Open AccessNaimark complements in the building of simplices
Author(s) -
S. Ya. Novikov,
М. Е. Федина
Publication year - 2021
Publication title -
journal of physics. conference series
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.21
H-Index - 85
eISSN - 1742-6596
pISSN - 1742-6588
DOI - 10.1088/1742-6596/1745/1/012108
Subject(s) - simplex , equiangular polygon , mathematics , combinatorics , hilbert space , norm (philosophy) , discrete mathematics , pure mathematics , geometry , monotone polygon , political science , law
We define frames for a finite dimensional Hilbert space ℍ as the complete systems in ℍ. An equiangular tight frame (ETF) is an equal norm tight frame with the same sharp angles between the vectors. A regular simplex is a special type of ETF in which the number of vectors is one more than the dimension of the space they span. A detailed and independent from other sources presentation of recent results by M. Fickus, J. Jasper, E. J. King and D. G. Mixon is given, in which a lower bound for the spark of the system of equal norm vectors is obtained using the restricted isometry property. The existence of the regular s -simplices for an arbitrary positive integer s is proved using Naimark complement. A review of recent results towards resolving the known Paulsen problem is given.