Search

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 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

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.

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.

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.

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

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

Let $G$ be a finite group and $k$ an algebraically closed field of characteristic $p>0$. We define the notion of a Dade $kG$-module as a generalization of endo-permutation modules for $p$-groups. We show that under a suitable equivalence relation, the set of equivalence classes of Dade $kG$-modules forms a group under tensor product, and the group obtained this way is isomorphic to the Dade group $D(G)$ defined by Lassueur. We also consider the subgroup $D^{\Omega} (G)$ of $D(G)$ generated by relative syzygies $\Omega_X$, where $X$ is a finite $G$-set. If $C(G,p)$ denotes the group of superclass functions defined on the $p$-subgroups of $G$, there are natural generators $\omega_X$ of $C(G,p)$, and we prove the existence of a well-defined group homomorphism $\Psi_G:C(G,p)\to D^\Omega(G)$ that sends $\omega_X$ to $\Omega_X$. The main theorem of the paper is the verification that the subgroup of $C(G,p)$ consisting of the dimension functions of $k$-orientable real representations of $G$ lies in the kernel of $\Psi_G$., Comment: Minor revision, 40 pages

In this paper, we study weight representations over the Schrodinger Lie algebra s n for any positive integer n. It turns out that the algebra s n can be realized by polynomial differential operators. Using this realization, we give a complete classification of irreducible weight s n -modules with finite dimensional weight spaces for any n. All such modules can be clearly characterized by the tensor product of so n -modules, sl 2 -modules and modules over the Weyl algebra.

In this article, we show that distributive law holds for non-abelian tensor product of Lie superalgebras under certain direct sums. Thereby we obtain a rule for non-abelian exterior square of a Lie...

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

Let F be an algebraically closed field of prime characteristic p. The modular McKay graph of G : = S L n ( p ) with respect to its standard FG-module W is the connected, directed graph whose vertices are the irreducible FG-modules and for which there is an edge from a vertex V 1 to V 2 if V 2 occurs as a composition factor of the tensor product V 1 ⊗ W . We show that the diameter of this modular McKay graph is 1 2 ( p − 1 ) ( n 2 − n ) .

Let A m , n be the tensor product of the Laurent polynomial algebra in m even variables and the exterior algebra in n odd variables over the complex field C , and the Witt superalgebra W m , n be the Lie superalgebra of superderivations of A m , n . In this paper, we classify the simple weight W m , n modules with finite-dimensional weight spaces with respect to the standard Cartan subalgebra of W m , 0 . Every such module is either a simple quotient of a tensor module or a module of highest weight type.

We study the problem when the product of two non-linear Galois conjugate characters of a finite group is irreducible. We also prove new results on irreducible tensor products of cross-characteristic Brauer characters of quasisimple groups of Lie type.

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 k,n1,…,nk be positive integers such that ni⩾2 for i=1,…,k and let Mni denote the algebra of ni×ni matrices over a field F for i=1,…,k. Let ⨂i=1kMni be the tensor product of Mn1,…,Mnk. We obtain...

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.

In this work we study some structural properties of the group ηq(G,H), q a non-negative integer, which is an extension of the q-tensor product G⊗qH, where G and H are normal subgroups of some group...

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

We study consequences and applications of the folklore statement that every double complex over a field decomposes into so-called squares and zigzags. This result makes questions about the associated cohomology groups and spectral sequences easy to understand. We describe a notion of `universal' quasi-isomorphism, investigate the behaviour of the decomposition under tensor product and compute the Grothendieck ring of the category of bounded double complexes over a field with finite cohomologies up to such quasi-isomorphism (and some variants). Applying the theory to the double complexes of smooth complex valued forms on compact complex manifolds, we obtain a Poincar\'e duality for higher pages of the Fr\"olicher spectral sequence, construct a functorial three-space decomposition of the middle cohomology, give an example of a map between compact complex manifolds which does not respect the Hodge filtration strictly, compute the Bott-Chern and Aeppli cohomology for Calabi-Eckmann manifolds, introduce new numerical bimeromorphic invariants, show that the non-K\"ahlerness degrees are not bimeromorphic invariants in dimensions higher than three and that the $\partial\overline{\partial}$-lemma and some related properties are bimeromorphic invariants if, and only if, they are stable under restriction to complex submanifolds., Comment: final version; to appear in J. London Math. Soc

In this paper, we construct a class of new modules for the quantum group U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) which are free of rank 1 when restricted to C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] . The irreducibility of these modules and submodule structure for reducible ones are determined. It is proved that any C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) -module of rank 1 is isomorphic to one of the modules we constructed, and their isomorphism classes are obtained. We also investigate the tensor products of the C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free modules with finite-dimensional simple modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) , and for the generic cases, we obtain direct sum decomposition formulas for them, which are similar to the well-known Clebsch–Gordan formula for tensor products between finite-dimensional weight modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) .

