February 2011 - mathfun@mailman.xmission.com

[math-fun] Boy do we need a good radical simplifier
by Bill Gosper 12 Feb '11

12 Feb '11

(Apologies if my suspicion of having already sent this is correct.) http://en.wikipedia.org/wiki/Theta_function gives several slightly unsimplified valuations of theta_3(0,e^-(n pi)), and then EllipticTheta[3, 0, E^-(6*\[Pi])] -> 3^(5/8)*(3*Sqrt[2] + 3^(5/4) + 2*Sqrt[3] - 3^(3/4) + 12^(3/4) - 4)^(1/3)/6/(1 + Sqrt[6] - Sqrt[2] - Sqrt[3])^(1/6)*\[Pi]^(1/4)/ Gamma[3/4] a cube root over a 6th root instead of just a square root: (Sqrt[1+Sqrt[2]+3^(1/4)+Sqrt[3]]*Pi^(1/4))/(2^(3/4)*3^(3/8)*Gamma[3/4]) i.e. 1/4 1/4 -6 Pi Sqrt[1 + Sqrt[2] + 3 + Sqrt[3]] Pi EllipticTheta[3, 0, E ] -> ---------------------------------------- 3/4 3/8 3 2 3 Gamma[-] 4 This is a respectable approximation to 1: N[%, 22] 1.000000013024824272160 -> 1.000000013024824272160 Mathematica has the requisite machinery, but requires constant manual struggling against its thuggish nesting of radicals and reversion to Root notation. It needs to distinguish Radical Number from Algebraic Number. It also needs some way to decache generated polynomials, which seem to be causing a storage leak. --rwg

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Re: [math-fun] Ah, FORTRAN and Overlay areas!
by Henry Baker 11 Feb '11

11 Feb '11

I don't know, but I don't think so. I think that each pass made modest changes to the internal representation, slowly converting it from source text into binary. For example, one pass might tokenize the text; the next pass might do a 1st level parse, etc. Many of the later passes might have had very specific tasks -- e.g., handling subscripts, doing a particular type of optimization, etc., so many of these passes might be no-ops for some programs. At 12:43 PM 2/11/2011, Fred lunnon wrote: >Was that the compiler that did everything via sorting? WFL > >On 2/11/11, Henry Baker <hbaker1(a)pipeline.com> wrote: >> The IBM 1401 (card & tape machine) Fortran compiler had something like 100 >> passes. It was easier to keep the user's Fortran program to be compiled in >> the random access memory (4k-16k 7-bit bytes) & read the compiler program >> passes in from tape. Since the compiler passes were arranged in order on >> the tape, when the tape was done reading, the compiled program was finished >> & arranged in the memory. I guess this was the mirror to the Unix "pipe" >> capability. >> >> I can't remember whether this compiler supported separate compilation -- >> probably not -- so a loader wasn't required. >> >> At 03:32 PM 2/10/2011, Guy Haworth wrote: >>>Nostalgia time! I remember delivering a large FORTRAN statistical package, >>> Package X, to the UK Cabinet Office's Central Statistical Office around >>> 1977. >>> >>>Functional decomposition was an architectural imperative and there were >>> many subroutines. They had to be mapped to Overlay areas, like cars 'hot >>> parking' on shared carparking spots. >>> >>>To avoid thrashing, we defined the lattice of subroutines in terms of what >>> calls what, and made sure subroutines on different levels of the lattice >>> went to different overlay areas. To make sure all was working ok, we >>> monitored the disc-channel to check for thrashing. We were amazed to >>> discover that there were 10 disc-transfer per overlay-swap. >>> >>>Our reaction to this was to reformat the binary file so that all the bits >>> that came in with the code ... came in first in one lump (sometimes with >>> the code if it wasn't the code that defined the size of the overlay) and >>> got split out to their own bits of space. >>> >>>Basically, we speeded up the program by a factor of 6 - but this unofficial >>> format never got to the customer. >>> >>>guy

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[math-fun] FORTRAN and Overlays ...
by Guy Haworth 11 Feb '11

