In a previous paper, the second author established that, given finite fields F < E and certain subgroups C ≤ E×, there is a Galois connection between the intermediate field lattice {L | F ≤ L ≤ E} and C’s subgroup lattice. Based on the Galois connection, the paper then calculated the irreducible, complex character degrees of the semi-direct product C⋊Gal(E/F ). However, the analysis when |F | i...

Problems of geometric probability for an arbitrary convex body of resolution in the euclidean space E3 has been investigated in [1]. In [9] Buffon’s problem is solved for a lattice of right-angled parallelepipeds in the 3-dimensional space. In this note we want to use the results in [5] for to solve problems of intersection for a particular lattice that we describe: the fundamental cell C0 of t...

Lattice Definition. (Lattice) Given n linearly independent vectors b1, , bn ∈ R, the lattice generated by them · · · is defined as L(b1, b2, · · · bn) = { xibi|xi ∈ Z}. We refer to b1, · · · , bn as a basis of the lattice. Equivalently, if we define B as the m × n matrix whose columns are b1, , bn, then the lattice generated by B is · · · L(B) = L(b1, b2, · · · , bn) = {Bx|x ∈ Z}. We say that t...

Abstract It is known that if G a group such the centre factor $$G/\zeta (G)$$ G / ζ ( ) polycyclic, then also commutator subgroup $$G'$$ ′ polycyclic. The aim of this paper to des...

Let G be a stratified Lie group and L be the sub-Laplacian on G. Let 0 6= f ∈ S(R). We show that Lf(L)δ, the distribution kernel of the operator Lf(L), is an admissible function on G. We also show that, if ξf(ξ) satisfies Daubechies’ criterion, then Lf(L)δ generates a frame for any sufficiently fine lattice subgroup of G.

We give a 2O(n)(1+1/ε)n time and poly(n)-space deterministic algorithm for computing a (1+ε) approximation to the volume of a general convex body K, which comes close to matching the (1+c/ε)n/2 lower bound for volume estimation in the oracle model by Bárány and Füredi (STOC 1986, Proc. Amer. Math. Soc. 1988). This improves on the previous results of Dadush and Vempala (Proc. Nat’l Acad. Sci. 20...

We show that all convex directed subgroups of a pl-group form distributive lattice under inclusions is Brouwer lattice. succeeded in extending some l-group results concerning rectifying and regular to the class $$ \mathcal{AO} -groups. Necessary sufficient conditions are given for an element be with unique value. In order prove this, properties lexicographic extensions -groups pl-groups investi...

In this paper, our purpose is twofold. Firstly, the tensor andresiduum operations on $L-$nested systems are introduced under thecondition of complete residuated lattice. Then we show that$L-$nested systems form a complete residuated lattice, which isprecisely the classical isomorphic object of complete residuatedpower set lattice. Thus the new representation theorem of$L-$subsets on complete re...