Sanity Phailed.me - FrontMotion Firefox. Don Cross - personal website - math, science, software, electronics - FrontMotion Firefox. 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. We should make a quick disclaimer before we get into the thick of things: this primer is not meant to connect probability theory to the real world.
So let us begin with probability spaces and random variables. An Interactive Guide To The Fourier Transform. 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? 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? Well, imagine you had a few filters lying around: Whoa. Oh! Lijst van grote getallen - Wikipedia - FrontMotion Firefox.
De termen zijn volgens de lange schaalverdeling.
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. 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. 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.