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.