Search

We construct a generalization of the Day convolution tensor product of presheaves that works for certain double $$\infty $$ ∞ -categories. Using this construction, we obtain an $$\infty $$ ∞ -categorical version of the well-known description of (one-object) operads as associative algebras in symmetric sequences; more generally, we show that (enriched) $$\infty $$ ∞ -operads with varying spaces of objects can be described as associative algebras in a double $$\infty $$ ∞ -category of symmetric collections.

The concept of interval-valued q-rung orthopair fuzzy graph (IVq-ROFG) is a major recent extension in the field of fuzzy graph. However, traditional IVq-ROFG imposes strict restrictions on the lower and upper bounds of membership degree and non-membership degree, which narrows its scope of applicability. This paper proposes a new graph called possibility degree-based interval-valued q-rung orthopair fuzzy graph (PDIVq-ROFG) and generalizes the traditional IVq-ROFG. Firstly, the concept of PDIVq-ROFG is given and some basic terms, such as complement, self-complement, bridge, degree and total degree, are discussed. Secondly, product operations on PDIVq-ROFG are presented, including tensor product, Cartesian product, semi-strong product and strong product. Moreover, some theorems about the degree and total degree under these product operations are given and elaborated with several examples. Finally, the application of PDIVq-ROFG to the network of oil pipelines is provided and the feasibility and effectiveness of the PDIVq-ROFG are verified.

For an edge-colored graph G, we call an edge-cut M of G monochromatic if the edges of M are colored with a same color. The graph G is called monochromatically disconnected if any two distinct vertices of G are separated by a monochromatic edge-cut. The monochromatic disconnection number, denoted by md(G), of a connected graph G is the maximum number of colors that are allowed to make G monochromatically disconnected. In this paper, we solve the Erdős-Gallai-type problems for the monochromatic disconnection, and give the monochromatic disconnection numbers for four graph products, i.e., Cartesian, strong, lexicographic, and tensor products.

A Cartesian decomposition of a coherent configuration $${\cal X}$$ is defined as a special set of its parabolics that form a Cartesian decomposition of the underlying set. It turns out that every tensor decomposition of $${\cal X}$$ comes from a certain Cartesian decomposition. It is proved that if the coherent configuration $${\cal X}$$ is thick, then there is a unique maximal Cartesian decomposition of $${\cal X}$$ , i.e., there is exactly one internal tensor decomposition of $${\cal X}$$ into indecomposable components. In particular, this implies an analog of the Krull-Schmidt theorem for the thick coherent configurations. A polynomial-time algorithm for finding the maximal Cartesian decomposition of a thick coherent configuration is constructed.

More and more simulation tools are being used in the development of gears in order to save development time and costs while improving the gears. BECAL is a comprehensive software tool for the tooth contact analysis (TCA) of bevel, hypoid, beveloid and spur gears. The gear geometry is provided by a manufacturing simulation or a geometry import. To determine the exact contact conditions in the TCA, the discrete flank points are converted into a continuous and differentiable surface representation. At present, it is an approximation by means of Bézier tensor product surfaces. With this surface representation, significant deviations to the target points can occur depending on the tooth geometry. In particular tip, root and end relief, strongly curved tooth root geometries or discontinuous topological measurement data due to e.g. micro-pitting can only be considered insufficiently.Hence, a new method for surface approximation with non-uniform rational b‑spline surfaces (NURBS) is presented. Its application can significantly improve the surface representation compared to the target geometry, leading to more realistic results regarding contact stress, tooth root stress and transmission error. To illustrate the advantages, NURBS-based surfaces are compared with the Bézier tensor product surfaces. Finally, the potential of the new approach regarding the prediction of lifetime and acoustics is demonstrated by application to different gear geometries.

Recently, we have proved that the rectangular pointwise Lipschitz regularity of a continuous function on the unit square is directly related with the local suprema of the coefficients of the function in the tensor product Faber–Schauder basis. In this paper, we provide print dimension information on the distribution at all bi-scales of these local suprema. We apply our results for self-affine functions associated to the Schauder product function and a particular type of Sierpinski carpets.

