scholarly journals Existence of prime elements in rings of generalized power series

2001 ◽  
Vol 66 (3) ◽  
pp. 1206-1216 ◽  
Author(s):  
Daniel Pitteloud
Keyword(s):  
Abelian Group ◽  
Power Series ◽  
Prime Ideal ◽  
Additive Group ◽  
Convex Subgroup ◽  
Generalized Power Series ◽  
Irreducible Elements ◽  
Closed Fields ◽  
Proper Convex ◽  
Prime Elements

AbstractThe field K((G)) of generalized power series with coefficients in the field K of characteristic 0 and exponents in the ordered additive abelian group G plays an important role in the study of real closed fields. Conway and Gonshor (see [2, 4]) considered the problem of existence of non-standard irreducible (respectively prime) elements in the huge “ring” of omnific integers, which is indeed equivalent to the existence of irreducible (respectively prime) elements in the ring K((G≤0)) of series with non-positive exponents. Berarducci (see [1]) proved that K((G≤0)) does have irreducible elements, but it remained open whether the irreducibles are prime i.e., generate a prime ideal. In this paper we prove that K((G≤0)) does have prime elements if G = (ℝ, +) is the additive group of the reals, or more generally if G contains a maximal proper convex subgroup.

Finitely continuous differentials on generalized power series

2014 ◽  
Vol 64 (2) ◽  
Author(s):  
I-Chiau Huang
Keyword(s):  
Abelian Group ◽  
Power Series ◽  
Vector Space ◽  
Inversion Formula ◽  
Finitely Generated ◽  
Generalized Power Series

AbstractLet κ[[eG]] be the field of generalized power series with exponents in a totally ordered Abelian group G and coefficients in a field κ. Given a subgroup H of G such that G/H is finitely generated, we construct a vector space ΩG/H of differentials as a universal object in certain category of κ[[eH]]-derivations on κ[[eG]]. The vector space ΩG/H together with logarithmic residues gives rise to a framework for certain combinatorial phenomena, including the inversion formula for diagonal delta sets.

CANCELLATION LAW AND UNIQUE FACTORIZATION THEOREM FOR STRING OPERATIONS

2006 ◽  
Vol 05 (02) ◽  
pp. 231-243
Author(s):  
DONGVU TONIEN
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Additive Group ◽  
New Product ◽  
Binary String ◽  
Factorization Theorem ◽  
General Criterion ◽  
Unique Factorization ◽  
Product Operation ◽  
Associative Law

Recently, Hoit introduced arithmetic on blocks, which extends the binary string operation by Jacobs and Keane. A string of elements from the Abelian additive group of residues modulo m, (Zm, ⊕), is called an m-block. The set of m-blocks together with Hoit's new product operation form an interesting algebraic structure where associative law and cancellation law hold. A weaker form of unique factorization and criteria for two indecomposable blocks to commute are also proved. In this paper, we extend Hoit's results by replacing the Abelian group (Zm, ⊕) by an arbitrary monoid (A, ◦). The set of strings built up from the alphabet A is denoted by String(A). We extend the operation ◦ on the alphabet set A to the string set String(A). We show that (String(A), ◦) is a monoid if and only if (A, ◦) is a monoid. When (A, ◦) is a group, we prove that stronger versions of a cancellation law and unique factorization hold for (String(A), ◦). A general criterion for two irreducible strings to commute is also presented.

scholarly journals Integral Cayley Graphs over Abelian Groups

10.37236/353 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Boolean Algebra ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Additive Group ◽  
Cyclic Groups ◽  
Vertex Set ◽  
A Finite Group ◽  
Gamma S

Let $\Gamma$ be a finite, additive group, $S \subseteq \Gamma, 0\notin S, -S=\{-s: s\in S\}=S$. The undirected Cayley graph Cay$(\Gamma,S)$ has vertex set $\Gamma$ and edge set $\{\{a,b\}: a,b\in \Gamma$, $a-b \in S\}$. A graph is called integral, if all of its eigenvalues are integers. For an abelian group $\Gamma$ we show that Cay$(\Gamma,S)$ is integral, if $S$ belongs to the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. The converse is proven for cyclic groups. A finite group $\Gamma$ is called Cayley integral, if every undirected Cayley graph over $\Gamma$ is integral. We determine all abelian Cayley integral groups.

On the Distribution of Sum of Independent Positive Binomial Variables

1970 ◽  
Vol 13 (1) ◽  
pp. 151-152 ◽  
Author(s):  
J. C. Ahuja
Keyword(s):  
Power Series ◽  
Inversion Formula ◽  
Probability Function ◽  
Random Variables ◽  
Random Variable ◽  
Characteristic Functions ◽  
Binomial Probability ◽  
Generalized Power Series ◽  
Alternative Derivation ◽  
Image Position

Let X1, X2, …, Xn be n independent and identically distributed random variables having the positive binomial probability function1where 0 < p < 1, and T = {1, 2, …, N}. Define their sum as Y=X1 + X2 + … +Xn. The distribution of the random variable Y has been obtained by Malik [2] using the inversion formula for characteristic functions. It appears that his result needs some correction. The purpose of this note is to give an alternative derivation of the distribution of Y by applying one of the results, established by Patil [3], for the generalized power series distribution.

scholarly journals Average of the first invariant factor of the reductions of abelian varieties of CM type

2016 ◽  
Vol 12 (02) ◽  
pp. 445-463 ◽  
Author(s):  
Sungjin Kim
Keyword(s):  
Abelian Group ◽  
Asymptotic Formula ◽  
Prime Ideal ◽  
Elliptic Curves ◽  
Short Range ◽  
Abelian Varieties ◽  
Prime Ideals ◽  
Good Reduction ◽  
Invariant Factor ◽  
Field Of Definition

For a field of definition [Formula: see text] of an abelian variety [Formula: see text] and prime ideal [Formula: see text] of [Formula: see text] which is of a good reduction for [Formula: see text], the structure of [Formula: see text] as abelian group is: [Formula: see text] where [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text]. We are interested in finding an asymptotic formula for the number of prime ideals [Formula: see text] with [Formula: see text], [Formula: see text] has a good reduction at [Formula: see text], [Formula: see text]. We succeed in proving this under the assumption of the Generalized Riemann Hypothesis (GRH). Unconditionally, we achieve a short range asymptotic for abelian varieties of CM type, and the full cyclicity theorem for elliptic curves over a number field containing the CM field.

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