Alternative math

I've claimed that there were other things you could do in high-school math besides learn how to divide polynomials in algebra.

One possibility: learn about the zoo of sequences.

If you have a rule for generating numbers: a first, second, third, fourth, and so son--what does the list look like? The order matters, so you can number the terms in the sequence if you need to: term 1, term 2, etc: T(1), T(2), T(3), etc. You usually need to number them.

The answer to that question has applications in measurement, chaos theory, control systems, and on and on.

Zeno knew about this one 2500 years ago: (1, 1/2, 1/4, 1/8, 1/16, 1/32 ...). The next term in the sequence is half the previous term. The numbers get small pretty quickly--closer and closer to 0 but never reaching it. This kind of sequence "converges:" it has a limit.

A boring example is (1, 1, 1, 1, ...).

You've probably heard of Fibonacci's sequence: (1, 1, 2, 3, 5, 8, 13, 21, 34, ...). The next term in the sequence is the sum of the previous two, and it gets larger without any limit ("diverging"). You can show that if you go farther and farther along, that the ratio of a term to the one before it is (1+√5)/2.

Some do neither: (1/2, 1/2, 1/4, 3/4, 1/8, 7/8, 1/16, 15/16, ...). The first, third, fifth, etc (odd) terms converge to 0, while the second, fourth, sixth,etc (even) terms converge to 1.

It isn't hard to predict what this will do: (1, -1/3, 1/9, -1/27, 1/81, ...)

Pick a rule. How about a continued fraction? Start with something something simple: T(n) = (1+1/(1+1/T(n-1)))

Let the first term be 1. The second is 1+1/(1+1/1) = 3/2. The third is 1+1/(1+1/(3/2)) = 8/5. Look at the Fibonacci sequence and see if you can guess that the fourth term of this sequence will be.

Suppose we use T(n) = 1 + 1/(2 + 1/T(n-1)). Start at 1. (1, 4/3, 15/11, 48/37, 173/133, ...). That converges to (1+√3)/2 or about 1.36602...

You can start at other numbers besides 1. How does the sequence change?

Things can get downright weird. Suppose you define a sequence T(1) =1, T(2) = 1, T(n) = T(n -T(n-1)) + T(n -T(n-2)). This is "Hofstader's Q Sequence." When a sequence gets a name, you know that there's either something very important or something very weird about it. The first numbers are pretty easy: (1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, ...). As you keep on going, though, there can be wild swings. Mostly the numbers stay close to T(n) ≈ n/2, but they jump around a lot. These aren't random numbers--they're all well-defined. However, it isn't even known if the sequence dies somewhere along the line, by trying to refer to an undefined term (like the -1'th term).

The heaviest algebra I've referred to so far is the quadratic equation.

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I'm not the first to think of different math directions; I got the idea from learning about some people who were trying to develop materials for exactly that purpose.

The New Mathematical Library series of books was "written by professional mathematicians in order to make some important mathematical ideas interesting and understandable to a large audience of high school students and laymen." The book from that series on continued fractions is available as a PDF.

Inflation

A few days ago when I commented on an announcement claiming that the universe's expansion is perhaps not accelerating. I mentioned in passing that I was not convinced of "initial inflation." After seeing posts defending dark energy from several directions, I thought I'd point you to a talk by one of the developers of the theory, who thinks it doesn't work. Hat tip to Peter Woit). That's an insider's take--I'm not at all expert on inflation. He thinks the theory inconsistent, that it doesn't work very well, and that it is not falsifiable. I took it to be an ad-hoc explanation that just appeared out of nowhere to explain why a model doesn't work right. ("In the "Deerslayer" tale Cooper has a stream which is fifty feet wide where it flows out of a lake; it presently narrows to twenty as it meanders along for no given reason, and yet when a stream acts like that it ought to be required to explain itself.")

How the other half lives

Chabuduo! An article from Aeon about life in China.