This paper introduces the generalized additive mixed model (GAMM) and the quantile generalized additive mixed model (QGAMM) through reanalyses of bilinguals’ lexical decision data from Dijkstra et al. (2010) and Miwa et al. (2014). We illustrate how regression splines can be used to test for nonlinear effects of cross-language similarity in form as well as for controlling experimental trial effects. We further illustrate the tensor product smooth for a nonlinear interaction between cross-language semantic similarity and word frequency. Finally, we show how the QGAMM helps clarify whether the effect of a particular predictor is constant across distributions of RTs.

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.

Let H be the twisted Heisenberg–Virasoro algebra. In this paper, we first present a method to produce a class of tensor product H-modules, by which we can obtain the known weight H-modules M(V,Aα,b...

A k-Zumkeller labeling for the graph G = (V, E) is an assignment f of a label to each vertices of G such that each edge uv ∈ E is assigned the label f (u) f (v), the resulting edge labels are k distinct Zumkeller numbers. In this paper, we prove that the graph Pm × Pn is k-Zumkeller graph for m, n ≥ 3 while Pm × Cn and Cm × Cn are k-Zumkeller graphs for n ≡ 4 (mod2). Also we show that the graphs Pm ⊗ Pn and Pm ⊗ Cn for m, n ≥ 3 admit k-Zumkeller labeling. Further, the graph Cm ⊗ Cn where m or n is even admit a k-Zumkeller labeling.

We study permutation orbifolds of the $2$-fold and $3$-fold tensor product for the Virasoro vertex algebra $\mathcal{V}_c$ of central charge $c$. In particular, we show that for all but finitely many central charges $\left(\mathcal{V}_c^{\otimes 3}\right)^{\mathbb{Z}_3}$ is a $W$-algebra of type $(2, 4, 5, 6^3 , 7, 8^3 , 9^3 , 10^2 )$. We also study orbifolds of their simple quotients and obtain new realizations of certain rational affine $W$-algebras associated to a principal nilpotent element. Further analysis of permutation orbifolds of the celebrated $(2,5)$-minimal vertex algebra $\mathcal{L}_{-\frac{22}{5}}$ is presented., Comment: 23 pages. v2: minor improvements and references added

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.

There are several tools, like algebraic polynomials, topological indices, graph energies, etc., to investigate the structural dependence of different properties and activities of chemical structure...

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 n>1 be a positive integer, { H 1 , … , H n } be a finite collection of complex Hilbert spaces with dim ⁡ ( H k ) ≥ 2 , and P 1 ( H k ) be the set of all rank-1 self-adjoint projections on H k ,...

Let p be a prime and let $J_r$ denote a full $r \times r$ Jordan block matrix with eigenvalue $1$ over a field F of characteristic p. For positive integers r and s with $r \leq s$ , the Jordan canonical form of the $r s \times r s$ matrix $J_{r} \otimes J_{s}$ has the form $J_{\lambda _1} \oplus J_{\lambda _2} \oplus \cdots \oplus J_{\lambda _{r}}$ . This decomposition determines a partition $\lambda (r,s,p)=(\lambda _1,\lambda _2,\ldots , \lambda _{r})$ of $r s$ . Let $n_1, \ldots , n_k$ be the multiplicities of the distinct parts of the partition and set $c(r,s,p)=(n_1,\ldots ,n_k)$ . Then $c(r,s,p)$ is a composition of r. We present a new bottom-up algorithm for computing $c(r,s,p)$ and $\lambda (r,s,p)$ directly from the base-p expansions for r and s.

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

The generalized quantum group U ( ϵ ) of type A is an affine analogue of quantum group associated to a general linear Lie superalgebra gl M | N . We prove that there exists a unique R matrix on the tensor product of fundamental type representations of U ( ϵ ) for arbitrary parameter sequence ϵ corresponding to a non-conjugate Borel subalgebra of gl M | N . We give an explicit description of its spectral decomposition, and then as an application, construct a family of finite-dimensional irreducible U ( ϵ ) -modules which have subspaces isomorphic to the Kirillov-Reshetikhin modules of usual affine type A M − 1 ( 1 ) or A N − 1 ( 1 ) .