11 Feb '11

100 passes? That IBM 1401 compiler must have been a salami slicer. I was operating on the ICL 1900 range, not a machine that made it over the pond. We had to split the source across 4 files - which fooled a remarkably unthinking civil servant into thinking that we were deliberately wasting computer time by compiling the same file four times. At the end, our team of four had a daily routine of getting our source 'ok', and then going off to a restaurant dinner while the whole thing compiled itself, created the binary and ran the test suite. We'd come back after dinner and look at the results, if not actually cure the remaining problems. G

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Re: [math-fun] Ah, FORTRAN and Overlay areas!
by Henry Baker 11 Feb '11

11 Feb '11

The IBM 1401 (card & tape machine) Fortran compiler had something like 100 passes. It was easier to keep the user's Fortran program to be compiled in the random access memory (4k-16k 7-bit bytes) & read the compiler program passes in from tape. Since the compiler passes were arranged in order on the tape, when the tape was done reading, the compiled program was finished & arranged in the memory. I guess this was the mirror to the Unix "pipe" capability. I can't remember whether this compiler supported separate compilation -- probably not -- so a loader wasn't required. At 03:32 PM 2/10/2011, Guy Haworth wrote: >Nostalgia time! I remember delivering a large FORTRAN statistical package, Package X, to the UK Cabinet Office's Central Statistical Office around 1977. > >Functional decomposition was an architectural imperative and there were many subroutines. They had to be mapped to Overlay areas, like cars 'hot parking' on shared carparking spots. > >To avoid thrashing, we defined the lattice of subroutines in terms of what calls what, and made sure subroutines on different levels of the lattice went to different overlay areas. To make sure all was working ok, we monitored the disc-channel to check for thrashing. We were amazed to discover that there were 10 disc-transfer per overlay-swap. > >Our reaction to this was to reformat the binary file so that all the bits that came in with the code ... came in first in one lump (sometimes with the code if it wasn't the code that defined the size of the overlay) and got split out to their own bits of space. > >Basically, we speeded up the program by a factor of 6 - but this unofficial format never got to the customer. > >guy

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[math-fun] about 33826 formulas,
by Simon Plouffe 11 Feb '11

11 Feb '11

Hello, I updated the list of 'known' formulas for the OEIS database. All of the formulas are conjectures, most of them are true, some are false positives, well, I have been doing this for a couple of years, it is true that there is a % of rejected, Here is a graphic of the results, http://www.plouffe.fr/simon/OEIS/conjectures/Percentage%20of%20formulas%20d… and here is the latest list : http://www.plouffe.fr/simon/OEIS/conjectures/OEIS_conjectured_formulas.txt So far, over 33826 formulas, I am still adding new ones, this gives a percentage of 18.26% of success. best regards, Simon Plouffe

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[math-fun] Ah, FORTRAN and Overlay areas!
by Guy Haworth 10 Feb '11

10 Feb '11

Nostalgia time! I remember delivering a large FORTRAN statistical package, Package X, to the UK Cabinet Office's Central Statistical Office around 1977. Functional decomposition was an architectural imperative and there were many subroutines. They had to be mapped to Overlay areas, like cars 'hot parking' on shared carparking spots. To avoid thrashing, we defined the lattice of subroutines in terms of what calls what, and made sure subroutines on different levels of the lattice went to different overlay areas. To make sure all was working ok, we monitored the disc-channel to check for thrashing. We were amazed to discover that there were 10 disc-transfer per overlay-swap. Our reaction to this was to reformat the binary file so that all the bits that came in with the code ... came in first in one lump (sometimes with the code if it wasn't the code that defined the size of the overlay) and got split out to their own bits of space. Basically, we speeded up the program by a factor of 6 - but this unofficial format never got to the customer. guy

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Re: [math-fun] Spherical trig question
by Bill Gosper 10 Feb '11

