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Graphs are essential to improve the performance of graph-based machine learning methods, such as spectral clustering. Various well-designed methods have been proposed to learn graphs that depict specific properties of real-world data. Joint learning of knowledge in different graphs is an effective means to uncover the intrinsic structure of samples. However, the existing methods fail to simultaneously mine the global and local information related to sample structure and distribution when multiple graphs are available, and further research is needed. Hence, we propose a novel intrinsic graph learning (IGL) with discrete constrained diffusion-fusion to solve the above problem in this article. In detail, given a set of the predefined graphs, IGL first obtains the graph encoding the global high-order manifold structure via the diffusion-fusion mechanism based on the tensor product graph. Then, two discrete operators are integrated to fine-prune the obtained graph. One of them limits the maximum number of neighbors connected to each sample, thereby removing redundant and erroneous edges. The other one forces the rank of the Laplacian matrix of the obtained graph to be equal to the number of sample clusters, which guarantees that samples from the same subgraph belong to the same cluster and vice versa. Moreover, a new strategy of weight learning is designed to accurately quantify the contribution of pairwise predefined graphs in the optimization process. Extensive experiments on six single-view and two multiview datasets have demonstrated that our proposed method outperforms the previous state-of-the-art methods on the clustering task.

In order to facilitate the representation of fuzzy relation matrix (FRM) models, a new system identification technique is proposed in this work to recognize the architecture and parameters of FRM models based on the semi-tensor product (STP) operation. Firstly, a fuzzy STP algorithm is defined for fuzzy inference. Secondly, a novel FRM framework is proposed for system parameter identification. Thirdly, the recognized FRM parameters are optimized to improve fuzzy system performance by the use of a hybrid training method based on the least squares estimator and the recursive Levenberg-Marquaedt algorithm. The effectiveness of the proposed structure and parameter identification techniques is verified by simulation of a multi-steps-ahead prediction modeling. Simulation results show that the proposed fuzzy STP technology is efficient for system identification, and the proposed matrix expression can be used to design multi-input multi-output (MIMO) systems with fuzzy FRM models.

The computational approach for solving the Faddeev-Merkuriev equations in total orbital momentum representation is presented. These equations describe a system of three quantum charged particles and are widely used in bound state and scattering calculations. The approach is based on the spline collocation method and exploits intensively the tensor product form of discretized operators and preconditioner, which leads to a drastic economy in both computer resources and time.

We show that if the cochain complex computing Ext groups (in the category of modules over Hopf algebroids) admits a cocyclic structure, then the noncommutative Cartan calculus structure on Tor over Ext dualises in a cyclic sense to a calculus on Coext over Cotor. More precisely, the cyclic duals of the chain resp. cochain spaces computing the two classical derived functors lead to complexes that compute the more exotic ones, giving a cyclic opposite module over an operad with multiplication that induce operations such as a Lie derivative, a cap product (or contraction), and a (cyclic) differential, along with higher homotopy operators defining a noncommutative Cartan calculus up to homotopy. In particular, this allows to recover the classical Cartan calculus from differential geometry or the Chevalley-Eilenberg calculus for Lie(-Rinehart) algebras without any finiteness conditions or the use of topological tensor products., Comment: 31 pages; to appear in Ann. Sc. Norm. Super. Pisa, Cl. Sci

The motivation behind 15 years of continuous development within the topic of the Tensor Product (TP) model transformation is that the greater the number of parameters or components of the Takagi Sugeno (TS) fuzzy model one can manipulate, the larger the complexity reduction or control optimization one can achieve. The paper proposes a radically new type of extension to the TP model transformation: while earlier variants of the TP model transformation focused on how the antecedent - consequent fuzzy set system of a given TS fuzzy model could be varied, the present work, in contrast, focuses on how the number of inputs to a given TS fuzzy model can be manipulated. The proposed extension is capable of changing the number of inputs or transforming the non-linearity between the fuzzy rules and the input dimensions. These new features considerably increase the modeling power of the TP model transformation, allowing for further complexity reduction and more powerful control optimisation to be achieved. The paper provides two examples to show how the proposed extension can be used in a routine-like fashion.

