scholarly journals Equality in Pollard's theorem on set addition of congruence classes

2007 ◽  
Vol 127 (1) ◽  
pp. 1-15 ◽  
Author(s):  
E. Nazarewicz ◽  
M. O'Brien ◽  
M. O'Neill ◽  
C. Staples
Keyword(s):  
Set Addition ◽  
Congruence Classes

scholarly journals SUMS OF PRODUCTS OF CONGRUENCE CLASSES AND OF ARITHMETIC PROGRESSIONS

2009 ◽  
Vol 05 (04) ◽  
pp. 625-634
Author(s):  
SERGEI V. KONYAGIN ◽  
MELVYN B. NATHANSON
Keyword(s):  
Arithmetic Progression ◽  
Congruence Class ◽  
Arithmetic Progressions ◽  
Sum Of Products ◽  
Sums Of Products ◽  
Positive Integers ◽  
Congruence Classes

Consider the congruence class Rm(a) = {a + im : i ∈ Z} and the infinite arithmetic progression Pm(a) = {a + im : i ∈ N0}. For positive integers a,b,c,d,m the sum of products set Rm(a)Rm(b) + Rm(c)Rm(d) consists of all integers of the form (a+im) · (b+jm)+(c+km)(d+ℓm) for some i,j,k,ℓ ∈ Z. It is proved that if gcd (a,b,c,d,m) = 1, then Rm(a)Rm(b) + Rm(c)Rm(d) is equal to the congruence class Rm(ab+cd), and that the sum of products set Pm(a)Pm(b)+Pm(c)Pm eventually coincides with the infinite arithmetic progression Pm(ab+cd).

scholarly journals Lehmer points and visible points on affine varieties over finite fields

2013 ◽  
Vol 156 (2) ◽  
pp. 193-207 ◽  
Author(s):  
KIT-HO MAK ◽  
ALEXANDRU ZAHARESCU
Keyword(s):  
Finite Fields ◽  
Affine Variety ◽  
Asymptotic Results ◽  
Visible Points ◽  
Congruence Classes

AbstractLet V be an absolutely irreducible affine variety over $\mathbb{F}_p$. A Lehmer point on V is a point whose coordinates satisfy some prescribed congruence conditions, and a visible point is one whose coordinates are relatively prime. Asymptotic results for the number of Lehmer points and visible points on V are obtained, and the distribution of visible points into different congruence classes is investigated.

scholarly journals Congruence Class Sizes in Finite Sectionally Complemented Lattices

2004 ◽  
Vol 47 (2) ◽  
pp. 191-205 ◽  
Author(s):  
G. Grätzer ◽  
E. T. Schmidt
Keyword(s):  
Congruence Class ◽  
Typical Result ◽  
Complemented Lattices ◽  
The Family ◽  
Complemented Lattice ◽  
Congruence Classes

AbstractThe congruences of a finite sectionally complemented lattice L are not necessarily uniform (any two congruence classes of a congruence are of the same size). To measure how far a congruence Θ of L is from being uniform, we introduce Spec Θ, the spectrum of Θ, the family of cardinalities of the congruence classes of Θ. A typical result of this paper characterizes the spectrum S = (mj | j < n) of a nontrivial congruence Θ with the following two properties:

scholarly journals Restricted set addition in groups, II. A generalization of the Erdős-Heilbronn conjecture

10.37236/1482 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Vsevolod F. Lev
Keyword(s):  
General Type ◽  
Sharp Estimate ◽  
Abelian Groups ◽  
Polynomial Method ◽  
Set Addition

