A Riemannian manifold is a differentiable manifold endowed with a Riemannian metric. Given a Riemannian metric, the lengths of curves can be defined via integration, and the angle of intersection of two curves is also well defined. However, there is a priori no direct way of aligning neighbouring tangent planes, hence no direct way of […]
With COVID-19 causing a transition to online teaching, I am now in the habit of making short videos… Consider two isolated capacitors, one with a charge on it, one without. Then suddenly bring them together. The charge will distribute equally on the two capacitors. But energy will have been lost! Where did it go?
This note elaborates on the “perturbation interpretation” of duality in convex optimisation. Caveat lector: regularity conditions are ignored. Recall that in convex analysis, a convex set is fully determined by its supporting hyperplanes. Therefore, any result about convex sets should be re-writable in terms of supporting hyperplanes. This is the basis of duality. A simple yet […]
Minimising subject to can be solved using Lagrange multipliers, where is referred to as the Lagrangian. The optimal value satisfies both and . Although was added to the problem, it turns out that has physical significance: it is the rate of change of the optimal value to perturbations around zero in the constraint . A […]
What is an attacking shot? Must an attacking shot be hit hard? What about a defensive shot? This essay will consider both the theory and the practice of attacking and defensive shots. And it will show that if you often lose 6-2 to an opponent then only a small amount of improvement might be enough […]
ElectronicsInformal Classroom Notesbasic electronic principlesdisconnecting an inductorinductor
Consider current flowing from a battery through an inductor then a resistor before returning to the battery again. What happens if the battery is suddenly removed from the circuit? Online browsing suggests that the voltage across the inductor reverses “to maintain current flow” but the explanations for this are either by incomplete analogy or by […]
Despite many online attempts at providing intuition behind Caratheodory’s criterion, I have yet to find an answer to why testing all sets should work. https://www.thestudentroom.co.uk/showthread.php?t=4284694 https://mathoverflow.net/questions/34007/demystifying-the-caratheodory-approach-to-measurability Therefore, I have taken the liberty of proffering my own intuitive explanation. For the impatient, here is the gist. Justification and background material are given later. We will […]
This note gives a very simple derivation of the Boltzmann distribution that avoids any mention of temperature or entropy. Indeed, the Boltzmann distribution can be understood as the unique distribution with the property that when a large (or even a small!) number of Boltzmann distributions are added together, all the different ways of achieving the […]
Informal Classroom Notesintuitionmeasure-theoretic probability
If is a -measurable random variable then there exists a Borel-measurable function such that . The standard proof of this fact leaves several questions unanswered. This note explains what goes wrong when attempting a “direct” proof. It also explains how the standard proof overcomes this difficulty. First some background. It is a standard result that […]
Informal Classroom Notescausalitydigital signal processingdiscrete-time signal processingstabilitytransfer functionz-transforms
Why should all the poles be inside the unit circle for a discrete-time digital system to be stable? And why must we care about regions of convergence (ROC)? With only a small investment in time, it is possible to gain a very clear understanding of exactly what is going on — it is not complicated if […]
