Science and math

What is Russell’s paradox?

Jovan Stosic August 19, 2023

Russell’s paradox is based on examples like this: Consider a group of barbers who shave only those men who do not shave themselves. Suppose there is a barber in this collection who does not shave himself; then by the definition of the collection, he must shave himself. But no barber in the collection can shave himself. (If so, he would be a man who does shave men who shave themselves.)

https://www.scientificamerican.com/article/what-is-russells-paradox/

What is Russell’s paradox? was last modified: August 19th, 2023 by Jovan Stosic

Price equation

Jovan Stosic August 19, 2023

In the theory of evolution and natural selection, the Price equation (also known as Price’s equation or Price’s theorem) describes how a trait or allele changes in frequency over time. The equation uses a covariance between a trait and fitness, to give a mathematical description of evolution and natural selection. It provides a way to understand the effects that gene transmission and natural selection have on the frequency of alleles within each new generation of a population. The Price equation was derived by George R. Price, working in London to re-derive W.D. Hamilton‘s work on kin selection. Examples of the Price equation have been constructed for various evolutionary cases. The Price equation also has applications in economics.

It is important to note that the Price equation is not a physical or biological law. It is not a concise or general expression of experimentally validated results. It is rather a purely mathematical relationship between various statistical descriptors of population dynamics. It is mathematically valid, and therefore not subject to experimental verification. In simple terms, it is a mathematical restatement of the expression “survival of the fittest” which is actually self-evident, given the mathematical definitions of “survival” and “fittest”.

https://en.wikipedia.org/wiki/Price_equation

Price equation was last modified: August 19th, 2023 by Jovan Stosic

Oskar Morgenstern

Jovan Stosic August 19, 2023

Oskar Morgenstern (January 24, 1902 – July 26, 1977) was a German-born economist. In collaboration with mathematician John von Neumann, he founded the mathematical field of game theory as applied to the social sciences and strategic decision-making (see von Neumann–Morgenstern utility theorem).

Companies he served as founder/co-founder of included Market Research Corporation of America, Mathematica and Mathematica Policy Research.

https://en.wikipedia.org/wiki/Oskar_Morgenstern

Oskar Morgenstern was last modified: August 19th, 2023 by Jovan Stosic

Hilbert space

Jovan Stosic July 11, 2023

In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space.

The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.

Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a linear subspace or a subspace (the analog of “dropping the altitude” of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to an orthonormal basis, in analogy with Cartesian coordinates in classical geometry. When this basis is countably infinite, it allows identifying the Hilbert space with the space of the infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space.

https://en.wikipedia.org/wiki/Hilbert_space

Hilbert space was last modified: July 11th, 2023 by Jovan Stosic

For all the self learners out there, there is a really good YouTube channel called eigenchris. He teachers Tensor Analysis He’s doing these videos for free, so please support him!

Jovan Stosic July 1, 2023

Hey guys! For all the self learners out there, there is a really good YouTube channel called eigenchris. He teachers Tensor Analysis (from the beginner level up to complex level) and is starting a series on special and general relativity. He’s doing these videos for free, so please support him!
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For all the self learners out there, there is a really good YouTube channel called eigenchris. He teachers Tensor Analysis He’s doing these videos for free, so please support him! was last modified: July 1st, 2023 by Jovan Stosic