We classify homogeneous four-dimensional manifolds satisfying a curvature condition which naturally generalizes the Einstein one. Our work leads to new examples of self-similar solutions two-loop renormalization group flow.

We prove that if all intersections of a convex body $B\subset \mathbb{R}^{4}$ with 3-dimensional linear subspaces are linearly equivalent then $B$ is centered ellipsoid. This gives an affirmative answer to the case $n=3$ following question by Banach from 1932: Is normed vector space $V$ whose $n$ -dimensional isometric, for fixed $2 \le n< \dim V$ , necessarily Euclidean? The dimensions and $\d...

A finite set of vectors \(\mathcal {X}\) in the d-dimensional Euclidean space \(\mathbb {R}^d\) is called a 2-distance set, if mutual distances between distinct elements has cardinality exactly 2. In this note we report, among other things, results computer-aided enumeration sets {R}^4\).