May 2012 - mathfun@mailman.xmission.com

[math-fun] shapes with constant surface - nice solution now?
by Warren Smith 09 May '12

09 May '12

>R.Munafo: If the annulus is acceptable as a 2-D solution (mentioned earlier in this discussion), then your annular torus would have to be acceptable in 3-D too. I don't think either should count. It has to be actually physically viable for fertilizer purposes. --well, ok. Stimulated by this complaint, it occurred to me (somewhat contrary to my mistaken claim before) that there is a 2D solution under Munafo's rules. Start with a star-shaped object S as in a previous post where the angle-condition equation was given for constant surface. Let said object erode for some timespan T>0 yielding smaller object S' with same topology. Now the object S-S' works: it will erode maintaining constant surface for 0<t<T, then instantly vanish. If star is mirror-symmetric then I also think we can make this 3D by torifying it -- torus with hollow-star cross section -- which would be a nice solution to my original problem! Further, with Munafo's attitude, if we demand S have SMOOTH measurable boundary, then I can prove no 2D solution exists. Here is proof sketch. Munafo is only permitting 2D shapes with the topology of a disk (or union of disjoint disks). Well, I suppose he would permit holes, but there are effectively no holes until such time as the erosion eliminates them by making them become exterior suddenly. If this sudden transition occurs on just part of the hole-boundary then we get a discontinuous increase-jump in surface, and game over, we lose. If it occurs on all of it simultaneously, then also discontinuous jump (now decrease) and we still lose, unless that instant was the final instant when the whole shape vanishes. OK, so, effectively during the timespan of existence of object, no holes exist. In that case by the Gauss-Bonnet theorem (which in 2D is a rather trivial statement) the total curvature of the boundary o is 2*pi radians, which means that after infinitesimal erosion time dt, the boundary shrinks by subtraction of 2*pi*dt from its length. (We only need this to work throughout an arbitrarily short time-interval.) But that contradicts constancy. Q.E.D. Objection to proof: the jump-decrease could be designed to cancel a jump-increase of exactly same size. Counter to objection: So what? You still are killed by the Gauss-Bonnet argument during timespan before the first hole is uncovered. You could make an infinite set of holes so that we get uncovering infinitely often, but that would not work since you'd need an UNCOUNTABLE infinity of holes, which'd contradict measurable boundary, or something, dammit. You also can use 3D Gauss-Bonnet theorem to see no 3D object with topology of a ball and smooth boundary can work... which is why my secret alleged messy 3D solution does have smooth boundary, but does not have the topology of a set of disjoint balls.

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[math-fun] Head zaps increase math skills
by ed pegg 09 May '12

09 May '12

http://news.discovery.com/human/brain-electricity-math.html It's barely enough to light a light bulb, but passing a very mild current of electricity through the brain can turn on a metaphorical light bulb in a person's brain. Scientists from the University of Oxford have shown that they can improve a person's math abilities for up to six months. The research could help treat the nearly 20 percent of the population with moderate to severe dyscalculia (math disability), and could probably aid students in other subjects as well. "I am certainly not advising people to go around giving themselves electric shocks," said Roi Cohen Kadosh, a scientist at the University of Oxford and a co-author of a new paper. "But we are extremely excited by the potential of our findings." The UK scientists used a method known as transcranial direct current stimulation, or TDCS. This non-invasive technique involves passing electricity through the skull to increase or decrease the activity of neurons, usually for less than 15 minutes. The amount of electricity is tiny, so small that most patients don't even know it is happening. In fact, many scientists were initially skeptical it would have any effect at all, said Jim Stinear, Director of the Neuralplasticity Laboratory at the Rehabilitation Institute of Chicago. For this experiment the scientists directed the current into the brain's parietal lobe, which is involved in number processing. Instead of learning familiar Arabic numerals, however, the scientists had the participants learn a new series of symbols that represented numbers. Then, while their brains were being stimulated, they tested the participants ability to organize those numbers. Patients who were on TDCS showed an improved ability to order the numbers. The electric current makes it subtly easier or more difficult to stimulate a particular group of nerves, depending on the needs of the researchers and the patient. For example, if researchers want to make it easier for a patient to learn, then the nerves will fire more readily. Other studies have shown that TDCS can improve a variety of brain functions, from pain management to rehabilitation after traumatic events, said Jim Stinear, Director of the Neuralplasticity Laboratory at the Rehabilitation Institute of Chicago. But what is "really remarkable," about this new research is how long the effects lasted: six months. If TDCS can improve number processing in normal people, it should be able to improve number processing in people who have lower than normal number processing skills, and that's who the Oxford scientists will be testing next. TDCS should be able to improve other types of learning, such as language, as long as they are near the surface of the brain. Structures like the hippocampus, which are buried under entire lobes of the brain, are likely beyond the reach of TDCS, said Cohen Kadosh. While the Oxford scientists don't advocate plugging yourself into a wall socket, they do eventually hope to create a device that will provide an appropriate amount of electrical current to the brain, and have filed a patent on such a device. Such a device won't instantly make you better at math, help you recover from a stroke faster, or manage pain better, said Stinear. Anybody using a device will still have to put in a significant amount of effort. Drawing a parallel between a popular stimulant, Stinear said that coffee can help you wake up,but if you just sit on the couch you still aren't being productive. The same goes for TDCS. "Electrical stimulation will most likely not turn you into Albert Einstein," said Kadosh, "but if we're successful it might be able to help some people to cope better with math."

