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In graph theory, graph labeling is an essential area of study because labeled graphs offer useful mathematical models for coding theory, cryptography, astronomy, radar, database administration and communication networks. Consider a bijection for a graph G of order n, f : V (G) → {1, 2, ..., n}. The weight of a vertex z of G, expressed as w(z), is defined as the sum of labels assigned to all vertices adjacent to vertex z in G. If the weights are distinct for every unique pair of vertices y, z in V (G), then the labeling f is referred to as distance antimagic. A distance antimagic graph is any graph G that accepts such a labeling. Distance antimagic labeling on various basic graph products are discussed in this paper. We explore results on (a, d)-distance antimagic labeling for the lexicographic product G * H and distance antimagic labeling for the cartesian product G * H, tensor product G x H and strong product G x H in this work, where the graphs G and H are cycle related graphs, paths or complete graphs. Also, computer-aided algorithms are designed to verify that vertex weights are distinct. [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]

A shortest [Formula: see text]-[Formula: see text] path between two vertices u and v of a graph G is a [Formula: see text]-[Formula: see text] geodesic of G. Let I[u, v] denote the set of all internal vertices lying on some [Formula: see text]-[Formula: see text] geodesic of G. For a nonempty subset S of [Formula: see text], let [Formula: see text]. If [Formula: see text], then S is a geodetic set of G. The cardinality of a minimum geodetic set of G is the geodetic number of G and it is denoted by [Formula: see text] In this paper, the exact geodetic numbers of the product graphs [Formula: see text] and [Formula: see text] are obtained, where T is a tree, [Formula: see text] denotes the complement of the complete graph [Formula: see text] and, [Formula: see text] and [Formula: see text] denote the tensor product and lexicographic product $($also called the wreath product$)$ of graphs, respectively.

In this paper, we aim to investigate the class of jointly hyponormal operators related to a positive operator $ A $ on a complex Hilbert space $ \mathcal{X} $, which is called jointly $ A $-hyponormal. This notion was first introduced by Guesba et al. in [Linear and Multilinear Algebra, 69(15), 2888–2907] for $ m $-tuples of operators that admit adjoint operators with respect to $ A $. Mainly, we prove that if $ \mathbf{B} = (B_1, \cdots, B_m) $ is a jointly $ A $-hyponormal $ m $-tuple of commuting operators, then $ \mathbf{B} $ is jointly $ A $-normaloid. This result allows us to establish, for a particular case when $ A $ is the identity operator, a sharp bound for the distance between two jointly hyponormal $ m $-tuples of operators, expressed in terms of the difference between their Taylor spectra. We also aim to introduce and investigate the class of spherically $ A $-$ p $-hyponormal operators with $ 0 < p < 1 $. Additionally, we study the tensor product of specific classes of multivariable operators in semi-Hilbert spaces.

Molecular descriptors are numerical graph invariants that are used to study the chemical structure of molecules. In this paper, we determine the upper bound of the Sombor index based on four operations involving the subdivision graph, semi-total point graph, semi-total line graph, and total graph related to the lexicographic and tensor product. The exact expressions of the first reformulated Zagreb index and the second hyper-Zagreb index of the tensor product are formulated on the basis of the four significant graphs. Further, the descriptors for certain standard graphs are obtained and the graphical comparison for the first reformulated Zagreb index has also been illustrated to understand the result better.

We consider the tensor products of square matrices of different sizes and introduce the stretching maps, which can be viewed as a generalized matricization. Stretching maps conserve algebraic properties of the tensor product, but are not necessarily injective. Dropping the injectivity condition allows us to construct examples of stretching maps with additional symmetry properties. Furthermore, this leads to the averaging of the tensor product and possibly could be used to compress the data.

