问:
Compute least hypothenuse of more than n distinct Pythagorean triples
I'm interested in a variation of https://oeis.org/A006339 (Least hypotenuse of n distinct Pythagorean triangles).

Basically, if A006339 is $u(n)$, I want $v(n) := \min_{n \leq i} u(i)$

In other words, $v(n)$ is increasing and for any $k \leq n$, there are less than $v(n)$ triples with hypothenuse $k$.

Are there tables for that sequence? How can I compute it?

A possible way would be to have lower bounds for A006339 and compute it until the lower bound is big enough. I'm not even sure about how I can compute A006339. There is an easy enumeration algorithm for Pythagorean triples due to a result by Hall (1970) and Roberts (1977) (https://stackoverflow.com/a/8263898/5133167), and it allows us to stop the search as any multiplication by $U$, $A$ or $D$ will increase the value of $c$. Is there a more performant way?

答:
I worked out a program with better printing. These are the first 95 numbers with extreme efficiency  in creating large numbers of representations. They get too long either written in decimal or factored; I stuck with factored. The number "reps"  is the count of
$$ a^2 + b^2 = n \; , \; \; a > b > 0 $$
The first is  $$ 5 = 2^2 + 1^2  $$
The second is  $$ 65 = 8^2 + 1^2 = 7^2 + 4^2   $$
The third  is  $$ 325 = 18^2 + 1^2 = 17^2 + 6^2 = 15^2+ 10^2    $$
The fourth  is  $$ 5525 = 74^2 + 7^2 = 73^2 + 14^2 = 71^2+ 22^2  = 70^2 + 25^2 = 62^2 + 41^2 = 55^2+ 50^2  $$
The relevant theorem is in Dickson's little book, 1929, Introduction to the Theory of Numbers   in which he says the exact number of representations of a number using the number of divisors of form $1 \pmod 4$ and subtracting off the number of divisors of form $3 \pmod 4,$  then multiplying by $4$ for all of them (including negatives, reversing order).  For us, the numbers are never squares, so zero is not involved. The result is that taking half of Dickson's number (he calls it $E$ for excess) is correct here.   Found it, page 80, section 51,  Theorem 65.

``` ( 0.4306765580733931 , 5 , 1 )   reps  1 prod   5
 ( 0.2702381544273197 , 13 , 1 )   reps  2 prod   5 13
 ( 0.2519296364125923 , 5 , 2 )   reps  3 prod   5^2 13
 ( 0.244650542118226 , 17 , 1 )   reps  6 prod   5^2 13 17
 ( 0.2058468324604344 , 29 , 1 )   reps  12 prod   5^2 13 17 29
 ( 0.1919587200065601 , 37 , 1 )   reps  24 prod   5^2 13 17 29 37
 ( 0.1866524112389434 , 41 , 1 )   reps  48 prod   5^2 13 17 29 37 41
 ( 0.1787469216608008 , 5 , 3 )   reps  64 prod   5^3 13 17 29 37 41
 ( 0.1745834300480449 , 53 , 1 )   reps  128 prod   5^3 13 17 29 37 41 53
 ( 0.1686130986895011 , 61 , 1 )   reps  256 prod   5^3 13 17 29 37 41 53 61
 ( 0.1615554674429964 , 73 , 1 )   reps  512 prod   5^3 13 17 29 37 41 53 61 73
 ( 0.158079186604075 , 13 , 2 )   reps  768 prod   5^3 13^2 17 29 37 41 53 61 73
 ( 0.1544226628011101 , 89 , 1 )   reps  1536 prod   5^3 13^2 17 29 37 41 53 61 73 89
 ( 0.1515171524096389 , 97 , 1 )   reps  3072 prod   5^3 13^2 17 29 37 41 53 61 73 89 97
 ( 0.150190483223688 , 101 , 1 )   reps  6144 prod   5^3 13^2 17 29 37 41 53 61 73 89 97 101
 ( 0.1477501131786861 , 109 , 1 )   reps  12288 prod   5^3 13^2 17 29 37 41 53 61 73 89 97 101 109
 ( 0.1466237184553111 , 113 , 1 )   reps  24576 prod   5^3 13^2 17 29 37 41 53 61 73 89 97 101 109 113
 ( 0.1431113929202641 , 17 , 2 )   reps  36864 prod   5^3 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113
 ( 0.1408841194731412 , 137 , 1 )   reps  73728 prod   5^3 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137
 ( 0.1386468838532138 , 5 , 4 )   reps  92160 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137
 ( 0.1385201075671774 , 149 , 1 )   reps  184320 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149
 ( 0.1370873187823978 , 157 , 1 )   reps  368640 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157
 ( 0.1345057169479844 , 173 , 1 )   reps  737280 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173
 ( 0.1333360731748201 , 181 , 1 )   reps  1474560 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181
 ( 0.1317096685686114 , 193 , 1 )   reps  2949120 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193
 ( 0.1311982683517524 , 197 , 1 )   reps  5898240 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197
 ( 0.1275639755045533 , 229 , 1 )   reps  11796480 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229
 ( 0.1271587398372755 , 233 , 1 )   reps  23592960 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233
 ( 0.1263760881150453 , 241 , 1 )   reps  47185920 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241
 ( 0.1249121781636255 , 257 , 1 )   reps  94371840 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257
 ( 0.1238932866290727 , 269 , 1 )   reps  188743680 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269
 ( 0.1232476925609442 , 277 , 1 )   reps  377487360 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277
 ( 0.1229342993142804 , 281 , 1 )   reps  754974720 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281
 ( 0.1220292459879827 , 293 , 1 )   reps  1509949440 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293
 ( 0.1206269875212659 , 313 , 1 )   reps  3019898880 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313
 ( 0.1204126778815846 , 29 , 2 )   reps  4529848320 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313
 ( 0.1203610007213705 , 317 , 1 )   reps  9059696640 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317
 ( 0.1190957566850307 , 337 , 1 )   reps  18119393280 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337
 ( 0.1183840591148901 , 349 , 1 )   reps  36238786560 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349
 ( 0.1181540875758708 , 353 , 1 )   reps  72477573120 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353
 ( 0.1170544627527939 , 373 , 1 )   reps  144955146240 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373
 ( 0.1162300592682939 , 389 , 1 )   reps  289910292480 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389
 ( 0.1158346526511383 , 397 , 1 )   reps  579820584960 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397
 ( 0.115640914438019 , 401 , 1 )   reps  1159641169920 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401
 ( 0.1152610595238196 , 409 , 1 )   reps  2319282339840 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409
 ( 0.1147094651682719 , 421 , 1 )   reps  4638564679680 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421
 ( 0.1141784098763921 , 433 , 1 )   reps  9277129359360 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433
 ( 0.1135000135583575 , 449 , 1 )   reps  18554258718720 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449
 ( 0.1132827525593783 , 5 , 5 )   reps  22265110462464 prod   5^5 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449
 ( 0.1131727366659431 , 457 , 1 )   reps  44530220924928 prod   5^5 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457
 ( 0.1130119349340791 , 461 , 1 )   reps  89060441849856 prod   5^5 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461
 ( 0.1122886528902697 , 37 , 2 )   reps  133590662774784 prod   5^5 13^2 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461
 ( 0.1121589678232448 , 13 , 3 )   reps  178120883699712 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461
 ( 0.1112158783706281 , 509 , 1 )   reps  356241767399424 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509
 ( 0.1108016106675626 , 521 , 1 )   reps  712483534798848 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521
 ( 0.1101384080494621 , 541 , 1 )   reps  1424967069597696 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541
 ( 0.1096306885348829 , 557 , 1 )   reps  2849934139195392 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557
 ( 0.1092623336390804 , 569 , 1 )   reps  5699868278390784 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569
 ( 0.109184661243966 , 41 , 2 )   reps  8549802417586176 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569
 ( 0.109022393369416 , 577 , 1 )   reps  17099604835172352 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577
 ( 0.1085553762730775 , 593 , 1 )   reps  34199209670344704 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593
 ( 0.1083280294582822 , 601 , 1 )   reps  68398419340689408 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601
 ( 0.1079943548673958 , 613 , 1 )   reps  136796838681378816 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613
 ( 0.1078850289369187 , 617 , 1 )   reps  273593677362757632 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617
 ( 0.1072480305374002 , 641 , 1 )   reps  547187354725515264 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641
 ( 0.1069411297181192 , 653 , 1 )   reps  1094374709451030528 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653
 ( 0.1067405999140516 , 661 , 1 )   reps  2188749418902061056 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661
 ( 0.1064456832186591 , 673 , 1 )   reps  4377498837804122112 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673
 ( 0.1063489015533347 , 677 , 1 )   reps  8754997675608244224 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677
 ( 0.1057834942757523 , 701 , 1 )   reps  17509995351216488448 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701
 ( 0.1056006150284899 , 709 , 1 )   reps  35019990702432976896 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709
 ( 0.1050677386679089 , 733 , 1 )   reps  70039981404865953792 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733
 ( 0.1045571277453435 , 757 , 1 )   reps  140079962809731907584 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757
 ( 0.1044740746281937 , 761 , 1 )   reps  280159925619463815168 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761
 ( 0.1043096601296809 , 769 , 1 )   reps  560319851238927630336 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769
 ( 0.1042282850833419 , 773 , 1 )   reps  1120639702477855260672 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773
 ( 0.103751273435192 , 797 , 1 )   reps  2241279404955710521344 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797
 ( 0.103519712755121 , 809 , 1 )   reps  4482558809911421042688 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809
 ( 0.1032925703785417 , 821 , 1 )   reps  8965117619822842085376 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821
 ( 0.1031435227654369 , 829 , 1 )   reps  17930235239645684170752 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829
 ( 0.1027073457992096 , 853 , 1 )   reps  35860470479291368341504 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853
 ( 0.1026361963957853 , 857 , 1 )   reps  71720940958582736683008 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857
 ( 0.1022867948546604 , 877 , 1 )   reps  143441881917165473366016 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877
 ( 0.1022181522110647 , 881 , 1 )   reps  286883763834330946732032 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881
 ( 0.1021247598253814 , 53 , 2 )   reps  430325645751496420098048 prod   5^5 13^3 17^2 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881
 ( 0.101539149197962 , 17 , 3 )   reps  573767527668661893464064 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881
 ( 0.1014246637053439 , 929 , 1 )   reps  1147535055337323786928128 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929
 ( 0.1012975688001153 , 937 , 1 )   reps  2295070110674647573856256 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937
 ( 0.1012345460491029 , 941 , 1 )   reps  4590140221349295147712512 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941
 ( 0.1010475355097117 , 953 , 1 )   reps  9180280442698590295425024 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953
 ( 0.1006824786789692 , 977 , 1 )   reps  18360560885397180590850048 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953 977
 ( 0.1003869949333126 , 997 , 1 )   reps  36721121770794361181700096 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953 977 997
 ( 0.1002133496204834 , 1009 , 1 )   reps  73442243541588722363400192 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953 977 997 1009
 ( 0.1001560585753007 , 1013 , 1 )   reps  146884487083177444726800384 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953 977 997 1009 1013
 ( 0.1000423464158209 , 1021 , 1 )   reps  293768974166354889453600768 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953 977 997 1009 1013 1021
```

