Cauchy Distribution Research Articles

Quasielastic Neutron Scattering Study on Thermal Gelation in Aqueous Solution of Agarose.

The dynamics of water and agarose molecules in an agarose aqueous solution has been studied by means of quasielastic neutron scattering (QENS). The dynamic structure factor S (Q,E) of the agarose aqueous solution was fitted well to the sum of the Lorentz and delta function. The former is attributed to the diffusive motion of water molecules and the latter to the local vibrational motion of agarose molecules. The self-diffusion coefficient D of water molecules was obtained from the Q-dependence of the width of the Lorentz function, while the mean square displacement <u2> of agarose molecules was obtained from the Q-dependence of the intensity of the delta term. In the cooling direction, both D and <u2> decreased with decreasing temperature and showed discontinuous changes around the thermal gelation temperature (around 314 K). In the heating direction, however, D and <u2> did not show the obvious change below 343 K, indicating a large hysteresis effect. The present results of <u2> and D revealed that the thermal gelation suppresses the motion of the polymer and accelerates the diffusion of water molecules. The activation energy Ea of the diffusion of water in the sol state is the same as that of bulk water, but the Ea in the gel state is clearly smaller than that of bulk water.

Cauchy distribution with Cuckoo search algorithms for solving job shop scheduling Problem

Cauchy distribution with Cuckoo search algorithms for solving job shop scheduling Problem

Color image restoration with mixed Gaussian–Cauchy noise and blur

Color image restoration with mixed Gaussian–Cauchy noise and blur

Simultaneous creatine and phosphocreatine mapping of skeletal muscle by CEST MRI at 3T.

To confirm that CrCEST in muscle exhibits a slow-exchanging process, and to obtain high-resolution amide, creatine (Cr), and phosphocreatine (PCr) maps of skeletal muscle using a POlynomial and Lorentzian Line-shape Fitting (PLOF) CEST at 3T. We used dynamic changes in PCr/CrCEST of mouse hindlimb before and after euthanasia to assign the Cr and PCr CEST peaks in the Z-spectrum at 3T and to obtain the optimum saturation parameters. Segmented 3D EPI was employed to obtain multi-slice amide, PCr, and Cr CEST maps of human skeletal muscle. Subsequently, the PCrCEST maps were calibrated using the PCr concentrations determined by 31 P MRS. A comparison of the Z-spectra in mouse hindlimb before and after euthanasia indicated that CrCEST is a slow-exchanging process in muscle (<150.7 s-1 ). This allowed us to simultaneously extract PCr/CrCEST signals at 3T using the PLOF method. We determined optimal B1 values ranging from 0.3 to 0.6 μT for CrCEST in muscle and 0.3-1.2 μT for PCrCEST. For the study on human calf muscle, we determined an optimum saturation time of 2 s for both PCr/CrCEST (B1 = 0.6 μT). The PCr/CrCEST using 3D EPI were found to be comparable to those obtained using turbo spin echo (TSE). (3D EPI/TSE PCr: (2.6 ± 0.3) %/(2.3 ± 0.1) %; Cr: (1.3 ± 0.1) %/(1.4 ± 0.07) %). Our study showed that in vivo CrCEST is a slow-exchanging process. Hence, amide, Cr, and PCr CEST in the skeletal muscle can be mapped simultaneously at 3T by PLOF CEST.

Electronic and structural aspects of the chiral helimagnetic compound Cu[formula omitted]OSeO[formula omitted

The electronic state and the structural properties of the helical magneto-electric material Cu2OSeO3 is investigated through temperature-dependent synchrotron based x-ray diffraction and x-ray absorption spectroscopy. The x-ray absorption spectra on Cu and Se-K edges indicate that Cu and Se ions are in the 2＋ and 4＋ charge states respectively. The pre-edge corresponding to K-line of Cu in the x-ray absorption spectra cannot be fitted using a single Lorentzian function. At least two different Lorentzian lines are required to fit the pre-edge, which are separated by ∼0.6±0.042 eV. This separation originates due to the difference in the crystallographic environment of Cu(1) and Cu(2) and the subsequent difference in the energy of the 3d orbitals of two Cu sites. The obtained energy difference reconfirms the value reported using resonant inelastic x-ray scattering and maximally localized Wannier function calculation. The synchrotron based powder x-ray diffraction indicates significant anomaly in the Cu-O-Cu bond angle around the magnetic transition temperature (TC∼ 59 K). Interestingly, the sample show negative thermal expansion in the lattice parameter below TC. Such anomalies can be connected to the exchange-striction mechanism.

