Klik hier om die volledige joernaal in PDF-formaat af te laai - LitNet

More documents

Recommendations

Info

LitNet Akademies Jaargang 9 (2), April 2012 Two Latin squares L and L ′ are orthogonal if the n 2 ordered pairs (L(i, j),L ′ (i, j)) are all distinct as i, j ∈ Zn vary. The transpose of a Latin square L, denoted by L T , is defined as in the ordinary matrix sense and L is symmetric if L = L T . If a Latin square L is orthogonal to its transpose, then L is called a self-orthogonal Latin square (SOLS). If a SOLS L is orthogonal to a symmetric Latin square S, then the pair (L,S) forms a SOLS with a symmetric orthogonal mate (SOLSSOM). If n is even, a SOLSSOM (L,S) is unipotent if S is unipotent. Furthermore, a SOLSSOM S = (L,S) is standard if L is idempotent and if S is idempotent (for odd n) or reduced (for even n). It is known that a SOLSSOM of order n exists for any positive integer n = 1,2,3,6,10, 14, while a SOLSSOM of order n does not exist if n = 1,2,3,6 (at the time of writing this paper the cases of orders n = 10,14 were still undecided). Unipotent SOLSSOMs have useful applications in the scheduling of spouse-avoiding mixed-doubles round-robin tennis tournaments [5, 9] and Whist tournaments [10, §III.5.10]. Two SOLSSOMs S = (L,S) and S ′ = (L ′ ,S ′ ) are (row, column)-paratopic if there exists a quadruple α = (p,ℓ,s,t) ∈ S 3 n × S2 of permutations (where Sn is the symmetric group of order n), called a (row, column)-paratopism mapping S to S ′ , such that ℓ(L(i, j)) = L ′ (p(i), p( j)) if t(0) = 0, L ′ (p( j), p(i)) if t(0) = 1, and s(S(i, j)) = S ′ (p(i), p( j)). Hence p is a permutation applied to the rows and columns of L and S, and ℓ is a permutation applied to the symbols of L. Furthermore, s is a permutation applied to the symbols of S, while t permutes the roles of the rows and columns of L (i.e. possibly achieves a switch between L and L T ). The notation S ′ = S α is henceforth used to denote the fact that α transforms S into S ′ . If S = S α , then α is called a (row, column)autoparatopism of S . If α = (p,ℓ,s,t) is a (row, column)-paratopism for which p = ℓ = s, then α is called a transpose-isomorphism, simply denoted by (p,t) ∈ Sn × S2, in which case S and S α are said to be transpose-isomorphic. If S α = S in this case, then α is called a transpose-automorphism of S . The notions of a (row, column)-paratopism and a transpose-isomorphism may be defined similarly for SOLS or for symmetric Latin squares (individually). For instance, if α = (p,ℓ,s,t) is a (row, column)-paratopism acting on a SOLSSOM S = (L,S), then (p,ℓ,t) is a (row, column)paratopism acting on L, while (p,s) is a (row, column)-paratopism acting on S. Various classes of SOLS of orders 4 n 10 have been enumerated by Graham and Roberts [11] and by Burger et al. [8, 7], and unipotent symmetric Latin squares have also been enumerated in the form of one-factorisations of the complete graph [2]. A problem that has not yet been addressed in the literature yet is that of the enumeration of SOLSSOMs. The availability of exhaustive repositories for non-(row,column)-paratopic symmetric Latin squares in [13] and non-transpose-isomorphic SOLS in [6] provide two different methods of generating SOLSS- OMs, namely finding, for each SOLS in the repository, its non-(row,column)-paratopic symmetric orthogonal mates, or finding, for each symmetric Latin square, its non-(row,column)paratopic SOLS-mates. In this paper we adopt both the enumeration methods described above (implemented as backtracking tree searches) with a view to enumerating distinct SOLSSOMs, standard SOLSS- OMs, transpose-isomorphism classes of SOLSSOMs, and (row, column)-paratopism classes of SOLSSOMs up to order 10, thereby also providing a means for validating our results. The numerical results obtained are summarised in Table 1. An important conclusion from these enumeration results is that there is no SOLSSOM of order 10; this settles a 34-year-old stan- 2
LitNet Akademies Jaargang 9 (2), April 2012 ding case with respect to orders for which SOLSSOMs exist. Since a SOLSSOM of order n in standard form is equivalent to a playing schedule for a spouse-avoiding mixed-doubles roundrobin tennis tournament involving n married couples, this shows that no such tournament can be scheduled for 10 married couples. There is an appendix at the end of the paper which contains lists of (i) the (row, column)paratopism class representatives of idempotent symmetric Latyn squares of orders n = 5,7,9 and (ii) the (row, column)-paratopism class representatives of standard SOLSSOMs of orders n = 4,5,7,8,9. n Distinct SOLSSOMs Standard SOLSSOMs Transposeisomorphism classes of SOLSSOMs (Row, column)paratopism classes of SOLSSOMs 2 0 0 0 0 3 0 0 0 0 4 1 152 2 31 1 5 172 800 12 749 1 6 0 0 0 0 7 12 192 768 000 480 1 210 622 2 8 608 662 978 560 000 374 400 7 547 904 042 32 9 464 573 723 443 200 000 3 528 000 640 121 719 688 26 10 0 0 0 0 Sloane #A166490 #A166489 #A166488 #A166487 Tabel 1: Enumeration of various equivalence classes of SOLSSOMs of orders 4 n 10. The last row of the table contains the sequence numbers in Sloane’s Online Encyclopedia of Integer Sequences [16] corresponding to the table columns. Keywords: Latin square, self-orthogonal Latin square, symmetric Latin square, SOLSSOM, enumeration. Inleiding In 1972 was die Briarcliff Racquet Club in New York op soek na ’n skedule vir ’n gemengdedubbels tennistoernooi vir getroude pare (m.a.w. waarin elke wedstryd uit twee opponerende spanne bestaan, waar elke span een man en een vrou bevat), sodat: (i) geen speler (as spanmaat of opponent) in dieselfde wedstryd as sy/haar eggenoot speel nie (ii) elke speler elke ander speler (behalwe sy/haar eggenoot) presies een keer teenstaan (iii) elke speler presies een keer saam met elke speler van die teenoorgestelde geslag (behalwe sy/haar eggenoot) in ’n span speel (iv) wedstryde in die minimum getal rondtes gegroepeer word waar geen speler meer as een keer per rondte speel nie en die rondtes elk ewe veel wedstryde bevat. 3
Page 1 and 2: LitNet Akademies Augustus 2012: jaa
Page 3 and 4: Adviesraadslede van LitNet Akademie
Page 5 and 6: Inhoudsopgawe Afdeling: Natuurweten
Page 7 and 8: Die impak van revisie op vertaalde
Page 9: LitNet Akademies Jaargang 9 (2), Ap
