A sandwich in thin lie algebras

A thin Lie algebra is a Lie algebra $L$ , graded over the positive integers, with its first homogeneous component $L_1$ of dimension two and generating $L$ , and such that each non-zero ideal of $L$ lies between consecutive terms of its lower central series. All homogeneous components of a thin Lie algebra have dimension one or two, and the two-dimensional components are called diamonds. Suppose the second diamond of $L$ (that is, the next diamond past $L_1$ ) occurs in degree $k$ . We prove that if $k>5$ , then $[Lyy]=0$ for some non-zero element $y$ of $L_1$ . In characteristic different from two this means $y$ is a sandwich element of $L$ . We discuss the relevance of this fact in connection with an important theorem of Premet on sandwich elements in modular Lie algebras.

Translation and modulation invariant Hilbert spaces

Let $$\mathcal H$$ H be a Hilbert space of distributions on $$\mathbf{R}^{d}$$ R d which contains at least one non-zero element of the Feichtinger algebra $$S_0$$ S 0 and is continuously embedded in $$\mathscr {D}'$$ D ′ . If $$\mathcal H$$ H is translation and modulation invariant, also in the sense of its norm, then we prove that $$\mathcal H= L^2$$ H = L 2 , with the same norm apart from a multiplicative constant.

Boolean Zero Square (BZS) Semigroups

We introduce a new class of semigroups, that we call BZS - Boolean Zero Square-semigroups. A semigroup S with a zero element, 0, is said to be a BZS semigroup if, for every , we have or . We obtain some properties that describe the behaviour of the Green’s equivalence relations , , and . Necessary and sufficient conditions for a BZS semigroup to be a band and an inverse semigroup are obtained. A characterisation of a special type of BZS completely 0-simple semigroup is presented.

The Source of Semiprimeness of Semigroups

In this study, we define new semigroup structures using the set S S = a ∈ S | a S a = 0 which is called the source of semiprimeness for a semigroup S with zero element. S S − idempotent semigroup, S S − regular semigroup, S S − reduced semigroup, and S S − nonzero divisor semigroup which are generalizations of idempotent, regular, reduced, and nonzero divisor semigroups in semigroup theory are investigated, and their basic properties are determined. In addition, we adapt some well-known results in semigroup theory to these new semigroups.

On the Achievability of Blind Source Separation for High-Dimensional Nonlinear Source Mixtures

For many years, a combination of principal component analysis (PCA) and independent component analysis (ICA) has been used for blind source separation (BSS). However, it remains unclear why these linear methods work well with real-world data that involve nonlinear source mixtures. This work theoretically validates that a cascade of linear PCA and ICA can solve a nonlinear BSS problem accurately—when the sensory inputs are generated from hidden sources via nonlinear mappings with sufficient dimensionality. Our proposed theorem, termed the asymptotic linearization theorem, theoretically guarantees that applying linear PCA to the inputs can reliably extract a subspace spanned by the linear projections from every hidden source as the major components—and thus projecting the inputs onto their major eigenspace can effectively recover a linear transformation of the hidden sources. Then subsequent application of linear ICA can separate all the true independent hidden sources accurately. Zero-element-wise-error nonlinear BSS is asymptotically attained when the source dimensionality is large and the input dimensionality is sufficiently larger than the source dimensionality. Our proposed “Data Availability” section just before the Acknowledgments is validated analytically and numerically. Moreover, the same computation can be performed by using Hebbian-like plasticity rules, implying the biological plausibility of this nonlinear BSS strategy. Our results highlight the utility of linear PCA and ICA for accurately and reliably recovering nonlinearly mixed sources and suggest the importance of employing sensors with sufficient dimensionality to identify true hidden sources of real-world data.

Efficient Zero Ring Labeling of Graphs

A zero ring is a ring in which the product of any two elements is zero, which is the additive identity. A zero ring labeling of a graph is an assignment of distinct elements of a zero ring to the vertices of the graph such that the sum of the labels of any two adjacent vertices is not the zero element in the ring. Given a zero ring labeling of a graph, if the cardinality of the set of distinct sums obtained from all adjacent vertices is equal to the maximum degree of the graph, then the zero ring labeling is efficient. In this paper, we showed the existence of an efficient zero ring labeling for some classes of trees and their disjoint union. In particular, we showed that an efficient zero ring labeling exists for some families of the following classes of trees: path graphs, star graphs, bistars, centipede graphs, caterpillars, spiders, lobsters, and rooted trees. We also showed results for other common classes of graphs.

