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Twisted generalized Weyl algebras (TGWAs) A(R,σ,t) are defined over a base ring R by parameters σ and t, where σ is an n-tuple of automorphisms, and t is an n-tuple of elements in the center of R. We show that, for fixed R and σ, there is a natural algebra map A (R , σ , t t ′) → A (R , σ , t) ⊗ R A (R , σ , t ′) . This gives a tensor product operation on modules, inducing a ring structure on the direct sum (over all t) of the Grothendieck groups of the categories of weight modules for A(R,σ,t). We give presentations of these Grothendieck rings for n = 1,2, when R = ℂ [ z ] . As a consequence, for n = 1, any indecomposable module for a TGWA can be written as a tensor product of indecomposable modules over the usual Weyl algebra. In particular, any finite-dimensional simple module over s l 2 is a tensor product of two Weyl algebra modules. [ABSTRACT FROM AUTHOR]

Pavarino proved that the additive Schwarz method with vertex patches and a low-order coarse space gives a p-robust solver for symmetric and coercive problems [Pavanno, Numer. Math., 66 (1993), pp. 493-515]. However, for very high polynomial degree it is not feasible to assemble or factorize the matrices for each patch. In this work we introduce a direct solver for separable patch problems that scales to very high polynomial degree on tensor product cells. The solver constructs a tensor product basis that diagonalizes the blocks in the stiffness matrix for the internal degrees of freedom of each individual cell. As a result, the nonzero structure of the cell matrices is that of the graph connecting internal degrees of freedom to their projection onto the facets. In the new basis, the patch problem is as sparse as a low-order finite difference discretization, while having a sparser Cholesky factorization. We can thus afford to assemble and factorize the matrices for the vertex-patch problems, even for very high polynomial degree. In the nonseparable case, the method can be applied as a preconditioner by approximating the problem with a separable surrogate. We demonstrate the approach by solving the Poisson equation and a H(div)-conforming interior penalty discretization of linear elasticity in three dimensions at p = 15. [ABSTRACT FROM AUTHOR]

An operator T acting on a Banach space X obeys property (t) if the isolated points of the spectrum σ(T) of T which are eigenvalues of finite multiplicity are exactly those points λ of the spectrum for which T −λ is an upper semi-Fredholm with index less than or equal to 0. In the present paper we examine the stability of property (t) under perturbations. We show that if T is an isoloid operator on a Banach space, that obeys property (t), and F is a bounded operator that commutes with T and for which there exists a positive integer n such that Fn is finite rank, then T + F obeys property (t). Further, we establish that if T is finite-isoloid, then property (t) is transmitted from T to T + R, for every Riesz operator R commuting with T. Property (t) does not transfer from operators T and S to their tensor product T ⊗S; we give necessary and/or sufficient conditions ensuring the passage of property (t) from T and S to T ⊗ S. Moreover, Perturbations by Riesz operators are considered. [ABSTRACT FROM AUTHOR]

Gaussian processes (GPs) are common components in Bayesian non‐parametric models having a rich methodological literature and strong theoretical grounding. The use of exact GPs in Bayesian models is limited to problems containing several thousand observations due to their prohibitive computational demands. We develop a posterior sampling algorithm using H‐matrix approximations that scales at O(nlog2n). We show that this approximation's Kullback–Leibler divergence to the true posterior can be made arbitrarily small. Although multidimensional GPs could be used with our algorithm, d‐dimensional surfaces are modelled as tensor products of univariate GPs to minimize the cost of matrix construction and maximize computational efficiency. We illustrate the performance of this fast increased fidelity approximate GP, FIFA‐GP, using both simulated and non‐synthetic data sets. [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.

