March 2014 - mathfun@mailman.xmission.com

Re: [math-fun] Lissajous physob
by Bill Gosper 08 Mar '14

08 Mar '14

JBudd...> On Sat, Mar 8, 2014 at 5:51 AM, Bill Gosper <billgosper(a)gmail.com> wrote: Alan Adler (gosper.org/alan.JPG . "I'm not an inventor, I'm an engineer, dammit," ) Is that the same Alan Adler who used to sign off sci.math or sci.math.research posts with "Ignorantly yours" ? [...snip...] Alan denies it. Harder to deny might be http://en.wikipedia.org/wiki/Alan_Adler --rwg

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Re: [math-fun] abysmal fish (was: blood)
by Bill Gosper 06 Mar '14

06 Mar '14

Anyone else with this problem? Ping me privately. --rwg On Wed, Mar 5, 2014 at 1:04 PM, Bill Gosper <billgosper(a)gmail.com> wrote: On Mon, Mar 3, 2014 at 10:51 AM, Bill Gosper <billgosper(a)gmail.com> wrote: [...snip...] Also, no coronary arteries, likes subzero seawater, faster than a speeding brinicle. (https://www.youtube.com/watch?v=r4cX2EPt2zE) --rwg _______________________________________________ I get this message: "The uploader has not made this video available in your country" when I try to play the video. :( Maybe you have to be in USA to see it.

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[math-fun] Aluminum economy
by Warren D Smith 06 Mar '14

06 Mar '14

My impression from examining chart below is that the "aluminum economy" is wanted: Refuel your car by turning in your aluminum oxide and getting back aluminum metal which you stick into your car's Al-air battery and off you go. The oxide is then re-converted to metal at a electrolytic smelting plant somewhere electricity is cheap. Stuff MJ/kg MJ/liter $Cost/kg (year 2014) ------------------------------------------------------------------------ Gasoline 46.4 34.2 4 Fat 37 Coal 30 20 (if one solid lump) 0.066 Batteries 0.1 to 0.9 0.3 to 4.3 TNT 4.6 Boron 58.9 137.8 5000 Silicon 32.2 75.1 3.1 Aluminum 31.0 83.8 1.9 Aluminum-air and Silicon-air batteries both have been made, and the former successfully commercially (were extensively used as backup power sources for the phone system). Oxygen, Silicon, Aluminum are the 3 most abundant elements in Earth's crust in decreasing order. Boron is about 90 times rarer than aluminum, Lithium is about 50 times rarer than Boron, and Beryllium about 10 times rarer than Lithium. The "boron economy" is stupid because boron is too rare & expensive to make. (It also has a very high melting point and is very hard, two other reasons it is hard to deal with.) It seems absolutely absurd to claim we each should run our car by refueling it with $200,000 worth of boron. The "hydrogen economy" is stupid since H is too dangerous/explosive and has far too low volumetric energy-density. There have been metal hydride storage ideas to get safer, but these are expensive and complex and low per-mass energy density. Either way running your car with hydrogen is absurdly unattractive. Rechargeable batteries are stupid since too expensive, do not last long enough, inefficient, low energy density, toxicity and environmental disposal problems. Beryllium is stupid since toxic and very rare. Lithium is stupid since rare. Silicon perhaps would be better than aluminum but I doubt it because it is more expensive, lower electric conductivity, harder, brittle, higher melting point (1687K versus Aluminum's 933K). -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] AIM 304
by Bill Gosper 06 Mar '14

