tag:blogger.com,1999:blog-67578052021-12-04T22:06:33.101ZKen's blogmostly on computers and mathematicsUnknownnoreply@blogger.comBlogger8457125tag:blogger.com,1999:blog-6757805.post-63821690465300261392021-12-04T22:06:00.000Z2021-12-04T22:06:00.244Z[owiwfzux] shuffling one bit<p>persons A, B, and C in separate rooms with point-to-point communication.</p> <ol><li>person C describes to person A how they intend to shuffle cards, e.g., 3 riffle shuffles.</li> <li>person A defines two decks, specifying the order of cards in each deck, and tells the information to person B.</li> <li>person B secretly randomly chooses one of person A's card orderings, constructs a deck with the chosen ordering, then gives the deck to person C. we go through this person B intermediary so that person A cannot secretly mark decks to tell them apart afterward.</li> <li>person C shuffles the received deck as declared in step 1, and gives the shuffled deck to person A.</li> <li>person A tries to determine which of the two decks person B gave to person C.</li> </ol> <p>I strongly suspect there are ways that person A can construct decks so that it is very difficult to shuffle them enough destroy the one bit of information that is encoded in them. you will need the full 7 riffle shuffles as proved by Diaconis. (practically perhaps more, if your riffle shuffles are not very good.)</p> <p>optionally omit step 1 to give person A less of an advantage.</p> <p>a "shuffling" step that person C can do to possibly make things difficult is the following: flip a coin. if and only if tails, reverse the order of the deck. (I am not aware of any way to do this quickly. perhaps cardistry magicians know of a way.) then, do other shuffles.</p> <p>variation: split the deck into 2. flip a coin. reverse one half or the other corresponding to the coin flip. then riffle shuffle.</p> <p>also consider person C <a href="/2019/07/rcvvgult-twisty-cube-password-generator.html">hand scrambling a speedcube</a> instead of shuffling a deck of cards. person B needs to be able to construct arbitrary (legal) Rubik's cube states given by person A.</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-41322426773190943952021-12-04T21:37:00.000Z2021-12-04T21:37:00.247Z[umgnxrbn] Xen networking on Debian bullseye<p>mostly following <a href="https://wiki.debian.org/Xen">Debian instructions for Xen</a>.</p><p>standard stuff added to /etc/network/interfaces:</p> <p>iface enp58s0f1 inet manual</p> <p>auto xenbr0<br> iface xenbr0 inet dhcp<br> bridge_ports enp58s0f1</p> <p>"ip a" does show xenbr0.</p> <p>(tangent: we would prefer to do "allow-hotplug" instead of "auto" to prevent boot from being slow if there is no ethernet cable plugged in, but allow-hotplug xenbr0 does not work (network does not come up; unsurprisingly udev cannot detect that there's a network cable plugged into "xenbr0"). with "auto", boot is slower, but not the full 5 minutes of delay as the boot message seems to suggest: boot continues after about 1 minute of delay.)</p> <p>somewhat nonstandard on this system is that NetworkManager is running: NetworkManager manages the wifi interface, and ifupdown (aka /etc/network/interfaces) manages ethernet. NetworkManager is supposed to avoid trying to manage interfaces in /etc/network/interfaces, as configured in the default /etc/NetworkManager/NetworkManager.conf:</p> <p>[main]<br> plugins=ifupdown,keyfile</p> <p>[ifupdown]<br> managed=false</p> <p>however, when bringing up a paraVM created with xen-create-image, we see this in journalctl:</p> <p>NetworkManager[68473]: <info> [1636093454.5580] settings: (vif7.0): created default wired connection 'Wired connection 1'</p> <p>and then later more references to vif and Wired connection 1. for some reason, NetworkManager is, I think, trying to bridge between "Wired connection 1" (which did not even exist!) and the vif for the VM. the solution is to prevent NetworkManager from touching vif devices:</p> <p>/etc/NetworkManager/conf.d/no-vif.conf :</p> <p>[keyfile]<br> unmanaged-devices=interface-name:vif*</p> <p>future work: let xenbr0 be a bridge to whichever network connection is active, ethernet or wifi.</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-32033649557383653012021-12-04T21:17:00.000Z2021-12-04T21:17:00.245Z[wluwvrvs] old style numerals with cfr-lm on Debian<p>one of the ways to get old style (lowercase) numerals in LaTeX is with the cfr-lm package.</p> <p>\usepackage{cfr-lm}</p> <p>On Debian (buster and bullseye), this package is in the texlive-fonts-extra package. however, installing the package with --no-install-recommends (or APT::Install-Recommends "false" in /etc/apt/apt.conf.d ) is insufficient; there are required dependencies among the package's Recommends (a bug).</p> <p>most of the additional needed packages can be tracked down with apt-file, but one error message is not so obvious:</p> <p>No file OMLlmm.fd.</p> <p>! LaTeX Error: This NFSS system isn't set up properly.</p> <p>the missing file is in the lmodern package with a file name without capitalization. somehow LaTeX knows to search for file names case-insensitively, even though its error messages give file names with case.</p> <p>lmodern: /usr/share/texmf/tex/latex/lm/omllmm.fd</p> <p>the complete set of packages needed for cfr-lm is</p> <p>apt install --no-install-recommends texlive-fonts-extra texlive-latex-recommended texlive-latex-extra texlive-fonts-recommended lmodern</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-7144311880379018382021-12-04T21:16:00.000Z2021-12-04T21:16:00.276Z[qlcpuedj] safe primes below many powers of 2<p>we give the largest safe prime below selected powers of 2. we also give additional safe primes below them until one which has 2 as a primitive root (generator). the exponents are <a href="/2021/04/lckoyipy-decimal-powers-of-2.html">rounded decimal powers of 2</a> (so the sequence is doubly exponential) (previously, <a href="/2021/07/oreozfqr-safe-primes-before-decimal.html">not rounding until the end which requires a large amount of floating point precision</a>.)</p> <p>the <a href="/2021/07/synhrdhu-artin-constant.html">amount of extra work required to find a safe prime with primitive root 2 is Artin's constant</a>.