March 2013 - mathfun@mailman.xmission.com

[math-fun] best tilt angle for solar panel
by Warren D Smith 11 Mar '13

11 Mar '13

Suppose you are on an airless spherical earth. You've got a flat solar panel and can put it facing at angle Q to the ground (and toward the equator of course). Assume Q is fixed, you cannot have a swiveling panel which changes orientation during the day. What is the best value of Q, maximizing the amount of solar energy hitting the panel, as a function of your latitude L and the Earth's axial tile angle A=23.4 degrees? You might naively imagine Q=L is best. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] best tilt angle for solar panel
by Henry Baker 11 Mar '13

11 Mar '13

I've thought about this a lot, since I've been thinking about getting some solar panels. The problem in the "real" world is somewhat more complicated: 1. You may have to ignore those days of the year that are so cloudy that you don't get enough sun. 2. You may have to factor in issues like what you're generating the power for; e.g., if you're generating it for air conditioning, you may care more about power at a particular hour of the day and certain days of the year than at other times & days. As a result of these & other calculations, I keep coming back to incorporating some sort of motor control on the panels. BTW, you don't have to swivel a huge panel (which could have problems in strong winds); you could have lots of little panels with little motors. For example, there are already lots of motor controllers for security camera applications, which could conceivably be repurposed for pointing small solar panels. Now architects have already done calculations about minimize heat load from the placement of windows relative to the sun. Since most homes & windows aren't on swivels, the calculation you propose is more relevant. I believe that I heard about some research at MIT's Media Lab in collaboration with MIT's architecture dept. They were developing shades & shutters that were "powered" merely by bimetallic metals responding to the temperature. At 06:58 PM 3/10/2013, Warren D Smith wrote: >Suppose you are on an airless spherical earth. You've got a flat solar panel >and can put it facing at angle Q to the ground (and toward the equator >of course). >Assume Q is fixed, you cannot have a swiveling panel which changes >orientation during the day. > >What is the best value of Q, maximizing the amount of solar energy hitting >the panel, as a function of your latitude L and the Earth's axial tile >angle A=23.4 degrees? > >You might naively imagine Q=L is best. > >-- >Warren D. Smith >http://RangeVoting.org <-- add your endorsement (by clicking >"endorse" as 1st step)

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Re: [math-fun] Physicists Discover 13 New Solutions To Three-Body Problem
by Henry Baker 10 Mar '13

10 Mar '13

The usual 2-body elliptical orbits are extremely robust, in the sense that any small (or even large) perturbations of any of the parameters (mass, energy, angular momentum, etc.) only make yet another ellipse with slightly different properties. Are there any 3-body configurations that show anything like this type of stability, other than a hierarchy like Sun, Earth, Moon ? The problem with many of these solutions is that the masses of the 3 objects must be identical. Even microscopic perturbations of the mass will screw up these solutions. This means that it is essentially impossible to find one of these solutions "in the wild". A truly stable solution will have to be able to tolerate deviations from mass (e.g., the impact of a meteor), deviations in energy (loss from tidal forces), deviations in angular momentum (more meteors or slight tugs from further away bodies), etc. Given the low cost of computation today, it might be simpler to do a brute force search with thousands of computers and millions of different initial configurations. If there are any somewhat stable configurations, surely one would pop up under such a brute force search. At 05:42 PM 3/9/2013, Allan Wechsler wrote: >I don't know whether I actually posted my conjecture a few years ago when >the "hey-for-three" solution was discovered. I'm not sure how to rigorize >the conjecture, but I am convinced that there is an >at-least-countably-infinite family of such stable (using the word in the >usual sense, pace Warren) configurations. > >Basically, if you start off three bodies in a random way, then the system >might spend some energy to eject one of the three, but if it doesn't, the >system will eventually come very close to one of its previous >configurations; if it does, then you can construct a stable solution by >tiny perturbations. > >Most of these stable modes are extremely shallow potential wells. > >On Sat, Mar 9, 2013 at 5:25 PM, Cris Moore <moore(a)santafe.edu> wrote: > >> You may enjoy these orbits as well: >> http://arxiv.org/pdf/math/0511219v2.pdf >> >> Cris >> >> On Mar 9, 2013, at 1:23 PM, Ray Tayek wrote: >> >> http://science.slashdot.org/story/13/03/09/1846253/physicists-discover-13-n… >> > >> > Posted by <http://unknownlamer.org/>Unknown Lamer on Saturday March 09, >> @02:33PM >> > from the mystical-spheres dept. >> > sciencehabit writes "It's the sort of abstract puzzle that keeps a >> scientist awake at night: Can you predict how three objects will orbit each >> other in a repeating pattern? In the 300 years since this '< >> http://en.wikipedia.org/wiki/Three-body_problem>three-body problem' was >> first recognized, just three families of solutions have been found. Now, < >> http://news.sciencemag.org/sciencenow/2013/03/physicists-discover-a-whoppin…>two >> physicists have discovered 13 new families. It's quite a feat in >> mathematical physics, and it could conceivably help astrophysicists >> understand new planetary systems." The <http://arxiv.org/abs/1303.0181>paper >> is available at arxiv. >> > >> > --- >> > co-chair http://ocjug.org/ >> >> Cristopher Moore >> Professor, Santa Fe Institute >> >> The Nature of Computation >> Cristopher Moore and Stephan Mertens >> Available now at all good bookstores, >> or through Oxford University Press >> http://www.nature-of-computation.org/