Page 13 and 14: LitNet Akademies Jaargang 9 (2), Ap
Page 17 and 18: en die SOLVSOM S2 = (L2,S2), met L2
Page 19 and 20: Vlak 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Page 29 and 30: Aanhangsel LitNet Akademies Jaargan
Page 33 and 34: LitNet Akademies Jaargang 9(2), Aug
Page 39 and 40: Figuur 2. Borrelbemiddelde elektros
Page 43 and 44: 3. Resultate en bespreking LitNet A
Page 51 and 52: Abstract Epigenetics: The link betw
Page 57 and 58: 2. Tipes epigenetiese modifikasies
Page 61 and 62:
LitNet Akademies Jaargang 9(2), Aug
Page 63 and 64:
Page 65 and 66:
Page 67 and 68:
Page 69 and 70:
Vetsug Veranderde geenuitdrukking v
Page 71 and 72:
ensieme gevolg deur PCR (MSRE-PCR)
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
Afdeling: Regte
Page 83 and 84:
Page 85 and 86:
1. Inleiding LitNet Akademies Jaarg
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
4.2. F-sake 4.2.1 Feite LitNet Akad
Page 93 and 94:
4.2.3. Uitspraak van die konstitusi
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
10 962-3. LitNet Akademies Jaargang
Page 103 and 104:
Page 105 and 106:
64 443. 65 2011:142. LitNet Akademi
Page 107 and 108:
93 F v Minister of Safety and Secur
Page 109 and 110:
Opsomming LitNet Akademies Jaargang
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
4. Dogmatiese grondslag LitNet Akad
Page 117 and 118:
5. Samevatting LitNet Akademies Jaa
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
43 Sien O’Donovan (2000:79-81); B
Page 125 and 126:
62 2003:690-1; sien ook Neethling,
Page 127 and 128:
1. Inleiding en tersaaklike wetgewi
Page 129 and 130:
3. Uitspraak en redes van die hof b
Page 131 and 132:
Page 133 and 134:
5. Bespreking 5.1 Die ondersteunend
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Eindnotas 1 Sien bv. De Jager v Gru
Page 141 and 142:
38 Par. 13-7. 39 1967 4 SA 459 (A).
Page 143 and 144:
Page 145 and 146:
Burial wishes LitNet Akademies Jaar
Page 147 and 148:
2. Die betekenis van die term begra
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
5.2.1 Geskrewe instruksies LitNet A
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
13 Joubert e.a. (2009:6). LitNet Ak
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
153 Trollip-saak 247. Sien ook Jans
Page 183 and 184:
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
Page 197 and 198:
Page 199 and 200:
Page 201 and 202:
Page 203 and 204:
Page 205 and 206:
Page 207 and 208:
19 Mbembe (2006:2). 20 Mbembe (2006
Page 209 and 210:
Page 211 and 212:
Page 213 and 214:
Page 215 and 216:
Page 217 and 218:
Page 219 and 220:
Page 221 and 222:
Page 223 and 224:
Page 225 and 226:
Page 227 and 228:
Page 229 and 230:
Page 231 and 232:
Page 233 and 234:
66 71 van 2008 art. 1: "Persoon: oo
Page 235 and 236:
112 Davis et al. 2010:241. 113 71 v
Page 237 and 238:
154 71 van 2008. 155 71 van 2008 ar
Page 239 and 240:
Page 241 and 242:
3. Die bestuursbenaderings LitNet A
Page 243 and 244:
Page 245 and 246:
Page 247 and 248:
Page 249 and 250:
Page 251 and 252:
Page 253 and 254:
8. Gevolgtrekking en samevatting Li
Page 255 and 256:
Page 257 and 258:
9 Kloppers 1999:430. 10 Bradstreet