A note on the generalized nilpotent elements of a module

In this paper, we introduce the notion of the generalized nilpotent element of a module. In \cite{Groenewald}, the notion of nilpotent element of a module is introduced in the following sense: a non-zero element $m$ of an $R$-module $M$ is said to be nilpotent if there exists some $a\in R$ such that $a^k m=0$ but $am\neq 0$ for some $k\in \mathbb N$. In our present work we aim to generalize this notion. We have extended this notion to the strongly nilpotent element of a module.

Quasi-automatic semigroups

Quasi-automatic semigroups are extensions of a Cayley graph of an automatic group. Of course, a quasi-automatic semigroup generalizes an automatic semigroup. We observe that a semigroup [Formula: see text] may be automatic only when [Formula: see text] is finitely generated, while a semigroup may be quasi-automatic but it is not necessary finitely generated. Similar to the usual automatic semigroups, a quasi-automatic semigroup is closed under direct and free products. Furthermore, a semigroup [Formula: see text] is graph automatic if and only if [Formula: see text] with a zero element adjoined is graph automatic, and also a semigroup [Formula: see text] is graph automatic if and only if [Formula: see text] with an identity element adjoined is graph automatic. However, the class of quasi-automatic semigroups is a much wider class than the class of automatic semigroups. In this paper, we show that every automatic semigroup is quasi-automatic but the converse statement is not true (see Example 3.6). In addition, we notice that the quasi-automatic semigroups are invariant under the changing of generators, while a semigroup may be automatic with respect to a finite generating set but not the other. Finally, the connection between the quasi-automaticity of two semigroups [Formula: see text] and [Formula: see text], where [Formula: see text] is a subsemigroup with finite Rees index in [Formula: see text] will be investigated and considered.

The periodic system of chemical elements solid model in the geological aspect

The topicality of the research comes from the need to obtain new knowledge about the manifestation of the periodic law in nature. Research aim is to associate the periodic system of chemical elements with the chemical composition and structure of natural objects. The research method suggests the creation of a solid model of the periodic system of chemical elements along with its comparison with ore formation objects as well as the manifestation of chemical elements isomorphism and some natural processes geochemistry. Research results. The solid version of the periodic table of the first 95 chemical elements together with a conventional zero element is proposed. Each volume cell characterizes a chemical element with an elementary crystal lattice of simple substance. Similar models can be composed of minerals and rocks associating with material substance of the earth's crust. 16 vertical groups in the model are arranged in a snake-like pattern. The model of the earth's crust with the “cubes” of chemical elements, minerals and mineral associations is proposed. The elements of adjacent spatial groups are naturally concentrated in combination, showing isomorphism while minerals enter the crystal lattice. The relative position of adjacent “cubes” follows the rule of translation in mutually perpendicular directions. The chemical elements of the first group can correspond spatially to volcanoes as well as mud volcanoes. The place of the zero chemical element is considered to be occupied by the elements of adjacent spatial groups. It is assumed that the faces of the “cube” of chemical elements are permeable areas through which chemical elements can be transferred. Summary. The confirmation of the model follows while considering ore formations, isomorphism of chemical elements in minerals and geochemistry of volcanic processes. 46 "Izvestiya vysshikh uchebnykh zavedenii. Gornyi zhurnal". No. 1. 2020 ISSN 0536-1028 Key words: chemical elements; solid model of periodic system; ore formations; isomorphism in crystals; geochemistry of volcanic processes.

Informal Complete Metric Space and Fixed Point Theorems

The concept of informal vector space is introduced in this paper. In informal vector space, the additive inverse element does not necessarily exist. The reason is that an element in informal vector space which subtracts itself cannot be a zero element. An informal vector space can also be endowed with a metric to define a so-called informal metric space. The completeness of informal metric space can be defined according to the similar concept of a Cauchy sequence. A new concept of fixed point and the related results are studied in informal complete metric space.