#mathematics
Literally a couple of years ago I posted some #AnimatedGifs of the simplest #NumericalIntegration techniques, namely the #RectangleMethod, the #TrapeziumRule and #SimpsonsRule. In the comments, @Miloslav Číž suggested that I show the errors of each method as the number of strips increases (or equivalently,the width of each decreases). I said I'd try to do it when I got time. In principle it is a simple but worthwhile request but getting a bit of time and motivation to do it is another matter. Also, I needed to a way to change the labels dynamically. Anyway, here we are.
The rectangle method is the most simple-minded and crudest method of finding the area below the curve. It converges slowly.
The trapezium rule is quite an improvement to the above and each area is approximated by joining up the ordinates at the beginning and end of each discrete interval with a straight line. This converges appreciably faster than the above method.
Finally the most sophisticated of the three is Simpson's rule in which pairs of adjacent strips use a #quadratic #parabola to interpolate between the ordinates. This even better at convergence than the trapezium rule. I've shown a different function here because for the function shown in the first two cases the difference between the approximation and the numerical #approximation is not discernible and the errors are very small.
#MyWork #Mathematics #Maths #Numerics #RectangleMethod #TrapeziumRule #SimpsonsRule #FreeSoftware #CCBYSA #Wikipedia #Wikimedia
#mathematics #mathsmanuelacasasoli #maths
Higher-order interactions: "from entanglement interactions in quantum mechanics to the trajectory of a disease spreading through a population.
Now we know that the network is just the shadow of the thing.
Josh Grochow
#mathematics #mathsmanuelacasasoli #maths
A visualized Kleinian group often presents a beautiful fractal structure and provides clues for understanding Möbius transformations the mathematical properties of the group.
Iterated inversion system: an algorithm for efficiently visualizing Kleinian groups and extending the possibilities of fractal art.
Journal of Mathematics and the Arts
#mathematics #mathsmanuelacasasoli #maths
Never say “I’m not a Math person”!
How math can be perceived as hard or easy, depending on who you are and the myth that one is or is not a “math person” by nature.
In the #2D #elasticity, #equilibrium of #stresses can be represented on an infinitesimal rectangular element with components of both #DirectStress and #ShearStress generally acting on all four edges. If you were to rotate the rectangle, the stresses change in a precisely orchestrated fashion. In the orientation where the shear stress components vanish, we get what are called #PrincipalStresses and they and their directions can be ascertained precisely through #eigenvalue analysis. This little graphical demonstration shows one example of such a state. The #AnimatedGif was produced a routine written in #WxMaxima.
#Mathematics #TheoryOfElasticity #Mechanics #SolidMechanics #Engineering #MyWork #CCBYSA #WorkInProgress
#mathematics #mathsmanuelacasasoli #maths
Designing flowers like geometrical structures...
The Sophia Construct is a Dodecahedron reflecting within itself.
Albion Rose from Mathematical Art Galleries
1-http://gallery.bridgesmathart.org/exhibitions/2021-joint-mathematics-meetings/albion
2-https://www.echoflowers.org/
The Simplest Math Problem No One Can Solve #maths #math #mathematics
https://www.youtube.com/watch?v=094y1Z2wpJg
#mathematics #mathsmanuelacasasoli #maths
Proof Assistant Makes Jump to Big-League Math
At last, a proof assistant has made a real contribution to the leading edge of mathematical research by verifying the correctness of a complicated, modern proof.
Further readings
1-https://bit.ly/378IHpp
2-https://leanprover-community.github.io/
Getting started with #Maxima in #Fedora #Linux - Fedora Magazine
https://fedoramagazine.org/getting-started-with-maxima-in-fedora-linux/
#mathematics #mathsmanuelacasasoli #maths
New Shape Opens ‘Wormhole’ Between Numbers and Geometry
The article by Kevin Hartnett (https://twitter.com/KSHartnett) expertly explains a mammoth 350-page paper by Fargues and Scholze about a new way of connecting number theory and geometry (Langlands program).
Fargues and Scholze (2021) Geometrization of the local Langlands correspondence: https://arxiv.org/abs/2102.13459
#mathematics #mathsmanuelacasasoli #maths
How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer.
Infinite mathematics
Cantor, Gödel, Cohen... Now, the new proof by Asperó and Schindler.
There is an infinite number of infinities. Which one corresponds to the real numbers?
Martin’s Maximum++ implies Woodin’s axiom
#mathematics #mathsmanuelacasasoli #maths
Curved ribbons or strips are suited to suggest a three-dimensionality… A system of four theoretically never-ending ribbons running over the surface of the sphere linking at the poles. Both poles are visible, one externally and one internally through the interspace between the two ribbons.
M.C. Escher
2021 Bridges Conference
Escher’s Sphere
Image Credit: Douglas Dwyer
#mathematics #mathsmanuelacasasoli #maths
In Einstein's general relativity the structure of space can change but not its topology. Topology is the property of something that doesn't change when you bend it or stretch it as long as you don't break anything.
Edward Witten
Knot Gallery
Tungsteno
https://twitter.com/74WTungsteno/status/1412049803106717702
#mathematics #mathsmanuelacasasoli #maths
Between mathematics and art, a new fractal image by “Occupy Math”
Image of the week 266
A conjugate Newton’s method fractal, a fairy river.
