Homogeneous Components Research Articles

Dependence between homogeneous components of polynomials with small degree of Poisson bracket

Dependence between homogeneous components of polynomials with small degree of Poisson bracket

Symmetric polynomials in free associative algebras

By a result of Margarete Wolf in 1936, we know that the algebra $K\langle X_d\rangle^{Sym(d)}$ of symmetric polynomials in noncommuting variables is not finitely generated. In 1984, Koryukin proved that if we equip the homogeneous component of degree $n$ with the additional action of $Sym(n)$ by permuting the positions of the variables, then the algebra of invariants $K\langle X_d\rangle^G$ of every reductive group $G$ is finitely generated. First, we make a short comparison between classical invariant theory of finite groups and its noncommutative counterpart. Then, we expose briefly the results of Wolf. Finally, we present the main result of our paper, which is, over a field of characteristic 0 or of characteristic $p>d$, the algebra $K\langle X_d\rangle^{Sym(d)}$ with the action of Koryukin is generated by the elementary symmetric polynomials.

Вполне инертные подгруппы вполне разложимых групп, имеющие однородные компоненты конечного ранга

Вполне инертные подгруппы вполне разложимых групп, имеющие однородные компоненты конечного ранга

Legend of Cadmus

During one hundred years of the development of archaeology in southeast Europe, one of the dominant issues was the origin of communities that were recognized by former researchers as a homogenous component named the Illyrians. The issue of their origin was observed through different interpretations of wethnogenetic processes which numerous researchers tried to use to create different developmental phases of creating communities named after the Illyrians. By combining the most recent interpretations of concepts of myths and legends of the past communities, written data containing live memories of these myths during the past two centuries, as well as archaeological and historical data, the relationship and complexities of the oldest myths and legends, such as the one of Cadmus and Harmonia, are reimposed as they suggest the origin of the entire term of the Illyrians as a geographic and religious concept entirely dichotomous compared to, seemingly, misinterpretations of the wider ethnic context of the prehistoric Balkans.

ON GRADED S-PRIME SUBMODULES OF GRADED MODULES OVER GRADED COMMUTATIVE RINGS

Let $G$ be an abelian group with identity $e$. Let $R$ be a $G$-graded commutative ring with identity, $M$ a graded $R$-module and $S\subseteq h(R)$ a multiplicatively closed subset of $R$. In this paper, we introduce the concept of graded $S$-prime submodules of graded modules over graded commutative rings. We investigate some properties of this class of graded submodules and their homogeneous components. Let $N$ be a graded submodule of $M$ such that $(N:_{R}M)\cap S=\emptyset $. We say that $N$ is \textit{a graded }$S$\textit{-prime submodule of }$M$ if there exists $s_{g}\in S$ and whenever $a_{h}m_{i}\in N,$ then either $s_{g}a_{h}\in (N:_{R}M)$ or $s_{g}m_{i}\in N$ for each $a_{h}\in h(R) $ and $m_{i}\in h(M).$

The Terwilliger algebra of the halved [formula omitted]-cube from the viewpoint of its automorphism group action

Let 12H(n,2) denote the halved n-cube with vertex set X and let T≔T(x0) denote the Terwilliger algebra of 12H(n,2) with respect to a fixed vertex x0∈X. In this paper, we assume n≥6. We first characterize T by considering the action of the automorphism group of 12H(n,2) on the set X×X×X. We show that T coincides with the centralizer algebra of the stabilizer of x0 in the automorphism group, and display three subalgebras of T further. Then we study the homogeneous components of V≔ℂX, each of which is a nonzero subspace of V spanned by the irreducible T-modules that are isomorphic. We give a computable basis for any homogeneous component of V. Finally, we describe the decomposition of T via its block-diagonalization and give a basis for the center of T by using the above homogeneous components of V.

