Search Results (7)

Let ${\left(k_{\mu }\right)}_{\mu =1}^4$ be a quartet of cyclic cubic number fields sharing a common conductor $c=pqr$ divisible by exactly three prime(power)s, $p,q,r$. For those components of the quartet whose 3-class group ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq {(\mathbb{Z}/3\mathbb{Z})}^2$ is elementary bicyclic, the automorphism group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$ of the maximal metabelian unramified 3-extension of $k_{\mu }$ is determined by conditions for cubic residue symbols between $p,q,r$ and for ambiguous principal ideals in subfields of the common absolute 3-genus field $k^{*}$ of all $k_{\mu }$. With the aid of the relation rank $d_2\left(\mathfrak{M}\right)$, it is decided whether $\mathfrak{M}$ coincides with the Galois group $\mathfrak{G}=\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k_{\mu }\right)/k_{\mu })$ of the maximal unramified pro-3-extension of $k_{\mu }$. Full article

For the characterization of multipliers of ${\mathit{L}}^p\left({\mathbb{R}}^d\right)$ or more generally, of ${\mathit{L}}^p\left(G\right)$ for some locally compact Abelian group G, the so-called Figa-Talamanca–Herz algebra ${\mathit{A}}_p\left(G\right)$ plays an important role. Following Larsen’s book, we describe multipliers as bounded linear operators that commute with translations. The main result of this paper is the characterization of the multipliers of ${\mathit{A}}_p\left(G\right)$. In fact, we demonstrate that it coincides with the space of multipliers of $\left({\mathit{L}}^p\left(G\right),{\parallel ·\parallel }_p\right)$. Given a multiplier T of $({\mathit{A}}_p\left(G\right),\parallel ·{\parallel }_{{\mathit{A}}_p\left(G\right)})$ and using the embedding $({\mathit{A}}_p\left(G\right){,\parallel ·\parallel }_{{\mathit{A}}_p\left(G\right)})\hookrightarrow \left({\mathit{C}}_0\left(G\right),{\parallel ·\parallel }_{\infty }\right)$, the linear functional $f\mapsto \left[T\right(f\left)\right(0\left)\right]$ is bounded, and T can be written as a moving average for some element in the dual $\mathit{P}{\mathit{M}}_p\left(G\right)$ of $({\mathit{A}}_p\left(G\right),\parallel ·{\parallel }_{{\mathit{A}}_p\left(G\right)})$. A key step for this identification is another elementary fact: showing that the multipliers from $\left({\mathit{L}}^p\left(G\right),{\parallel ·\parallel }_p\right)$ to $\left({\mathit{C}}_0\left(G\right),{\parallel ·\parallel }_{\infty }\right)$ are exactly the convolution operators with kernels in $\left({\mathit{L}}^q\left(G\right),{\parallel ·\parallel }_q\right)$ for $1<p<\infty$ and $1/p+1/q=1$. The proofs make use of the space of mild distributions, which is the dual of the Segal algebra $\left({\mathit{S}}_0\left(G\right),{\parallel ·\parallel }_{{\mathit{S}}_0}\right)$, and the fact that multipliers T from ${\mathit{S}}_0\left(G\right)$ to ${\mathit{S}}_0^{\prime }\left(G\right)$ are convolution operators of the form $T:f\mapsto \sigma \ast f$ for some uniquely determined $\sigma \in {\mathit{S}}_0^{\prime }$. This setting also allows us to switch from the description of these multipliers as convolution operators (by suitable pseudomeasures) to their description as Fourier multipliers, using the extended Fourier transform in the setting of $\left({\mathit{S}}_0^{\prime }\left(G\right),{\parallel ·\parallel }_{{\mathit{S}}_0^{\prime }}\right)$. The approach presented here extends to other function spaces, but a more detailed discussion is left to future publications. Full article

Let $\Gamma$ be a graph and $G⩽Aut(\Gamma )$. A graph $\Gamma$ can be called G-arc-transitive (GAT) if G acts transitively on its arc set. A regular covering projection $p:\overline{\Gamma }\to \Gamma$ is arc-transitive (AT) if an AT subgroup of $Aut(\Gamma )$ lifts under p. In this study, by applying a number of concepts in linear algebra such as invariant subspaces (IVs) of matrix groups (MGs), we discuss regular AT elementary abelian covers (R-AT-EA-covers) of the $C13$ graph. Full article

Let M be a commutative cancellative monoid with an element of infinite order. The binary operation can be extended to all finite subsets of M by the pointwise definition. So, we can consider the theory of finite subsets of M. Earlier, we have proved the following result: in the theory of finite subsets of M elementary arithmetic can be interpreted. In particular, this theory is undecidable. For example, the free monoid (the sets of all words with concatenation) has this property, the corresponding algebra of finite subsets is the theory of all finite languages with concatenation. Another example is an arbitrary Abelian group that is not a torsion group. But the method of proof significantly used an element of infinite order, hence, it can’t be immediately generalized to torsion groups. In this paper we prove the given theorem for Abelian torsion groups that have elements of unbounded order: for such group, the theory of finite subsets allows interpreting the elementary arithmetic. Full article

