Householder projections to obtain the BioKGs model. The paper validates HSTrHouse’s efficacy by comparing it with advanced models on three standard BioKGs for DDIs research.

Vector Equilibrium Problems—A Unified Approach and Applications

We present existing results and properties for the solutions of some vector equilibrium problems with set-valued functions in the case of a vector space ordered by a cone with some “interiority” properties. Some applications concerning the existence of equilibrium for abstract economies and vector optimization problems are given.

Möbius function of the subgroup lattice of a finite group and Euler characteristic

AbstractThe Möbius function of the subgroup lattice of a finite group has been introduced by Hall and applied to investigate several questions. In this paper, we consider the Möbius function defined on an order ideal related to the lattice of the subgroups of an irreducible subgroup G of the general linear group $$\textrm{GL}(n,q)$$ GL ( n , q ) acting on the n-dimensional vector space $$V=\mathbb {F}_q^n$$ V = F q n , where $$\mathbb {F}_q$$ F q is the finite field with q elements. We find a relation between this function and the Euler characteristic of two simplicial complexes $$\Delta _1$$ Δ 1 and $$\Delta _2$$ Δ 2 , the former raising from the lattice of the subspaces of V, the latter from the subgroup lattice of G.

Fast and efficient context-aware embedding generation using fuzzy hashing for in-vehicle network intrusion detection

In the rapidly advancing field of automotive cybersecurity, the protection of In-Vehicle Networks (IVNs) against cyber threats is crucial. Current deep learning solutions offer robustness but at the cost of high computational demand and potential privacy breaches due to the extensive IVN data required for model training. Our study proposes a novel intrusion detection system (IDS) specifically designed for IVNs that prioritizes computational efficiency and data privacy. Utilizing fuzzy hashing techniques, we generate context-aware embeddings that effectively preserve the privacy of IVN data. Among the machine learning algorithms evaluated, the Support Vector Machine (SVM) emerged as the most effective, particularly when paired with TLSH hash embeddings. This combination achieved notable detection performance, as substantiated by T-SNE visualizations that demonstrate a distinct segregation of normal and attack traffic within the vector space. To validate the effectiveness and practicality of our proposed IDS, we conducted exhaustive experiments on the well-known car-hacking dataset and the more complex ROAD dataset, which includes diverse and sophisticated attack scenarios. Our findings reveal that the proposed lightweight IDS not only demonstrates high detection accuracy but also maintains this performance within the computational constraints of current IVN systems. The system's capability to operate effectively in real-time environments makes it a viable solution for modern automotive cybersecurity needs.

Invariant Grassmannians and a K3 surface with an action of order 192*2

Given a complex vector space V of finite dimension, its Grassmannian variety parametrizes all subspaces of V of a given dimension. Similarly, if a finite group G acts on V, its invariant Grassmannian parametrizes all the G−invariant subspaces of V of a given dimension. Based on this fact, we develop an algorithm for finding equations of G−invariant projective varieties arising as an intersection of hypersurfaces of the same degree.We apply the algorithm to find equations describing a polarized K3 surface with a faithful action of T192⋊μ2 and some further symmetric K3 surfaces with a degree 8 polarization.