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Given Banach space operators A, B, let δ A , B denote the generalised derivation δ (X) = (L A - R B) (X) = A X - X B and ▵ A , B the length two elementary operator ▵ A , B (X) = (I - L A R B) (X) = X - A X B . This note considers the structure of m-symmetric operators δ ▵ A 1 , B 1 , ▵ A 2 , B 2 m (I) = (L ▵ A 1 , B 1 - R ▵ A 2 , B 2 ) m (I) = 0 . It is seen that there exist scalars λ i ∈ σ a (B 1) , 1 ≤ i ≤ 2 , such that δ λ 1 A 1 , λ 2 A 2 m (I) = 0 . Translated to Hilbert space operators A and B this implies that if δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 , then there exists λ ¯ ∈ σ a (B ∗) such that δ (λ A) ∗ , λ A m (I) = 0 = δ λ ¯ B , λ B ∗ m (I) . We prove that the operator δ ▵ A ∗ , B ∗ , ▵ A , B m is compact if and only if (i) there exists a real number α and finite sequnces (i) { a j } j = 1 n ⊆ σ (A) , { b j } j = 1 n ⊆ σ (B) such that a j b j = 1 - α , 1 ≤ j ≤ n ; (ii) decompositions ⊕ j = 1 n H j and ⊕ j = 1 n H J of H such that ⊕ j = 1 n (A - a j I) | H j and ⊕ j = 1 n (B - b j I) | H j are nilpotent. If δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 implies δ ▵ A ∗ , B ∗ , ▵ A , B (I) = 0 , then A and B satisfy a (Putnam-Fuglede type) commutativity theorem; conversely, a sufficient condition for δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 to imply δ ▵ A ∗ , B ∗ , ▵ A , B (I) = 0 is that λ A and λ ¯ B satisfy the commutativity property for scalars lambda ¯ ∈ σ a (B ∗) . An analogous result is seen to hold for the operators ▵ δ A ∗ , B ∗ , δ A , B m and ▵ δ A ∗ , B ∗ , δ A , B m (I) . Perturbation by commuting nilpotents is considered. [ABSTRACT FROM AUTHOR]

A graph G with p vertices and q edges has antimagic labeling if there is a bijection from the edge set of the graph to the label set { 1 , 2 , ... , q } such that p vertices must have distinct vertex sums, where the vertex sums are determined by adding up all the edge labels incident to each vertex v in V (G). Hartsfield and Ringel conjectured that every connected graph is antimagic, except P 2 . In this study, we identified a class of connected graphs that lend credence to this conjecture. In this paper, we proved that the tensor product of a Wheel Graph, Helm Graph, and Flower graph with Star is antimagic. [ABSTRACT FROM AUTHOR]

Let sl (2) ⋉ h n , n ≥ 1 , be the Galilean Lie algebra over a field of characteristic zero, here h n is the Heisenberg Lie algebra of dimension 2 n + 1 , and sl (2) acts on h n so that, sl (2) -modules, h n ≃ V (2 n - 1) ⊕ V (0) (here V(k) denotes the irreducible sl (2) -module of highest weight k). In this paper, we study the tensor product of two uniserial representations of sl (2) ⋉ h n . We obtain the sl (2) -module structure of the socle of V ⊗ W and we describe the space of intertwining operators Hom sl (2) ⋉ h n (V , W) , where V and W are uniserial representations of sl (2) ⋉ h n . The structure of the radical of V ⊗ W follows from that of the socle of V ∗ ⊗ W ∗ . The result is subtle and shows how difficult is to obtain the whole socle series of arbitrary tensor products of uniserials. In contrast to the serial associative case, our results for sl (2) ⋉ h n reveal that these tensor products are far from being a direct sum of uniserials; in particular, there are cases in which the tensor product of two uniserial (sl (2) ⋉ h n) -modules is indecomposable but not uniserial. Recall that a foundational result of T. Nakayama states that every finitely generated module over a serial associative algebra is a direct sum of uniserial modules. This article extends a previous work in which we obtained the corresponding results for the Lie algebra sl (2) ⋉ a m where a m is the abelian Lie algebra of dimension m + 1 and sl (2) acts so that a m ≃ V (m) as sl (2) -modules. [ABSTRACT FROM AUTHOR]