Slow reading

When possible, read the originals instead of the commenters. I just checked the UW library system, and found 11,627 title that include the name Shakespeare. I'll bet Shakespeare is more fun to read than any of them.

So, when I heard about The Theology of the Body, of course I tried to read Male and Female He Created Them.

I am bald enough already, and I'm only 44% of the way through, according to the kindle meter. The intro was mostly interesting (explanations of where his philosophical themes came from), but the body of the work is the individual weekly addresses. He tells you what he's going to tell you, then he tells you, then he tells you that he told you. Then he recaps during the next week's talk.

Unfortunately skimming doesn't pay, since scattered here and there are some good brief insights and some "say what?" moments.

For example, in one section he mentions a model of the union with God in Heaven as though the beatific vision were all-absorbing, with no room for anything else. That's a plausible assumption, but you can easily model other possibilities. Orthodox doctrine says that after the resurrection we will be incarnate bodies again, not bodiless spirits. The whole point of creating incarnation and divinization seems to be to bring the spirit thoroughly in union with the world. While we're here we're to try to incarnate love and beauty and other good things from God; it seems plausible that in Heaven we are to do the same.

Nobody can know God in Himself as He is--no finite creature can--we can only know Him in ways we are designed for. (Exodus 33:20, anybody?) Here we are given the opportunity to love God through our neighbors. How would that be different in Heaven? And if we are to be interacting with each other, it seems plausible that we would also be reflecting God to each other, enlightening each other. Divinization would be both directly with God and in community.

But I've not been there, so I'm just guessing too.

Pressing on.

UPDATE: In a slightly later address, JohnPaul affirms something closer to my model than the one he mentioned earlier.

Learn something new every day.

Apparently when tossed a stuffed kitten by a 4-year-old at close range, my startled efforts to grab it out of the air (knocking it this way and that rather than closing on it) resemble a cat trying to catch a feather on a string. So she informs me. And wanted to see it again.

Once in a lifetime

I gather Lucifer is looking quizzically at the thermostat. For once, no blues...

Just yesterday the story included "What did Jesus say to the Cubs?" says a popular T-shirt in Chicago. "Don’t do nothin’ til I get back."

Internet

The UW has decided to go to VOIP. The plans aren't finalized--or very far along, from what I hear. There's a deadline for contract renewal for the old system, and it isn't going to budge whether we're ready or not.

Pretty much everything else runs on the net, why not the phones too?

That makes the net a single point of failure. I hope yesterday's attacks suggested something to the powers that be. We have dumb phones, but also have a tablet, which doesn't handle phone calls but which can handle Skype. Provided the net is up. At least mobile phones aren't run over the net yet.

All the wonderful ideas: pay your bills with smartphones, pay bus fare with smartphones, use smartphone for ID, use smartphone for RFID key entry--do the inventors have any notion how fragile their system is?

Dark Energy

I've always been leery of claims about dark energy and the universe expanding more rapidly (and of the initial inflation). Do we really understand the behavior and distribution of Type 1A supernovae well enough to draw conclusions? " The teams found that more than 50 distant type 1a supernovae are fainter than expected for their measured redshift." 50 is not a very big number.

A recent paper says no; with more supernovae and a different statistical analysis approach that doesn't treat them all the same finds that the distribution is consistent with a constant expansion. Please follow that last link, and look at the second figure. The blue curve is the dark energy model and the red dashed line is the constant expansion model.

I don't know if this group made any major mistakes--I'm not expert enough on supernovae. But it looks like the foundation for dark energy was worse than I thought. (And I'm not convinced about initial inflation.) "Laws are like sausages, it is better not to see them being made."

Math as seen through a glass, darkly

A professor of education and mathematics education (but not, I notice, of mathematics) announces that "mathematics can be decolonized." Oh joy. She can't see any obvious ways of changing the content, so she looks for the "human aspect:" how do students see themselves.

"what mathematics actually is... Mathematician and academic Jo Boaler points out that mathematics is the only subject where students and mathematicians give very different answers to this question."