Body:
I worked out a program with better printing. These are the first 95 numbers with extreme efficiency  in creating large numbers of representations. They get too long either written in decimal or factored; I stuck with factored. The number "reps"  is the count of
$$ a^2 + b^2 = n \; , \; \; a > b > 0 $$
The first is  $$ 5 = 2^2 + 1^2  $$
The second is  $$ 65 = 8^2 + 1^2 = 7^2 + 4^2   $$
The third  is  $$ 325 = 18^2 + 1^2 = 17^2 + 6^2 = 15^2+ 10^2    $$
The fourth  is  $$ 5525 = 74^2 + 7^2 = 73^2 + 14^2 = 71^2+ 22^2  = 70^2 + 25^2 = 62^2 + 41^2 = 55^2+ 50^2  $$
The relevant theorem is in Dickson's little book, 1929, Introduction to the Theory of Numbers   in which he says the exact number of representations of a number using the number of divisors of form $1 \pmod 4$ and subtracting off the number of divisors of form $3 \pmod 4,$  then multiplying by $4$ for all of them (including negatives, reversing order).  For us, the numbers are never squares, so zero is not involved. The result is that taking half of Dickson's number (he calls it $E$ for excess) is correct here.   Found it, page 80, section 51,  Theorem 65.

 ( 0.4306765580733931 , 5 , 1 )   reps  1 prod   5
 ( 0.2702381544273197 , 13 , 1 )   reps  2 prod   5 13
 ( 0.2519296364125923 , 5 , 2 )   reps  3 prod   5^2 13
 ( 0.244650542118226 , 17 , 1 )   reps  6 prod   5^2 13 17
 ( 0.2058468324604344 , 29 , 1 )   reps  12 prod   5^2 13 17 29
 ( 0.1919587200065601 , 37 , 1 )   reps  24 prod   5^2 13 17 29 37
 ( 0.1866524112389434 , 41 , 1 )   reps  48 prod   5^2 13 17 29 37 41
 ( 0.1787469216608008 , 5 , 3 )   reps  64 prod   5^3 13 17 29 37 41
 ( 0.1745834300480449 , 53 , 1 )   reps  128 prod   5^3 13 17 29 37 41 53
 ( 0.1686130986895011 , 61 , 1 )   reps  256 prod   5^3 13 17 29 37 41 53 61
 ( 0.1615554674429964 , 73 , 1 )   reps  512 prod   5^3 13 17 29 37 41 53 61 73
 ( 0.158079186604075 , 13 , 2 )   reps  768 prod   5^3 13^2 17 29 37 41 53 61 73
 ( 0.1544226628011101 , 89 , 1 )   reps  1536 prod   5^3 13^2 17 29 37 41 53 61 73 89
 ( 0.1515171524096389 , 97 , 1 )   reps  3072 prod   5^3 13^2 17 29 37 41 53 61 73 89 97
 ( 0.150190483223688 , 101 , 1 )   reps  6144 prod   5^3 13^2 17 29 37 41 53 61 73 89 97 101
 ( 0.1477501131786861 , 109 , 1 )   reps  12288 prod   5^3 13^2 17 29 37 41 53 61 73 89 97 101 109
 ( 0.1466237184553111 , 113 , 1 )   reps  24576 prod   5^3 13^2 17 29 37 41 53 61 73 89 97 101 109 113
 ( 0.1431113929202641 , 17 , 2 )   reps  36864 prod   5^3 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113
 ( 0.1408841194731412 , 137 , 1 )   reps  73728 prod   5^3 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137
 ( 0.1386468838532138 , 5 , 4 )   reps  92160 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137
 ( 0.1385201075671774 , 149 , 1 )   reps  184320 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149
 ( 0.1370873187823978 , 157 , 1 )   reps  368640 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157
 ( 0.1345057169479844 , 173 , 1 )   reps  737280 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173
 ( 0.1333360731748201 , 181 , 1 )   reps  1474560 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181
 ( 0.1317096685686114 , 193 , 1 )   reps  2949120 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193
 ( 0.1311982683517524 , 197 , 1 )   reps  5898240 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197
 ( 0.1275639755045533 , 229 , 1 )   reps  11796480 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229
 ( 0.1271587398372755 , 233 , 1 )   reps  23592960 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233
 ( 0.1263760881150453 , 241 , 1 )   reps  47185920 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241
 ( 0.1249121781636255 , 257 , 1 )   reps  94371840 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257
 ( 0.1238932866290727 , 269 , 1 )   reps  188743680 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269
 ( 0.1232476925609442 , 277 , 1 )   reps  377487360 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277
 ( 0.1229342993142804 , 281 , 1 )   reps  754974720 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281
 ( 0.1220292459879827 , 293 , 1 )   reps  1509949440 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293
 ( 0.1206269875212659 , 313 , 1 )   reps  3019898880 prod   5^4 13^2 17^2 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313
 ( 0.1204126778815846 , 29 , 2 )   reps  4529848320 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313
 ( 0.1203610007213705 , 317 , 1 )   reps  9059696640 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317
 ( 0.1190957566850307 , 337 , 1 )   reps  18119393280 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337
 ( 0.1183840591148901 , 349 , 1 )   reps  36238786560 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349
 ( 0.1181540875758708 , 353 , 1 )   reps  72477573120 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353
 ( 0.1170544627527939 , 373 , 1 )   reps  144955146240 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373
 ( 0.1162300592682939 , 389 , 1 )   reps  289910292480 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389
 ( 0.1158346526511383 , 397 , 1 )   reps  579820584960 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397
 ( 0.115640914438019 , 401 , 1 )   reps  1159641169920 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401
 ( 0.1152610595238196 , 409 , 1 )   reps  2319282339840 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409
 ( 0.1147094651682719 , 421 , 1 )   reps  4638564679680 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421
 ( 0.1141784098763921 , 433 , 1 )   reps  9277129359360 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433
 ( 0.1135000135583575 , 449 , 1 )   reps  18554258718720 prod   5^4 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449
 ( 0.1132827525593783 , 5 , 5 )   reps  22265110462464 prod   5^5 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449
 ( 0.1131727366659431 , 457 , 1 )   reps  44530220924928 prod   5^5 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457
 ( 0.1130119349340791 , 461 , 1 )   reps  89060441849856 prod   5^5 13^2 17^2 29^2 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461
 ( 0.1122886528902697 , 37 , 2 )   reps  133590662774784 prod   5^5 13^2 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461
 ( 0.1121589678232448 , 13 , 3 )   reps  178120883699712 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461
 ( 0.1112158783706281 , 509 , 1 )   reps  356241767399424 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509
 ( 0.1108016106675626 , 521 , 1 )   reps  712483534798848 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521
 ( 0.1101384080494621 , 541 , 1 )   reps  1424967069597696 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541
 ( 0.1096306885348829 , 557 , 1 )   reps  2849934139195392 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557
 ( 0.1092623336390804 , 569 , 1 )   reps  5699868278390784 prod   5^5 13^3 17^2 29^2 37^2 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569
 ( 0.109184661243966 , 41 , 2 )   reps  8549802417586176 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569
 ( 0.109022393369416 , 577 , 1 )   reps  17099604835172352 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577
 ( 0.1085553762730775 , 593 , 1 )   reps  34199209670344704 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593
 ( 0.1083280294582822 , 601 , 1 )   reps  68398419340689408 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601
 ( 0.1079943548673958 , 613 , 1 )   reps  136796838681378816 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613
 ( 0.1078850289369187 , 617 , 1 )   reps  273593677362757632 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617
 ( 0.1072480305374002 , 641 , 1 )   reps  547187354725515264 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641
 ( 0.1069411297181192 , 653 , 1 )   reps  1094374709451030528 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653
 ( 0.1067405999140516 , 661 , 1 )   reps  2188749418902061056 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661
 ( 0.1064456832186591 , 673 , 1 )   reps  4377498837804122112 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673
 ( 0.1063489015533347 , 677 , 1 )   reps  8754997675608244224 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677
 ( 0.1057834942757523 , 701 , 1 )   reps  17509995351216488448 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701
 ( 0.1056006150284899 , 709 , 1 )   reps  35019990702432976896 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709
 ( 0.1050677386679089 , 733 , 1 )   reps  70039981404865953792 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733
 ( 0.1045571277453435 , 757 , 1 )   reps  140079962809731907584 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757
 ( 0.1044740746281937 , 761 , 1 )   reps  280159925619463815168 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761
 ( 0.1043096601296809 , 769 , 1 )   reps  560319851238927630336 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769
 ( 0.1042282850833419 , 773 , 1 )   reps  1120639702477855260672 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773
 ( 0.103751273435192 , 797 , 1 )   reps  2241279404955710521344 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797
 ( 0.103519712755121 , 809 , 1 )   reps  4482558809911421042688 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809
 ( 0.1032925703785417 , 821 , 1 )   reps  8965117619822842085376 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821
 ( 0.1031435227654369 , 829 , 1 )   reps  17930235239645684170752 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829
 ( 0.1027073457992096 , 853 , 1 )   reps  35860470479291368341504 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853
 ( 0.1026361963957853 , 857 , 1 )   reps  71720940958582736683008 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857
 ( 0.1022867948546604 , 877 , 1 )   reps  143441881917165473366016 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877
 ( 0.1022181522110647 , 881 , 1 )   reps  286883763834330946732032 prod   5^5 13^3 17^2 29^2 37^2 41^2 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881
 ( 0.1021247598253814 , 53 , 2 )   reps  430325645751496420098048 prod   5^5 13^3 17^2 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881
 ( 0.101539149197962 , 17 , 3 )   reps  573767527668661893464064 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881
 ( 0.1014246637053439 , 929 , 1 )   reps  1147535055337323786928128 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929
 ( 0.1012975688001153 , 937 , 1 )   reps  2295070110674647573856256 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937
 ( 0.1012345460491029 , 941 , 1 )   reps  4590140221349295147712512 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941
 ( 0.1010475355097117 , 953 , 1 )   reps  9180280442698590295425024 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953
 ( 0.1006824786789692 , 977 , 1 )   reps  18360560885397180590850048 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953 977
 ( 0.1003869949333126 , 997 , 1 )   reps  36721121770794361181700096 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953 977 997
 ( 0.1002133496204834 , 1009 , 1 )   reps  73442243541588722363400192 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953 977 997 1009
 ( 0.1001560585753007 , 1013 , 1 )   reps  146884487083177444726800384 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953 977 997 1009 1013
 ( 0.1000423464158209 , 1021 , 1 )   reps  293768974166354889453600768 prod   5^5 13^3 17^3 29^2 37^2 41^2 53^2 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 233 241 257 269 277 281 293 313 317 337 349 353 373 389 397 401 409 421 433 449 457 461 509 521 541 557 569 577 593 601 613 617 641 653 661 673 677 701 709 733 757 761 769 773 797 809 821 829 853 857 877 881 929 937 941 953 977 997 1009 1013 1021