</p> <p>this work expands on <a href="/2011/01/cvogqzhd-some-large-safe-primes.html">previous work</a> and OEIS A181356.</p><a href="/2021/11/cvdvpjmf-safe-primes-below-doubly.html">also previously, safe primes below powers of 10.</a> <p>for example, in the list below, 2^3.1 ~= 8.57, and round(2^3.1) = 9, so 2^round(2^3.1) = 2^9 = 512. the largest safe primes less than 512 are 2^9-9 = 512-9 = 503 (which has least primitive root 5), 2^9-33 = 512-33 = 479 (which has least primitive root 13), and 2^9-45 = 512-45 = 467 (which has least primitive root 2), where we stop, having found a safe prime with primitive root 2. the offsets are -9, -33, and -45.</p> <p>future post (srzrbviv): source code and implementation discussion.</p> <p>currently, the recommendation for cryptographic security for Diffie-Hellman key exchange when using the integer discrete logarithm problem is safe prime moduli of size at least 2^11 = 2048 bits.</p> <p>2^round(2^1.4) = 2^3 : -1 -3<br> 2^round(2^1.5) = 2^3 : -1 -3<br> 2^round(2^1.6) = 2^3 : -1 -3<br> 2^round(2^1.7) = 2^3 : -1 -3<br> 2^round(2^1.8) = 2^3 : -1 -3<br> 2^round(2^1.9) = 2^4 : -5<br> 2^round(2^2.0) = 2^4 : -5<br> 2^round(2^2.1) = 2^4 : -5<br> 2^round(2^2.2) = 2^5 : -9 -21<br> 2^round(2^2.3) = 2^5 : -9 -21<br> 2^round(2^2.4) = 2^5 : -9 -21<br> 2^round(2^2.5) = 2^6 : -5<br> 2^round(2^2.6) = 2^6 : -5<br> 2^round(2^2.7) = 2^6 : -5<br> 2^round(2^2.8) = 2^7 : -21<br> 2^round(2^2.9) = 2^7 : -21<br> 2^round(2^3.0) = 2^8 : -29<br> 2^round(2^3.1) = 2^9 : -9 -33 -45<br> 2^round(2^3.2) = 2^9 : -9 -33 -45<br> 2^round(2^3.3) = 2^10 : -5<br> 2^round(2^3.4) = 2^11 : -9 -21<br> 2^round(2^3.5) = 2^11 : -9 -21<br> 2^round(2^3.6) = 2^12 : -17 -89 -149<br> 2^round(2^3.7) = 2^13 : -45<br> 2^round(2^3.8) = 2^14 : -161 -197<br> 2^round(2^3.9) = 2^15 : -165<br> 2^round(2^4.0) = 2^16 : -269<br> 2^round(2^4.1) = 2^17 : -285<br> 2^round(2^4.2) = 2^18 : -17 -1265 -1661<br> 2^round(2^4.3) = 2^20 : -233 -449 -989<br> 2^round(2^4.4) = 2^21 : -9 -285<br> 2^round(2^4.5) = 2^23 : -321 -729 -1101<br> 2^round(2^4.6) = 2^24 : -317<br> 2^round(2^4.7) = 2^26 : -677<br> 2^round(2^4.8) = 2^28 : -437<br> 2^round(2^4.9) = 2^30 : -1385 -1697 -2645<br> 2^round(2^5.0) = 2^32 : -209 -1409 -3509<br> 2^round(2^5.1) = 2^34 : -641 -1001 -1637<br> 2^round(2^5.2) = 2^37 : -45<br> 2^round(2^5.3) = 2^39 : -381<br> 2^round(2^5.4) = 2^42 : -2201 -3737 -5417 -12581<br> 2^round(2^5.5) = 2^45 : -573<br> 2^round(2^5.6) = 2^49 : -2709<br> 2^round(2^5.7) = 2^52 : -473 -2729 -2933<br> 2^round(2^5.8) = 2^56 : -2249 -2837<br> 2^round(2^5.9) = 2^60 : -3677<br> 2^round(2^6.0) = 2^64 : -1469<br> 2^round(2^6.1) = 2^69 : -165<br> 2^round(2^6.2) = 2^74 : -545 -9521 -22745 -23777 -26045<br> 2^round(2^6.3) = 2^79 : -2001 -2709<br> 2^round(2^6.4) = 2^84 : -5297 -7013<br> 2^round(2^6.5) = 2^91 : -81 -5661<br> 2^round(2^6.6) = 2^97 : -6909<br> 2^round(2^6.7) = 2^104 : -15773<br> 2^round(2^6.8) = 2^111 : -429<br> 2^round(2^6.9) = 2^119 : -3981<br> 2^round(2^7.0) = 2^128 : -15449 -21509<br> 2^round(2^7.1) = 2^137 : -849 -1785 -2289 -30189<br> 2^round(2^7.2) = 2^147 : -2601 -15201 -30249 -38145 -44841 -49761 -92565<br> 2^round(2^7.3) = 2^158 : -665 -14117<br> 2^round(2^7.4) = 2^169 : -20493<br> 2^round(2^7.5) = 2^181 : -20265 -83793 -96093<br> 2^round(2^7.6) = 2^194 : -6641 -37961 -38057 -42257 -74681 -86801 -139565<br> 2^round(2^7.7) = 2^208 : -4973<br> 2^round(2^7.8) = 2^223 : -28929 -44901<br> 2^round(2^7.9) = 2^239 : -87429<br> 2^round(2^8.0) = 2^256 : -36113 -188069<br> 2^round(2^8.1) = 2^274 : -54605<br> 2^round(2^8.2) = 2^294 : -184181<br> 2^round(2^8.3) = 2^315 : -51321 -61245<br> 2^round(2^8.4) = 2^338 : -35861<br> 2^round(2^8.5) = 2^362 : -169805<br> 2^round(2^8.6) = 2^388 : -17297 -156377 -222113 -365369 -375737 -497693<br> 2^round(2^8.7) = 2^416 : -222749<br> 2^round(2^8.8) = 2^446 : -324857 -835781<br> 2^round(2^8.9) = 2^478 : -699821<br> 2^round(2^9.0) = 2^512 : -38117<br> 2^round(2^9.1) = 2^549 : -1005069<br> 2^round(2^9.2) = 2^588 : -313793 -406997<br> 2^round(2^9.3) = 2^630 : -104585 -636521 -813917<br> 2^round(2^9.4) = 2^676 : -322229<br> 2^round(2^9.5) = 2^724 : -52517<br> 2^round(2^9.6) = 2^776 : -356033 -1710317<br> 2^round(2^9.7) = 2^832 : -1281293<br> 2^round(2^9.8) = 2^891 : -1926369 -2217081 -4672149<br> 2^round(2^9.9) = 2^955 : -1077885<br> 2^round(2^10.0) = 2^1024 : -1093337 -1370753 -1428353 -1503509<br> 2^round(2^10.1) = 2^1097 : -3161265 -3560373<br> 2^round(2^10.2) = 2^1176 : -2056193 -2069357<br> 2^round(2^10.3) = 2^1261 : -2634393 -4262745 -4668993 -4944945 -5565909<br> 2^round(2^10.4) = 2^1351 : -25905 -1138665 -3569025 -3720261<br> 2^round(2^10.5) = 2^1448 : -549269<br> 2^round(2^10.6) = 2^1552 : -551729 -6104357<br> 2^round(2^10.7) = 2^1663 : -611625 -774609 -1455849 -1719765<br> 2^round(2^10.8) = 2^1783 : -1233705 -3645345 -4163709<br> 2^round(2^10.9) = 2^1911 : -3231309<br> 2^round(2^11.0) = 2^2048 : -1942289 -4801589<br> 2^round(2^11.1) = 2^2195 : -3502989<br> 2^round(2^11.2) = 2^2353 : -1678485<br> 2^round(2^11.3) = 2^2521 : -4968453<br> 2^round(2^11.4) = 2^2702 : -246797<br> 2^round(2^11.5) = 2^2896 : -761717<br> 2^round(2^11.6) = 2^3104 : -4210193 -8338889 -10214933<br> 2^round(2^11.7) = 2^3327 : -775149<br> 2^round(2^11.8) = 2^3566 : -16802105 -37290401 -56582861<br> 2^round(2^11.9) = 2^3822 : -9543617 -11776781<br> 2^round(2^12.0) = 2^4096 : -10895177 -13463237<br> 2^round(2^12.1) = 2^4390 : -77336201 -79007045<br> 2^round(2^12.2) = 2^4705 : -24442605<br> 2^round(2^12.3) = 2^5043 : -1559505 -12588225 -40350225 -44478165<br> 2^round(2^12.4) = 2^5405 : -81935433 -86932389<br> 2^round(2^12.5) = 2^5793 : -11998425 -33851673 -36961425 -50712993 -83738889 -236665449 -254515869<br> 2^round(2^12.6) = 2^6208 : -20057957<br> 2^round(2^12.7) = 2^6654 : -11162645<br> 2^round(2^12.8) = 2^7132 : -7052309<br> 2^round(2^12.9) = 2^7643 : -44120649 -123495021<br> 2^round(2^13.0) = 2^8192 : -43644929 -49930517<br> 2^round(2^13.1) = 2^8780 : -22581833 -25207709<br> 2^round(2^13.2) = 2^9410 : -10428521 -103539077<br> 2^round(2^13.3) = 2^10086 : -29814917<br> 2^round(2^13.4) = 2^10809 : -6562329 -216052833 -262814913 -415344465 -429846909<br> 2^round(2^13.5) = 2^11585 : -370812549<br> 2^round(2^13.6) = 2^12417 : -103708449 -294381693<br> 2^round(2^13.7) = 2^13308 : -10956077<br> 2^round(2^13.8) = 2^14263 : -397079661<br> 2^round(2^13.9) = 2^15287 : -214216065 -312159765<br> 2^round(2^14.0) = 2^16384 : -364486013<br> 2^round(2^14.1) = 2^17560 : -141642533<br> 2^round(2^14.2) = 2^18820 : -339964313 -571288529 -898757417 -1027256537 -1156399229<br>2^round(2^14.3) = 2^20171 : -276625749
