April 2018 - mathfun@mailman.xmission.com

Re: [math-fun] Miss those IBM/DEC blinking lights?
by Henry Baker 08 Apr '18

08 Apr '18

Unlike the DEC blinking lights, the IBM blinking lights were trivially replaceable -- simply grasp one between your thumb and index finger and the incandescent bulb would pop out, and you could then push back a new one into place. There was also a "lamp test" button on the console which you could press to light up *all* of the lights, to see if any were burned out. If I recall correctly, I think that the real-time clock was one of the registers always displayed in the console lights. One day, we noticed that one of the bits of this clock had its light burned out. After a couple of seconds of contemplation, one of my associates swapped the light from the highest order bit of the real-time clock with the light that failed. He correctly computed that we would never see that high order bit turn on (except via the lamp test button) during the lifetime of this computer. He was correct! At 09:17 AM 4/8/2018, Victor Miller wrote: >In the summer of 1966 I was working at the Rome (NY) Air Development >Center, on a Univac 1218 computer (a militarized verson of the 418). It >was an 18 bit ones-complement machine. On the console were two banks of >lights each in a 6 x 3 format. The machine had a primitive time sharing >system, and the lights on the console normally displayed a pattern which >was I L (for idle loop). I found out where the pattern was stored and one >day I changed it to read FU. The Univac engineer came in, saw it, and said >"things aren't that bad, arey they?". I just stared at him, and he said, >well, I guess they are. > >On Sun, Apr 8, 2018 at 10:11 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> Robert Baruch is building a 32-bit RISC-V processor >> out of 3.3v MSI logic chips. >> >> This cool video describes his circuit board for a >> single 32-bit register (out of 32 such registers) >> of this modern RISC architecture. He uses two >> standard PCIE-style busses & connectors for his >> sink + 2 source busses. >> >> In 1950's/1960's style, the register comes with its >> own set of LED blinking lights! (BTW, the LED's >> cost more and take substantially more power than >> the register itself.) >> >> For the cognescenti: I find it curious that he >> built a full 32-bit register board instead of >> a *bit slice* of the entire register-cum-ALU >> a la the PDP-8 or the old AMD bit-slice chips. >> >> https://www.youtube.com/watch?v=yLs_NRwu1Y4

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[math-fun] Miss those IBM/DEC blinking lights?
by Henry Baker 08 Apr '18

08 Apr '18

Robert Baruch is building a 32-bit RISC-V processor out of 3.3v MSI logic chips. This cool video describes his circuit board for a single 32-bit register (out of 32 such registers) of this modern RISC architecture. He uses two standard PCIE-style busses & connectors for his sink + 2 source busses. In 1950's/1960's style, the register comes with its own set of LED blinking lights! (BTW, the LED's cost more and take substantially more power than the register itself.) For the cognescenti: I find it curious that he built a full 32-bit register board instead of a *bit slice* of the entire register-cum-ALU a la the PDP-8 or the old AMD bit-slice chips. https://www.youtube.com/watch?v=yLs_NRwu1Y4

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[math-fun] Vacationing in France
by David Wilson 08 Apr '18

08 Apr '18

The wife and I are thinking about a vacation to Paris, which is one of her bucket list items. The only people in France with whom I have correspondence are on seqfan, e.g. M. Gerard or Dr. Plouffe, others? I was hoping I might get some inside information on when, why and how to do Paris.

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Re: [math-fun] Gerrymandering
by Keith F. Lynch 08 Apr '18

08 Apr '18

Tom Knight <tk(a)csail.mit.edu> wrote: > If the region is borderless, how do I decide which district a new > person belongs in? By, for each district, summing the squares of the distances between the new person and every person in the district. Whichever district gives the smallest sum is the district the new person is assigned to. Of course there would be implicit borders, the boundaries between where a new person would be assigned to one district rather than another, but nobody needs to think in terms of them. Anyhow, these implicit boundaries would change whenever anyone moved, was born, or died. It might be interesting to study how such boundaries would move in response to such events, what kinds of shapes the boundaries would tend to have, and whether these boundaries can ever be counterintuitive or pathological. Not very practical in the US as it exists, since apparently the census department assigns everyone to a census block with explicit borders, rather than assigning everyone a point location. And not necessarily desirable even if possible, though arguably better than the current system which is vulnerable to gerrymandering. But then this is the math list, not the politics list. Interesting math problems can be inspired by all sort of things. Similarly, the question of what proportion of trees are visible in a forest, where there's a tree at every point x,y where x and y are integers and at no other points, where you're at the origin, and where a tree is visible iff there's no other tree directly between you and it, limit as the forest grows large, is of little relevance to actual forestry.