Mathematicians view the subject as an exciting, creative endeavour in which problem solving, curiosity, excitement, intuition and perseverance play important roles - albeit in relation to abstract objects of study.

For school and even undergraduate mathematics students, these aspects of mathematics are often not experienced and remain opaque. Students tend to believe that mathematics is a set of procedures to be followed. They think only particularly gifted people can do and understand these procedures.

Not true. IIRC Latin was another subject that the skilled and the students had wildly different opinions about. PE is another: humiliating agony for some of us, fun for others. I saw premeds struggling with physics.

One reason for this is given by a study in the US, which showed that the more a field attributes success to giftedness rather than effort, the fewer female and black academics are in that field. This is because the field perpetuates stereotypes about who belongs in the field. The same study found that mathematics professors hold the most fixed ideas about giftedness.

But this view of giftedness versus effort is not borne out by research. A number of scholars have argued that all people are capable of learning mathematics, to high levels.

Go ahead, follow that link. It's a 2-page PDF book review, and it doesn't say what Prof Brodie says it says. The book discusses human mathematical ability in general, not in degree. Nearly everybody can abstract to 1+2=3. Most of us can master the times tables and get some feel for fractions (not always taught well). Somewhat fewer, but still a lot, can learn algebra and proofs. Fewer of those are skilled at it, but that's not a problem--they've been trained in the rudiments of a new language and way of knowing things. I didn't use high school French much for years afterwards either.

If Prof Brodie actually did a little math, instead of math education, she'd know that accomplishment is about giftedness. I've a BS in math, and my only solo paper is in mathematical physics. John Baez is a mathematical physicist. I'm not in his league. He's not in Erdos' league, and would be the first to say so.

Whether stereotypes have any significant effect is open to question. For this or that individual, perhaps. I've seen nothing convincing that shows any causal link of the right order of magnitude. On the contrary, elementary textbooks (perhaps not in South Africa) are drenched in "diversity." The problem-solving children in the explanations are carefully mixed in race and sex. Side-bars extol the accomplishments of 2nd tier mathematicians in order to achieve the right sex ratio. The way is made as friendly as possible--but it turns out there's no royal road to math after all.

That's not to say that we can't do better in math education. There are plenty of subfields. Some kids do fine in algebra but bomb proof-based geometry; some do the reverse. At least the rudiments of algebra are very widely used, but some aspects of topology or knot theory might be accessible in place of more "advanced" algebra (e.g polynomials) or geometry. I tried to teach some TAG 3rd graders a little bit about abstract groups, and I think some of it stuck for at least a week.

But "decolonization" of something as thoroughly abstract as math? There's no ethnic culture associated with it, no economic culture--there are a few mathematical cultures, if you like, but nothing that screams "dead white men."

No, I'm afraid that her real goal is in the tail:

Everybody deserves access to its beauty and its power - and everybody should be able to push back when the discipline is used to destroy and oppress.

That sounds very much like "if the numbers mean something I don't like, I get to reject them." And reject them in good conscience, because labels like decolonization and liberation automatically put you on the side of the angels. Perhaps she means she doesn't like certain technologies, but she puts the blame squarely on math, and I assume she means what she says.

Time for Kremlinology?

Humor me in a little Kremlinology.

Never mind the D and R honorifics for now. Most politicking never shows up in the news--it consists in jockeying for empires within and between bureaucracies.

Even in the armed forces, when fighting a common enemy, infighting and empire-building go on, and have been known to even involve assassinations. Not just the Japanese: we also lost some people in WWII because of inter-service rivalry (and politician stupidity).

So suppose the FBI and CIA are having a spat, or the NSA and the Pentagon, or NASA and the DOE. Or suppose there's an internal squabble in State. What might you expect to see?