In this paper we study the modular structure of the Foulkes module H (2 m) of the symmetric group S 2 m acting on set partitions of a set of size 2m into m sets each of size 2, defined over a field of positive characteristic p. We aim to understand the decomposition of the tensor powers of H (2 m) . In particular, we study the asymptotics of the non-projective part of tensor powers of these modules by computing the gamma invariant as defined by Dave Benson and Peter Symonds. [ABSTRACT FROM AUTHOR]

Kobayashi–Pevzner discovered in [Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs, Selecta Math. (N.S.) 22(2) (2016) 847–911, MR 3477337] that the failure of the multiplicity-one property in the fusion rule of Verma modules of 2 occurs exactly when the Rankin–Cohen brackets vanish, and classified all the corresponding parameters. In this paper, we provide yet another characterization for these parameters, and give a precise description of indecomposable components of the tensor product. Furthermore, we discuss when the tensor products of two Verma modules are isomorphic to each other for semisimple Lie algebras . [ABSTRACT FROM AUTHOR]

In this paper, we focus on studying two classes of finite dimensional -associative algebras, which are extensions of a family of -dimensional Kac-Paljutkin type semi-simple Hopf algebras . All their indecomposable modules are classified. Furthermore, their representation rings are described by generators with suitable relations. [ABSTRACT FROM AUTHOR]

In this paper, the connection between the ϵ -pseudospectrum of the tensor product operator A1 ⊗ A2 and the ϵ -pseudospectrums of operators A1 and A2 has been investigated and some results are given about this connection under certain conditions. [ABSTRACT FROM AUTHOR]

We investigate two different ways of recovering a Grothendieck category as a filtered bicolimit of small categories and the compatibility of both with the tensor product of Grothendieck categories. Firstly, we show that any locally presentable linear category (and in particular any Grothendieck category) can be recovered as the filtered bicolimit of its subcategories of α-presentable objects, with α varying in the family of small regular cardinals. We then prove that the tensor product of locally presentable linear categories (and in particular the tensor product of Grothendieck categories) can be recovered as a fltered bicolimit of the Kelly tensor product of α-cocomplete linear categories of the corresponding subcategories of α-presentable objects. Secondly, we show that one can recover any Grothendieck category as a filtered bicolimit of its linear site presentations. We then prove that the tensor product of Grothendieck categories, in contrast with the first case, cannot be recovered in general as a filtered bicolimit of the tensor product of the corresponding linear sites. Finally, we show how the first presentation can be helpful when computing tensor products of cocontinuous linear functors between Grothendieck categories. [ABSTRACT FROM AUTHOR]

This paper is committed in characterizing the preserving maps for the class of separable matrices and a subclass of infinitely divisible matrices which are also separable. For this, an association between the classes of separable matrices and infinitely divisible matrices is established. Also, several properties and attributes of the above mentioned classes are stated and proved. [ABSTRACT FROM AUTHOR]

In this paper we introduce some results on hyponormal operators acting on a complex Hilbert space H. We give sufficient conditions for which the sum and product of two hyponormal operators is a hyponormal operator. Also, we study the sum direct and tensor product of this class. [ABSTRACT FROM AUTHOR]

This paper characterize important concepts such as the Riesz bases and pair of dual frames for tensor product of Hilbert spaces and introduce some special properties of pair of dual frames in these spaces. We show that Riesz bases and pair of dual frames in H ⊗ K are preserved under invertible and unitary operators on H . Finally, We present a representation for Hilbert–Schmidt operators by a pair of dual frames. [ABSTRACT FROM AUTHOR]

In this paper, we present a new analytical scheme to generate atomic solutions of partial differential equations. The theory of tensor product in Banach spaces coupled with some features of atomics operators are utilized to attain our results. Some demonstrative examples are given for completeness. [ABSTRACT FROM AUTHOR]

We address the problem of when the tensor product of two finitely generated modules over a Cohen-Macaulay local ring is Ulrich in the generalized sense of Goto et al., and in particular in the original sense from the 80's. As applications, besides freeness criteria for modules, characterizations of complete intersections, and an Ulrich-based approach to the long-standing Berger's conjecture, we give simple proofs that two celebrated homological conjectures, namely the Huneke-Wiegand and the Auslander-Reiten problems, are true for the class of Ulrich modules. [ABSTRACT FROM AUTHOR]

