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The spectral abscissa is the largest real part of an eigenvalue of a matrix and the spectral radius is the largest modulus. Both are examples of spectral max functions—the maximum of a real-valued function over the spectrum of a matrix. These mappings arise in the control and stabilization of dynamical systems. In 2001, Burke and Overton characterized the regular subdifferential of the spectral abscissa and showed that the spectral abscissa is subdifferentially regular in the sense of Clarke when all active eigenvalues are nonderogatory. In this paper we develop new techniques to obtain these results for the more general class of convexly generated spectral max functions. In particular, we extend the Burke–Overton subdifferential regularity result to this class. These techniques allow us to obtain new variational results for the spectral radius.

The rodeo algorithm is an efficient algorithm for eigenstate preparation and eigenvalue estimation for any observable on a quantum computer. This makes it a promising tool for studying the spectrum and structure of atomic nuclei as well as other fields of quantum many-body physics. The only requirement is that the initial state has sufficient overlap probability with the desired eigenstate. While it is exponentially faster than well-known algorithms such as phase estimation and adiabatic evolution for eigenstate preparation, it has yet to be implemented on an actual quantum device. In this work, we apply the rodeo algorithm to determine the energy levels of a random one-qubit Hamiltonian, resulting in a relative error of 0.08% using mid-circuit measurements on the IBM Q device Casablanca. This surpasses the accuracy of directly-prepared eigenvector expectation values using the same quantum device. We take advantage of the high-accuracy energy determination and use the Hellmann-Feynman theorem to compute eigenvector expectation values for a different random one-qubit observable. For the Hellmann-Feynman calculations, we find a relative error of 0.7%. Here, we conclude by discussing possible future applications of the rodeo algorithm for multi-qubit Hamiltonians.

A three-step treatment of Si wafers by gas cluster ion beam with decreasing energy was used to improve the performance of surface smoothing. First, a high energy treatment at 15 keV and an ion fluence of 2 × 1016 cm−2 was used to remove initial surface features (scratches). Next, treatments at 8 and 5 keV with the same fluences reduced the roughness that arose due to the formation of morphological features induced by the surface sputtering at the first high energy step. The surface morphology was characterized by the atomic force microscopy. The root mean square roughness Rq and 2D isotropic power spectral density functions were analyzed. For comparison, the smoothing performances of single-step treatments at 15, 8, and 5 keV were also studied. The lowest roughness values achieved for the single and three-step treatments were 1.06 and 0.65 nm, respectively.

We develop the necessary methodology to conduct principal component analysis at high frequency. We construct estimators of realized eigenvalues, eigenvectors, and principal components, and provide the asymptotic distribution of these estimators. Empirically, we study the high-frequency covariance structure of the constituents of the S&P 100 Index using as little as one week of high-frequency data at a time, and examines whether it is compatible with the evidence accumulated over decades of lower frequency returns. We find a surprising consistency between the low- and high-frequency structures. During the recent financial crisis, the first principal component becomes increasingly dominant, explaining up to 60% of the variation on its own, while the second principal component drives the common variation of financial sector stocks. Supplementary materials for this article are available online.

This study theoretically investigates the coupling efficiency of multi-Gaussian Shell-mode (MGSM) beams in ocean turbulence. The expression for the fiber-coupling efficiency of the MGSM beams propagating through oceanic turbulent media is derived using the cross-spectral density function. Numerical simulations are performed to examine the relationship between fiber-coupling efficiency and the beam order, and the scintillation index of the MGSM beams in ocean turbulence is also examined. In the analysis of transmission efficiency, the effects of the receiving aperture and source coherence on transmission efficiency are investigated, taking into account ocean turbulence induced by salinity and temperature fluctuations. The analysis of the fiber-coupling efficiency for MGSM beams presented in this work provides insights for optimizing the design of free-space optical communication systems.

A linear combination, with negative weights, of two continuous covariance functions has been analyzed by a few authors just for special cases and only in the real domain. However, a covariance is a complex valued function: for this reason, the general problem concerning the difference of two covariance functions in the complex domain needs to be analyzed, while it does not yet seem to have been addressed in the literature; hence, exploring the conditions such that the difference of two covariance functions is again a covariance function can be considered a further property. Therefore, this paper yields a contribution to the theory of correlation, hence the results cannot be restricted to the particular field of application. Starting from the difference of two complex covariance functions defined in one dimensional Euclidean space, wide families of models for the difference of two complex covariance functions can be built in any dimensional space, utilizing some well known properties. In particular, the difference of two real covariance functions has been considered; moreover, the difference between some special isotropic covariance functions has also been analyzed. A detailed analysis of the parameters of the models involved has been proposed; this kind of analysis opens a gate for modeling, in any dimensional space, the correlation structure of a peculiar class of complex valued random fields, as well as the subset of real valued random fields. Some relevant hints about how to construct the subset of real covariance functions characterized by negative values have also been given.

A Fan-Theobald-von Neumann system [7] is a triple (V,W,λ), where V and W are real inner product spaces and λ:V→W is a norm-preserving map satisfying a Fan-Theobald-von Neumann type inequality together with a condition for equality. Examples include Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decomposition systems (Eaton triples). The present article is a continuation of [9] where the concepts of commutativity, automorphisms, majorization, and reduction were introduced and elaborated. Here, we describe some transfer principles and present Fenchel conjugate and subdifferential formulas.

A key aspect of ultracold bosonic quantum gases in deep optical lattice potential wells is the realization of the strongly interacting Mott insulating phase. Many characteristics of this phase are well understood, however little is known about the effects of a random external potential on its gapped quasiparticle and quasihole low-energy excitations. In the present study we investigate the effect of disorder upon the excitations of the Mott insulating state at zero temperature described by the Bose–Hubbard model. Using a field-theoretical approach we obtain a resummed expression for the disorder ensemble average of the spectral function. Its analysis shows that disorder leads to an increase of the effective mass of both quasiparticle and quasihole excitations. Furthermore, it yields the emergence of damped states, which exponentially decay during propagation in space and dominate the whole band when disorder becomes comparable to interactions. We argue that such damped-localized states correspond to single-particle excitations of the Bose-glass phase.

Since published in 1988, the FFT Accumulation Method (FAM) has been used extensively to compute the Spectral Correlation Function (SCF) and the Spectral Coherence Function (SCoF) to obtain or detect cyclic features of cyclostationary signals. When the input is a Gaussian random variable (r.v.), the SCF (or SCoF) estimates are also random variables with some probability density function (pdf). Although the FAM is considered the most computationally efficient method, there has been no in-depth statistical analysis of the algorithm. This paper analyzes the statistics of spectral estimates of the SCF using the FAM algorithm by obtaining the pdf for the points covering the frequency and cycle frequency f ; α plane, and application examples with simulation results are provided. The method proposed in the paper can be extended to other algorithms, provided they can be given by a quadratic form.