问:
Finding $\lim_{x \to 0} \left( \frac{\sin(x)-3\sinh(x)+2x}{x^2(\tanh(2x)+ \sin(x))} \right)$
I am trying to evaluate the following limit:
$$L=\lim_{x \to 0} \left( \frac{\sin(x)-3\sinh(x)+2x}{x^2(\tanh(2x)+\sin(x))} \right)$$

Begin by rewriting the limit as:
$$L=\frac{\lim\limits_{x \to 0}\left(\cfrac{\sin(x)-3\sinh(x)+2x}{x^2} \right)}{\lim\limits_{x \to 0}(\tanh(2x)+\sin(x))} \tag{1}$$
Applying L'Hospital's Rule to the numerator only:
$$L=\frac{\lim\limits_{x \to 0}\left(\cfrac{\cos(x)-3\cosh(x)+2}{2x} \right)}{\lim\limits_{x \to 0}(\tanh(2x)+\sin(x))} \tag{2}$$
The numerator is still in an indeterminate form, applying L'Hopital to the numerator again:
$$L=\frac{\lim\limits_{x \to 0}\left(\cfrac{-\sin(x)-3\sinh(x)}{2} \right)}{\lim\limits_{x \to 0}(\tanh(2x)+\sin(x))} \tag{3}$$

Rewriting as a single limit:
$$L=-\frac{1}{2}\lim_{x \to 0}\frac{\sin(x)+3\sinh(x)}{\tanh(2x)+\sin(x)} \tag{4}$$
And applying  L'Hospital's Rule...
$$L=-\frac{1}{2}\lim_{x \to 0}\left(\frac{\cos(x)+3\cosh(x)}{2\operatorname{sech}^2(2x)+\cos(x)} \right)=-\frac{2}{3} \tag{5}$$
But according to Wolfram Alpha, $L=-\frac{2}{9}$

So something must be wrong in my calculation (I guess it's the limit of a product bit)?

答:
To make it as simple as possible, I use equivalents  via Taylor-Young's formula at the relevant order:

$\sin x-2\sinh x+2x=x-\frac{x^3}6+o(x^3)-3x-\frac{3x^3}6+o(x^3)+2x=-\frac{2x^3}3+o(x^3)$.

Therefore $\:\sin x-2\sinh x+2x\sim_0-\dfrac{2x^3}3.$

$\tanh 2x+\sin x=2x+o(x)+x+o(x)=3x+o(x)\sim_0 3x$

As a consequence
$$\frac{\sin x-2\sinh x+2x}{x^2(\tanh 2x+\sin x)}\sim_0\frac{-\cfrac{2x^3}3}{x^2\cdot 3x}=-\frac29.$$

ContentLicense: CC BY-SA 4.0


问:
Solve trigonometric equation $\tan\theta + \sec\theta =2\cos \theta$

$$\tan\theta + \sec\theta =2\cos \theta,\quad 0\le \theta\le 2\pi$$Find all the possible solutions for the equations.

Multiply both sides by $\sec\theta - \tan \theta$.
$$\implies (\tan\theta + \sec\theta)(\sec\theta - \tan\theta) = (\sec\theta -\tan\theta)2\cos \theta$$
$$\implies 1 = 2 -2\sin \theta$$ $$\implies \sin \theta=\frac12 \implies \theta = \arcsin\frac12$$Such a solution gets me two solutions $\frac{\pi}6$ and $\frac{5\pi}6$. But when I Wolfram it, I am supposed to get one more solution i.e $\frac{3\pi}2$, but at $\frac{3\pi}2$ $\tan \theta$ and $\sec\theta$ aren't defined.

答:
$$\sec\theta+\tan\theta=2\cos\theta=\frac2{\sec\theta}$$

$$\implies \sec\theta-\tan\theta=\frac1{\sec\theta+\tan\theta}=\frac{\sec\theta}2$$

$$\implies \sec\theta\left(1-\frac12\right)=\tan\theta$$

$$\implies \frac{\tan\theta}{\sec\theta}=\frac12\iff \sin\theta=\frac12=\sin\frac\pi6$$

$$\implies \theta=n\pi+(-1)^n\frac\pi6\text{ where }n \text{ is any integer}$$

If $n$ is even $=2m$(say), $\theta=2m\pi+\frac\pi6$

As $0\le \theta\le2\pi, 0\le2m\pi+\frac\pi6\le2\pi\implies 0\le 12m+1\le12\implies m=0$

Similarly, if $n$ is odd $=2m+1$(say), $\theta=(2m+1)\pi-\frac\pi6\implies m=0$

ContentLicense: CC BY-SA 3.0


问:
Counts on spanning tree with G and its complement graph G'
I got a really hard problem that is there any graphs G(G is not a self-complementary graph) that has the same count of spanning tree with its complement tree G'?

I tried multiple methods to get the result but got nothing.

I tried to use Java to generate all graphs and get its complement graph and compare their counts of spanning trees. It is aged so that I stopped it when generating order-10 graphs. That is, now I know it doesn't exist with graphs that have less than 10 vertices.
I tried to use the degree matrix of G to show if there can or cannot be the same co-factor with a graphic matrix with its complement graph's matrix, but as the calculate was based on the matrixes, I didn't get a reasonable result that can prove anything.
Thus, are there any graphs that have the same count of spanning tree with its complement graph and why?

Thanks!

答:
There are such graphs. Indeed, there are infinitely many non self-complementary graphs that have the same Tutte polynomial as their complement; see this mathoverflow post. Since the Tutte polynomial encodes the number of spanning forests of a graph (i.e. if $T_G$ is the Tutte polynomial of $G$, $T_G(1,1)$ is the number of spanning forests), this answers your question affirmatively.

Well, except you were asking about spanning trees and I'm talking about spanning forests. But at a cursory glance it seems that the construction cited in the link gives connected graphs with connected complements, in which case the two notions are the same.


问:
Are the graphs of inverse functions always reflections of the function's graph in the line $y = x$?
For an invertible function, is the graph of its inverse always the mirror image of the function's graph in the line $y=x$? And, if yes, then why is it so?

答:
Yes,
I used to find the same thing confusing, this is the best way I found to think about it:
the first step is to believe that swapping x and y is the same as reflecting wrt $y=x$ (this should not be difficult).

Then, if $g$ is the inverse of $f:A\to B$, it means that $x = g(y)$. In other words, I imagine that I can write a formula $g(y_0)$ that gives me the $x$ corresponding to a given $y_0$ for every possible $y_0$. If I consider the points satisfying my original equation $y=f(x)$ and the new $x=g(y)$, these are the same curves in the plane.

However, if I want to graph the function corresponding to "formula" $g$, I will do as I always do, and plot $y = g(x)$. This, compared to the original graph $x=g(y)$, corresponds to switching x and y, i.e. the reflection of the original graph.

Note that I didn't discuss details on domain and codomain, the above cosiderations assumes they are "good enough".


问:
how to integrate $\int x^ae^xdx$ where a is non-integer
It's not hard to evaluate $\int x^ne^xdx$ if n is integer, but how to do it if n is non-integer ?

答:
By parts,

$$\int x^ne^x\,dx=x^ne^x-n\int x^{n-1}e^x\,dx
\\=x^ne^x-nx^{n-1}e^x+n(n-1)\int x^{n-1}e^x\,dx
\\=\cdots
\\=e^x\sum_{k=0}^\infty(-1)^k(n)_kx^{n-k}.$$

where $(n)_k$ denotes the falling factorial.

Unfortunately, this series diverges for all $x$.


问:
For which a sequence has a limit in $\mathbb{R}$
Let $x_1 = a$. And $x_{n + 1} = x_n^2 + 3x_n + 1$.

So, the question is: "For which a this sequence has a limit in $\mathbb{R}$.

My idea is to prove that this sequence is monotonous and limited. But looks like that's not true because it has quadratic dependence from a previous member.

The second idea is to use Cauchy convergence test, but looks like its harder to prove it's fundamentality.

答:
By induction, it is easy to see that for all $n \geq 2$, one has $x_n > 1$. Then for $n \geq 2$, you have $x_{n+1} \geq x_n^2 + 1 \geq x_n + 1$, so
$$x_{n+1}-x_n \geq 1$$
so $x_n \geq n-1 + x_1$ which tends to $+\infty$, so $(x_n)$ tends to $+\infty$.


问:
Does the following identity hold?
I encountered a claim in the proof a a claim in a paper that seemed to imply/use the following:

Suppose we have two subgroups $K$ and $H$ of a larger group $G$.  Let $T$ be a translation of $K$, which is defined as a subset of $G$ such that every $g \in G$ can be uniquely written as $t k$, for some $t \in T$ and $k \in K$. For example, if $G = \mathbb{Z} / 12 \mathbb{Z}$ and $K = \{0, 2, 4, \cdots, 10\}$, then $T = \{0, 1\}$ is a possible translation of $K$.

Let us also for simplicity define the notation $t K = \{tk ~|~ k \in K\}$.

