Sailing the Last Voyage with Newton and Pascal
And why not? Wild bottlenose dolphins bond over their use of tools, with distinct cliques and classes forming over decades as a result of their skills, scientists have found. The communities, which have been compared with societies such as the Bullingdon Club in humans, mean the aquatic animals share their knowledge only with those in …
Continue reading "The Palm Beach Way of Exclusivity is in the Animal Kingdom, Too"
Next month, Lord willing, I start the second year of my PhD adventure in Computational Engineering. "Adventure" is a good way to describe it, especially in my superannuated state. In the midst of getting the coursework started last fall (yes, Europeans, sad to say we have coursework with PhD programmes) we had a "freshman" orientation. …
Our goal is to demonstrate that, for the Hermite polynomial $latex H_{2n+1}(x)=\sum_{j=0}^{n}f(x_{j})H_{j}(x)+\sum_{j=0}^{n}f'(x_{j})\hat{H}_{j}(x) $ where $latex H_{j}(x)=[1-2(x-x_{j})L'_{j}(x_{j})]L_{j}^{2}(x) $ $latex \hat{H}_{j}(x)=(x-x_{j})L_{j}^{2}(x) $ the error function is given by the equation $latex f(x)-H_{2n+1}(x)=\frac{f^{\left(2n+2\right)}\left(\eta\left(x\right)\right)}{\left(2n+2\right)!}\overset{n}{\underset{j=0}{\prod}}\left(x-x_{j}\right)^{2} $ where $latex f\in C^{2n+2}[a,b] $ Let us begin by considering a point $latex \hat{x}\in[a,b]$ where $latex \hat{x}\neq x_{j}$, i.e., it is not equal to …
The objective is to show that back substitution is backward stable. Consider the system $latex Rx=c $ where $latex R$ is an upper triangular $latex m\times m$ matrix and $latex x,c$ are $latex m$ column vectors. For a $latex 3\times3$ matrix system, this would look like: $latex \left[\begin{array}{ccc} r_{1,1} & r_{1,2} & r_{1,3}\\ 0 & …
Continue reading "Stability of Back Substitution for Matrix Systems"
The purpose of this piece is to document the numerical integration of a function in a two-dimensional space using the conjugate gradient method. Basic Problem Statement and Closed Form Solution The boundary value problem is as follows: $latex {\frac{\partial^{2}}{\partial{x}^{2}}}u(x,y)+{\frac{\partial^{2}}{\partial{y}^{2}}}u(x,y)=-5\, y\sin(x)\sin(2\, y)+4\,\sin(x)\cos(2\, y) $ where $latex u(x,0)=0 $ $latex u(x,\pi)=0 $ $latex u(0,y)=0 $ $latex u(\pi,y)=0 …
The object of this is to solve the differential equation $latex {\frac{d^{3}}{d{x}^{3}}}y(x)-\mu\,\left(1-\left(y(x)\right)^{2}\right){\frac{d^{2}}{d{x}^{2}}}y(x)+2\,\mu\, y(x)\left({\frac{d}{dx}}y(x)\right)^{2}+{\frac{d}{dx}}y(x)=0 $ for the following boundary conditions and parameters: $latex \mu=\frac{1}{2}$ $latex y\left(0\right)=0$ $latex {\frac{d}{dx}}y(0)=\frac{1}{2}$ $latex y\left(2\right)=1$ Conventional wisdom would indicate that, because of the high order of the derivatives, this problem cannot be solved using a scalar implementation of simple shooting. However, …
While reading Grabau's 1913 Principles of Stratigraphy, I came across this fascinating description of the geology of Florida and its coral reefs. This is for the most part a summary of expeditions in the early 1850's by two of this country's pioneering geologists: Louis Agassiz and Joseph LeConte. In addition to the outline of the …
Continue reading "A Geologic View of Old Florida and its Coral Reefs"
Generally speaking I stop this time of year and commemorate an event of which you, the reader of this blog, are the beneficiary (?). On 1 July 1997 I uploaded the pages of my first website, called then The Wave Equation Page for Piling and now vulcanhammer.net. I duly editorialised the tenth anniversary of that …
This from the "godfather of global warming", among other things: Lovelock blasted greens for treating global warming like a religion. “It just so happens that the green religion is now taking over from the Christian religion,” Lovelock observed. “I don’t think people have noticed that, but it’s got all the sort of terms that religions …
Continue reading "We Really Don't Need to Make a Religion Out of Science"
This, from A.W. Grabau's 1913 classic Principles of Stratigraphy: The chief agents in retaining the sun's heat within the earth's atmosphere are the carbon dioxide and the water vapor, which act as thermal blankets. It has been estimated by Arrhenius (2) that if the amount of CO2 in the atmosphere were increased 2.5 to 3 …
Continue reading "Carbon Dioxide and Climate Change: An Old View"
Positive Infinity 