Congruence relation

Résumé

In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation. Basic example The prototypical example of a congruence relation is congruence modulo n on the set of integers. For a given positive integer n, two integers a and b are called congruent modulo n, written : a \equiv b \pmod{n} if a - b is divisible by n (or equivalently if a and b have the same remainder when divided by n). For example, 37 and 57 are congruent mod

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https://en.wikipedia.org/wiki/Congruence_relation

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Elliptic curves over a finite field and the trace formula

Ian Nicholas Petrow

We prove formulas for power moments for point counts of elliptic curves over a finite field k such that the groups of k-points of the curves contain a chosen subgroup. These formulas express the moments in terms of traces of Hecke operators for certain congruence subgroups of SL 2(Z). As our main technical input we prove an Eichler-Selberg trace formula for a family of congruence subgroups of SL 2(Z) which include as special cases the groups 1(N) and (N). Our formulas generalize results of Birch and Ihara (the case of the trivial subgroup and the full modular group), and previous work of the authors (the subgroups Z/2Z and (Z/2Z)2 and congruence subgroups 0(2),0(4)). We use these formulas to answer statistical questions about point counts for elliptic curves over a fixed finite field, generalizing results of Vladu, Gekeler, Howe and others.

Wiley2017

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An improvement on the number of simplices in F-q(d)

Van Thang Pham

Let epsilon be a set of points in F-q(d). Bennett et al. (2016) proved that if \epsilon\ >> [GRAHICS] then epsilon determines a positive proportion of all k-simplices. In this paper, we give an improvement of this result in the case when epsilon is the Cartesian product of sets. Namely, we show that if kd epsilon is the Cartesian product of sets and [GRAHICS] = o(\epsilon), the number of congruence classes of k-simplices determined by epsilon is at least (1 - omicron(1)) [GRAPHICS] , and in some cases our result is sharp. (C) 2017 Elsevier B.V. All rights reserved.

Elsevier Science Bv2017

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One-dimensional models for blood flow in arteries

Luca Formaggia, Alfio Quarteroni

In this paper a family of one-dimensional nonlinear systems which model the blood pulse propagation in compliant arteries is presented and investigated. They are obtained by averaging the Navier-Stokes equation on each section of an arterial vessel and using simplified models for the vessel compliance. Different differential operators arise depending on the simplifications made on the structural model. Starting from the most basic assumption of pure elastic instantaneous equilibrium, which provides a well-known algebraic relation between intramural pressure and vessel section area, we analyse in turn the effects of terms accounting for inertia, longitudinal prestress and viscoelasticity. The problem of how to account for branching and possible discontinuous wall properties is addressed, the latter aspect being relevant in the presence of prosthesis and stents. To this purpose a domain decomposition approach is adopted and the conditions which ensure the stability of the coupling are provided. The numerical method here used in order to carry out several test cases for the assessment of the proposed models is based on a finite element Taylor-Galerkin scheme combined with operator splitting techniques

2003

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