What is Domain Theory?

Domain theory studies type of partially ordered sets, also known as domains.

Partial orders are relations characterised by the RAST property of being reflexive, anti-symmetric and transitive.  They are often depicted by a <= sign.

A <= relation on a set of real numbers is clearly reflexive (a<=a) due to the allowance of equality. Antisymmetry implies equality if a<=b and b<=a (think of a Venn diagram of elements
A set with a partial order defined on it is called a poset.

The idea of domain theory is to use recursion to find successively better approximations to some desired object.

Deep Learning with Lex Fridman of MIT

Lex Fridman is a researcher and lecture at MIT who has developed immense resources on deep learning for those looking to learn.

Why are "Chak" Equations So-Called?

"Cha-Mo" equations are so-named as they were independently discovered by Sydney Chapman and Andrey Kolmogorov.

They are pivotal equations in the theory of Markovian stochastic processes.

Both Chapman and Kolmogorov were mathematicians, with Chapman doubling up as a notable geophysicist studying, amongst other things, the ozone layer and the magnetosphere.

What are the Elements of Modern Analysis? How do they figure in Quantum Mechanics?

Modern analysis is more than Riemann Integration (which is more akin to "classical" analysis).

Modern analysis includes Lebesgue Integration (far more widely applicable than Riemann) and the notion of Hilbert Spaces (henceforth abbreviated to H-Spaces).

H-Spaces  are a generalisation of the notion of Euclidean space and used significantly in partial differential equations and quantum mechanics; the Sturm-Liouville theory, for example, which studies a particular second-order differential equation, utilises Hilbert spaces intrinsically.

H-Spaces have the very pleasant property that they are complete - any sequence of points in the space are guaranteed to converge to a point that actually lives within the space.

Spectral theory is another "advanced" analysis concept introduced by Mr Hilbert. It extends the eigenvalue-eigenvector theory into a broader theory on the structure of operators on a wide variety of mathematical spaces. It later found application in quantum mechanics.

What is a semiprime or biprime?

A semiprime, or biprime, is the product of two prime numbers. Clearly the result is not a prime number, hence the term "semi" prime.

The constituent primes of a semiprime may equal each other, so semiprimes may be squares of prime numbers. An example is the number 4 which is the product of two instances of the same prime number 2. The OEIS (Online Encyclopedia of Integer Sequences) has a listing of semiprimes.

The number 106 is a semiprime, as is 111, and 123.

A Very Good Graphing Tool

Desmos is a very good graphing tool.

What is LQE? Why is LQE called LQE?

Linear quadratic estimation (LQE) is better known by its adopted "trade name", the Kalman filter (widely used in GNC i.e. guidance, navigation and control, applications).

Kalman was fundamental in establishing the so-called "state space representation of dynamical systems".

Due to the scepticism against Kalman's ideas on filtering, he was forced to initially publish his results in the field of mechanical engineering.

His ideas were subsequently applied in space navigation for the Apollo program of the 1960s.

Extensions of Kalman's ideas to nonlinear systems have led to the development of the Extended Kalman Filter and so-called "unscented" Kalman filter (UKF).

What are Vine Copulas an Extension Of?

An interesting topic in data science is vine copulas. They are an extension of the Gaussian Copula.

Don't know what a copula is? No Problem.

A copula is simply a multivariate cumulative probability distribution with some special probabilistic properties.

Vine copulas have been used in portfolio optimization problems to model tail risk.

Quantum Science Studies for Pythonistas

A high level introduction to quantum science can be found here.

What is a z-score?

A z-score is the number of standard deviations from a mean data point. "x minus mu over sigma" is the typical formula for a z-score.

Python Programmers and Physics

Python programmers who like physics read Physics World.

Pi calculated to 31 trillion digits

A Google employee from Japan has calculated pi to 31 trillion digits. The calculation took 25 virtual machines 121 days to complete. The value pi appears all over mathematics, from the area of a circle to the evaluation of the Riemann zeta function.

What did Nikolay Krasovsky Study?

Nikolay Krasovsky, born in Yekaterinburg (on the Iset River), worked in the mathematical theory of control, dynamical systems and differential games.

A Lookback at 17th Century Maths

A very good "lookback" at 17th Century maths can be found on the following website. This includes Edmonton-born Brook Taylor's treatise: "Methodus Incrementorum Directa & Inversa". Taylor was famed for the Taylor series and Taylor's theorem. John Napier's treatise on logarithms is also presented in translated form: "Mirifici Logarithmorum Canonis Descriptio".

Latin is one of the "hazards" of reading 17th Century maths. As a taster, Mirifici (above) is the plural of Mirificus (wonderful, miraculous). Canonis is the genitive of canon (referring to a catalog of sacred writings). Descriptio means simply description.

As well as discovering logarithms, John Napier was also known for inventing the so-called "Napiers Bones" which was a calculating device for multiples and quotients. It was based on lattice multiplication.

Bifurcation Theory for PyCryptonistas

What is bifurcation theory? An abstract definition is possible, but the most common usage is in dynamical systems, when a small change to parameters creates a major change to the "topology" of a solution (typically of a family of differential equations). It is informally part of the "qualitative" theory of differential equations.