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Re: [math-fun] Curved space paradox
by Don Reble 19 Feb '18

19 Feb '18

> Suppose we have a 3-dimensional space of constant, negative Gaussian > curvature. > > For some large enough R, construct the locus of points at distance R > from a fixed center C. ... an ordinary sphere. ... > Now, this sphere must itself have constant Gaussian curvature. Yes. > Select a point P on this sphere, and begin drawing circles of > increasing radius r on the sphere centered at P. > I mean r to be measured along the surface of the sphere, not straight > through the hyperbolic 3-space in which the sphere is embedded. > ...the circumference of a circle, centered at P, changes with its > radius r. For small r, it looks like 2 pi r. But as r gets larger, the > negative curvature of the ambient hyperbolic space begins to make > itself felt, and the circumference goes above 2 pi r... No. But it would, if r measures the radius that goes straight through the hyperbolic 3-space. > (If it doesn't, you didn't make R big enough -- that is exactly > what I meant by "large enough R" above.) Well: if the sphere is instead a horosphere, then it's surface is Euclidean, and the circumference is 2 pi r. If a hypersphere, the surface is negatively curved, and the circumference is above 2 pi r. A horo- or hyper-sphere has constant Gaussian curvature, but isn't a locus of points at distance R from a fixed center. --- Hmm... Let theta be the angle subtended by that curved radius at the sphere center C. Then the circumference is 2 pi r sin(theta) / theta regardless of the space's (uniform) curvature. -- Don Reble djr(a)nk.ca

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Re: [math-fun] a numerical approximation close to 7 digit pi
by Dan Asimov 19 Feb '18

19 Feb '18

----- 355/113 is 6.57 decimal digits ----- Please explain how the / is .57 digits. (More seriously: Where does the 6.57 come from?) —Dan

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Re: [math-fun] a numerical approximation close to 7 digit pi
by Don Reble 17 Feb '18

17 Feb '18

Then there are the Heegner approximations, log(744 + 96^3) / sqrt(19) 7.7 digits log(744 + 960^3) / sqrt(43) 13.9 log(744 + 5280^3) / sqrt(67) 18.4 log(744 + 640320^3) / sqrt(163) 31.1 which AFAIK might go by some other name. -- Don Reble djr(a)nk.ca

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[math-fun] a numerical approximation close to 7 digit pi
by Bill Gosper 16 Feb '18

16 Feb '18

On 2018-02-16 14:54, françois mendzina essomba2 via math-fun wrote: > Hello > > I noticed that: > > (9*(34+(3^7)/(10^6)))^(1/5) =3.14159264992426 Log[23 + 25/9 (√8 - 25/9)] = 3.141592653489906 (ries) --rwg

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[math-fun] Analog arithmetic saves power?
by Henry Baker 16 Feb '18