A variation model of the gradient anisotropic elasticity theory is constructed. Its distinctive feature is the fact that the density of potential energy, along with symmetric fourth- and sixth-rank stiffness tensors, also contains a nonsymmetric fifth-rank stiffness tensor. Accordingly, the stresses in it also depend on the curvatures and the couple stresses depend on distortions. The Euler equations are three fourth-order equilibrium equations. The spectrum of boundary-value problems is determined by six pairs of alternative boundary conditions at each nonsingular surface point. At each special point of the surface (belonging to surface edges), in the general case, additional conditions arise for continuity of the displacement vector and the meniscus force vector when crossing the surface through the edge. In order to reduce the number of the physical parameters requiring experimental determination, particular types of the fifth- and sixth-rank stiffness tensors are postulated. Along with the classical tensor of anisotropic moduli, it is proposed to introduce a first-rank stiffness tensor (a vector with a length dimension), with the help of which the fifth-rank tensor is constructed from the classical fourthrank tensor by means of tensor multiplication. The sixth-rank tensor is constructed as the tensor product of the classical fourth-rank tensor and two first-rank tensors.

The basic formulas for the unitary principal series representations of the group SL(2, ℂ) are given, and the decomposition of a tensor product of two representations into irreducible representations is considered. A simple proof of the completeness of the 3j-symbols is given.

Given a discrete semigroup $$(S,\cdot )$$ , there is a natural operation on the Stone–Cech compactification $$\beta S$$ of S which extends the operation of S and makes $$(\beta S,\cdot )$$ a compact right topological semigroup with S contained in its topological center. If S and T are discrete semigroups, $$p\in \beta S$$ , and $$q\in \beta T$$ , then the tensor product $$p\otimes q$$ is a member of $$\beta (S\times T)$$ . It is known that tensor products are both algebraically and topologically rare in $$\beta (S\times T)$$ . We investigate when the algebraic product of two tensor products is again a tensor product. We get a simple characterization for a large class of semigroups. The characterization is in terms of a notion of cancellation. We investigate where that notion sits among standard cancellation notions.

Given two operator spaces E and F, injectivity of the canonical map from the $$\lambda $$ -tensor product $$E\bigotimes ^\lambda F$$ into the operator space injective tensor product $$E\bigotimes ^{\min } F$$ is investigated, where the $$\lambda $$ -tensor product is a generalization of the Haagerup, the operator space projective and the Schur tensor products. Properties related to the spectra of elements in $${\mathcal {A}}\bigotimes ^\lambda {\mathcal {B}}$$ , when $${\mathcal {A}} $$ and $${\mathcal {B}}$$ are two $$\hbox {C}^*$$ -algebras, are explored. One of the two important questions that are discussed is regarding the spectrum of an elementary tensor $$a\otimes b$$ in terms of the spectra of $$a\in {\mathcal {A}}$$ and $$b\in {\mathcal {B}}$$ , and the other is about the possibility of determining elements of the tensor product from their spectra in the sense that two elements $$u,v\in {\mathcal {A}}\bigotimes ^\lambda {\mathcal {B}}$$ are equal if and only if the nonzero spectral values of ux and vx coincide for every $$x\in {\mathcal {A}}\bigotimes ^\lambda {\mathcal {B}}$$ .