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 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]

AbstractIn this paper, we discuss the A-vertex magicness of some products of graphs, where A is a non-trivial additive Abelian group with identity 0. We characterize A-vertex magicness of the Cartesian product of Pn and Pm, where A has at least three elements, and we also discuss the A -vertex magicness of co-normal and tensor product of two graphs. Finally, we give a new method to construct an infinite number of A-vertex magic graphs from existing ones using the join operation. As a consequence we obtain an earlier result in [Balamoorthy et al. in AKCE Int. J. Graphs Comb. (2022) 19 (3), 268–275]

Let $X$ be a Banach space, $\mathcal{B}(X)$ the algebra of bounded linear operators on $X$ and $(J, \|{\cdot}\|_J)$ an admissible Banach ideal of $\mathcal{B}(X)$. For $T\in\mathcal{B}(X)$, let $L_{J, T}$ and $R_{J, T}\in\mathcal{B}(J)$ denote the left and right multiplication defined by $L_{J, T}(A)=TA$ and $R_{J, T}(A)=AT$, respectively. In this paper, we study the transmission of some concepts related to recurrent operators between $T\in\mathcal{B}(X)$, and their elementary operators $L_{J, T}$ and $R_{J, T}$. In particular, we give necessary and sufficient conditions for $L_{J, T}$ and $R_{J, T}$ to be sequentially recurrent. Furthermore, we prove that $L_{J, T}$ is recurrent if and only if $Tøplus T$ is recurrent on $Xøplus X$. Moreover, we introduce the notion of a mixing recurrent operator and we show that $L_{J, T}$ is mixing recurrent if and only if $T$ is mixing recurrent.

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]

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]

We study projective and injective tensor products of Banach L 0 -modules over a σ -finite measure space. En route, we extend to Banach L 0 -modules several technical tools of independent interest, such as quotient operators, summable families, and Schauder bases. [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]

The state of moderate and severe food insecurity in Ethiopia has not been significantly reduced for a long time due to cultural, natural, and manmade shocks, which cover most part of the country with considerable magnitude and have adverse effects on the health and economy. This temporal evolution of the wider geographic distribution of the food insecurity levels has not been investigated for the feeding culture and shocks effect in Ethiopia, though previous studies have indicated significant geographic distribution and related factors. In addition, the longtime zone-specific comprehensive drivers were not assessed. Therefore, this study focused on investigating the feeding culture-adjusted food insecurity levels (FCSL) and their evolutional sustainability across zones by identifying the factors causing each level of household's food insecurity tailored to a specific zone.In this study, Ethiopian socio-economic longitudinal data from years 2012, 2014, and 2016 with a sample size of 3835 households were analyzed. An ordinal spatiotemporal model with different interactions under empirical Bayes estimation was adopted, and the type III interaction with Markov random field was selected to reveal the evolution of FCSL sustainability across zones and find-out the causing factors of households to each level of food insecurity tailored to zone-level by analyzing the effects of diverse factors and shocks. The result reveals that a greater portion (52.07 %) of households’ population is moderately and chronically food insecure. Basically, households living in neighboring zones have significantly similar food insecurity levels, and slight improvements were observed over time. This transition over time within neighboring zones, revealed that chronic food insecurity in neighboring zones has transited to moderate food insecurity, particularly in most of the northern and southwestern areas of Ethiopia. However, the interaction term showed that the change in food insecurity levels for households living in neighboring zones is similar at one time point, but the evolution is not sustainable. Therefore, working on major zone-specific factors, such as enhancing zone-level urbanization through improving market and road linkage to rural areas, education, employment, non-agricultural businesses, and promoting integrated farming by conserving soil with consideration of additional confounding factors, will bring change and can sustainably eradicate moderate and chronic food insecurity in zone households through controlling dependency ratios, shocks, and small land size farming. This study brings resilient, manageable, and zone-specific mitigation for households living in food-insecure and vulnerable zones.

The bicategory of Landau-Ginzburg models denoted by ℒ𝒢K possesses adjoints and this helps in explaining a certain duality that exists in the setting of Landau-Ginzburg models in terms of some specified relations. The construction of ℒ𝒢K is reminiscent of, but more complex than, the construction of the bicategory of associative algebras and bimodules. In this paper, we review this complex but very inspiring construction in order to expose it more to pure mathematicians. In particular, we spend some time explaining the intricate construction of unit morphisms in this bicategory from a new vantage point. Besides, we briefly discuss how this bicategory could be constructed in more than one way using the variants of the Yoshino tensor product. Furthermore, without resorting to Atiyah classes, we prove that the left and right unitors in this bicategory have direct right inverses but do not have direct left inverses.