The Dunkl--Dirac operator is a deformation of the Dirac operator by means of Dunkl derivatives. We investigate the symmetry algebra generated by the elements supercommuting with the Dunkl--Dirac operator and its dual symbol. This symmetry algebra is realised inside the tensor product of a Clifford algebra and a rational Cherednik algebra associated with a reflection group or root system. For reducible root systems of rank three, we determine all the irreducible finite-dimensional representations and conditions for unitarity. Polynomial solutions of the Dunkl--Dirac equation are given as a realisation of one family of such irreducible unitary representations., Comment: v3 40p. Final version accepted in J. Algebra. See v2 for proof of Thm 4.1

In this paper, we propose a real vector representation of reduced quaternion matrix and study its properties. By using this real vector representation, Moore-Penrose inverse, and semi-tensor product of matrices, we study some kinds of solutions of reduced biquaternion matrix equation (1.1). Several numerical examples show that the proposed algorithm is feasible at last.

Considering multidimensional structure of the multi-exposure images,a new Tensor product and Tensorsingular value decomposition based Multi-Exposure image Fusion (TT-MEF) method is proposed. The main innovation of this work is to explore a new feature representation of multi-exposure images in the new tensor domain and design the fusion strategy on this basis. Specifically,the luminance and the chrominance channels are fused separately to maintain color consistency. For the luminance fusion,the luminance channel of multi-exposure images is divided into two parts,that is,de-mean term and mean term. The de-mean term is represented as a tensor to extract the feature. Then,the tensor product and tensor-singular value decomposition (T-SVD) are used to design a tensor feature extractor. Furthermore,a fusion strategy of the de-mean term is presented according to the visual saliency model,and a fusion strategy of the mean term is defined by the local and the global visual weights to control counterpoise between the local and global luminance. For the chrominance fusion,a new fusion strategy is also designed by the tensor product and T-SVD,similar to the luminance fusion. Finally,the fused image is obtained by combining the luminance and chrominance fusion. Experimental results show that the proposed TT-MEF method generally outperforms the existing state-of-the-art in terms of subjective visual quality and objective evaluation.

Feedback shift registers (FSRs), which have two configurations: Fibonacci and Galois, are a primitive building block in stream ciphers. In this paper, a transformation between Fibonacci FSRs and Galois FSRs is improved based on semi-tensor product (STP) of matrices. It is verified that a weakly equivalent Galois FSR with fewer stages cannot be found for a Fibonacci FSR with n stages, but not vice versa. Furthermore, for a given Fibonacci FSR with n stages, there are totally ( 2 n − 1 ) ! 2 − 1 weakly equivalent Galois FSRs. Additionally, an effective algorithm is developed to reduce the number of variables of the Galois FSRs while keeping it weakly equivalent to the given Fibonacci FSR. Finally, the feasibility of the proposed strategies is demonstrated by numerical examples.

We study finiteness conditions and closure properties for the box-tensor product G ⊠ H and related constructions. The content of this paper extends some results concerning the non-abelian tensor product of groups G ⊗ H . In particular, we deduce a quantitative version of the finiteness criterion for the non-abelian tensor product. Moreover, we obtain finiteness conditions for some functors that arise out of the non-abelian tensor square of groups, such as the second homology group H 2 ( G ) , the non-abelian exterior square G ∧ G and the second stable homotopy group of an Eilenberg-MacLane space π 2 S ( K ( G , 1 ) ) .

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.

By computing the completely bounded norm of the flip map on the Haagerup tensor product $C_0 Y_1\otimes_{C_0 X} C_0 Y_2$ associated to a pair of continuous mappings of locally compact Hausdorff spaces $Y_1\rightarrow X\leftarrow Y_2$, we establish a simple characterisation of the Beck-Chevalley condition for base change of operator modules over commutative $C^*$-algebras, and a descent theorem for continuous fields of Hilbert spaces., Comment: 10 pages. Minor changes to the exposition; corrected an error in Remarks 6 not affecting the rest of the paper

For an integral, commutative unital quantale L, we show that there exists a tensor product between stratified L-semiuniform convergence spaces and, with respect to this tensor product, that the category of stratified L-semiuniform convergence spaces and uniformly continuous maps is a symmetric monoidal category. As a result, we obtain the monoidal closedness of the category of stratified L-semiuniform convergence spaces and uniformly continuous maps.

By exploiting the subblock structure of a tensor product code, we provide an upper bound on the maximum correctable burst length of tensor product codes used with interleavers designed based on only the minimum distance and the maximum correctable burst length of its component codes, without taking the specific code geometry and soft decoding into consideration. We provide necessary and sufficient conditions for an interleaver to achieve this upper bound. Further, for two cases of the tensor product codes, we design optimal intracodeword interleavers that satisfy the proposed conditions.