06 Mar '14

Also, by http://dspace.mit.edu/bitstream/handle/1721.1/6088/AIM-304.pdf p25: HarmonicNumber[p]==-Sum[(3*k + p)* Binomial[k - p - 1, k]/k/(k + p)/Binomial[2*k, k], {k, ∞}] (Terminates after (integer) p), which Mma converts to two useless 3F2s unless integer 2p. --rwg Ouch, there are way more typos in AIM 304 than I thought. Even "some" instead of "sum". I wonder if MIT lost the errata. Repairing one egregium, p66: Sum[1/Fibonacci[n], {n, ∞}] == Sum[(I^((-1 + k)*k)* ((-1)^k*Fibonacci[2 + 2*k] + Fibonacci[3 + 4*k]))/ (Fibonacci[1 + 2*k]*Fibonacci[2 + 2*k]* Product[LucasL[1 + 2*j], {j, k}]),{k, 0, ∞}] For some reason, this (rapidly convergent) rhs completely stymies Mma's numerics. It doesn't even yell False if you change LucasL[2j+1] to LucasL[2j-1]. There shortly follows an acceleration of Sum 1/(x^n+cy^n) which divides by zero if there is a nonnegative integer j s.t. x^j=y^(j+1) or y^j=x^(j+1). PUZZLE 1 (from back in my smarter days): rewrite Sum(1/(x^n-1)) (= Sum[DivisorSigma[0,n]/x^n]) so as to avoid the 1/0. SPOILER Sum[1/(x^n - 1), {n, ∞}] == Sum[1/(x^(n/4) - (-1)^n) + (-1)^n/(x^(n/4) - 1), {n, ∞}]/2 PUZZLE 2: Prove it. --rwg

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[math-fun] Fwd: Why people don't like mathematicians
by meekerdb 05 Mar '14

05 Mar '14

Lol: http://joyreactor.com/post/1189625

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[math-fun] abysmal fish (was: blood)
by Bill Gosper 05 Mar '14

05 Mar '14

On Mon, Mar 3, 2014 at 10:51 AM, Bill Gosper <billgosper(a)gmail.com> wrote: > Various sites claim our body creates (and destroys) about 2 million > erythrocytes/sec. > Yesterday, I had a kid compute the correction increment required for his > growth. I > think we overshot by a factor of 2 by using a (presumably adult) count of > 2*10^13, > getting about 63000/sec. This roughly coincides with the total of > vertebrate species, > all but 25 of which have erythrocytes! Do see > http://en.wikipedia.org/wiki/Channichthyidae. It's enough to warm the > subzero heart > of a Creationist. > --rwg > Also, no coronary arteries, likes subzero seawater, faster than a speeding brinicle. (https://www.youtube.com/watch?v=r4cX2EPt2zE) http://newsinfo.inquirer.net/582509/how-deep-can-fish-go-scientists-may-hav… suggests fundamental biochemistry limits fish to shallower than 8.3km. In light of Channichthyidae <http://en.wikipedia.org/wiki/Channichthyidae>, I would be amazed. --rwg

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[math-fun] A Powerful New Way to Edit DNA
by Henry Baker 04 Mar '14