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[math-fun] An infinite product for the logarithm
by françois mendzina essomba2 07 Apr '18

07 Apr '18

(2/sqrt((x+1)/sqrt(x)+2))*(2/sqrt(sqrt((x+1)/sqrt(x)+2)+2))*(2/sqrt(sqrt(sqrt((x+1)/sqrt(x)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt((x+1)/sqrt(x)+2)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(sqrt((x+1)/sqrt(x)+2)+2)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(sqrt(sqrt((x+1)/sqrt(x)+2)+2)+2)+2)+2)+2))*... =(sqrt(x)*log(x))/(x-1); approximate values for any x greater than 1 for a product of ten terms. (1024)/(sqrt((x+1)/sqrt(x)+2)*sqrt(sqrt((x+1)/sqrt(x)+2)+2)*sqrt(sqrt(sqrt((x+1)/sqrt(x)+2)+2)+2)*sqrt(sqrt(sqrt(sqrt((x+1)/sqrt(x)+2)+2)+2)+2)*sqrt(sqrt(sqrt(sqrt(sqrt((x+1)/sqrt(x)+2)+2)+2)+2)+2)*sqrt(sqrt(sqrt(sqrt(sqrt(sqrt((x+1)/sqrt(x)+2)+2)+2)+2)+2)+2)* sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt((x+1)/sqrt(x)+2)+2)+2)+2)+2)+2)+2)*sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt((x+1)/sqrt(x)+2)+2)+2)+2)+2)+2)+2)+2)*sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt((x+1)/sqrt(x)+2)+2)+2)+2)+2)+2)+2)+2)+2)* sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt((x+1)/sqrt(x)+2)+2)+2)+2)+2)+2)+2)+2)+2)+2))=(sqrt(x)*log(x))/(x-1);

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Re: [math-fun] Gerrymandering
by Keith F. Lynch 07 Apr '18

