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An interesting story about industrial rail in the United States. About 20 mins. From The Economist.

commercial railways in the United States

  • Europe has an impressive and growing network of high-speed passenger links
  • America’s freight railways are one of the unsung transport successes of the past 30 years.
  • Before deregulation America’s railways were going bust. … By 1980 a fifth of rail mileage was owned by bankrupt firms.
     
  • Since 1981 productivity has risen by 172%, after years of stagnation. Adjusted for inflation, rates are down by 55% 
  • Coal is the biggest single cargo, accounting for 45% by volume and 23% by value.
  • since 1990 the average horsepower of their fleet has risen by 72%
  • [since 1990] the number of ton-miles per (American) gallon of fuel [rose] from 332 to 457—an improvement of 38%
  • But the fastest-growing part of rail freight has been "intermodal" traffic: containers or truck trailers loaded on to flat railcars. The number of such shipments rose from 3m in 1980 to 12.3m in 2006, before the downturn caused a slight falling back.
  • one freight train can carry as much as 280 lorries can

(Source: )





Distribution of primes up to 19# (9699690).

Distribution of primes up to 19# (9699690).


hi-res




Readers of isomorphismes, you might enjoy powers of two tumblr.

powersoftwo:

2100 = 1,267,650,600,228,229,401,496,703,205,376 — one nonillion, two hundred sixty-seven octillion, six hundred fifty septillion, six hundred sextillion, two hundred twenty-eight quintillion, two hundred twenty-nine quadrillion, four hundred one trillion, four hundred ninety-six billion, seven hundred three million, two hundred five thousand, three hundred seventy-six (31 digits, 320 characters)

I think I’ve been subscribed since the 30’s. Never a letdown. And of course it’s only going to get more exciting.




Real numbers are imaginary, and imaginary numbers are real.


[I]maginary numbers describe a physical state of something, so as much as a number can exist, these do. But … real numbers, [being ideal], are imaginary.

David Manheim

(I changed some parts that I don’t agree with but the phrasing and initiative are his.)

The “rational” numbers are ratios and the “counting” numbers are, um, what you get when you count. But “real” and “imaginary” numbers have nothing to do with reality or imagination (each is both real and ideal in the same sense).

 

How about we start referring to them this way?

  • ℝ = the complete numbers. ℝ is the Cauchy-completion of the integers, meaning that ℝ has completely fills in enough options so that any sequential pattern will be able to dance wherever it wants and never need to step its shoe on another element outside the system in order to fulfill its pattern.
  • Any field adjoined to the √−1 becomes "twisting numbers". This derives from the “twisting” feeling one gets when multiplying numbers from ℂ. For example 3exp{i 10°} • 5exp{i 20°} = 15exp{i 30°}, they spiral as they multiply outwards. Keep multiplying numbers off the zero line and they keep twisting. Tristan Needham coined the word “amplitwist” for use in ℂ.
  • ℂ = the complete, twisting numbers. Since ℂ=ℝ adjoin √−1.
  • "Complete spiral numbers" sounds nice as well.

Just to give a few examples of other acceptable numbers systems:

  • ℚ adjoin √2
  • the algebraics
  • ℚ adjoin √[a+√[b+c]]
  • sets
  • DAGs
  • square matrices … with many kinds of stuff inside
    Magma to group2.svg
  • special matrix families
  • certain polynomials (sequences) … taking many kinds of things (not just “regular numbers”) as the inputs
  • clock numbers (modulo numbers)
  • Archimedean fields and non-Archimedean fields
  • functions themselves … and the number of things that functions can represent boggles the mind. Especially when the range can be different than the domain. (Declarative sentences can have a codomain of truth value. Time series have a domain of an interval. Rotations of an object map the object to itself in a space. And more….)
  • And many, many more! Imagination is the limiting reagent here.




Mathematical matrices are blocks of numbers. So they don’t necessarily have definite characteristics the way plain vanilla, singleton numbers do. They’re freer. For example 131 ∈ positive and −9 ∈ negative. But is

[[5,0], [3,8]]

positive, negative, or zero? Well, it has all three within it. How can a matrix be “just” +, or 0?

image

Likewise, is the function sin (x) ℝ→ℝ positive, negative, or zero? It maps to an infinite number (ℶ) of positives, an infinite number (ℶ) of negatives, and to a lesser-but-still-infinite number (ℵ) of zeroes. So how could it be just one of the three.

image

But James Mercer realised in 1909 that some functions “act like” regular singleton numbers, being positive, negative, or zero. And if it walks like a duck, quacks like a duck, smells like a duck — it is a duck. For example if you multiply any ℝ→ℝ function by a Schwartz function, you won’t change its sign. Hmm, that’s just like multiplying by a positive number — never changes the sign of its mate.

 

Matrices

Similarly, some matrices “act like” single solitary numbers, being positive, negative, or zero.

  • Matrices that act like a single positive number are called positive definite.
  • Matrices that act like a single negative number are called negative definite.
  • Matrices that act like a single non-negative (≥0) number are called positive semidefinite.
  • Matrices that act like a single non-positive (≤0) number are called negative semidefinite.

