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.

Morse Theory a Must Know for the PyCrypto

Morse Theory is the theory made up by Marston Morse from Maine, an American mathematician known for his work on "calculus of variations in the large".

The Morse-Palais Lemma is named after him and Richard Palais.

Eisenstein and Irreducibility

Gotthold Eisenstein gives his name to Eisenstein's criterion, a famous theorem on the irreducibility of polynomials with coefficients in Z to polynomial products over Q (the rationals being a larger "field" than the integers). German mathematician Theodor Schönemann was in fact the first to publish it.

Fisher Information (Info about a PopPar in an RV X)

At some point a statistician will come head-to-head with the Fisher information, which measures the "information" about a population parameter contained in an observable random variable X. 

It is the expected value of the observed information (sounds a bit like "maximum likelihood", right?).

Formally it is what statisticians call the "variance of the score".  

The notion of "score" is something quite peculiar and distinctive to this branch of mathematics and is also known as the "informant". It is the rate of change of the log likelihood function of the population parameter, with respect to the parameter.

It is named after British polymath, Sir Ronald Aylmer Fisher, FRS.

Combinatorics

Combinatorics is the mathematics of counting. Its origins lie in the ancient world.

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.