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[math-fun] shapes with constant surface
by Warren Smith 09 May '12

09 May '12

Actually, a nice 3D solution occurs to me... which unfortunately seems useless and is kind of a cheat. Make a torus, i.e. the points within distance <=1 from a large circle. Also, make that circle itself be a "surface" (of area=0, or you could make it be another torus so the area was positive). The torus erodes from both the outside-in and the inside-out and always has constant surface until the instant it vanishes. Cute... except not useful for fertilizer purposes since only the surfaces ACCESSIBLE from the outside are allowed to erode in real life. (As I said originally, I believe I do have a genuine 3D solution, but it is messy.)

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Re: [math-fun] [EXTERNAL] shapes with constant surface area
by Henry Baker 09 May '12

09 May '12

It is possible to unify these two subjects, as some fertilizers are also solid rocket fuel... At 08:08 AM 5/9/2012, Cordwell, William R wrote: >In solid rocket motors, it can be desirable to have a roughly constant burning (surface) area. One solution is to start out with a star shaped hole down the middle; the star erodes as it burns, but the hole also gets larger. Of course, the outside surface is not burning in this case, so it doesn't directly apply to the fertilizer question, but it seems as though you could have a cylinder with an inside hole that gets larger at a similar rate as the outside surface gets smaller. > >-----Original Message----- >From: math-fun-bounces(a)mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Warren Smith >Sent: Wednesday, May 09, 2012 8:58 AM >To: math-fun; David Langer; Monty McGovern; Holt, Robert >Subject: [EXTERNAL] [math-fun] shapes with constant surface area > >The surface of "slow release fertilizer" beads gradually erode, say at >1 centimeter per year. >With a spherical bead, the resulting curve of "fertilization rate" >would decline with time e.g. >it would be proportional to (1-t)^2 for 0<t<1 for a 1cm-radius bead. >It might be better to have beads with constant fertilization rate 1. > >QUESTION: is there a shape (or set of shapes) whose surface area stays constant as they erode? > >The answer in 1 and in 2 dimensions is "yes." >In 3D the answer is less obvious, but I have a (presently rather messy) argument the answer also is yes. > >More advanced question: >you might want the rate to increase with time (as your plants grow bigger they need more)... >or be some other curve... WHICH curves F(t) are achievable fertilization-rate curves? > >-- >Warren D. Smith

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[math-fun] shapes with constant surface area
by Warren Smith 09 May '12