A very short note on the (rational) graded Hori map

The graded Hori map has been recently introduced by Han-Mathai in the context of T-duality as a -graded transform whose homogeneous components are the Hori-Fourier transforms in twisted cohomology associated with integral multiples of a basic pair of T-dual closed 3-forms. We show how in the rational homotopy theory approximation of T-duality, such a map is naturally realized as a pull-iso-push transform, where the isomorphism part corresponds to the canonical equivalence between the left and the right gerbes associated with a T-duality configuration.

Spatio-temporal bands of coronal bright points and their relation to solar torsional oscillations

ABSTRACT Using automatic processing of daily fits files from SOHO_EIT, we studied the temporal evolution of the latitudinal distribution of bright coronal points from 1995 to 2020. The study revealed homogeneous and inhomogeneous components of the latitudinal distribution. The homogeneous component has cyclical variations with a significant phase shift compared to the number of sunspots. The residual inhomogeneous component manifests itself as migration of the centre of increased density from high to low latitudes. The bands of the residual density of coronal bright points (CBP) reflect solar torsional oscillations. It should be associated with the strengthening of the local alpha-omega dynamo. However, the cyclical nature of the number of CBP is explained by the diffusion of large-scale magnetic fluxes.

Diamond distances in Nottingham algebras

Nottingham algebras are a class of just-infinite-dimensional, modular, [Formula: see text]-graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. Homogeneous components of a Nottingham algebra have dimension one or two, and in the latter case they are called diamonds. The first diamond occurs in degree [Formula: see text], and the second occurs in degree [Formula: see text], a power of the characteristic. Many examples of Nottingham algebras are known, in which each diamond past the first can be assigned a type, either belonging to the underlying field or equal to [Formula: see text]. A prospective classification of Nottingham algebras requires describing all possible diamond patterns. In this paper, we establish some crucial contributions towards that goal. One is showing that all diamonds, past the first, of an arbitrary Nottingham algebra [Formula: see text] can be assigned a type, in such a way that the degrees and types of the diamonds completely describe [Formula: see text]. At the same time we prove that the difference in degrees of any two consecutive diamonds in any Nottingham algebra equals [Formula: see text]. As a side-product of our investigation, we classify the Nottingham algebras where all diamonds have type [Formula: see text].

Stochastic comparisons of series and parallel systems with dependent log-logistic components

This paper studies the usual stochastic, star, convex transform, and dispersive orders of both series and parallel systems comprised of dependent log-logistic components. We establish that, without any restriction on the parameters, the lifetime of a parallel or series system with dependent components is smaller than that with dependent homogeneous components in the sense of the dispersive orders. Sufficient conditions are established for the star ordering between the lifetimes of series and parallel systems consisting of multiple outlier dependent log-logistic distributed components. Under certain conditions on the Archimedean copula and the parameter, we also discuss convex transform order of series and parallel systems.

Alteration of power law scaling of spontaneous brain activity in schizophrenia

Nonlinear dynamical analysis has been used to quantify the complexity of brain signal at temporal scales. Power law scaling is a well-validated method in physics that has been used to describe the dynamics of a system in the frequency domain, ranging from noisy oscillation to complex fluctuations. In this research, we investigated the power-law characteristics in a large-scale resting-state fMRI data of schizophrenia and healthy participants derived from Taiwan Aging and Mental Illness cohort. We extracted the power spectral density (PSD) of resting signal by Fourier transform. Power law scaling of PSD was estimated by determining the slope of the regression line fitting to the logarithm of PSD. t-Test was used to assess the statistical difference in power law scaling between schizophrenia and healthy participants. The significant differences in power law scaling were found in six brain regions. Schizophrenia patients have significantly more positive power law scaling (i.e., more homogenous frequency components) at four brain regions: left precuneus, left medial dorsal nucleus, right inferior frontal gyrus, and right middle temporal gyrus and less positive power law scaling (i.e., more dominant at lower frequency range) in bilateral putamen compared with healthy participants. Moreover, significant correlations of power law scaling with the severity of psychosis were found. These findings suggest that schizophrenia has abnormal brain signal complexity linked to psychotic symptoms. The power law scaling represents the dynamical properties of resting-state fMRI signal may serve as a novel functional brain imaging marker for evaluating patients with mental illness.

