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On the beta distribution, the nonlinear Fourier transform and a combinatorial problem
Saksida, Pavle
V članku obravnavamo nekaj verjetnostnih in kombinatoričnih aspektov nelinearne Fourierove trasformacije, prirejene problemom AKNS-ZS. Dokažemo, da so volumni družine politopov, ki nastopajo v ...potenčnem razvoju nelinearne Fourierove transformacije, porazdeljeni z verjetnostno distribucijo beta. Do tega rezultata pridemo s pomočjo opazovanja diskretizacije Eulerjevega tipa naše nelinearne Fourierove transformacije. Ta pristop nas pripelje do kombinatoričnega problema, ki sprašuje po številu alternirajočih urejenih particij naravnega števila v med seboj različne člene. Izpeljemo eksplicitno formulo za števila alternirajočih particij. Pokažemo, da so ta števila porazdeljena po verjetnostni porazdelitvi, ki je naravna diskretizacija distribucije beta pri primernih parametrih. Izpeljemo tudi rodovne funkcije za števila alternirajočih particij. Te rodovne funkcije izrazimo s pomočjo naše diskretizacije nelinearne Fourierove transformacija.
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