09 May '12

> Cordwell, William R: > In solid rocket motors, it can be desirable to have a roughly constant burning (surface) area. One solution is to start out with a star shaped hole down the middle; the star erodes as it burns, but the hole also gets larger. Of course, the outside surface is not burning in this case, so it doesn't directly apply to the fertilizer question, but it seems as though you could have a cylinder with an inside hole that gets larger at a similar rate as the outside surface gets smaller. --actually, the "obvious" 2D solution I had in mind was an annulus (ring between two concentric circles) which erodes from the inside and outside simultaneously to have constant surface. But this solution would not be good as a solid rocket since for that we want erosion from inside ONLY. But stimulated by Cordwell, I realize it actually is possible to have a 2D "rocket" solution in 2D which really only erodes from inside and really does have constant surface area for some positive timespan. Make the inward (concave) points of the star-shaped hole have angles A1,A2,...An and make the outward-pointing (convex) vertices have angles B1,B2,...,Bn. Then constant surface happens if and only if 2 * sum_k cot(Ak / 2) = sum_k (2*pi - Bk). Note that necessarily sum_k (Ak-Bk) = 2*pi. It similarly is possible to erode a star-shaped polygon from the outside only while enjoying constant surface for a positive timespan, but then the constancy CANNOT persist the whole way (until it is eroded away to nothing); that ONLY is possible with a topology (e.g. an annulus) more complicated than just a union of disjoint disks. Also, if we restrict ourselves to smooth surfaces (polygons due to corners are disqualified) then constancy for even a positive timespan, no matter how small, is impossible in 2D unless you have annulus topology.

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[math-fun] shapes with constant surface area
by Warren Smith 09 May '12

09 May '12

The surface of "slow release fertilizer" beads gradually erode, say at 1 centimeter per year. With a spherical bead, the resulting curve of "fertilization rate" would decline with time e.g. it would be proportional to (1-t)^2 for 0<t<1 for a 1cm-radius bead. It might be better to have beads with constant fertilization rate 1. QUESTION: is there a shape (or set of shapes) whose surface area stays constant as they erode? The answer in 1 and in 2 dimensions is "yes." In 3D the answer is less obvious, but I have a (presently rather messy) argument the answer also is yes. More advanced question: you might want the rate to increase with time (as your plants grow bigger they need more)... or be some other curve... WHICH curves F(t) are achievable fertilization-rate curves? -- Warren D. Smith

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[math-fun] Harding's Theorem
by rcs＠xmission.com 08 May '12

08 May '12

Puzzle: How much can you learn about hyperfactorials modulo a prime P? What is 1! 2! ... (P-1)! mod P? mod P^2? What about 1^1 * 2^2 * 3^3 * ... * (P-1)^(P-1)? What looks true, and what can you prove? To give everyone a chance to play, lets hold off reporting until noon tomorrow. That's Wednesday, May 9, noon Mountain Daylight Time. (God's own time zone :-).) Rich

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[math-fun] Gosper's notes on higher factorials
by Robert Munafo 08 May '12

08 May '12

What is the date (year of authorship) of these ancient (mostly) POX notes? I want to add this as a reference for OEIS sequence A2109: https://oeis.org/A002109 It would also be nice to add a sequence for the (Gosper's) second factorial with integer arguments, which I take to be 1, 1, 16, 122009559759792, 16358..68032 (169 digits, approx. 1.63588 x 10 ^ 168) but I don't like to add sequences without a reference. - Robert On 5/7/12, Bill Gosper <billgosper(a)gmail.com> wrote: > By way of preservation, Neil has kindly scanned in my ancient, > (mostly) POX (pre-Tex), brief notes <http://gosper.org/facfun.pdf> > (3.7M) on higher factorials. > These stem from my astounding discovery that the dot at the > bottom of an exclamation point is actually hollow, representing > the number 0, and what happens when you change it to a 1. > --rwg -- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com

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[math-fun] Gamma function (formulas collection)
by Bill Gosper 08 May '12

08 May '12

wds> many formulas about gamma function are in my paper:http://rangevoting.org/WarrenSmithPages/homepage/gammprox.pdf ------ By way of preservation, Neil has kindly scanned in my ancient, (mostly) POX (pre-Tex), brief notes <http://gosper.org/facfun.pdf> (3.7M) on higher factorials. These stem from my astounding discovery that the dot at the bottom of an exclamation point is actually hollow, representing the number 0, and what happens when you change it to a 1. --rwg

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[math-fun] Jimmy Wales: all taxpayer-funded academic research available for free online
by Henry Baker 07 May '12