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[math-fun] Physicists Discover 13 New Solutions To Three-Body Problem
by Ray Tayek 10 Mar '13

10 Mar '13

http://science.slashdot.org/story/13/03/09/1846253/physicists-discover-13-n… Posted by <http://unknownlamer.org/>Unknown Lamer on Saturday March 09, @02:33PM from the mystical-spheres dept. sciencehabit writes "It's the sort of abstract puzzle that keeps a scientist awake at night: Can you predict how three objects will orbit each other in a repeating pattern? In the 300 years since this '<http://en.wikipedia.org/wiki/Three-body_problem>three-body problem' was first recognized, just three families of solutions have been found. Now, <http://news.sciencemag.org/sciencenow/2013/03/physicists-discover-a-whoppin…>two physicists have discovered 13 new families. It's quite a feat in mathematical physics, and it could conceivably help astrophysicists understand new planetary systems." The <http://arxiv.org/abs/1303.0181>paper is available at arxiv. --- co-chair http://ocjug.org/

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[math-fun] Property lists on INT's and FLONUM's ???
by Henry Baker 10 Mar '13

10 Mar '13

I just came across the 1960 "Lisp 1 Programmer's Manual", which I'd never seen before: http://history.siam.org/sup/Fox_1960_LISP.pdf As this manual clearly describes, both integers and floats had _property lists_ and were _uniquized_. Q: When did Lisp numbers lose their property lists ? Q: Did any other Lisps keep property lists for numbers ? Q: Were Lisp property lists ever used for interesting things: e.g., storing primality, factors, etc. ? Q: Also, who is "K. Maling", who is credited with writing the Lisp 1 READ function ? --- My notes from the above manual: Note that in the original Lisp 1, _integers had property lists_ !!! This must have required that integers be INTERN'ed, although the Lisp 1 manual doesn't mention INTERN or OBLIST at all (INTERN/OBLIST apparently arrived with Lisp 1.5). "On the association lists for integers the indicator INT points to the integral value. The integral value is given as a full machine word with the value pushed to the right-hand end of the register." In Lisp 1, _floating point numbers had property lists_ !!! "Floating-point numbers are listed in a separate list. A floating-point number is put on this list either as it is read in or _as it is generated within a calculation_. _Neither this list nor the list of atomic symbols has any duplications._ An association list is created for a floating-point number when the number is put on the list of numbers." "Only _positive_ numerical values are put on the association lists for floating-point numbers; if a number is negative the association list for the number is preceded by the indicator MINUS, and the entry in the list of floating-point numbers points to the MINUS indicator." The representations of the integers +1.0 and -1.0 are given on page 92. In modern notation, the property list of +1.0 is '(NUMB FLO (1.0)), and -1.0 is the list '(MINUS +1.0). Floating point numbers only acquired a 'PNAME' (print name) when they were first printed out. Apparently, only positive floats existed as atoms; negative floats were the list `(MINUS ,num). The representation of the integer 1 is given on page 94 of the Lisp 1 manual. In modern notation, the property list of +1 is '(PNAME ("1") INT 1 APVAL (1)). --- Interesting _builtin_ capabilities for 1960 Lisp 1: gensym() !! It isn't mentioned, but I presume that these symbols were uniquized. compab(x,y,z) = |x-y|<z, where x,y,z are all floats. diff(y,x) "The function diff differentiates the algebraic expression y with respect to x. ... Gradients must be provided on the association lists of all the functions used in the expression y, except for PLUS and TIMES." matrixmultiply(x,y). In modern notation, x = '(matrix (row a11 a12 a13) (row a21 a22 a23) (row a31 a32 a33)) y = '(matrix (col b11 b21 b31) (col b12 b22 b32) (col b13 b23 b33)) Curiously, there wasn't a builtin transpose operation. Even modern Common Lisp, with all of its other bloat, doesn't have matrix multiplication built in!