We describe the class (semigroup) of quantum channels mapping states with finite entropy into states with finite entropy. We show, in particular, that this class is naturally decomposed into three convex subclasses, two of them are closed under concatenations and tensor products. We obtain asymptotically tight universal continuity bounds for the output entropy of two types of quantum channels: channels with finite output entropy and energy-constrained channels preserving finiteness of the entropy., 21 pages, preliminary version, any comments are welcome. arXiv admin note: substantial text overlap with arXiv:1704.01905, arXiv:1907.02458

This paper presents a boundary moving least squares method (BMLS) for solving 3D linear elasticity problems. In the proposed method, moving least squares interpolant is used to construct the 1D shape function employing the boundary discrete points in each axis. Then, the Kronecker product is introduced to generate the tensor product points in the regular region coving the problem domain, which simply the 3D shape function of the conventional meshless method into 1D shape function for raising the efficiency of calculation in the BMLS. In addition, building a virtual boundary is employed to solve irregular domain problems. Numerical results show that the BMLS method gives accurate solutions and desirable convergence rates when comparing with the analytical solutions and the finite element method (FEM) solutions. Furthermore, the choices of the basis function and the weight functions are also discussed in this work.

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.

This article studies a high-dimensional inference problem involving the matrix tensor product of random matrices. This problem generalizes a number of contemporary data science problems including the spiked matrix models used in sparse principal component analysis and covariance estimation and the stochastic block model used in network analysis. The main results are single-letter formulas (i.e., analytical expressions that can be approximated numerically) for the mutual information and the minimum mean-squared error (MMSE) in the Bayes optimal setting where the distributions of all random quantities are known. We provide non-asymptotic bounds and show that our formulas describe exactly the leading order terms in the mutual information and MMSE in the high-dimensional regime where the number of rows $n$ and number of columns $d$ scale with $d = O(n^{\alpha })$ for some $\alpha . On the technical side, this article introduces some new techniques for the analysis of high-dimensional matrix-valued signals. Specific contributions include a novel extension of the adaptive interpolation method that uses order-preserving positive semidefinite interpolation paths, and a variance inequality between the overlap and the free energy that is based on continuous-time I-MMSE relations.

We analyze families of Markov chains that arise from decomposing tensor products of irreducible representations. This illuminates the Burnside-Brauer theorem for building irreducible representations, the McKay correspondence, and Pitman's 2 M − X theorem. The chains are explicitly diagonalizable, and we use the eigenvalues/eigenvectors to give sharp rates of convergence for the associated random walks. For modular representations, the chains are not reversible, and the analytical details are surprisingly intricate. In the quantum group case, the chains fail to be diagonalizable, but a novel analysis using generalized eigenvectors proves successful.

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

Let K be an algebraically closed field and G = SL 2 ( K ) . Let E be the natural module and S r E the rth symmetric power. We consider here, for r , s ≥ 0 , the tensor product of S r E and the dual of S s E . In characteristic zero this tensor product decomposes according to the Clebsch-Gordan formula. We consider here the situation when K is a field of positive characteristic. We show that each indecomposable component occurs with multiplicity one and identify which modules occur for given r and s.

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.

We present a B-spline based semi-analytical technique for solving 2D convection-diffusion-reaction equations. The main feature of the presented technique is to separate the satisfaction of the conditions on the boundary and the elliptic partial differential equation inside. To be more precise, we transform the original equation into the problem with homogeneous boundary conditions and seek the approximate solution as a sum of the modified B-spline tensor products which satisfy the homogeneous boundary conditions of the problem. The cubic and quintic B-spline are used in the framework of the method. The coefficients of linear combination are determined to satisfy the governing equation. Eight numerical examples have been studied to demonstrate the high effectivity of the presented technique in solving 2D convection-diffusion-reaction problems in single and multi-connected domains.

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.

Let Sk and Kk respectively denote a path, a star and a complete graph on k vertices. By a -decomposition of a graph G, we mean a decomposition of G into r copies of and s copies of In this paper, it shown that the graph admits a -decomposition if and only if where denotes a tensor product of complete graphs. Also we extend the existence of such a decomposition in complete m-partite graphs.

We propose two conjectures on the tensor product of Weil representations with irreducible representations in finite symplectic and unitary groups.