Let F be a field of characteristic zero and let R be an algebra that admits a regular grading by an abelian group H. Moreover, we consider G a group and let A be an algebra with a grading by the group G × H, we define the R-hull of A as the G × H-graded algebra given by $\mathfrak {R}(A)=\oplus _{(g,h)\in G\times H}A_{(g,h)}\otimes R_{h}$ . In this paper we provide a basis for the graded identities (resp. central polynomials) of the R-hull of A, assuming that a (suitable) basis for the graded identities (resp. central polynomials) of the G × H-graded algebra A is known. In particular, for any a, $b\in \mathbb {N}$ , we find a basis for the graded identities and the graded central polynomials for the algebra Ma,b(E), graded by the group $G\times \mathbb {Z}_{2}$ . Here E is the Grassmann algebra of an infinite dimensional F-vector space, equipped with its natural $\mathbb {Z}_{2}$ -grading and the matrix algebra Ma+b(F) is equipped with an elementary grading by the group $G\times \mathbb {Z}_{2}$ , so that its neutral component coincides with the subspace of the diagonal matrices. We describe the isomorphism classes of gradings on Ma,b(E) that arise in this way and count the isomorphism classes of such gradings. Moreover, we give an alternative proof of the fact that the tensor product Ma,b(E) ⊗ Mr,s(E) is PI-equivalent to Mar+bs,as+br(E). Finally, when the grading group is $\mathbb {Z}_{3}\times \mathbb {Z}_{2}$ (resp. $\mathbb {Z}\times \mathbb {Z}_{2}$ ), we present a complete description of a basis for the graded central polynomials for the algebra M2,1(E) (resp. Ma,b(E) in the case b = 1).

The tensor product of $${\mathbb {A}}^1$$ -invariant sheaves with transfers introduced by Voevodsky is generalized to reciprocity sheaves via the theory of modulus presheaves with transfers. We prove several general properties of this construction and compute it in some cases. In particular we obtain new (motivic) presentations of the absolute Kahler differentials and the first infinitesimal neighborhood of the diagonal.

This article proposes a new method for the fast and efficient calculation of 3D Tchebichef moments,which are an essential tool for the characterization and analysis of 3D objects. This method integrates the Kronecker tensor product to the computation of 3D Tchebichef moments for higher orders with the advantage of being parallelizable. The experimental results clearly show the benefits and efficacy of the proposed method compared to existing methods.

We study the $$\mathfrak {gl}_{1|1}$$ supersymmetric XXX spin chains. We give an explicit description of the algebra of Hamiltonians acting on any cyclic tensor products of polynomial evaluation $$\mathfrak {gl}_{1|1}$$ Yangian modules. It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the Drinfeld polynomials. In particular our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. We also give dimensions of the generalized eigenspaces. We show that when the tensor product is irreducible, then all eigenvectors can be constructed using Bethe ansatz. We express the transfer matrices associated to symmetrizers and anti-symmetrizers of vector representations in terms of the first transfer matrix and the center of the Yangian.

Twisted generalized Weyl algebras (TGWAs) $A(R,\sigma,t)$ are defined over a base ring $R$ by parameters $\sigma$ and $t$, where $\sigma$ 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 $\sigma$, there is a natural algebra map $A(R,\sigma,tt')\to A(R,\sigma,t)\otimes_R A(R,\sigma,t')$. 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 $A(R,\sigma,t)$. We give presentations of these Grothendieck rings for $n=1,2$, when $R=\mathbb{C}[z]$. 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 $\mathfrak{sl}_2$ is a tensor product of two Weyl algebra modules., Comment: 31 pages, 3 figures

This paper is concerned with convergence and superconvergence properties of the local discontinuous Galerkin (LDG) method for two-dimensional semilinear second-order elliptic problems of the form $$-\Delta u=f(x,y,u)$$ on Cartesian grids. By introducing special Gauss-Radau projections and using duality arguments, we obtain, under some suitable choice of numerical fluxes, the optimal convergence order in $$L^2$$ -norm of $${O}(h^{p+1})$$ for the LDG solution and its gradient, when tensor product polynomials of degree at most p and grid size h are used. Moreover, we prove that the LDG solutions are superconvergent with an order $$p+ 2$$ toward particular Gauss-Radau projections of the exact solutions. Finally, we show that the error between the gradient of the LDG solution and the gradient of a special Gauss-Radau projection of the exact solution achieves $$(p+1)$$ -th order superconvergence. Some numerical experiments are performed to illustrate the theoretical results.

