We introduce the algebras satisfying the $(ℬ,n)$ condition. If $\Lambda$, $\Gamma$ are algebras satisfying the $(ℬ,n)$, $(ℰ,m)$ condition, respectively, we give a construction of $(m+n)$-almost split sequences in some subcategories ${(ℬ\otimes ℰ)}^{(i_0,j_0)}$ of $mod(\Lambda \otimes \Gamma )$ by tensor products and mapping cones. Moreover, we prove that the tensor product algebra $\Lambda \otimes \Gamma$ satisfies the $({(ℬ\otimes ℰ)}^{(i_0,j_0)},n+m)$ condition for some integers $i_0$, $j_0$; this construction unifies and extends the work of A. Pasquali (2017), (2019).

We give sufficient conditions implying that the projective tensor product of two Banach spaces $X$ and $Y$ has the $p$-sequentially Right and the $p$-$L$-limited properties, $1\le p<\infty$.

We show that each general Haar system is permutatively equivalent in $L^p\left([0,1]\right)$, 1 < p < ∞, to a subsequence of the classical (i.e. dyadic) Haar system. As a consequence, each general Haar system is a greedy basis in $L^p\left([0,1]\right)$, 1 < p < ∞. In addition, we give an example of a general Haar system whose tensor products are greedy bases in each $L^p\left({[0,1]}^d\right)$, 1 < p < ∞, d ∈ ℕ. This is in contrast to [11], where it has been shown that the tensor products of the dyadic Haar system are not greedy bases...

We give a classification of all linear natural operators transforming $p$-vectors (i.e., skew-symmetric tensor fields of type $(p,0)$) on $n$-dimensional manifolds $M$ to tensor fields of type $(q,0)$ on $T^{A}M$, where $T^A$ is a Weil bundle, under the condition that $p\ge 1$, $n\ge p$ and $n\ge q$. The main result of the paper states that, roughly speaking, each linear natural operator lifting $p$-vectors to tensor fields of type $(q,0)$ on $T^A$ is a sum of operators obtained by permuting the indices of the tensor products of linear natural...

Let $G$ be a group generated by a set of finite order elements. We prove that any bicrossed product $H_{m,d}\left(q\right)\bowtie k\left[G\right]$ between the generalized Taft algebra $H_{m,d}\left(q\right)$ and group algebra $k\left[G\right]$ is actually the smash product $H_{m,d}\left(q\right)♯k\left[G\right]$. Then we show that the classification of these smash products could be reduced to the description of the group automorphisms of $G$. As an application, the classification of $H_{m,d}\left(q\right)\bowtie k[C_{n_1}\times C_{n_2}]$ is completely presented by generators and relations, where $C_n$ denotes the $n$-cyclic group.

Let d ≥ 2 be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products ${\prod }_{\genfrac{}{}{0pt}{}{k=1}{U_{d^k}\ne -a_i}}^{\infty }(1+\left(a_i\right)/\left(U_{d^k}\right))$ (i=1,...,m) or ${\prod }_{\genfrac{}{}{0pt}{}{k=1}{V_{d^k}\ne -a_i}}^{\infty }(1+\left(a_i\right)\left(V_{d^k}\right)$ (i=1,...,m) to be algebraically dependent, where $a_i$ are non-zero integers and $U_n$ and $V_n$ are generalized Fibonacci numbers and Lucas numbers, respectively. The purpose of this paper is to relax the condition on the non-zero integers $a_1,...,a_m$ to non-zero real algebraic numbers, which gives new cases where the infinite products above are algebraically...

We investigate whether the projective tensor product of two Banach spaces $X$ and $Y$ has the reciprocal Dunford–Pettis property of order $p$, $1\le p<\infty$, when $X$ and $Y$ have the respective property.

We completely determine the ${\ell }_q$ and C(K) spaces which are isomorphic to a subspace of ${\ell }_p\otimes {̂}_{\pi }C\left(\alpha \right)$, the projective tensor product of the classical ${\ell }_p$ space, 1 ≤ p < ∞, and the space C(α) of all scalar valued continuous functions defined on the interval of ordinal numbers [1,α], α < ω₁. In order to do this, we extend a result of A. Tong concerning diagonal block matrices representing operators from ${\ell }_p$ to ℓ₁, 1 ≤ p < ∞. The first main theorem is an extension of a result of E. Oja and states...

Assume that S is a commutative complete discrete valuation domain of characteristic p, S* is the unit group of S and $G=G_p\times B$ is a finite group, where $G_p$ is a p-group and B is a p’-group. Denote by $S^{\lambda }G$ the twisted group algebra of G over S with a 2-cocycle λ ∈ Z²(G,S*). We give necessary and sufficient conditions for $S^{\lambda }G$ to be of OTP representation type, in the sense that every indecomposable $S^{\lambda }G$-module is isomorphic to the outer tensor product V W of an indecomposable $S^{\lambda }{G}_p$-module V and an irreducible...

