More about false positives

This feels like a similar topic to the previous post: If someone asks me if such-and-such is correct and I can only answer in terms of good or bad (e.g. my mouth is full and I can give a "thumbs up" but cannot say "correct") then I say "good" as a substitute for "correct". The good/bad axis and the correct/incorrect axis are neither orthogonal nor co-linear.

Was that a blue bird? Ignoring observations during classification

I have a little story that illustrates possible approaches to handling false positives in a classification system with limited alternative categories. 

Driving in Bedford a small bird with a rust-red underbelly flies past the car and I get a brief (.3 sec?) glimpse.

Analysis: It was too small to be a blue bird and there are no birds that small around here with rust-red breasts. 

• If the bird had been a bit bigger it could have been a blue bird or, even bigger, it could have been a robin.
• If it wasn't really red then it could have been a Junco or sparrow.
• If it was an accidental European bird, it might be one of those
• If I misjudged size as well as position of the red color, it  could have been a Towhee or Catbird.

No small birds like that? Maybe the bird was a bit larger than I thought. For some reason I am more willing to dismiss a magnitude observation than a color one. Anyway, the mind obviously tries to loosen the observed 'measurements'; doubting each, in turn, in order to fit to a broader set of categories.

Who let the algebraists write the Wikipedia geometry definitions???

It must take a certain type of personality to slave for hours over a hot keyboard to give Wikipedia and the world a clear definition of some mathematical term - hard to format and complicated. But apparently this personality type tends towards the "French" formalism of - what was it? ...yes! - the Erlanger program[OOPS! THAT NOT IT - try - Bourbaki]. I see little effort made in Wikipedia to make simple math ideas accessible, simply.

I am reading Wikipedia for their definition of "fiber bundle" and it seems to me it used to be a bit more comprehensible. Then I look up "double fibration" and hit a bunch of stuff about Twistor space, like:

Ω(x)=ωA−ixAA′πA′

and I think - geez! what happened to the geometry? I mean pictures. In general, Wikipedia mathematical definitions are abundantly formal, often failing to give the common sense meaning. For example do they show the picture for Lagrange multipliers? Actually they do, but not the right picture.

Lagrange multipliers find where the optimum valued isotherm touches the constraint for the last time. The key idea being that you can look for where the red and green vectors coincide:

Continuous Croissants

My croissant recipe - derived from the Julia Childs's video - is no big deal. What is fun is that although at least 8 hours long and involving multiple repeated steps, there are many opportunities to pause the sequence by putting dough in the fridge or the freezer. As a result you can start a batch almost anytime of day and, with a bit of planning, have them be fresh baked at any time of day as well. Since I love to eat croissant and am enjoying baking them, I could easily go off my diet and have non-stop heartburn.

Note: croissant are the same as bread, with two risings. The dough includes milk, sugar, and oil [which is a bit different from bread] and there is a whole bunch of messing around with butter and rolling pins between the two risings, but it is still the same basic idea - just a variation.

Recipe:
MIXING and FIRST RISING
Take out a big bowl and a small one. 
Put 1tsp yeast [or use 1/2 and double the rise times below] in small bowl and add 1/2 cup of warm milk, 1/4 cup warm water, 1 tsp sugar.

While you have the sugar and tablespoon in hand, start adding to the big bowl:
Put 1 tsp sugar in big bowl , add 1 tsp salt, 3-4 tbs of cooking oil, 2 cups of flour.
[I used King Arthur, tried mixing in some pastry flour - which is a disaster, and got good results with Gold flour. Am now trying mixing King Arthur and Gold].

Add small bowl contents to big  bowl. Mix thoroughly until it is all one lump of dough stuck to the fork (or mixer). Add liquid/oil if it is too crumbly to form a single lump. Now kneed the heck out of it: maybe three minutes by hand until it becomes silky.Let this dough rise at room temperature until the ball is about 2 times as wide. Takes a bit less than 2 hours at 70F room temperature. 

After this rising, punch it down and put aside in the fridge, briefly, while you soften the butter.

SOFTEN BUTTER
[Following Julia C]
Take one stick of unsalted butter out from fridge and hammer it out with a rolling pin, then use the heel of your hand to squash down the bumps and warm it slightly. Scape it together a few times. Use the heel of your hand a second time...you kind of want the butter and dough to be equally soft, maybe the butter a little harder than the dough. Scrape it together into a 5"x4" rectangle.

Key points: kneading enough, and softening enough - with no lumps in the butter.

FOLDING and ROLLING
Take dough from fridge, flatten into a 14" diameter circle. Put the butter rectangle in the middle and fold up the edges (without pulling on them) to make an envelope around the butter. Pinch the seams. Roll carefully into 14"x7" rectangle, turning it over a few times, and fold the long dimension in 3.  Then turn the dough 90 degrees clockwise and, in the direction that was previously the 7" width, roll it out again into 14"x7" rectangle, fold in 3 again. That is two "turns".

Let dough rest in fridge for 2 hours. Then repeat the rolling and folding two more times. That is two more "turns". Now let dough rest another 2 hours in fridge. 

SECOND RISING AND BAKING
Dough in now ready to be rolled out 1/8" thick. Following Julia, I cut the dough in half, put one in fridge while rolling out other half. Cut that half again and put a quarter in fridge while rolling out the other quarter to 1/8" thickness. While working with a small amount of dough, the rest remains cool.

At 1/8" thickness, cut into triangles, roll them up and bend into crescents. When rolling dough into a tube towards the point of the triangle, stretch the dough a little in the direction you are rolling. 

Put on baking sheet and allowed to rise again for 2-3 hours. When ready, they become jiggly.

Heat oven  (note: you can use a hotter over without moisture, or a cooler one with more moisture. I have settled on  450 with a small splash of water on the coil, to make steam]
Bake for 9 minutes. Just before they are ready, they smell good. If you are burning the bottoms you can start to smell it - so keep a nose out for these things.

Let cool. They are best about 1/2 hour later.

WHERE TO PAUSE: 
 - any time dough is frozen it stops the sequence. It resumes when it dough is back at fridge temperature. 
 - any time it is rising you can slow it down longer than 2 hours by putting in fridge. For example, form the croissant before bed, put in fridge, use warm oven to accelerate rising in the morning. You can have em for breakfast.

Form follows content

The language theory is that: meanings are in your head and find expression in what is said and what is heard. In other words the meaning content of an expression is used to parse and articulate the form and structure of an expression.

Having a "theory" doesn't mean you know all the answers. I am implementing a program that allows meaning to evolve on the way through a sentence and I can see that there would be different approaches to implement such a thing - some more atomic than others.

The black hole at the birth of the universe

http://phys.org/news/2014-08-black-hole-birth-universe.html
I wrote about something like this in a college term paper in cosmology - a generalization of the donut shape in 4D.

Traffic Equations

If you are like me, during a long drive you'll drift into thinking about traffic. Seems to me there is an important function that relates speed and density of the cars, this is the minimum_safe_distance( s ) for a speed s. Below this distance you cannot stop in time. The average distance between cars is >= min_safe_dist(s). If we think of density as the reciprocal of average distance then

• (1/traffic density) >= min_safe_dist(s)

  We can add a few other rules of thumb:

• higher density lanes tend to merge into lower density ones
• as many cars enter an interchange as leave it. 
• people usually drive as fast as they can