This article proposed an improved 2D-logistic-adjusted-Sine map, which has better ergodicity, pseudo-randomness, unpredictability and wider chaotic range than many existing 2D-chaotic maps. Utilizing improved 2D-logistic-adjusted-Sine map generates chaotic sequences, where the initial values of the system are assigned based on SHA 512 hash function. Compared with traditional encryption scheme, the size of cipher image is different to plain image, because multiply can be applied between mismatched dimensional matrices through semi-tensor product, and therefore, attackers cannot obtain the size of plain image from encryption image. In order to lighten transmission and storage burden, we reduce the results of semi-tensor product by data process. Keys security, correlation coefficient, information entropy and ability of defending differential attack are analyzed to confirm the security of our encryption scheme. Experimental results confirm that the proposed algorithm can encrypt image effectively and securely.

The centralizer of the image of the diagonal embedding of $U_q(\mathfrak{sl}_2)$ in the tensor product of three irreducible representations is examined in a Schur-Weyl duality spirit. The aim is to offer a description in terms of generators and relations. A conjecture in this respect is offered with the centralizers presented as quotients of the Askey-Wilson algebra. Support for the conjecture is provided by an examination of the representations of the quotients. The conjecture is also shown to be true in a number of cases thereby exhibiting in particular the Temperley-Lieb, Birman-Murakami-Wenzl and one-boundary Temperley-Lieb algebras as quotients of the Askey-Wilson algebra., 26 pages

We give an integral formulation for the length frequencies in classical Coxeter groups. From this we derive a precise asymptotic for the peak of length frequency taken among elements in a Coxeter group. An interpretation of the peak of frequency in terms of an integral of characters afforded by tensor products of unitary representations.

In this paper, we introduce polynomial versions of the weak Dunford–Pettis property and the weak Dunford–Pettis $$^{*}$$ property for Banach lattices. By using Fremlin projective Banach lattice tensor products, we obtain several characterizations of the weak Dunford–Pettis property and the weak Dunford–Pettis $$^{*}$$ property in terms of regular homogeneous polynomials on Banach lattices.

We identify the modulation spaces associated to tensor products of amalgam spaces having a large class of Banach spaces as their local component. As consequences of the main results, we describe the modulation spaces associated to tensor products of various $L^p$ spaces., 28 pages

We compute global sections of tensor products and symmetric products of several secant bundles of a nonsingular projective curve. We closely follow the approach of Gentiana Danila’s work on the similar problem for nonsingular projective surfaces.

Let $\mathcal {C}$ C be a decomposable plane curve over an algebraically closed field k of characteristic 0. That is, $\mathcal {C}$ C is defined in k2 by an equation of the form g(x) = f(y), where g and f are polynomials of degree at least two. We use this data to construct three affine pointed Hopf algebras, A(x, a, g), A(y, b, f) and A(g, f), in the first two of which g [resp. f ] are skew primitive central elements, with the third being a factor of the tensor product of the first two. We conjecture that A(g, f) contains the coordinate ring $\mathcal {O}(\mathcal {C})$ O ( C ) of $\mathcal {C}$ C as a quantum homogeneous space, and prove this when each of g and f has degree at most five or is a power of the variable. We obtain many properties of these Hopf algebras, and show that, for small degrees, they are related to previously known algebras. For example, when g has degree three A(x, a, g) is a PBW deformation of the localisation at powers of a generator of the downup algebra A(− 1,− 1,0). The final section of the paper lists some questions for future work.

We consider the irreducible representations each of dimension 2 of the necklace braid group $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ). We then consider the tensor product of the representations of $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ) and determine necessary and sufficient condition under which the constructed representations are irreducible. Finally, we determine conditions under which the irreducible representations of $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ) of degree 2 are unitary relative to a hermitian positive definite matrix.