<br>2^round(2^14.4) = 2^21619 : -674266449 -833225181
<br>2^round(2^14.5) = 2^23170 : -259994441 -366289061
<br>2^round(2^14.6) = 2^24834 : -415810805
<br>2^round(2^14.7) = 2^26616 : -2909149877
<br>2^round(2^14.8) = 2^28526 : -520866521 -811123361 -990961337 -1485278921 -1487483597
<br>2^round(2^14.9) = 2^30574 : -568083185 -1447030925
<br>2^round(2^15.0) = 2^32768 : -718982153 -2543122349
<br>2^round(2^15.1) = 2^35120 : -881530949</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-52885733878857473152021-12-04T21:10:00.000Z2021-12-04T21:10:00.234Z[dclecbyo] 20 centers of projection<p>assume a map projection in which distortion is small near the center of projection and large toward the edges, e.g., orthographic. depict a sphere as a collection of 12 or 20 maps centered at the vertices of an icosahedron or dodecahedron respectively. the maps will overlap, perhaps by a lot.</p> <p>12 = 3*4 and 20 = 4*5, so both numbers are good for putting the maps in an elegant rectangular array. if one avoids placing centers of projection <a href="/2015/04/dwqjanwz-12-tribes.html">at the poles</a>, north can always be up.</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-42638912894716003542021-12-04T21:05:00.001Z2021-12-04T21:05:00.265Z[bhxtejtu] x and 1-x proportions<p>order individuals by wealth and compute the running sum: Lorenz curve. there is a special proportion p such that the top p of the population holds 1-p of the wealth, e.g., the top 1% holds 99% of the wealth.</p> <p>hypothesize that smaller p means greater inequality. another similar measure is the Gini coefficient.</p> <p>is it guaranteed that there is only one special p?</p> <p>can p be read off a Lorenz curve?</p> <p>what if the poorest have negative wealth?</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-73246446873511861852021-12-04T21:05:00.000Z2021-12-04T21:05:00.265Z[tkhqrixn] hash of primes<p>print primes in decimal, one per line. partition into intervals from 2^n to 2^(n+1)-1. hash each partition with MD5.</p> <p>goal is to verify whether a prime number generator is working properly. here are results from primesieve version 6.3 packaged on Ubuntu 18.04 . future work: repeat with some other way of generating primes.</p> <p>the first result is the MD5 hash of the empty string:</p> <p>primesieve -p 1 1 | md5sum<br> d41d8cd98f00b204e9800998ecf8427e -</p> <p>the next result is the hash of the string "2\n3\n":</p> <p>primesieve -p 2 3 | md5sum<br> 19283599a9866154a20cbb0be6adc1bc -</p> <p>the next result is the hash of the string "5\n7\n":</p> <p>primesieve -p 4 7 | md5sum<br> ec4aab475ce80bfd5469640c71b17108 -</p> <p>primesieve -p 8 15 | md5sum<br> ea40c553d35c3769e4563356e5571312 -</p> <p>primesieve -p 16 31 | md5sum<br> 389ee0bc459bb738682302eaeeffdb9e -</p> <p>primesieve -p 32 63 | md5sum<br> 4625d9d99b82ddd64b2a7f744e81b5a2 -</p> <p>primesieve -p 64 127 | md5sum<br> 0d59f9c405e9f4ddd6a9dff34d995747 -</p> <p>primesieve -p 128 255 | md5sum<br> 24ac923d7d3fcb9ee30f660db48a896a -</p> <p>primesieve -p 256 511 | md5sum<br> 0def87cc99e15b8537b4fa735e9e173e -</p> <p>primesieve -p 512 1023 | md5sum<br> 09f3b02a4e83b28eb353da6b143173e5 -</p> <p>primesieve -p 1024 2047 | md5sum<br> 9a0e27c6dd0eb9c6f16896b1a19504f9 -</p> <p>primesieve -p 2048 4095 | md5sum<br> f8812a2f97f570cc893c089aafacf7df -</p> <p>primesieve -p 4096 8191 | md5sum<br> 4dabac6164589732ef6f9ffa843c1125 -</p> <p>primesieve -p 8192 16383 | md5sum<br> 62b0b80451cc9e02de892c93cd5cac34 -</p> <p>primesieve -p 16384 32767 | md5sum<br> 4969d628ab6f72295afa9e663542cb59 -</p> <p>primesieve -p 32768 65535 | md5sum<br> a90e03d5ce780a57bd962016af6a88b9 -</p> <p>primesieve -p 65536 131071 | md5sum<br> 7e9e7de2250a57a7aab73d8a347234c0 -</p> <p>primesieve -p 131072 262143 | md5sum<br> 725762c6e5d246e32a71fff4490eb345 -</p> <p>primesieve -p 262144 524287 | md5sum<br> b87228cac6ba574ce059b06a89cf743b -</p> <p>primesieve -p 524288 1048575 | md5sum<br> 9db8a44d1018406bed9ddd810fdf7ed7 -</p> <p>primesieve -p 1048576 2097151 | md5sum<br> 00220343780b1453c9b7a634c4bcf395 -</p> <p>primesieve -p 2097152 4194303 | md5sum<br> 0b007f8da982f6bb8405bc1a987479ab -</p> <p>primesieve -p 4194304 8388607 | md5sum<br> fbf768c73a8b3808ad987616df137937 -</p> <p>primesieve -p 8388608 16777215 | md5sum<br> cbcd6f4bba8e755e38fec96f700bc6e4 -</p> <p>primesieve -p 16777216 33554431 | md5sum<br> 6be429185d33bf792dec21a9e2f838a8 -</p> <p>primesieve -p 33554432 67108863 | md5sum<br> ba6d3bfb1b1fd482c2f6c2068e6def69 -</p> <p>primesieve -p 67108864 134217727 | md5sum<br> 5690fcae89b922e83149cde919d13b08 -</p> <p>primesieve -p 134217728 268435455 | md5sum<br> e01b4b2f2665faa58a6c7c0e570c5623 -</p> <p>primesieve -p 268435456 536870911 | md5sum<br> 5e73b7013ea260fbc89689d196867c7f -</p> <p>primesieve -p 536870912 1073741823 | md5sum<br> baacef4b4e06204e8c70e9e7b70caf51 -</p> <p>primesieve -p 1073741824 2147483647 | md5sum<br> 72922f6383a3a6003c3a7d648bb8e3da -</p> <p>primesieve -p 2147483648 4294967295 | md5sum<br> 9c1e25fef3b6abc9a1dca782bf5d422c -</p> <p>primesieve -p 4294967296 8589934591 | md5sum<br> b45d191f562952acb0fe37aff3d92733 -</p> <p>primesieve -p 8589934592 17179869183 | md5sum<br> e1ab401038abbef1283e57ca6f058e4b -</p> <p>primesieve -p 17179869184 34359738367 | md5sum<br> 153ae06a21b0fedce43b4fcd43c5ff6b -</p> <p>primesieve -p 34359738368 68719476735 | md5sum<br> b0c7f4f0c04ffca5c3bb22bc48a8e5e1 -</p> <p>primesieve -p 68719476736 137438953471 | md5sum<br> d0f12246675c9b5c9c3a373a509d42a1 -</p> <p>primesieve -p 137438953472 274877906943 | md5sum<br> 64e90d67029fc0e6deed56a0f564aad4 -</p> <p>primesieve -p 274877906944 549755813887 | md5sum<br> 26b043e2db96fb2c5e80da3898507372 -</p> <p>primesieve -p 549755813888 1099511627775 | md5sum<br> 6b7104f5a6e8dcac949350888a9df58a -</p> <p>primesieve -p 1099511627776 2199023255551 | md5sum<br> 71ee6df793118e85829074b152e59c73 -</p> <p>primesieve -p 2199023255552 4398046511103 | md5sum<br> c420df38251cada4f31fb9df35dde6f1 -</p> <p>primesieve -p 4398046511104 8796093022207 | md5sum<br> 95c931c24f7a99483ac6d16b5530d838 -</p> <p>primesieve -p 8796093022208 17592186044415 | md5sum<br> 157266729337657753d52198f32a2698 -</p> <p>primesieve -p 17592186044416 35184372088831 | md5sum<br> 5d98aa14014d80dc646409f9103c9949 -</p> <p>primesieve -p 35184372088832 70368744177663 | md5sum<br> 6aa4896128011565848ccfad6f00418b -</p> <p>the final result above is the MD5 checksum of all 46-bit primes that have their most significant bit set, that is, primes between 2^45 = 35184372088832 and 2^46 - 1 = 70368744177663 .