Leaks, for one. Stuff that's supposed to be secret that somehow shows up. It might be embarrassing, or it might be supposed to be top secret and the agency responsible for the secret gets egg on its face for not keeping secrets. Or it might kill an ongoing operation that was Not Invented Here.

Embarrassing incidents, like the infamous "Reset" button. That whole idea was so tone-deaf that it had to be an enthusiasm from higher-ups, but the mis-translation on top of everything else really put the icing on the cake. Unless Hillary used the dictionary herself, somebody probably got fired for that little bit of carelessness--and if so, I'll bet it was the wrong person.

Strings of replacements of leadership. If one category of administrator or general officer seems to be systematically replaced by another, probably somebody lost the political struggle big time. One big problem is that you won't understand what you see, because outsiders don't always know what the categories are.

Noisy leaders or wannabe's or whistle-blowers that suddenly decide to retire to be with their families.

Agencies starting to arm up. Just in case, of course. Not for shooting, but so that the Department of Education doesn't let the Bureau of Land Management intimidate them when they show up in force at some contested site.

All these things happen all the time at some level, of course. How do you know when the activity is significant? When does it strike you that (e.g.) there've been a lot of high profile suicides lately? Is that random chance, greater popularity, more reporting, or more intense pressure? Note that a national database of phone contacts can with difficulty find a few terrorist needles in the haystack, but it is ideal for tracking the contacts of somebody you want to get the dirt on.

No, I'm not looking for conspiracies. You don't have to. This just seems like the kind of behavior you expect from competing bureaucracies. It might even be more bitter at the University level ("the stakes are lower"), but when the organizations get really big the troubles for the rest of us get bigger too.

What next

"Our real and immediate interest is to rebuild the character of our civilization; to recover that common understanding of up and down; of right and wrong; of what is worthy and what is unworthy; of what is godly and what is ungodly; along with the telling power of example. Let the world titter in its cynicism: the recovery begins when we fast and pray."

Paleo diet

It turns out that 400,000 year old skulls had tartar. Tartar is porous, and stuff can get entrapped. Researchers found "charcoal from indoor fires; evidence for the ingestion of essential plant-based dietary components; and fibers that might have been used to clean teeth or were remnants of raw materials." "Within the calculus, the researchers also discovered small plant fibers, which they suspect may have been used to clean teeth—prehistoric tooth picks."

There are also traces of fatty acids and bits of starch, presumably from nuts and other plants.

The real paleo diet didn't prevent tartar buildup. I wonder what else it didn't prevent. Attacks by cave bears, ...

Reporting from Russia

Another headline: Putin ally tells Americans: vote Trump or face nuclear war. A headline calculated to get your back up, no?

Vladimir Zhirinovsky, a flamboyant veteran lawmaker known for his fiery rhetoric, told Reuters in an interview that Trump was the only person able to de-escalate dangerous tensions between Moscow and Washington.

That's way less dramatic, and a lot less impertinent.

But for still more information, it might help to check somebody who knows the players a little better: a Czech string theorist.

You know, he is a boss of a nationalist party in Russia – one of the four parties in the Parliament. No self-declared anti-Putin party has made it to the Russian Parliament. So Reuters calls him "a Putin ally". It's a very problematic label, of course, because he is still leading a different party that competes with Putin's. Putin is mainly an old-fashioned conservative politician not too different from Helmut Kohl and many others. Zhirinovsky is the head of a highly idiosyncratic nationalist party. To say that they're "the same" means to show the lack of understanding for Politics 101.

Also, I've been watching Zhirinovsky for some 25 years. He's been a part-time clown. All of his famous propositions that have made it to the media were tainted by some conspiracy theories or immense exaggeration – and, I believe that in many cases, intentional humor (which is a good thing in Russia because they usually have a shortage of it). In some cases, I could feel a sympathy with the "core" of his proclamations. Sometimes, I disagreed. But I don't remember that his propositions could have been considered as accurate, trustworthy predictions or realistic plans.