We solve a long-standing conjecture by Barker, proving that the minimal and maximal tensor products of two finite-dimensional proper cones coincide if and only if one of the two cones is generated by a linearly independent set. Here, given two proper cones $C_1$, $C_2$, their minimal tensor product is the cone generated by products of the form $x_1 \otimes x_2$, where $x_1 \in C_1$ and $x_2 \in C_2$, while their maximal tensor product is the set of tensors that are positive under all product functionals $f_1 \otimes f_2$, where $f_1$ is positive on $C_1$ and $f_2$ is positive on $C_2$. Our proof techniques involve a mix of convex geometry, elementary algebraic topology, and computations inspired by quantum information theory. Our motivation comes from the foundations of physics: as an application, we show that any two non-classical systems modelled by general probabilistic theories can be entangled., v2: added more background, several minor corrections v3: minor improvements

Parallelization schemes on many-core architectures, in this case, CUDA and OpenMP, are used to accelerate and improve the accuracy of adaptive multidimensional integration algorithms. The one-dimensional Gauss–Kronrod adaptive method is generalized to 3, 4, 5 and 6 dimensions. The implementation of the traditional tensor product construction of the grid and weights for multidimensional integration is revisited and reformulated taking advantages of the multi and many-core architectures. Tests performed in a set of benchmark functions show that the algorithm is numerically accurate, with accelerations as high as 800X in CUDA and 300X in the OpenMP implementation both compared to a sequential multidimensional integration algorithm.

In this paper we introduce a novel family of Markov chains on the simple representations of SL2$\left ({\mathbb {F}_p}\right )$ F p in defining characteristic, defined by tensoring with a fixed simple module and choosing an indecomposable non-projective summand. We show these chains are reversible and find their connected components and their stationary distributions. We draw connections between the properties of the chain and the representation theory of SL2$\left ({\mathbb {F}_p}\right )$ F p , emphasising symmetries of the tensor product. We also provide an elementary proof of the decomposition of tensor products of simple SL2$\left ({\mathbb {F}_p}\right )$ F p -representations.

We investigate tensor products of random matrices, and show that independence of entries leads asymptotically to$$\varepsilon $$ε-free independence, a mixture of classical and free independence studied by Młotkowski and by Speicher and Wysoczański. The particular$$\varepsilon $$εarising is prescribed by the tensor product structure chosen, and conversely, we show that with suitable choices an arbitrary$$\varepsilon $$εmay be realized in this way. As a result, we obtain a new proof that$$\mathcal {R}^\omega $$Rω-embeddability is preserved under graph products of von Neumann algebras, along with an explicit recipe for constructing matrix models.

Region visual features enhance the generative capability of the machines based on features, however they lack proper interaction attentional perceptions and thus ends up with biased or uncorrelated sentences or pieces of misinformation. In this work, we propose Attribute Interaction-Tensor Product Representation (aiTPR) which is a convenient way of gathering more information through orthogonal combination and learning the interactions as physical entities (tensors) and improving the captions. Compared to previous works, where features are added up to undefined feature spaces, TPR helps in maintaining sanity in combinations and orthogonality helps in defining familiar spaces. We have introduced a new concept layer that defines the objects and also their interactions that can play a crucial role in determination of different descriptions. The interaction portions have contributed heavily for better caption quality and has out-performed different previous works on this domain and MSCOCO dataset. We introduced, for the first time, the notion of combining regional image features and abstracted interaction likelihood embedding for image captioning., 11 pages

Let R be a Noetherian local k-algebra whose derivation module $${\mathrm{Der}}_k(R)$$ is finitely generated. Our main goal in this paper is to investigate the impact of assuming that $${\mathrm{Der}}_k(R)$$ has finite projective dimension (or finite Gorenstein dimension), mainly in connection with freeness, under a suitable hypothesis concerning the vanishing of (co)homology or the depth of a certain tensor product. We then apply some of our results towards the critical case $${\mathrm{depth}}\,R=3$$ of the Herzog–Vasconcelos conjecture and consequently to the strong version of the Zariski–Lipman conjecture.

