April 2011 - mathfun@mailman.xmission.com

[math-fun] Balancing chemical equations by hand
by Marc LeBrun 24 Apr '11

24 Apr '11

Can anyone come up with a nice way to balance chemical equations manually? All the web seems to offer is either vague "fiddle around until it works" or the nuclear option "translate into a simultaneous linear system and solve". Is there anything in between? It need not be theoretically optimal, just easy to apply by hand to small solvable cases. I'm imagining a well-defined procedure repeatedly "adjusting" coefficients until "done", then dividing out their common factor, perhaps akin to an n-D raster line drawing algorithm that somehow manages to hill climb onto a scaled solution. It should be more clever than, say, mindlessly trying all the possible cases in some fixed order, yet stay grounded in the problem domain. It might even be "morally equivalent" to Gaussian elimination but performed directly on the chemical equations. Longhand division is kind of like this. There's a little eyeballing and maybe even some backtracking estimating the digits, but it's a reasonably effective way to arrive at the answer by hand. Crunching determinants for simple chemistry is analogous to using Newton's method on everyday division problems. Any ideas?

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[math-fun] Garbage collection considered a 'bottleneck'
by Henry Baker 24 Apr '11

24 Apr '11

FYI -- This article amused me, because I had assumed people were generally happy with web pages that were prone to seizures. The whole GC/noGC debate seems to ebb and swell with each order of magnitude increase in memory. How many papers have been written by now about how -- if you just have enough memory -- you won't need garbage collection anymore. Ask any New York City resident circa the late 1970's about how well that idea works out! Maybe I need to go back to doing GC research again... http://java.dzone.com/terracotta-releases-bigmemory Terracotta Says 'Goodbye Java GC, Hello BigMemory' Submitted by Mitchell Pronsc... on Tue, 2010/09/14 - 7:18am Last month, Terracotta told DZone that they were working on a module for Ehcache and Hibernate users that would *** bypass the Java garbage collection bottleneck ***. The severely limited heap size allowed by the Java garbage collector (if you don't want to have significant performance issues) is a problem that has plagued the industry for years. Terracotta believes that today's memory requirements are finally forcing many to address this issue, and now Terracotta has their own answer - BigMemory. DZone interviewed Terracotta CEO Amit Pandey about his company's solution to the GC problem in Java. BigMemory is a snap-in module that provides off-heap memory storage. It is a pure Java solution that works with all JVMs and bypasses the Java garbage collector. You don't even need to be in distributed caching mode to use BigMemory, but it works with both standalone and distributed caches. Like most Ehcache features, it requires only a few lines of configuration. BigMemory can also handle hundreds of millions of objects. Here are the two steps needed to implement BigMemory (when it is released) in Ehcache: 1. Update the Java classpath and startup arguments to include the new ehcache-2.3 jar and allocate sufficient memory for the BigMemory off-heap store. 2. Update the cache configuration in ehcache.xml to set the size of the BigMemory off-heap store. <ehcache> <cache name="mycache" maxElementsInMemory="10000" overflowToOffHeap="true" maxMemoryOffHeap="4G"/> </ehcache> "People have been trying to make better garbage collectors for years and years. *** Sun/Oracle have been putting a lot of money into solving this problem, but they're only getting incremental improvements." *** Terracotta, Pandey says, is able to solve the problem more elegantly because they they "own the cache." So they decided to write their own memory manager and *** bypass Java GC completely, freeing developers from difficult and time consuming GC tuning. *** Currently in the late beta stages, Pandey told DZone that the responses to BigMemory were very positive from early beta users. Testing of the memory manager revealed that the responses to increased memory were fast and flat, meaning that it could be counted on to meet performance requirements in a Service Level Agreement. The greater memory utilization with BigMemory also allows server consolidation. Terracotta has tested heap sizes up to 350GB without running into a wall like the 2GB threshold in Java GC. Pandey says Terracotta is avoiding many of the problems (fragmentation, poor memory management techniques, etc.) that others have encountered when they tried to build their own memory manager. Pandey says the tagline for BigMemory will be: "Use all the memory you need to get blazingly fast performance." BigMemory will be available for Ehcache and Hibernate users in October of this year.

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Re: [math-fun] How can I accurately calculate (p/q)^n mod 1, for large n
by Dan Asimov 23 Apr '11

23 Apr '11

Yes, Julian was right, in fact what I was calculating was Floor[100*Mod[(3/2)^k,1]] for long sequences of consecutive k. the error was just a typo in the math-fun query. --Dan Sometimes the brain has a mind of its own.