Is Lubos more accurate? Maybe. He's been a bit of a Putin apologist: sometimes even a lot of an apologist. He wasted a lot of bits arguing that Russians didn't shoot down Flight 17 when obviously they did and were seen high-tailing it out of there. He's worth paying attention to, though.

Physics had some visitors from Poland back when Solidarity was making headlines. They read emails from back home, and told us that even the best US papers generally missed the point of the events they reported on. A local may have a bias in his views, but he probably knows the situation better than foreign reporters.

Some years back Rantburg had a Greek commenter who often had very contrary views about the EU. The Greeks had dictatorship and near civil-war within living memory, and it would hardly be surprising if they thought that, on the whole, even a suffocating central government was better than what they'd had on their own. I was sorry when he dropped off the site. I didn't have to agree with his take, but it was good to have somebody near the scene who understood other aspects of what was going on.

Headlines vs body

"Electronic spoons which make food taste sweeter on horizon". Or on Drudge: "veggies taste like chocolate."

Wow! Except here's what's in the body of the story

For sweet tastes there is a channel called TRPM5 which is temperature sensitive, so people taste more sweetness when the food is hot than cold. So to mimic sweeter tastes the device changes the temperature of the tongue rapidly from 77F (25C) to 104F (40C.)

So it warms up the tongue just a bit to make some foods taste sweeter. What a let-down.

"The team say that they are also working on producing different tastes which they claim is time consuming because even the difference between a lemon and a lime is vast."

Preparing for elections

AVI has posted a series on the election and our role and influence. As of this writing he's presumably got a few more in the pipe.

Somebody pointed out that while contempt of congress is a crime, contempt for congress comes naturally. That's fine--if you insist on being in the public eye, you'll be judged. Not always wisely, perhaps, but that's life.

It is interesting to hear what people say about this election. Not pleasant, but interesting. This is Dane county--Trump comes in for matter-of-fact abuse. (Not many Hillary signs out, BTW--more for senator or representative.) The vicious invective is reserved not for the candidate, but for his supporters. I had a brief email exchange with one of the local reporters asking him who they were (I didn't know any personally), and his final letter casually dismissed them as stupid and unworthy of attention because they were Trump supporters.

"Everybody knows" that Trump supporters are racists, terrified of change and hating people different from them. Hmm. I could with equal (perhaps better) justice claim that Sanders supporters are envious ignoramuses, who just want free stuff at everybody else's expense; and profess myself duly horrified at how many fellow Americans have descended to this level. However, I think the Sanders phenomenon wasn't really an endorsement of socialist economics, but a kick in the pants for the Democratic party and an assertion that the status quo is broken--the big financial institutions are making the rest of us assume all their risks while they rake it in.

Likewise I think the Trump phenomenon is a kick in the pants for the Republicans, and an assertion that that the status quo is broken--the governing groups are disconnected from the public interest and the public will.

The critiques are similar, and the differences have more to do with the values the groups hold. BTW, I doubt that Sanders voters will sit out the election--they'll hold their noses and vote for Hillary even though they despise her, because she, even if distantly, claims to support their tribe's values.

So why the viciousness? Is this the default setting? A radio show suggested that WW-II produced a generation of politicians used to fighting on the same side to preserve the US, and suggested that though the politics might be wildly divergent, they had a sense of being in the same boat. Maybe so.

OK, assume this is the true normal. I don't think this kind of bitterness is something Jesus would approve of. So how do we combat the bitterness and do better than normal? I can hear some folks already: "Compromise--do it my way." Or words to that effect. Maybe instead "Dine with a Democrat" or "Share a Repast with a Republican" or "Lift a Libation with a Libertarian?" Find something in common--but that may be harder than it looks, given the way preferences seem to cluster. Better might be "Haul sandbags together while the river rises." Which I suppose is the WW-II unification hypothesis with a different enemy.

Something about the name

On our outing yesterday, Better Half noticed in a restaurant a sign advertising the use of Kronos Foods. Does that make you think twice about eating there?