16 Feb '18

Flash memories have been storing 2-3 bits/cell for several years now, which requires distinguishing 4-8 different levels when reading. Perhaps it's time to reconsider using *quatenary* (base 4) or *octal* (base 8) arithmetic in computer arithmetic circuits? I hadn't realized that A/D converters were so simple and might save so much power (and perhaps even chip space?). https://news.mit.edu/2018/chip-neural-networks-battery-powered-devices-0214 MIT researchers have developed a special-purpose chip that increases the speed of neural-network computations by three to seven times over its predecessors, while reducing power consumption 93 to 96 percent. That could make it practical to run neural networks locally on smartphones or even to embed them in household appliances. New chip reduces neural networks' power consumption by up to 95 percent, making them practical for battery-powered devices. Larry Hardesty | MIT News Office February 13, 2018 Most recent advances in artificial-intelligence systems such as speech- or face-recognition programs have come courtesy of neural networks, densely interconnected meshes of simple information processors that learn to perform tasks by analyzing huge sets of training data. But neural nets are large, and their computations are energy intensive, so they're not very practical for handheld devices. Most smartphone apps that rely on neural nets simply upload data to internet servers, which process it and send the results back to the phone. Now, MIT researchers have developed a special-purpose chip that increases the speed of neural-network computations by three to seven times over its predecessors, while reducing power consumption 94 to 95 percent. That could make it practical to run neural networks locally on smartphones or even to embed them in household appliances. "The general processor model is that there is a memory in some part of the chip, and there is a processor in another part of the chip, and you move the data back and forth between them when you do these computations," says Avishek Biswas, an MIT graduate student in electrical engineering and computer science, who led the new chip's development. "Since these machine-learning algorithms need so many computations, this transferring back and forth of data is the dominant portion of the energy consumption. But the computation these algorithms do can be simplified to one specific operation, called the dot product. Our approach was, can we implement this dot-product functionality inside the memory so that you don't need to transfer this data back and forth?" Biswas and his thesis advisor, Anantha Chandrakasan, dean of MIT's School of Engineering and the Vannevar Bush Professor of Electrical Engineering and Computer Science, describe the new chip in a paper that Biswas is presenting this week at the International Solid State Circuits Conference. Back to analog Neural networks are typically arranged into layers. A single processing node in one layer of the network will generally receive data from several nodes in the layer below and pass data to several nodes in the layer above. Each connection between nodes has its own "weight," which indicates how large a role the output of one node will play in the computation performed by the next. Training the network is a matter of setting those weights. A node receiving data from multiple nodes in the layer below will multiply each input by the weight of the corresponding connection and sum the results. That operation -- the summation of multiplications -- is the definition of a dot product. If the dot product exceeds some threshold value, the node will transmit it to nodes in the next layer, over connections with their own weights. A neural net is an abstraction: The "nodes" are just weights stored in a computer's memory. Calculating a dot product usually involves fetching a weight from memory, fetching the associated data item, multiplying the two, storing the result somewhere, and then repeating the operation for every input to a node. Given that a neural net will have thousands or even millions of nodes, that's a lot of data to move around. But that sequence of operations is just a digital approximation of what happens in the brain, where signals traveling along multiple neurons meet at a "synapse," or a gap between bundles of neurons. The neurons' firing rates and the electrochemical signals that cross the synapse correspond to the data values and weights. The MIT researchers' new chip improves efficiency by replicating the brain more faithfully. In the chip, a node's input values are converted into electrical voltages and then multiplied by the appropriate weights. Summing the products is simply a matter of combining the voltages. Only the combined voltages are converted back into a digital representation and stored for further processing. The chip can thus calculate dot products for multiple nodes -- 16 at a time, in the prototype -- in a single step, instead of shuttling between a processor and memory for every computation. All or nothing One of the keys to the system is that all the weights are either 1 or -1. That means that they can be implemented within the memory itself as simple switches that either close a circuit or leave it open. Recent theoretical work suggests that neural nets trained with only two weights should lose little accuracy -- somewhere between 1 and 2 percent. Biswas and Chandrakasan's research bears that prediction out. In experiments, they ran the full implementation of a neural network on a conventional computer and the binary-weight equivalent on their chip. Their chip's results were generally within 2 to 3 percent of the conventional network's. "This is a promising real-world demonstration of SRAM-based in-memory analog computing for deep-learning applications," says Dario Gil, vice president of artificial intelligence at IBM. "The results show impressive specifications for the energy-efficient implementation of convolution operations with memory arrays. It certainly will open the possibility to employ more complex convolutional neural networks for image and video classifications in IoT [the internet of things] in the future."

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[math-fun] a numerical approximation close to 7 digit pi
by françois mendzina essomba2 16 Feb '18

16 Feb '18

Hello I noticed that: (9*(34+(3^7)/(10^6)))^(1/5) =3.14159264992426

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Re: [math-fun] Locker puzzle
by Bill Gosper 15 Feb '18

