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We define and study cohomological tensor functors from the category

The Cuntz semigroup of a Given a We establish the existence of tensor products in the category As a main tool for our approach we introduce the category For every (local) We also develop a theory of If

We generalize Jones’ planar algebras by internalising the notion to a pivotal braided tensor category Building on our previous work on categorified traces, we prove that there is an equivalence of categories between anchored planar algebras in

We study symmetric and antisymmetric tensor products of Hilbert-space operators, focusing on norms and spectra for some well-known classes favored by function-theoretic operator theorists. We pose many open questions that should interest the field.

We define the We study both the projective and injective tensor products, and their relation. Moreover, in order to be general and flexible, we study three different types of expectations: Bochner integrals, Pettis integrals and Dunford integrals. One of the problems studied is whether two random variables with the same injective moments (of a given order) necessarily have the same projective moments; this is of interest in applications. We show that this holds if the Banach space has the approximation property, but not in general. Several chapters are devoted to results in special Banach spaces, including Hilbert spaces, One of the main motivations of this paper is the application to Zolotarev metrics and their use in the contraction method. This is sketched in an appendix.

This paper investigates bisimulation relations of delayed switched Boolean control networks (DSBCNs), using the semi‐tensor product (STP) approach. Firstly, the notion of bisimulation in DSBCNs is formalized. Subsequently, based on the skeleton matrix of DSBCN, two necessary and sufficient conditions are derived for checking this notion. Moreover, as an application of bisimulation relations, this paper explores the propagation of controllability for DSBCNs through a bisimulation relation. It shows the possibility that the controllability of a DSBCN can be inferred by studying a potentially simpler DSBCN. At last, two examples are proposed to illustrate the efficiency of the obtained results. (i)The notion of bisimulation of DSBCNs are given, which extends the corresponding bisimulations notion of Boolean control networks and switching linear systems to DSBCNs. (ii)Two necessary and sufficient conditions are derived for checking this notion, which characterizes the bisimulation relations of DSBCNs from two different aspects. (iii)The propagation of controllability for DSBCNs is esxplored through a bisimulation relation.

If we go through the literature, one can find many matrices that are derived for a given simple graph. The one among them is the anti-adjacency matrix which is given as follows; The anti-adjacency matrix of a simple undirected graph $G$ with vertex set   $V (G) \\,= \\,\\{\\,v_1,\\,v_2,\\\ \\dots, v_n\\}$   is an $n \\times n$ matrix $B(G) = (b_{ij} )$, where $b_{ij} = 0$ if there exists an edge between $v_i$ and $v_j$ and $1$ otherwise. In this paper, we try to bring out an expression, which establishes a connection between the anti-adjacency matrices of the two graphs $G_1$ and $G_2$ and the   anti-adjacency matrix of their tensor product, $G_1 \\otimes G_2$. In addition, an expression for the anti-adjacency matrix of the disjunction of two graphs, $G_1\\lor G_2$, is obtained in a similar way. Finally, we obtain an expression for the anti-adjacency matrix for the generalized tensor product and generalized disjunction of two graphs.  Adjacency and anti-adjacency matrices are square matrices that are used to represent a finite graph in graph theory and computer science. The matrix elements show whether a pair of vertices in the graph are adjacent or not.

Suppose that E and F are Banach lattices. It is known that there are several norms on the Fremlin tensor product E ⊗ ¯ F that turn it into a normed lattice; in particular, the projective norm | π | (known as the Fremlin projective norm) and the injective norm | ϵ | (known as the Wittstock injective norm). Now, assume that E and F are locally convex-solid vector lattices. Although we have a suitable vector lattice structure for the tensor product E and F (known as the Fremlin tensor product and denoted by E ⊗ ¯ F ), there is a lack of topological structure on E ⊗ ¯ F , in general. In this note, we consider a linear topology on E ⊗ ¯ F that makes it into a locally convex-solid vector lattice, as well; this approach can be taken as a generalization of the projective norm of the Fremlin tensor product between Banach lattices.

In this paper, semi-tensor product (STP) and related properties of dimension keeping semi-tensor product (DK-STP) are analyzed. The commutativity and anticommutativity of DK-STP are studied by means of matrix mapping, and sufficient conditions for both are obtained. The structure matrix of the Lie bracket of non-square matrices (NSM) is discussed, and some properties are derived. The correspondences between the special Lie subalgebras of square matrix and Lie subalgebras of NSM are discussed through a homomorphism.

Matrix and tensor nuclear norms have been successfully used to promote the low-rankness of tensors in low-rank tensor completion. However, singular value decomposition (SVD), which is computationally expensive for large-scale matrices, frequently appears in solving those nuclear norm minimization models. Based on the tensor-tensor product (T-product), in this paper, we first establish the equivalence between the so-called transformed tubal nuclear norm for a third-order tensor and the minimum of the sum of two factor tensors’ squared Frobenius norms under a general invertible linear transform. Gainfully, we introduce a mode-unfolding (often named as “spatio-temporal” in the internet traffic data recovery literature) regularized tensor completion model that is able to efficiently exploit the hidden structures of tensors. Then, we propose an implementable alternating minimization algorithm to solve the underlying optimization model. It is remarkable that our approach does not require any SVDs and all subproblems of our algorithm enjoy closed-form solutions. A series of numerical experiments on traffic data recovery, color images and videos inpainting demonstrate that our SVD-free approach takes less computing time to achieve satisfactory accuracy than some state-of-the-art tensor nuclear norm minimization approaches.