ЗАСТОСУВАННЯ МЕТОДІВ ДЕКОМПОЗИЦІЇ У КОРОТКОСТРОКОВОМУ ПРОГНОЗУВАННІ СУМАРНОГО ЕЛЕКТРИЧНОГО НАВАНТАЖЕННЯ ЕНЕРГОСИСТЕМИ

Based on the performed researches the method of decomposition of graphs of total electric loading of power system with application of a method of Hilbert-Huang is improved. This approach allows obtaining a homogeneous basic component of electrical load and temperature component, which has a close correlation with air temperature, which improves the accuracy of short-term forecasting. The results of testing the developed mathematical model are given. Ref. 9, fig. 1, table.

Deformation behavior and fracture mechanism of a novel laminated Ni/Al sheets during in-situ tensile process

For integrated forming and reaction preparation of NiAl complex thin-walled component, a novel laminated Ni/Al sheets (LNAS) with diffused intermediate zone are fabricated using hot pressing in vacuum. Based on the tensile tests at different temperatures and strain rates, the LNAS show considerable deformability. To study the deformation process and fracture mechanism of LNAS, in-situ tensile test has been utilized. The results show that the initiation and propagation of microcracks mainly occur in diffused intermediate zone and the microcracks are perpendicular to the Ni (Al) foils. The necking and fracture of LNAS initially take place in the Ni foils. Moreover, a W-shaped NiAl component is obtained using LNAS at room-temperature. In particular, the microcracks generated in the diffused intermediate zone of deformed LNAS can self-repair through subsequent high temperature reaction after forming and homogeneous NiAl metallic component can be prepared.

Graded φ-2-absorbing hyperideals in graded multiplicative hyperrings

Let G be an abelian group with identity e. Let R be a graded multiplicative hyperring and [Formula: see text] be a function where [Formula: see text] is the set of graded hyperideals of [Formula: see text] and [Formula: see text] for every graded hyperideal I of R. In this paper, we introduce and study the concepts of graded [Formula: see text]-2-absorbing hyperideals and graded [Formula: see text]-2-absorbing primary hyperideals of [Formula: see text]. Moreover, we give a number of main results and the basic properties concerning these classes of graded hyperideals and their homogeneous components.

Constituents of graded Lie algebras of maximal class and chains of 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 nonzero 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. If L 1 is the only diamond, then L is a graded Lie algebra of maximal class. We present simpler proofs of some fundamental facts on graded Lie algebras of maximal class, and on thin Lie algebras, based on a uniform method, with emphasis on a polynomial interpretation. Among else, we determine the possible values for the most fundamental parameter of such algebras, which is one less than the dimension of their largest metabelian quotient.

Manipulating Chemical Processes by Pseudosolid Spatial Limitation.

It is always preferred to perform chemical processes in liquid or gas phases because of the merits of operation convenience, reaction efficiency, and component homogeneity. However, tremendous efforts have to be made to purify the final product and minimize procedure losses unless a well-defined chemical mechanism is found. Herein, an unconventional chemical functioning system accommodating molecule-in-pseudosolid manipulation is reported. It entails the properties of enhanced molecular effective collision and directional guidance for delicate chemical reaction spatial controls. This design achieves facilitated rates on multicomponent chemical reactions with pros of unique simultaneous final product separation through intrapseudosolid spatial limitation. Localized homogeneous component mixing, pronounced molecular collision, and pure product separation happening in this action surmount the obstacles of conventional chemical operation with a straightforward sketch. A path toward fine chemistry is therefore paved, where traditional thoughts on beneficial reaction environments may be reconsidered.

