scholarly journals The rational field is not universally definable in pseudo-exponentiation

2016 ◽  
Vol 232 (1) ◽  
pp. 79-88
Author(s):  
Jonathan Kirby
Keyword(s):  
2013 ◽  
Vol 21 (4) ◽  
pp. 273-282 ◽  
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama

Abstract In this article, we formalize a free Z-module and its property. In particular, we formalize the vector space of rational field corresponding to a free Z-module and prove formally that submodules of a free Z-module are free. Z-module is necassary for lattice problems - LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm and cryptographic systems with lattice [20]. Some theorems in this article are described by translating theorems in [11] into theorems of Z-module, however their proofs are different.


1985 ◽  
Vol 37 (2) ◽  
pp. 238-259 ◽  
Author(s):  
John D. Dixon

Let G be a subgroup of the general linear group GL(n, Q) over the rational field Q, and consider its action by right multiplication on the vector space Qn of n-tuples over Q. The present paper investigates the question of how we may constructively determine the orbits and stabilizers of this action for suitable classes of groups. We suppose that G is specified by a finite set {x1, …, xr) of generators, and investigate whether there exist algorithms to solve the two problems:(Orbit Problem) Given u, v ∊ Qn, does there exist x ∊ G such that ux = v; if so, find such an element x as a word in x1, …, xr and their inverses.(Stabilizer Problem) Given u, v ∊ Qn, describe all words in x1, …, xr and their inverses which lie in the stabilizer


1987 ◽  
Vol 24 (2) ◽  
pp. 47-50
Author(s):  
Yu. M. Pridatko ◽  
V. L. Shabrov ◽  
E. V. Svetinskii ◽  
R. Kh. Valeev

2019 ◽  
Vol 15 (3) ◽  
pp. 540-561
Author(s):  
Sylvia Moretzsohn

This paper seeks to situate historically the production of what is now called “fake news” and points out the misconception of establishing a dividing line in which the traditional press would be the sole source for reliable information, even though it was and still is the origin of much untrue or biased information. It criticizes the methods of the fact-checking agencies, which end up selling a false idea of objectivity. But above all, it points out the need to deepen the discussion about credibility at a time when reference information standards are challenged and beliefs seem to be allowed to prevail over the evidences. If arguments are useless in face of convictions, and if journalism is more than never necessary, the way to recover its role would have to be sought outside the rational field, in order to deactivate the affections that lead to the formation of bubbles refractoryto all criticism.Este artigo procura situar historicamente a produção do que hoje se chama “fake news” e assinala o equívoco de se estabelecer uma linha divisória na qual a imprensa tradicional seria a exclusiva fonte para a informação confiável, mesmo porque ela própria foi e continua a ser a origem de muita informação inverídica ou deturpada. Critica os métodos das agências de checagem, que acabam por vender uma falsa ideia de objetividade. Mas, principalmente, aponta a necessidade de um aprofundamento da discussão sobre credibilidade, em um tempo em que os padrões da informação de referência são contestados e as crenças parecem autorizadas a prevalecer sobre as evidências. Se os argumentos são inúteis diante das convicções, e se apesar disso o jornalismo é mais do que nunca necessário, a saída para recuperar o seu papel precisaria ser buscada fora do campo racional para depois recuperá-lo, de modo a desativar os afetos que levam à formação das bolhas refratárias a qualquer crítica.Este artículo busca situar históricamente la producción de lo que hoy se llama “fake news” y señala el equívoco de establecer una línea divisoria en la que la prensa tradicional sería la única fuente para la información confiable, incluso porque ella misma fué y sigue siendo el origen de mucha información falsa o engañosa. Critica los métodos de las agencias de chequeo, que acaban por vender una errónea idea de objetividad. Pero, principalmente, apunta la necesidad de una profundización de la discusión sobre credibilidad, en un tiempo en que los parámetros de la información de referencia son contestados y las creencias parecen autorizadas a prevalecer sobre las evidencias. Si los argumentos son inútiles ante las convicciones, y si a pesar de ello el periodismo sigue siendo necesario, la salida para recuperar su papel necesitaría ser buscada fuera del campo racional, para desactivar los afectos que llevan a la formación de las burbujas refractarias a cualquier crítica.


1951 ◽  
Vol 3 ◽  
pp. 44-51 ◽  
Author(s):  
Loo-Keng Hua

Let K be an algebraic field of degree n over the rational field, and let b be the ground ideal (différente) of the field. Let


1980 ◽  
Vol 45 (2) ◽  
pp. 359-362 ◽  
Author(s):  
Chris Ash ◽  
John Rosenthal

We consider the effect on decidability of adding, to the decidable theory of algebraically closed fields of characteristic zero, relation symbols for algebraic independence or function symbols for differentiation. Our results show that the corresponding theories are usually undeeidable.Let k and K be algebraically closed fields of characteristic zero. Let K be an extension of k of transcendence degree n over k. Since k has characteristic 0, we may assume that the rational field, Q, is a subfield of k.Let Indn be the n-ary relation on K which holds for exactly those n-tuples from K which are algebraically independent over k.Let x1, …, xn be a transcendence base for K over k. For i = 1, 2, …, n, let Di: K → K be the partial differentiation function with respect to xi and this base.Let KnInd = (K, +, ·, Indn), n ≤ 1 and let KnDiff = (K, +, ·, D1, …, Dn), n ≤ 1 where K has transcendence degree n over k.We show that the theories of these structures are independent of k when k has infinite transcendence degree over Q, that KnDiff has undeeidable theory for n ≤ 1 and that KnInd has undeeidable theory for n ≤ 2. The theory of K1Ind is decidable.


2018 ◽  
Vol 17 (07) ◽  
pp. 1850127
Author(s):  
H. Behravesh ◽  
M. Delfani

For a finite group [Formula: see text], we denote by [Formula: see text] the minimal degree of faithful permutation representations of [Formula: see text], and denote by [Formula: see text] and [Formula: see text], the minimal degree of faithful representation of [Formula: see text] by quasi-permutation matrices over the rational field [Formula: see text] and the complex field [Formula: see text], respectively. In this paper, we calculate [Formula: see text], [Formula: see text] and [Formula: see text] for the groups of order [Formula: see text], where [Formula: see text] is an odd prime.


Let H be a class group— in the sense of class-field theory— in the rational field P, whose order is some power of a prime l . With H there is associated an Abelian extension K of P. The purpose of this paper is to determine in rational terms and for all fields K given in the described manner, the set T(K/P) of cyclic extensions A of K of relative degree l , which are absolutely normal. In particular we shall find the ramification laws for these fields A, and the possible extension types of a group of order l by the Galois group of K, which are realized in Galois groups of fields in T(K/P). It is fundamental to the programme outlined, that we aim at obtaining purely rational criteria of determination.


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