The claim is then the following: for any translation of $T$ and any subgroup $H \leq G$ which is neither a subset nor a superset of $K$, we have the following:
$$\sum_{t \in T} |tK \cap H|^2 = |H| \cdot |K \cap H|$$

Is this claim true? It does seem to work out for example with $G, T$ as given above and $H = \{0, 3, 6, 9\}$, but I can't seem to find a proof of this fact or even an intuition as to why this should be true. If it is not true, then are there any other conditions we can impose (maybe I have missed them in reading the paper) so that it is true?

EDIT: The paper in question can be found on ArXiv at this link. The question stems from a step in the proof of lemma $4$.

答:
This is basically the same argument as Benjamin's, but I dislike putting group elements in lists.

Let $T'=\{t\in T\mid tK\cap H=\varnothing\}$ as in the other answer.

Not only is $|tK\cap H|$ always either $0$ or $|K\cap H|$, but it's a coset of $K\cap H$. That is, if $s\in tK\cap H$ (i.e. the intersection is nonempty) then $s\in tK$ means $tK=sK$ and $s\in H$ means $H=sH$ so

$$ tK\cap H=sK\cap sH=s(K\cap H). $$

This yields a map $T'\to H/(K\cap H)$. It's actually a one-to-one correspondence.

Since the $tK$s (hence the $tK\cap H$s) are disjoint, this map is one-to-one, and it's also onto since $G/K$ is a partition so every coset of $H/(K\cap H)$ intersects some $tK$ (hence some $tK\cap H$).

This means we have

$$ \sum_{t\in T'} |tK\cap H|^2=\left|\frac{H}{K\cap H}\right||K\cap H|^2 = |H||K\cap H|. $$

Title: Does the following identity hold?


问:
Every solution of $x'= f(x)$ is trivial or regular
How can I prove that every solution of $x'= f(x)$, ($x$ is a function from an open interval to $\mathbb R^n$) is trivial or regular?

I need a hint or something to begin to solve this question.

Thanks a lot

EDIT

Note that we have to prove that a solution of an ODE of this form
hasn't singularities, except in the trivial solution.

$f$ is a $C^1$ class function.

Regular in this context means $f(x)\neq 0$ for each $x$.

Trivial in this context means $f(x)=0$ for each $x$

答:
Thanks for fixing the question. The starting point is the Picard uniqueness theorem. It applies here because a $C^1$ satisfies the Lipschitz condition on every closed interval.

The uniqueness of solution means:

If two solutions agree at some moment, they agree at all times.

Let's rephrase this in the special case when the second solution is a trivial one.

If a  solution  is at a singular point at some moment, it is there at all times.

Which is what you were asked to show.


问:
What does the notation f ∈ {0, 1}^N mean?
I'm trying to solve a problem in which this notation occurs. On the one hand, I have the question of what it means that a function f is an element of {0,1} ^ N and what {0,1} ^ N means.

Thank you in advance!

答: $X^Y$ denotes the set of functions from $Y$ to $X$. If you have $\lbrace 0,1 \rbrace^{\mathbb{N}}$, this denotes "functions" from $\mathbb{N}$ to $\lbrace 0,1 \rbrace$, i.e. the set of sequences of $0$ and $1$.


问:
If $G/H$ and $H$ are finitely generated, then so is $G$
I'm trying to prove that if $H$ is a normal subgroup of a group $G$ such that $H$ and $G/H$ are finitely generated, then G is finitely generated also. I'm trying to find a finite set $X$ such that $G$ is generated by $X$, but I have no ideal how to find this set using the finite generator sets of $H$ and $G/H$.

I need help

Thanks in advance

答:
Let $S$ be a finite generating set for $H$. By the normality of $H$, we have for all $g \in G$ that $g = h_{1}gh^{-1}$ for some $h,h_{1} \in H$.

Using the generalized associative law for $G$ and reindexing if necessary we have that

$g = gh_{1_S}=g($finite product in $S)$.

Since $G/H$ partitions $G$ we know that $gh_{1_S}$ is in one of the cosets of $G$ in $H$, meaning that

$gh_{1_S} \in ($finite product of elements in $G$ taken from a (finite subset of $G)=G_{1} )H$ using the fact that in our case, $(gH)^{k} = g^{k}H$ as well as a similar technique to the one above to put $H$ at the end.

Then
$g = ($finite product in $G_{1})($finite product in $S)$ for all $g \in G$ where $|S|,|G_{1}|$ are finite.

Hence, one can take $S \cup G_{1}$ to be a finite generating set for $G$.


答:
It means that $f$ is a boolean function [the image is $\{0,1\}$], where the domain $S$ has $N$ elements.

The function $f$ can be thought of as a vector of length $N$ as well, where writing $S=\{s_1,\ldots, s_n\}$, then $f$ can be written $f=(f(s_1),f(s_2),\ldots, f(s_n))$.

Title: What does the notation f ∈ {0, 1}^N mean?

Body:
I'm trying to solve a problem in which this notation occurs. On the one hand, I have the question of what it means that a function f is an element of {0,1} ^ N and what {0,1} ^ N means.

Thank you in advance!


问:
Why functions can be thought as tuples?
I found these two posts but they don’t explain in detail why functions can be thought as tuples. The autor wants to show that $\mathbb R^n$ and $\mathbb R^{\{1,2,...,n\} }$ are actually the same thing.

Note: if $S$ is a set then $\mathbb R^S$ is the set of all functions from $S$ to  $\mathbb R $.

Intrepreting tuples as functions

help understanding a paragraph from Linear Algebra Done Right

答:
The spaces $\Bbb R^{\{1,\dots,n\}}$ and $\Bbb R^n$ are isomorphic as vector spaces. In particular, the map $\Phi:\Bbb R^{\{1,\dots,n\}} \to \Bbb R^n$ defined by
$$
\Phi(f) = (f(1),f(2),\dots,f(n))
$$
is an isomorphism.

Title: Why functions can be thought as tuples?


问:
Proving a condition on a cont-differentiable function on positive real numbers.
Let f be a continuously differentiable function on [0,infinity) such that $f '(x) \le f(x)$ for all $x$.Suppose $f(0)=5$. Show that $f(x) \le 5e^{x} ~~\forall x$.

I am not getting how we will proceed with this one.Can anyone plz help me to figure it out (I tried using mean value theorem but to no avail).Thanks.

答:
Note that we have
$$f'(x) - f(x) \leq 0 \implies e^{-x}\left(f'(x) - f(x)\right) \leq 0\implies \dfrac{d}{dx}\left(f(x) e^{-x}\right) \leq 0$$
Hence, $f(x) e^{-x}$ is a decreasing function. Hence,
$$f(x) e^{-x} \leq f(0)e^{-0} = f(0) = 5 \implies f(x) \leq 5 e^x$$</p>


问:
How to interpret the convexity condition for functions?
A set $C$ is called convex, if for any two of its points $x$ and $y$ points of the form $tx+(1-t)y$ for $t\in[0,1]$ belong to $C$ as well. This can be easily interepreted as the condition that for any two points in the set, the line joining these points belongs entirely to the set.

On a convex set $C$ one can define a convex function $f:C\to\Bbb R$, that is a function such that $\forall x,y\in C, t\in[0,1]$ the condition $f(tx+(1-t)y)\le tf(x)+(1-t)f(y)$ holds. Is there an interepretion for this condition akin to one for the definition of convex sets? Best I can do is "image of a line is less than or equal (whatever that means) to the line connecting the images of the endpoints". Is there anything neater/cleaner?

答: This is equivalent to the epigraph of $f$, which is $\{(x,t):f(x)\leq t,x\in Dom(f)\}$, being a convex set.


问:
How to estimate join probability?
Problem:
a fair die is rolled at the same time as a fair coin is tossed. Let A be the number on the upper surface of the die and let B describe the outcome of the coin toss, where B is equal to 1 if the result is “head” and it is equal to 0 if the result if “tail”. The random variables X and Y are given by X = A + B and Y = A − B, respectively. Calculate P(X,Y)?