This paper is concerned with studying a new class of multivariable commuting operators know as left (m , C) -invertible p-tuples and related to a given conjugation transformation C on a Hilbert space Y . Some structural properties of some members of this class are given. [ABSTRACT FROM AUTHOR]

Let A = f1; 2; 3; ...; ng and Pk(A) denotes the set of all k-element subsets of A. The Kneser graph KGn,2 has the vertex set V (KGn,2)= P2(A) and edge set E(KGn,2) = {XY | X, Y ∈ P2(A) and X Ո Y = ∅ }. A star with k edges is denoted by Sk. In this paper, we show that the graph KGn,2 can be decomposed into S5 if and only if n ≥ 7 and n ≡ 0; 1; 2; 3(mod 5). [ABSTRACT FROM AUTHOR]

For two given graphs, we construct an augmentation graph. We obtain an injective algebra homomorphism from the Leavitt path algebra of the augmentation graph to the tensor product of the Leavitt path algebras of the given graphs, and give a necessary and sufficient condition on the surjectivity of the homomorphism. This gives rise to a realization of the tensor product of two Leavitt path algebras as a Leavitt path algebra. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the dominion of any full orthodox subsemigroup of a medial orthodox semigroup is described by the Isbell zigzag theorem in the category of medial orthodox semigroups. As a consequence, the dominions of any full completely regular subsemigroup of a medial completely regular semigroup as well as that of any full Clifford subsemigroup of a medial Clifford semigroup are also described by the Isbell zigzag theorem in the categories of all medial completely regular semigroups and of all medial Clifford semigroups, respectively. [ABSTRACT FROM AUTHOR]

If R is a commutative unital ring and M is a unital R -module, then each element of End R (M) determines a left End R (M) [ X ] -module structure on End R (M) , where End R (M) is the R -algebra of endomorphisms of M and End R (M) [ X ] = End R (M) ⊗ R R [ X ]. These structures provide a very short proof of the Cayley-Hamilton theorem, which may be viewed as a reformulation of the proof in Algebra by Serge Lang. Some generalisations of the Cayley-Hamilton theorem can be easily proved using the proposed method. [ABSTRACT FROM AUTHOR]

This paper investigates the Sylvester-transpose matrix equation A ⋉ X + X T ⋉ B = C , where all mentioned matrices are over an arbitrary field. Here, ⋉ is the semi-tensor product, which is a generalization of the usual matrix product defined for matrices of arbitrary dimensions. For matrices of compatible dimensions, we investigate criteria for the equation to have a solution, a unique solution, or infinitely many solutions. These conditions rely on ranks and linear dependence. Moreover, we find suitable matrix partitions so that the matrix equation can be transformed into a linear system involving the usual matrix product. Our work includes the studies of the equation A ⋉ X = C , the equation X ⋉ B = C , and the classical Sylvester-transpose matrix equation. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with unity, M be a unitary R -module and Γ be a simple connected graph. We examine different equivalence relations on subsets A f (M) \ { 0 } , A s (M) \ { 0 } and A t (M) \ { 0 } of M , where A f (M) is the set of full-annihilators, A s (M) is the set of semi-annihilators and A t (M) is the set of star-annihilators in M. We prove that elements x , y ∈ M are neighborhood similar in the annihilating graph ann f (Γ (M)) if and only if the submodules ann (x) M and ann (y) M of M are equal. We study the isomorphism of annihilating graphs arising from M and the tensor product M ⊗ R T − 1 R , where T = R \ C (M) , C (M) = { r ∈ R   ∣   r m = 0 for some 0 ≠ m ∈ M }. [ABSTRACT FROM AUTHOR]

In the analysis of longitudinal or functional data, a time‐varying additive model (tvAM) has been introduced that is effective at avoiding the curse of dimensionality and capturing dynamic features. The present article focuses on the unified two‐step estimators of a tvAM with sparse or dense longitudinal or functional data. It is proved that the two‐step estimators have the same asymptotic distribution as that of oracle estimators. Furthermore, a unified convergence theory is established, based on which a unified inference is proposed without deciding whether the data are sparse or dense. Also, a testing statistic that can adapt to the sparse and dense cases in a unified framework is proposed to check whether the bivariate nonparametric functions are time varying, and the asymptotic distribution of the proposed test statistic is derived. Simulation studies are conducted to assess the finite‐sample performance of the proposed model and methods, and two different types of data are considered to illustrate the proposed method. [ABSTRACT FROM AUTHOR]

