We answer in the affirmative the question posed by Conti and Rossi [7,8] on the existence of nilpotent Lie algebras of dimension 7 with an Einstein pseudo-metric of nonzero scalar curvature. Indeed, we construct a left-invariant pseudo-Riemannian metric $g$ of signature $(3, 4)$ on a nilpotent Lie group of dimension 7, such that $g$ is Einstein and not Ricci-flat. We show that the pseudo-metric $g$ cannot be induced by any left-invariant closed $G_2^*$-structure on the Lie group. Moreover, some results on closed and harmonic $G_2^*$-structures on an arbitrary 7-manifold $M$ are given. In particular, we prove that the underlying pseudo-Riemannian metric of a closed and harmonic $G_2^*$-structure on $M$ is not necessarily Einstein, but if it is Einstein then it is Ricci-flat.

Since the appearance of the so-called colorful Carathéodory and Helly theorems in $\mathbb{R}^d$, many papers have been published with colorful generalizations of classical results in combinatorial geometry. The growing number of such publications is a good motivation to consider colorful analogs of known results in a more abstract setting. This is why we introduce in this note the colorful Carathéodory, Helly and Sierksma numbers of a convexity space. We study relationships between these new numbers and the classical (monochromatic) Carathéodory, Helly and exchange numbers of convexity spaces. Special attention is paid to the new numbers in product convexity and convex sum spaces.

We study completeness of a topological vector space with respect to different filters on $\mathbb N$. In the metrizable case all these kinds of completeness are the same, but in non-metrizable case the situation changes. For example, a space may be complete with respect to one ultrafilter on $\mathbb N$, but incomplete with respect to another. Our study was motivated by [Aizpuru, Listán-García and Rambla-Barreno; Quaest. Math., 2014] and [Listán-García; Bull. Belg. Math. Soc. Simon Stevin, 2016] where for normed spaces the equivalence of the ordinary completeness and completeness with respect to $f$-statistical convergence was established.

Let $F_g$ be the free group functor, left adjoint to the forgetful functor between the category of groups $\mathsf{Grp}$ and the category of sets $\mathsf{Set}$. Let $f\colon A \to B$ and $h\colon A \to C$ be two functions in $\mathsf{Set}$ and let $\mathrm{Ker}(\mathrm{F}_g(f))$ and $\mathrm{Ker}(\mathrm{F}_g(h))$ be the kernels of the induced morphisms between free groups. Provided that the kernel pairs $Eq(f)$ and $Eq(h)$ of $f$ and $h$ permute (such as it is the case when the pushout of $f$ and $h$ is a double extension in $\mathsf{Set}$), this short article describes a method to rewrite a general element in the intersection $\mathrm{Ker}(\mathrm{F}_g(f)) \cap \mathrm{Ker}(\mathrm{F}_g(g))$ as a product of generators in $A$ which is $\langle f,h \rangle$-symmetric in the sense of the higher covering theory of racks and quandles.

Similarly like classical topological groups, the point-free counterparts, localic groups, possess natural uniformities (see e.g. [4,2,10]) obtained from an involutive binary operation on $L$ (roughly corresponding to the algebra of subsets of a classical group). It is the operation that naturally induces the uniformities (even if it would not result from a group) and a study of this aspect of the construction is the main topic of this article. We have here a functor associating with the localic groups quantales of a special type (and with frame group homomorphisms quantale homomorphisms) which are shown to create the uniformities, in fact as a special case of the natural uniformities connected with metric structures. Also, we present a condition under which the quantale allows a reconstruction of the localic group.

We employ a fuzzy optimality condition for the Fréchet subdifferential and some advanced techniques of variational analysis such as formulae for the subdifferentials of an infinite family of nonsmooth functions and the coderivative scalarization to investigate robust optimality condition and robust duality for a nonsmooth/nonconvex multiobjective optimization problem dealing with uncertain constraints in arbitrary Asplund spaces. We establish necessary optimality conditions for weakly and properly robust efficient solutions of the problem in terms of the Mordukhovich subdifferentials of the related functions. Further, sufficient conditions for weakly and properly robust efficient solutions as well as for robust efficient solutions of the problem are provided by presenting new concepts of generalized. Finally, we formulate a Mond-Weir-type robust dual problem to the reference problem, and examine weak, strong, and converse duality relations between them under the pseudo convexity assumptions.

The celebrated Touchard congruence states that $B_{n+p}\equiv B_n+B_{n+1}$ (mod $p$), where $p$ is a prime number and $B_n$ denotes the $n$-th Bell number. In this paper we study divisibility properties of $B_{n-p}$ and their generalizations involving higher powers of $p$ as well as the $r$-Bell numbers. In particular, we show a close relation of the considered problem to the Sun-Zagier congruence, which is additionally improved by deriving \mbox{a new} relation between $r$-Bell and derangement numbers. Finally, we conclude some results on the period of the Bell numbers modulo $p$.