15 Feb '18

I slighted the kid. He waited several days before working on it. His father wrote: Hi Bill, Just wanted to add that when Gabe was explaining the answer to me, he said "I can't remember if Bill said that all the lockers were opened at the beginning or not. If they were, then it's the perfect squares, but if they weren't then it's everything BUT the perfect squares." Gabe then wrote: welllllllll I had heard of it before, but I can explain. first, think about that a locker is swi[t]ched every time one of its factors goes by. that means the numbers w/ an odd no. of factors will be open. and the numbers w/ odd numbers of factors are squares, and that gives us our answer. ------- So DanA was scooped! Gabe's mom wrote: Ok, after much prodding...it's from 'How to be a Math Genius', by Dr. Mike Goldsmith. See the problem and answer below. Note that solution doesn't have an explanation, Gabe said he figured that out by himself. ----- Problem: Prison break At the prison, 20 prisoners are locked in 20 cells. The 20 prison guards looking after them have a strange way of locking up. The first guard unlocks all the cells. The second guard then locks every second door (2, 4, 6, etc.). The third turns the key in every third door, locking it if it is unlocked and unlocking it if it is locked. The fourth guard turns the key in every fourth door, and so on until all 20 guards have left. Which cells remain open allowing the prisoners to escape? Can you spot a pattern in these numbers? Answer: Prison break Doors 1, 4, 9 and 16 remain open, so 4 prisoners escape. These are all square numbers. Knowing this pattern, you can quickly work out the answer for 50 guards and 50 prisoners, or even 100. ----------- The young Neil Bickford likewise had a huge math collection, and knew it by heart. Conjecture: Lame books are good for smart kids. I'm reminded of that photo of the very young Terry Tao reading C.D. Olds's Continued Fractions. Gabe only got 620 on the math SAT, and asked what "normal" 7 yr olds got. Stay tuned for his AMC10 results. --rwg On Tue, Feb 13, 2018 at 8:49 AM, Bill Gosper <billgosper(a)gmail.com> wrote: > On 2018-02-04 06:31, Hans Havermann wrote: > >Dan Asimov wrote > Suppose there is a school with an infinity of lockers, numbered 1, 2, 3, > .... > > Initially all lockers are closed. > > Definition: A locker is "switched" if it is changed from shut to open, or > vice versa. > > At 1/2 minute to noon, every 2nd locker 2, 4, 6, ... is switched. > > at 1/3 minute to noon, every 3rd locker 3, 6, 9, ... is switched. > > Or in general: For each positive integer n >= 2: > > * At 1/n of a minute to noon, every nth locker n, 2n, 3n, ... is > switched. > > No other changes are made to the lockers. > > Question: > --------- > > > Which lockers are open at noon? > > Unfortunately, Dan failed to open all the lockers at 1 minute to noon, > > > > > The lockers that are open should be the ones where the number of > > divisors of their locker number less one (because we don't switch for > > divisibility by 1) is odd. > > I think Dan should've opened 'em all at 1 minute to noon, > > >Equivalently, the lockers that are closed > > are the ones where the number of divisors of their locker number is > > odd. > So, e,g., #1 never opens, and the answer is the non-squares, no? > (A certain 7yr old missed this and told me "The square numbers!" > without a word to his dumbfounded parents.) > --rwg > > > > DivisorSigma[0,Range[25]] > > {1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5,2,6,2,6,4,4,2,8,3} > > > > Seemingly, an odd number of divisors happens for squares. This must be > > so because in non-squares, for every divisor greater than the > > square-root there is a complement smaller than the square-root. But in > > squares the square-root is its own complement. > > > > >

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[math-fun] a very interesting page (with lots of formulas)
by Simon Plouffe 15 Feb '18

15 Feb '18

Hello, I stumbled on this page : http://www-elsa.physik.uni-bonn.de/~dieckman/InfProd/InfProd.html My impression is that many of these results are new, especialy the ones with the LerchPhi function. A nice and impressive piece of work. Best regards, Simon Plouffe

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[math-fun] The sin halfangle formula
by Bill Gosper 14 Feb '18

14 Feb '18

is always(?) taught as √(1-cos x)/√2. This fails outright for negative x, and numerically for small positive x. And impertinently imposes cosine for sine. Exercise: Find a nice formula for sin(x/2). Only in terms of sin x, good for -π/2 < x < π/2 (at least), and stable near 0. I had a he<<uva time rederiving it. I got a triangle wave when I plotted what I remembered. I was going to send you guys "Hey look at this triangle wave." After a slow, geriatric rederivation, I got the same bleeping triangle wave! The segments are imperceptibly curved! --rwg Plot this spoiler for -2π<x<2π: StringReverse@"\)]\)]\)]x[niS + 1\([xoBtrqS\* + ]\)]x[niS - 1\([xoBtrqS\*\( ,\)]x[niS\([xoBnoitcarF\*\(\!" Then plot the difference from sin(x/2).

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Re: [math-fun] Origin information
by Keith F. Lynch 14 Feb '18

14 Feb '18

Dan Asimov <dasimov(a)earthlink.net> wrote: > I don't know about internet protocols and security, but: > How feasible would it be to ensure that every post or ad to any > website anywhere (or at least in the U.S.) is correctly identified > with its geographic location (to some reasonable precision)? I'm no expert, but I can't see how it could ever be feasible. This email is a "post to a website" in the sense that all posts to this list are archived on a website (xmission.com) And I'm evading my location without even trying. First, KeithLynch.net is hosted in Minnesota, a state I've never even visited. Second, I'm not really posting from KeithLynch.net, I'm posting from Panix, which is in New York City. But I'm not actually in NYC. I've visted that city, most recently in 2004, but I've never lived or worked there. I'm actually in Virginia. A friend of mine who is more skilled at such things than I am was quite active on the net for several months in 2012, and the US government couldn't figure out his location, despite trying really hard. He was a federal fugitive at the time. They only caught him when he was injured in a bike crash and was hospitalized. In principle, though not in practice, the net could be completely rebuilt on different principles. But I doubt that would help. Even with the 19th century telegraph lines, it would have been trivial to link a telegraph sounder to a telegraph key, to relay messages from one line to another. Similarly with telephones. Similarly with letters -- just get someone to forward a letter to someone else. Public key cryptography can be used (until quantum computers break it) to prove who wrote a message. (Unless someone steals a private key, or beats it out of its owner.) But keys are associated with people or organizations. I can't see any way to associate one with a location, i.e. a key that anyone visiting my home town will somehow learn, but will forget the moment they leave.

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