Being able to identity these simple subtypes of matrices makes Théorie much easier. Instead of talking about linear operators in general and not getting many facts to reason from, the person coming up with the theory can build with smaller, easier-to-handle blocks, and later on prove that the small blocks can build up anything.

I first saw semi-definite matrices in economic theory, where a p.s.d. Hessian indicates a local minimum of a value function. (A Hessian is like a Jacobian but filled with second derivatives instead of first. And a value function maps from {all the complicated choices of life} → utility ∈ ℝ. So value functions have a Holy Grail status.)

But semi-definite & definite functions are used in functional data analysis as well.

 

Functions

\int 3x - 5x^2 + 13 x^3 dx

Positive semi-definite functions are used as kernels in

  • landmark regression
  • data smoothing, especially of high-frequency time series
  • audio transformations
  • photoshop transformations
  • BS stock price prediction

At some point in the 20th century we learned to expand the definition of number to include any corpus of things that behaves like numbers. Nowadays you can say corpora of functions and well-behaved corpora of matrices effectively are numbers. (I mean “commutative rings with identity and multiplicative inverses”.) That is a story for another time.

 

Uses

Besides being philosophically interesting as an example of a generalised number, positive semi-definite functions and matrices have practical uses. Because they are simple building blocks of more complicated things,




image

  • I like Indonesian food better than Japanese food i ⪰ j, and
  • I like Japanese food better than English food j ⪰ e.
  • I also like French food better than English food f ⪰ e, but
  • I see French food as so different from the “exotic Eastern” foods that I can’t really say whether I prefer French food to Indonesian f≹i or Japanese f≹j.

    I would just be in a different mood if I wanted French food than if I wanted “exotic Eastern” food.

So my restaurant preferences are shaped like a poset. In a poset some things are comparable  and some things ain’t . Popularity is shaped as a poset and so is sexiness. Taste in movies is a poset too. The blood types have the same mathematical form as a poset but only if you reinterpret the relation  as “can donate to” rather than “is better than”. So not really the same as ethnic food.

image

Partial rankings | orders are transitive, so

  • (indonesian ⪰ japanese and japaneseenglish) implies indonesian ⪰ english.

That means I can use the “I prefer " symbol to codify what I said at the outset:

  • Indonesian  Japanese English
  • French English
  • neither⪰j nor⪰ f … nor⪰ f nor⪰ j (no comparison possible )


Posets correspond nicely to graphs since posets are multitrees.

image

image

Total orders — where any two things can be ranked  — also correspond to graphs, but the edges always line up the nodes into a one-dimensional path. So their graphs look less interesting and display less weird dimensional behaviour. Multitrees (posets) can have fractional numbers of dimensions, like 1.3 dimensions. That’s not really surprising since there are so many kinds of food / movies / attractiveness, and you probably haven’t spent the mental effort to precisely figure out what you think about how you rate all of them.

Rankings | orders are a nice way to say something mathematical without having to use traditional numbers.

I don’t need to score Indonesian at 95 and score Japanese at 85. Scores generated that way don’t mean as much as Zagat and US News & World Report would like you to think, anyway — certainly they don’t have all the properties that the numbers 85 and 95 have.

It’s more honest to just say Indonesian ⪰ Japanese lexicographically, and quantify no more.




This is for my homies in maths class.

Mathematical matrices are blocks of numbers, arrayed in 2-D. (Higher-dimensional array-verbs are called tensors.)

  1. image
    Left “times” right equals target. Each entry in the target is the result of a series of +'s and ×'s along the red and blue. A long sum of pairwise products.

  2. Your left hand goes across and your right hand goes up/down.
    imagewhere
    image.
  3. There need to be as many abcdefg's as there are 1234567's or else the operation can't be done.
  4. Also you can tell how big the output matrix will be. There can be three blue rows so the output has three rows. There can be four red columns so the output has four columns.
  5. This is the “inner product” because multiplying vector-shaped blocks (tall blocks) like Aᵀ•B results in an equal or smaller sized output.

    (There is also an “outer product" which is a different way of combining the info from the two matrices. That gives you an equal or larger shaped result when you multiply vector/list-shaped tall blocks A∧B.)
  6. Try playing around with this one or that one.

Matrix multiplication is the simplest example of a linear operator, the broad class of which explains quantum mechanics and ODE’s. You can also apply different matrices at different points as in a vector field — on a flat surface or a curvy, holey surface.




!0 === 1

Only tested this in Javascript and Ruby, but I bet it works in C like it did in Javascript.

mars@scheherazade:~$ js
Rhino 1.7 release 2 2010 01 20

js> 3+!0
4

Ruby is safeguarded against such hijinks:

mars@scheherazade:~$ irb
irb(main):001:0> 3+!0
TypeError: false can't be coerced into Fixnumfrom (irb):1:in `+'from (irb):1from :0

(Source: farside.ph.utexas.edu)