10 Feb '11

wfl>On 2/7/11, Bill Gosper <billgosper(a)gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> wrote: rwg> Ah, from 3.5 yrs ago, > ------------- >From gosper(a)alum.mit.edu <http://gosper.org/webmail/src/compose.php?send_to=gosper%40alum.mit.edu> Mon Jul 16 08:47:39 2007 Date: Mon, 16 Jul 2007 10:39:48 -0400 (EDT) Message-ID: <23173122.9411184596788302.JavaMail.gbourne(a)brunch.mit.edu <http://gosper.org/webmail/src/compose.php?send_to=23173122.9411184596788302…>> From: Bill Gosper <gosper(a)alum.mit.edu <http://gosper.org/webmail/src/compose.php?send_to=gosper%40alum.mit.edu>> To: math-fun(a)mailman.xmission.com <http://gosper.org/webmail/src/compose.php?send_to=math-fun%40mailman.xmissi…> >Lemma: The apical solid angle of an isosceles trapezoidal pyramid with vertex angles a,b,a,c is 2 c b c 2 b - sin (-) + (cos(a) - 1) sin(-) sin(-) - sin (-) + cos(a) + 1 2 2 2 2 (d79) 2 acos(-------------------------------------------------------------) b c (cos(a) + 1) cos(-) cos(-) 2 2 I derived this by splitting the general a,b,a,c hedral vertex into trihedrals a,b,d and d,a,c, then maximizing over d, getting b c (d38) cos(d) = cos(a) - 2 sin(-) sin(-) 2 2 Clearer might have been the difference between two isosceles trihedrals: solid_angle(d,d,b)-solid_angle(d-a,d-a,c) with d chosen to equalize the dihedrals, using dihedral(a,b,c):=acos(csc(a)*csc(b)*cos(c)-cot(a)*cot(b)), which gives sin(a) d = atan(----------------------), b c cos(a) - csc(-) sin(-) 2 2 and eventually the same solid angle formula. But the max technique generalizes from trapezoidal pyramids to arbitrary quadrilateral pyramids with apex angles a,b,c,d, whose maximal solid angle is cos(d) + cos(c) + cos(b) + cos(a) a b c d --------------------------------- - tan(-) tan(-) tan(-) tan(-) a b c d 2 2 2 2 4 cos(-) cos(-) cos(-) cos(-) 2 2 2 2 Note we get the old formula when d=0. Note also the symmetry, giving the same result for a,b,d,c. In this maximal case, the "diagonal angles", i.e. the apex angles exposed by splitting into two triangular pyramids, are a d b c b d a c 2 (sin(-) sin(-) + sin(-) sin(-)) (sin(-) sin(-) + sin(-) sin(-)) 2 2 2 2 2 2 2 2 acos(1 - -----------------------------------------------------------------) c d a b sin(-) sin(-) + sin(-) sin(-) 2 2 2 2 and same(a<->c). ------------ WFL>If there is a connection with the original thread, I'm afraid it has escaped me. Then I'm confused--I thought this was exactly what David requested. For the unit sphere, solid angle := area. If you place the apex of a pyramid at the center, the vertex angles will equal the great arcs on the surface. The base of an isosceles trapezoidal pyramid will project onto David's trapezoidal area. Is it clear that the area of a hinged quadrilateral abac is maximal when flexed into the isosceles configuration? Is this all wrong? Irrelevant? DW>When I worked at ComputerVision, I implemented an algorithm for polygonal area. For each edge of the polygon, I computed the area of the quadrilateral between the edge and its shadow on the x-axis via orthogonal projection. The edges were directed counterclockwise around the polygon, allowing computation of directed normals, the areas were added or subtracted according to whether the normal pointed toward or away from the x-axis. Adding up all these areas gave the area of the polygon. (rwg)>I sent this to the kids a few months ago. Probably shoulda cc'ed math-fun: -------------- Subj: That Im[z w*] trick is really handy. Here's the area of an arbitrary polygon: In[100]:=area[L_List]:=Total[MapThread[Im[#1\[Conjugate]*#2]&,{L,RotateLeft[L]}]]/2 E.g., the b x h triangle in general position: In[104]:= FullSimplify[ ComplexExpand[area[z+E^(I*t)*{0,b,h*I+x}]],{b,h,x,t}\[Element]Reals] Out[104]= (b h)/2 In[99]:= Area[Pentagon] Out[99]= 5/(4 Sqrt[5 - 2 Sqrt[5]]) In[106]:= area[Table[Exp[2*k*I*\[Pi]/5]/2/Sin[\[Pi]/5], {k, 5}]] Out[106]= (5 Sqrt[5/8 + Sqrt[5]/8])/(8 (5/8 - Sqrt[5]/8)) In[107]:= FullSimplify[% - %99] Out[107]= 0 -------------- --rwg

