scholarly journals Torsion Part of ℤ-module

2015 ◽  
Vol 23 (4) ◽  
pp. 297-307 ◽  
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Finite Rank ◽  
Rational Numbers ◽  
Reduction Algorithm ◽  
Definition Of ◽  
Torsion Part ◽  
Torsion Free

Summary In this article, we formalize in Mizar [7] the definition of “torsion part” of ℤ-module and its properties. We show ℤ-module generated by the field of rational numbers as an example of torsion-free non free ℤ-modules. We also formalize the rank-nullity theorem over finite-rank free ℤ-modules (previously formalized in [1]). ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [23] and cryptographic systems with lattices [24].

scholarly journals Divisible ℤ-modules

2016 ◽  
Vol 24 (1) ◽  
pp. 37-47 ◽  
Author(s):  
Yuichi Futa ◽  
Yasunari Shidama
Keyword(s):  
Vector Space ◽  
Coding Theory ◽  
Reduction Algorithm ◽  
Definition Of ◽  
Coefficient Ring ◽  
Torsion Free

Summary In this article, we formalize the definition of divisible ℤ-module and its properties in the Mizar system [3]. We formally prove that any non-trivial divisible ℤ-modules are not finitely-generated.We introduce a divisible ℤ-module, equivalent to a vector space of a torsion-free ℤ-module with a coefficient ring ℚ. ℤ-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [15], cryptographic systems with lattices [16] and coding theory [8].

scholarly journals Regulating hulls of almost completely decomposable groups

1993 ◽  
Vol 54 (2) ◽  
pp. 143-155 ◽  
Author(s):  
A. Mader ◽  
C. Vinsonhaler
Keyword(s):  
Finite Index ◽  
Finite Rank ◽  
Abelian Groups ◽  
Additive Group ◽  
Rational Numbers ◽  
Direct Sum ◽  
Free Abelian Groups ◽  
Decomposable Groups ◽  
The Relationship ◽  
Torsion Free

AbstractThis note investigates torsion-free abelian groups G of finite rank which embed, as subgroups of finite index, in a finite direct sum C of subgroups of the additive group of rational numbers. Specifically, we examine the relationship between G and C when the index of G in C is minimal. Some properties of Warfield duality are developed and used (in the case that G is locally free) to relate our results to earlier ones by Burkhardt and Lady.

scholarly journals Torsion Z-module and Torsion-free Z-module

2014 ◽  
Vol 22 (4) ◽  
pp. 277-289
Author(s):  
Yuichi Futa ◽  
Hiroyuki Okazaki ◽  
Kazuhisa Nakasho ◽  
Yasunari Shidama
Keyword(s):  
Finite Rank ◽  
Coding Theory ◽  
Reduction Algorithm ◽  
Finitely Generated ◽  
Torsion Free

Summary In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov´asz) base reduction algorithm [20], cryptographic systems with lattice [21], and coding theory [11].

scholarly journals Locally irreducible rings

1985 ◽  
Vol 32 (1) ◽  
pp. 129-145 ◽  
Author(s):  
C. Vinsonhaler ◽  
W. Wickless
Keyword(s):  
Finite Rank ◽  
Abelian Groups ◽  
Free Groups ◽  
Additive Group ◽  
Infinite Rank ◽  
Field Of Definition ◽  
Free Abelian Groups ◽  
Definition Of ◽  
Torsion Free

In the study of torsion-free abelian groups of finite rank the notions of irreducibility, field of definition and E-ring have played significant rôles. These notions are tied together in the following theorem of R. S. Pierce:THEOREM. Let R be a ring whose additive group is torsion free finite rank irreducible and let Γ be the centralizer of QR as a QE(R) module. Then Γ is the unique smallest field of definition of R. Moreover, Γ ∩ R is an E-ring, in fact, it is a maximal E-subring of R.In this paper we consider extensions of Pierce's result to the infinite rank case. This leads to the concept of local irreducibility for torsion free groups.

New definition of MV-semimodules

2021 ◽  
pp. 1-12
Author(s):  
Simin Saidi Goraghani ◽  
Rajab Ali Borzooei ◽  
Sun Shin Ahn
Keyword(s):  
Basic Properties ◽  
Mv Algebra ◽  
Definition Of ◽  
Torsion Free

In recent years, A. Di Nola et al. studied the notions of MV-semiring and semimodules and investigated related results [9, 10, 12, 26]. Now in this paper, by using an MV-semiring and an MV-algebra, we introduce the new definition of MV-semimodule, study basic properties and find some examples. Then we study A-ideals on MV-semimodules and Q-ideals on MV-semirings, and by using them, we study the quotient structures of MV-semimodule. Finally, we present the notions of prime A-ideal, torsion free MV-semimodule and annihilator on MV-semimodule and we study the relations among them.

Signal Recovery by Linear Estimation

2019 ◽  
pp. 260-385
Author(s):  
John Stillwell
Keyword(s):  
Human Activities ◽  
Mathematical Concept ◽  
Rational Numbers ◽  
Precise Definition ◽  
Computable Analysis ◽  
Signal Recovery ◽  
Linear Estimation ◽  
Definition Of ◽  
Computable Sequence ◽  
Informal Description

This chapter develops the basic results of computability theory, many of which are about noncomputable sequences and sets, with the goal of revealing the limits of computable analysis. Two of the key examples are a bounded computable sequence of rational numbers whose limit is not computable, and a computable tree with no computable infinite path. Computability is an unusual mathematical concept, because it is most easily used in an informal way. One often talks about it in terms of human activities, such as making lists, rather than by applying a precise definition. Nevertheless, there is a precise definition of computability, so this informal description of computations can be formalized.

Optimization Analysis on Dynamic Reduction Algorithm

2018 ◽  
Vol 6 (5) ◽  
pp. 447-458
Author(s):  
Yizhou Chen ◽  
Jiayang Wang
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Reduction Process ◽  
New Method ◽  
Reduction Algorithm ◽  
Optimization Analysis ◽  
Sampling Process ◽  
Dynamic Methods ◽  
Definition Of

Abstract On the basis of rough set theory, the strengths of dynamic reduction are elaborated compared with traditional non-dynamic methods. A systematic concept of dynamic reduction from sampling process to the generation of the reduct set is presented. A new method of sampling is created to avoid the defects of being too subjective. And in order to deal with the over-sized time consuming problem in traditional dynamic reduction process, a quick algorithm is proposed within the constraint conditions. We have also proved that dynamic core possesses the essential characteristics of a reduction core on the basis of the formalized definition of the multi-layered dynamic core.

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