The performance of a wide area damping controller (WADC) can be affected considerably if a photovoltaic (PV) power plant is integrated with the power system. A WADC using state feedback control law is proposed in this paper to eliminate the frequency oscillation of interconnected power system integrated with solar PV cell. Output power of the solar PV cell varies with irradiation. This variation in power degrades the performance of the WADC. Communication delay, which is always present in case of wide area measurement system (WAMS) also has a detrimental effect on the effectiveness of WADC. Another non-linear phenomenon which is always present in the WADC is the saturation of the actuator which reduces the effectiveness of the controller. The damping controller developed in this paper contemplates the effect of saturation of the actuator, time varying delay and changing operating condition using linear matrix inequality (LMI) criterion. The changing operating conditions are considered using a linear parameter varying (LPV) modelling of the system. After developing the LPV model of the system, the model is converted into a tensor product (TP) model. A supplementary damping controller (SDC) is considered in this paper as the WADC. The actuator of the damping controller is a thyristor controlled series capacitor (TCSC). A modified two area 11-bus power system with PV power plant and modified IEEE 39-bus power system with PV power plant is used as a test system to understand the usefulness of the designed controller. [ABSTRACT FROM AUTHOR]

A dynamic phasor estimation technique based on the tensor product is presented to enhance the accuracy and robustness of phasor measurement units (PMUs) in the smart electric grid. The performance of the proposed tensor product-based dynamic phasor estimation algorithm (TDPE) is demonstrated with various test signals, i.e., DC decaying, noise, harmonics, power swings, frequency ramps, abrupt change, off-nominal frequency events as per Standard IEEE C37.118. 1a-2014 to represent peculiar dynamic scenarios of the modern power system. The simulation results reveal that the performance factor, i.e., total vector error (%TVE) of the proposed TDPE approach, is below 3 % during various simulated dynamic environments. Thus, the proposed tensor-based dynamic phasor estimation methodology can be effective to extract the phasors of voltage/current signals during peculiar dynamics in the smart electric grids and enhance the several applications based on phasor measurement units (PMUs) information for the stable and secure power system. [ABSTRACT FROM AUTHOR]

Large agricultural field trials may display irregular spatial trends that cannot be fully captured by a purely randomization‐based analysis. For this reason, paralleling the development of analysis‐of‐variance procedures for randomized field trials, there is a long history of spatial modeling for field trials, starting with the early work of Papadakis on nearest neighbor analysis, which can be cast in terms of first or second differences among neighboring plot values. This kind of spatial modeling is amenable to a natural extension using splines, as has been demonstrated in recent publications in the field. Here, we consider the P‐spline framework, focusing on model options that are easy to implement in linear mixed model packages. Two examples serve to illustrate and evaluate the methods. A key conclusion is that first differences are rather competitive with second differences. A further key observation is that second differences require special attention regarding the representation of the null space of the smooth terms for spatial interaction, and that an unstructured variance–covariance structure is required to ensure invariance to translation and rotation of eigenvectors associated with that null space. We develop a strategy that permits fitting this model with ease, but the approach is more demanding than that needed for fitting models using first differences. Hence, even though in other areas, second differences are very commonly used in the application of P‐splines, our conclusion is that with field trials, first differences have advantages for routine use. [ABSTRACT FROM AUTHOR]

The primary goal of this article is devoted to the study of fractal bases and fractal frames for L 2 (I × J) , the collection of all square integrable functions on the rectangle I × J . The fractal function when recognized as an internal binary operation paved way for the construction of right and left partial fractal convolution operators on L 2 (I) , for any real compact interval I. The aforementioned operators defined on one dimensional space have been applied to obtain operators on the space L 2 (I × J) by the identification of L 2 (I × J) with the tensor product space L 2 (I) ⊗ L 2 (J) . In this paper, we establish properties of this bounded linear operator which eventually helps to prove that L 2 (I × J) admits Bessel sequences, Riesz bases and frames consisting of products of fractal (self-referential) functions in a nice way. [ABSTRACT FROM AUTHOR]

In this paper, we construct all the indecomposable modules of a class of non-pointed Hopf algebras, which are quotient Hopf algebras of a class of prime Hopf algebras of GK-dimension one. Then the decomposition formulas of the tensor product of any two indecomposable modules are established. Based on these results, the representation ring of the Hopf algebras is characterized by generators and some relations.

