Welcome to the basic ideas of optimal transport. From Monge to Kantorovich.
Oct 19, 2022
We record some very simple comments on the difficulties of computing Fermat Weber points of probability measures. These are points x which minimize the average euclidean distance from the points supporting a probability measure mu on Rn. The problem is that mathematicians need to stop asking for formulas for naive geometric questions. Besides formulas are not as useful as they might initially appear. Instead fast approximation and convergence are much more useful, although the resultant is not known a priori.
Oct 18, 2022
Brief remark on stabilizing the medial axis transform M(A) of open subsets A of Rn. The basic idea is to use the Reduction-to-Singularity method from our thesis to define homotopy-reductions M(A) to W(A). This is speculative and not formally proven at this stage.
Oct 15, 2022
In this post we present what we believe Weber and Kirchoff's solution to Veritasium's video on electricity. This video caused surprising uproar from physicists and engineers. The surprising degree of controversy is indicative a deep misunderstanding underlying electromagnetism, and this is where Weber-Kirchoff's original idea of surface charge distributions proportional to voltage being the cause of steady state currents in resistive wires. We recall that Weber and Kirchoff are the first scientists to derive the telegraphy equation, which predicts that electrical signals in conductive wires travel at the speed of light $c$. This is important work even prior to Maxwell's equations.
Oct 8, 2022
We continue to review and reflect on questions suggested by Craig Alan Feinstein's proposed negative solution to P versus NP. We are specifically reflecting on a signed subset sum problem, which is effectively a linearization of the unsigned subset sum. The question then reduces to deciding the injectivity of a map, namely from a large dimensional finite object to a much smaller dimensional but infinite object, namely integers. This raises the question of what is the computational complexity of deciding injectivity of maps? Is it really ever possible to deterministically decide the injectivity of a map without necessarily running through every evaluation? Along the way we discuss several aspects of CAF's work.
Oct 6, 2022
We revisit the well known subject of the computational complexity of Fibonacci's sequence 1, 1, 2, 3, 5, 8, 13, etc.. However we are looking at the question with Wolfram's idea of computational irreducibility. Yet Wolfram does not appear to define irreducibility, and this suggests the question of relating topological irreducibility to questions of computation, and this is somewhat speculative, as we are looking for strategies to prove that O(log(n)) is the minimal time complexity of computing fibonacci sequences. Can we formalize Wolfram's definition and obtain a strategy for proving the minimality of these running times? We're curious and this is exploration.
Oct 4, 2022
In this post we present the basic python and gekko code for simulating two negatively charged electrons acting under Weber's force. The challenge is to obtain bounded orbits, i.e. to witness electrons passing the Weber critical radius and becoming attractive. As we've discussed in previous posts, this requires high energy levels comparable to the famous $E=m c^2$ formula.
Sep 13, 2022
Our goal in this post is to integrate the equations of motion for three charged bodies under Weber's force law. Why? Because we want to test the Sansbury proposal that the electron $e^{-}$ is structured, specifically as a 3-body system with net charge -1=-1-1+1. Yes, this is strange hypothesis and very important to investigate. But there is a challenge: the Newtonian equations of motion combined with Weber force are not the usual second order ODEs. This means we need to introduce a Differential Algebraic Equation (DAE) solver named GEKKO, and this has delayed the publication of this post. With GEKKO we can solve ODEs of the form $F(x, x', x'')=0$. This post is theoretical. But we are working on the python notebooks for the actual code for Weber 3-body problem with net charge -1 = -1 -1 + 1.
Sep 11, 2022
Bohr and quantum mechanics never explained the apparent nonradiation of the hydrogen atom. It was instead postulated, taken as given, and led to the quantum discontinuities, etc... But the great Wilhelm Weber (1804-1891) and the quiet softspoken Ralph Sansbury have a very interesting alternative, which answers everything that Bohr and Quantum could not, and succeeds where they surrendered, which is explaining the stability of atoms.
Sep 4, 2022
E=mc^2 is the most famous formula in history, but what does it mean? Here we emphasize that Einstein derived the formula from mathematics in 1905, but Wilhelm Weber derived the formula from physics in 1860. With Einstein there is no meaning, no physics; but Weber has an immediate meaning in his particle electrodynamics, and E=mc^2 represents an incredible fact about Weber's critical radius and the possibility of 2=+1+1 as a stable electronic system!
Sep 3, 2022
Our attention has recently been drawn to Craig Alan Feinstein's work on P vs NP, and we record some ideas below. Basically we find that IF P is not equal to NP, then it will be unprovable because one cannot formally prove lower bounds on worst case complexity of algorithms. Or this is our impression at the moment.
Sep 3, 2022
We record some ideas based on rigid disk packings to potentially construct open sets with arbitarily small volume and which cannot be expanded-embedded into the unit disk.
Aug 31, 2022
Mapping Class Group, Bieri-Eckmann duality, spines, Teichmueller space
Aug 30, 2022
Brief update. Have not posted in long time! Not good for readers.
Aug 29, 2022
(Under construction) Here we record some observations and ideas regarding Ahlfor's measure conjecture. We begin writing basic python code to study limit points of divergent subsequences in Kleinian-type groups in higher dimensions. For the purposes of this article however we restrict ourselves to small dimensions.
Jun 16, 2022