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[math-fun] stable 3-body solutions?
by Warren D Smith 10 Mar '13

10 Mar '13

>From: Allan Wechsler <acwacw(a)gmail.com> I don't know whether I actually posted my conjecture a few years ago when the "hey-for-three" solution was discovered. I'm not sure how to rigorize the conjecture, but I am convinced that there is an at-least-countably-infinite family of such stable (using the word in the usual sense, pace Warren) configurations. Basically, if you start off three bodies in a random way, then the system might spend some energy to eject one of the three, but if it doesn't, the system will eventually come very close to one of its previous configurations; if it does, then you can construct a stable solution by tiny perturbations. --You may be right, but that reasoning is not necessarily valid. Consider the geodesic flow on a negative curvature compact hyperbolic manifold. There are an infinite set of closed orbits, which is known (also seems shown basically by Wechsler's reasoning to get a nearly-perfect closed orbit then "pull the string taut") -- but every one is unstable in the sense tiny perturbation to initial conditions grows exponentially with time. In fact every geodesic trajectory is unstable. And this is of course a Hamiltonian system in the sense energy is conserved.

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[math-fun] "memoization"
by Henry Baker 09 Mar '13

09 Mar '13

The term "memoization" in computer science typically refers to the somewhat magical ability of a simple RAM lookup table of previous function results to transform a slow recursive implementation of a function into a fast implementation of the function. Thus, the following extremely slow Lisp implementation of Fibonacci is suddenly transformed into a high performance implementation by the simple usage of memoization which remembers the results of previous calls to fib so they don't have to be laboriously recomputed. (defun fib (n) (if (< n 2) n (+ (fib (- n 1)) (fib (- n 2))))) We now note that it doesn't harm performance "too" much if we sometimes forget a particular result and have to recompute it; this is somewhat akin to a "page fault" in a virtual memory system. In fact, for many applications, a finite, fixed-size "memo table" is often good enough for government work. Question: does anyone here have an academic _reference_ for this last statement that a fixed-size memo table is good enough? I seem to recall reading a short paper (in English) from the 1960's or 1970's which made precisely this point about fixed-size memo tables. Sadly, Google hasn't been able to find this paper for me, and the Elsevier & other paywalls won't allow me to search some of these old journals.

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[math-fun] Go language
by Warren Smith 06 Mar '13

06 Mar '13

Go is an open-source programming language being developed by google. It looks pretty good. http://golang.org/ -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] graph laplacians
by Warren Smith 05 Mar '13

05 Mar '13

http://arxiv.org/abs/1301.6628 Jonathan A. Kelner, Lorenzo Orecchia, Aaron Sidford, Zeyuan Allen Zhu: A Simple, Combinatorial Algorithm for Solving SDD Systems in Nearly-Linear Time (Submitted on 28 Jan 2013) In this paper, we present a simple combinatorial algorithm that solves symmetric diagonally dominant (SDD) linear systems in nearly-linear time. --They reduce the problem to solving Poisson's equation on a graph. which is, essentially, a circuit made of resistors and each node attached to a current source. Task is to find all the voltages. The solve method is to repeatedly randomly sample a cycle from the set of cycles in the graph (under a certain probability distribution) and then modify to make the total current flowing round the cycle become 0. Near linear runtime bounds are proved. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] The Scottish Book
by Stuart Anderson 05 Mar '13

05 Mar '13

I came across this translation of the "Scottish Book" 1935-1941 by Stan Ulam, http://kielich.amu.edu.pl/Stefan_Banach/pdf/ks-szkocka/ks-szkocka3ang.pdf. at the webpage http://kielich.amu.edu.pl/Stefan_Banach/archiwalia.html. In the introduction Ulam wrote, "Perhaps some of the problems will still present an actual interest to mathematicians", does anyone know if any of the problems are unsolved, unsolvable, related to current problems? Ulam also wrote "The tradition of the Scottish Book continues. Since 1945 new problems have been formulated and inscribed and a new volume is in progress", does anyone know of the 'new' volume? Stuart

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