A Symmetric Band Lanczos Process Based on Coupled Recurrences and Some Applications

2001 ◽  
Vol 23 (2) ◽  
pp. 542-562 ◽  
Author(s):  
Zhaojun Bai ◽  
Roland W. Freund
Keyword(s):  
Lanczos Process ◽  
Symmetric Band

Amino group stretching vibrations in primary aliphatic amines

1967 ◽  
Vol 45 (14) ◽  
pp. 1605-1610 ◽  
Author(s):  
P. J. Krueger ◽  
D. W. Smith
Keyword(s):  
Alkyl Chain ◽  
Inductive Effect ◽  
Lone Pair ◽  
Aliphatic Amines ◽  
Lone Pair Orbital ◽  
First Approximation ◽  
Symmetric Band ◽  
Integrated Intensity ◽  
Stretching Vibrations ◽  
Primary Aliphatic Amines

Primary aliphatic amines can be characterized by infrared spectroscopy as to the nature and degree of branching of the alkyl chain, using the fundamental antisymmetric and symmetric NH2 stretching frequencies measured in dilute C2Cl4 solution. Where steric and direct conjugation effects are absent both frequencies and the integrated intensity of the antisymmetric band increase linearly with decreasing inductive electron-donating power of the alkyl chain. The integrated intensity of the symmetric band, which is smaller than that for the antisymmetric band in all the 26 compounds studied, is independent of the inductive nature of the alkyl chain as a first approximation. These results indicate that the N-atom lone-pair orbital in RNH2 plays a negligibly small role in modifying the inductive effect of R on NH2 frequencies and intensities as long as the N-atom hybridization remains sp3.

scholarly journals Evidence of the Poisson/Gaudin–Mehta phase transition for band matrices on global scales

2018 ◽  
Vol 07 (02) ◽  
pp. 1850002
Author(s):  
Sheehan Olver ◽  
Andrew Swan
Keyword(s):  
Phase Transition ◽  
Boundary Conditions ◽  
Rate Of Convergence ◽  
Critical Behavior ◽  
Level Density ◽  
Boundary Effects ◽  
Fourth Moment ◽  
Band Matrices ◽  
Band Matrix ◽  
Symmetric Band

We prove that the Poisson/Gaudin–Mehta phase transition conjectured to occur when the bandwidth of an [Formula: see text] symmetric band matrix grows like [Formula: see text] is naturally observable in the rate of convergence of the level density to the Wigner semi-circle law. Specifically, we show for periodic and non-periodic band matrices the rate of convergence of the fourth moment of the level density is independent of the boundary conditions in the localized regime [Formula: see text] with a rate of [Formula: see text] for both cases, whereas in the delocalized regime [Formula: see text] where boundary effects become important, the rate of convergence for the two ensembles differs significantly, slowing to [Formula: see text] for non-periodic band matrices. Additionally, we examine the case of thick non-periodic band matrices [Formula: see text], showing that the fourth moment is maximally deviated from the Wigner semi-circle law when [Formula: see text], and provide numerical evidence that the eigenvector statistics also exhibit critical behavior at this point.

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