Answer:
since both events A and B are independent,  P(A=a,B=b) can be found; P(A=a,B=b) = 1/6*1/2 = 1/12. With the provided information, P(X) and P(Y) can also be calculated. I am not sure how to calculate P(X,Y)? Any help will be appreciated?

```<code>X = {1,2,3,4,5,6,7} -> pX(x) = {1/12, 1/6, 1/6, 1/6, 1/6, 1/12}

Y = {0,1,2,3,4,5,6} -> pY(y) = {1/12, 1/6, 1/6, 1/6, 1/6, 1/12}
</code>```

答:
It is difficult to get to $P(X = x, Y = y)$ directly from $P(X = x)$ and $P(Y = y)$. It is simpler to get there from $P(A = a, B = b)$.

Take an example,

$P(X = 1) = \frac{1}{12}$, $P(Y = 1) = \frac{1}{6}$.

There is only one possible way for $X = 1$ which is die shows $1$ and you get a tail on the coin. So the probability of $\frac{1}{12}$.

But when it comes to $P(Y = 1)$, you have two possible ways: $1$ on the die and tail on the coin OR $2$ on the die and head on the coin. So the probability of $\frac{1}{6}$.

Now when it comes to finding $P(X = 1, Y = 1)$, please note that you are looking for probability where the sum and difference are both $1$ which is only possible if die showed $1$ and you got a tail on the coin.

So to find $P(X = 1, Y = 1)$, we focus on $1$ on the die and tail on the coin which is nothing but $P(A = 1, B = 0) = \frac{1}{12}$.

If $A = 1, B = 1$, given $X = A + B, Y = A - B \implies A = \frac{X + Y}{2} = 1, B = \frac{X - Y}{2} = 0$ and that is why we found $P(A = 1, B = 0)$.

Valid combinations of X, Y are

i) $\{1, 1\}, \{2, 2\}, \{3, 3\}, \{4, 4\}, \{5, 5\}, \{6, 6\}$ that correspond to $A = a, B = 0$.

ii) $\{7, 5\}, \{6, 4\}, \{5, 3\}, \{4, 2\}, \{3, 1\}, \{2, 0\}$ that correspond to $A = a, B = 1$.

All of the above combinations will have probability of $\frac{1}{12}$. Rest of them will have probability of zero. For example , $P(X = 1, Y = 0) = 0$.

Title: How to estimate join probability?


答:
Imagine a function $f$ from the set $\{1,2,..,n\}$ to the real numbers. To define such a function we must specify one value for each of the inputs $1,2,..,n$. Therefore, we get $f(1), f(2), .., f(n)$ which we can put in a tuple as $(f(1), f(2),..,f(n))$. Therefore, we can map any function to a tuple in $\mathbb R^n$.

Given any tuple $(x_1,.x_n)\in\mathbb R^n$, we can define the function $f$ by $f(i)=x_i$

Therefore, we can view any tuple as a function from $\{1,2,..,n\}$ to $\mathbb R$.

Title: Why functions can be thought as tuples?

Body:
I found these two posts but they don’t explain in detail why functions can be thought as tuples. The autor wants to show that $\mathbb R^n$ and $\mathbb R^{\{1,2,...,n\} }$ are actually the same thing.

Note: if $S$ is a set then $\mathbb R^S$ is the set of all functions from $S$ to  $\mathbb R $.

Intrepreting tuples as functions

help understanding a paragraph from Linear Algebra Done Right


问:
maximum number of edges to be removed to possess a property
I am working on a problem. We know that on squaring a cycle, degree of every vertex is 4. For squares of cycles, we know if we delete any arbitrary edge then still eccentricity is same for all vertices. And if we delete 3 edges incident on a single vertex, it loses its property that eccentricity is same for all vertices, because eccentricity is not same for those graphs having a vertex of degree1. How to find out the maximum number of edges to be deleted so that eccentricity is still same.  Any idea or suggestion will be helpful. Thanks for your help.</p>

答:
Here is what I can say with certainty

Let $n\ge 5$.  The maximum number of edges that can be removed from a squared cycle so that the eccentricity of each vertex remains unchanged is at least $n-2-(n+2)\operatorname{mod}4$.

To show this is true, I will give an explicit set of removed edges in each case.  Label consecutive vertices of the cycle $1,2,3,\ldots, n$ and call an edge 'outer' if it connects adjacent vertices in the cycle, and 'inner' if it connects vertices a distance of two from each other.  Splitting into four cases, depending on the residue class of $n$ modulo $4$, we can remove the following edges:

$n\equiv 0\operatorname{mod}4$: $n-6$ outer edges and $2$ inner edges.  Remove all outer edges except those connecting $(1,2),(2,3),(3,4),(\frac{n}{2}+1,\frac{n}{2}+2),(\frac{n}{2}+2,\frac{n}{2}+3),$ and $(\frac{n}{2}+3,\frac{n}{2}+4)$.  Remove inner edges connecting $(1,3)$ and $(\frac{n}{2}+2,\frac{n}{2}+4).$

$n\equiv 1\operatorname{mod}4$: $n-5$ outer edges and $0$ inner edges.  Remove all outer edges except those connecting $(1,2),(2,3),(3,4),(\frac{n+3}{2},\frac{n+3}{2}+1),$ and $(\frac{n+3}{2}+1,\frac{n+3}{2}+2)$.  Remove no inner edges.

$n\equiv 2\operatorname{mod}4$: $n-4$ outer edges and $2$ inner edges.  Remove all outer edges except those connecting $(1,2),(2,3),(\frac{n}{2}+1,\frac{n}{2}+2),$ and $(\frac{n}{2}+2,\frac{n}{2}+3)$.  Remove inner edges connecting $(1,3)$ and $(\frac{n}{2}+1,\frac{n}{2}+3)$.

$n\equiv 3\operatorname{mod}4$: $n-5$ outer edges and $2$ inner edges.  Remove all outer edges except those connecting $(1,2),(2,3),(3,4),(\frac{n+3}{2},\frac{n+3}{2}+1),$ and $(\frac{n+3}{2}+1,\frac{n+3}{2}+2)$.  Remove inner edges connecting $(1,3)$ and $(\frac{n+3}{2},\frac{n+3}{2}+2)$.

This should all match up with the conjecture.  The above is proved case-by-case by induction.  Notice that the remaining graphs in all cases have two gaps of outer edges.  In each gap, we can add $2$ vertices and $2$ edges so that the eccentricity of each vertex is raised by $1$ (note that the eccentricity of a squared cycle with $n+4$ edges is $1$ greater than the eccentricity of a squared cycle with $n$ edges).

The hard part is getting rid of the words 'at least' in the above claim.  It can be shown that when $n\equiv 1\operatorname{mod}4$, we cannot remove any inner edges, and we need at least $5$ outer edges to preserve eccentricity.  This establishes the result for $n\equiv 1\operatorname{mod}4$.

When $n\equiv 0\operatorname{mod}4$, we can remove at most $2$ inner edges, but I can't seem to show that $6$ outer edges are necessary.

The other cases are even trickier.  I have reason to believe my result is true for all cases, but a formal proof eludes me.


问:
What is the time of small oscillation in this question?
Question :

A particle rests in equilibrium under the attraction of two centres of force which attract directly as the distance, their attractions per unit of mass at unit distance being u and u'; the particle is slightly displaced towards one of them;

Show that the time of small oscillation is
$$
{2\pi \over \left(\,{u + u'}\,\right)^{1/2}}
$$

Solution
Force diagram
-In this picture F1 is the force on the particle due to B and
F2 is the force on the particle due to A.

$$F1 = m.u.(a - x) $$ and

$$F2 = m.u'.(a + x) $$
where m = mass of the particle

u = force per unit mass per unit distance

Net Resultant force $$= F2 - F1 $$
$$=m.(u'a + u'x - ua + ux) $$
$$=m.(u + u')[x - a.(u-u')/(u+u')]$$

Accelaration $$=(u + u')[x - a.(u-u')/(u+u')] $$

Time period = $${2\pi \over \left( {u + u'}\, \right)^{1/2} }$$

答:
Let's make sure we get all the signs right.
Your origin is at the middle of the distance between the two objects, With the positive direction towards $B$. Then $$\frac{F_A}m=-u(a+x)\\\frac{F_B}m=+u'(a-x)$$
Then the acceleration is $$\ddot x=-u(a+x)+u'(a-x)=-(u+u')x+(u'-u)a$$
This seems similar to the equation for a mass/spring system:
$$\ddot x=-\frac km x$$
The period for this is $$T=\frac{2\pi}{\sqrt{\frac km}}$$
We can see a similarity, if we write $u+u'=\frac km$. So what's the difference? Is that extra term at the end. We can rewrite that equation as $$\ddot x=(u+u')(x-x_0)$$where $$x_0=\frac{u'-u}{u'+u}a$$
Then change the variable $y=x-x_0$, with $\ddot y=\ddot x$, so the period for $y$ is then $$T=\frac{2\pi}{(u+u')^{1/2}}$$
Wondering what that $x_0$ is? It's the equilibrium position, where the two forces cancel. Going back to the first formula for acceleration, and setting it to be $0$ at $x_0$ you get
$$-(u+u')x_0+(u'-u)a=0$$
Solve for $x_0$, and you get the formula that I've put in before:$$x_0=\frac{u'-u}{u'+u}a$$


问:
Multivariable mean value theorem for proving contraction
Let $f : \mathcal{I} \subset \mathbb{R}^n \rightarrow \mathcal{I}$ is continuous, and $\mathcal{I}$ is a closed $\ell_{\infty}$ norm ball. From Brouwer fixed point theorem (p. 240, Real Mathematical Analysis by C. Pugh), a fixed point of $f$ exists in $\mathcal{I}$. To give a sufficient condition for $f$ to be a contraction on $\mathcal{I}$, can I then simply require $\big\Vert \frac{\partial f}{\partial x}\big\Vert_{\infty} < 1$? It seems from this post Mean value theorem for vector valued multivariable function such an approach is valid for $\ell_2$ norm, but I'm not sure if that's the case for $\ell_{\infty}$ norm.

This condition is from a proof in the appendix of a paper I'm trying to understand, but not much detail is provided. I'd appreciate any suggestions/counterexamples.

答:
If I understand your question correctly:

Let $f: \mathcal{I} \to \mathbb{R}^n$ be continuously differentiable. Then the Frechet derivative at $x$ is the same, whether we use the $\ell^\infty$ or $\ell^2$. The reason is the following. Write $x=(x_1,\dots,x_n)$. Then $|x_k|=\sqrt{|x_k|^2}\leq \sqrt{|x_1|^2+\dots+|x_n|^2}=\|x\|_2$ for all $k$. Hence, $\|x\|_\infty \leq \|x\|_2$.

