The main lemma

1991 ◽  
pp. 54-59
Author(s):  
Carlos Simpson
Keyword(s):  
Main Lemma

Energy Approximation

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Main Lemma ◽  
The Other ◽  
Coarse Scale ◽  
Continuous Solutions ◽  
Material Derivative ◽  
Energy Variation ◽  
Energy Approximation ◽  
The Difference ◽  
Derivatives Of

This chapter presents the equations and calculations for energy approximation. It establishes the estimates (261) and (262) of the Main Lemma (10.1) for continuous solutions; these estimates state that we are able to accurately prescribe the energy that the correction adds to the solution, as well as bound the difference between the time derivatives of these two quantities. The chapter also introduces the proposition for prescribing energy, followed by the relevant computations. Each integral contributing to the other term can be estimated. Another proposition for estimating control over the rate of energy variation is given. Finally, the coarse scale material derivative is considered.

Transport-Elliptic Estimates

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Elliptic Equation ◽  
Transport Equation ◽  
Transport Properties ◽  
Error Term ◽  
Main Lemma ◽  
Spatial Derivative ◽  
Material Derivative ◽  
Divergence Operator ◽  
Spatial Derivatives ◽  
Elliptic Estimates

This chapter solves the underdetermined, elliptic equation ∂ⱼQsuperscript jl = Usuperscript l and Qsuperscript jl = Qsuperscript lj (Equation 1069) in order to eliminate the error term in the parametrix. For the proof of the Main Lemma, estimates for Q and the material derivative as well as its spatial derivatives are derived. The chapter finds a solution to Equation (1069) with good transport properties by solving it via a Transport equation obtained by commuting the divergence operator with the material derivative. It concludes by showing the solutions, spatial derivative estimates, and material derivative estimates for the Transport-Elliptic equation, as well as cutting off the solution to the Transport-Elliptic equation.

The Divergence Equation

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Vector Fields ◽  
Classical Mechanics ◽  
Main Lemma ◽  
Constant Vector ◽  
Smooth Vector ◽  
Continuous Solutions ◽  
All Solutions ◽  
Divergence Equation ◽  
Conservation Of Momentum ◽  
Special Solutions

This chapter introduces the divergence equation. A key ingredient in the proof of the Main Lemma for continuous solutions is to find special solutions to this divergence equation, which includes a smooth function and a smooth vector field on ³, plus an unknown, symmetric (2, 0) tensor. The chapter presents a proposition that takes into account a condition relating to the conservation of momentum as well as a condition that reflects Newton's law, which states that every action must have an equal and opposite reaction. This axiom, in turn, implies the conservation of momentum in classical mechanics. In view of Noether's theorem, the constant vector fields which act as Galilean symmetries of the Euler equation are responsible for the conservation of momentum. The chapter shows proof that all solutions to the Euler-Reynolds equations conserve momentum.

On Diophantine equations solvable in models of open induction

1990 ◽  
Vol 55 (2) ◽  
pp. 779-786 ◽  
Author(s):  
Margarita Otero
Keyword(s):  
Diophantine Equations ◽  
Main Lemma ◽  
Peano Arithmetic ◽  
The Real ◽  
Real Closure ◽  
Open Induction ◽  
Induction Scheme

AbstractWe consider IOpen, the subsystem of PA (Peano Arithmetic) with the induction scheme restricted to quantifier-free formulas.We prove that each model of IOpen can be embedded in a model where the equation has a solution. The main lemma states that there is no polynomial f{x,y) with coefficients in a (nonstandard) DOR M such that ∣f(x,y) ∣ < 1 for every (x,y) Є C, where C is the curve defined on the real closure of M by C: x2 + y2 = a and a > 0 is a nonstandard element of M.

scholarly journals On Hau's lemma

1994 ◽  
Vol 50 (3) ◽  
pp. 451-458 ◽  
Author(s):  
W.K.A. Loh
Keyword(s):  
Prime Power ◽  
Main Lemma ◽  
Precise Form ◽  
Image Position ◽  
Maximum Multiplicity

Let f ∈ ℤ[X] and let q be a prime power pl(l ≥ 2). Hua stated and proved thatfor some unspecified constant C > 0 depending on the derivative f′ of f; M denoting the maximum multiplicity of the roots of the congruence p−tf′(x) ≡ 0 (mod p), where t is an integer chosen so that the polynomial p−tf′(x) is primitive. An explicit value for C was given by Chalk for p ≥ 3. Subsequently, Ping Ding (in two successive articles) obtained better estimates for p ≥ 2.This article provides a better result, based upon a more precise form of Hua's main lemma, previously overlooked.

Structure of the Book

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Euler Equations ◽  
Weak Solutions ◽  
Energy Levels ◽  
Main Lemma ◽  
Optimal Regularity ◽  
Continuous Solutions ◽  
Hölder Continuous ◽  
Energy Part ◽  
Error Terms ◽  
Incompressible Euler

This chapter provides an overview of the book's structure. Section 3 deals with the error terms which need to be controlled, whereas Part III explains some notation of the book and presents a basic construction of the correction. The goal is to clarify how the scheme can be used to construct Hölder continuous weak solutions—continuous in space and time—to the incompressible Euler equations that fail to conserve energy. Part IV shows how to iterate the construction of Part III to obtain continuous solutions to the Euler equations. It then discusses the concept of frequency energy levels, along with the Main Lemma. It also highlights some additional difficulties which arise as one approaches the optimal regularity and illustrates how these difficulties can be overcome. Parts V–VII verify all the estimates needed for the proof of the Main Lemma.

Sign in / Sign up

Export Citation Format

Share Document