The Machado–Bishop theorem for weighted vector-valued functions vanishing at infinity has been extensively studied. In this paper, we give an analogue of Machado's distance formula for bounded weighted vector-valued functions. A number of applications are given; in particular, some types of the Bishop–Stone–Weierstrass theorem for bounded vector-valued continuous spaces in the uniform topology are discussed; the splitting of C (I × J , X ⊗ Y) as the closure of C (I , X) ⊗ C (J , Y) in different senses and the extension of continuous vector-valued functions are studied. [ABSTRACT FROM AUTHOR]

For p ∈ [ 1 , ∞) , we show that every unital L p -operator algebra contains a unique maximal C ⁎ -subalgebra, which is always abelian if p ≠ 2. Using this, we canonically associate to every unital L p -operator algebra A an étale groupoid G A , which in many cases of interest is a complete invariant for A. By identifying this groupoid for large classes of examples, we obtain a number of rigidity results that display a stark contrast with the case p = 2 ; the most striking one being that of crossed products by topologically free actions. Our rigidity results give answers to questions concerning the existence of isomorphisms between different algebras. Among others, we show that for the L p -analog O 2 p of the Cuntz algebra, there is no isometric isomorphism between O 2 p and O 2 p ⊗ p O 2 p , when p ≠ 2. In particular, we deduce that there is no L p -version of Kirchberg's absorption theorem, and that there is no K -theoretic classification of purely infinite simple amenable L p -operator algebras for p ≠ 2. Our methods also allow us to recover a folklore fact in the case of C*-algebras (p = 2), namely that no isomorphism O 2 ≅ O 2 ⊗ O 2 preserves the canonical Cartan subalgebras. [ABSTRACT FROM AUTHOR]

This paper presents a new data driven control for a class of discrete-time multiple input and multiple output nonlinear systems which uses the principles of model-free adaptive control and the tensor product. The design of a linearization model using the notion of the tensor product overcomes reset problems in the calculation of the pseudo partial derivative and also improves control performance. The stability of the proposed method is guaranteed by theoretical analysis. Numerical simulation results and an experimental trial are presented to validate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

For bounded linear operators defined on complex infinite-dimensional Banach space, H. Zariouh, in an article [1] introduced and studied the property (g a z) . In this study, through techniques using the local spectral theory of operators, we discover the sufficient conditions that allow the transfer of the property (g a z) from two tensor factors T and S to their tensor product T ⊗ S . The stability of the property (g a z) in the tensor product under perturbations is also investigated. The theory is exemplified by considering suitable classes of operators such as shift operators, convolution operators, and m-invertible contractions. [ABSTRACT FROM AUTHOR]

We prove various results in infinite-dimensional differential calculus that relate the differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: (1) in the theory of infinite-dimensional vector bundles, to construct new bundles from given ones, such as dual bundles, topological tensor products, infinite direct sums, and completions (under suitable hypotheses); (2) in the theory of locally convex Poisson vector spaces, to prove continuity of the Poisson bracket and continuity of passage from a function to the associated Hamiltonian vector field. Topological properties of topological vector spaces are essential for the studies, which allow the hypocontinuity of bilinear mappings to be exploited. Notably, we encounter k R -spaces and locally convex spaces E such that E × E is a k R -space. [ABSTRACT FROM AUTHOR]

Consider a simple graph G with vertex set V (G) and edge set E (G) . A graph invariant for G is a number related to the structure of G, which is invariant under the symmetry of G. The Sombor index of G is a new graph invariant defined as S O (G) = ∑ u v ∈ E (G) (d u) 2 + (d v) 2 . In this work, we connected the theory of the Sombor index with abstract algebra. We computed this topological index over the tensor and Cartesian products of a monogenic semigroup graph by presenting two different algorithms; the obtained results are illustrated by examples. [ABSTRACT FROM AUTHOR]

In this paper we study the cancellation law for the tensor product in the category Sup of complete lattices and join-preserving maps. First, we investigate the tensor product of generalized power-set lattices. Based on which, we prove that the cancellation law for the tensor product has a close relation to that for the cartesian product of posets, and give a class of complete lattices which do not satisfy the cancellation law for the tensor product. Then, we also investigate the cancellation law for particular subclasses of complete lattices. [ABSTRACT FROM AUTHOR]