We consider several notions of regularity, including strong regularity, bounded relative units, and Ditkin's condition, in the setting of vector-valued function algebras. Given a commutative Banach algebra A and a compact space X, let A be a Banach A-valued function algebra on X and let A be the subalgebra of A consisting of scalar-valued functions. This paper is about the connection between regularity conditions of the algebra A and the associated algebras A and A. That A inherits a certain regularity condition P to A and A is the easy part of the problem. We investigate the converse and show that, under certain conditions, A receives P form A and A. The results apply to tensor products of commutative Banach algebras as they are included in the class of vector-valued function algebras. [ABSTRACT FROM AUTHOR]

The notion of Cauchy dual for left-invertible covariant representations was studied by Trivedi and Veerabathiran. Using the Moore-Penrose inverse, we extend this notion for the covariant representations having closed range and explore several useful properties. We obtain a Wold-type decomposition for regular completely bounded covariant representation whose Moore-Penrose inverse is regular. Also, we discuss an example related to the non-commutative bilateral weighted shift. We prove that the Cauchy dual of the concave covariant representation (σ , V) modulo N (V ~) is hyponormal modulo N (V ~) . [ABSTRACT FROM AUTHOR]

Tensor ring (TR) decomposition has been widely applied as an effective approach in a variety of applications to discover the hidden low‐rank patterns in multidimensional and higher‐order data. A well‐known method for TR decomposition is the alternating least squares (ALS). However, solving the ALS subproblems often suffers from high cost issue, especially for large‐scale tensors. In this paper, we provide two strategies to tackle this issue and design three ALS‐based algorithms. Specifically, the first strategy is used to simplify the calculation of the coefficient matrices of the normal equations for the ALS subproblems, which takes full advantage of the structure of the coefficient matrices of the subproblems and hence makes the corresponding algorithm perform much better than the regular ALS method in terms of computing time. The second strategy is to stabilize the ALS subproblems by QR factorizations on TR‐cores, and hence the corresponding algorithms are more numerically stable compared with our first algorithm. Extensive numerical experiments on synthetic and real data are given to illustrate and confirm the above results. In addition, we also present the complexity analyses of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this note, considering the main properties of closed ideals of C* ­-algebras, we will determine the structure of closed ideals of the C* ­-algrbra C(X, A), the space of all continuous functions from compact Hausdorff space X to C* ­-algebra A. Indeed, we will show that for every closed ideal I of C(X, A), there is some closed subset F of topological space X × prim(A), such that I = {f ∈ C(X, A) : ∀ (x, P) ∈ F, f(x) ∈ P}. [ABSTRACT FROM AUTHOR]

A m-quasi class A k , denoted as M ∈ Q (A k , m) , is defined as an operator M acting on a complex Hilbert space H , provided that the following condition holds true: M ∗ m | M k + 1 | 2 k + 1 M m ≥ M ∗ m | M | 2 M m , where both k and m are positive integers. In this research, we unveil fundamental structural characteristics of these operators. Leveraging these findings, we can establish that P is self-adjoint if P represents the Riesz idempotent associated with the isolated point λ in the spectrum of M ∈ Q (A k , m) . Additionally, we provide a necessary and sufficient condition for M ⊗ N to belong to Q A k when both M and N are non-zero. [ABSTRACT FROM AUTHOR]

Using the Wold–von Neumann decomposition for the isometric covariant representations due to Muhly and Solel, we prove an explicit representation of the commutant of a doubly commuting pure isometric representation of the product system over N 0 k. As an application we study a complete characterization of invariant subspaces for a doubly commuting pure isometric representation of the product system. This provides us a complete set of isomorphic invariants. Finally, we classify a large class of an isometric covariant representations of the product system. [ABSTRACT FROM AUTHOR]

The geometric Satake correspondence can be regarded as a geometric construction of rational representations of a complex connected reductive group G. In their study of this correspondence, Mirkovi'c and Vilonen introduced algebraic cycles that provide a linear basis in each irreducible representation. Generalizing this construction, Goncharov and Shen define a linear basis in each tensor product of irreducible representations. We investigate these bases and show that they share many properties with the dual canonical bases of Lusztig. [ABSTRACT FROM AUTHOR]

In this work, we introduce the concept of *-operator frame, which is a generalization of *-frames in Hilbert pro-C*-modules, and we establish some results. We also study the tensor product of *-operator frame for Hilbert pro-C*-modules. [ABSTRACT FROM AUTHOR]

