Let $r\ge 3$, $n\ge r$ and $\pi =(d_1,d_2,...,d_n)$ be a non-increasing sequence of nonnegative integers. If $\pi$ has a realization $G$ with vertex set $V\left(G\right)=\{v_1,v_2,...,v_n\}$ such that $d_G\left(v_i\right)=d_i$ for $i=1,2,...,n$ and $v_1{v}_2\cdots v_r{v}_1$ is a cycle of length $r$ in $G$, then $\pi$ is said to be potentially $C_r^{\text{'}\text{'}}$-graphic. In this paper, we give a characterization for $\pi$ to be potentially $C_r^{\text{'}\text{'}}$-graphic.

We prove that eight dihedral angles in a pyramid with an arbitrary quadrilateral base always sum up to a number in the interval $(3\pi ,5\pi )$. Moreover, for any number in $(3\pi ,5\pi )$ there exists a pyramid whose dihedral angle sum is equal to this number, which means that the lower and upper bounds are tight. Furthermore, the improved (and tight) upper bound $4\pi$ is derived for the class of pyramids with parallelogramic bases. This includes pyramids with rectangular bases, often used in finite element mesh generation...

Let $S\left(RG\right)$ be a normed Sylow $p$-subgroup in a group ring $RG$ of an abelian group $G$ with $p$-component $G_p$ and a $p$-basic subgroup $B$ over a commutative unitary ring $R$ with prime characteristic $p$. The first central result is that $1+I(RG;B_p)+I(R\left(p^i\right)G;G)$ is basic in $S\left(RG\right)$ and $B[1+I(RG;B_p)+I(R\left(p^i\right)G;G)]$ is $p$-basic in $V\left(RG\right)$, and $[1+I(RG;B_p)+I(R\left(p^i\right)G;G)]G_p/G_p$ is basic in $S\left(RG\right)/G_p$ and $[1+I(RG;B_p)+I(R\left(p^i\right)G;G)]G/G$ is $p$-basic in $V\left(RG\right)/G$, provided in both cases $G/G_p$ is $p$-divisible and $R$ is such that its maximal perfect subring $R^{p^i}$ has no nilpotents whenever $i$ is natural. The second major result is that $B(1+I(RG;B_p))$ is $p$-basic in $V\left(RG\right)$ and $(1+I(RG;B_p))G/G$ is $p$-basic...

We consider initial boundary problems of a two-chemical substances chemotaxis system. In the four-dimensional setting, it was shown that solutions exist globally in time and remain bounded if the total mass is less than ${\left(8\pi \right)}^2$, whereas the solution emanating from some initial data of large magnitude may blows up. This result can be regarded as a generalization of the well-known $8\pi$ problem in the Keller–Segel system to higher dimensions. We will compare mathematical structures of the Keller–Segel...