04 Mar '14

FYI -- So much for the 'junk DNA' hypothesis... "If a previously unseen virus attacks, a new spacer, a new mug shot, is made and put at the end of the chain." http://www.nytimes.com/2014/03/04/health/a-powerful-new-way-to-edit-dna.html A Powerful New Way to Edit DNA By ANDREW POLLACKMARCH 3, 2014 In the late 1980s, scientists at Osaka University in Japan noticed unusual repeated DNA sequences next to a gene they were studying in a common bacterium. They mentioned them in the final paragraph of a paper: ÂThe biological significance of these sequences is not known.Â Now their significance is known, and it has set off a scientific frenzy. The sequences, it turns out, are part of a sophisticated immune system that bacteria use to fight viruses. And that system, whose very existence was unknown until about seven years ago, may provide scientists with unprecedented power to rewrite the code of life. In the past year or so, researchers have discovered that the bacterial system can be harnessed to make precise changes to the DNA of humans, as well as other animals and plants. This means a genome can be edited, much as a writer might change words or fix spelling errors. It allows Âcustomizing the genome of any cell or any species at will,Â said Charles Gersbach, an assistant professor of biomedical engineering at Duke University. Already the molecular system, known as Crispr, is being used to make genetically engineered laboratory animals more easily than could be done before, with changes in multiple genes. Scientists in China recently made monkeys with changes in two genes. Scientists hope Crispr might also be used for genomic surgery, as it were, to correct errant genes that cause disease. Working in a laboratory Â not, as yet, in actual humans Â researchers at the Hubrecht Institute in the Netherlands showed they could fix a mutation that causes cystic fibrosis. But even as it is stirring excitement, Crispr is raising profound questions. Like other technologies that once wowed scientists Â like gene therapy, stem cells and RNA interference Â it will undoubtedly encounter setbacks before it can be used to help patients. It is already known, for instance, that Crispr can sometimes change genes other than the intended ones. That could lead to unwanted side effects. The technique is also raising ethical issues. The ease of creating genetically altered monkeys and rodents could lead to more animal experimentation. And the technique of altering genes in their embryos could conceivably work with human embryos as well, raising the specter of so-called designer babies. ÂIt does make it easier to genetically engineer the human germ line,Â said Craig C. Mello, a Nobel laureate at the University of Massachusetts Medical School, referring to making genetic changes that could be passed to future generations. Still, Crispr is moving toward commercial use. Five academic experts recently raised $43 million to start Editas Medicine, a company in Cambridge, Mass., that aims to treat inherited disease. Other start-ups include Crispr Therapeutics, which is being formed in London, and Caribou Biosciences in Berkeley, Calif. Agricultural companies might use Crispr to change existing genes in crops to create new traits. That might sidestep the regulations and controversy surrounding genetically engineered crops, which generally have foreign DNA added. The development of the new tool is an example of the unanticipated benefits of basic research. About 15 years ago, after it became possible to sequence the entire genomes of bacteria, scientists noticed that many species had those repeated DNA sequences that were first noticed a decade earlier in Osaka. They were called Âclustered regularly interspaced short palindromic repeatsÂ Â Crispr for short. But what was their purpose? In 2007, researchers at Danisco, a company that supplies bacterial cultures used in making cheese and yogurt, confirmed hypotheses that Crispr protects bacteria from viruses. It is part of an adaptive immune system Â one that remembers a pathogen so it is ready the next time that same invader appears. The human adaptive immune system is why people get measles only once and why vaccines work. But it was not imagined that single-cell organisms like bacteria had such systems. Here is how it works. The repeated DNA sequences in the bacterial genome are separated from one another by other sequences. These ÂspacersÂ are excerpts from the sequences of viruses that have attacked the bacterium or its ancestors. They are like genetic mug shots, telling the bacterium which bad guys to watch for. The Crispr defense system will slice up any DNA with that same sequence, so if the same virus invades again, it will be destroyed. If a previously unseen virus attacks, a new spacer, a new mug shot, is made and put at the end of the chain. That means the Crispr region Âis like a tape recording of exposure to prior invaders,Â said Erik J. Sontheimer, a Northwestern University professor who helped unravel the mechanism. And it provides a way to tell two bacterial strains apart, because even two strains from the same species are likely to have encountered different viruses. This is already being used to identify sources of food-poisoning outbreaks. Cheese and yogurt companies can examine Crispr regions to see if their bacterial cultures are immunized against particular viruses that could slow production. ÂNow you can extend the shelf life of that great strain,Â said Rodolphe Barrangou of North Carolina State University, who previously worked at Danisco and was the lead author on the 2007 paper. ÂThat has changed the game quite a bit for the dairy industry.