Biquadratic tensors play a central role in many areas of science. Examples include elastic tensor and Eshelby tensor in solid mechanics, and Riemannian curvature tensor in relativity theory. The singular values and spectral norm of a general third order tensor are the square roots of the M-eigenvalues and spectral norm of a biquadratic tensor, respectively. The tensor product operation is closed for biquadratic tensors. All of these motivate us to study biquadratic tensors, biquadratic decomposition, and norms of biquadratic tensors. We show that the spectral norm and nuclear norm for a biquadratic tensor may be computed by using its biquadratic structure. Then, either the number of variables is reduced, or the feasible region can be reduced. We show constructively that for a biquadratic tensor, a biquadratic rank-one decomposition always exists, and show that the biquadratic rank of a biquadratic tensor is preserved under an independent biquadratic Tucker decomposition. We present a lower bound and an upper bound of the nuclear norm of a biquadratic tensor. Finally, we define invertible biquadratic tensors, and present a lower bound for the product of the nuclear norms of an invertible biquadratic tensor and its inverse, and a lower bound for the product of the nuclear norm of an invertible biquadratic tensor, and the spectral norm of its inverse.

Let k be a field of characteristic p≠ 0. In 1968, M. E. Sweedler revealed for the first time, the usefulness of the concept of modularity. This notion, which plays an important role especially for Galois theory of purely inseparable extensions, was used to characterize purely inseparable extensions of bounded exponent which were tensor products of simple extensions. A natural extension of the definition of modularity is to say that K/k is q-modular (quasi-modular) if K is modular up to some finite extension. In subsequent papers, M. Chellali and the author have studied various property of q-modular field extensions, including the questions of q-modularity preservation in case [k : kp] is finite. This paper grew out of an attempt to find analogue results concerning the preservation of q-modularity, without the hypothesis on k but with extra assumptions on K/k. In particular, we investigate existence conditions of lower (resp. upper) quasi-modular closures for a given q-finite extension.

In this work, by combining a class of local support and infinitely differentiable functions together with the sinc function, we construct a new class of univariate blending functions with three local shape parameters αi, βi, and λi. The new blending functions have the properties of $C^{\infty }$ smoothness, compact support, and partition of unity. The shape parameter αi has tension property, and βi can adjust the support of the blending functions. With λi, the given blending functions can be used to interpolate sets of points partly or entirely without solving a linear system of equations. Some simple conditions for the blending functions possessing nonnegativity and/or linear independence are developed. Based on the new univariate blending functions, tensor product blending functions and local tensor product blending functions are also developed.

This is an expository paper on tensor products where the standard approaches for constructing concrete instances of algebraic tensor products of linear spaces, via quotient spaces or via linear maps of bilinear maps, are reviewed by reducing them to different but isomorphic interpretations of an abstract notion, viz. the universal property, which is based on a pair of axioms.

We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module and sheaf tensor product and Hom bifunctors as well as their derived functors, Tor and Ext, and give explicit computations for interval modules. We give a classification of injective, projective, and flat interval modules. We state Kunneth theorems and universal coefficient theorems for the homology and cohomology of chain complexes of persistence modules in both the sheaf and graded modules settings and show how these theorems can be applied to persistence modules arising from filtered cell complexes. We also give a Gabriel-Popescu theorem for persistence modules. Finally, we examine categories enriched over persistence modules. We show that the graded module point of view produces a closed symmetric monoidal category that is enriched over itself., 41 pages, accepted by Foundations of Computational Mathematics

A Trace-Finite-Cell-Method for the numerical analysis of thin shells is presented combining concepts of the TraceFEM and the Finite-Cell-Method. As an underlying shell model we use the Koiter model, which we re-derive in strong form based on first principles of continuum mechanics by recasting well-known relations formulated in local coordinates to a formulation independent of a parametrization. The field approximation is constructed by restricting shape functions defined on a structured background grid on the shell surface. As shape functions we use on a background grid the tensor product of cubic splines. This yields $C^1$-continuous approximation spaces, which are required by the governing equations of fourth order. The parametrization-free formulation allows a natural implementation of the proposed method and manufactured solutions on arbitrary geometries for code verification. Thus, the implementation is verified by a convergence analysis where the error is computed with an exact manufactured solution. Furthermore, benchmark tests are investigated., submitted to Computational Mechanics