AbstractFor a bounded linear operator, by local spectral theory methods, we study the property [Formula: see text] 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 [Formula: see text] 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.

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]

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]

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]

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]

The use of classical methods to solve some fractional partial differential equations, usually, leads to dead ends with no possible solutions as the separation of variables. Hence, in this paper, we use the theory of tensor product of Banach spaces to find certain exact solutions, called atomic solutions which will give us more than one solution to the proposed problems. As an application, we will find the all possible solutions for some fractional partial differential Burger’s type equations and see the behavior of these solutions.

In this paper, all simple Yetter-Drinfeld modules and indecomposable projective Yetter-Drinfeld modules over the $ 2 $-rank Taft algebra $ \mathcal{\bar{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 $ \mathcal{\bar{A}} $ is described explicitly by generators and relations.

BackgroundFood insecurity and vulnerability in Ethiopia are historical problems due to natural- and human-made disasters, which affect a wide range of areas at a higher magnitude with adverse effects on the overall health of households. In Ethiopia, the problem is wider with higher magnitude. Moreover, this geographical distribution of this challenge remains unexplored regarding the effects of cultures and shocks, despite previous case studies suggesting the effects of shocks and other factors. Hence, this study aims to assess the geographic distribution of corrected-food insecurity levels (FCSL) across zones and explore the comprehensive effects of diverse factors on each level of a household's food insecurity.MethodThis study analyzes three-term household-based panel data for years 2012, 2014, and 2016 with a total sample size of 11505 covering the all regional states of the country. An extended additive model, with empirical Bayes estimation by modeling both structured spatial effects using Markov random field or tensor product and unstructured effects using Gaussian, was adopted to assess the spatial distribution of FCSL across zones and to further explore the comprehensive effect of geographic, environmental, and socioeconomic factors on the locally adjusted measure.ResultDespite a chronological decline, a substantial portion of Ethiopian households remains food insecure (25%) and vulnerable (27.08%). The Markov random field (MRF) model is the best fit based on GVC, revealing that 90.04% of the total variation is explained by the spatial effects. Most of the northern and south-western areas and south-east and north-west areas are hot spot zones of food insecurity and vulnerability in the country. Moreover, factors such as education, urbanization, having a job, fertilizer usage in cropping, sanitation, and farming livestock and crops have a significant influence on reducing a household's probability of being at higher food insecurity levels (insecurity and vulnerability), whereas shocks occurrence and small land size ownership have worsened it.ConclusionChronically food insecure zones showed a strong cluster in the northern and south-western areas of the country, even though higher levels of household food insecurity in Ethiopia have shown a declining trend over the years. Therefore, in these areas, interventions addressing spatial structure factors, particularly urbanization, education, early marriage control, and job creation, along with controlling conflict and drought effect by food aid and selected coping strategies, and performing integrated farming by conserving land and the environment of zones can help to reduce a household's probability of being at higher food insecurity levels.

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]

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]

AbstractSome of the most comprehensively studied degree-based topological indices are the Zagreb indices. In this article, the pair of Zagreb indices have been determined for five product graphs namely tensor product, Cartesian product, lexicographic product, strong product, symmetric difference of G1 and G2 in terms of the Zagreb indices of the signed graphs of the factor graphs G1 and G2 and their orders and sizes.

Given Banach space operators Si{S}_{i} and Ti{T}_{i}, i=1,2i=1,2, we use elementary properties of the left and right multiplication operators to prove, that if the tensor products pair (S1⊗S2,T1⊗T2)\left({S}_{1}\otimes {S}_{2},{T}_{1}\otimes {T}_{2}) is strictly mm-isometric, i.e., ΔS1⊗S2,T1⊗T2m(I⊗I)=∑j=0m(−1)jmj(S1⊗S2)m−j(T1⊗T2)m−j=0{\Delta }_{{S}_{1}\otimes {S}_{2},{T}_{1}\otimes {T}_{2}}^{m}\left(I\otimes I)={\sum }_{j=0}^{m}{\left(-1)}^{j}\left(\begin{array}{c}m\\ j\end{array}\right){\left({S}_{1}\otimes {S}_{2})}^{m-j}{\left({T}_{1}\otimes {T}_{2})}^{m-j}=0, then there exist a non-zero scalar cc and positive integers m1,m2≤m{m}_{1},{m}_{2}\le m such that m=m1+m2−1m={m}_{1}+{m}_{2}-1, (S1,cT1)\left({S}_{1},c{T}_{1}) is strict-m1{m}_{1}-isometric and S2,1cT2\left({S}_{2},\frac{1}{c}{T}_{2}\right) is strict m2{m}_{2}-isometric.