Verification of different Fizeau fringe analysis algorithms based on airborne wind lidar data in support of ESA's Aeolus mission.

The Aeolus mission by the European Space Agency was launched in August 2018 and stopped operations in April 2023. Aeolus carried the direct-detection Atmospheric LAser Doppler INstrument (ALADIN). To support the preparation of Aeolus, the ALADIN Airborne Demonstrator (A2D) instrument was developed and applied in several field campaigns. Both ALADIN and A2D consist of so-called Rayleigh and Mie channels used to measure wind from both molecular and particulate backscatter signals. The Mie channel is based on the fringe-imaging technique, which relies on determining the spatial location of a linear interference pattern (fringe) that originated from multiple interference in a Fizeau spectrometer. The accuracy of the retrieved winds is among others depending on the analytic algorithm used for determining the fringe location on the detector. In this paper, the performance of two algorithms using Lorentzian and Voigt fit functions is investigated by applying them to A2D data that were acquired during the AVATAR-I airborne campaign. For performance validation, the data of a highly accurate heterodyne detection wind lidar (2-µm DWL) that was flown in parallel are used as a reference. In addition, a fast and non-fit-based algorithm based on a four-pixel intensity ratio approach (R 4) is developed. It is revealed that the Voigt-fit-based algorithm provides 50% more data points than the Lorentzian-based algorithm while applying a quality control that yields a similar random error of about 1.5m/s. The R 4 algorithm is shown to deliver a similar accuracy as the Voigt-fit-based algorithms, with the advantage of a one to two orders of magnitude faster computation time. Principally, the R 4 algorithm can be adapted to other spectroscopic applications where sub-pixel knowledge of the location of measured peak profiles is needed.

Computed tomography dose index determination in dose modulation prospectively involving the third-generation iterative reconstruction and noise index.

Optimizing CT protocols is challenging in the presence of automatic dose modulation because the CT dose index (CTDIvol) at different patient sizes is unknown to the operator. The task is more difficult when both the image quality index and iterative reconstruction prospectively affect the dose determination. It is of interest in practice to be informed of the CTDIvol during the protocol initialization and evaluation. It was our objective to obtain a predictive relationship between CTDIvol, the image quality index, and iterative reconstruction strength at various patient sizes. Dose modulation data were collected on a GE Revolution 256-slice scanner utilizing a Mercury phantom and selections of the noise index (NI) from 8 to 17, the third generation iterative reconstruction (ASIR-V) from 0% to 80%, and phantom diameters from 16 to 36cm. The fixed parameters were 120kVp, a pitch of .984, and a collimation of 40mm with a primary slice width of 2.5mm. The CTDIvol per diameter was based on the average tube current over three adjacent slices (same or similar diameter) multiplied by a conversion factor between the average mA of the series and the reported CTDIvol. The relationship between CTDIvol, NI, and ASIR-V for each diameter was fitted with a 2nd order polynomial of ASIR-V multiplied by a power law of NI. The ASIR-V fit parameters versus diameter followed a Lorentz function while the NI exponent versus diameter followed an exponential growth function. The CTDIvol predictions were accurate within 15% compared to phantom results on a separate GE Revolution. For clinical relevance, the phantom diameter was converted to an abdomen or chest equivalent diameter and was well matched to patient data. The fitted relationship for CTDIvol. for given values of NI and ASIR-V blending for a range of phantom sizes was a good match to phantom and patient data. The results can be of direct help for selecting adequate parameters in CT protocol development.