Let X be a Banach space, B(X) the algebra of bounded linear operators on X and (J, ||·||J) an admissible Banach ideal of B(X). For T ∈ B(X), LJ,T and RJ,T ∈ B(J) denote the left and right multiplication defined by LJ,T(A) = TA and RJ,T (A) = AT, respectively. In this paper, we study the transmission of some concepts related to recurrent operators between T ∈ B(X), and their elementary operators LJ,T and RJ,T. In particular, we give necessary and sufficient conditions for LJ,T and RJ,T to be sequentially recurrent. Furthermore, we prove that LJ,T is recurrent if and only if T ... T is recurrent on X ... X. Moreover, we introduce the notion of a mixing recurrent operator and we show that LJ,T is mixing recurrent if and only if T is mixing recurrent. [ABSTRACT FROM AUTHOR]

We consider the BGG category O of a quantized universal enveloping algebra Uq(g). It is well-known that M ⊗ N ∈ O if M or N is finite dimensional. When g is simple and of type ADE, we prove in this paper that M ⊗N ∈ O if M and N are both infinite dimensional. [ABSTRACT FROM AUTHOR]

This paper is an extension of the research on (cartesian) product of c-spaces. This paper demonstrates that the finite (tensor)product of quotients of c-spaces can be represented as a quotient of its (tensor) product. Some properties of the tensor product of c-spaces have been investigated in this context. Properties of the space of c-continuous functions have been probed and the relevance of the standard c-structure on it has been established. [ABSTRACT FROM AUTHOR]

Let E be a Banach space and A be a commutative Banach algebra with identity. Let P(E, A) be the space of A-valued polynomials on E generated by bounded linear operators (an n-homogenous polynomial in P(E, A) is of the form P = ∑i=1∞ Tin, where Ti : E → A, 1 ≤ i < ∞, are bounded linear operators and ∑i=1∞ ||Ti||n < ∞). For a compact set K in E, we let P(K, A) be the closure in C(K, A) of the restrictions P|K of polynomials P in P(E, A). It is proved that P(K, A) is an A-valued uniform algebra and that, under certain conditions, it is isometrically isomorphic to the injective tensor product PN(K) ⊗∈ A, where PN(K) is the uniform algebra on K generated by nuclear scalar-valued polynomials. The character space of P(K, A) is then identified with K̂N × M(A), where K̂N is the nuclear polynomially convex hull of K in E, and M(A) is the character space of A. [ABSTRACT FROM AUTHOR]

We study the category Osh of representations over a shifted Yangian. This category has a tensor product structure and contains distinguished modules, the positive prefundamental modules and the negative prefundamental modules. Motivated by the representation theory of the Borel subalgebra of a quantum affine algebra and by the relevance of quantum integrable systems in this context, we prove that tensor products of prefundamental modules with irreducible modules are either cyclic or cocyclic. This implies the existence and uniqueness of morphisms, the R-matrices, for such tensor products. We prove the R-matrices are polynomial in the spectral parameter, and we establish functional relations for the R-matrices. As applications, we prove the Jordan-Holder property in the category Osh. We also obtain a proof, uniform for any finite type, that any irreducible module factorizes through a truncated shifted Yangian. [ABSTRACT FROM AUTHOR]

For a bounded linear operator, by local spectral theory methods, we study the property (b z) , which means that the difference of the approximate point spectrum with the upper semi-Fredholm spectrum coincides with the set of all finite-range left poles. We will investigate this property under closed proper subspaces of X , also under the tensor product. In addition, the relationships of this property with other spectral properties are studied. Among others, we will obtain several characterizations for the operators that verify the property (bz) and show that the set of these operators is closed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of tensor product between intuitionistic fuzzy submodules. We establish a formal framework for the tensor product operation, examining its properties and applications within the context of intuitionistic fuzzy modules. We then establish a relationship between the Hom functor and the tensor product in the category of intuitionistic fuzzy modules. The connection between tensor products and hom-functors in some algebraic structures, such as modules, is made possible via a natural isomorphism known as the Hom- Tensor adjunction and it establishes a relationship between HomCR-IFM(B⊗A,C) and HomCR-IFM(A,HomCR-IFM(B,C)). An application of tensor product of intuitionistic fuzzy modules can be used in decision-making processes by embracing ambiguity and vagueness, making it a valuable tool when exact data is lacking. [ABSTRACT FROM AUTHOR]

