Every simple quadrangulation of the sphere is generated by a graph called a pseudo-double wheel with two local expansions (Brinkmann et al. "Generation of simple quadrangulations of the sphere.'' Discrete Math., Vol. 305, No. 1-3, pp. 33--54, 2005). So, toward a classification of the spherical tilings by congruent quadrangles, we propose to classify those with the tiles being convex and the graphs being pseudo-double wheels. In this paper, we verify that a certain series of assignments of edge-lengths to pseudo-double wheels does not admit a tiling by congruent convex quadrangles. Actually, we prove the series admits only one tiling by twelve congruent concave quadrangles such that the symmetry of the tiling has only three perpendicular 2-fold rotation axes, and the tiling seems to be new.

This paper establishes some existence and nonexistence results of self-similar radial symmetric solutions to some class of strongly coupled reaction-diffusion systems with cross-diffusion. The considered class of systems allows to reduce the problem to a single equation with exponential source terms. Using the famous Mountain Pass Theorem and some smallness conditions on the system parameters it is possible to generalize wellknown results on self-similar radial solutions for a related problem that have been established by Y. Mizutani, N. Muramoto and K. Yoshida in 1999. As an application of the results derived in the present paper it is possible to conclude the existence and nonexistence of self-similar radial solutions for multi-species chemotaxis-model in the conflict-free setting and in the presence of a conflict of interests.

For a group $G$ with a certain positive presentation, we provide a criterion for two tuples of generators of $G$ to be Hurwitz equivalent. Based on this criterion, we present an algorithmic approach to solve the Hurwitz equivalence and the Hurwitz search problems by using the word reversing method.

We show that a generic $C^{1}$ expanding map on a compact Riemannian manifold has a unique measure of maximum total exponent which is fully supported and of zero entropy. We also show that for $r\ge 2$ a generic $C^{r}$ expanding map does not have fully supported measures of maximum total exponent.

We introduce two notions of equivalence for rational quadratic forms. Two $n$-ary rational quadratic forms are commensurable if they possess commensurable groups of automorphisms up to isometry. Two $n$-ary rational quadratic forms $F$ and $G$ are projectivelly equivalent if there are nonzero rational numbers $r$ and $s$ such that $rF$ and $sG$ are rationally equivalent. It is shown that if $F$\ and $G$\ have Sylvester signature $\{-,+,+,...,+\}$ then $F$\ and $G$\ are commensurable if and only if they are projectivelly equivalent. The main objective of this paper is to obtain a complete system of (computable) numerical invariants of rational $n$-ary quadratic forms up to projective equivalence. These invariants are a variation of Conway's $p$-excesses. Here the cases $n$ odd and $n$ even are surprisingly different. The paper ends with some examples