Suppose $f\in L^1(\Bbb{R}^d)$, $\Lambda\subset\Bbb{R}^d$ is a finite union of translated lattices such that $f+\Lambda$ tiles with weight. We prove there exists lattice $L\subset\Bbb{R}^d$ $f+L$ also tiles, possibly different As corollary, together result Kolountzakis, it implies any convex polygon multi-tiles the plane by translations admits multi-tiling, multiplicity.

For a function defined on a convex set in a Euclidean space, midpoint convexity is the property requiring that the value of the function at the midpoint of any line segment is not greater than the average of its values at the endpoints of the line segment. Midpoint convexity is a well-known characterization of ordinary convexity under very mild assumptions. For a function defined on the integer...

Let o(G) be the average of element orders a finite group G. A research topic concerning this quantity is understanding relation between and o(H), where H subgroup N class nilpotent groups let L(G) lattice In paper, we show that set {o(G)o(H) | G∈N,H∈L(G)} dense in [0,∞). Other density results are outlined throughout paper.

ABSTRACT In this article an attempt has been made to study some new algebraic structures of intuitionistic multi-anti fuzzy normal subgroups under homomorphism are discussed. Keywords Intuitionistic multi-fuzzy set (IMFS), subgroup (IMAFSG), (IMAFNSG). Mathematics Subject Classification 20N25, 03E72, 08A72, 03F55, 06F35, 03G25, 08A05

Let H be a group, let {G(: i e 1} be a set of groups and, for each i, let 9t be a a monomorphism: H -* Gt, with H9t = Hi. We call such a system of groups and monomorphisms an amalgam and denote it by [G{; H; 9,; / / , ] . By an embedding of the amalgam into a group G is meant a set of monomorphisms (p; : G; -*• G such that 9i<pi = 0j<Pj-, for all i,j and G,<P; n Gjcpj — H9k<pk, for all i,j, k. ...

I am trying to understand a beautiful work of Pierre Schapira [5]. Notations and Conventions • For any set S we denote by IS the identity map S → S. For any subset A ⊂ S we denote by 1A the characteristic function of A. • In the sequel all manifolds will be assumed real analytic. A morphism of manifolds will be a real analytic map between two manifolds. Introduction Suppose we are given a simpl...

In this note, we present a simple geometric argument to determine a lower bound on the split rank of intersection cuts. As a first step of this argument, a polyhedral subset of the lattice-free convex set that is used to generate the intersection cut is constructed. We call this subset the restricted lattice-free set. It is then shown that dlog2(l)e is a lower bound on the split rank of the int...

quasinormal subgroups have been studied for nearly 80 years. in finite groups, questions concerning them invariably reduce to p-groups, and here they have the added interest of being invariant under projectivities, unlike normal subgroups. however, it has been shown recently that certain groups, constructed by berger and gross in 1982, of an important universal nature with regard to the existen...