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In this work, we study non-rough norms in L(X, Y), the space of bounded linear operators between Banach spaces X and Y. We prove that L(X, Y) has non-rough norm if and only if X ∗ and Y have non-rough norm. We show that the injective tensor product X ⊗ ^ ε Y has non-rough norm if and only if both X and Y have non-rough norm. We also give an example to show that non-rough norms are not stable under projective tensor product. We also study a related concept namely the small diameter properties in the context of L (X , Y) ∗ . These results lead to a discussion on stability of the small diameter properties for projective and injective tensor product spaces. [ABSTRACT FROM AUTHOR]

We introduce a simple algorithm that efficiently computes tensor products of Pauli matrices. This is done by tailoring the calculations to this specific case, which allows to avoid unnecessary calculations. The strength of this strategy is benchmarked against state-of-the-art techniques, showing a remarkable acceleration. As a side product, we provide an optimized method for one key calculus in quantum simulations: the Pauli basis decomposition of Hamiltonians. [ABSTRACT FROM AUTHOR]

For a standard graded ring R of dimension ≥ 2 over a perfect field of characteristic p > 0 and a homogeneous ideal I of finite colength, the HK density function of R with respect to I is a compactly supported continuous function f R , I : [ 0 , ∞) ⟶ [ 0 , ∞) , whose integration yields the HK multiplicity e HK (R , I) . Here we answer a question of V. Trivedi about the Hilbert–Kunz density function of the tensor product of standard graded rings and show that it is the convolution of the Hilbert–Kunz density function of the factor rings. Using Fourier transform, as a corollary, we get that HK multiplicity of the tensor product of rings is a product of the HK multiplicity of the factor rings. We compute the Fourier transform of the HK density function of a projective curve. [ABSTRACT FROM AUTHOR]

We introduce the algebras satisfying the (ℬ , n) condition. If Λ, Γ are algebras satisfying the (ℬ , n), (ℰ , m) condition, respectively, we give a construction of (m+n)-almost split sequences in some subcategories (B ⊗ E) (i 0 , j 0) of mod(Λ ⊗ Γ) by tensor products and mapping cones. Moreover, we prove that the tensor product algebra Λ ⊗ Γ satisfies the ((B ⊗ E) (i 0 , j 0) , n + m) condition for some integers i0, j0; this construction unifies and extends the work of A. Pasquali (2017), (2019). [ABSTRACT FROM AUTHOR]

As is well known, a class of 2 n 2 -dimensional Kac-Paljutkin Hopf algebras H 2 n 2 was introduced by Pansera. It is the generalization of the 8-dimensional Kac-Paljutkin Hopf algebra H8. In this paper, the classification of Yetter-Drinfeld modules over H 2 n 2 is given by Radford's method of constructing Yetter-Drinfeld modules for a Hopf algebra. Furthermore, the tensor product of Yetter-Drinfeld modules over H 2 n 2 is established, and the Grothendieck ring r ( H 2 n 2 H 2 n 2 Y D) is described explicitly by generators and relations. Communicated by Julia Plavnik [ABSTRACT FROM AUTHOR]

All rings throughout this paper are commutative and Noetherian. Semidualizing modules were studied independently by Foxby [4], Golod [5], and Vasconcelos [11]. A finite R-module C is called semidualizing if the natural homothety map XCR : R → HomR (C, C) is an isomorphism and ExtRi(C, C) = 0 for all i > 0. If a semidualizing R-module has finite injective dimension, it is called dualizing and is denoted by D. The ring itself is an example of a semidualizing R-module. Many researchers, in particular Sather-Wagstaff [12], have studied the semidualizing modules. Let M be an R-module. The trace ideal of M, denoted by trR(M), is the sum of images of all homomorphisms from M to R. Trace ideals have attracted the attention of many researchers in recent years. In particular, Herzog et al. [6] and Dao et al. [3] studied the trace ideals of canonical modules. Also, the trace ideals of semidualizing modules were studied in [1]. In this paper, we study the trace ideals of tensor product of two arbitrary modules. We prove some known facts with a different approach via trace ideals. For example, let C and C' be two semidualizing R-modules. We show that C⊗R C' is projective if and only if C and C' are projective R-modules of rank 1. Also, we study the trace ideals of maximal Cohen-Macaulay modules over a Gorenstein local ring. Material and Methods The content of this paper is organized as follows. First, we present some definitions, lemmas, and basic results that will be used in the proofs of our theorems. Then, we use the trace ideals of modules for proving the results. Results and discussion The main objectives of this article are as follows: studying the trace ideals of tensor product of projective modules, proving some known facts with a different approach via trace ideals, generalizing some results about semidualizing modules, and discussing trace ideals of maximal Cohen- Macaulay modules over Gorenstein rings. Conclusion First, we prove the following Proposition. Proposition 1: Let M, N be two arbitrary R-modules. Then trR(M⊗N) = R if and only if trR (M) = R and trR (N) = R. [ABSTRACT FROM AUTHOR]