In this article, left {g, h}-derivation and Jordan left {g, h}-derivation on algebras are introduced. It is shown that there is no Jordan left {g, h}-derivation over $${{\cal M}_n}\left(C \right)$$ and ℍℝ, for g ≠ h. Examples are given which show that every Jordan left {g, h}-derivation over $${{\cal T}_n}\left(C \right)$$ , $${{\cal M}_n}\left(C \right)$$ and ℍℝ are not left {g, h}-derivations. Also, the Jordan left {g, h}-derivations over $${{\cal T}_n}\left(C \right)$$ , $${{\cal M}_n}\left(C \right)$$ and ℍℝ are right centralizers, where C is a 2-torsionfree commutative ring. Moreover, we prove the result of Jordan left {g, h}-derivation to be a left {g, h}-derivation over tensor products of algebras as well as for algebra of polynomials.

In this paper, we examine properties of the tensor powers of quantum mappings Φ. In particular, we review positivity properties of unitary and nonunitary qubit mappings Φ⊗2. For arbitrary finite-dimensional systems, we present the relationship between the positive and completely positive divisibility of dynamical mappings $$ {\Phi}_t^{\otimes 2} $$ and Φt. A criterion of annihilation of entanglement by an arbitrary qubit mapping Φ⊗2 is found.

Based on the n-ary counting system, combined with the matrix semi-tensor product theory and Hilbert curve, a chaotic image encryption algorithm is designed. Different from the traditional encryption method, the algorithm proposed in this paper is an encryption algorithm with scrambling and diffusion at the same time. First, the pixel value is converted from decimal to n-ary. In the n-ary counting system, the plaintext image is randomly divided into some groups, and the Hilbert curve is used for scrambling to each group. The blocks are converted into scrambled images, so that the scrambling and diffusion can be carried out at the same time. Then, in order to improve the security of the algorithm, another round of diffusion is carried out based on matrix semi-tensor product mechanism. Chaotic sequence is generated by Chen system. This chaotic sequence performs matrix semi-tensor product operation with the first round of encrypted image, and generate second encrypted images. Finally, this encryption method is applied to color image encryption. Compared with some representative algorithms, the experimental results show that the algorithm proposed in this paper is secure and it can resist common attacks.

The oscillator Racah algebra $\mathcal{R}_n(\mathfrak{h})$ is realized by the intermediate Casimir operators arising in the multifold tensor product of the oscillator algebra $\mathfrak{h}$. An embedding of the Lie algebra $\mathfrak{sl}_{n-1}$ into $\mathcal{R}_n(\mathfrak{h})$ is presented. It relates the representation theory of the two algebras. We establish the connection between recoupling coefficients for $\mathfrak{h}$ and matrix elements of $\mathfrak{sl}_n$-representations which are both expressed in terms of multivariate Krawtchouk polynomials of Griffiths type., 28 pages

Quantum resource theories offer formal frameworks for quantification and manipulation of intrinsic resources associated in quantum systems. Each resource, such as quantum entanglement, coherence, contextuality and many others, has attracted considerable interest encompassing the characterization, quantification, and its operational application. Despite the fundamental importance of quantum resource, the development of a unified framework of resource theories has been initiated only recently, especially in the framework of additive quantum resources. Monogamy is a fundamental property of many entanglement measures, and a unified view to monogamy inequalities of bipartite entanglement in multi-party quantum systems has recently been established. By investigating the monogamy property of the additive quantum resource, general monogamy inequalities given by the α th power of additive quantum resource measures are presented: any additive resource measure satisfy the polynomial relation for α ≥ 1 and obey monogamy inequalities for α ≤ 0. However, the relation is not clear for 0 < α

We define and study a pro-p version of Sidki’s weak commutativity construction. This is the pro-p group $$\mathfrak {X}_p(G)$$ generated by two copies G and $$G^{\psi }$$ of a pro-p group, subject to the defining relators $$[g,g^{\psi }]$$ for all $$g \in G$$ . We show for instance that if G is finitely presented or analytic pro-p, then $$\mathfrak {X}_p(G)$$ has the same property. Furthermore we study properties of the non-abelian tensor product and the pro-p version of Rocco’s construction $$\nu (H)$$ . We also study finiteness properties of subdirect products of pro-p groups. In particular we prove a pro-p version of the $$(n-1)-n-(n+1)$$ Theorem.

