A generalization of the ingleton—Main lemma and a class of non-algebraic matroids

COMBINATORICA ◽  
1988 ◽  
Vol 8 (1) ◽  
pp. 87-90 ◽  
Author(s):  
B. Lindström
Keyword(s):  
Main Lemma ◽  
Algebraic Matroids

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.

scholarly journals Identifiability in phylogenetics using algebraic matroids

2021 ◽  
Vol 104 ◽  
pp. 142-158
Author(s):  
Benjamin Hollering ◽  
Seth Sullivant
Keyword(s):  
Algebraic Matroids

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.

scholarly journals Algebraic matroids and Frobenius flocks

2018 ◽  
Vol 323 ◽  
pp. 688-719 ◽  
Author(s):  
Guus P. Bollen ◽  
Jan Draisma ◽  
Rudi Pendavingh
Keyword(s):  
Algebraic Matroids
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