07 Apr '18

Dan Asimov <dasimov(a)earthlink.net> wrote: > 2) refers to the fact that any simply-connected bounded region has > a well-defined "eccentricity" ? the eccentricity of the unique > best-approximating filled ellipse. Requiring that the eccentricity > lie below some maximum value makes the region "sufficiently circular". I for one have no idea how to find the best-approximating filled ellipse. Anyhow, why not just look at the ratio of the square of the perimeter to the enclosed area? Or maybe the ratio of the square of the perimeter of the convex hull to the original area? But I think you missed my point. My new approach is explicitly borderless, which is something I don't think anyone has looked at before, at least not in this list or anywhere else I've looked. Brent Meeker <meekerdb(a)verizon.net> wrote: > I don't think bad boundaries is the problem. In many cases > geometric proximity isn't a measure of common interests. Sure it > is for some local issues like getting pot holes fixed and managing > elementary schools. But for Congress critters? Engineers, > teachers, investors, doctors,...all living in the same area > may have quite different interests at the national level. You seem to be advocating syndicalism, in which people are grouped by occupation rather than location. That opens even more cans of worms than geography. How large is each occupation? For instance is "math teacher" the same occupation as "geography teacher"? Should "math teacher" be further subdivided into which branch of math they teach? What about people with multiple occupations or no occupation? And why occupation anyway, rather than, say, race, gender, age, net wealth, sexual orientation, IQ score, preferred temperature, urban vs. rural, favorite TV show, or countless other ways to categorize people. Maybe even political affiliation. Personally, I don't think anyone should ever represent a competent adult without the latter's explicit individual consent, revocable at any time for any reason. Just like hiring a lawyer. But I was hoping to discuss math, not politics. Henry Baker <hbaker1(a)pipeline.com> wrote: > Suppose that one had a very long thin state which forced all of the > districts into *line segments* -- i.e., each person lives "closer" > (under the chosen metric) to a state border than to any other person. > .... If we have enough population for 3 Representatives, then we > need to find 2 dividing lines, which provides for 2 sources of > non-uniqueness. You're not going to address my proposal at all? I think in the general case (e.g. the people are not uniformly or symmetrically distributed) there would always be a unique solution, even in your unlikely case of a one-dimensional state. Similarly if the state was in disconnected or oddly-shaped pieces. Just so long as the state border isn't so convoluted that it becomes meaningless to ask whether any person of non-zero size is inside or outside the state. Mike Speciner <ms(a)alum.mit.edu> wrote: > So measuring the "goodness" of a districting probably requires > various kinds of map data in addition to population data. Of > course, if we had a way of secure remote voting, some of these > considerations could go away. Good point. Thanks for recognizing that we don't. As the paper "On Trusting Trust" showed, no computer can be trusted unless you personally designed, built, and programmed it, from the logic gates on up. Of course there could be non-computerized remote voting. Thane Plambeck <tplambeck(a)gmail.com> wrote: > this is a good resource created (i believe) by moon duchin, a > mathematician at tufts .... > https://sites.tufts.edu/gerrymandr/resources/ Thanks. I see that there's no mention of anything like my idea there. Michael Kleber <michael.kleber(a)gmail.com> wrote: > It's worth remembering that congressional districts *must* be a > union of census tracts (since otherwise the decennial "population" > is undefined). Thanks. I had wrongly assumed that they kept track of the home location of each individual. Wikipedia says that census tracts are divided into census block groups, which in turn are divided into census blocks. Whichever is the smallest for which population numbers are available to the public could be used in my suggestion in place of individuals. Of course they'd be weighted in proportion to how many people they contained, and all the people would be assumed to be in the center of the tract, block, or whatever. That would have the advantage that when a person moved, nobody would have to calculate for which district the sum of the squares of the distances from that person's new location to each of the people in the district was smallest, but would have the disadvantage that there are explicit borders. I was hoping to keep it completely borderless, just to see if it was possible, and just to be different.

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Re: [math-fun] Geometry problem suggested by gerrymandering
by Dan Asimov 07 Apr '18

07 Apr '18

That's very cool, Gene, to have a physical method of solving the problem! I'm not worried (yet) about tiny regions forming, because they won't persist in minima of the total length function. —Dan ----- You may wish to add the additional condition that the areas must be equal. Otherwise, you can make all but one region be very tiny. Here's a physical solution. Fill the manifold with N mutually immiscible liquids with equal surface tensions between the liquids. Shake, and let surface tension minimize the total boundary. -- Gene On Saturday, April 7, 2018, 2:46:25 PM PDT, Dan Asimov <dasimov(a)earthlink.net> wrote: Consider a smooth surface M like the unit sphere S^2 or the square torus T^2 = R^2/Z^2. Question: ----- For which integers N > 0 can M be divided into N regions of positive area such that the total length of the regions' boundaries is a local minimum? ----- The question becomes interesting when N is big enough for the surface to be divided into N *simply connected* regions. (E.g., the torus cannot be cut into fewer than N >= 4 simply connected pieces. In that case the total boundary length (counting each edge only once) seems to be sqrt(8).) Can we find the optimal configuration for M=S^2 and M=T^2 for N <= 10 (and N >= 4 if M=T^2). Optimal meaning that the total length of the boundary is a global minimum. For further study: What is the minimum number of simply connected pieces that the surface of genus g can be cut into?) —Dan -----

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[math-fun] Geometry problem suggested by gerrymandering
by Dan Asimov 07 Apr '18

07 Apr '18

Consider a smooth surface M like the unit sphere S^2 or the square torus T^2 = R^2/Z^2. Question: ----- For which integers N > 0 can M be divided into N regions of positive area such that the total length of the regions' boundaries is a local minimum? ----- The question becomes interesting when N is big enough for the surface to be divided into N *simply connected* regions. (E.g., the torus cannot be cut into fewer than N >= 4 simply connected pieces. In that case the total boundary length (counting each edge only once) seems to be sqrt(8).) Can we find the optimal configuration for M=S^2 and M=T^2 for N <= 10 (and N >= 4 if M=T^2). Optimal meaning that the total length of the boundary is a global minimum. For further study: What is the minimum number of simply connected pieces that the surface of genus g can be cut into?) —Dan