</p><p>future work: this seems amenable to parallelization and tree hashing. let the leaves of the tree be the bitstring of prime and composite. reuse previously hashed results, especially of long strings of consecutive composites.</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-48466702071630096982021-12-04T21:00:00.000Z2021-12-04T21:00:00.240Z[tfzwneso] cave system outside non-overlapping cubes<p>start with some random points. grow circles around them, all circles growing at the same rate. a circle stops growing when it touches another circle. this results in a circle packing. (not a very tight circle packing: what is a better way to generate denser random circle packings? maybe Apollonian gasket.)</p> <p>same thing but with axis-aligned growing squares instead of circles. what is the nature of the space outside of all squares after they have grown as much as possible? is it typically connected?</p> <p>similar thing with rectangles. when a side touches another side, growth stops only in that direction. the other 3 sides keep growing until they too are each impeded. there needs to be an exterior bounding box. vaguely has a similar feeling to Voronoi partition. with this much additional freedom to grow compared to squares, there probably will not be much remaining space, and the remaining space probably won't be very connected.</p> <p>same thing except with bricks (rectangular parallelepipeds) in 3D. 3D probably has enough space for the gaps between bricks to typically remain connected.</p> <p>motivation is to generate a cave system for a game, a collection of rooms connected by twisty passages. with bricks, everything axis-aligned is like Minecraft. we want rooms and passages to have distinctive shapes. probably best if character can climb everywhere, including ceiling.</p> <p>define a room to be a gap large enough to fit a cube of given size. (or sphere?) what is the extent of a room? define a center as a point surrounded by an empty cube of the given size, then define a room from the connected component of centers. devilish detail remains of choosing the size of the probe cube.</p> <p>bricks could grow at different rates, or not all start growing at the same time. inspired by crystal growth in the Cave of the Crystals. the cave is the free space not occupied by crystals.</p> <p>"epoxy" to fill in gaps. tunneling to make more passages.</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-82630298606550782602021-12-04T20:54:00.000Z2021-12-04T20:54:00.270Z[mcbkqach] those who will never be vaccinated<ol type="1"> <li><p>if I am an illegal immigrant and choose to be vaccinated against COVID-19, might the information I provide on vaccination paperwork be used to discover my illegality and hasten my deportation?</p></li> <li><p>if I am evading law enforcement, perhaps for a crime that hasn't been discovered yet, might the information I provide or leak during vaccination be used to apprehend me? along with information directly provided on vaccination paperwork, leaked information might include becoming filmed on security cameras or automated license-plate recording cameras, or being tracked by cell phone tracking systems.</p></li> </ol> <p>these are rhetorical questions; the answers to both is "obviously yes". we do not have any mechanisms to credibly convince people of these groups that the answer is "no". law enforcement, including immigration, cannot be stopped in accessing (via subpoena) any of the above information.</p> <p>included in group 2 are people who are evading paying child support.</p> <p>also included in group 2 might be a very large number of people trying to keep a low profile because of "three felonies a day".</p> <p>also included in group 2 might be people who self-recognize that they have a predilection toward "antisocial behavior", and might do something criminally antisocial, or might do something antisocial that attracts attention of law enforcement leading to arrest for a different crime, during the course of being vaccinated. antisocial behavior, as judged by our racist society, might include "being black".</p> <p>what is the total population of the people in these categories? assuming they rationally choose not be vaccinated because they don't want to be deported or end up in jail or have wealth confiscated, are they alone enough to prevent general herd immunity? people in these groups likely geographically cluster. regions with high concentrations of such people are even less likely to locally reach herd immunity.</p> <p>we have created a society with a large number of such people. benefits perhaps have included cheap labor, law and order. but we now pay its cost -- pay the piper -- in situations like this, being unable to stop a disease -- those who are not vaccinated will breed new COVID-19 variants -- and being unable to return to normalcy.</p> <p>it seems it will require a mind-bogglingly vast restructuring of society to significantly decrease the population of such people.</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-48671800181080601102021-12-04T19:07:00.000Z2021-12-04T19:07:00.236Z[bekvhdwz] ECC signature QR code<p>elliptic curve cryptography (ECC) offers signatures considerably shorter than RSA or integer Diffie-Hellman. we examine the signature size of Ed25519. (constructing an Ed25519 collision requires 2^128 work.)