On groupoids having a linear congruence class geometry

1982 ◽  
Vol 180 (2) ◽  
pp. 395-411 ◽  
Author(s):  
Thomas Ihringer
Keyword(s):  
Congruence Class ◽  
Linear Congruence ◽  
Class Geometry

On finite algebras having a linear congruence class geometry

1984 ◽  
Vol 19 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Thomas Ihringer
Keyword(s):  
Congruence Class ◽  
Linear Congruence ◽  
Finite Algebras ◽  
Class Geometry

scholarly journals RSA Cryptography Algorithm Using linear Congruence Class.

2016 ◽  
Vol 4 (5) ◽  
pp. 1335-1347
Author(s):  
PurnaChandra Sethi ◽  
◽  
PrafullaKumar Behera ◽  
Keyword(s):  
Congruence Class ◽  
Linear Congruence ◽  
Rsa Cryptography

Congruence relations on almost distributive lattices-II

2019 ◽  
pp. 2150005
Author(s):  
Gezahagne Mulat Addis
Keyword(s):  
Distributive Lattice ◽  
Congruence Relation ◽  
Congruence Class ◽  
Distributive Lattices ◽  
Ideal Formula ◽  
Congruence Relations

For a given ideal [Formula: see text] of an almost distributive lattice [Formula: see text], we study the smallest and the largest congruence relation on [Formula: see text] having [Formula: see text] as a congruence class.

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 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 Deformations of reducible Galois representations to Hida-families

2021 ◽  
pp. 1-24
Author(s):  
Anwesh Ray
Keyword(s):  
Deformation Theory ◽  
Galois Representations ◽  
Congruence Class ◽  
Representation Formula ◽  
Theoretic Approach ◽  
Hida Families ◽  
Global Deformation

The global deformation theory of residually reducible Galois representations with fixed auxiliary conditions is studied. We show that [Formula: see text] lifts to a Hida line for which the weights range over a congruence class modulo-[Formula: see text]. The advantage of the purely Galois theoretic approach is that it allows us to construct [Formula: see text]-adic families of Galois representations lifting the actual representation [Formula: see text], and not just the semisimplification.

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 Linear Congruence with Side Conditions

1963 ◽  
Vol 70 (8) ◽  
pp. 837-840 ◽  
Author(s):  
David Rearick
Keyword(s):  
Linear Congruence
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