The Beauty of Surjective Homomorphisms

What is a surjective homomorphism?

Simply put together the two definitions of surjection and homomorphism. 

A surjection is an "onto" mapping. The function maps elements of a set "onto" the entirety of elements in a set B. The elements of the set B are "fully recruited" in the action of the surjection on set A.

A possible alternative description for a surjection is a "fully range-engaged mapping" as we are engaging the totality of the function's range in the action of the mapping.

A homomorphism is a function that maps a "binary combo" of elements in the "domain", to the "binary combo" of the mapped or modified ("homomorphism-applied") elements in the range.

If this homomorphism captures the entirety of the range-set, it is surjective. A little bit of thought will assure the PyCryptonista that surjectivity in homomorphisms is quite a desirable property.

Cauchy's Kingship of Complex Function Theory

Baron Cauchy is oft celebrated as the developer extraordinaire of complex function theory.  

The first influential theorem he proved was Cauchy's integral theorem, also known as the Cauchy-Goursat Theorem (CGT).

(Goursat, though lesser known than Augustin-Louis Cauchy, was also an extraordinary mathematician where mathematical analysis was concerned)

The CGT is an important statement about integrals for holomorphic functions in the complex plane. A holomorphic function is a complex valued function, that may have one or more complex variable arguments, which is broadly speaking "complex differentiable". The notion of complex differentiability in this context has a specific interpretation. 

Who was Nobuo Yoneda?

Nobua Yoneda was a Japanese mathematician and computer scientist (1930-1996) from whom the famed Yoneda Lemma gets its name.  It is ultimately a grand generalization of Cayley's Theorem in group theory.

His doctoral advisor was Shokichi Iyanaga. Iyanaga in turn studied under Teiji Takagi who was instrumental in developing Japanese encryption systems in World War 2.

He has done work on dialects of ALGOL (formerly known as International Algebraic Language, or IAL).

The "Mediant" Series (aka "Farey" Series)

The Farey series gets its name from British geologist Farey.

It is notable that a geologist had such an impact on the theory of numbers.

In 1816, Farey published a statement that the middle of three successive terms is the "mediant" of the other two. 

The proof was supplied by Cauchy (or Baron Augustin-Louis Cauchy, to give him his full title).

PyCryptoing on the Physics Maths Border

Studies of nonlinearity often bring us to the badlands between physics and mathematics. 

One such exponent is Professor Pierre Raphael who specializes in the study of nonlinear waves.

Pierre Raphael joined Cambridge's Department of Pure Mathematics and Mathematical Statistics (DPMMS) in 2019. He has a PhD from Cergy Pontoise (since January 2020 known as CY Cergy Paris university).

At Cambridge, PR is the Herchel Smith Professor of Pure Mathematics (named after organic chemist Herchel Smith who was an undergraduate at Emmanuel College, Cambridge).

Mathematical Expositors

Super-able mathematical expositors are valuable to PyCryptos requiring mathematical intuition to be ingested speedily in order to achieve success in algorithmic implementation.

Examples of such worthy expositors include:

Walter Warwick Sawyer - lots of good essays and expository book on Numerical Functional Analysis. Essays include "AMA Revisted".

Harold Edwards - lots of good essays and expository book on Riemann Zeta Function (1974)

Considerations in Natural Language Generation

An article slightly tangential to natural language processing but provides useful background considerations to PyCryptos operating in this space, as well as to programmers designing conversational interfaces, is this article on how phrasing affects memorability. Cornell professor Jon Kleinberg co-authors.