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[math-fun] Spherical trig question
by Bill Gosper 10 Feb '11

10 Feb '11

David Wilson>On the unit sphere, measuring angles in radians. Place an arc with measure A on the equator. Place arcs with measures B and C at each end of A running north along meridians (perpendicular to A). Join the northern endpoints of B and C with a great circle arc. What is the area of the spherical quadrilateral thus formed? rwg>If you know the angles between those arc at those endpoints to be a, isn't it just the angular excess 2 a + 2 pi - 2 pi = 2a? If you know all four arcs, max_solid_angle(a,b,c,d):=2*acos((cos(d)+cos(c)+cos(b)+cos(a))/(4*cos(a/2)*cos(b/2)*cos(c/2)*cos(d/2))-tan(a/2)*tan(b/2)*tan(c/2)*tan(d/2)) --rwg

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[math-fun] Primative Pythagorean Tetrahedra
by Richard Hess 09 Feb '11

09 Feb '11

Dear Funsters, Can anyone help with the following problems? (1) Do there exist tetrahedra with each face a Primative Pythagorean Triangle? I have discovered one with 3 of the four faces PPTs having edges 42601, 41449, 9840, 11849, 40920 and 6601. Unfortunately the face with edges 9840, 11849 and 6601 has a common factor of 41. I think I've eliminated all cases with edges less that 10^8. (2) Do there exist tetrahedra with each face a Pythagorean Triangle such that the ratio of shortest to longest edge is more than 410688/743665 = .55224866? This occurs for a tetrahedron with edges of 743665, 609960, 425425, 591313, 450984 and 410688. I think I've eliminated all cases with edges less that 2x10^7. Best, Dick Hess

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[math-fun] DEC founder Ken Olsen dies at age 84
by Eugene Salamin 09 Feb '11