Similarly, $\|x\|_2=\sqrt{|x_1|^2+\dots+|x_n|^2}\leq \sqrt{n\|x\|_\infty^2}=\sqrt{n}\|x\|_\infty.$

Denote the Frechet derivative in the $\ell_2$ norm of $f$ at $x$ by $Df(x)$. Then $Df(x)$ is also the Frechet derivative with respect to the $\ell^\infty$ norm since
$$
\frac{\|f(x+h)-f(x)-Df(x)h\|_\infty}{\|h\|_\infty}\leq \sqrt{n}\frac{\|f(x+h)-f(x)-Df(x)h\|_2}{\|h\|_2}
$$
As $\|h\|_\infty \to 0$, we also have $\|h\|_2 \to 0$. The right hand side goes to zero. Therefore, the left hand side does as well by squeeze theorem so $Df(x)$ is the same in both norms.

In order to be a contraction, we use the fact that
$$
|f(x)-f(y)|_\infty\leq \sup_{x\in \mathcal{I}}\|Df(x)\|\|x-y\|_\infty
$$
If for all $x$, we have $\|Df(x)\|\leq \alpha <1$, then $f$ will be a contraction.


问:
How is this average possible?
I am studying atomic physics from Physics of Atoms and Molecules (B. H. Bransden) and in some part he says:

\begin{equation}
\left\langle\cos^2 \theta\right\rangle=\dfrac{1}{3}
\end{equation}

But from everything I know this should be equals to $\dfrac{1}{2}$.

Visual proof (page 168 of the book)

答: Depends on what $\langle \rangle$ and $\theta$ mean.  If $\theta$ is colatitude on a sphere, where the area element is $\sin(\theta)\; d\theta d\phi/(4\pi)$, then they are right.

Title: How is this average possible?


问:
Derivative of a piecewise function at its endpoints
I have a function $f(x)= \begin{cases} 3-(x+1)^2 & \text{if } x\leq0, \\ 1+(x-1)^2 & \text{if } x \geq 0. \end{cases}$ and I want to evaluate $f'(0)$. If I differentiate,$f'(x)= \begin{cases} -2x-2 & \text{if } x\leq0, \\ 2x-2 & \text{if } x \geq 0. \end{cases}$ and we can see at $f'(0)$, both are $-2$. Does this mean I can conclude that $f'(0)=-2$?

Can I take this to the general case, such that if I have two  sufficiently well-behaved cases in my piece wise function(continuous at endpoint etc), and I want to evaluate the derivative of the function at their endpoint, is it sufficient for the derivative of both "pieces" to be the same at that point for me to conclude that the derivative of the function at that point exists and is equal to the value calculated?

答: Yes, since $f$ is continuous at $0$ and since your computations show that the left derivative of $f$ at $0$ is equal to the right derivative of $f$ at $0$.


问:
Number of prime ideals in Dedekind domain containing a given prime.
I got stuck on the following problem:

Let $L/\mathbb{Q}$ be an extension of degree $4$ such that $\mathcal{O}_L=\mathbb{Z}[\alpha]$ is generated by a single element. Show that $\mathcal{O}_L$ has at most $3$ prime ideals that contain $3$.

I know that $\mathcal{O}_L$ is a Dedekind ring and thus the number of its prime ideals which contain $3$ is finite. Why it is at most $3$? Does it have something to do with the dimension of $L/\mathbb{Q}$?

答: By the Kummer-Dedekind theorem, if $3$ is contained in more than $3$ distinct prime ideals of $\Bbb{Z}[\alpha]$, then the minimal polynomial of $\alpha$ over $\Bbb{Z}$ splits into more than $3$ distinct irreducible factors over $\Bbb{F}_3$.


Title: Proving a condition on a cont-differentiable function on positive real numbers.

Body:
Let f be a continuously differentiable function on [0,infinity) such that $f '(x) \le f(x)$ for all $x$.Suppose $f(0)=5$. Show that $f(x) \le 5e^{x} ~~\forall x$.

I am not getting how we will proceed with this one.Can anyone plz help me to figure it out (I tried using mean value theorem but to no avail).Thanks.

Body: Let $g(x) = e^{-x}f(x)$. Then $g'(x) = e^{-x}(f'(x)-f(x))$. Since $g'(x) \le 0$, we have $g(x) \le g(0)$ for all $x \ge 0$. Hence $f(x) \le e^x f(0)$.


问:
Newton-Raphson Method - need help understanding an example
I am currently trying to finish an assignment regarding Newton-Raphson method. Anyone able to explain an example to me? The function $f(x) = \sin(x) + \sin(\frac{10x}{3})$ is shown on the picture. The red dot is the starting point and the cross is a local extrema found by using the NR method. My question is why isn't the cross on the local extremas near the starting point, specifically where $x \approx 1.5$ and $x \approx 0.5$ which is visible on the picture. Why did the point end up where it did and passed two extremas on its way? Could this be an implementation issue, or is this how it's supposed to be ?

答:
Newton-Raphson gives no guarantee to converge to the root nearest to the starting point. It just converges to some root, when it does.

When you are far from a root, the linear approximation on which the method is based does not hold and the iterations can wander randomly.

Anyway, the results that you show do not seem to match this situation.

On this complex plot, you see your curve in blue, and the first derivative in green. Then the magenta curve is the first NR iterate, starting from the initial approximation $x$, and the black curve is the second approximation.

You can see various plateaus corresponding to the extrema of the blue curve (roots of  the green one), and partially drawn asymptotic curves in between, corresponding to erratic behavior.


问:
There is a unique binary relation $<$ which turn the integral domain $(\mathbb{Z},+,\times)$ into an ordered integral domain.
Axioms for order relation:

$x \not\lt x$

$[x \lt y \land y \lt z] \Rightarrow x \lt z$

$[x \lt y \lor x = y \lor y \lt x]$

$x \lt y \Rightarrow x+z \lt y+z$

$[x <y \land 0 < z] \Rightarrow x\times z < y \times z$

Question:
We have that $(\mathbb{Z}, +, \times, <)$ is the ordered integral domain of integers with the usual order relation defined as $x<y \Leftrightarrow y-x \in Z^+$, we must show that if $(\mathbb{Z}, +, \times, <')$ is an ordered integral domain too, then $<'$ is identical with $<$.

Here is my short atempt:

$$(x<y) \Leftrightarrow (0 <y-x) \Leftrightarrow (y-x \in Z^+) \Leftrightarrow (0 <' y-x) \Leftrightarrow (x<'y)$$

Since $Z^+$ is unique and the order relation on integral domain should give $0<x$ for any positive $x$, I tried this approach after failing in all other paths, some of my other ideas was to assume $<$ and $<'$ as ordering relations satisfying the axioms for ordering and then trying to reach a contradiction if the relations are not identical, but I got stucked.

My atempt is correct?

答:
It is enough to prove $1>0$ since this would imply that $x>0>-x$ for $x\in Z^+$, using $1>0$ implies $2>1$ implies ... implies $x>x-1$ and then using the second axiom, while for the other direction we get $1+(-1)>0+(-1)$ using the fourth axiom again.
Since 1 is not 0 in an integral domain, we have that $1>0\vee 1<0$ and if $1<0$, using the fourth axiom, we get that $-1>0$ and using the last axiom we get $0<-1\wedge 0<-1\Rightarrow 0<1$ which is a contradiction. So $1>0$ and we are done.

EDIT: Showing the explicit induction.

Basis for $x=1$ is trivial since we showed $0<1$.

Let $x\in Z^+$ s.t. $x>1$, then $x-1\in Z^+$. Let us assume $x-1>0$ and show that $x>0$: Indeed, we get
$$x=(x-1)+1>^*0+1=1>0$$ where we used the fourth axiom in the starred inequality, and $1>0$ in the last inequality. We finish by using transitivity (2nd axiom) and concluding that $x>0$.


问:
Tensor product of Galois extension
Let $K/k$ be a finite Galois extension of fields with Galois group $G$. How to show that the (n+1)-fold tensor product
$$K \otimes_k K \otimes_k K \cdots \otimes_k K$$  is isomorphic to $$\prod\limits_{G^n} K$$ (where the index set $G^n$ is the n-fold product of $G$) by the map  $$x_0 \otimes x_1 \otimes \cdots x_n \mapsto (x_0g_1(x_1)g_1g_2(x_2) \ldots g_1g_2\ldots g_n(x_n))_{(g_1, \ldots, g_n)}$$

It is claimed as part of a proof of something else) in Stephen Shatz’s book Profinite Groups, Arithmetic, Geometry.

答:
Reuns's answer is short and great. I'll give a down to earth proof so we can see what happenes at the level of elements.

As $K/k$ is finite Galois, it is finite and separable, by primitive element theorem, there exists $\alpha\in K$ with its minimal polynomial $f(x)$ over $k$ s.t $\frac{k[x]}{(f(x))}\cong K,x\mapsto \alpha$. And for each $\sigma\in G$, there exists a polynomial $f_\sigma (x)$ over $k$ s.t. $\frac{k[x]}{(f(x))}\to \frac{k[x]}{(f(x))},x\mapsto f_\sigma (x)$ correpsonds to $\sigma:K\to K$.

We first deal with the case $n=2$:

$$ \begin{align}
K\otimes_k K & \cong \frac{k[x_0]}{(f(x_0))}\otimes  \frac{k[x_1]}{(f(x_1))} \\
& \cong \frac{k[x_0,x_1]}{(f(x_0),f(x_1))}\\
& \cong \frac{K[x_0,x_1]}{(f(x_0),\prod_{\sigma\in G}(x_1-f_\sigma (x_0)))}\\
& \cong \prod_{\sigma\in G} \frac{\frac{k[x_0]}{(f(x_0))}[x_1]}{(x_1-f_\sigma (x_0))}\\
& \cong \prod_{\sigma\in G} \frac{k[x_0,x_1]}{(f(x_0),x_1-f_\sigma(x_0))}\\
& \cong \prod_{\sigma\in G} K\\
\end{align} $$
$$x_0\otimes x_1 \mapsto (x_0 f_\sigma (x_0))_{\sigma \in G}$$
$$\alpha(x_0)\otimes \beta(x_1)\mapsto (\alpha(x_0)\beta(f_\sigma(x_0)))_{\sigma\in G}$$
$$\alpha\otimes \beta\mapsto (\alpha \cdot\sigma(\beta))_{\sigma\in G}$$
If we tensor with another $K\cong \frac{k[x_2]}{(f(x_2))}$, let $s$ be any of the primitive element in $\frac{k[x_0,x_1]}{(f(x_0),x_1-f_\sigma (x_0))}$, it could be $x_0,f_\sigma(x_0)$ or any $f_\tau (x_0)$ where $\tau\in G$. Then we have