Let $\mathbb{Z}{̂}_p$ be the ring of p-adic integers, $U\left(\mathbb{Z}{̂}_p\right)$ the unit group of $\mathbb{Z}{̂}_p$ and $G=G_p\times B$ a finite group, where $G_p$ is a p-group and B is a p’-group. Denote by $\mathbb{Z}{̂}_p^{\lambda }G$ the twisted group algebra of G over $\mathbb{Z}{̂}_p$ with a 2-cocycle $\lambda \in Z²(G,U\left(\mathbb{Z}{̂}_p\right))$. We give necessary and sufficient conditions for $\mathbb{Z}{̂}_p^{\lambda }G$ to be of OTP representation type, in the sense that every indecomposable $\mathbb{Z}{̂}_p^{\lambda }G$-module is isomorphic to the outer tensor product V W of an indecomposable $\mathbb{Z}{̂}_p^{\lambda }{G}_p$-module V and an irreducible $\mathbb{Z}{̂}_p^{\lambda }B$-module W.

Let $H_4$ and $H_8$ be the Sweedler’s and Kac-Paljutkin Hopf algebras, respectively. We prove that any Hopf algebra which factorizes through $H_8$ and $H_4$ (equivalently, any bicrossed product between the Hopf algebras $H_8$ and $H_4$) must be isomorphic to one of the following four Hopf algebras: $H_8\otimes H_4,H_{32,1},H_{32,2},H_{32,3}$. The set of all matched pairs $(H_8,H_4,▹,◃)$ is explicitly described, and then the associated bicrossed product is given by generators and relations.

We discuss the existence of the current $g^{k}T$, $k\in \mathbb{N}$ for positive and closed currents $T$ and unbounded plurisubharmonic functions $g$. Furthermore, a new type of weighted Lelong number is introduced under the name of weight $k$ Lelong number.

Melham discovered the Fibonacci identity $F_{n+1}{F}_{n+2}{F}_{n+6}-F{³}_{n+3}=(-1)ⁿFₙ$. He then considered the generalized sequence Wₙ where W₀ = a, W₁ = b, and $Wₙ=p{W}_{n-1}+q{W}_{n-2}$ and a, b, p and q are integers and q ≠ 0. Letting e = pab - qa² - b², he proved the following identity: $W_{n+1}{W}_{n+2}{W}_{n+6}-W{³}_{n+3}=e{q}^{n+1}(p³W_{n+2}-q²W_{n+1})$. There are similar differences of products of Fibonacci numbers, like this one discovered by Fairgrieve and Gould: $FₙF_{n+4}{F}_{n+5}-F{³}_{n+3}={(-1)}^{n+1}{F}_{n+6}$. We prove similar identities. For example, a generalization of Fairgrieve and Gould’s identity is $WₙW_{n+4}{W}_{n+5}-W{³}_{n+3}=eqⁿ(p³W_{n+4}-q{W}_{n+5})$.

Let ℓ be a Banach sequence space with a monotone norm $\parallel ·{\parallel }_{\ell }$, in which the canonical system $\left(e_i\right)$ is a normalized symmetric basis. We give a complete isomorphic classification of Cartesian products $E_0^{\ell }\left(a\right)\times E_{\infty }^{\ell }\left(b\right)$ where $E_0^{\ell }\left(a\right)=K^{\ell }\left(exp(-p^{-1}{a}_i)\right)$ and $E_{\infty }^{\ell }\left(b\right)=K^{\ell }\left(exp\left(p{a}_i\right)\right)$ are finite and infinite ℓ-power series spaces, respectively. This classification is the generalization of the results by Chalov et al. [Studia Math. 137 (1999)] and Djakov et al. [Michigan Math. J. 43 (1996)] by using the method of compound linear topological invariants developed by...

We study sums and products in a field. Let $F$ be a field with $\mathrm{ch}\left(F\right)\ne 2$, where $\mathrm{ch}\left(F\right)$ is the characteristic of $F$. For any integer $k\ge 4$, we show that any $x\in F$ can be written as $a_1+\cdots +a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1\cdots a_k=1$, and that for any $\alpha \in F\setminus \left\{0\right\}$ we can write every $x\in F$ as $a_1\cdots a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1+\cdots +a_k=\alpha$. We also prove that for any $x\in F$ and $k\in \{2,3,\cdots \}$ there are $a_1,\cdots ,a_{2k}\in F$ such that $a_1+\cdots +a_{2k}=x=a_1\cdots a_{2k}$.