On the Kolmogorov Distance for the Maximum Likelihood Estimator in the Explosive Ornstein-Uhlenbeck Process

The paper estimates the Kolmogorov distance between the distribution of the normalized maximum likelihood estimator of the positive drift parameter in the nonergodic Ornstein-Uhlenbeck process and the standard Cauchy distribution and shows exponential error rate for large time limit.

Traveling spiral wave chimeras in coupled oscillator systems: emergence, dynamics, and transitions

Systems of coupled nonlinear oscillators often exhibit states of partial synchrony in which some of the oscillators oscillate coherently while the rest remain incoherent. If such a state emerges spontaneously, in other words, if it cannot be associated with any heterogeneity in the system, it is generally referred to as a chimera state. In planar oscillator arrays, these chimera states can take the form of rotating spiral waves surrounding an incoherent core, resembling those observed in oscillatory or excitable media, and may display complex dynamical behavior. To understand this behavior we study stationary and moving chimera states in planar phase oscillator arrays using a combination of direct numerical simulations and numerical continuation of solutions of the corresponding continuum limit, focusing on the existence and properties of traveling spiral wave chimeras as a function of the system parameters. The oscillators are coupled nonlocally and their frequencies are drawn from a Lorentzian distribution. Two cases are discussed in detail, that of a top-hat coupling function and a two-parameter truncated Fourier approximation to this function in Cartesian coordinates. The latter allows semi-analytical progress, including determination of stability properties, leading to a classification of possible behaviors of both static and moving chimera states. The transition from stationary to moving chimeras is shown to be accompanied by the appearance of complex filamentary structures within the incoherent spiral wave core representing secondary coherence regions associated with temporal resonances. As the parameters are varied the number of such filaments may grow, a process reflected in a series of folds in the corresponding bifurcation diagram showing the drift speed s as a function of the phase-lag parameter α.

Stochastic Volatility Modeling based on Doubly Truncated Cauchy Distribution and Bayesian Estimation for Chinese Stock Market

Stochastic Volatility Modeling based on Doubly Truncated Cauchy Distribution and Bayesian Estimation for Chinese Stock Market

Different classical estimation methods using ranked set sampling and data analysis for the inverse power Cauchy distribution

A recent two-parameter distribution that offers additional flexibility in modeling a real lifetime dataset is the inverse power Cauchy distribution (IPCD). The IPCD is user-friendly and analytically attractive, and it can be effectively utilized to analyze actual datasets. On the other hand, ranked set sampling (RSS) is a method of gathering data that first combines simple random sampling (SRS) with the expertise and judgment of the field investigator in determining where to take samples. The RSS is known to be more efficient than the SRS as well as less expensive since fewer sample units need to be procured and evaluated. In this article, various estimation techniques for IPCD are examined. The parameter estimates are determined utilizing the RSS and SRS of seven frequentist approaches, which are the maximum likelihood, Anderson-Darling, least squares, Cramér-von Mises, maximum product spacing, weighted least squares, and right-tail Anderson-Darling approaches. Using accuracy criteria, a simulation analysis is done to assess the efficacy of the provided RSS-based estimates. For further clarification, one real data analysis is provided.

On stability estimations without any conditions of symmetry

Let X, X1, X2, ..., Xn be i.i.d. random variables. B. Ramachandran and C.R. Rao have proved that if distributions of sample mean ‾X = ‾X(n) = (X1 + ⋯ + Xn)/n and monomial X are coincident at least at two points n = j1 and n = j2 such that log j1/ log j2 is irrational, then X follows a Cauchy law. Assuming that condition of coincidence of \bar X(n) and X are fulfilled at least for two n values, but only approximately, with some error ε in metric λ, we prove (without any conditions of symmetry) that, in certain sense, characteristic function of X is close to the characteristic function of the Cauchy distribution.