Let F be an infinite field of characteristic different from two and E be the unitary Grassmann algebra of an infinite dimensional F-vector space L. Denote by Egr an arbitrary $\mathbb {Z}_{2}$ -grading on E such that the subspace L is homogeneous. We consider Egr ⊗ E⊗n as a $(\mathbb {Z}_{2}\times {\mathbb {Z}_{2}^{n}})$ -graded algebra, where the grading on E is supposed to be the canonical one, and we find its graded ideal of identities.

De Casteljau algorithm and degree elevation of Bezier and NURBS curves/surfaces are two important techniques in computer aided geometric design. This paper presents the de Casteljau algorithm and degree elevation of toric surface patches, which include tensor product and triangular rational Bezier surfaces as special cases. Some representative examples of toric surface patches with common shapes are illustrated to verify these two algorithms. Moreover, the authors also apply the degree elevation of toric surface patches to isogeometric analysis. And two more examples show the effectiveness of proposed method.

Let λ be a reflexive Banach sequence lattice and X be a Banach lattice. In this paper, we show that the positive injective tensor product $$\lambda {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over \otimes } _{\left| \varepsilon \right|}}X$$ is a Grothendieck space if and only if X is a Grothendieck space and every positive linear operator from λ* to X** is compact.

We deduce a formula for the product of two Γ-series in four variables, connected with the lattice B = ℤ〈(1, −1, −1, 1)〉. As a consequence, we obtain a formula for the product of Gauss hypergeometric functions F2,1, which can be interpreted as a part of the explicit decomposition of the tensor product of two representations of 𝔤𝔩3 into the direct sum of irreducible representations. Bibliography: 12 titles.

Let $$L_G$$ be the Laplacian matrix of a graph G with n vertices, and let $${\mathbf {b}}$$ be a binary vector of length n. The pair $$(L_G, {\mathbf {b}})$$ is said to be controllable (and we also say that G is Laplacian controllable for $${\mathbf {b}}$$ ) if $$L_G$$ has no eigenvector orthogonal to $${\mathbf {b}}$$ . In this paper we study the Laplacian controllability of joins, Cartesian products, tensor products and strong products of two graphs. Besides some theoretical results, we give an iterative construction of infinite families of controllable pairs $$(L_G, {\mathbf {b}})$$ .

In this paper we are concerned with the following question: if the tensor product of finitely generated modules $M$ and $N$ over a local complete intersection domain is maximal Cohen-Macaulay, then must $M$ or $N$ be a maximal Cohen-Macaulay? Celebrated results of Auslander, Lichtenbaum, and Huneke and Wiegand, yield affirmative answers to the question when the ring considered has codimension zero or one, but the question is very much open for complete intersection domains that have codimension at least two, even open for those that are one-dimensional, or isolated singularities. Our argument exploits Tor-rigidity and proves the following, which seems to give a new perspective to the aforementioned question: if $R$ is a complete intersection ring which is an isolated singularity such that dim($R$) > codim($R$), and the tensor product $M\otimes_R N$ is maximal Cohen-Macaulay, then $M$ is maximal Cohen-Macaulay if and only if $N$ is maximal Cohen-Macaulay., This is a pre-print of an article published in Annali di Matematica. The final authenticated version is available online at: https://doi.org/10.1007/s10231-020-01019-9

In this paper we prove that the $$\ell $$ -group tensor product of two Archimedean f-rings is again an f-ring. We will use this result to characterize multiplicative $$\ell $$ -bimorphisms between unital f-rings.