Signature Gr\"obner bases in free algebras over rings

We generalize signature Gr\"obner bases, previously studied in the free algebra over a field or polynomial rings over a ring, to ideals in the mixed algebra $R[x_1,...,x_k]\langle y_1,\dots,y_n \rangle$ where $R$ is a principal ideal domain. We give an algorithm for computing them, combining elements from the theory of commutative and noncommutative (signature) Gr\"obner bases, and prove its correctness. Applications include extensions of the free algebra with commutative variables, e.g., for homogenization purposes or for performing ideal theoretic operations such as intersections, and computations over $\mathbb{Z}$ as universal proofs over fields of arbitrary characteristic. By extending the signature cover criterion to our setting, our algorithm also lifts some technical restrictions from previous noncommutative signature-based algorithms, now allowing, e.g., elimination orderings. We provide a prototype implementation for the case when $R$ is a field, and show that our algorithm for the mixed algebra is more efficient than classical approaches using existing algorithms.Comment: 10 page

Direct Observation of Trapping Motion in Elongated Fourier-Transform Mass Spectrometry Trapped Ion Cells

AbstractMeasurements by Fourier-transform mass spectrometry (FTMS) have been used to measure trapping oscillation profiles in elongated trapped ion cells of length 10–43 cm. Trapping periods extracted from these profiles are found to vary linearly with cell length of elongated cells. This is in contrast with the prediction based on a quadrupolar approximation of the electric field that trapping period should increase exponentially with increased cell length. An alternate analytical expression for trapping motion is derived that better accounts for the motion of ions with sufficient energy to approach the trap plates. Calculated trapping frequencies are within a few percent of values determined from ion trajectory simulations for any combination of cell length, trap potential, ion mass, and ion kinetic energy. The new expression also explains the experimentally determined trapping data obtained in elongated cells. This expression predicts an average axial energy near 0.6 eV for the ions that are preferentially detected by FTMS with the specific pulse sequence employed

Computing elements of certain form in ideals to prove properties of operators

Proving statements about linear operators expressed in terms of identities often leads to finding elements of certain form in noncommutative polynomial ideals. We illustrate this by examples coming from actual operator statements and discuss relevant algorithmic methods for finding such polynomials based on noncommutative Gr\"obner bases. In particular, we present algorithms for computing the intersection of a two-sided ideal with a one-sided ideal as well as for computing homogeneous polynomials in two-sided ideals and monomials in one-sided ideals. All methods presented in this work are implemented in the Mathematica package OperatorGB.Comment: 26 page

Compatible rewriting of noncommutative polynomials for proving operator identities

The goal of this paper is to prove operator identities using equalities between noncommutative polynomials. In general, a polynomial expression is not valid in terms of operators, since it may not be compatible with domains and codomains of the corresponding operators. Recently, some of the authors introduced a framework based on labelled quivers to rigorously translate polynomial identities to operator identities. In the present paper, we extend and adapt the framework to the context of rewriting and polynomial reduction. We give a sufficient condition on the polynomials used for rewriting to ensure that standard polynomial reduction automatically respects domains and codomains of operators. Finally, we adapt the noncommutative Buchberger procedure to compute additional compatible polynomials for rewriting. In the package OperatorGB, we also provide an implementation of the concepts developed.Comment: 17 page

Flourishing or floundering? : Factors contributing to CLIL Primary Teachers' Wellbeing in Austria

CLIL approaches to teaching are part of the general increase in plurilingual approaches to education. Recent work on CLIL in primary education has shown how demanding this can be for teachers. To better understand the nature of these challenges and their potential impact on teacher wellbeing, this study investigated the situation of CLIL primary teachers in Austria. Data were collected via in-depth interviews and coded in an inductive manner using QCAmap software. The analysis of the data revealed that despite considerable individual variation in terms of how CLIL was put into practice, all the wellbeing of all of the teachers' in this study appeared to be threatened by a lack of teaching material, negative public perceptions of teachers generally, low societal appreciation of primary teachers specifically, and language-related challenges in teaching CLIL. In contrast, the teachers' relationships with their students and their conviction in the CLIL approach seemed to contribute positively to their sense of wellbeing in their professional roles. We conclude with some considerations for practice to ensure all primary CLIL teachers flourish in their professional roles.Els enfocaments AICLE formen part de l'increment generalitzat dels enfocaments plurilingües en educació. Treballs recents sobre l'educació primària han revelat com d'exigent pot arribar a ser l'AICLE per al professorat que l'imparteix. Amb la finalitat de millorar la nostra comprensió d'aquests reptes i del seu impacte potencial en el benestar del professorat, el present estudi investiga la situació de mestres AICLE a Àustria. Les dades obtingudes per mitjà d'entrevistes en profunditat van ser codificades mitjançant el programari QCAmap, seguint un procediment inductiu. L'anàlisi va revelar que malgrat les considerables variacions individuals en la forma en la qual l'AICLE s'impartia, el benestar global de totes les mestres participants en l'estudi semblava amenaçat per l'escassetat de material didàctic, les percepcions negatives d'altres docents en general, i específicament, la baixa estima mostrada cap als mestres de primària per la societat, així com els reptes lingüístics que la docència AICLE comporta. Per contra, la relació de les mestres amb els seus alumnes, i la seva confiança en l'enfocament AICLE semblen contribuir positivament a una sensació de benestar com a docents. Es conclou amb un conjunt de consideracions pràctiques amb la finalitat d'assegurar que tots els docents AICLE de primària avancin en la seva labor com a docents.Los enfoques AICLE forman parte del incremento generalizado de los enfoques plurilingües en educación. Trabajos recientes sobre la educación primaria han revelado cómo de exigente puede llegar a ser el AICLE para el profesorado que lo imparte. Con la finalidad de mejorar nuestra comprensión de estos retos y de su impacto potencial en el bienestar del profesorado, el presente estudio investiga la situación de maestras AICLE en Austria. Los datos obtenidos por medio de entrevistes en profundidad fueron codificados mediante el software QCAmap, siguiendo un procedimiento inductivo. El análisis reveló que a pesar de las considerables variaciones individuales en la forma en la que el AICLE se impartía, el bienestar global de todas las maestras participantes en el estudio parecía amenazado por la escasez de material didáctico, las percepciones negativas de otros profesores en general, y específicamente, la baja estima mostrada hacia los maestros de primaria por la sociedad, así como los retos lingüísticos que la docencia AICLE conlleva. Por el contrario, la relación de las maestras con sus alumnos, y su confianza en el enfoque AICLE parecen contribuir positivamente a una sensación de bienestar como docentes. Se concluye con un conjunto de consideraciones prácticas con la finalidad de asegurar que todos los docentes AICLE de primaria salen adelante en su labor como docentes

Consistency of randomized integration methods

For integrable functions we provide a weak law of large numbers for structured Monte Carlo methods, such as estimators based on randomized digital nets, Latin hypercube sampling, randomized Frolov point sets as well as Cranley-Patterson rotations. Moreover, we suggest median modified methods and show that for integrands in $L^p$ with $p>1$ a strong law of large numbers holds.Comment: 12 page

Estudo da região etmoidal de Liophis jaegeri jaegeri, Philodryas olfersii e Thamnodynastes strigatus, a partir da reconstrução de modelos tridimensionais digitais e as adaptações morfológicas ao hábitat utilizado

The Neotropical region shows an extremely diverse snake fauna. These animals present great morphological and ecological variation in relation to the microhabitat analyzed. Nevertheless, few studies can qualify the use of skull development in the relation between animal and habitat. The results of such studies can contribute to about the research on adaptation to the habitat use and morphology. This work aims to reconstruct a threedimensional virtual model through histological sections of Liophis jaegeri jaegeri’s, Philodryas olfersii’s and Thamnodynastes strigatus’s ethmoidal region, to contribute to the knowledge of colubrids’ ontogeny, and to describe comparatively the ethmoidal region of the skull of these three species trying to elucidate aspects of its development in response to selection for adaptive features to the habitat. The material was obtained from the herpetological scientific collection of the Natural Sciences Zoobotanic Foundation of Rio Grande do Sul . To the reconstruction of the three-dimensional models of the ethmoidal region were used 15 sections of the snake Liophis jaegeri jaegeri, 22 sections of Philodryas olfersii and 21 sections of Thamnodynastes strigatus. The identified modifications in the examined ethmoidal region in relation to the environment occur mainly in the narial fenestra, nasal vestibule and nasal cavity. Key words: histology of the ethmoidal region, virtual models, snakes, habitats.A região neotropical apresenta uma fauna de serpentes extremamente diversificada, com representantes de várias famílias. Esses animais apresentam grande variação morfológica e ecológica em relação ao tipo de microhábitat utilizado. Entretanto, poucos trabalhos podem qualificar o uso de estudo do desenvolvimento craniano nas relações do animal com seu habitat. Os objetivos deste trabalho são: reconstruir um modelo tridimensional digital por meio de séries histológicas da região etmoidal do crânio das espécies Liophis jaegeri jaegeri, Philodryas olfersii e Thamnodynastes strigatus; contribuir para o conhecimento da ontogenia dos colubrídeos; descrever comparativamente a região etmoidal do crânio dessas espécies e relacionar os diferentes caracteres morfológicos da região etmoidal das espécies com a adaptação ao uso de diferentes habitats. O material foi obtido da Coleção Científica Herpetológica do Museu de Ciências Naturais da Fundação Zoobotânica do Rio Grande do Sul. Para reconstrução dos modelos tridimensionais da região etmoidal foram utilizados 15 cortes histológicos da serpente Liophis jaegeri jaegeri, 22 de Philodryas olfersii e 21 de Thamnodynastes strigatus. As modificações identificadas na região etmoidal dos espécimes examinados em relação ao seu meio ambiente ocorrem principalmente na fenestra narina, vestíbulo nasal e cavidade nasal. Palavras-chave: histologia da região etmoidal, modelos virtuais, serpentes, hábitat