Page 259 and 260:
Page 261 and 262:
89 71 van 2008 art. 140(1)(a). 90 7
Page 263 and 264:
131 Kloppers 1999:430. 132 71 van 2
Page 265 and 266:
Page 267 and 268:
Page 269 and 270:
Page 271 and 272:
Page 273 and 274:
Page 275 and 276:
Page 277 and 278:
Page 279 and 280:
Page 281 and 282:
Page 283 and 284:
Page 285 and 286:
Page 287 and 288:
Page 289 and 290:
Page 291 and 292:
Page 293 and 294:
Page 295 and 296:
Page 297 and 298:
Page 299 and 300:
Page 301 and 302:
90 Grondwet artt. 7(2) en 8(1). 91
Page 303 and 304:
Page 305 and 306:
Page 307 and 308:
5.2 Vereistes vir kennisgewing van
Page 309 and 310:
Page 311 and 312:
Page 313 and 314:
Page 315 and 316:
5 Sien art. 8; Otto en Otto (2010,
Page 317 and 318:
Page 319 and 320:
93 Par. 14 (my kursivering). LitNet
Page 321 and 322:
Page 323 and 324:
Page 325 and 326:
Page 327 and 328:
Page 329 and 330:
Page 331 and 332:
Page 333 and 334:
Universiteit van die Witwatersrand
Page 335 and 336:
Page 337 and 338:
Page 339 and 340:
11. Gevolgtrekking LitNet Akademies
Page 341 and 342:
Page 343 and 344:
Page 345 and 346:
Page 347 and 348:
Page 349 and 350:
Page 351 and 352:
Page 353 and 354:
Page 355 and 356:
Page 357 and 358:
Page 359 and 360:
Page 361 and 362:
Page 363 and 364:
Page 365 and 366:
Page 367 and 368:
Page 369 and 370:
Page 371 and 372:
Page 373 and 374:
Page 375 and 376:
Page 377 and 378:
Page 379 and 380:
Page 381 and 382:
Page 383 and 384:
Page 385 and 386:
Page 387 and 388:
Page 389 and 390:
Page 391 and 392:
Page 393 and 394:
Page 395 and 396:
Page 397 and 398:
Page 399 and 400:
Page 401 and 402:
Page 403 and 404:
Page 405 and 406:
Page 407 and 408:
Page 409 and 410:
Page 411 and 412:
Page 413 and 414:
Page 415 and 416:
Page 417 and 418:
liggaam be-leef, oor my liggaam so
Page 419 and 420:
Page 421 and 422:
Page 423 and 424:
Page 425 and 426:
Page 427 and 428:
Page 429 and 430:
Page 431 and 432:
Page 433 and 434:
Page 435 and 436:
Page 437 and 438:
Page 439 and 440:
Page 441 and 442:
Page 443 and 444:
Page 445 and 446:
Page 447 and 448:
Page 449 and 450:
Page 451 and 452:
Page 453 and 454:
Page 455 and 456:
Page 457 and 458:
Page 459 and 460:
Page 461 and 462:
Page 463 and 464:
Page 465 and 466:
4.2 Wat is kuns? Wat is die funksie
Page 467 and 468:
Page 469 and 470:
Page 471 and 472:
Page 473 and 474:
Page 475 and 476:
Page 477 and 478:
Page 479 and 480:
Page 481 and 482:
Page 483 and 484:
Page 485 and 486:
Page 487 and 488:
Page 489 and 490:
Page 491 and 492:
Page 493 and 494:
Page 495 and 496:
Page 497 and 498:
Kaapkolonie 46 Nieu-Engeland Jones
Page 499 and 500:
Kaapkolonie 2 36 Kaapkolonie 3 51 K
Page 501 and 502:
Page 503 and 504:
Page 505 and 506:
Page 507 and 508:
Page 509 and 510:
Page 511 and 512:
Page 513 and 514:
Page 515 and 516:
Page 517 and 518:
Page 519 and 520:
Page 521 and 522:
Page 523 and 524:
Page 525 and 526:
Page 527 and 528:
Figuur 7. Uitgeweryprofiel van Brey
Page 529 and 530:
Page 531 and 532:
Page 533 and 534:
Page 535 and 536:
Page 537 and 538:
Page 539 and 540:
Page 541 and 542:
Page 543 and 544:
Page 545 and 546:
Page 547 and 548:
Page 549 and 550:
Page 551 and 552:
Page 553 and 554:
wat soggens met hul helder kleur gl
