Quasigroups satisfying Stein’s third law with a specified number of idempotents

Quasigroups satisfying Stein’s third law (QSTL for short) have been associated with other types of combinatorial configurations, such as cyclic orthogonal arrays. These have been studied quite extensively over the years by various researchers, including Curt Lindner. An idempotent model of a QSTL of order $v$ (briefly QSTL$\left(v\right)$), corresponds to a perfect Mendelsohn design of order $v$ with block size four (briefly a $(v,4,1)$-PMD) and these are known to exist if and only if $v\equiv 0,1\left(\mathrm{mod}4\right)$, except for $v=4,8$. There is a QSTL$\left(4\right)$ with two idempotents and it is known that a QSTL$\left(8\right)$ contains either $0$ or $4$ idempotents. In this paper, we formally investigate the existence of a QSTL$\left(v\right)$ with a specified number $n$ of idempotent elements, briefly denoted by QSTL$(v,n)$. The necessary conditions for the existence of a QSTL$(v,n)$ are $v\equiv 0,1\left(\mathrm{mod}4\right)$, $0\le n\le v$, and $v?n$ is even. We show that these conditions are also sufficient with few definite exceptions and a handful of possible exceptions. Holey perfect Mendelsohn designs of type $4^n{u}^1$ with block size four (HPMD$\left(4^n{u}^1\right)$ for short) are useful to establish the spectrum of QSTL$(v,n)$. In particular, we show that for $0\le u\le 8$, an HPMD$\left(4^n{u}^1\right)$ exists if and only if $n\ge max(4,?u/2?+1)$, except possibly $(n,u)=(12,1)$.     

On the asymptotic growth of positive solutions to a nonlocal elliptic blow-up system involving strong competition

For a competition-diffusion system involving the fractional Laplacian of the form

$?{(?\mathrm{\Delta })}^{s}u=u{v}^2,?{(?\mathrm{\Delta })}^{s}v=v{u}^2,u,v>0\text{in}{\mathbb{R}}^N,$

with $s\in (0,1)$, we prove that the maximal asymptotic growth rate for its entire solutions is 2s. Moreover, since we are able to construct symmetric solutions to the problem, when $N=2$ with prescribed growth arbitrarily close to the critical one, we can conclude that the asymptotic bound found is optimal. Finally, we prove existence of genuinely higher dimensional solutions, when $N\ge 3$. Such problems arise, for example, as blow-ups of fractional reaction-diffusion systems when the interspecific competition rate tends to infinity.     

Boundedness in a three-dimensional chemotaxis–haptotaxis model with nonlinear diffusion

The quasilinear chemotaxis–haptotaxis system

$\{\begin{array}{cc}{u}_t=\mathrm{?}?(D(u)\mathrm{?}u)?\chi \mathrm{?}?(u\mathrm{?}v)?\xi \mathrm{?}?(u\mathrm{?}w)\hfill & \hfill \\ \phantom{u_t=}+\mu u(1?u?w),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=\mathrm{\Delta }v?v+u,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ w_t=?vw,\hfill & x\in \mathrm{\Omega },t>0,\hfill \end{array}$

is considered under homogeneous Neumann boundary conditions in a bounded and smooth domain $\mathrm{\Omega }?{\mathbb{R}}^3$. Here $\chi >0$, $\xi >0$ and $\mu >0$, $D(u)\ge c_D{u}^{m?1}$ for all $u>0$ with some $c_D>0$ and $D(u)>0$ for all $u\ge 0$. It is shown that if the ratio $\frac{\chi }{\mu }$ is sufficiently small, then the system possesses a unique global classical solution that is uniformly bounded. Our result is independent of m.     

Parametric CR-umbilical locus of ellipsoids in C2

For every real numbers $a?1$, $b?1$ with $(a,b)\ne (1,1)$, the curve parametrized by $\theta \in \mathbb{R}$ valued in ${\mathbb{C}}^2?{\mathbb{R}}^4$

$\gamma :\theta ?(x(\theta )+\sqrt{?1}y(\theta ),u(\theta )+\sqrt{?1}v(\theta ))$

with components:

$\begin{array}{c}x(\theta ):=\sqrt{\frac{a?1}{a(ab?1)}}\cos ?\theta ,y(\theta ):=\sqrt{\frac{b(a?1)}{ab?1}}\sin ?\theta ,\hfill \\ u(\theta ):=\sqrt{\frac{b?1}{b(ab?1)}}\sin ?\theta ,v(\theta ):=?\sqrt{\frac{a(b?1)}{ab?1}}\cos ?\theta ,\hfill \end{array}$

has image contained in the CR-umbilical locus:

$\gamma (\mathbb{R})?\mathsf{UmbCR}\left({\mathsf{E}}_{a,b}\right)?{\mathsf{E}}_{a,b}$

of the ellipsoid ${\mathsf{E}}_{a,b}?{\mathbb{C}}^2$ of equation $a{x}^2+y^2+b{u}^2+v^2=1$, where the CR-umbilical locus of a Levi nondegenerate hypersurface $M^3?{\mathbb{C}}^2$ is the set of points at which the Cartan curvature of M vanishes.