The nonzero component graph of a near-vector space

The combination of algebraic structures and graphs was carried out by investigating thoroughly the relation among the algebraic structure and the graph theoretic properties. Moreover, it needs to explore algebraic structure. The results of the combination of algebraic structures and graphs have many applications in the fields of Internet modeling, coding, etc. For example, the famous Cayley graph was constructed from groups and widely used in network models. Das introduced the subspace inclusion graph on finite-dimensional vector space over a finite filed, and studied that the graph is bipartite and some special properties if the dimension of the vector space is 3. In this paper, this bipartite inclusion graph in the case of 3-dimensional is extended to more general dimensional bipartite inclusion graph. The diameter, girth, clique number, covering number, independence number and matching number are studied and the properties are shown, such as regular, planar and Eulerian. Moreover, the authors also introduce a new results about the structure and the number of 1-dimensional and n-1-dimensional subspaces on n-dimensional vector space.

In this paper, nonzero component graphs and nonzero component union graphs of finite-dimensional vector spaces are studied using the zero-divisor graph of a specially constructed 0–1-distributive lattice and the zero-divisor graph of rings. Furthermore, we define an equivalence relation on nonzero component graphs and nonzero component union graphs to deduce that these graphs are the graph join of zero-divisor graphs of Boolean algebras and complete graphs. The last section characterizes the perfect and chordal nonzero component and nonzero component union graphs. Additionally, we observe that the nonzero component graph and reduced nonzero component union graph of free semi-modules could be treated as the zero-divisor graph of a 0–1-distributive lattice.

The non-zero component graph of a vector space with finite dimension over a finite field F is the graph \(G~ = ~\left( {V,E} \right)\), where vertices of G are the non-zero vectors in V, two of which are adjacent if they have at least one basis vector with non-zero coefficient common in their basic representation. In this paper, we discuss certain properties of the non-zero component graphs of vector spaces with finite dimension over finite fields and their graph invariants.

Chapter 5 - Vector Spaces, Hilbert Spaces, and the L2 Space

In this paper, we introduce the intersection graph of k-subspaces of a finite dimensional vector space which denoted by Gk,b (V, E). We obtain necessary and sufficient condition on k such that these graphs are complete and characterize the vector spaces for which the GK,B (V, E) are isomorphic. Also we show that all of such graphs are connected for k ≥ 2.

In a recent paper, Das introduced the graph $$\mathcal {I}n(\mathbb {V})$$ , called subspace inclusion graph on a finite dimensional vector space $$\mathbb {V}$$ , where the vertex set is the collection of nontrivial proper subspaces of $$\mathbb {V}$$ and two vertices are adjacent if one is properly contained in another. Das studied the diameter, girth, clique number and chromatic number of $$\mathcal {I}n(\mathbb {V})$$ when the base field is arbitrary, and he also studied some other properties of $$\mathcal {I}n(\mathbb {V})$$ when the base field is finite. At the end of the above paper, the author posed the open problem of determining the automorphisms of $$\mathcal {I}n(\mathbb {V})$$ . In this paper, we give the answer to the open problem.

In this paper, we continue the study of the subspace inclusion graph on a finite-dimensional vector space with dimension n where the vertex set is the collection of non-trivial proper subspaces of a vector space and two vertices are adjacent if one is contained in other. It is shown that is perfect and non-planar. Moreover, a necessary and sufficient condition is provided for to be Eulerian. For , it is shown that is bipartite, vertex and edge-transitive and has a perfect matching. We also provide exact value of the independence number and bounds on the domination number of for .