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.

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.

In this article, we find an α-valuation for disjoint union of some bipartite graphs and the tensor product of paths and even cycles.

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]

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]

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]

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]

Given Banach space operators Ai, Bi (i = 1, 2), let δi denote (the generalised derivation) δi(X) = (LAi − RBi )(X) = AiX − XBi. If 0 ∈ σa(Bi), i = 1, 2, and if Δδ1,δ2n(I)=(Lδ1Rδ1-I)n(I)=0\Delta _{{\delta _1},\delta 2}^n\left( I \right) = {\left( {{L_{{\delta _1}}}{R_{{\delta _1}}} - I} \right)^n}\left( I \right) = 0, then ΔA1,A2n(I)=0\Delta _{{A_1},A2}^n\left( I \right) = 0. For Hilbert space pairs (A, B) such that 0 ∈ σa(B*) and Δδ*,δn(I)=0(i.e., δ is n-isometric)\Delta _{{\delta ^*},\delta }^n\left( I \right) = 0\left( {i.e.,\,\delta \,is\,n - isometric} \right), where δ= δA,B and δ* = δA* ,B*, this implies ΔA*,An(I)=0\Delta _{{A^*},A}^n\left( I \right) = 0 (and hence there exists a positivie integer m ≤ n such that A is strictly m-isometric). If Δδ*,δn(I)=0\Delta _{{\delta ^*},\delta }^n\left( I \right) = 0, then there exists a scalar λ such that 0 ∈ σa((B − λI)*) and, given δ is strictly n-isometric, there exists a positive integer m ≤ n such that A − λI is strictly m-isometric. Furthermore, there exist decompositions ℋ = ℋ1 ⊕ ℋ2 and ℋ = ℋ11 ⊕ ℋ22 of ℋ and ti-nilpotent operators Ni (i = 1, 2) such that either A − λI = αI + N1 and B − λI = (0I|ℋ1 ⊕ 2eit I|ℋ2 ) + N2, or, A − λI = αI + N1, α = eit, 0 ≤ t < 2π, and B − λI = (0I|ℋ11 ⊕ 2eit I|ℋ22 ) + N2, or, A − λ = (α1I|ℋ1 ⊕ α2I|ℋ2 ) + N1 and Bλ= (0I|ℋ11 ⊕ μI|ℋ22 ) + N2, where μ = eit|μ|, 0 ≤ t < 2π, 0 < |μ| < 2, α1=eit|μ|+i4-|μ|22{\alpha _1} = {e^{it}}{{\left| \mu \right| + i\sqrt {4 - {{\left| \mu \right|}^2}} } \over 2} and α2=eit|μ|-i4-|μ|22{\alpha _2} = {e^{it}}{{\left| \mu \right| - i\sqrt {4 - {{\left| \mu \right|}^2}} } \over 2}.

Algebras of distributions of binary isolating and semi-isolating formulae are derived objects for a given theory and reflect binary formula relations between 1-type realizations. These algebras are related to the following natural classification questions: 1) for a given class of theories, determine which algebras correspond to theories from that class, and classify those algebras; 2) classify theories from the class according to the isolating and semi-isolating formulae algebras defined by those theories. The description of a finite algebra of binary isolating formulas unambiguously implies the description of an algebra of binary semi-isolating formulas, which makes it possible to trace the behavior of all binary formula relations of a given theory. The paper describes algebras of binary formulas for tensor products. The Cayley tables are given for the obtained algebras. Based on these tables, theorems are formulated describing all algebras of binary formulae distributions for tensor multiplication theory of regular polygons on an edge. It is shown that they are completely described by two algebras

AbstractLet [Formula: see text] be a simple undirected graph and let A be an additive Abelian group with identity 0. A mapping [Formula: see text] is said to be a A-vertex magic labeling of G if there exists a μ in A such that [Formula: see text] for any vertex v of G. If G admits such a labeling, then it is called an A-vertex magic graph. If G is A-vertex magic for any non-trivial Abelian group A, then G is called a group vertex magic graph. In this paper, we consider A-vertex magic and group vertex magic labeling of different products of graphs.

