Law of Pure Types and Some Exotic Spectra of Fractal Spectral Measures

2014 ◽  
pp. 47-64 ◽  
Author(s):  
Xin-Rong Dai ◽  
Xing-Gang He ◽  
Chun-Kit Lai
Keyword(s):  
Spectral Measures

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

ON THE SPECTRAL MEASURES OF SOME WEAKLY STATIONARY SEQUENCES INVOLVING RANDOMLY SPACED OBSERVATIONS

2012 ◽  
Vol 12 (01) ◽  
pp. 1150004
Author(s):  
RICHARD C. BRADLEY
Keyword(s):  
Spectral Density ◽  
Density Function ◽  
Spectral Density Function ◽  
Strong Mixing ◽  
Spectral Measures ◽  
Mixing Conditions ◽  
Stationary Sequences ◽  
Second Moments ◽  
The Central Limit Theorem ◽  
Complex Valued

In an earlier paper by the author, as part of a construction of a counterexample to the central limit theorem under certain strong mixing conditions, a formula is given that shows, for strictly stationary sequences with mean zero and finite second moments and a continuous spectral density function, how that spectral density function changes if the observations in that strictly stationary sequence are "randomly spread out" in a particular way, with independent "nonnegative geometric" numbers of zeros inserted in between. In this paper, that formula will be generalized to the class of weakly stationary, mean zero, complex-valued random sequences, with arbitrary spectral measure.

Multivariate extremes, aggregation and dependence in elliptical distributions

2002 ◽  
Vol 34 (03) ◽  
pp. 587-608 ◽  
Author(s):  
Henrik Hult ◽  
Filip Lindskog
Keyword(s):  
Random Vector ◽  
Positive Constant ◽  
Tail Dependence ◽  
Constant Factor ◽  
Dispersion Matrix ◽  
Elliptical Distributions ◽  
Spectral Measures ◽  
Lower Tail ◽  
Explicit Formulae ◽  
Dependence Properties

In this paper, we clarify dependence properties of elliptical distributions by deriving general but explicit formulae for the coefficients of upper and lower tail dependence and spectral measures with respect to different norms. We show that an elliptically distributed random vector is regularly varying if and only if the bivariate marginal distributions have tail dependence. Furthermore, the tail dependence coefficients are fully determined by the tail index of the random vector (or equivalently of its components) and the linear correlation coefficient. Whereas Kendall's tau is invariant in the class of elliptical distributions with continuous marginals and a fixed dispersion matrix, we show that this is not true for Spearman's rho. We also show that sums of elliptically distributed random vectors with the same dispersion matrix (up to a positive constant factor) remain elliptical if they are dependent only through their radial parts.

Acoustic classification of coronal stops of Eastern Punjabi

Phonetica ◽  
2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Qandeel Hussain ◽  
Alexei Kochetov
Keyword(s):  
Standard Deviation ◽  
Voice Onset Time ◽  
Voice Quality ◽  
Onset Time ◽  
Quality Measures ◽  
Spectral Measures ◽  
Cepstral Peak Prominence ◽  
Laryngeal Configurations ◽  
Acoustic Classification

Abstract Punjabi is an Indo-Aryan language which contrasts a rich set of coronal stops at dental and retroflex places of articulation across three laryngeal configurations. Moreover, all these stops occur contrastively in various positions (word-initially, -medially, and -finally). The goal of this study is to investigate how various coronal place and laryngeal contrasts are distinguished acoustically both within and across word positions. A number of temporal and spectral correlates were examined in data from 13 speakers of Eastern Punjabi: Voice Onset Time, release and closure durations, fundamental frequency, F1-F3 formants, spectral center of gravity and standard deviation, H1*-H2*, and cepstral peak prominence. The findings indicated that higher formants and spectral measures were most important for the classification of place contrasts across word positions, whereas laryngeal contrasts were reliably distinguished by durational and voice quality measures. Word-medially and -finally, F2 and F3 of the preceding vowels played a key role in distinguishing the dental and retroflex stops, while spectral noise measures were more important word-initially. The findings of this study contribute to a better understanding of factors involved in the maintenance of typologically rare and phonetically complex sets of place and laryngeal contrasts in the coronal stops of Indo-Aryan languages.

scholarly journals Closed spectral measures in Fréchet spaces

1984 ◽  
Vol 7 (1) ◽  
pp. 15-21 ◽  
Author(s):  
W. Ricker
Keyword(s):  
Desirable Property ◽  
Fréchet Spaces ◽  
Spectral Measures

Closed spectral measures, which are often used in the theory of operators, have the desirable property that theirL1-space is complete. In this note criteria are given which assure the closedness of spectral measures acting in Fréchet spaces.

Sign in / Sign up

Export Citation Format

Share Document