Schur's Theorem and its generalisation, Baer's Theorem, are distinguished results in group theory, connecting the upper central quotients with the lower central series. The aim of this paper is to generalise these results in two different directions, using novel methods related with the non-abelian tensor product. In particular, we prove a version of Schur-Baer Theorem for finitely generated groups. Then, we apply these newly obtained results to describe the $k$-nilpotent multiplier, for $k\geq 2$, and other invariants of groups.

We consider finite-dimensional Hopf algebras $u$ which admit a smooth deformation $U\to u$ by a Noetherian Hopf algebra $U$ of finite global dimension. Examples of such Hopf algebras include small quantum groups over the complex numbers, restricted enveloping algebras in finite characteristic, and Drinfeld doubles of height $1$ group schemes. We provide a means of analyzing (cohomological) support for representations over such $u$, via the singularity categories of the hypersurfaces $U/(f)$ associated to functions $f$ on the corresponding parametrization space. We use this hypersurface approach to establish the tensor product property for cohomological support, for the following examples: functions on a finite group scheme, Drinfeld doubles of certain height 1 solvable finite group schemes, bosonized quantum complete intersections, and the small quantum Borel in type $A$., 52 pages, minor changes to text

Semidefinite programming is based on optimization of linear functionals over convex sets defined by linear matrix inequalities, namely, inequalities of the form $$L_A(X)=I-A_1X_1-\dots-A_g X_g\succeq0.$$ Here the $X_j$ are real numbers and the set of solutions is called a spectrahedron. These inequalities make sense when the $X_i$ are symmetric matrices of any size, $n\times n$, and enter the formula though tensor product $A_i\otimes X_i$: The solution set of $L_A(X)\succeq0$ is called a free spectrahedron since it contains matrices of all sizes and the defining ``linear pencil" is ``free" of the sizes of the matrices. In this article, we report on empirically observed properties of optimizers obtained from optimizing linear functionals over free spectrahedra restricted to matrices $X_i$ of fixed size $n\times n$. The optimizers we find are always classical extreme points. Surprisingly, in many reasonable parameter ranges, over 99.9\% are also free extreme points. Moreover, the dimension of the active constraint, $\ker(L_A(X^\ell))$, is about twice what we expected. Another distinctive pattern regards reducibility of optimizing tuples $(X_1^\ell,\dots,X_g^\ell)$. We give an algorithm for representing elements of a free spectrahedron as matrix convex combinations of free extreme points; these representations satisfy a very low bound on the number of free extreme points neede, 46 pages body. Includes table of contents and index

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.

The Littlewood-Richardson coefficients $c^{\lambda}_{\mu\nu}$ give the multiplicity of an irreducible polynomial ${\rm GL}_n$-representation $F^{\lambda}_n$ in the tensor product of polynomial representations $F^{\mu}_n\otimes F^{\nu}_n$. In this paper, we generalize these coefficients to an $r$-fold tensor product of rational representations, and give a new method for computing them using an analogue of statistical contingency tables. We demonstrate special cases in which our method reduces to counting statistical contingency tables with prescribed margins. Finally, we extend our result from the general linear group to both the orthogonal and symplectic groups.

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.

In this paper we find certain solutions of some fractional partial differential equations. Tensor product of Banach spaces is used where separation of variables does not work.

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.

In the present paper, we introduce the notion of non-abelian tensor product of a given precrossed module by one of its ideals. We use this concept to study the second homology and the nilpotent mul...

Real stabilizer operators, which are also known as real Clifford operators, are generated, through composition and tensor product, by the Hadamard gate, the Pauli Z gate, and the controlled-Z gate. We introduce a normal form for real stabilizer circuits and show that every real stabilizer operator admits a unique normal form. Moreover, we give a finite set of relations that suffice to rewrite any real stabilizer circuit to its normal form., In Proceedings QPL 2021, arXiv:2109.04886

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.

The problem of whether the cohomological support map of a finite dimensional Hopf algebra has the tensor product property has attracted a lot of attention following the earlier developments on representations of finite group schemes. Many authors have focussed on concrete situations where positive and negative results have been obtained by direct arguments. In this paper we demonstrate that it is natural to study questions involving the tensor product property in the broader setting of a monoidal triangulated category. We give an intrinsic characterization by proving that the tensor product property for the universal support datum is equivalent to complete primeness of the categorical spectrum. From these results one obtains information for other support data, including the cohomological one. Two theorems are proved giving compete primeness and non-complete primeness in certain general settings. As an illustration of the methods, we give a proof of a recent conjecture of Negron and Pevtsova on the tensor product property for the cohomological support maps for the small quantum Borel algebras for all complex simple Lie algebras., 20 pages

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 $$x\in {\mathcal {A}}\bigotimes ^\lambda {\mathcal {B}}$$ .