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[math-fun] Fwd: How can I accurately calculate (p/q)^n mod 1, for large n
by Bill Gosper 23 Apr '11

23 Apr '11

Oops, luckily, Julian was eavesdropping! ---------- Forwarded message ---------- From: Julian Ziegler Hunts <julianj.zh(a)gmail.com> Date: Sat, Apr 23, 2011 at 8:40 PM Subject: Re: [math-fun] How can I accurately calculate (p/q)^n mod 1, for large n To: billgosper(a)gmail.com > Timing[Do[10^16*Floor[Mod[1.5^k, 1]], {k, 99999}]] What? That's always 0. Mod[x,1] is ≥0 and <1, so Floor[Mod[x,1]]=0 for all x. The two methods should never differ, since even the numerics know that 0≤Mod[x,1]<1. Maybe you want Floor[10^16*Mod[1.5^k,1]], which diverges from the exact computation at k=21, or, with 10000 (decimal) digits of precision, at k=56758. Julian On Sat, Apr 23, 2011 at 7:32 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > > *From: * > > At 05:20 PM 4/23/2011, Dan Asimov wrote: > >Maybe I should be more specific. > > > > >Mathematica can do a loop of length 10000, > >say calculating > > > > 100*Floor[Mod[(3/2)^k,1]] > > > > >for 10001 <= k < 20000, > > > >but when I try to do something like that for > > >a loop of size 100,000 or 1,000,000, it seems > >to take forever (or hang, I'm not sure which). > > > >--Dan > > > It's computing the numerators and denominators, e.g., 2^100000, exactly. > > In[622]:= Timing[Do[If[10^16*Floor[Mod[(3/2)^k, 1]] != > 10^16*Floor[Mod[1.5^k, 1]],Print[k]], {k, 99999}]] > > Out[622]= {59.0911, Null} > > About 50 times faster: > > In[623]:= Timing[Do[10^16*Floor[Mod[1.5^k, 1]], {k, 99999}]] > > Out[623]= {1.21972, Null} > > Or 60 this way: > In[625]:= Block[{t = 10^16},Timing[Do[t*Floor[Mod[1.5^k, 1]], {k, 99999}]]] > > Out[625]= {0.997455, Null} > > At some point, the two methods will probably differ by 1 (in 10^16). > --rwg > (Boy, Neil must be busy, not to have beaten me to this one.) >

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[math-fun] OEIS movie
by Stuart Anderson 23 Apr '11

23 Apr '11

Nice, graphs of log scaled OEIS sequences static animated, with soundtrack by tonydnoe; The 30 second version; http://www.youtube.com/watch?v=wB0EeBmkrY0 The full 8 minute version; http://www.youtube.com/watch?v=LCWglXljevY cheers, Stuart

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[math-fun] How can I accurately calculate (p/q)^n mod 1, for large n
by Bill Gosper 23 Apr '11

23 Apr '11

*From: * At 05:20 PM 4/23/2011, Dan Asimov wrote: >Maybe I should be more specific. > >Mathematica can do a loop of length 10000, >say calculating > > 100*Floor[Mod[(3/2)^k,1]] > >for 10001 <= k < 20000, > >but when I try to do something like that for >a loop of size 100,000 or 1,000,000, it seems >to take forever (or hang, I'm not sure which). > >--Dan It's computing the numerators and denominators, e.g., 2^100000, exactly. In[622]:= Timing[Do[If[10^16*Floor[Mod[(3/2)^k, 1]] != 10^16*Floor[Mod[1.5^k, 1]],Print[k]], {k, 99999}]] Out[622]= {59.0911, Null} About 50 times faster: In[623]:= Timing[Do[10^16*Floor[Mod[1.5^k, 1]], {k, 99999}]] Out[623]= {1.21972, Null} Or 60 this way: In[625]:= Block[{t = 10^16},Timing[Do[t*Floor[Mod[1.5^k, 1]], {k, 99999}]]] Out[625]= {0.997455, Null} At some point, the two methods will probably differ by 1 (in 10^16). --rwg (Boy, Neil must be busy, not to have beaten me to this one.)