Low-rank matrix models have been universally useful for numerous applications, from classical system identification to more modern matrix completion in signal processing and statistics. The nuclear norm has been employed as a convex surrogate of the low-rankness since it induces a low-rank solution to inverse problems. While the nuclear norm for low rankness has an excellent analogy with the |$\ell _1$| norm for sparsity through the singular value decomposition, other matrix norms also induce low-rankness. Particularly as one interprets a matrix as a linear operator between Banach spaces, various tensor product norms generalize the role of the nuclear norm. We provide a tensor-norm-constrained estimator for the recovery of approximately low-rank matrices from local measurements corrupted with noise. A tensor-norm regularizer is designed to adapt to the local structure. We derive statistical analysis of the estimator over matrix completion and decentralized sketching by applying Maurey's empirical method to tensor products of Banach spaces. The estimator provides a near-optimal error bound in a minimax sense and admits a polynomial-time algorithm for these applications. [ABSTRACT FROM AUTHOR]

The study of algebraic structures endowed with the concept of symmetry is made possible by the link between Lie algebras and symmetric monoidal categories. This relationship between Lie algebras and symmetric monoidal categories is useful and has resulted in many areas, including algebraic topology, representation theory, and quantum physics. In this paper, we present analogous definitions for Lie algebras within the framework of whiskered structures, bimorphisms, crossed complexes, crossed differential graded algebras, and tensor products. These definitions, given for groupoids in existing literature, have been adapted to establish a direct correspondence between these algebraic structures and Lie algebras. We show that a 2-truncation of the crossed differential graded Lie algebra, obtained from our adapted definitions, gives rise to a braided crossed module of Lie algebras. We also construct a functor to simplicial Lie algebras, enabling a systematic mapping between different Lie algebraic categories, which supports the validity of our adapted definitions and establishes their compatibility with established categories. [ABSTRACT FROM AUTHOR]

In this paper, all simple Yetter-Drinfeld modules and indecomposable projective Yetter-Drinfeld modules over the 2 -rank Taft algebra A ¯ are construted and classified by Radford's method of constructing Yetter-Drinfeld modules over a Hopf algebra. Furthermore, the projective class ring of the category of Yetter-Drinfeld modules over A ¯ is described explicitly by generators and relations. [ABSTRACT FROM AUTHOR]

Explaining the foundation of cognitive abilities in the processing of information by neural systems has been in the beginnings of biophysics since McCulloch and Pitts pioneered work within the biophysics school of Chicago in the 1940s and the interdisciplinary cybernetists meetings in the 1950s, inseparable from the birth of computing and artificial intelligence. Since then, neural network models have traveled a long path, both in the biophysical and the computational disciplines. The biological, neurocomputational aspect reached its representational maturity with the Distributed Associative Memory models developed in the early 70 s. In this framework, the inclusion of signal-signal multiplication within neural network models was presented as a necessity to provide matrix associative memories with adaptive, context-sensitive associations, while greatly enhancing their computational capabilities. In this review, we show that several of the most successful neural network models use a form of multiplication of signals. We present several classical models that included such kind of multiplication and the computational reasons for the inclusion. We then turn to the different proposals about the possible biophysical implementation that underlies these computational capacities. We pinpoint the important ideas put forth by different theoretical models using a tensor product representation and show that these models endow memories with the context-dependent adaptive capabilities necessary to allow for evolutionary adaptation to changing and unpredictable environments. Finally, we show how the powerful abilities of contemporary computationally deep-learning models, inspired in neural networks, also depend on multiplications, and discuss some perspectives in view of the wide panorama unfolded. The computational relevance of multiplications calls for the development of new avenues of research that uncover the mechanisms our nervous system uses to achieve multiplication. [ABSTRACT FROM AUTHOR]

