The Norm of a Skew Polynomial

Let D be a finite-dimensional division algebra over its center and R = D[t;σ,δ] a skew polynomial ring. Under certain assumptions on δ and σ, the ring of central quotients D(t;σ,δ) = {f/g|f ∈ D[t;σ,δ],g ∈ C(D[t;σ,δ])} of D[t;σ,δ] is a central simple algebra with reduced norm N. We calculate the norm N(f) for some skew polynomials f ∈ R and investigate when and how the reducibility of N(f) reflects the reducibility of f.

A generalisation of Amitsur’s A-polynomials

We find examples of polynomials f ∈ D [t; σ, δ] whose eigenring ℰ(f) is a central simple algebra over the field F = C ∩ Fix(σ) ∩ Const(δ).

A brief guide to producing a national population projection

Background There are surprisingly few resources available which offer an introductory guide to preparing a national population projection using a cohort-component model. Many demography textbooks cover projections quite briefly, and many academic papers on projections focus on advanced technical issues. Aims The aim of this paper is to provide a short and accessible guide to producing a national-scale population projection using the cohort-component model. Data and methods The paper describes the cohort-component model from a population accounting perspective, presents all the necessary projection calculations, and covers the key steps which form part of the projections preparation process – from gathering input data to validating outputs. An accompanying Excel workbook implements the model and contains example projections for Australia. Conclusions Calculating a national population projection using a cohort-component model involves fairly simple algebra, but the broader projections preparation process is more complex, and requires careful consideration and judgement.

Splitting quaternion algebras defined over a finite field extension

We study systems of quadratic forms over fields and their isotropy over [Formula: see text]-extensions. We apply this to obtain particular splitting fields for quaternion algebras defined over a finite field extension. As a consequence we obtain that every central simple algebra of degree [Formula: see text] is split by a [Formula: see text]-extension of degree at most [Formula: see text].

The descending central series of higher commutators for simple algebras

This paper characterizes the potential behaviors of higher commutators in a simple algebra.

Analisis Kesalahan Mahasiswa PGSD pada Tugas Aljabar Sederhana Mata Kuliah Konsep Dasar Mapel Matematika SD

This study aimed to identify the errors of PGSD students on simple algebra assignments on Basic Concept of Elementary Mathematics Subject at STAHN Mpu Kuturan Singaraja. The subjects of this study were Semester II PGSD students of STAHN Mpu Kuturan Singaraja in the academic year 2019/2020 as many as 11 people. Errors in mathematics in this study were divided into factual errors, procedural errors, and conceptual errors. This type of research was a quantitative descriptive study. The data collection method used was a test. The test instrument used was a concept understanding test to be able to find students' errors. A total of 20 items in the test were validated by two experts with the Lawshe’s CVR technique. The results showed that (1) the general error rate of PGSD study program students on Simple Algebra assignment on Basic Concept of Elementary Mathematics Subject was 30.26% in the low error category, (2) there was the highest error of 43% (error category moderate) and lowest 16% (very low error category), 3) factual errors ranged from 10% - 20% (very low category), 4) procedural errors ranged from 7% - 53% (very low category to medium category), and 5) conceptual errors ranged from 35% - 65% (low to high categories).

Multiplicity One for Pairs of Prasad–Takloo-Bighash Type

Let ${\textrm{E}}/{\textrm{F}}$ be a quadratic extension of non-archimedean local fields of characteristic different from $2$. Let ${\textrm{A}}$ be an ${\textrm{F}}$-central simple algebra of even dimension so that it contains ${\textrm{E}}$ as a subfield, set ${\textrm{G}}={\textrm{A}}^\times $ and ${\textrm{H}}$ for the centralizer of ${\textrm{E}}^\times $ in ${\textrm{G}}$. Using a Galois descent argument, we prove that all double cosets ${\textrm{H}} g {\textrm{H}}\subset{\textrm{G}}$ are stable under the anti-involution $g\mapsto g^{-1}$, reducing to Guo’s result for ${\textrm{F}}$-split ${\textrm{G}}$ [14], which we extend to fields of positive characteristic different from $2$. We then show, combining global and local results, that ${\textrm{H}}$-distinguished irreducible representations of ${\textrm{G}}$ are self-dual and this implies that $({\textrm{G}},{\textrm{H}})$ is a Gelfand pair $$\begin{equation*}\operatorname{dim}_{\mathbb{C}}(\operatorname{Hom}_{{\textrm{H}}}(\pi,\mathbb{C}))\leq 1\end{equation*}$$for all smooth irreducible representations $\pi $ of ${\textrm{G}}$. Finally we explain how to obtain the multiplicity one statement in the archimedean case using the criteria of Aizenbud and Gourevitch ([1]), and we then show self-duality of irreducible distinguished representations in the archimedean case too.

Fast 2D full-waveform modeling and inversion using the Schur complement approach

In most full-waveform inversion (FWI) problems, sufficient prior information is available to constrain the velocity of certain parts of the model, e.g., the water column or, in some cases, near-surface velocities. We take advantage of this situation and develop a fast Schur-complement-based forward modeling and inversion approach by partitioning the velocity model into two parts. The first part consists of the constrained zone that does not change during the inversion, whereas the second part is the anomalous zone to be updated during the inversion. For this decomposition, we partially factorize the governing system of linear equations by computing a Schur complement for the anomalous zone. The Schur complement system is then solved to compute the fields in the anomalous zone, which are then back substituted to compute the fields in the constrained region. For each successive modeling steps with new anomalous zone velocities, the corresponding Schur complement is easily computed using simple algebra. Because the anomalous part of the model is comparatively smaller than the whole model, considerable computational savings can be achieved using our Schur approach. Additionally, we showed that the Schur complement method maintains the accuracy of standard frequency-domain finite difference formulations, but this comes at a slightly higher peak memory requirement. Our FWI workflow shows reduced runtime by 15%–57% depending upon the depth of the water column without losing any accuracy compared to the standard method.

On H-simple not necessarily associative algebras

An algebra [Formula: see text] with a generalized [Formula: see text]-action is a generalization of an [Formula: see text]-module algebra where [Formula: see text] is just an associative algebra with [Formula: see text] and a relaxed compatibility condition between the multiplication in [Formula: see text] and the [Formula: see text]-action on [Formula: see text] holds. At first glance, this notion may appear too general, however, it enables to work with algebras endowed with various kinds of additional structures (e.g. comodule algebras over Hopf algebras, graded algebras, algebras with an action of a semigroup by anti-endomorphisms). This approach proves to be especially fruitful in the theory of polynomial identities. We show that if [Formula: see text] is a finite dimensional (not necessarily associative) algebra over a field of characteristic [Formula: see text] and [Formula: see text] is simple with respect to a generalized [Formula: see text]-action, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of polynomial [Formula: see text]-identities of [Formula: see text]. In particular, if [Formula: see text] is a finite dimensional (not necessarily group graded) graded-simple algebra, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of graded polynomial identities of [Formula: see text]. In addition, we study the free-forgetful adjunctions corresponding to (not necessarily group) gradings and generalized [Formula: see text]-actions.