$$\frac{k[x_0,x_1]}{(f(x_0),x_1-f_\sigma (x_0))}\otimes_k \frac{k[x_2]}{(f(x_2))}\cong \prod_{\tau\in G}\frac{k[x_0,x_1,x_2]}{(f(x_0),x_1-f_\sigma (x_0),x_2-f_\tau(s))}\cong \prod_{\tau\in G}K$$
Then $x_0\otimes x_1 \otimes x_2$ sends to $x_0 \cdot f_\sigma (x_0) \cdot f_\tau (s)$, if we take $s=x_1=f_\sigma (x_0)$, we have $\alpha_0\otimes \alpha_1 \otimes \alpha_2 \mapsto \alpha_0 \cdot \sigma (\alpha_1)\cdot \tau \sigma (\alpha_2)$

In the general case, in
$${\bigotimes_k}^n K\cong \frac{k[x_0,\cdots,x_n]}{(f(x_0),...,f(x_n))}$$
given an element $(\sigma_1,...,\sigma_n)\in G^n$, the correspoindg $K$-factor is
$$\frac{k[x_0,\cdots,x_n]}{(f(x_0),x_1-f_{\sigma_1}(x_0),x_2-f_{\sigma_2}(x_1),\cdots,x_n-f_{\sigma_n}(x_{n-1}))}$$
$$\cong \frac{k[x_0,\cdots,x_n]}{(f(x_0),x_1-f_{\sigma_1}(x_0),x_2-f_{\sigma_2}f_{\sigma_2}(x_0),\cdots,x_n-f_{\sigma_n}\cdots f_{\sigma_1}(x_0))}\cong K$$
So $x_0\otimes\cdots \otimes x_n\mapsto x_0\cdot [f_{\sigma_1}(x_0)]\cdots [ f_{\sigma_n}\cdots f_{\sigma_1}(x_0)]$ which further induces the map of abstract elements of $K$,
$$\alpha_0\otimes\cdots \otimes \alpha_n\mapsto  \alpha_0 [\sigma_1(\alpha_1)][\sigma_2 \sigma_1(\alpha_2)]\cdots [(\sigma_n\cdots\sigma_1)(\alpha_n)]$$
The result follows. As we can see, the above form is not necessary, if we take the primitive element to be $x_0$, then we can get the following nice written identification:
$$\alpha_0\otimes\cdots \otimes \alpha_n\mapsto  \alpha_0 [\sigma_1(\alpha_1)][\sigma_2 (\alpha_2)]\cdots [\sigma_n(\alpha_n)]$$

Title: Tensor product of Galois extension


问:
Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$
I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$.

I found a solution by myself 10 hours after I posted it, here it is:

$$f(x)=\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{nx^n}{1+x^n},\quad\quad g(x)=\displaystyle\sum_{n=1}^{\infty}\frac{nx^n}{1-x^n},$$

then I must prove that $f(e^{-\pi})=\frac1{24}$. It was not hard to find the relation between $f(x)$ and $g(x)$, namely $f(x)=g(x)-4g(x^2)+4g(x^4)$.

Note that $g(x)$ is a Lambert series, so by expanding the Taylor series for the denominators and reversing the two sums, I get

$$g(x)=\sum_{n=1}^{\infty}\sigma(n)x^n$$

where $\sigma$ is the divisor function $\sigma(n)=\sum_{d\mid n}d$.

I then define for complex $\tau$ the function
$$G_2(\tau)=\frac{\pi^2}3\Bigl(1-24\sum_{n=1}^{\infty}\sigma(n)e^{2\pi in\tau}\Bigr)$$ so that
$$f(e^{-\pi})=g(e^{-\pi})-4g(e^{-2\pi})+4g(e^{-4\pi})=\frac1{24}+\frac{-G_2(\frac i2)+4G_2(i)-4G_2(2i)}{8\pi^2}.$$

But it is proven in Apostol "Modular forms and Dirichlet Series", page 69-71 that $G_2\bigl(-\frac1{\tau}\bigr)=\tau^2G_2(\tau)-2\pi i\tau$, which gives $\begin{cases}G_2(i)=-G_2(i)+2\pi\\ G_2(\frac i2)=-4G_2(2i)+4\pi\end{cases}\quad$. This is exactly was needed to get the desired result.

Hitoshigoto oshimai !

I find that sum fascinating. $e,\pi$ all together to finally get a rational. This is why mathematics is beautiful!

Thanks to everyone who contributed.

答:
We will use the Mellin transform technique. Recalling the Mellin transform and its inverse

$$ F(s) =\int_0^{\infty} x^{s-1} f(x)dx, \quad\quad f(x)=\frac{1}{2 \pi i} \int_{c-i \infty}^{c+i \infty} x^{-s} F(s)\, ds.  $$

Now, let's consider the function

$$ f(x)= \frac{x}{e^{\pi x}+1}. $$

Taking the Mellin transform of $f(x)$, we get

$$ F(s)={\pi }^{-s-1}\Gamma  \left( s+1 \right)  \left(1- {2}^{-s} \right)
\zeta  \left( s+1 \right),$$

where $\zeta(s)$ is the zeta function . Representing the function in terms of the inverse Mellin Transform, we have

$$ \frac{x}{e^{\pi x}+1}=\frac{1}{2\pi i}\int_{C}{\pi }^{-s-1}\Gamma  \left( s+1 \right)  \left( 1-{2}^{-s} \right)
\zeta  \left( s+1 \right) x^{-s}ds. $$

Substituting $x=2n+1$ and summing yields

$$\sum_{n=0}^{\infty}\frac{2n+1}{e^{\pi (2n+1)}+1}=\frac{1}{2\pi i}\int_{C}{\pi}^{-s-1}\Gamma  \left( s+1 \right)\left(1-{2}^{-s} \right)
\zeta\left( s+1 \right) \sum_{n=0}^{\infty}(2n+1)^{-s}ds$$

$$ = \frac{1}{2\pi i}\int_{C}{\pi }^{-s-1}\Gamma  \left( s+1 \right)  \left(1-{2}^{-s} \right)^2\zeta\left( s+1 \right) \zeta(s)ds.$$

Now, the only contribution of the poles comes from the simple pole $s=1$ of $\zeta(s)$ and the residue equals to $\frac{1}{24}$. So, the sum is given by

$$ \sum_{n=0}^{\infty}\frac{2n+1}{e^{\pi (2n+1)}+1}=\frac{1}{24} $$

Notes: 1)

$$ \sum_{n=0}^{\infty}(2n+1)^{-s}= \left(1- {2}^{-s} \right) \zeta  \left( s \right).  $$

2) The residue of the simple pole $s=1$, which is the pole of the zeta function, can be calculated as

$$ r = \lim_{s=1}(s-1)({\pi }^{-s-1}\Gamma  \left( s+1 \right)  \left({2}^{-s}-1 \right)^2\zeta\left( s+1 \right) \zeta(s))$$

$$  =  \lim_{s\to 1}(s-1)\zeta(s)\lim_{s\to 1}  {\pi }^{-s-1}\Gamma  \left( s+1 \right)  \left({2}^{-s}-1 \right)^2\zeta\left( s+1 \right)  = \frac{1}{24}. $$

For calculating the above limit, we used the facts

$$ \lim_{s\to 1}(s-1)\zeta(s)=1, \quad \zeta(2)=\frac{\pi^2}{6}. $$

3) Here is the technique for computing the Mellin transform of $f(x)$.


答:
Suppose you have
$$
f(x)=\begin{cases}
h(x)  \ , \ x \geq 0,\\
g(x)  \ , \ x \leq 0,
\end{cases}
$$
where $h$ and $g$ are differentiable at $0$. This means that there exist the following limits
$$
h'(0)=\lim\limits_{x\to 0} \frac{h(x)-h(0)}{x},
$$
$$
g'(0)=\lim\limits_{x\to 0} \frac{g(x)-g(0)}{x}.
$$
Notice now that
$$
\lim\limits_{x \to 0^+}\frac{f(x)-f(0)}{x}=\lim\limits_{x \to 0^+}\frac{h(x)-h(0)}{x}=\lim\limits_{x\to 0} \frac{h(x)-h(0)}{x}=h'(0)
$$
$$
\lim\limits_{x \to 0^-}\frac{f(x)-f(0)}{x}=\lim\limits_{x \to 0^-}\frac{g(x)-g(0)}{x}=\lim\limits_{x\to 0} \frac{g(x)-g(0)}{x}=g'(0).
$$
Since both limits exist and we are supposing $h'(0)=g'(0)$ we have that the following limit exist
$$
f'(0)=\lim\limits_{x \to 0}\frac{f(x)-f(0)}{x}=h'(0)=g'(0).
$$

Title: Derivative of a piecewise function at its endpoints

Body:
I have a function $f(x)= \begin{cases} 3-(x+1)^2 & \text{if } x\leq0, \\ 1+(x-1)^2 & \text{if } x \geq 0. \end{cases}$ and I want to evaluate $f'(0)$. If I differentiate,$f'(x)= \begin{cases} -2x-2 & \text{if } x\leq0, \\ 2x-2 & \text{if } x \geq 0. \end{cases}$ and we can see at $f'(0)$, both are $-2$. Does this mean I can conclude that $f'(0)=-2$?

Can I take this to the general case, such that if I have two  sufficiently well-behaved cases in my piece wise function(continuous at endpoint etc), and I want to evaluate the derivative of the function at their endpoint, is it sufficient for the derivative of both "pieces" to be the same at that point for me to conclude that the derivative of the function at that point exists and is equal to the value calculated?


问:
What is an interpretation of the rank of a probability matrix?
Let $X$ and $Y$ be random variables taking values in the finite sets $\mathcal X=\{1,\dots,m\}$, $\mathcal Y=\{1,\dots,n\}$. Without loss of generality assume $m \ge n$.

Let
$$p_{ij} = \Pr(X=i, Y=j)$$
be the joint probability mass function of $X$ and $Y$ and assume that $p_{ij} > 0$ for all $i \in \mathcal X, j \in \mathcal Y$.

Consider the matrix $P = (p_{ij}) \in \mathbb R^{m \times n}$.

Question: Is there an interpretation for $\operatorname{rank}P$?

I'm especially interested in an interpretation of $P$ being of full rank. If one of $n$ or $m$ equals 2 then $P$ is of full rank iff $X$ and $Y$ are independent (suppose $m \ge n = 2$. Divide the columns of $P$ by their sums. This leaves the rank of the resulting matrix unchanged at 1. Hence its two columns are identical. But the first column contains $\Pr(X=\cdot\mid Y=1)$ and the second column $\Pr(X=\cdot\mid Y=2)$. Thus $X$ and $Y$ are independent). But what about the case $m \ge n \ge 3$? Is there an interpretation for $P$ being of full rank in that case?

答:
A partial answer is that this matrix has rank $1$ if and only if $X$ and $Y$ are independent. This is seen in the fact that in case of independence, and only then, we have
\begin{align}
& \big( \Pr(X=x\ \&\  Y=y) : x=1,\ldots, n \big) \\[8pt]
= {} & \Pr(Y=y)\cdot\big( \Pr(X=x) : x=1,\ldots, n\big) \\[8pt]
= {} & \Big( \text{a scalar multiple of } \big( \Pr(X=x) : x=1,\ldots, n\big) \Big)
\end{align}
So all columns of the matrix are scalar multiples of just one column.


问:
Solution of wave equation
I posted this question for the first time on Physics Stack Exchange more than one year ago. The question was closed as off topic. Even if I reworked the question no one considered the possibility ro reopen if. For this reason I decided to poste the question on Math Stack Exchange. Even if my question is related to a physical problem, the difficulty I am facing are related to the math only.

I am dealing with the solution of the wave equation in two different cases represented in figure by Case A and Case B. The two figures were obtained by shining a slab with an incident wave coming from the right (Case A) and from the left (Case B). I verified the solution for Case A. It appears to be correct. Thus I used the same methodology to derive the solution for Case B. However I suspect the solution for Case B is wrong, even if I am not able to understand why.
Independently of the correct/wrong result of the wave equation in case B, I would like to know if there is a way to derive the solution of the wave equation in Case B from that of Case A by applying a kind of symmetry.
