A Field of Sets - A Powerful Data Structure for the Pythonista

A field of sets is a collection or system of sets, where sum, product and difference of any two sets, forms another set in the same field. Thus a field of sets is characterised by closure under standard set operations.

Mathematical Analogies That Made Probability Theory A Reality

Probability, likelihood, chance - these concepts have an intuitive existence - linked to our perception of the "odds" of particular events.

But in today's world, probability also has a formal definition to go with its intuitive meaning, and this definition was an axiomatic one, spelled out by Kolmogorov. He achieved this by appealing to Lebesgue's  theories of measure and integration. These ideas are linked in the form of the Lebesgue integral. Riemann's ideas alone are not enough to support the axiomatisation of the probability concept.

The relevant ideas and analogies are as follows.

Events as sets
Measure of a set -> Probability of an event
Integral of a function -> Mathematical expectation of a random variable

Analogies led fortuitously to further analogies; for example independence of random variables paralleled harmoniously with the corresponding properties of orthogonal functions.

Quite fundamental to Kolmogorov's axiomatisation is the idea of a field of sets (not to be confused with fields in ring theory e.g. field of real numbers, rationals etc.), where the field of interest, so-to-speak, is the set of events which are (physically) subsets of elementary events.  It is denoted Curly-F. The elements of this set, Curly-F, are called random events.

The idea of a field of sets is defined in Hausdorff's Mengenlehre (a German idiom for set theory, with Mengen literally meaning quantity and lehre meaning teaching). Hausdorff was known for his involvement in topology.

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.