This paper is supposed to form a keystone towards a new and alternative approach to Fourier analysis over LCA (locally compact Abelian) groups G. In an earlier paper the author has already shown that one can introduce convolution and the Fourier–Stieltjes transform on $(\mathit{M}\left(G\right),\parallel ·{\parallel }_{\mathit{M}})$, the space of bounded measures (viewed as a space of linear functionals) in an elementary fashion over ${\mathbb{R}}^d$. Bounded uniform partitions of unity (BUPUs) are easily constructed in the Euclidean setting (by dilation). Moving on to general LCA groups, it becomes an interesting challenge to find ways to construct arbitrary fine BUPUs, ideally without the use of structure theory, the existence of a Haar measure and even Lebesgue integration. This article provides such a construction and demonstrates how it can be used in order to show that any so-called homogeneous Banach space$(\mathit{B},\parallel ·{\parallel }_{\mathit{B}})$ on G, such as $({\mathit{L}}^p\left(G\right),{\parallel ·\parallel }_p)$, for $1\le p<\infty$, or the Fourier–Stieltjes algebra $\mathcal{F}\mathit{M}\left(G\right)$, and in particular any Segal algebra is a Banach convolution module over $(\mathit{M}\left(G\right),\parallel ·{\parallel }_{\mathit{M}})$ in a natural way. Via the Haar measure we can then identify $\left({\mathit{L}}^1\left(G\right),{\parallel ·\parallel }_1\right)$ with the closure (of the embedded version) of ${\mathit{C}}_c\left(G\right)$, the space of continuous functions with compact support, in $(\mathit{M}\left(G\right),\parallel ·{\parallel }_{\mathit{M}})$, and show that these homogeneous Banach spaces are essential${\mathit{L}}^1\left(G\right)$-modules. Thus, in particular, the approximate units act properly as one might expect and converge strongly to the identity operator. The approach is in the spirit of Hans Reiter, avoiding the use of structure theory for LCA groups and the usual techniques of vector-valued integration via duality. The ultimate (still distant) goal of this approach is to provide a new and elementary approach towards the (extended) Fourier transform in the setting of the so-called Banach–Gelfand triple$({\mathit{S}}_0,{\mathit{L}}^2,{\mathit{S}}_0^{\prime })\left(G\right)$, based on the Segal algebra ${\mathit{S}}_0\left(G\right)$. This direction will be pursued in subsequent papers. Full article

In this paper, we mainly count the number of subgroup chains of a finite nilpotent group. We derive a recursive formula that reduces the counting problem to that of finite p-groups. As applications of our main result, the classification problem of distinct fuzzy subgroups of finite abelian groups is reduced to that of finite abelian p-groups. In particular, an explicit recursive formula for the number of distinct fuzzy subgroups of a finite abelian group whose Sylow subgroups are cyclic groups or elementary abelian groups is given. Full article

The Banach Gelfand Triple $({\mathit{S}}_0,{\mathit{L}}^2,{\mathit{S}}_0^{\prime })\left({\mathbb{R}}^d\right)$ consists of $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),{\parallel ·\parallel }_{{\mathit{S}}_0}\right)$ , a very specific Segal algebra as algebra of test functions, the Hilbert space $\left({\mathit{L}}^2\left({\mathbb{R}}^d\right),{\parallel ·\parallel }_2\right)$ and the dual space ${\mathit{S}}_0^{\prime }\left({\mathbb{R}}^d\right)$ , whose elements are also called “mild distributions”. Together they provide a universal tool for Fourier Analysis in its many manifestations. It is indispensable for a proper formulation of Gabor Analysis, but also useful for a distributional description of the classical (generalized) Fourier transform (with Plancherel’s Theorem and the Fourier Inversion Theorem as core statements) or the foundations of Abstract Harmonic Analysis, as it is not difficult to formulate this theory in the context of locally compact Abelian (LCA) groups. A new approach presented recently allows to introduce $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),{\parallel ·\parallel }_{{\mathit{S}}_0}\right)$ and hence $({\mathit{S}}_0^{\prime }\left({\mathbb{R}}^d\right),\parallel ·{\parallel }_{{\mathit{S}}_0^{\prime }})$ , the space of “mild distributions”, without the use of the Lebesgue integral or the theory of tempered distributions. The present notes will describe an alternative, even more elementary approach to the same objects, based on the idea of completion (in an appropriate sense). By drawing the analogy to the real number system, viewed as infinite decimals, we hope that this approach is also more interesting for engineers. Of course it is very much inspired by the Lighthill approach to the theory of tempered distributions. The main topic of this article is thus an outline of the sequential approach in this concrete setting and the clarification of the fact that it is just another way of describing the Banach Gelfand Triple. The objects of the extended domain for the Short-Time Fourier Transform are (equivalence classes) of so-called mild Cauchy sequences (in short ECmiCS). Representatives are sequences of bounded, continuous functions, which correspond in a natural way to mild distributions as introduced in earlier papers via duality theory. Our key result shows how standard functional analytic arguments combined with concrete properties of the Segal algebra $\left({\mathit{S}}_0\left({\mathbb{R}}^d\right),{\parallel ·\parallel }_{{\mathit{S}}_0}\right)$ can be used to establish this natural identification. Full article