09 Feb '11

Kenneth Olsen, a computer industry pioneer and co-founder of Digital Equipment Corp., has died. He was 84. His death Sunday was announced by Gordon College in Wenham, Mass., where he was a trustee and benefactor. The college did not release a cause of death. DEC, which Olsen launched in 1957, is considered an icon in technology circles today. The company attracted top engineers and helped usher in a technology revolution that changed the way people interact with computers. In the 1960s and 1970s, Digital played a central role in creating the market for "minicomputers," powerful, refrigerator-sized machines that appealed to scientists, engineers and other number crunchers who did not need the bigger, multimillion-dollar mainframes used by big corporations. At its peak in the 1980s, DEC was the second-largest computer maker behind International Business Machines Corp. "In the heady days of Bill Gates and Steve Jobs, it's too easy to forget that it was Ken Olsen's vision of interactivity that took computing away from the centralized mainframe and into the hands of the people," said Gordon Bell, who joined DEC in 1960 and headed the company's engineering operations for more than 20 years. Ultimately, DEC lost its way in the Internet-era transformations of the technology industry, which shrunk computers down to pocket-sized gadgets that people carry wherever they go. And Olsen is still remembered for his 1977 prediction that "there is no reason for any individual to have a computer in their home." He later insisted the quote was taken out of context and that he simply meant he could not envision a day when computers would run people's lives. Born in Bridgeport, Conn., Olsen grew up in the neighboring town of Stratford. His father designed machine tools and Olsen and his brothers spent hours tinkering with gadgets in the family basement. After being drafted during World War II, Olsen attended the Navy's electronics school, where he learned how to maintain radars, sonars and navigation systems. He went on to earn undergraduate and masters degrees in engineering from the Massachusetts Institute of Technology. At MIT, Olsen worked in the university's Lincoln Laboratory, a federally funded research center created in 1951 to develop technology to improve the nation's air defense system. That technology, powered by MIT's advanced Whirlwind computers, grew into the Air Force's Semi-Automatic Ground Environment defense system, which was used to track and intercept enemy aircraft. One of Olsen's roles at Lincoln Laboratory was to serve as a liaison with IBM, a major contractor on the project. Olsen also worked on Lincoln Lab's TX-2 computer, which helped break new ground in computer-aided drafting. In 1957, Olsen teamed with MIT colleague Harlan Anderson to start Digital Equipment Corp. with $70,000 from American Research and Development, an early venture capital firm. The company was headquartered in an old wool mill in Maynard, Mass. DEC named its first computer the PDP-1, for Programmed Data Processor. But it was the PDP-8, which was introduced in 1965 and became a building block for computer systems made by other companies, that really established minicomputers as a major new industry. The PDP-11 — and later DEC's Virtual Address eXtension, or VAX, series — offered a serious alternative to IBM's central mainframe approach. By the mid-1980s, many other companies had tried to enter the business. Digital was also a pioneer in the use of networking technology to link its computers together and enable DEC engineers around the world to communicate electronically almost instantly. DEC's innovative machines helped bring computers out from glass-enclosed rooms inside big corporations, where they were operated by men in white lab coats, and made them accessible to small and medium-sized operations and even individual users. "The computers we built were of a cost and size that they brought computing down a level," said Bell, now a principal researcher in Microsoft Corp.'s Silicon Valley Research Group. DEC computers also trained and influenced many key players in the technology industry. Microsoft co-founders Bill Gates and Paul Allen used the PDP-10 to create the first version of the BASIC programming language for a personal computer. And Dave Cutler, who developed several key operating systems for DEC, went on to develop the Windows NT and Azure operating systems for Microsoft. For many years, the company's sophisticated technology drove rapid corporate growth and inspired deep loyalty. That growth came even as Olsen discouraged his salesmen from selling products that customers didn't need and shied away from traditional advertising, convinced that good products would sell themselves. In 1986, Fortune Magazine called Olsen "America's most successful entrepreneur." By the late 1980s, DEC had more than 120,000 employees worldwide. Sales peaked at $14 billion in 1992. According to Edgar Schein, an emeritus professor at the MIT Sloan School of Management and author of "DEC is Dead, Long Live DEC," Olsen had a distinctive management philosophy. His corporate culture valued creativity, ingenuity and open communication. And while he had a legendary temper and demanded top-notch work, Olsen empowered his employees with enormous freedom and responsibility. "Ken Olsen built a company that encouraged innovation and rewarded people with good ideas," said Win Hindle, a former DEC senior vice president who spent 32 years at the company. Olsen was also fiercely loyal to his employees and he abhorred the prospect of layoffs. Dan Tymann, executive vice president of Gordon College in Wenham, Mass., where Olsen was a trustee, said Olsen's management style reflected a devout Christian faith. Olsen constantly implored his employees to "do the right thing," Tymann said. Digital's fortunes had begun to decline by the early 1990s. The company was late to recognize the growing popularity of smaller personal computers and desktop workstations for business use. DEC also resisted the market's shift away from proprietary technology to open systems, including PCs powered by Intel microprocessors and generic servers running UNIX software. "Olsen continued to believe in innovation while the market became more of a commodity market," Schein said. "People wanted simpler, cheaper desktop computers, while DEC continued to produce sophisticated computers for the technical market." Even as DEC tried to catch up with new products, including a line of personal computers, it never regained its footing. The company posted its first quarterly loss in 1990. Faced with struggling product lines, Olsen had no choice but to start cutting Digital's work force through buyouts, early retirements and eventually layoffs. In 1992, Olsen left the company at the request of the board. Robert Palmer, a DEC vice president, took over and set about trying to turn things around. But the heyday of the minicomputer — and Digital Equipment Corp. — was over. In 1998, Compaq Computer Corp. bought what was left of DEC for $9.6 billion. Four years later, Compaq and the remnants of DEC were acquired by Hewlett-Packard Co. A memorial service at Gordon College is set for May 14.

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