scholarly journals Perpendicularity in an Abelian Group

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Pentti Haukkanen ◽  
Mika Mattila ◽  
Jorma K. Merikoski ◽  
Timo Tossavainen
Keyword(s):  
Number Theory ◽  
Abelian Group

We give a set of axioms to establish a perpendicularity relation in an Abelian group and then study the existence of perpendicularities in(ℤn,+)and(ℚ+,·)and in certain other groups. Our approach provides a justification for the use of the symbol⊥denoting relative primeness in number theory and extends the domain of this convention to some degree. Related to that, we also consider parallelism from an axiomatic perspective.

THE ISOPERIMETRIC METHOD IN NON-ABELIAN GROUPS WITH AN APPLICATION TO OPTIMALLY SMALL SUMSETS

2008 ◽  
Vol 04 (06) ◽  
pp. 927-958 ◽  
Author(s):  
ÉRIC BALANDRAUD
Keyword(s):  
Number Theory ◽  
Abelian Group ◽  
Abelian Groups ◽  
Solvable Groups ◽  
Additive Number Theory ◽  
Alternating Groups ◽  
Additive Number ◽  
New Interpretation ◽  
Set Addition

Set addition theory is born a few decades ago from additive number theory. Several difficult issues, more combinatorial in nature than algebraic, have been revealed. In particular, computing the values taken by the function: [Formula: see text] where G is a given group does not seem easy in general. Some successive results, using Kneser's Theorem, allowed the determination of the values of this function, provided that the group G is abelian. Recently, a method called isoperimetric, has been developed by Hamidoune and allowed new proofs and generalizations of the classical theorems in additive number theory. For instance, a new interpretation of the isoperimetric method has been able to give a new proof of Kneser's Theorem. The purpose of this article is to adapt this last proof in a non-abelian group, in order to give new values of the function μG, for some solvable groups and alternating groups. These values allow us in particular to answer negatively a question asked in the literature on the μG functions.

scholarly journals Direct and inverse problems in additive number theory and in non-abelian group theory

2014 ◽  
Vol 40 ◽  
pp. 42-54 ◽  
Author(s):  
G.A. Freiman ◽  
M. Herzog ◽  
P. Longobardi ◽  
M. Maj ◽  
Y.V. Stanchescu
Keyword(s):  
Number Theory ◽  
Group Theory ◽  
Abelian Group ◽  
Inverse Problems ◽  
Additive Number Theory ◽  
Direct And Inverse Problems ◽  
Additive Number

On Normal Sequence in Abelian Group Cn⊕Cn

2014 ◽  
Vol 981 ◽  
pp. 255-257
Author(s):  
Hai Yan Zhang ◽  
Xin Song Yang
Keyword(s):  
Information Technology ◽  
Number Theory ◽  
Abelian Group ◽  
Sufficient Conditions ◽  
Information Coding ◽  
Electronic Information ◽  
Construction Problem ◽  
Normal Sequence ◽  
Combinatorial Number Theory ◽  
Combination Number

Electronic information technology is based on expression, storage, transmission, and process of information. Information expression is usually generalized as information coding technology, one of whose heart theories is combinatorial number theory in algebra. In the process of storing, transmitting, and processing of electronic information technology, problems about encryption and safety need to use algebra theories of group, ring, and domain. It is thus clear that algebra is very important in electronic information theory. This paper makes use of abelian group basic theory of algebra, together with combination number theory, discusses construction problem of normal sequence in of abelian group, and gives several sufficient conditions for a guess establishment of W.D.Gao.

On some weighted zero-sum constants II

2018 ◽  
Vol 14 (02) ◽  
pp. 383-397 ◽  
Author(s):  
Mohan N. Chintamani ◽  
Prabal Paul
Keyword(s):  
Number Theory ◽  
Abelian Group ◽  
Positive Integer ◽  
Upper Bound ◽  
Structure Theorem ◽  
Finite Abelian Group ◽  
Zero Sum

For a finite abelian group [Formula: see text] with exponent [Formula: see text], let [Formula: see text]. The constant [Formula: see text] (respectively [Formula: see text]) is defined to be the least positive integer [Formula: see text] such that given any sequence [Formula: see text] over [Formula: see text] with length [Formula: see text] has a [Formula: see text]-weighted zero-sum subsequence of length [Formula: see text] (respectively at most [Formula: see text]). In [M. N. Chintamani and P. Paul, On some weighted zero-sum constants, Int. J. Number Theory 13(2) (2017) 301–308], we proved the exact value of this constant for the group [Formula: see text] and proved the structure theorem for the extremal sequences related to this constant. In this paper, we prove the similar results for the group [Formula: see text] and we obtained an upper bound when [Formula: see text] is replaced by any integer [Formula: see text].

