Here’s the Wikipedia article on this topic.

The freeze-frame can be considered a species of fixed point, including verbal fixed points, such as the one that occurs in the song ‘The House of the Rising Sun’ (“one foot on the platform, one foot on the train”).

# Tag: Mathematics

# lorem ipsum

Mathematics is what keeps you from losing your moorings.

# lorem ipsum

Mathematics is whatever keeps you from being a gypsy.

# lorem ipsum

Learning Algebra is the cyber equivalent of getting on your feet.

# lorem ipsum

A Jordan curve is to a polygon what a power series is to a polynomial.

# Russian proverb

“Lead a simple life, so that you can do complicated mathematics.”

# Chisanbop (Finger Math)

# Small fines, and faint damnation

In USA Today for 22.Apr.2019, columnist Ken Fisher pointed out that small fines are really a buy signal.

So, adding my two cents: The dynamic of imposing a small fine is similar to the dynamic of praising via faint damnation.

The ‘small’ fines Fisher refers to are actually large sums of money – sometimes billions of dollars, but are small in relation to the wealth of the companies concerned. This is a good example of the nature of ratio data (the four types of data – or, ‘levels of measurement’ as they are more technically called – being nominal, ordinal, interval, and ratio). That is, items of ratio data are most appropriately compared, just as their designation says, via ratios. Since we are talking ‘small’ billions here, it is apropos to include a couple famous references along that line:

1. ‘Only a Billion’ is a title used somewhere by Isaac Asimov (as a chapter title, I believe, in his book ‘On Numbers’);

2. “A billion here, a billion there, and pretty soon you’re talking real money.”

— Senator Dirksen (attributed)

# On the sum of the first n primes

# Technical Dissociation

As we move upwards socially, things come together. This is a process of integration, famously described by Abraham Maslow’s hierarchy of needs. The shape is like a pyramid, an inverted ‘V’. However, as we move upwards technically, things come apart. This is a process of dissociation, of a parting of ways of things that were previously conflated, or synonymous. The shape is like a normal ‘V’. (Superimposing the inverted ‘V’ and the normal ‘V’ gives a simultaneous picture of what is happening.)

A few examples of technical dissociation:

* loss of unique-factorization as you move from the reals to the complex

numbers – e.g., 26 = (2)(3) and 26 = (1 – 5i)(1 + 5i)

* apparent astronomical movement versus actual astronomical movement

* in foraging theory, time minimization versus energy maximization

(pp. 8-9 of the book ‘Foraging Theory’ by Stephens and Krebs)

* The subgroup identity is equal to the group identity, but when we move up to rings,

we find that a subring can have an identity different from the ring.

* In foraging, path depletion not equal to negative acceleration of the energy gain function.

* complete information versus perfect information

* a function being analytic versus being infinitely differentiable

* two types of paraboloid (elliptic and hyperbolic)

* ‘heavy-tailed’ distribution has several meanings

* MAD (median absolute deviation) has more than one meaning

* non-unique generalization of the single-variable derivative

* The domain of a partial function is ambiguous, depending on the discipline

(Logic or Mathematics).

* multiple, and only partially satisfactory, definitions of tortuosity

* general life situation (gls) versus momentary life situation (mls) (terminology of Kurt Lewin)

* singularities of solutions not necessarily occurring only at singularities of the equation

* inequality of the types of cardinality for surface area and volume (e.g.: Gabriel’s horn)

* Homeomorphism type is not necessarily determined by homotopy type.

* Coverage probability splits into ‘actual’ and ‘nominal’.

* utility versus exactness – e. g., Agresti and Coull’s 1998 paper “Approximate is Better

than ‘Exact’ for Interval Estimation of Binomial Proportions.” (cited in the Wikipedia

article on binomial proportion confidence intervals)

* having to choose between a statistical estimator that is unbiased or which has better

mean squared error

* There are two types of Hermite polynomials: the “probabilists’ Hermite polynomials”

and the “physicists’ Hermite polynomials”.

* canonical form versus normal form (see the Wikipedia article on computer algebra)

* A subgroup of a finitely generated group need not be finitely generated.

* exploiting prey versus exploiting patches

* the zero-one law in foraging theory versus Kolmogorov’s zero-one law – the former

being prescriptive, and the latter being descriptive

* how the product topology is defined for finitely many spaces versus how it is defined

for infinitely many spaces

* a series converging versus getting arbitrarily many digits correct

* for organizations, normative control and a regime of collective interest versus rational

control and a regime of self-interest (as noted in ‘Metaphor and the Embodied Mind’ –

Boland and Tenkasi)

* dice equivalence versus dice winning against each other with equal probability

* whether energy is present versus whether it is available

* sidereal time versus solar time

* a removable versus a non-removable discontinuity

* continuity versus differentiability

* perception controlling behavior versus behavior controlling perception

* radiant energy versus heat

* topological convergence versus convergence in measure

* sub-sonic versus super-sonic explosions

* cycloid versus circular arc

* longitudinal versus transversal waves

* traditional versus public-key cryptography

* Nash equilibrium for a game repeated finitely many times versus infinitely many times

* conceptual simplicity versus computational simplicity (e.g., n! versus Stirling’s formula)

* Spheroidal coordinates are of two types: oblate and prolate.

* how symmetric groups behave on finite versus on infinite sets

* optimal behavior in the Prisoners’ Dilemma in the short run (betrayal)

versus in the long run (cooperation)

* If W is a generalized complex subspace of a generalized complex vector space V,

then V/W is not necessarily a generalized complex quotient of V.

* topological definition of an object versus geometrical definition

* defining fields by polynomials giving different results in the finite and infinite cases

* temperature versus conductivity

* intensive versus extensive properties

* amortized update time of an algorithm versus worst-case update time

* impossibility versus probability of 0 (Things of probability 0 happen all the time.)

* conservation as wilderness preservation versus as resource management

* duality in terms of polar reciprocation versus topological duality

* powdered chocolate mix for a cold drink versus for a hot drink

* non-uniqueness of tetration (i.e., repeated, or iterated, exponentiation)

* inequality between the Hausdorff dimension of a set and its

topological dimension

Here is my big list of such:

href=”https://aconeyislandofthemindblog.files.wordpress.com/2019/06/technical-dissociation-big-list.pdf” title=”Technical Dissociation Big List”>Technical Dissociation Big List