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Re: [math-fun] How can I accurately calculate (p/q)^n mod 1, for large n
by Henry Baker 23 Apr '11

23 Apr '11

I seem to recall that Pari GP is pretty fast at this sort of thing. I haven't tried it in a while, but it's easly accessible at: http://pari.math.u-bordeaux.fr/ At 05:20 PM 4/23/2011, Dan Asimov wrote: >Maybe I should be more specific. > >Mathematica can do a loop of length 10000, >say calculating > > 100*Floor[Mod[(3/2)^k,1]] > >for 10001 <= k < 20000, > >but when I try to do something like that for >a loop of size 100,000 or 1,000,000, it seems >to take forever (or hang, I'm not sure which). > >--Dan > >Sometimes the brain has a mind of its own.

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Re: [math-fun] How can I accurately calculate (p/q)^n mod 1, for large n
by Dan Asimov 23 Apr '11

23 Apr '11

Maybe I should be more specific. Mathematica can do a loop of length 10000, say calculating 100*Floor[Mod[(3/2)^k,1]] for 10001 <= k < 20000, but when I try to do something like that for a loop of size 100,000 or 1,000,000, it seems to take forever (or hang, I'm not sure which). --Dan Sometimes the brain has a mind of its own.

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[math-fun] How can I accurately calculate (p/q)^n mod 1, for large n
by Dan Asimov 23 Apr '11

23 Apr '11

Short of using a multi-precision package, is there some hack for calculating high powers of a rational number mod 1, to at least 3 decimal places (but ideally more like 10). Mma seems to be able to do one such calculation quickly, but takes forever, if not longer, in a large loop. So I'm wondering about shortcuts. --Dan Sometimes the brain has a mind of its own.

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[math-fun] Mars Discovery: Dry Ice Lake
by Henry Baker 22 Apr '11

22 Apr '11

FYI -- Wow! A frozen reservoir of dry ice (CO2) as big as Lake Superior at the south pole of Mars. Maybe Earth might have such a reservoir at its south pole, as well? http://www.huffingtonpost.com/2011/04/21/mars-discovery-dry-ice-lake-dust-b… Mars Discovery: Dry Ice Lake Suggests Planet Used To Have Intense 'Dust Bowl' Climate By ALICIA CHANG 04/21/11 03:35 PM ET AP LOS ANGELES -- Think Mars today is a hostile place? It was worse 600,000 years ago, according to new research that suggests the planet had a dustier, stormier atmosphere. "It was an unpleasant place to hang out," said lead researcher Roger Phillips of the Southwest Research Institute. He said Mars' climate was probably a lot like the American Dust Bowl of the 1930s Â but a lot worse. The evidence comes from the discovery of a huge underground reservoir of dry ice, or frozen carbon dioxide, at its south pole Â much more than scientists realized. They suspect some of that store of carbon dioxide was once in Mars' atmosphere, making it denser. In the recent geologic past, when Mars' axis tilted, sunlight reached the southern polar cap, melting some of the frozen carbon dioxide. This release would have made the atmosphere thicker and caused more dust to loft into the air, creating severe storms. Other times, carbon dioxide cycled back into the ground as part of a seasonal cycle. There is an upside to that stormier climate: The thicker atmosphere back then meant there were more regions on the planet where liquid water probably existed. Water is considered an essential ingredient for life. Still, "it was not the balmy, tropical Mars" that existed even billions of years earlier, Phillips said. Mars today is frigid, arid and constantly bombarded by lethal radiation. Its atmosphere, made up mostly of carbon dioxide, is many times thinner than the top of Mount Everest on Earth. In fact, the Martian atmosphere is less than 1 percent of Earth's. The red planet wasn't always this unforgiving. A maze of gullies, canyons and river channels on the surface points to a warmer and wetter past very early in the planet's history. The underground dry ice deposit, roughly the size of Lake Superior, was discovered using ground-piercing radar aboard the NASA Mars Reconnaissance Orbiter designed to probe below the crust. Researchers estimate it represents 30 times more carbon dioxide than previously believed. Its presence may help explain how most of the Martian atmosphere disappeared. Advertisement "It really is a buried treasure," said Jeffrey Plaut of the NASA Jet Propulsion Laboratory who was part of the discovery team reporting online Thursday in the journal Science. "We found something underground that no one else realized was there." Though the newfound store sounds like a lot, it's only enough carbon dioxide to double the mass of the feeble Martian atmosphere if released Â not enough to warm up the planet substantially or allow water to pool. "The atmosphere would still be quite thin and would not have the density necessary to warm things up enough to have liquid water stable on the surface," said Peter Thomas of Cornell University who had no role in the mission. The mystery of what happened to Mars' atmosphere has long intrigued scientists. NASA plans to explore the upper atmosphere and study how gases are lost to space with a new spacecraft in 2013.

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