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[math-fun] Gerrymandering
by Keith F. Lynch 06 Apr '18

06 Apr '18

It's been nearly four years since we discussed gerrymandering. I've been thinking about it lately, since it's been back in the news, and since I've come up with what I think is an original way to measure it. Last time we discussed it, the discussion was all about the borders, e.g. minimizing the variation in their curvature, or minimizing the ratio between the square of the perimeter and the area enclosed. But that's obviously not the right measure, since making a border frizzy on a centimeter scale would greatly change such a measure without changing the real amount of gerrymandering at all. My proposal is to start with a database listing, to high precision (e.g. 10 meters) the home location of everyone in the state. The gerrymandering in a district would be measured by summing the squares of the distances between each two people in the district. The gerrymandering in a state would be measured by summing the squares of the gerrymandering numbers for each district in the state. (Of course the number of districts is kept constant, and their populations are kept as equal as possible, i.e. the most populous district can have at most one more person than the least populous district.) There's probably no practical algorithm for finding the unique lowest score. But everyone would be free to draw boundaries however they liked, so long the populations were equal, and submit their scores. Whoever came up with the lowest score before the deadline would have their submission accepted. Simulated annealing would probably be a good way to find low-score redistricting plans. This is fairly CPU-intensive, but I think today even laptop computers could easily handle it. With N people in a district, there would be N*(N-1)/2 comparisons to make. Is census data available that gives everyone's home location to within ten meters? Of course all other information (name, age, income, citizenship, etc.) would be stripped for privacy reasons.

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Re: [math-fun] Gerrymandering
by Dan Asimov 06 Apr '18

06 Apr '18

It's occurred to me that two conditions might be enough to attain some kind of fair division: 1) Each region should be convex (or spherically convex on earth); and 2) Each region should be sufficiently circular. 2) refers to the fact that any simply-connected bounded region has a well-defined "eccentricity" — the eccentricity of the unique best-approximating filled ellipse. Requiring that the eccentricity lie below some maximum value makes the region "sufficiently circular". —Dan -----Original Message----- >From: "Keith F. Lynch" <kfl(a)KeithLynch.net> >Sent: Apr 5, 2018 7:52 PM >To: math-fun(a)mailman.xmission.com >Subject: [math-fun] Gerrymandering > >It's been nearly four years since we discussed gerrymandering. > >I've been thinking about it lately, since it's been back in the >news, and since I've come up with what I think is an original way >to measure it. > >Last time we discussed it, the discussion was all about the borders, >e.g. minimizing the variation in their curvature, or minimizing the >ratio between the square of the perimeter and the area enclosed. > >But that's obviously not the right measure, since making a border >frizzy on a centimeter scale would greatly change such a measure >without changing the real amount of gerrymandering at all. > >My proposal is to start with a database listing, to high precision >(e.g. 10 meters) the home location of everyone in the state. The >gerrymandering in a district would be measured by summing the squares >of the distances between each two people in the district. The >gerrymandering in a state would be measured by summing the squares of >the gerrymandering numbers for each district in the state. (Of course >the number of districts is kept constant, and their populations are >kept as equal as possible, i.e. the most populous district can have >at most one more person than the least populous district.) > >There's probably no practical algorithm for finding the unique lowest >score. But everyone would be free to draw boundaries however they >liked, so long the populations were equal, and submit their scores. >Whoever came up with the lowest score before the deadline would have >their submission accepted. > >Simulated annealing would probably be a good way to find low-score >redistricting plans. > >This is fairly CPU-intensive, but I think today even laptop computers >could easily handle it. With N people in a district, there would be >N*(N-1)/2 comparisons to make. > >Is census data available that gives everyone's home location to within >ten meters? Of course all other information (name, age, income, >citizenship, etc.) would be stripped for privacy reasons. > >_______________________________________________ >math-fun mailing list >math-fun(a)mailman.xmission.com >https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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