scholarly journals Algebraic Numbers as Product of Powers of Transcendental Numbers

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 887 ◽  
Author(s):  
Pavel Trojovský

The elementary symmetric functions play a crucial role in the study of zeros of non-zero polynomials in C [ x ] , and the problem of finding zeros in Q [ x ] leads to the definition of algebraic and transcendental numbers. Recently, Marques studied the set of algebraic numbers in the form P ( T ) Q ( T ) . In this paper, we generalize this result by showing the existence of algebraic numbers which can be written in the form P 1 ( T ) Q 1 ( T ) ⋯ P n ( T ) Q n ( T ) for some transcendental number T, where P 1 , … , P n , Q 1 , … , Q n are prescribed, non-constant polynomials in Q [ x ] (under weak conditions). More generally, our result generalizes results on the arithmetic nature of z w when z and w are transcendental.

1888 ◽  
Vol 7 ◽  
pp. 41-42
Author(s):  
R. E. Allardice

The theorem that any rational symmetric function of n variables x1, x2, … xn is expressible as a rational function of the n elementary symmetric functions, Σx1, Σx1x2, Σx1x2x3, etc., is usually proved by means of the properties of the roots of an equation. It is obvious, however, that the theorem has no necessary connection with the properties of equations; and the object of this paper is to give an elementary proof of the theorem, based solely on the definition of a symmetric function.


2012 ◽  
Vol 60 (2) ◽  
pp. 219-224 ◽  
Author(s):  
Alexander Kovačec ◽  
Salma Kuhlmann ◽  
Cordian Riener

2021 ◽  
Vol 60 (3-4) ◽  
pp. 363-398

Abstract The Roman father and son of the same name, P. Decius Mus, became paragon heroes by deliberately giving their lives in battle that Rome might win over a fierce enemy. Both engaged in a special ritual called devotio (from which our word “devotion” derives) to offer themselves to the gods of the Underworld, with whom regular people have very little interaction and to whom they rarely sacrifice. While the Mus family is the most famous for this act, it turns out the willingness to sacrifice oneself for Rome frequently occurs within stories of great patriots, including the story of Horatius Cocles, Mettius Curtius, Atilius Regulus, and even the traitors Coriolanus and Tarpeia. Romans regarded self-sacrifice as a very high, noble endeavor, whereas they loathed and persecuted practitioners of human sacrifice. It is therefore quite amazing to read that the Romans thrice engaged in state-sponsored human sacrifice, a fact they rarely mention and generally forget. The most famous enemy practitioners of human sacrifice were the Druids, whom the Romans massacred on Mona Island on Midsummer Night's Eve, but the Carthaginians, the Germans, the Celts, and the Thracians all infamously practiced human sacrifice. To Romans, the act of human sacrifice falls just short of cannibalism in the spectrum of forbidden practices, and was an accusation occasionally thrown against an enemy to claim they are totally barbaric. On the other hand, Romans recognized their own who committed acts of self-sacrifice for the good of the society, as heroes. There can be no better patriot than he who gives his life to save his country. Often the stories of their heroism have been exaggerated or sanitized. These acts of heroism often turn out to be acts of human sacrifice, supposedly a crime. It turns out that Romans have a strong legacy of practicing human sacrifice that lasts into the historic era, despite their alleged opposition to it. Numerous sources relate one story each. Collecting them all makes it impossible to deny the longevity of human sacrifice in Rome, although most Romans under the emperors were probably unaware of it. The paradox of condemning but still practicing human sacrifice demonstrates the nature of Roman religion, where do ut des plays a crucial role in standard sacrifice as well as in unpleasant acts like human sacrifice. Devotio was an inverted form of sacrifice, precisely because it was an offering to the gods of the Underworld, rather than to Jupiter or the Parcae. Romans may have forsaken devotio, but they continued to practice human sacrifice far longer than most of us have suspected, if one widens the current narrow definition of human sacrifice to include events where a life is taken in order to bring about a better future for the commonwealth, appease the gods, or ensure a Roman victory in battle.


10.37236/1877 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
J. Bell ◽  
A. M. Garsia ◽  
N. Wallach

We introduce here a new approach to the study of $m$-quasi-invariants. This approach consists in representing $m$-quasi-invariants as $N^{tuples}$ of invariants. Then conditions are sought which characterize such $N^{tuples}$. We study here the case of $S_3$ $m$-quasi-invariants. This leads to an interesting free module of triplets of polynomials in the elementary symmetric functions $e_1,e_2,e_3$ which explains certain observed properties of $S_3$ $m$-quasi-invariants. We also use basic results on finitely generated graded algebras to derive some general facts about regular sequences of $S_n$ $m$-quasi-invariants


10.37236/1547 ◽  
2000 ◽  
Vol 8 (1) ◽  
Author(s):  
Leigh Roberts

Recently Lapointe et. al. [3] have expressed Jack Polynomials as determinants in monomial symmetric functions $m_\lambda$. We express these polynomials as determinants in elementary symmetric functions $e_\lambda$, showing a fundamental symmetry between these two expansions. Moreover, both expansions are obtained indifferently by applying the Calogero-Sutherland operator in physics or quasi Laplace Beltrami operators arising from differential geometry and statistics. Examples are given, and comments on the sparseness of the determinants so obtained conclude the paper.


Author(s):  
José Ferraz-Caetano ◽  
João Paiva ◽  
Francisco Malta Romeiras

Resumo No final do século XIX, a química ganhou notoriedade como uma das principais “ciências ao serviço” da nação. O surgimento de novos tópicos, métodos e práticas úteis contribuíram para a valorização da química e para a definição de medidas governamentais em temas como saúde pública, educação e proteção ambiental. Lente na Academia Politécnica do Porto entre 1877 e 1910, António Ferreira da Silva (1853–1923) desempenhou um papel central na modernização do ensino e da investigação em química em Portugal. Ferreira da Silva foi responsável pela introdução de cursos suplementares de química, pela reformulação do ensino prático, e pela elaboração de novos procedimentos e regulamentos de ensino “que em muito engrandeceram a educação científica” em Portugal. Enquanto lente da Academia Politécnica do Porto, Ferreira da Silva privilegiou ainda a articulação entre o Laboratório da Academia e as indústrias nacionais, contribuindo, em larga medida, para emergência da Química Analítica como uma nova disciplina.Palavras-chave: António Ferreira da Silva; Academia Politécnica do Porto; Química Analítica. Abstract By the turn of the nineteenth century, chemistry had become a “science at the service” of the nation. The emergence of useful topics, methods, and practices contributed to the valorization of chemistry and to the definition of new governmental directives on issues such as public health, education and environment. Lecturer at the Academia Politécnica do Porto between 1877 and 1911, António Ferreira da Silva (1853–1923) played a crucial role in the modernization of the teaching and practice of chemistry in Portugal. Ferreira da Silva created new supplementary chemistry courses, reformed the practical teaching of chemistry, and drafted new proceedings and syllabi “that glorified scientific education” in Portugal. As lecturer of the Academia Politécnica do Porto, he made important steps in the establishment of collaborations between the Academia’s Laboratory and national industries, which largely contributed to the emergence of Analytical Chemistry as an autonomous discipline. Keywords: António Ferreira da Silva; Academia Politécnica do Porto; Analytical Chemistry.


2017 ◽  
Vol 2 (4) ◽  
pp. 682-691 ◽  
Author(s):  
Wanxi Yang ◽  
◽  
Mao Li ◽  
Yulu Feng ◽  
Xiao Jiang ◽  
...  

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