In this paper we Önd certain solutions of some fractional partial di§erential equations. Tensor product of Banach spaces is used to Önd some solutions where separation of variables does not work. We solve the fractional wave type equation using fractional Fourier Series

In this paper, we are interested in studying when an element $z$ in the projective tensor product $X \widehat{\otimes}_\pi Y$ turns out to be a Daugavet point. We prove first that, under some hypothesis, the assumption of $X \widehat{\otimes}_\pi Y$ having the Daugavet property implies the existence of a great amount of isometries from $Y$ into $X^*$. Having this in mind, we provide methods for constructing non-trivial Daugavet points in $X \widehat{\otimes}_\pi Y$. We show that $C(K)$-spaces are examples of Banach spaces such that the set of the Daugavet points in $C(K) \widehat{\otimes}_\pi Y$ is weakly dense when $Y$ is a subspace of $C(K)^*$. Finally, we present some natural results on when an elementary tensor $x \otimes y$ is a Daugavet point., Comment: 16 pages

We extend the diffusion-map formalism to data sets that are induced by asymmetric kernels. Analytical convergence results of the resulting expansion are proved, and an algorithm is proposed to perform the dimensional reduction. A coordinate system connected to the tensor product of Fourier basis is used to represent the underlying geometric structure obtained by the diffusion-map, thus reducing the dimensionality of the data set and making use of the speedup provided by the two-dimensional Fast Fourier Transform algorithm (2-D FFT). We compare our results with those obtained by other eigenvalue expansions, and verify the efficiency of the algorithms with synthetic data, as well as with real data from applications including climate change studies.

The linear non-polynomial spline is used here to solve the fractional partial differential equation (FPDE). The fractional derivatives are described in the Caputo sense. The tensor products are given for extending the one-dimensional linear non-polynomial spline to a two-dimensional spline to solve the heat equation. In this paper, the convergence theorem of the method used to the exact solution is proved and the numerical examples show the validity of the method. All computations are implemented by Mathcad15.

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.

In this paper, we study two properties viz., property-U and property-SU of a subspace Y of a Banach space X, which correspond to the uniqueness of the Hahn–Banach extension of each linear functional in (Formula presented.) and when this association forms a linear operator of norm-1 from (Formula presented.) to (Formula presented.). It is proved that, under certain geometric assumptions on (Formula presented.) these properties are stable with respect to the injective tensor product; Y has property-U (SU) in Z if and only if (Formula presented.) has property-U (SU) in (Formula presented.). We prove that when (Formula presented.) has the Radon–Nikod (Formula presented.) m Property for (Formula presented.), (Formula presented.) has property-U (property-SU) in (Formula presented.) if and only if Y is so in X. We show that if (Formula presented.) and Y has property-U (SU) in X then Y/Z has property-U (SU) in X/Z. On the other hand, Y has property-SU in X if Y/Z has property-SU in X/Z and (Formula presented.) is an M-ideal in X. This partly solves the 3-space problem for property-SU. We characterize all hyperplanes in (Formula presented.) which have property-SU. We derive necessary and sufficient conditions for all finite codimensional proximinal subspaces of (Formula presented.) which have property-U (SU).

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

Based on the diagonally compensated reduction, the preconditioned progressive iterative approximation (PPIA) for tensor product Bezier patches is presented. Due to the effectiveness of the preconditioner, the convergence rate of progressive iterative approximation (PIA) is accelerated significantly. To improve the robustness and reduce the computational complexity of PPIA, the inexact PPIA format for tensor product Bezier patches is presented. Several numerical examples are presented to illustrate the effectiveness of the proposed methods.

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.

In this paper, we consider C10-decompositions of the tensor product of complete graphs, Km × Kn, m, n ≥ 3. It is proved that the necessary conditions for such a decomposition are also sufficient. T...

In this paper, algebraic relations among four kinds of right semi-tensor product (STP) are discussed. Firstly, this paper provides definitions of right STPs, consisting of the first right matrix-matrix STP, the second right matrix-matrix STP, the first right matrix-vector STP, and the second right matrix-vector STP. Secondly, relations among these right STPs are proposed. Finally, the main results show the convertibility of these right STPs.

We define new norms for symmetric tensors over ordered normed spaces; these norms are defined by considering linear combinations of tensor products or powers of positive elements only. Relations between the different norms are studied. The results are applied to the problem of representing a finitely exchangeable distribution as a mixture of powers, i.e, mixture of distributions of i.i.d. sequences, using a signed mixing measure., 48 pages