Continuous frames and tensor products are important topics in theoretical physics. This paper combines those concepts. We derive fundamental properties of continuous frames for tensor product of Hilbert spaces. This includes, for example, the consistency property, i.e. preservation of the frame property under the tensor product, and the description of the canonical dual tensors by those on the Hilbert space level. We show the full characterization of all dual systems for a given continuous frame, a result interesting by itself, and apply this to dual tensor frames. Furthermore, we discuss the existence on non-simple tensor product (dual) frames. Continuous frame multipliers and their Schatten class properties are considered in the context of tensor products. In particular, we give sufficient conditions for obtaining partial trace multipliers of the same form, which is illustrated with examples related to short-time Fourier transform and wavelet localization operators. As an application, we offer an interpretation of a class of tensor product continuous frame multipliers as density operators for bipartite quantum states, and show how their structure can be restricted to the corresponding partial traces. [ABSTRACT FROM AUTHOR]

In this paper we find a certain solution of Poisson type fractional differential equation using theory of tensor product of Banach spaces. [ABSTRACT FROM AUTHOR]

In a recent paper, Dave Benson and Peter Symonds defined a new invariant γG(M) for a finite dimensional module M of a finite group G which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for permutation modules of the symmetric group corresponding to two-part partitions using tools from representation theory and combinatorics. [ABSTRACT FROM AUTHOR]

• Analysis of Hanoi graphs , which arise from the classic Tower of Hanoi puzzle. • Nearly tight upper and lower bounds on the treewidth of these graphs. • An extension of the Kneser graph and a proof that its treewidth is nearly linear. • A new proof of a known lower bound on the treewidth of tensor products of graphs. The objective of the well-known Tower of Hanoi puzzle is to move a set of discs one at a time from one of a set of pegs to another, while keeping the discs sorted on each peg. We propose an adversarial variation in which the first player forbids a set of states in the puzzle, and the second player must then convert one randomly-selected state to another without passing through forbidden states. Analyzing this version raises the question of the treewidth of Hanoi graphs. We find this number exactly for three-peg puzzles and provide nearly-tight asymptotic bounds for larger numbers of pegs. [ABSTRACT FROM AUTHOR]

A claw is a star with three edges. The Kneser graph KGn,2 is the graph whose vertices are the 2-subsets of an n-set, in which two vertices are adjacent if and only if their intersection is empty. In this paper, we prove that KGn,2 is claw-decomposable, for all n ≥ 6. [ABSTRACT FROM AUTHOR]

Solving linear systems of equations is a fundamental problem in mathematics. When the linear system is so large that it cannot be loaded into memory at once, iterative methods such as the randomized Kaczmarz method excel. Here, we extend the randomized Kaczmarz method to solve multi-linear (tensor) systems under the tensor–tensor t-product. We present convergence guarantees for tensor randomized Kaczmarz in two ways: using the classical matrix randomized Kaczmarz analysis and taking advantage of the tensor–tensor t-product structure. We demonstrate experimentally that the tensor randomized Kaczmarz method converges faster than traditional randomized Kaczmarz applied to a naively matricized version of the linear system. In addition, we draw connections between the proposed algorithm and a previously known extension of the randomized Kaczmarz algorithm for matrix linear systems. [ABSTRACT FROM AUTHOR]

In this article, it is shown that the notion of tensor sum semigroups introduced in [8] is not valid. Further, a C0-semigroup generated by the tensor sum of generators of two C0-semigroups is developed. [ABSTRACT FROM AUTHOR]

By a {C3α, C3β}\{ C_3^\alpha ,\,C_3^\beta \} -decomposition of a graph G, we mean a partition of the edge set of G into α cycles of length 3 and β cycles of length.6. In this paper, necessary and sufficient conditions for the existence of a {C3α, C3β}3\alpha + 6\beta = {{\lambda m(m - 1)n(n - 1)} \over 2} -decomposition of (Km × Kn)(λ), where × denotes the tensor product of graphs and λ is the multiplicity of the edges, is obtained. In fact, we prove that for λ ≥ 1, m, n ≥ 3 and (m, n) ≠ (3, 3), a {C3α, C3β} -decomposition of (Km × Kn)(λ) exists if and only if λ(m − 1)(n − 1) ≡ 0 (mod 2) and 3α+6β=λm(m-1)n(n-1)2 .