Page 555 and 556:
Page 557 and 558:
Page 559 and 560:
Page 561 and 562:
Page 563 and 564:
Page 565 and 566:
Page 567 and 568:
Page 569 and 570:
Page 571 and 572:
Page 573 and 574:
Page 575 and 576:
Page 577 and 578:
Page 579 and 580:
Page 581 and 582:
Page 583 and 584:
Page 585 and 586:
Page 587 and 588:
Page 589 and 590:
Page 591 and 592:
Page 593 and 594:
Page 595 and 596:
Page 597 and 598:
Page 599 and 600:
Page 601 and 602:
Page 603 and 604:
Page 605 and 606:
3. Die makrostrukturele faktore 3.1
Page 607 and 608:
Page 609 and 610:
Page 611 and 612:
Page 613 and 614:
Page 615 and 616:
Page 617 and 618:
Page 619 and 620:
Page 621 and 622:
Abstract LitNet Akademies Jaargang
Page 623 and 624:
Page 625 and 626:
Page 627 and 628:
Page 629 and 630:
Page 631 and 632:
Die volgende is ’n uiteensetting
Page 633 and 634:
Page 635 and 636:
Page 637 and 638:
Groepplasing ten opsigte van kultuu
Page 639 and 640:
Page 641 and 642:
5.2 Beperkinge van die studie LitNe
Page 643 and 644:
Page 645 and 646:
Page 647 and 648:
1 66 Sesotho Engels 1.5 1.5 2 53.5
Page 649 and 650:
Page 651 and 652:
Page 653 and 654:
Page 655 and 656:
Figuur 1. Sillabusontwerprosedures
Page 657 and 658:
Page 659 and 660:
Page 661 and 662:
A Interaksionele aktiwiteite 1. Int
Page 663 and 664:
Inligtingsg aping Probleemo plossin
Page 665 and 666:
Page 667 and 668:
Page 669 and 670:
Page 671 and 672:
Page 673 and 674:
Page 675 and 676:
Page 677 and 678:
Page 679 and 680:
Page 681 and 682:
Page 683 and 684:
Page 685 and 686:
Page 687 and 688:
Page 689 and 690:
Page 691 and 692:
Page 693 and 694:
Page 695 and 696:
Page 697 and 698:
Page 699 and 700:
Page 701 and 702:
Page 703 and 704:
Page 705 and 706:
Page 707 and 708:
Page 709 and 710:
Met vergunning van Iziko Museums va
Page 711 and 712:
Somerset West (We) LitNet Akademies
Page 713 and 714:
Bibliografie LitNet Akademies Jaarg
Page 715 and 716:
Page 717 and 718:
Page 719 and 720:
Page 721 and 722:
Page 723 and 724:
Page 725 and 726:
Page 727 and 728:
Page 729 and 730:
Page 731 and 732:
Page 733 and 734:
Page 735 and 736:
Page 737 and 738:
Page 739 and 740:
Abstract LitNet Akademies Jaargang
Page 741 and 742:
Page 743 and 744:
Page 745 and 746:
Page 747 and 748:
Page 749 and 750:
Page 751 and 752:
Figuur 4. /i/ in hiet en /y/ in huu
Page 753 and 754:
4.2 Die egte (ronde) agtervokale Li
Page 755 and 756:
Figuur 9. /A/;/α/ in hat en /a/ in
Page 757 and 758:
Page 759 and 760:
Page 761 and 762:
Page 763 and 764:
Page 765 and 766:
Page 767 and 768:
Figuur 20. Formantstruktuur van die
Page 769 and 770:
Page 771 and 772:
Page 773 and 774:
Page 775 and 776:
Page 777 and 778:
Page 779 and 780:
21 Die eerste skrywer van die boek
Page 781 and 782:
Page 783 and 784:
Page 785 and 786:
Page 787 and 788:
2. Die tafereel Van Dale (1975:947)
Page 789 and 790:
Page 791 and 792:
5. New York LitNet Akademies Jaarga
Page 793 and 794:
7. Suid-Afrika LitNet Akademies Jaa
Page 795 and 796:
Page 797 and 798:
Page 799 and 800:
Page 801 and 802:
show all

Klik hier om die volledige joernaal in PDF-formaat af te laai - LitNet

Create successful ePaper yourself

Delete template?

Save as template?