Consider $\mathcal T_n(F)$ - the ring of all $n\times n$ upper triangular matrices defined over some field $F$. A map $\phi$ is called a zero product preserver on ${\mathcal T}_n(F)$ in both directions if for all $x,y\in{\mathcal T}_n(F)$ the condition $xy=0$ is satisfied if and only if $\phi(x)\phi(y)=0$. In the present paper such maps are investigated. The full description of bijective zero product preservers is given. Namely, on the set of the matrices that are invertible, the map $\phi$ may act in any bijective way, whereas for the zero divisors and zero matrix one can write $\phi$ as a composition of three types of maps. The first of them is a conjugacy, the second one is an automorphism induced by some field automorphism, and the third one transforms every matrix $x$ into a matrix $x'$ such that $\{y\in\mathcal T_n(F) xy=0\}=\{y\in\mathcal T_n(F) x'y=0\}$, $\{y\in\mathcal T_n(F) yx=0\}=\{y\in\mathcal T_n(F) yx'=0\}$., Roksana Słowik., and Obsahuje bibliografii
We consider the dynamics of spatially periodic nematic liquid crystal flows in the whole space and prove existence and uniqueness of local-in-time strong solutions using maximal L^{p} regularity of the periodic Laplace and Stokes operators and a local-intime existence theorem for quasilinear parabolic equations à la Clément-Li (1993). Maximal regularity of the Laplace and the Stokes operator is obtained using an extrapolation theorem on the locally compact abelian group G: = \mathbb{R}^{n - 1} \times \mathbb{R}/L\mathbb{Z} to obtain an R-bound for the resolvent estimate. Then, Weis’ theorem connecting R-boundedness of the resolvent with maximal L^{p} regularity of a sectorial operator applies., Jonas Sauer., and Obsahuje seznam literatury
A necessary and sufficient condition for the Reeb vector field of a three dimensional non-Kenmotsu almost Kenmotsu manifold to be minimal is obtained. Using this result, we obtain some classifications of some types of (k, μ, v)-almost Kenmotsu manifolds. Also, we give some characterizations of the minimality of the Reeb vector fields of (k, μ, v)-almost Kenmotsu manifolds. In addition, we prove that the Reeb vector field of an almost Kenmotsu manifold with conformal Reeb foliation is minimal., Yaning Wang., and Obsahuje bibliografii
Our aim in this paper is to study Musielak-Orlicz-Sobolev spaces on metric measure spaces. We consider a Hajłasz-type condition and a Newtonian condition. We prove that Lipschitz continuous functions are dense, as well as other basic properties. We study the relationship between these spaces, and discuss the Lebesgue point theorem in these spaces. We also deal with the boundedness of the Hardy-Littlewood maximal operator on Musielak-Orlicz spaces. As an application of the boundedness of the Hardy-Littlewood maximal operator, we establish a generalization of Sobolev’s inequality for Sobolev functions in Musielak-Orlicz-Hajłasz-Sobolev spaces., Takao Ohno, Tetsu Shimomura., and Obsahuje seznam literatury
We define and study Musielak-Orlicz-Sobolev spaces with zero boundary values on any metric space endowed with a Borel regular measure. We extend many classical results, including completeness, lattice properties and removable sets, to Musielak-Orlicz-Sobolev spaces on metric measure spaces. We give sufficient conditions which guarantee that a Sobolev function can be approximated by Lipschitz continuous functions vanishing outside an open set. These conditions are based on Hardy type inequalities., Takao Ohno, Tetsu Shimomura., and Obsahuje seznam literatury
Geiss, Keller and Oppermann (2013) introduced the notion of n-angulated category, which is a “higher dimensional” analogue of triangulated category, and showed that certain (n-2)-cluster tilting subcategories of triangulated categories give rise to n-angulated categories. We define mutation pairs in n-angulated categories and prove that given such a mutation pair, the corresponding quotient category carries a natural n-angulated structure. This result generalizes a theorem of Iyama-Yoshino (2008) for triangulated categories., Zengqiang Lin., and Obsahuje seznam literatury