A cubic fuzzy graph is a fuzzy graph that simultaneously supports fuzzy membership and interval-valued fuzzy membership. This simultaneity leads to a better flexibility in modeling problems regarding uncertain variables. The cubic fuzzy graph structure, as a combination of cubic fuzzy graphs and graph structures, shows better capabilities in solving complex problems, especially where there are multiple relationships. Since many problems are a combination of different relationships, as well, applying some operations on them creates new problems; therefore, in this article, some of the most important product operations on cubic fuzzy graph structure have been investigated and some of their properties have been described. Studies have shown that the product of two strong cubic fuzzy graph structures is not always strong and sometimes special conditions are needed to be met. By calculating the vertex degree in each of the products, a clear image of the comparison between the vertex degrees in the products has been obtained. Also, the relationships between the products have been examined and the investigations have shown that the combination of some product operations with each other leads to other products. At the end, the cubic fuzzy graph structure application in the diagnosis of brain lesions is presented. [ABSTRACT FROM AUTHOR]

In this paper, we characterize when, for any infinite cardinal α , the Fremlin tensor product of two Archimedean Riesz spaces (see Fremlin in Am J Math 94:777–798, 1972) is Dedekind α -complete. We also provide an example of an ideal I in an Archimedean Riesz space E such that the Fremlin tensor product of I with itself is not an ideal in the Fremlin tensor product of E with itself. [ABSTRACT FROM AUTHOR]

We consider the family P of n-tuples P consisting of polynomials P 1 , ... , P n with nonnegative coefficients, which satisfy ∂ i P j (0) = δ i , j , i , j = 1 , ... , n. With any such P, we associate a Reinhardt domain ▵ P n that we call the generalized Hartogs triangle. We are particularly interested in the choices P a = (P 1 , a , ... , P n , a) , a ⩾ 0 , where P j , a (z) = z j + a ∏ k = 1 n z k , j = 1 , ... , n. The generalized Hartogs triangle associated with P a is given by: ▵ a n = { z ∈ C × C ∗ n - 1 : | z j | 2 < | z j + 1 | 2 (1 - a | z 1 | 2) , j = 1 , ... , n - 1 , | z n | 2 + a | z 1 | 2 < 1 }. The domain ▵ 0 2 is the Hartogs triangle. Unlike most domains relevant to the multi-variable operator theory, the domain ▵ P n , n ⩾ 2 , is never polynomially convex. However, ▵ P n is always holomorphically convex. With any P ∈ P and m ∈ N n , we associate a positive semi-definite kernel K P , m on ▵ P n. This, combined with the Moore's theorem, yields a reproducing kernel Hilbert space H m 2 (▵ P n) of holomorphic functions on ▵ P n. We study the space H m 2 (▵ P n) and the multiplication n-tuple M z acting on H m 2 (▵ P n). It turns out that M z is never rationally cyclic, but H m 2 (▵ P n) admits an orthonormal basis consisting of rational functions on ▵ P n. Although the dimension of the joint kernel of M z ∗ - λ is constant of value 1 for every λ ∈ ▵ P n , it has jump discontinuity at the serious singularity 0 of the boundary of ▵ P n with the value equal to ∞. We capitalize on the notion of joint subnormality to define a Hardy space H 2 (▵ 0 n) on the n-dimensional Hartogs triangle ▵ 0 n. This in turn gives an analog of the von Neumann's inequality for ▵ 0 n. [ABSTRACT FROM AUTHOR]

In this paper, we give several examples of n-perinormal operators for each n ≥ 3 such as (1) n-perinormal whose restriction to its invariant subspace is not n-perinormal, (2) n-perinormal which is not (n - 1)-perinormal and (3) an invertible n-perinormal operator whose inverse is not n-perinormal. There are several papers studying n-perinormal operators which are using the assertions that a restriction of n-perinormal operator to its invariant subspace, the inverse of n-perinormal operator is also n-perinormal even if n ≥ 3. We remark that if n = 2 then 2-perinormal is equal to quasihyponormal, and since a restriction of quasihyponormal to any invariant subspace is always quasihyponormal, so it is 2-perinormal. And every invertible 2-perinormal is invertible hyponormal, so the inverse of it is also hyponormal and 2-perinormal. We also show that Weyl's theorem holds for every n-perinormal and some results related to the Riesz idempotent of n-perinormal. Moreover, We study fundamental structural characteristics of class of (n; k)-quasiperinormal operators. Also, we show that, if T is (n; k)-quasiperinormal, then T -≥ has finite ascent for all λ ≥ C. Further, we give a necessary and sufficient condition for T S to be in a class of (n; k)-quasiperinormal. [ABSTRACT FROM AUTHOR]