Â The real frenzy, however, started in 2012, when a team led by Emmanuelle Charpentier, then at Umea University in Sweden, and Jennifer A. Doudna of the University of California, Berkeley, demonstrated a way for researchers to use Crispr to slice up any DNA sequence they choose. Scientists must synthesize a strand of DNAÂs chemical cousin RNA, part of which matches the DNA sequence to be sliced. This Âguide RNAÂ is attached to a bacterial enzyme called Cas9. When the guide RNA binds to the corresponding DNA sequence, Cas9 cuts the DNA at that site. The cell tries to repair the cut but often does so imperfectly, which is enough to disable, or knock out a gene. To change a gene, scientists usually insert a patch Â a bit of DNA similar to where the break occurred but containing the desired change. That patch is sometimes incorporated into the DNA when the cell repairs the break. Would this work in organisms besides bacteria? ÂI knew it was like firing a starting gun in a race,Â Dr. Doudna said, but sure enough, by early 2013 scientists had shown it would work in human cells, and those of many other animals and plants, even though these species are not known to have Crispr-based immune systems. ÂI donÂt know any species of plant or animal where it has been tried and it failed,Â said George Church, a professor of genetics at Harvard Medical School. ÂIt allows you to do genome engineering on organisms that are very hard to do otherwise.Â In the past, making an animal with multiple genetic changes usually required creating separate animals with single changes and then crossbreeding them to produce offspring with multiple changes. With Crispr, multiple genetic changes can be made in one step, by putting multiple guide RNAs into the cell. ÂIt just completely changes the landscape,Â Dr. Doudna said. Berkeley scientists used to farm out that work to specialized laboratories or companies. Now, she said, Âpeople are able to make mice in their own labs.Â There are other techniques that can do what Crispr does, though Crispr is Âthe easiest by far,Â Dr. Church said. RNA interference, for instance, can silence particular genes. It is similar to Crispr in that it also uses RNA that matches the gene to be silenced. But RNA interference works by inhibiting messenger RNA, which translates a gene into a protein. That usually provides only a partial and temporary disabling of the gene, because the cell can make new messenger RNA. Crispr disables the gene itself, potentially a more complete and permanent inactivation. There are also already ways to change genes, namely zinc-finger nucleases and transcription activator-like effector nucleases, or Talens. The biotechnology company Sangamo BioSciences is already conducting a clinical trial of a treatment for H.I.V. that uses zinc fingers to alter patientsÂ immune cells to make them resistant to the virus. Both techniques use proteins to guide where the DNA is cut; it is more difficult to develop a protein that binds to a specific DNA sequence than it is to make a piece of RNA with the matching sequence. With zinc fingers Âit might take you months or years to get something to work well for one gene,Â said Dr. Gersbach at Duke. With Crispr, Âit takes days to weeks.Â Quick is not always accurate, however. While Crispr is generally precise, it can have off-target effects, cutting DNA at places where the sequence is similar but not identical to that of the guide RNA. Crispr Âmay not yet have adequate specificity to completely displaceÂ the older techniques, Dana Carroll, a biochemistry professor at the University of Utah, wrote in a commentary in Nature Biotechnology in September. Still, scientists are already figuring out how to make Crispr more specific. Another obstacle for treating diseases will be the delivery of the genetic changes to all the cells in the body that need it. For some diseases, it may be possible to extract blood stem cells from the body, alter them using Crispr, and put them back. If that is not possible, the DNA needed to make Cas9, the guide RNA and the corrective patch might be put into a disabled virus. This technique is used for gene therapy, but does not always work well. It is likely to be a few years before Crispr is tested in people. For now, there is a lot more to learn about it. Chase L. Beisel at North Carolina State reported that Crispr could be used to kill one strain of bacteria in a mixture of strains, by targeting a sequence unique to that strain. That might one day lead to antibiotics that can kill the bad bugs without also killing the good ones. David S. Weiss of Emory University found that some bacteria use Cas9 to silence one of their own genes, rather than that of a virus, to help them evade detection by their hostÂs immune system. The pace of new discoveries and applications is dizzying. ÂAll of this has basically happened in a year,Â Dr. Weiss said. ÂItÂs incredible.Â A version of this article appears in print on March 4, 2014, on page D1 of the New York edition with the headline: A Powerful New Way to Edit DNA.