07 May '12

FYI -- http://www.guardian.co.uk/technology/2012/may/01/wikipedia-research-jimmy-w… Wikipedia founder to help in government's research scheme Academic spring campaign aims to make all taxpayer-funded academic research available for free online Alok Jha, science correspondent guardian.co.uk, Tuesday 1 May 2012 16.04 EDT Jimmy Wales, co-founder of Wikipedia, is to help the government in its bid to make taxpayer-funded research available for free online. The government has drafted in the Wikipedia founder Jimmy Wales to help make all taxpayer-funded academic research in Britain available online to anyone who wants to read or use it. The initiative, which has the backing of No 10 and should be up and running in two years, will be announced by the universities and science minister, David Willetts, in a speech to the Publishers Association on Wednesday. The move will embolden what has been dubbed the "academic spring" Â a growing campaign among academics and research funders for open access in academic publishing. They want to unlock the results of research from behind the lucrative paywalls of journals controlled by publishing companies. Almost 11,000 researchers have signed up to a boycott of journals owned by the huge academic publisher Elsevier. Subscriptions to the thousands of research journals can cost a big university library millions of pounds each year Â costs that have started to bite as budgets are squeezed. Harvard University, frustrated by the rising costs of journal subscriptions, recently encouraged its faculty members to make their research freely available through open access journals and to resign from publications that keep articles behind paywalls. "Giving people the right to roam freely over publicly funded research will usher in a new era of academic discovery and collaboration, and will put the UK at the very forefront of open research," Willetts writes in the Guardian. Willetts said he recognised the value that academic publishers brought to the research process. "But, as the world changes, both cultural and technological change, their business model is going to change. I want to work with the Publishers Association as we move to the new model." Wales is a vocal supporter of free and open access to information on the web and he was brought in by No 10 earlier this year as an unpaid adviser to government on crowdsourcing and opening up policymaking. On open access, he will assist the Department for Business, Innovation and Skills and the UK Research Councils to develop new ways to store and distribute research data and articles. He will initially advise the research councils on its Â£2m Gateway to Research project, a website that will act as a portal, linking to publicly funded UK research all over the web. "Jimmy Wales can make sure that we maximise the collaborative potential, the added value from that portal," Willetts added. "Wikipedia has become a crucial part of our cultural landscape and having the advice from the person who created Wikipedia as we embark on this big project will be incredibly helpful." Wales will also feed ideas into the work of Dame Janet Finch, a former vice-chancellor of Keele University, who was asked by Willetts to convene academics, librarians and publishers to work out how an open-access scheme for publicly funded research might work in the UK. Her recommendations to government are expected in June this year. A government source said that, in the longer term, Wales would help to set up the next generation of open-access platforms for British researchers. "He's also going to be advising us on the format in which academic papers should be published and data standards. One of the big opportunities is, right now, a journal article might be published but the underlying data isn't and we want to move into a world where the data is published alongside an article in an open format, available free of charge." This initiative is most likely to result in a central repository that will host all research articles that result from public funding. The aim is that, even if an academic publishes their work in a traditional subscription journal, a version of their article would simultaneously appear on the freely available repository. The repository would also have built-in tools to share, comment and discuss articles. One of the biggest challenges in achieving full open access for research will be the resistance of journal publishers to changing their lucrative business models. The majority of the world's scientific research, estimated at about 1.5m new articles a year, is published in journals owned by a small number of large publishing companies including Elsevier, Springer and Wiley. Scientists submit manuscripts to the journals, which are sent out for peer review before publication. The work is then available to other researchers by subscription, usually through their libraries. Publishers of the academic journals, which can cost universities up to Â£16,500 a year each to access, argue the price is necessary to sustain a high-quality peer review process. David Prosser, executive director of Research Libraries UK, which represents academic libraries, welcomed the plans in principle and said the details of their implementation would be crucial. A parallel system that runs alongside the journals might be difficult to operate, he said. "What would an author put into this parallel system, are they putting in a different type of research output other than the paper?" Making research data standardised and more available would be valuable, he added. "The worry is that there's all this data out there and it's in lots of different formats and it's not interoperable and it's not being archived properly and it's going to disappear and there's a danger of a data black hole. The fact that the government is talking about doing something for that is absolutely fabulous."

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