The aim of this paper is to illustrate how degree sequences may successfully be used over some graph products. Moreover, by taking into account the degree sequence, we will expose some new distinguishing results on special graph products. We will first consider the degree sequences of tensor and cartesian products of graphs and will obtain the omega invariant of them. After that we will conclude that the set of graphs forms an abelian semigroup in the case of tensor product whereas this same set is actually an abelian monoid in the case of cartesian product. As a consequence of these two operations, we also give a result on distributive law which would be important for future studies. [ABSTRACT FROM AUTHOR]

The two well-known numerical radius inequalities for the tensor product A ⊗ B acting on H ⊗ K , where A and B are bounded linear operators defined on complex Hilbert spaces H and K , respectively are 1 2 ‖ A ‖ ‖ B ‖ ≤ w (A ⊗ B) ≤ ‖ A ‖ ‖ B ‖ and w (A) w (B) ≤ w (A ⊗ B) ≤ min { w (A) ‖ B ‖ , w (B) ‖ A ‖ }. In this article, we develop new lower and upper bounds for the numerical radius w (A ⊗ B) of the tensor product A ⊗ B and study the equality conditions for those bounds. [ABSTRACT FROM AUTHOR]

Let E be a Banach space. For a topological space X, let C b (X , E) be the space of all bounded continuous E-valued functions on X, and let C K (X , E) be the subspace of C b (X , E) consisting of all functions having a pre-compact image in E. We show that C K (X , E) is isometrically isomorphic to the injective tensor product C b (X) ⊗ ^ ε E , and that C b (X , E) = C b (X) ⊗ ^ ε E if and only if E is finite dimensional. Next, we consider the space Lip(X, E) of E-valued Lipschitz operators on a metric space (X, d) and its subspace LipK(X, E) of Lipschitz compact operators. Utilizing the results on C b (X , E) , we prove that LipK(X, E) is isometrically isomorphic to a tensor product Lip (X) ⊗ ^ α E , and that Lip (X , E) = Lip (X) ⊗ ^ α E if and only if E is finite dimensional. Finally, we consider the space D1(X, E) of continuously differentiable functions on a perfect compact plane set X and show that, under certain conditions, D1(X, E) is isometrically isomorphic to a tensor product D 1 (X) ⊗ ^ β E . [ABSTRACT FROM AUTHOR]

We introduce the notion of continuous frame in n-Hilbert space which is a generalization of discrete frame in n-Hilbert space. The tensor product of Hilbert spaces is a very important topic in mathematics. Here we also introduce the concept of continuous frame for the tensor products of n-Hilbert spaces. Further, we study dual continuous frame and continuous Bessel multiplier in n-Hilbert spaces and their tensor products. [ABSTRACT FROM AUTHOR]

In this paper, all simple Yetter-Drinfeld modules and indecomposable projective Yetter-Drinfeld modules over a family of non-pointed 8m-dimension Hopf algebras of tame type with rank two, are construted and classified. The technique is Radford's method of constructing Yetter-Drinfeld modules over a Hopf algebra. Furthermore, the projective class rings of the category of Yetter-Drinfeld modules over this class of Hopf algebras are described explicitly by generators and relations. [ABSTRACT FROM AUTHOR]

We show that the Carathéodory space of place functions on the free product of two Boolean algebras is Riesz isomorphic with Fremlin's Archimedean Riesz space tensor product of their respective Carathéodory spaces of place functions. We provide a solution to Fremlin's problem 315Y(f) [ Measure Theory , Torres Fremlin, Colchester, 2004] concerning completeness in the free product of Boolean algebras by applying our results on the Archimedean Riesz space tensor product to Carathéodory spaces of place functions. [ABSTRACT FROM AUTHOR]