Solutions of the KdV Equation through Analysis of Regular Symmetries

We investigate a case of the generalized Korteweg – De Vries Burgers equation. Our aim is to demonstrate the need for the application of further methods in addition to using Lie Symmetries. The solution is found through differential topological manifolds. We apply Lie’s theory to take the PDE to an ODE. However, this ODE is of third order and not easily solvable. It is through differentiable topological manifolds that we are able to arrive at a solution

Enlargeable length-structure and scalar curvatures

We define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a closed n-manifold with an enlargeable Riemannian length-structure with an arbitrary closed smooth manifold carries no Riemannian metrics with positive scalar curvature. We show that closed smooth manifolds with a locally CAT(0)-metric which is strongly equivalent to a Riemannian metric are examples of closed manifolds with an enlargeable Riemannian length-structure. Moreover, the result is correct in arbitrary dimensions based on the main result of a recent paper by Schoen and Yau. We define the positive MV-scalar curvature on closed orientable topological manifolds and show the compactly enlargeable length-structures are the obstructions of its existence.

Topological Manifolds and Polish Spaces

We give,as a preliminary result, some topological characterizations of locally compact second-countable Hausdorff spaces. Then we show that a topological manifold, with boundary or not,is precisely a Polish space with a coordinate open cover; this connects geometry with descriptive set theory.

Embedding Topological $n$-Manifolds into Compact Nonstandard Topological $n$-Manifolds with Boundary

We show as a main message that there is a simple dimension-preserving way to openly and densely embed every topological manifold into a compact ``nonstandard'' topological manifold with boundary.This class of ``nonstandard'' topological manifolds with boundary contains the usual topological manifolds with boundary.In particular,the Alexandroff one-point compactification of every given topological $n$-manifold is a ``nonstandard'' topological $n$-manifold with boundary.

Elementary Proof of Existence of a Subordinate Smooth Partition of Unity on Smooth Manifolds

We give in particular an elementary proof of the existence of a smooth partition of unity subordinate to any given open cover for smooth manifolds. As a side note, given are also two elementary proofs of the existence of a subordinate partition of unity for topological manifolds. These in particular fill a gap in the related literature.

Uniformly Embedding Topological Manifolds into $L^{2}$ Spaces

With a simple argument, we show as a main note that every locally compact second-countableHausdorff space is topologically embeddable into some $L^{2}$ space with respect to some finite nonzero Borel measure, where the embedding may be chosen so that it is uniform and its range is included in some open proper subset of the $L^{2}$ space.

On Solutions of the Nonlinear Heat Equation Using Modified Lie Symmetries and Differentiable Topological Manifolds

In this paper, we present three simple analytical techniques for obtaining solutions of the nonlinear heat equations. The heat equations, both linear and nonlinear, are very important to the mathematical sciences. This is because they are reduced forms of many models, hard to solve directly. The techniques are based on Lie’ symmetry group theoretical methods. The first is the pure Lie approach, followed by our modified Lie approach. The third is our differentiable topological manifolds approach. As an application, we determine the separation distance, in the quantum superposition principle, relevant to nanoscience.