New lower bounds are obtained for the number N_B(x) of Novák numbers not exceeding the given quantity x. In addition, conditioned on the generalized Riemann Hypothesis, upper bounds are found for the number of prime factors of Novák numbers and a description of the prime factors of Novák numbers N such that 2N is a Novák-Carmichael number is presented.
We solve a technical problem related to adeles on an algebraic surface. Given a finite set of natural numbers up to two, one associates an adelic group. We show that this operation commutes with taking intersections if the surface is defined over an uncountable field and we provide a counterexample otherwise.
We study regular global attractors of dissipative dynamical semigroups with discrete and continuous time and we investigate attractors for non-autonomous perturbations of such semigroups. The main theorem states that regularity of global attractors preserves under small non-autonomous perturbations. Besides, non-autonomous regular global attractors remain exponential and robust. We apply these general results to model non-autonomous reaction-diffu\-sion systems in a bounded domain of R^3 with time-dependent external forces.
A family of neighbourly polytopes in R2d with N=2d+4 vertices is constructed. All polytopes in the family have a planar Gale diagram of a special type, namely, with exactly d+3 black points in convex position. These Gale diagrams are parametrized by 3-trees (trees with a certain additional structure). For all polytopes in the family, the number of faces of dimension m containing a given vertex A depends only on d and m.
For every finite-dimensional vector space V and every V-flag variety X we list all connected reductive subgroups in GL(V) acting spherically on X.
We put forward a method for constructing semiorthogonal decompositions of the derived category of G-equivariant sheaves on a variety X under the assumption that the derived category of sheaves on X admits a semiorthogonal decomposition with components preserved by the action of the group G on X. This method is used to obtain semiorthogonal decompositions of equivariant derived categories for projective bundles and blow-ups with a smooth centre as well as for varieties with a full exceptional collection preserved by the group action. Our main technical tool is descent theory for derived categories.