To establish a multi-dimensional representation solely on structural MRI (sMRI) for early diagnosis of AD. A total of 3377 participants' sMRI from four independent databases were retrospectively identified to construct an interpretable deep learning model that integrated multi-dimensional representations of AD solely on sMRI (called s2MRI-ADNet) by a dual-channel learning strategy of gray matter volume (GMV) from Euclidean space and the regional radiomics similarity network (R2SN) from graph space. Specifically, the GMV feature map learning channel (called GMV-Channel) was to take into consideration spatial information of both long-range spatial relations and detailed localization information, while the node feature and connectivity strength learning channel (called NFCS-Channel) was to characterize the graph-structured R2SN network by a separable learning strategy. The s2MRI-ADNet achieved a superior classification accuracy of 92.1% and 91.4% under intra-database and inter-database cross-validation. The GMV-Channel and NFCS-Channel captured complementary group-discriminative brain regions, revealing a complementary interpretation of the multi-dimensional representation of brain structure in Euclidean and graph spaces respectively. Besides, the generalizable and reproducible interpretation of the multi-dimensional representation in capturing complementary group-discriminative brain regions revealed a significant correlation between the four independent databases (p < 0.05). Significant associations (p < 0.05) between attention scores and brain abnormality, between classification scores and clinical measure of cognitive ability, CSF biomarker, metabolism, and genetic risk score also provided solid neurobiological interpretation. The s2MRI-ADNet solely on sMRI could leverage the complementary multi-dimensional representations of AD in Euclidean and graph spaces, and achieved superior performance in the early diagnosis of AD, facilitating its potential in both clinical translation and popularization.

In this chapter we review the main ideas of linear algebra. The one twist is that we allow our scalars to come from any field instead of just the field of real numbers. In particular, the notion of a vector space over the field \(\mathbb{Z}_{2}\) will be essential in our study of coding theory. We will also need to look at other finite fields when we discuss Reed–Solomon codes. One benefit to working with finite dimensional vector spaces over finite fields is that all sets in question are finite, and so computers can be useful in working with them.

Generalized incidence theorems, homogeneous forms and sum–product estimates in finite fields

In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual frames in order to obtain and study the different representations of the elements of the vector space provided by a frame. We relate the introduced theory with the classical one of dual frames for Hilbert spaces and apply it to study dual frames for three types of vector spaces: for vector spaces over conjugate closed subfields of the complex numbers (in particular, for cyclotomic fields), for metric vector spaces, and for ultrametric normed vector spaces over complete non-archimedean valued fields. Finally, we consider the matrix representation of operators using dual frames and its application to the solution of operators equations in a Petrov-Galerkin scheme.

In this paper, a construction of Cartesian authentication codes from vector space over finite fields is presented and its size parameters are computed. Under the assumption that the choose of encoding rules and source states obeys uniform probability distribution, the probabilities P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</sub> of a successful impersonation attack and P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</sub> of a successful substitution attack of these codes are also computed.

The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold M with the Grothendieck group of constructible sheaves on M. When M is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of $\mathbf {k}$ -vector spaces on M. In this paper, we characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. We formulate consequences of our result for Topological Data Analysis: there cannot exist nontrivial additive invariants of persistence modules that are continuous for the interleaving distance.

Industry is evolving towards Industry 4.0, which holds the promise of increased flexibility in manufacturing, better quality and improved productivity. A core actor of this growth is using sensors, which must capture data that can used in unforeseen ways to achieve a performance not achievable without them. However, the complexity of this improved setting is much greater than what is currently used in practice. Hence, it is imperative that the management cannot only be performed by human labor force, but part of that will be done by automated algorithms instead. A natural way to represent the data generated by this large amount of sensors, which are not acting measuring independent variables, and the interaction of the different devices is by using a graph data model. Then, machine learning could be used to aid the Industry 4.0 system to, for example, perform predictive maintenance. However, machine learning directly on graphs, needs feature engineering and has scalability issues. In this paper we discuss methods to convert (embed) the graph in a vector space, such that it becomes feasible to use traditional machine learning methods for Industry 4.0 settings.

Let P(V) be the partial linear transformation semigroup of a vector space V under composition. Given a fixed subspace W of V, define the following subsemigroups of P(V): $$\begin{aligned} PT(V,W)&=\{\alpha \in P(V)\ |\ V\alpha \subseteq W\},\\ T(V,W)&=\{\alpha \in P(V,W)\ |\ \mathrm{dom}\ \alpha =V\},\\ I(V,W)&=\{\alpha \in P(V,W)\ |\ \alpha \text { is injective}\}. \end{aligned}$$ In this paper, we prove certain isomorphism theorems and compute the ranks of these three semigroups for any proper subspace W of V when V is a finite-dimensional vector space over a finite field. Gaussian binomial coefficients play an essential role in these computations.