BTW, Frog Alley Road connects to Pheasant Fields Drive, which connects to Turtle Run and Shallow Waters Circle. Yes, there are houses there. Pheasant Fields I understand an attraction for, but for me Shallow Waters bring mosquitoes.

Cosmic rays and Cerencov light

This Forbes article by Siegal is a nicely illustrated explanation of how these telescopes looking for cosmic rays work.

The devil is in the details, of course. How much energy did the thing have? You can sort-of reconstruct an estimate based on how much light you saw on the ground and what angle you reconstructed the cosmic ray came in at. But--the light produced will depend a bit on the type of cosmic ray it was (proton, oxygen nucleus, iron nucleus, whatever). So you wind up with different numbers depending on what you think the particle was. The proportions of nuclei seen seem to vary with cosmic ray energy, too. We have no way of identifying what a cosmic ray was before it interacts. You can guess--for example there's a strong probability that the very highest energy particles were iron nuclei.

And the interactions in the atmosphere are random. If you happen to produce more muons early on in the shower, they'll carry more of the energy--but not show up any brighter in your detector. Result--you see less light, and think the original energy is lower than it really was. The only really good handles you have are statistical. When you see a lot of the cosmic rays you can get a distribution whose shape will tell you things about the original energy distribution--the outliers won't matter so much.

By the time the shower reaches ground level there's not much left but muons and photons (and neutrinos--lots of neutrinos).

FWIW, the highest energy core of a cosmic ray shower can sometimes (1/few-thousand events can have hundreds of muons) be surprisingly intense and narrow--10meters or even smaller. (muon bundles). And you, gentle reader, were probably struck by at least one cosmic ray shower muon while you were reading this.

Mosquitoes

We had a family rule: "Bugs in the house get stomped; bugs outside you leave alone." We had clear exceptions: fireflies and ladybugs got escorted outside, and mosquitoes were fair game anywhere.

"God in His wisdom made the fly, and then forgot to tell us why." I gather He wanted us to figure it out for ourselves.

JSTOR reports that mosquitoes may actually be good for something. Two things.

MacDonald cites a study that found that mosquito larvae were part of the ecological balance in stagnant water: without them protists multiplied like mad and some bacteria which the protists ate dropped in numbers. They're not sure yet what the consequences of that could be.

And apparently they are sometimes pollinators too.

Only female mosquitoes suck blood and that phase only lasts while they are breeding. Most of the time, mosquitoes of both sexes eat plant nectar, making them important pollinators as they move from plant to plant. Some orchids, for example, rely heavily on mosquitoes, and these rare plants would be at risk without their buzzing partners.

I'm still going to swat them. I've had malaria already, thank you, and I don't want West Nile. Or even a tiny welt.

16" shells

The deadline for submitting proposals is past, and unfortunately "Open Burning and/or Open Detonation (OB/OD) are not permitted technologies for any end item(s) or component(s)."

There might be a loophole there, though. If the detonation were underground... You'd need to keep people at a substantial distance, but perhaps it might be an alternative to Burning Man.

If you insist on being boring, try to imagine how to safely disassemble something full of Explosive D: a combination of picric acid and other substances. Maybe it isn't so boring...

A little here, a little there

Mistakes add up sometimes.

A gas explosion in Fitchburg:

said the first oversight occurred in 2011 when his client, ... had his gas-powered dryer replaced with an electric one, and the dryer installer failed to cap the gas line during the job, ...

The second mistake happened on Aug. 25, the day of the explosion, Rottier said, when tree trimmers working on Grittner’s property needed water and asked Grittner to open a water valve in the basement to allow for that.

But Grittner, according to what Rottier told 27 News, first accidentally opened a gas valve attached to the uncapped line instead, allowing gas to accumulate in the house and some hours later trigger the explosion at 6:45 pm.

Nothing disastrous in itself, but together...

Looking back, I think I've done as many two-step as one-step screw-ups.