In the first part of this paper we consider partially ordered sets for which the intrinsic ‘interval topology’ ((1), p. 60) is Hausdorffian. The main result of this section is the following: the interval topology of a lattice is Hausdorffian if and only if convergence with respect to the interval topology is equivalent to strong o*-convergence. This may be regarded as an answer either to Birkhoff's problem 23 ((1), p. 62) or to his problem 25 ((1), p. 64). In the case of a complete lattice, we have an alternative formulation. The interval topology of a complete lattice is Hausdorffian if and only if every filter (or, alternatively, every directed net) in the lattice has an o-convergent refinement. This condition may be regarded as a strong type of compactness.
In this paper, we derive several differential Harnack estimates (also known as Li-Yau-Hamilton-type estimates) for positive solutions of Fisher's equation. We use the estimates to obtain lower bounds on the speed of traveling wave solutions and to construct classical Harnack inequalities.Date: October 7, 2016. 1 2 XIAODONG CAO, BOWEI LIU, IAN PENDLETON, AND ABIGAIL WARD Our work introduces and proves three Li-Yau-Hamilton-type Harnack inequalities which constrain positive functions satisfying the Fisher-KPP equation on an arbitrary Riemannian manifold M n . Depending on the setting we obtain different inequalities. The study of differential Harnack inequalities was first initiated by P. Li and S.-T. Yau in [10] (also see [1]). Harnack inequalities have since played an important role in the study of geometric analysis and geometric flows (for example, see [7,13]). Applications have also been found to the study of nonlinear parabolic equations, e.g. in [8]. One of these is a recent reproof of the classical result of H. Fujita [6], which states that any positive solution to the Endangered Species Equation in dimension n,blows up in finite time provided 0 < n(p − 1) < 2; see [3].When the dimension falls into a certain range we can integrate our differential Harnack inequality along any space-time curve to obtain a classical Harnack inequality which allows us to compare the values of positive solutions at any two points in space-time when time is large.The organization for the paper is as follows: In Section 2 we present the precise formulations and the proofs of our two inequalities governing closed manifolds. In Section 3 we state and prove a similar Harnack inequality for complete noncompact manifolds. In Section 4, we end the paper with the aforementioned results on the minimum speed of traveling wave solutions and classical Harnack inequalities.
In 1952 W. E. Roth published two theorems, one of which has come to be known as Roth’s removal rule and (slightly generalised) goes as follows. [Recall that square matrices M, N are similar when there is an invertible matrix R such that RMR-1 = N. The matrix entries can be elements from any field, although for simplicity we shall call them ‘numbers’.]
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