Microstructure and Superconducting Properties of Rapid Heating, Quenching, and Transformation (RHQT) Powder‐in‐Tube Nb3Al Wires Doped with Sn Element

Properties of rapid heating, quenching, and transformation (RHQT)‐treated powder‐in‐tube (PIT) Nb3Al1−xSnx wires are studied herein. After the RHQ process, the body‐centered cubic (bcc) phase is formed in the Sn‐doped Nb3Al wires. The transformed superconducting A15 phase shows a uniform microstructure and homogenous component distribution feature. With the increase of the Sn doping level, the critical transition temperature (Tc), critical current density (Jc), and irreversibility field (Birr) of the superconducting wires increase first and then reduce, achieving the highest Tc value of 17.6 K and the smallest ΔTc of 0.6 K in the 4% Sn doped Nb3Al wire. The 4% Sn addition helps to improve the layer Jc of the sample to 2.1 × 105 A cm−2 at 8 K and 5 T, which is about 2.5 times that of the pure Nb3Al sample. Furthermore, the Nb3Al1−xSnx (x = 0.04) wire gives the highest Birr value of 17.22, 13.23, and 9.11 T at 8, 10, and 12 K, respectively. The addition of Sn helps to achieve stoichiometric Nb3Al A15 phase and formation of precipitations with the size of 10 nm, which might act as the flux‐pinning center.

A Basic Study on the Absorption Properties and Their Prediction of Heterogeneous Micro-Perforated Panels: A Case Study of Micro-Perforated Panels with Heterogeneous Hole Size and Perforation Ratio

Micro-perforated panels (MPPs) are one of the most promising alternatives to conventional porous sound-absorbing materials. Traditionally, the theory of the sound absorption properties of MPPs is based on the assumption that MPPs are a homogeneous material with identical pores at regular intervals. However, in recent years, some MPPs have not met these conditions, and although a variety of designs have been created, their properties and prediction methods were studied in only fewer works. In this paper, considering the wide variety of MPP designs, we made a trial production of heterogeneous MPPs, which are MPPs with holes of different diameters, and studied the prediction method applicable to these MPPs. We measured the normal incidence sound absorption characteristics of those MPPs, backed by a rigid backing and air-cavity in-between, in an impedance tube. The prediction method proposed in this work is to treat the heterogeneous MPPs as combinations of several homogeneous components, and to combine them after applying the existing theory on homogeneous MPPs to each component. As a result, except in a few cases, the measured and predicted values of the absorption properties agreed relatively well. We also found that the arrangement of the holes in the material and the depth of the back cavity affected the agreement between the measured and predicted results.

Stochastic comparisons of residual lives of series-parallel and parallel-series systems with independent subsystems consisting of dependent components

In this article, we consider residual lives of series-parallel and parallel-series systems with independent subsystems consisting of dependent homogeneous components with the joint distribution of their lifetimes being modeled by an Archimedean copula. By considering two such systems with different numbers of components within each subsystem, we establish various stochastic orderings and aging results for the residual lives of series-parallel and parallel-series systems. The results established here are quite general. Several examples are presented to illustrate all the established results and their reliability implications.

A Preliminary Study on Microplastic Occurrences in Surface Waters of Ousudu Lake, Pondicherry, India.

Microplastics (MP) in both freshwater and marine ecosystem is the next issue that has been recently the fore. However, freshwaters are the potential source and pathway to transport MPs to the marine environment. There is a lack of understanding about the presence and analysis of MP in the freshwater system in India, one of the leading global plastic producers and consumers responsible for 5.6 million tonnes of waste every year. The present attempt is the first in Ousudu Lake, Puducherry, Tamil Nadu, to study the MPs occurrence and examine its properties in the surface water of the largest lake and an important wetland. The concentration range of MP in Ousudu Lake was 0.0039 particles/m2. Fibre was dominant, homogenous with the size fraction of ≤100µm comprised of total MPs collected in Lake. Raman spectroscopy identified plastics with polyethylene as a homogenous component. This study indicated that the presence of MP in water might influence the environment.