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[math-fun] Harmonic numbers
by Bill Gosper 03 Mar '14

03 Mar '14

WDS> H(n) = SUM(1/j, j=1..n) is the nth "harmonic number." QUESTION: what if we want to generalize this to real or complex n, not merely integer n, similarly to how Euler converted factorial to gamma function? ANSWER: Let psi(x) = d/dx ( ln Gamma(x) ) = Gamma'x) / Gamma(x) be the "psi function"; then H(x) = psi(x) + EulerMascheroniConstant = 0.5772156649 is the sought-for generalization. Euler also had found the integral representation H(x) = INTEGRAL( (1 - t^x) / (1-t), t=0..1 ) which is equivalent to this. http://en.wikipedia.org/wiki/Harmonic_number#Special_values_for_fractional_… discusses further, and many other things are "continuizable" in similar way (also discussed in the wikipedia article). <WDS------- Conway prefers "Euler's constant". This wiki seems to lack the general formula for H(p/q), the last entry in the numeric tables ending each tAoCP. Also, that first H_x formula could be enlivened: H_x = lim y->∞ H_{x,y}, where H{x,y} = H{y,x} := Sum(y/k/(k+y))_{k=1}^x (not Mma's braindead Sum semantics). Also, by http://dspace.mit.edu/bitstream/handle/1721.1/6088/AIM-304.pdf p25: HarmonicNumber[p]==-Sum[(3*k + p)* Binomial[k - p - 1, k]/k/(k + p)/Binomial[2*k, k], {k, ∞}] (Terminates after (integer) p), which Mma converts to two useless 3F2s unless integer 2p. --rwg

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[math-fun] Harmonic numbers
by Warren D Smith 03 Mar '14

03 Mar '14

H(n) = SUM(1/j, j=1..n) is the nth "harmonic number." QUESTION: what if we want to generalize this to real or complex n, not merely integer n, similarly to how Euler converted factorial to gamma function? ANSWER: Let psi(x) = d/dx ( ln Gamma(x) ) = Gamma'x) / Gamma(x) be the "psi function"; then H(x) = psi(x) + EulerMascheroniConstant = 0.5772156649 is the sought-for generalization. Euler also had found the integral representation H(x) = INTEGRAL( (1 - t^x) / (1-t), t=0..1 ) which is equivalent to this. http://en.wikipedia.org/wiki/Harmonic_number#Special_values_for_fractional_… discusses further, and many other things are "continuizable" in similar way (also discussed in the wikipedia article).

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[math-fun] Gelfond-Schneider, Lindemann-Weierstrass, and formatting
by Keith F. Lynch 03 Mar '14

03 Mar '14

Eugene Salamin <gene_salamin(a)yahoo.com> wrote: > Yahoo mail messed up the formatting, so here is a corrected version. > Gelfond?Schneider theorem:? If a and b are algebraic numbers with > a???0,1 and if b is not a rational number, then any value of a^b is > a transcendental number. The digest option turns all non-ASCII characters into question marks. As such, I recommend avoiding non-ASCII characters on this list. As you say, the Gelfond-Schneider theorem involves raising an algebraic number to an algebraic power. But we were discussing raising a transcendental number (e) to an algebraic power, so Lindemann-Weierstrass is the relevant theorem. No, wait. We were discussing raising a transcendental number (e) to a transcendental power (pi*sqrt(163)). So I don't know what theorem applies, if any. Maybe e^(pi*sqrt(n)) really can be an integer for some nonzero positive integer n? It is an integer for n = -1, after all. Dan Asimov <dasimov(a)earthlink.net> wrote: > P.S. This reminder by Keith led me to review that April 1975 column > (which I have on my computer from a CD I got from the MAA). Fun! I too reviewed it, mostly to make sure I was remembering the year correctly. (I was.) But I still have the original magazine. I'm such a packra^H^H^H archivist.

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