</p> <p>an Ed25519 signature is 64 bytes, 512 bits, or 155 digits. this fits within a version 4 (size) QR code at ECC (error correcting code, <a href="/2016/04/ccszcytw-fps.html">an unfortunate collision of acronyms</a>) level L, or version 8 at level H. note: the QR code examples below do not encode real signatures or public keys. instead, they merely encode digit strings with the right lengths (future post duartbli). although QR codes can encode raw binary data, we've chosen base-10 digit strings with the same amount of information because many QR code readers don't do so well with binary data. (imagine that <a href="/2016/11/wisnmyni-data-as-number.html">radix conversion</a> has been done.)</p> <p><img src="https://www.mit.edu/~kenta/one/qr-ed25519/bekvhdwz/qr512l.png" width="123" height="123" alt="QR code 155 digits, level L"> <img src="https://www.mit.edu/~kenta/one/qr-ed25519/bekvhdwz/qr512h.png" width="171" height="171" alt="QR code 155 digits, level H"></p> <p>it's helpful to identify the public key corresponding to the signature. rather than a key identifier (as usually done with RSA or DSA, discussed below), public keys for ECC are short enough to be included in their entirety. Ed25519 public keys are 32 bytes, 256 bits, or 78 digits. here are QR codes encoding a string consisting of "ED25519:", 78 digits (representing a public key), a separator ":", and 155 digits (representing a signature). it is version 6 at ECC L, version 10 at H. (we use <a href="/2010/03/pbafqzwx-notes-on-qr-code.html">colon as a separator</a> for maximum efficiency, though it turns out not to matter.)</p> <p><img src="https://www.mit.edu/~kenta/one/qr-ed25519/bekvhdwz/qr256-512l.png" width="147" height="147" alt="QR code 71 digits :ED25519: 155 digits, level L"> <img src="https://www.mit.edu/~kenta/one/qr-ed25519/bekvhdwz/qr256-512h.png" width="195" height="195" alt="QR code 71 digits :ED25519: 155 digits, level H"></p> <p>note that the payload, i.e., the message which was signed, is not included in the above examples. given the payload and the public key, the signature can be verified in place; no additional information is required.</p> <p>the problem of establishing that an included public key can be trusted is the giant open problem of PKI. (additional information is required.)</p> <p>here is a nice illustration of what Ed25519 private and public keys and signatures contain: <a href="https://blog.mozilla.org/warner/2011/11/29/ed25519-keys/" class="uri">https://blog.mozilla.org/warner/2011/11/29/ed25519-keys/</a></p> <p>for reference, here are possible sizes of public key identifiers in PGP (gnupg, gpg, openpgp). all of these could be applied to Ed25519 but would require standardization.</p><p> 32-bit key ID (4 bytes, 10 digits)<br/>64-bit key ID (8 bytes, 20 digits)<br/>MD5 key fingerprint (128 bits, 16 bytes, 39 digits)<br/>SHA-1 key fingerprint (160 bits, 20 bytes, 49 digits)</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-86174068014500634062021-12-04T10:32:00.000Z2021-12-04T10:32:00.251Z[avmeugtp] it's raining snakes<p>as might be expected for someone who has "had enough", Samuel L. Jackson snaps, suicidally doing something that causes mid-air destruction of the aircraft. perhaps using a purple lightsaber.</p> <p>down below, as humor or horror, the running gag is snakes falling from above when you least expect it.</p><p>snakes from a plane.</p> <p>previously: <a href="/2011/04/sqvprqme-snakes-on-plane-2.html">(2)</a>, <a href="/2011/04/kjfgqjzg-snakes-on-plane-3.html">(3)</a>.</p>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-19729914167665902102021-12-04T10:15:00.000Z2021-12-04T10:15:00.255Z[mcrpntzv] downfall<p>entitle the famous scene of Yitzhak Rabin and Yassar Arafat shaking hands, with Bill Clinton behind them, "Downfall" (13 September 1993, Oslo Accords).</p> <p>Rabin would go on to be assassinated by an Israeli citizen unhappy with Israel making peace with the Palestinians. Israelis generally agreed with the assassination, choosing to replace Rabin (actually Rabin's successor) with a hard-line government bent on making war with Palestine.</p> <p>Arafat's PLO would go on go be supplanted by Hamas, a hard-line government bent on making war with Israel. along the way, Arafat dies, possibly assassinated by that Isareli government bent on making war with Palestine.</p> <p>Clinton's political party would quickly go on to lose both houses of Congress (Contract with America), and later the Presidency, to a hard-line party bent on antisemitism and Islamophobia.</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-21789498326889360782021-12-04T10:02:00.000Z2021-12-04T10:02:00.260Z[hzwtbcrh] slow-growing integer exponentials<p>the sequence a[n] = 2^n, the powers of 2, has growth rate 2.</p> <p>a[n] = a[n-1] + a[n-2], Fibonacci sequence (also Lucas sequence), powers of the matrix [1 1 ; 1 0], growth rate = golden ratio = 1.618 = Roots[1+x-x^2==0,x] = (1+sqrt(5))/2.</p> <p>growth rate is the <a href="/2021/03/nfefxuyy-symbolic-eigenvalues.html">largest eigenvalue of the matrix</a>. golden ratio seems to be the slowest constant growth possible with 2x2 integer matrix. (but doubling every two iterations, average growth sqrt(2) = 1.4 is also possible.)</p> <p>longer lag a[n] = a[n-1] + a[n-3], OEIS A000930 (also A179070), powers of matrix [1 0 1; 1 0 0 ; 0 1 0], growth rate = supergolden ratio = 1.46557123187676802665673122522 = Roots[1+x^2-x^3==0,x] = (1 + (29/2 - (3*sqrt(93))/2)^(1/3) + ((29 + 3*sqrt(93))/2)^(1/3))/3</p> <p>longer lag a[n] = a[n-1] + a[n-4], powers of matrix [1 0 0 1; 1 0 0 0 ; 0 1 0 0 ; 0 0 1 0], growth rate 1.38027756909761411567330169182 = Roots[1+x^3-x^4==0,x] = 1/4 + sqrt(1 - 16*(2/(3*(-9 + sqrt(849))))^(1/3) + 2*(2/3)^(2/3)*(-9 + sqrt(849))^(1/3))/4 + sqrt(1/2 + 4*(2/(3*(-9 + sqrt(849))))^(1/3) - ((-9 + sqrt(849))/2)^(1/3)/3^(2/3) + 1/(2*sqrt(1 - 16*(2/(3*(-9 + sqrt(849))))^(1/3) + 2*(2/3)^(2/3)* (-9 + sqrt(849))^(1/3))))/2</p> <p>the quintic case surprisingly has a closed-form growth rate: a[n] = a[n-1] + a[n-5], powers of matrix [1 0 0 0 1; 1 0 0 0 0; 0 1 0 0 0; 0 0 1 0 0 ; 0 0 0 1 0], growth rate 1.3247179572447460260 = Roots[1+x^4-x^5==0,x] = (27/2 - (3*sqrt(69))/2)^(1/3)/3 + ((9 + sqrt(69))/2)^(1/3)/3^(2/3). the polynomial factors 1 + x^4 - x^5 = (1 - x + x^2) * (1 + x - x^3) . the real root (that of the cubic factor), the growth rate, is the plastic number: see Padovan sequence below.