In this paper, we consider the frameworks for quantum-classical and classical-quantum cryptographic translation by (Nimbe et al. Int. J. Theor. Phys. 60, 793–818 2021). The high or low level framework for quantum-classical cryptographic translation has three major phases with the framework for quantum-classical conversion being the intermediary phase. The high or low level framework for classical-quantum cryptographic translation also has three major phases with the framework for classical-quantum conversion being the intermediary phase. Mathematical steps are presented for these conversion frameworks and are shown to be implementable. Implementations of these intermediary phases are presented using c++ programming language. [ABSTRACT FROM AUTHOR]

In this work we introduce a three-party quantum conference where the involved parties initially hold asymmetric quantum information which are finally shared amongst themselves by the execution of a quantum communication protocol. The three conferencing parties along with a controller are connected amongst themselves through a multipartite quantum entanglement which is used as quantum resource in the protocol. The role of the controller is confined to a final action without which the conference can not be performed. The present work is an instance of applications of multipartite entangled resources in communication problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of left(right)-G sets by external hyperoperations and some examples presented. We con- struct quotient left(right)-G sets by regular (strongly) regular relations. Also, we consider fundamental relation as a smallest strongly regular relations and by complete parts concepts introduce an equivalence re- lation that is coincide with fundamental relation. The main purpose of this paper is to introduce the concepts of tensor product and direct limit on G-sets of n-ary semihypergroups that are non-additive modification of classical construction in module theory. This concept is crucially important in homological algebra and several properties are found and examples are presented. [ABSTRACT FROM AUTHOR]

Recent work by Baum et al. ['Expanders, exact crossed products, and the Baum–Connes conjecture', Ann. K-Theory 1(2) (2016), 155–208], further developed by Buss et al. ['Exotic crossed products and the Baum–Connes conjecture', J. reine angew. Math. 740 (2018), 111–159], introduced a crossed-product functor that involves tensoring an action with a fixed action $(C,\unicode[STIX]{x1D6FE})$ , then forming the image inside the crossed product of the maximal-tensor-product action. For discrete groups, we give an analogue for coaction functors. We prove that composing our tensor-product coaction functor with the full crossed product of an action reproduces their tensor-crossed-product functor. We prove that every such tensor-product coaction functor is exact, and if $(C,\unicode[STIX]{x1D6FE})$ is the action by translation on $\ell ^{\infty }(G)$ , we prove that the associated tensor-product coaction functor is minimal, thereby recovering the analogous result by the above authors. Finally, we discuss the connection with the $E$ -ization functor we defined earlier, where $E$ is a large ideal of $B(G)$. [ABSTRACT FROM AUTHOR]

Given a graph G = (V , E) and a finite set T of positive integers containing 0 , a T -coloring of G is a function f : V (G) → Z + ∪ { 0 } for all u ≠ w in V (G) such that if u w ∈ E (G) then | f (u) − f (w) | ∉ T. For a T -coloring f of G , the f -span sp T f (G) is the maximum value of | f (u) − f (w) | over all pairs u , w of vertices of G. The T -span sp T (G) is the minimum f -span overall T -colorings of G. The f -edge span of a T -coloring esp T f (G) is the maximum value of | f (u) − f (w) | over all edges u w of G. The T -edge span esp T (G) is the minimum f -edge span overall T -colorings of G. This paper discusses the T -span and T -edge span of Cartesian, Join, Union and Tensor products of graphs. Also, we discuss the relation between Restricted span and Restricted edge span of graphs. [ABSTRACT FROM AUTHOR]

Consider the space B[X...αY] of operators on the tensor product X...αY of normed spaces X and Y equipped with a uniform crossnorm ||·||α. Take the induced uniform norm on B[X...αY] and consider its restriction to the tensor product B[X] ... B[Y] of the algebra of operators B[X] and B[Y]. It is proved that such a restriction is a reasonable crossnorm onB[X] ... B[Y]. [ABSTRACT FROM AUTHOR]

. As an essential tool in homological algebra, tensor products play a basic role in classifying and studying modules. Since hypermodules are generalization of modules, it is important to generalize the concept of the tensor products of modules to the hypermodules. In this paper, in order to achieve this goal, we present a more general form of the definition of hypermodule. Based on this new definition, some of the required concepts and properties have been studied. By obtaining a free object in the category of hypermodules, the notion of tensor product of hypermodules is provided and some of its properties are studied. [ABSTRACT FROM AUTHOR]