In 1980, Erdős and Heilbronn posed the problem of estimating (from below) the number of sums $a+b$ where $a\in A$ and $b\in B$ range over given sets $A,B\subseteq{\Bbb Z}/p{\Bbb Z}$ of residues modulo a prime $p$, so that $a\neq b$. A solution was given in 1994 by Dias da Silva and Hamidoune. In 1995, Alon, Nathanson and Ruzsa developed a polynomial method that allows one to handle restrictions of the type $f(a,b)\neq 0$, where $f$ is a polynomial in two variables over ${\Bbb Z}/p{\Bbb Z}$. In this paper we consider restricting conditions of general type and investigate groups, distinct from ${\Bbb Z}/p{\Bbb Z}$. In particular, for $A,B\subseteq{\Bbb Z}/p{\Bbb Z}$ and ${\cal R}\subseteq A\times B$ of given cardinalities we give a sharp estimate for the number of distinct sums $a+b$ with $(a,b)\notin\ {\cal R}$, and we obtain a partial generalization of this estimate for arbitrary Abelian groups.

scholarly journals ACDC: Analysis of Congruent Diversification Classes

2022 ◽  
Author(s):  
Sebastian Hoehna ◽  
Bjoern Tore Kopperud ◽  
Andrew F Magee
Keyword(s):  
Congruence Class ◽  
R Package ◽  
Rate Function ◽  
Diversification Rate ◽  
Extinction Rate ◽  
Diversification Rates ◽  
Common Features ◽  
Alternative Hypotheses ◽  
Rate Functions ◽  
Congruence Classes

Diversification rates inferred from phylogenies are not identifiable. There are infinitely many combinations of speciation and extinction rate functions that have the exact same likelihood score for a given phylogeny, building a congruence class. The specific shape and characteristics of such congruence classes have not yet been studied. Whether speciation and extinction rate functions within a congruence class share common features is also not known. Instead of striving to make the diversification rates identifiable, we can embrace their inherent non-identifiable nature. We use two different approaches to explore a congruence class: (i) testing of specific alternative hypotheses, and (ii) randomly sampling alternative rate function within the congruence class. Our methods are implemented in the open-source R package ACDC (https://github.com/afmagee/ACDC). ACDC provides a flexible approach to explore the congruence class and provides summaries of rate functions within a congruence class. The summaries can highlight common trends, i.e. increasing, flat or decreasing rates. Although there are infinitely many equally likely diversification rate functions, these can share common features. ACDC can be used to assess if diversification rate patterns are robust despite non-identifiability. In our example, we clearly identify three phases of diversification rate changes that are common among all models in the congruence class. Thus, congruence classes are not necessarily a problem for studying historical patterns of biodiversity from phylogenies.

A Note on Fibonacci Type Groups

1975 ◽  
Vol 18 (2) ◽  
pp. 173-175 ◽  
Author(s):  
C. M. Campbell ◽  
E. F. Robertson
Keyword(s):  
Free Group ◽  
Normal Closure ◽  
Index Set ◽  
Congruence Classes

Let Fn be the free group on {ai:i ∊ ℤ n} where the set of congruence classes mod n is used as an index set for the generators. The permutation (1, 2, 3, …, n) of ℤn induces an automorphism θ of Fn by permuting the subscripts of the generators. Suppose w is a word in Fn and let N(w) denote the normal closure of {wθi-1:l ≤i≤n}. Define the group Gn(w) by Gn(w)=Fn/N(w) and call wdi-1=l the relation (i) of Gn(w).

scholarly journals Partial semigroups and primality indicators in the fractal generation of binomial coefficients to a prime square modulus

2009 ◽  
Vol 42 (1) ◽  
Author(s):  
D. Doan ◽  
B. Kivunge ◽  
J. J. Poole ◽  
J. D. H Smith ◽  
T. Sykes ◽  
...  
Keyword(s):  
Summer Programs ◽  
Lattice Path ◽  
Binomial Coefficients ◽  
Generation Process ◽  
Research Experience ◽  
Original Function ◽  
Partial Semigroups ◽  
Congruence Classes

AbstractThis paper, resulting from two summer programs of Research Experience for Undergraduates, examines the congruence classes of binomial coefficients to a prime square modulus as given by a fractal generation process for lattice path counts. The process depends on the isomorphism of partial semigroup structures associated with each iteration. We also consider integrality properties of certain critical coefficients that arise in the generation process. Generalizing the application of these coefficients to arbitrary arguments, instead of just to the prime arguments appearing in their original function, it transpires that integrality of the coefficients is indicative of the primality of the argument.

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