Math Problems | Sanity Phailed.me - FrontMotion Firefox. Don Cross - personal website - math, science, software, electronics - FrontMotion Firefox. Probability Theory — A Primer | Math ∩ Programming - FrontMotion Firefox. It is a wonder that we have yet to officially write about probability theory on this blog. Probability theory underlies a huge portion of artificial intelligence, machine learning, and statistics, and a number of our future posts will rely on the ideas and terminology we lay out in this post. Our first formal theory of machine learning will be deeply ingrained in probability theory, we will derive and analyze probabilistic learning algorithms, and our entire treatment of mathematical finance will be framed in terms of random variables. And so it’s about time we got to the bottom of probability theory.
In this post, we will begin with a naive version of probability theory. That is, everything will be finite and framed in terms of naive set theory without the aid of measure theory. This has the benefit of making the analysis and definitions simple. The downside is that we are restricted in what kinds of probability we are allowed to speak of.
Finite Probability Spaces must be 1. and for all. An Interactive Guide To The Fourier Transform | BetterExplained - FrontMotion Firefox. The Fourier Transform is one of deepest insights ever made. Unfortunately, the meaning is buried within dense equations: Yikes. Rather than jumping into the symbols, let's experience the key idea firsthand. Here's a plain-English metaphor: What does the Fourier Transform do? Given a smoothie, it finds the recipe.How? Here's the "math English" version of the above: The Fourier Transform takes a time-based pattern, measures every possible cycle, and returns the overall "cycle recipe" (the strength, offset, & rotation speed for every cycle that was found).
Time for the equations? If all goes well, we'll have an aha! This isn't a force-march through the equations, it's the casual stroll I wish I had. From Smoothie to Recipe A math transformation is a change of perspective. The Fourier Transform changes our perspective from consumer to producer, turning What did I see?
In other words: given a smoothie, let's find the recipe. Why? So... given a smoothie, how do we find the recipe? Whoa. Stop. Oh! Lijst van grote getallen - Wikipedia - FrontMotion Firefox. De termen zijn volgens de lange schaalverdeling. Als de waarde van de termen volgens de korte schaalverdeling gevonden moet worden, kan vanaf biljoen de macht van 10 berekend worden door de helft van de exponent volgens de lange schaal te nemen en er 3 bij op te tellen. Bijvoorbeeld: 1 biljoen volgens de lange schaal is (zie tabel) 1012, volgens de korte schaal 109 (9 = 12÷2 + 3). Spirals. What is a spiral? A spiral is a curve in the plane or in the space, which runs around a centre in a special way.
Different spirals follow. Most of them are produced by formulas. Spirals by Polar Equations top Archimedean Spiral topYou can make a spiral by two motions of a point: There is a uniform motion in a fixed direction and a motion in a circle with constant speed. You get formulas analogic to the circle equations. SpiralThe radius r(t) and the angle t are proportional for the simpliest spiral, the spiral of Archimedes. If you connect both spirals by a straight (red) or a bowed curve, a double spiral develops. Equiangular Spiral (Logarithmic Spiral, Bernoulli's Spiral) top More Spirals topIf you replace the term r(t)=at of the Archimedean spiral by other terms, you get a number of new spirals.
I chose equations for the different spiral formulas suitable for plotting. Clothoide (Cornu Spiral) top Spirals Made of Arcs topHalf circle spirals Spirals Made of Line Segments top German D.H.O. Top. Fibonacci. As you probably know by now, the Fibonacci sequence shows up in the most unexpected places. Scientists are still attempting to figure out the mysteries of our planet. Some have even begun the investigation of our solar system, and even our galaxy. Our Milky Way even has the Fibonacci spiral imbedded into it. As shown previously in the explanation of the Golden Ratio, the following spiral can be obtained using the Fibonacci sequence. As you know, our Milky Way looks like the image below. Place your mouse over this image, and you will see the Fibonacci spiral at work!
Our world is filled with many mysteries that scientists and mathematicians alike are working to figure out if this was just a coincidence, or something else. Logarithmic spiral. This is the spiral for which the radius grows exponentially with the angle. The logarithmic relation between radius and angle leads to the name of logarithmic spiral or logistique (in French). The distances where a radius from the origin meets the curve are in geometric progression. The curve was the favorite of Jakob (I) Bernoulli (1654-1705). On his request his tombstone, in the Munster church in Basel, was decorated with a logarithmic spiral (bottom side). The curve, which looks by the way more like an Archimedes' spiral, has the following Latin text accompanied: eadem mutata resurgo.
However, Rene Descartes (1638) was the first to study the curve. What are the remarkable qualities of the equiangular spiral? Other qualities of the spiral are the following: the radius of curvature is equal to the arc length: R = s. Remarkable! The logarithmic spiral is the curve for which the angle between the tangent and the radius (the polar tangent) is a constant. Notes. Wiskunde.startpagina.nl.