This article shows that although EU Member States are quite unsuccessful when they plead many different grounds before the ECJ to justify the non-transposition of EU directives, there are still strikingly many grounds that can possibly justify non-transposition. This article also argues that although there are many good reasons for the ECJ to be very strict in accepting possible grounds justifying the non-transposition of EU directives, there was no good reason for the ECJ to reject as justifying basis one special case of the pointlessness of the transposition. Namely the pointlessness because an activity referred to in a directive does not yet exist in a Member State due to an EU law compatible national legal obstacle for such activity in that State.
Adult head structures of Lepicerus inaequalis were examined in detail and interpreted functionally and phylogenetically. The monogeneric family clearly belongs to Myxophaga. A moveable process on the left mandible is an autapomorphy of the suborder. Even though Lepiceridae is the "basal" sistergroup of the remaining three myxophagan families, it is likely the group which has accumulated most autapomorphic features, e.g. tuberculate surface structure, internalised antennal insertion, and a specific entognathous condition. Adults of Lepiceridae and other myxophagan groups possess several features which are also present in larvae (e.g., premental papillae, semimembranous mandibular lobe). This is probably related to a very similar life style and has nothing to do with "desembryonisation". Lepiceridae and other myxophagans share a complex and, likely, derived character of the feeding apparatus with many polyphagan groups (e.g., Staphyliniformia). The mandibles are equipped with large molae and setal brushes. The latter interact with hairy processes or lobes of the epi- and hypopharynx. This supports a sistergroup relationship between both suborders.
We maximize the total height of order ideals in direct products of finitely many finite chains. We also consider several order ideals simultaneously. As a corollary, a shifting property of some integer sequences, including digit sum sequences, is derived.
A power digraph, denoted by $G(n,k)$, is a directed graph with $\mathbb Z_{n}=\{0,1,\dots ,n-1\}$ as the set of vertices and $E=\{(a,b)\colon a^{k}\equiv b\pmod n\}$ as the edge set. In this paper we extend the work done by Lawrence Somer and Michal Křížek: On a connection of number theory with graph theory, Czech. Math. J. 54 (2004), 465–485, and Lawrence Somer and Michal Křížek: Structure of digraphs associated with quadratic congruences with composite moduli, Discrete Math. 306 (2006), 2174–2185. The heights of the vertices and the components of $G(n,k)$ for $n\geq 1$ and $k\geq 2$ are determined. We also find an expression for the number of vertices at a specific height. Finally, we obtain necessary and sufficient conditions on $n$ such that each vertex of indegree $0$ of a certain subdigraph of $G(n,k)$ is at height $q\geq 1$.
In this paper we introduce and investigate a Henstock-Kurzweil-type integral for Riesz-space-valued functions defined on (not necessarily bounded) subintervals of the extended real line. We prove some basic properties, among them the fact that our integral contains under suitable hypothesis the generalized Riemann integral and that every simple function which vanishes outside of a set of finite Lebesgue measure is integrable according to our definition, and in this case our integral coincides with the usual one.
It is well known by results of Golod and Shafarevich that the Hilbert $2$-class field tower of any real quadratic number field, in which the discriminant is not a sum of two squares and divisible by eight primes, is infinite. The aim of this article is to extend this result to any real abelian $2$-extension over the field of rational numbers. So using genus theory, units of biquadratic number fields and norm residue symbol, we prove that for every real abelian $2$-extension over $\mathbb Q$ in which eight primes ramify and one of theses primes $\equiv -1\pmod 4$, the Hilbert $2$-class field tower is infinite.