</p> <p>sextic: <span style="font-family: sans-serif;">a[n] = a[n-1] + a[n-6], growth</span> rate 1.28519903324534936790726046413 = N[Roots[1+x^5-x^6==0,x],30]</p><p>the next polynomial that factors is 1 + x^10 - x^11 = (1 - x + x^2) * (1 + x - x^3 - x^4 + x^6 + x^7 - x^9) . the quadratic factor is the same as in the quintic case.</p> <p>what is the growth rate as function of lag?</p> <p>because these sequences are powers of a matrix, one can compute high terms very quickly using exponentiation by squaring. the matrices contain repeated values. how can this be exploited to compute quicker?</p> <p>if we take the two oldest numbers, growth rate is slower:</p> <p>Padovan sequence OEIS A000931 (also Perrin sequence A001608): a[n] = a[n-2] + a[n-3], powers of matrix [0 1 1; 1 0 0 ; 0 1 0], growth rate = plastic number = 1.32471795724474602596090885448 = Roots[0==1+x-x^3,x] = (27/2 - (3*sqrt(69))/2)^(1/3)/3 + ((9 + sqrt(69))/2)^(1/3)/3^(2/3)</p> <p>a[n] = a[n-3] + a[n-4], OEIS A017817, growth rate = 1.22074408460575947536168534911 = Roots[0==1+x-x^4,x] = sqrt(-4*(2/(3*(9 + sqrt(849))))^(1/3) + ((9 + sqrt(849))/2)^(1/3)/3^(2/3))/2 + sqrt(4*(2/(3*(9 + sqrt(849))))^(1/3) - ((9 + sqrt(849))/2)^(1/3)/3^(2/3) + 2/sqrt(-4*(2/(3*(9 + sqrt(849))))^(1/3) + ((9 + sqrt(849))/2)^(1/3)/3^(2/3)))/2</p> <p>quintic case a[n] = a[n-4] + a[n-5], growth rate = 1.16730397826141868425604589985 = Roots[0==1+x-x^5,x].</p> <p>growth rate seems to be 1+ln(2)/lag for large lag (via Mathematica and Inverse Symbolic Calculator). more series terms?</p> <p>previously, <a href="/2014/08/zljfhron-lagged-fibonacci-digits.html">evaluating recursive sequences modulo N</a>.</p><p>motivation was to test something on integer inputs varying in scale exponentially, but the inputs should not have easy factorizations. a first attempt, the sequence interleaving 2^n and 3*2^n grows more slowly than 2^n but has all easy factorizations. Fibonacci is better (though it does have algebraic factorizations) but has no degrees of freedom to control growth rate. a^n+1 and a^n-1 for integer a and n have algebraic factorizations: Cunningham project. <a href="/2021/04/lckoyipy-decimal-powers-of-2.html">rounded decimal powers of two.</a></p>Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-71564618149009337932021-12-04T09:56:00.000Z2021-12-04T09:56:00.251Z[dzyomchb] symmetrically distributing points on a hypersphere<p>the vertices of a geodesic polyhedron distribute points evenly and symmetrically around a sphere (2-sphere, embedded in 3 dimensions). it is pretty.</p> <p>is there an analogous family of pretty distributions of points on the 3-sphere, embedded in 4 dimensions?</p> <p>when there are many vertices on a 2-sphere, the vertex figure around most vertices is a hexagon, reflecting the optimality of hexagonal close packing of discs on a plane.</p> <p>on the 3-sphere, we analogously expect most vertices to be packed as if at the centers of spheres in a fcc or hcp close packing. assuming fcc because it has more symmetry, most Voronoi cells around vertices will be approximate rhombic dodecahedra.</p> <p>on the 2-sphere, for any geodesic polyhedron based on the icosahedron, there are 12 special vertices surrounded by only 5 triangles, not 6. the Voronoi cell around such a vertex is a pentagon.</p> <p>what is the maximum symmetry the analogous 4D geodesic polytope can have? is it the symmetry of the 600-cell? or the tesseract? what polyhedron is the Voronoi cell around weird vertices, and how does that polyhedron connect to the rhombic dodecahedra that make up most of the cells? it probably needs quadrilateral faces.</p> <p>maybe it has to be the symmetry of the tesseract, and the weird vertices are surrounded by cubes, and the rhombic faces of the rhombic dodecahedra get distorted into squares to meet the faces of the cubes. all this is wild speculation.</p> <p>what happens on 4-spheres (in 5D) and beyond? we remain interested only in pretty distributions of points. high density lattice packings of identical hyperspheres in Euclidean spaces of various (low) dimensions are known. for large numbers of points on an N-sphere, distortions of these sphere packings will form "most" of the points on the hypersphere.</p><p>in higher dimensions, regular polytopes become boring: only the symmetry of the hypercube is available. but maybe asking that the symmetry of a point distribution have the symmetry of a regular polytope asks too much. for example, the E8 polytope and its associated symmetry are not regular.</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-33275508715170082522021-12-04T08:54:00.000Z2021-12-04T08:54:00.272Z[hrocucwq] hyperaccelerated bongcloud proof game<p>the main line way to play the <a href="https://www.reddit.com/r/AnarchyChess/comments/pcxipc/top_comment_picks_the_next_move_legal_or_not_day_0/">Hyperaccelerated Bongcloud chess opening</a> is</p> <p>1. Ke1xe2</p> <p>however, those who are sticklers about "rules" may prefer reaching the position via transposition, for example:</p> <p>1. Nf3 Nc6 2. Ng1 Nd4 3. Nf3 Nxe2 4. Ng1 Nd4 5. Nf3 Nc6 6. Ke2 Nb8 7. Ng1</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-10610585421593552552021-12-04T08:51:00.000Z2021-12-04T08:51:00.247Z[tpywqclf] mazes with doors<p>let buttons in the maze control internal doors, closing or opening them. this is a maze with state.</p> <p>straightforward: only 1 button reachable initially. pushing it opens a (probably distant) door making the next button accessible, and so forth. pushing a "used" button again has no effect. the last button opens a door making the exit accessible. design the locations of the buttons so that one necessarily visits a large portion of the maze.