633 KB88 KB200820252022Bannai, EiichiBannai, EtsukoTanaka, HajimeZhu, Yanštevilka:2letnik:22P2.01 (43 str.)COBISSID_HOST:115874307ISSN:1855-3966URN:URN:NBN:SI:doc-PG1SIQBZenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneaasociativna shemaassociation schemecoherent configurationHahn polynomialHahnov polinomHermite polynomialHermiteov polinomkoherentna konfiguracijarelative t-designrelativni t-dizajnTerwilliger algebraTerwilligerjeva algebraTight relative t-designs on two shells in hypercubes, and Hahn and Hermite polynomials|Relative ?$t$?-designs in the ?$n$?-dimensional hypercube ?$\mathcal{Q}_n$? are equivalent to weighted regular ?$t$?-wise balanced designs, which generalize combinatorial ?$t-(n, k, \lambda)$? designs by allowing multiple block sizes as well as weights. Partly motivated by the recent study on tight Euclidean ?$t$?-designs on two concentric spheres, in this paper we discuss tight relative ?$t$?-designs in ?$\mathcal{Q}_n$? supported on two shells. We show under a mild condition that such a relative ?$t$?-design induces the structure of a coherent configuration with two fibers. Moreover, from this structure we deduce that a polynomial from the family of the Hahn hypergeometric orthogonal polynomials must have only integral simple zeros. The Terwilliger algebra is the main tool to establish these results. By explicitly evaluating the behavior of the zeros of the Hahn polynomials when they degenerate to the Hermite polynomials under an appropriate limit process, we prove a theorem which gives a partial evidence that the non-trivial tight relative ?$t$?-designs in ?$\mathcal{Q}_n$? supported on two shells are rare for large ?$t$?Relativni ?$t$?-dizajni v ?$n$?-dimenzionalni hiperkocki ?$\mathcal{Q}_n$? so ekvivalentni uteženim regularnim ?$t$?-kratnim uravnoteženim dizajnom, ki posplošujejo kombinatorne ?$t-(n, k, \lambda)$? dizajne na ta način, da dovoljujejo večkratne velikosti blokov, pa tudi uteži. Motivirani tudi z nedavno študijo o tesnih evklidskih ?$t$?-dizajnih na dveh koncentričnih sferah, v članku obravnavamo tesne relativne ?$t$?-dizajne v hiperkocki ?$\mathcal{Q}_n$?, podprte na dveh lupinah. Pokažemo, da, pod blagim pogojem, velja, da takšen relativen ?$t$?-dizajn inducira strukturo koherentne konfiguracije z dvema vlaknoma. Iz te strukture sklepamo tudi, da mora imeti polinom iz družine Hahnovih hipergeometrijskih ortogonalnih polinomov samo celoštevilske enostavne ničle. Glavno orodje pri izpeljavi teh rezultatov je Terwilligerjeva algebra. Z eksplicitno oceno vedenja ničel Hahnovih polinomov, ki pri ustreznem limitnem procesu degenerirajo v Hermiteove polinome, dokažemo izrek, ki deloma dokazuje, da so netrivialni tesni relativni ?$t$?-dizajni v ?$\mathcal{Q}_n$?, podprti na dveh lupinah, za velike ?$t$? redkiTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije