The Riemann Surface Explained it Terms of Manifolds

A Riemann surface is a connected, complex one-dimensional manifold. 

Dessin d'enfant

A dessin d'enfant is a graph embedding used to study Riemann surfaces.

Read up on graph embedding and NP-hardness to understand this more.

The Fresh Air of Mathematics

"It is a pleasure to constantly inhale the fresh air of mathematics" - so remarked A.J.M. Taylor-Woodhead in the 1920s at Kings College, Cambridge, where he studied and lectured on the "divine art" of mathematics aka. "language of the universe". 

Taylor-Woodhead was known as saying that walking was a useful way for the mathematician to reframe their relationship with reality and emphasised in these "walks of truth" that the focus should not be on mathematical problems but meditation on mathematical perfection which, in his belief, would spur more fruitful development of one's innate mathematical talents.

Machine Learny Language: Exchangeable RVs

The concept of exchangeable random variables is commonplace but the terminology might seem new.  

An exchangeable sequence of random variables is one in which any re-ordering of any finite sequence results in the same joint probability function.

An alternative phrasing of this is the joint distribution is invariant to finite permutation.

The term exchangeable was a neologism at the time of its inception in the 1920s in a book on Logic written by an alumnus of Kings College, Cambridge, where said coiner of said term studied mathematics.

The Kantarovich Paradox, Expressed in FAAM

Leonid Vitaliyevich Kantarovich was an innovator in mathematical programming. His autobiography describing the evolution of his ideas is extraordinary, exploring the paradox of deep theory and application in numerical analysis: "In those days, my theoretical and applied research had nothing in common. But later, especially in the postwar period, I succeeded in linking them and showing broad possibilities for using the ideas of functional analysis in Numerical Mathematics. This I proved in my paper, the very title of which, 'Functional Analysis and Applied Mathematics' (Russian), seemed, at that time, paradoxical. In 1949, the work was awarded the State Prize and later was included in the book, 'Functional Analysis in Normed Spaces' (Russian), written with G P Akilov (1959)."

Revisiting Map Reduce

Map reduce is very prevalent in distributed data processing. 

It's an idea from functional programming. The basic idea is you have a map function that can apply a filter to a list,  and then apply a summary operation, which is the reduce.

A Mickey Mouse example to bring this to life could be - you have list of fishing vessels, you want to filter for foreign flags, and then you want to add them all up, and you want to do this every 6 hours and create a time series database of this data.

A popular open source implementation is Apache Hadoop.

The Autoregressive Nature of LLM Operations

LLMs are AUTOREGRESSIVE generative models.

An autoregression is a regression of a variable against its own lagged values.

For example, an AR(1) predicts the current value based on the immediately preceding value, AR(n) uses the n most recent values.  

One remarkable feature of these models is in-context learning, which has been hypothesised as being Bayesian in nature.

The Python Datasets library

Hugging Face has a Python datasets library which has natural language training data sets amongst others.

Designing an Experiment - Single and Double Blind Trials

Single-blind and double-blind trials relate to how experiments are set up.

• Double-blind - where neither subjects nor experimenters know who is in TEST or CONTROL group
• Single-blind - only the subjects (e.g.. patients in a medical study) know which group they are in

This is about experimental research methods rather than mathematical experimental design (hence it does not lend itself well of the bat to Python automation).

The idea is to prevent bias/personal beliefs from influencing the results of the experiment/clinical trial. Both setups should remove the so-called placebo effect.

Coding with Bilinear Maps

When Pycryptonistas advance beyond the familiar realm of linear algebra to the new realm of multilinear algebra there are terms that must be mastered post haste to make sense of the new environs.

Amongst these terms is the term bilinear map a simple sounding term but oddly not widely socialised even among linear algebra aficionados.

A bilinear map is a function that combines elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.

To be confident in bilinear maps you should be able to product multiple examples of bilinear maps from basic mathematics.

Surjective or Non Surjective - That is the Question (to the Homomorphism)

Augment the definitions of surjection and homomorphism.

However, before we do so, we must clearly state: a surjection operates on a set, and maps elements of said set, to elements of another set.  It is a relation between sets. A homomorphism too is a relation on sets. However those sets are part of binary structures with associated binary operations. We can call those special sets bicombinable sets.

A  SUR-jection is an "onto" mapping. 

The SUR-jection or SUR-jective function maps elements of a set "onto" the entirety/totality of elements in a set B (every B is included). The elements of the set B are "fully recruited"/"fully engaged"/"fully employed" in the action of the surjection on set A. A SUR-jection is thus also an "enveloping function", in its action of "enveloping" all elements of the target set/codomain.

Other names for a SUR-jection include "all you can eat" function, or "codomain covering function".

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

A homomorphism acts on binary structures, not necessarily groups.

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.

And so begins the tale of what to call a surjective homomorphism. Remember a homomorphism is just a function with a special property which only makes sense on bi-combinable sets.

In a way the homomorphism function is just a tool to express a relation between the binary operators in two bi-combinable sets.

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, born in the Occitanie region in France)

The CGT is an important statement about integrals for holomorphic functions ("specially differentiable") 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.

Arithmetic Dynamics

Arithmetic dynamics is a fascinating area which amalgamates number theory and dynamical systems. Number-theoretic properties of integer points (and integral-related points e.g. rational points) under the repeated application of an algorithmic rule that expresses a polynomial or rational function are studied. Joseph Silverman from Brown University is one of the field's exponents.

Essential Prerequisites for Competency in Calculus of Variations

It is not difficult to understand the need for and the objective of the calculus of variations. 

The classical brachistochrone problem and the isoperimetric problem associated with Dido of Carthage are cases in point.

However, the prerequisites for competency are not as apparent. 

A suggested list of pre-competencies are listed below.

An nonpareil familiarity with function spaces is essential to make the theoretical arguments stick. One can even say that function spaces are the biosphere in which the calculus of variations exists and flourishes.

If you don't live and breathe function spaces you may find the journey down the road of calculus of variations somewhat tough.

We can start with a simple example of C[0,1] as a warm-up example. This is the set of all functions defined and continuous on the closed interval bounded by 0 and 1.

Quantum Key Distribution

Quantum key distribution is a problem in quantum cryptography.