</p><p>variations:</p> <p>all buttons always accessible. maze is always solvable no matter what the button state. implement this by having the combined button state be the seed for a random (solvable) maze generator: pushing any button (probably) radically changes the maze. perhaps goal is shortest solution. or shortest solution that turns on all buttons.</p><p>a way to generate that doesn't radically change the maze for each button press: partially generate the maze with a fixed PRNG, then finish it off with a new PRNG seeded by button state. how well this works depends heavily on the chosen maze-generating algorithm. "doesn't radically change the maze" is subjective.</p> <p>count the number of "on" buttons. let this number be the seed for the maze generator. it feels a little bit like a 3D maze with weird elevators: an elevator only goes one direction and a distance of one floor no matter what floor, but after being used, only goes the other direction. maybe a weird stairwell or ladder.</p> <p>partition the buttons into two sets (perhaps visually). the tuple of the count of on buttons for each set is the seed for a maze generator. one wanders a "meta" plane of mazes. perhaps this meta plane is itself a maze with obstacles, so sometimes buttons cannot be pushed even if you reach them.</p> <p>buttons are not optional. being in a cell with a button automatically pushes or unpushes it. it's more like a motion sensor than a button. or, movement between two adjacent cells triggers the change in doors of the maze. or, movement in a particular direction. all of these result in mazes in which you can't easily go backward, though a UI could provide undo.</p> <p>pushing a button flips the state (xor) of a set of doors associated with the button.</p><p>some of these will require effort to ensure that the maze is solvable.</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-45117884302686321772021-12-04T08:48:00.000Z2021-12-04T08:48:00.255Z[honhnynk] endgames on infinite boards<p>investigate chess endgames on infinite boards (including half infinite, quarter infinite, 1/8 infinite) by investigating them on sufficiently large but finite boards. if the defending king makes it to certain edges, it is deemed to have a strategy that escapes to infinity so the game is drawn.</p><p>because everything is finite, tablebases are possible.</p><p>these finite regions only approximate infinity, but perhaps a sufficiently large regions can approximate infinity close enough for the endgame evaluation to be exact.</p><p>this probably has a chance of working only if the defending side does not have ranged pieces (bishop, rook, queen).</p><p>many tricky details remain. might not work at all.</p><p>vaguely similar to angel versus square eater.</p> <p>this will not work on a <a href="/2021/02/xqeokrwx-perpetual-check-wins.html">chess variant in which perpetual check wins</a>.</p><p>previously, <a href="/2021/06/xzlyepzm-programming-general-endgames.html">endgames on arbitrary but finite boards</a>.</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-75736855897338873652021-12-04T08:27:00.000Z2021-12-04T08:27:00.246Z[lhcbianh] twice fun<p>having fun is inherently enjoyable. then, having fun, especially in circumstances with adversity, also signals good mental health, high self-qi.</p> <p>maybe having fun isn't ever inherently enjoyable: maybe it's this signaling that's only ever being enjoyed. would it be as much fun if you were alone and couldn't tell anyone?</p><p>what things require adversity to be fun? it wouldn't be as fun if the adversity were removed.</p><p>possible examples: scary amusement park rides, horror movies.</p><p>hazing.</p><p>previously similar: <a href="/2016/05/twltdatf-sex-as-validation.html">validation is fun.</a></p><p>future post: sex is fun.</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-44082871737753632892021-12-02T08:27:00.000Z2021-12-02T08:27:00.248Z[abjimsnm] gaze into the abyss<p>some German spelling variations in a famous quote from Beyond Good and Evil (Jenseits von Gut und Böse) (1886) by Friedrich Nietzsche:</p> <p>Wer mit Ungeheuern kämpft, mag zusehn/zusehen, dass/daß er nicht dabei zum Ungeheuer wird. Und wenn du lange in einen Abgrund blickst, blickt der Abgrund auch in dich hinein.</p> <p>dass/daß is eszett (sharp s). daß is probably more correct, but dass seems more common in collections of quotes. what was German orthography in 1886?</p> <p>zusehen seems to be more correct, but zusehn seems to be a valid archaic spelling.</p> <p>the quote comes from a chapter of sayings and provides no further context. in reality, becoming a monster is complicated (investigated by Zimbardo and many others), and probably requires much, much more than just gazing into the abyss. though perhaps Zimbardo gazed into the abyss too long himself, becoming evil.</p> <p>I like the alternate translation "stare into the void; the void stares back", though that changes the meaning. "abyss" implies looking downward, toward evil, whereas "void" makes no ethical or moral judgment. and I like the weirdness of the void (nothingness) staring back, in contrast to the (implied) monsters of the abyss staring back. does spending mental effort on a subject seemingly devoid of ethical and moral issues (e.g., mathematics) turn you evil?</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-48123954662850273592021-11-28T20:35:00.000Z2021-11-28T20:35:45.612Z[znqttiii] non-polynomials<p>here are some subjectively interesting functions that are not polynomials, accompanied with a range over which the function does not do anything weird (no singularities).</p> <p>motivation was to test numerical integration. many quadrature schemes are exact for low-degree polynomials (future post), so we avoid polynomials.</p> <p>we give the functions in Mathematica syntax for convenience of integration, followed by notes. some integrals Mathematica cannot do. all of these can be evaluated quickly to high precision by some way other than numerical integration, sometimes a special function, given in notes.</p> <p>do some quadrature schemes work poorly if the first derivative is infinite (i.e., function is vertical) at an endpoint?