THE MAXIMUM SIZE OF -SUM-FREE SETS IN CYCLIC GROUPS

2018 ◽  
Vol 99 (2) ◽  
pp. 184-194
Author(s):  
BÉLA BAJNOK ◽  
RYAN MATZKE
Keyword(s):  
Number Theory ◽  
Abelian Group ◽  
Explicit Formula ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Maximum Size ◽  
Critical Pair ◽  
Cyclic Groups ◽  
Free Subset ◽  
Free Sets

A subset$A$of a finite abelian group$G$is called$(k,l)$-sum-free if the sum of$k$(not necessarily distinct) elements of$A$never equals the sum of$l$(not necessarily distinct) elements of $A$. We find an explicit formula for the maximum size of a$(k,l)$-sum-free subset in$G$for all$k$and$l$in the case when$G$is cyclic by proving that it suffices to consider$(k,l)$-sum-free intervals in subgroups of $G$. This simplifies and extends earlier results by Hamidoune and Plagne [‘A new critical pair theorem applied to sum-free sets in abelian groups’,Comment. Math. Helv. 79(1) (2004), 183–207] and Bajnok [‘On the maximum size of a$(k,l)$-sum-free subset of an abelian group’,Int. J. Number Theory 5(6) (2009), 953–971].

scholarly journals Class Number and Ramification in Number Fields

1963 ◽  
Vol 23 ◽  
pp. 97-101 ◽  
Author(s):  
Armand Brumer ◽  
Michael Rosen
Keyword(s):  
Number Theory ◽  
Abelian Group ◽  
Algebraic Number ◽  
Class Number ◽  
Finite Abelian Group ◽  
Number Fields ◽  
Algebraic Number Theory ◽  
Outstanding Problem ◽  
Principal Ideals

In the ring Ok of algebraic integers of a number field K the group Ik of ideals of Ok modulo the subgroup Pk of principal ideals is a finite abelian group of order hk, the class number of K. The determination of this number is an outstanding problem of algebraic number theory.

A SIMPLIFIED PROOF OF HESSELHOLT’S CONJECTURE ON GALOIS COHOMOLOGY OF WITT VECTORS OF ALGEBRAIC INTEGERS

2012 ◽  
Vol 86 (3) ◽  
pp. 456-460
Author(s):  
WILSON ONG
Keyword(s):  
Number Theory ◽  
Abelian Group ◽  
Residue Field ◽  
Galois Cohomology ◽  
Algebraic Integers ◽  
Original Proof ◽  
Cambridge Philos ◽  
Witt Vectors ◽  
Residue Fields ◽  
Valuation Field

AbstractLet K be a complete discrete valuation field of characteristic zero with residue field kK of characteristic p>0. Let L/K be a finite Galois extension with Galois group G=Gal(L/K) and suppose that the induced extension of residue fields kL/kK is separable. Let 𝕎n(⋅) denote the ring of p-typical Witt vectors of length n. Hesselholt [‘Galois cohomology of Witt vectors of algebraic integers’, Math. Proc. Cambridge Philos. Soc.137(3) (2004), 551–557] conjectured that the pro-abelian group {H1 (G,𝕎n (𝒪L))}n≥1 is isomorphic to zero. Hogadi and Pisolkar [‘On the cohomology of Witt vectors of p-adic integers and a conjecture of Hesselholt’, J. Number Theory131(10) (2011), 1797–1807] have recently provided a proof of this conjecture. In this paper, we provide a simplified version of the original proof which avoids many of the calculations present in that version.

scholarly journals Some problems of analytic number theory III

1981 ◽  
Vol Volume 4 ◽  
Author(s):  
R Balasubramanian ◽  
K Ramachandra
Keyword(s):  
Number Theory ◽  
Abelian Group ◽  
Analytic Number Theory ◽  
Divisor Problem ◽  
Main Theme ◽  
Group Problem ◽  
Circle Problem ◽  
Analytic Number ◽  
International Audience

International audience The main theme of this paper is to systematize the Hardy-Landau $\Omega$ results and the Hardy $\Omega_{\pm}$ results on the divisor problem and the circle problem. The method of ours is general enough to include the abelian group problem and the results of Richert and the later modifications by Warlimont, and in fact theorem 6 of ours is an improvement of their results. All our results are effective as in our earlier paper II with the same title. Some of our results are new.

Multiplicative Number Theory I

2006 ◽  
Author(s):  
Hugh L. Montgomery ◽  
Robert C. Vaughan
Keyword(s):  
Number Theory
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