The slab is defined between $x=0$ and $x=L$, and a solution for a slab between  $x=-L/2$ and $x=L/2$ is not acceptable.
Please note that $\eta(x)$ does not necessarely any symmetry with respect to  $x$.

This is my solution. Given the wave equation:

\begin{equation}
\frac{d^2y(x)}{dx^2}+k^2 \left(1+\eta(x)\right) y(x)=0
\end{equation}
Its solution can be found by using the Green function method. Details of the method are provided for Case B. When applied to Case A the wave equation has the solution:
\begin{equation}
y(x)=c_1 e^{-j k x}+c_2 e^{j k x}+\frac{j k}{2} \int_0^Le^{j k |x-t|} \eta(t) y(t) dt
\end{equation}
where $j^2 = -1$ and $c_1$ and $c_2$ are the integration constant. In order to determine the constants we have to consider the boundary conditions.
When referring to the case A in figure it is possible to distinguis an incident wave and a reflected wave on the right hand side, and a transmitted wave on the left hand side. In this case the following boundatry conditions hold:
\begin{align}
y(L)=1+R_L\\
y'(L)=-jk(2-y(L))\\
y(0)=T_L\\
y'(0)=-jky(0)
\end{align}
By applying these boundary condition while distinguish the two cases $t-x>0$ (left side of the picture) and $t-x<0$ (right side of the picture) it is possibler to find a differential equation in $T_L$.

Now I would like to repeat the same kind of computation on the second picture (Case B). In this picture the incident wave is coming from the left side, while the transmitted wave is on the right side. In the case A a wave propagating in the same direction of the x axis has a positive exponential term. I have a doubt concerning figure B. In this case a wave travelling in the same direction of the x axis has a dependance on $-x$. Is this correct?
I suppose that it's just a question of conventions. The boundary condition for the case in figure B are of course switched:
\begin{align}
y(0)=1+R_L\\
y'(0)=-jk(2-y(0))\\
y(L)=T_L\\
y'(L)=-jky(L)
\end{align}
Concerning the case A, I first differentiate the general solution:
\begin{equation}
y'(x)=-jk c_1 e^{-jk x}+jk c_2 e^{+jk x}+\frac{j k}{2} \int_0^L\frac{d}{dx}\left(e^{j k |x-t|} \right) \eta(t) y(t) dt
\end{equation}
Now I considered two different cases: when $t-x>0$ (left) and $t-x<0$ (right).
For $t-x>0$ I can write:
\begin{equation}
y'(x)=-jk c_1 e^{-jk x}+jk c_2 e^{+jk x}-jk U(x)
\end{equation}
where
\begin{equation}
U(x)= \frac{j k}{2} \int_0^L\left(e^{-j k (x-t)} \right) \eta(t) y(t) dt
\end{equation}
Then
\begin{equation}
y'(x)=-jk \left( c_1 e^{-jk x}- c_2 e^{+jk x} + U(x) \right)
\end{equation}
While when $t-x<0$ I have:
\begin{equation}
y'(x)=-jk \left( c_1 e^{-jk x}- c_2 e^{+jk x}- K(x) \right)
\end{equation}
where
\begin{equation}
K(x)= \frac{j k}{2} \int_0^L\left(e^{j k (x-t)} \right) \eta(t) y(t) dt
\end{equation}
The boundary conditions in $x=0$ correspond to the case $t-x>0$, while the boundary conditions in $x=L$ correspond to the case $t-x<0$. For $x=0$ we have
\begin{align}
y(0)=c_1+c_2+U(0)\\
y'(0)=-jk(c_1-c_2+U(0))\\
\end{align}
So that $c_2=0$. By repeating the same computation in $x=L$ it is possible to determine that $c_1=e^{jkL}$. Then the solution is:
\begin{equation}
y(x)=e^{j k(L- x)}+\frac{j k}{2} \int_0^Le^{j k |x-t|} \eta(t) y(t) dt
\end{equation}
Now the case in figure B. This is where I'm not sure. According to the picture $e^{-jkx}$ is the forward propagating wave. The Green functions are for $x>L$:
\begin{equation}
\begin{cases}
y_1(x)=C_1 e^{-jkx}\\
y_1(L)=A'
\end{cases}
\end{equation}
While for $x<0$:
\begin{equation}
\begin{cases}
y_2(x)=c_2 e^{jkx}\\
y_2(0)=B'
\end{cases}
\end{equation}
Then
\begin{equation}
\begin{cases}
y_1(L)=c_1 e^{-jkL}=A' & \Rightarrow y_1(x)=A' e^{jk(L-x)}\\
y_2(0)=c_2=B' & \Rightarrow y_2(x)=B' e^{jkx}
\end{cases}
\end{equation}
The Wronskian is
\begin{equation}
W=
\begin{vmatrix}
A' e^{jk(L-x)} & B' e^{jkx}\\
-jk A' e^{jk(L-x)} & jk B' e^{jkx}\\
\end{vmatrix}
=2 jk A'B' e^{jkL}
\end{equation}
So that the Green function is
\begin{equation}
G(x,t)=
\begin{cases}
\frac{A' e^{jk(L-x)} B' e^{jkt}}{2 jk A'B' e^{jkL}}\\
\frac{A' e^{jk(L-t)} B' e^{jkx}}{2 jk A'B' e^{jkL}}\\
\end{cases}
\end{equation}
Which reduces to
\begin{equation}
G(x,t)=\frac{1}{2jk} e^{jk|t-x|}
\end{equation}
So that the solution of the differential equation for the case in figure B is
\begin{equation}
y(x)=c_1 e^{-j k x}+c_2 e^{j k x}+\frac{j k}{2} \int_0^Le^{j k |t-x|} \eta(t) y(t) dt
\end{equation}

答:
Case A and B cannot be directly compared. We have to distinguish the following cases:

Propagation from Left To Right (LTR) when the field propagating in the positive z direction is given by $e^{-jkx}$. This is the Case B.

Propagation from Left To Right (LTR) when the field propagating in the positive z direction is given by $e^{+jkx}$.

Propagation from Right To Left (RTL) when the field propagating in the positive z direction is given by $e^{-jkx}$.

Propagation from Right To Left (RTL) when the field propagating in the positive z direction is given by $e^{+jkx}$. This is the Case A.

As it is possible to see, to be consistent, we can only compare cases (1) and (3) or cases (2) and (4), i.e. we cannot mix the engineering and physic sign conventions. The engineering convention uses a minus sign for forward propagating waves, while the physic convention uses a positive sign. The sign convention has a direct implication on the wave vector ($k=\beta - j \alpha$ while applying the engineering sign convention, and $k=\beta + j \alpha$ while applying the physic sign convention), on the complex permittivity ($\epsilon=\epsilon' - j \epsilon''$ while applying the engineering sign convention, and $\epsilon=\epsilon' + j \epsilon''$ while applying the physic sign convention) as well as on the temporal and spatial Fourier transform.

Title: Solution of wave equation


问:
Prove that there are infinitely many natural numbers $n$, such that $n(n+1)$ can be expressed as sum of two positive squares in two distinct ways.

Prove that there are infinitely many natural numbers $n$, such that $n(n+1)$ can be expressed as sum of two positive squares in two distinct ways.($a^2+b^2$, is same $b^2+a^2$), $n \in \mathbb{N}.$

I have proved the above question which appeared in one of the Math-Olympiad. And I do know the solution. Sharing the question only because the question has a cute solution.

Body:
Let $n = 4x^4$, we have:
$$n(n+1) = (4x^4)^2 + (2x^2)^2 = (4x^4-2x^2)^2 + (4x^3)^2$$

Let $n = (u^2 + v^2)^2$, we have
$$\begin{align}
n(n+1) = & ((u^2 + v^2)^2)^2 + (u^2+v^2)^2\\
       = & (u^4 - 2uv - v^4)^2 + (2uv^3-v^2+2u^3v+u^2)^2\\
       = & (u^4 + 2uv - v^4)^2 + (2uv^3+v^2+2u^3v-u^2)^2
\end{align}$$

Let $n = (x+y)^2 + (2xy)^2$, we finally have an example that $n$ is not a square:
$$\begin{align}
n(n+1) = & (4x^2y^2+2xy+y^2-x^2)^2 + (4x^2y+y+x)^2\\
       = & (4x^2y^2+2xy-y^2+x^2)^2 + (4y^2x+y+x)^2
\end{align}$$


问:
Conjugacy of factors and semidirect products
Let $G = N \rtimes H$ be the semidirect product of two cyclic groups $N$ and $H$, and assume that there exists a cylic group $K \leq G$ such that $G \simeq N \rtimes K$. Is it true that $H$ and $K$ are conjugate in $G$ by an element of $N$?

I feel this should be true and that it should have a very straightfoward demonstration. However, all I've managed to prove so far is that both $H$ and $K$ act on $N$ through the same automorphism of $N$, that is, if $H = \langle h\rangle$ and $K = \langle k \rangle$ then $$hnh^{-1} = knk^{-1}\,,\quad \forall n \in N\,.$$

I was mostly interested in the infinite cyclic case $N \simeq H \simeq K \simeq \mathbb{Z}$, but I also feel that this should not really depend on the isomorphism classes of the subgroups involved.

答:
After playing with this for a while, I think the following is a counter-example:

Let $$G = \mathbb{Z}/4\mathbb{Z} \rtimes_\varphi \mathbb{Z}/2\mathbb{Z} = \langle x, y \mid x^4 = y^2 = 1\,, yxy^{-1} = x^3\rangle\,,$$ and take $N = \langle x \rangle$, $H = \langle y \rangle$ and $K = \langle xy \rangle$. We have both $G \simeq N \rtimes H$ and $G \simeq N \rtimes K$. However, any conjugate of $y$ by $x^i \in N$ satisfies: $$x^iyx^{-i} = x^ix^{-3i}y = x^{-2i}y\,.$$

Hence, $K$ is not a conjugate of $H$ by an element of $N$.


答:
Wolfram Alpha agrees with your picture.

So indeeed when running Newton's method, you get approximately 1.346, the closest root, not like you were obtaining.

However, even if you would not get the closest extremum, the reason for that phenomenon would be that Newton's method gives no guarantees to which specific local extremum it will converge to.

Title: Newton-Raphson Method - need help understanding an example

Body: I am currently trying to finish an assignment regarding Newton-Raphson method. Anyone able to explain an example to me? The function $f(x) = \sin(x) + \sin(\frac{10x}{3})$ is shown on the picture. The red dot is the starting point and the cross is a local extrema found by using the NR method. My question is why isn't the cross on the local extremas near the starting point, specifically where $x \approx 1.5$ and $x \approx 0.5$ which is visible on the picture. Why did the point end up where it did and passed two extremas on its way? Could this be an implementation issue, or is this how it's supposed to be ?