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]

We consider a functional regression model in the framework of reproducing kernel Hilbert spaces, where the interaction effect of two functional predictors, as well as their main effects, over the functional response is of interest. The regression component of our model is expressed by one trivariate coefficient function, the functional ANOVA decomposition of which yields the main and interaction effects. The trivariate coefficient function is estimated by optimizing a penalized least squares objective with a roughness penalty on the function estimate. The estimation procedure can be implemented easily using standard numerical tools. Asymptotic results for the proposed model, with or without functional measurement errors, are established under the reproducing kernel Hilbert space (RKHS) framework. Extensive numerical studies show the advantages of the proposed method over existing methods in terms of the prediction and estimation of the coefficient functions. An application to the histone modifications and gene expressions of a liver cancer cell line further demonstrates the better prediction accuracy of the proposed method over that of its competitors. [ABSTRACT FROM AUTHOR]

The generalized resolution was introduced and justified as a criterion for selecting nonregular factorial designs. Although there has been extensive research conducted on other aspects of nonregular designs, few works have investigated the construction of nonregular designs with maximum generalized resolutions, as we do in this study. To date, our knowledge of nonregular designs with maximum generalized resolutions is predominantly computational, except for very few theoretical results. We derive lower bounds on relevant J-characteristics and present the construction results. With the assistance of the lower bounds, many of the constructed designs are shown to have maximum generalized resolutions. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with studying a new class of multivariable commuting operators know as left ( m , C ) $(m,C)$ -invertible p-tuples and related to a given conjugation transformation C on a Hilbert space Y $\mathcal{Y}$ . Some structural properties of some members of this class are given.

This paper presents a polytopic linear parameter varying (LPV) controller design methodology for nonlinear anti-vibration suspension based on stochastic road identification. As the main nonlinear component in semi-active suspension, the continuous damping control (CDC) damper value is mapped by damping velocity and control current through a bench experiment. linear parameter varying gain-scheduled method is applied for dealing with the nonlinear deterministic model. As the uncertainties caused by road stochastic excitation to suspension cannot be ignored, it is essential to identify the excitation signal as a measurable variable parameter. Empirical-mode-decomposition (EMD) and support vector machine (SVM) are utilized to identify and classify the road stochastic signal input into standard level, so polytopic LPV controller can be designed with CDC damper velocity, control current, and road level as variable parameters. Meanwhile the exponential increase of high-order plant elements in multi-parameter LPV model makes it too complicated for real-time computing. Hence the tensor product model transformation is utilized to identify the orthogonal basis of system plant so that low-rank approximation of it can be implemented by truncated high-order singular value decomposition (T−HOSVD) method. Real-time simulation shows the performance and feasibility of controller, which can efficaciously identify the road stochastic excitation as measurable variable parameters for nonlinear suspension polytopic LPV control. [ABSTRACT FROM AUTHOR]

Recent developments in commutative algebra, linear algebra, and graph theory allow us to approach various issues in several fields. Circulant graphs now have a wider range of practical uses, including as the foundation for optical networks, discrete cellular neural networks, small-world networks, models of chemical reactions, supercomputing and multiprocessor systems. Herein, we are concerned with the decompositions of the bipartite circulant graphs. We propose the Cartesian and tensor product approaches as helping tools for the decompositions. The proposed approaches enable us to decompose the bipartite circulant graphs into many categories of graphs. We consider the use cases of applying the described theory of bipartite circulant graph decomposition to the problems of finding new topologies and deadlock-free routing in them when building supercomputers and networks-on-chip. [ABSTRACT FROM AUTHOR]