</p> <p>Integrate[Exp[-x^2/2]/Sqrt[2*Pi],{x,0,2}] : standard normal distribution, Erf, inflection at 1</p> <p>Integrate[Sin[x],{x,0,Pi}] : integrating sine over a much larger range would even more strongly avoid polynomiality</p> <p>Integrate[Cos[x],{x,0,Pi/2}]</p> <p>Integrate[Cos[n*tau-x*Sin[tau]]/Pi,{tau,0,Pi}] : BesselJ[n,x], limits fixed, let n=1 x=10</p> <p>Integrate[x^x,{x,0,1}] : sophomore's dream, limits fixed, defined as limit at 0, <a href="/2021/07/dwjgaijg-sophomore-dream-guitar-pick.html">vertical at 0</a></p> <p>Integrate[x^-x,{x,0,1}] : sophomore's dream, limits fixed, defined as limit at 0, <a href="/2021/07/dwjgaijg-sophomore-dream-guitar-pick.html">vertical at 0</a></p> <p>Integrate[Sqrt[x],{x,0,1}] : vertical at 0</p> <p>Integrate[x^((1+Sqrt[5])/2-1),{x,0,1}] : vertical at 0, <a href="/2021/09/stxggztu-irrational-powers.html">exponent is golden ratio</a></p> <p>Integrate[Log[x],{x,1,E}]</p> <p>Integrate[1/x,{x,1,E}]</p> <p>Integrate[Exp[x],{x,0,1}]</p> <p>Integrate[Exp[x]/x,{x,1/4,2}] : exponential integral</p><p>Integrate[Exp[-1/x],{x,0,1}] : exponential integral, defined as limit at 0, <a href="/2021/11/jbxyjdmz-exp-1x.html">flat at zero</a></p> <p>Integrate[1/Log[x],{x,0,1/2}] : logarithmic integral, defined as limit at 0</p> <p>Integrate[Sqrt[1-x^2],{x,-1,1}] : semicircle area, vertical at endpoints</p> <p>Integrate[Sqrt[1/(1-x^2)],{x,-1/2,1/2}] : 1/6 circle perimeter, we avoid integrating from -1 and 1 because function is undefined at endpoints</p><p>Integrate[1/(1+x^2),{x,0,1}] : atan(1) = pi/4</p> <p>Integrate[Sqrt[1-k^2*Sin[x]^2], {x,0,Pi/2}] : complete elliptic integral of the second kind, <a href="/2012/08/kbitlnmx-quadratically-converging.html">arithmetic-geometric mean</a>, limits fixed, <a href="/2004/08/ramanujan-and-ellipses.html">ellipse perimeter</a>, let k=255/256, we have used trigonometric form to avoid the function being undefined at endpoints</p> <p>Integrate[1/Sqrt[1-k^2*Sin[x]^2], {x,0,Pi/2}] : complete elliptic integral of the first kind, limits fixed, let k=15/16, arithmetic-geometric mean, we have used trigonometric form to avoid the function being undefined at endpoints</p> <p>Integrate[x^a*(1-x)^b, {x,0,1}] : beta function, limits fixed, let a=1/3 b=1/3, vertical at endpoints</p> <p>future work: rescale, compute Taylor expansions around the midpoint of the interval.</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-43638294100511996752021-11-20T22:21:00.000Z2021-11-20T22:21:42.513Z[nbejauna] Lovers' Waltz cadenza<p>The Lover's Waltz by Jay Ungar and Molly Mason has a brief cadenza before its final note. could the cadenza be expanded?</p><p>dancing to a non-rhythmic cadenza requires significant departure from basic waltz.</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-18102421724621412342021-11-20T22:20:00.000Z2021-11-20T22:20:54.338Z[xiyniyeo] Ocean's 8 acrobat<p>in Ocean's 8, Qin Shaobo reprises his role from previous films of the franchise. but in having him, the producers missed a great opportunity to keep it an "all-female crew": there are a great many female gymnasts, acrobats, and contortionists who could have performed that (small) role, probably more available than male. some of them (e.g., Olympic gymnasts, Youtube stars) are even celebrities, if they wanted to follow the movie's theme of celebrity cameos all over the place.</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-74321593236957277252021-11-20T18:00:00.000Z2021-11-20T18:00:00.234Z[dndvumus] most annoying country to map<p>given a connected region on a sphere, project it onto a plane. some shapes of regions necessarily have more distortion than others, no matter what map projection is used. ("amount of distortion" needs to be precisely defined.) which shapes?</p> <p>it's not strictly area: a long thin band almost circling a great circle can be flattened into a rectangle without much distortion. or a star-shaped union of such bands. we only care about the internal distortion within a shape after projection, and not, say, that the distance between consecutive star endpoints is wrong.</p> <p>perhaps the badness of a shape is proportional to the size of its largest inscribed circle (spherical cap).</p><p>consider only regions without internal holes. (merge the internal hole into the region if necessary.) map non-contiguous regions by treating each connected component separately (as done with Alaska and Hawaii on typical U.S. maps.)</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-7243821232602288812021-11-20T11:50:00.000Z2021-11-20T11:50:15.306Z[lrfhagot] alcohol and evolution of mental health<p>beer is proof that God wants us to self-medicate for mental health problems.</p> <p>is it God's love, or evolution? that is, are the evolution of mental health diseases and the development of alcohol as a technology of human civilization linked? perhaps mental health issues that could not be treated (somewhat) with alcohol were evolutionarily bred out.</p> <p>other recreational drugs: have any been cultivated (genetically engineered) for as long as alcohol has been brewed? long enough to affect human evolution?</p> Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-6757805.post-62752206583625542172021-11-19T20:21:00.000Z2021-11-19T20:21:35.264Z[vonobuym] astronomical dust to dust<p>in astronomy, dust is interstellar material larger than a molecule. how large is a dust particle permitted to be?</p><p>a human corpse floating in space would probably be considered by astronomers as dust. the proverb "dust to dust" is literally true as scientific jargon.</p><p>dust absorbs radiation and reemits with a black body spectrum. a living human body maintains a constant body temperature (homeostasis), and plants do photosynthesis, so, in general, living things are more complicated than dust in terms of radiation.</p><p>can planets and asteroids be dust? geologically active planets emit radiation unrelated to insolation, and atmospheres turn radiation into weather, kinetic energy. radiation can also induce phase changes, e.g., comets. maybe some of these processes have equilibrated. if none or not much of these processes are happening, is a planet dust?</p>Unknownnoreply@blogger.com0