Maximum order-index of matrices over commutative inclines: an answer to an open problem

AbstractThis paper proves that the maximum order-index of n×n matrices over an arbitrary commutative incline equals (n−1)2+1. This is an answer to an open problem “Compute the maximum order-index of a member of Mn(L)”, proposed by Cao, Kim and Roush in a monograph Incline Algebra and Applications, 1984, where Mn(L) is the set of all n×n matrices over an incline L

Fuzzy Sets, Fuzzy Logic and Their Applications

The present book contains 20 articles collected from amongst the 53 total submitted manuscripts for the Special Issue “Fuzzy Sets, Fuzzy Loigic and Their Applications” of the MDPI journal Mathematics. The articles, which appear in the book in the series in which they were accepted, published in Volumes 7 (2019) and 8 (2020) of the journal, cover a wide range of topics connected to the theory and applications of fuzzy systems and their extensions and generalizations. This range includes, among others, management of the uncertainty in a fuzzy environment; fuzzy assessment methods of human-machine performance; fuzzy graphs; fuzzy topological and convergence spaces; bipolar fuzzy relations; type-2 fuzzy; and intuitionistic, interval-valued, complex, picture, and Pythagorean fuzzy sets, soft sets and algebras, etc. The applications presented are oriented to finance, fuzzy analytic hierarchy, green supply chain industries, smart health practice, and hotel selection. This wide range of topics makes the book interesting for all those working in the wider area of Fuzzy sets and systems and of fuzzy logic and for those who have the proper mathematical background who wish to become familiar with recent advances in fuzzy mathematics, which has entered to almost all sectors of human life and activity

Small representations, string instantons, and Fourier modes of Eisenstein series (with an appendix by D. Ciubotaru and P. Trapa)

This paper concerns some novel features of maximal parabolic Eisenstein series at certain special values of their analytic parameter s. These series arise as coefficients in the R4 and D4R4 interactions in the low energy expansion of scattering amplitudes in maximally supersymmetric string theory reduced to D=10-d dimensions on a torus T^d, d<8. For each d these amplitudes are automorphic functions on the rank d+1 symmetry group E_d+1. Of particular significance is the orbit content of the Fourier modes of these series when expanded in three different parabolic subgroups, corresponding to certain limits of string theory. This is of interest in the classification of a variety of instantons that correspond to minimal or next-to-minimal BPS orbits. In the limit of decompactification from D to D+1 dimensions many such instantons are related to charged 1/2-BPS or 1/4-BPS black holes with euclidean world-lines wrapped around the large dimension. In a different limit the instantons give nonperturbative corrections to string perturbation theory, while in a third limit they describe nonperturbative contributions in eleven-dimensional supergravity. A proof is given that these three distinct Fourier expansions have certain vanishing coefficients that are expected from string theory. In particular, the Eisenstein series for these special values of s have markedly fewer Fourier coefficients than typical ones. The corresponding mathematics involves showing that the wavefront sets of the Eisenstein series are supported on only certain coadjoint nilpotent orbits - just the minimal and trivial orbits in the 1/2-BPS case, and just the next-to-minimal, minimal and trivial orbits in the 1/4-BPS case. Thus as a byproduct we demonstrate that the next-to-minimal representations occur automorphically for E6, E7, and E8, and hence the first two nontrivial low energy coefficients are exotic theta-functions.Comment: v3: 127 pp. Minor changes. Final version to appear in the Special Issue in honor of Professor Steve Ralli