Analysis of a cracked neo-Hookean substrate with initial stress under a rigid punch

The coupling problem of a cracked neo-Hookean substrate with initial stress under a rigid punch is examined in the current work. Fourier series is used to transform the mixed boundary value problem(MBVP) into singular integral equations (SIEs). The obtained SIEs is solved by using Gauss-Chebyshev collocation method. The stress distributions on the contact zone and crack surface, and related singularities are present. The numerical experiments examine how the punch position, punch width, crack length and propagation parameter affect the stress field. The result showed that: for k = 1 and unchanged material property parameters, the incremental stress distribution on the initially stressed neo-Hookean base is very similar to the Cauchy stress distribution of the isotropic linear elastic base.

Spectral deconvolution of matrix models: the additive case

Abstract We implement a complex analytic method to build an estimator of the spectrum of a matrix perturbed by the addition of a random matrix noise in the free probabilistic regime. This method, which has been previously introduced by Arizmendi, Tarrago and Vargas, involves two steps: the first step consists in a fixed point method to compute the Stieltjes transform of the desired distribution in a certain domain, and the second step is a classical deconvolution by a Cauchy distribution, whose parameter depends on the intensity of the noise. This method thus reduces the spectral deconvolution problem to a classical one. We provide explicit bounds for the mean squared error of the first step under the assumption that the distribution of the noise is unitary invariant. In the case where the unknown measure is sparse or close to a distribution with a density with enough smoothness, we prove that the resulting estimator converges to the measure in the $1$-Wasserstein distance at speed $O(1/\sqrt{N})$, where $N$ is the dimension of the matrix.

Inhomogeneity identification by measuring magnetic quantum oscillations

This study explores the identification of sample inhomogeneity via magnetic quantum oscillations analysis in semimetal NbSb2. By doping Bi and Cr, we obtained a homogenous Bi-doped sample and an inhomogeneous Cr-doped sample, whose homogeneity was confirmed by comparing the magnetic quantum oscillation before and after grinding the samples. The magnetic quantum oscillations in the inhomogeneous sample exhibited a distinct phase shift and unusual field-dependent amplitude, believed to result from a non-uniform Fermi energy. The analysis of the magnetic quantum oscillations demonstrated that the homogeneous Bi-doped sample can be interpreted by the symmetric and Lorentzian effective Fermi energy distribution, while the inhomogeneous Cr-doped sample exhibited an asymmetric distribution, illustrating an unconventional violation of the Lifshitz-Kosevich formula. This research provides a novel method for identifying material inhomogeneity and mitigating potential misinterpretations of magnetic quantum oscillations' unusual phase, commonly seen as a nontrivial Berry phase indicator in topological materials studies.

FTIR spectroscopic investigations of protein conformation provide clues of radioadaptation in the kidney of low-dose irradiated rats

Proteins comprise almost half of the weight of plasma membrane and are responsible for preforming various membrane functions. Functionality of the protein is sensitive to its structure. Therefore, modification of the protein conformation can abrogate its function. We demonstrate the induction of radioadaptation in the kidney of low-dose irradiated rats in the form of diminishing the alterations of the protein secondary structure. The FTIR amide I absorption band fitting using Lorentz function was used to identify the various components of the protein secondary structure. FTIR data show changes in amide I band position, amide I/amide II area ratio, FWHM of amide I band, the shifts in the β-structure bands at 1611, 1623, 1634 cm−1 and the decrease in the β/α ratio. 2Gy x-irradiation lead to an increase in the β-structure at the expense of the α-helix. The effect of 2Gy x-irradiation was diminished when preceded with 100 mGy x-rays 24 h earlier. The data provide clue of induction of radioadaptation in the kidneys of whole-body irradiated rats following receiving a low priming dose 24 h prior to a challenging dose.

Circular-Statistics-Based Estimators and Tests for the Index Parameter α of Distributions for High-Volatility Financial Markets

The distributions for highly volatile financial time-series data are playing an increasingly important role in current financial scenarios and signal analyses. An important characteristic of such a probability distribution is its tail behaviour, determined through its tail thickness. This can be achieved by estimating the index parameter of the corresponding distribution. The normal and Cauchy distributions, and, sometimes, a mixture of the normal and Cauchy distributions, are suitable for modelling such financial data. The family of stable distributions can provide better modelling for such financial data sets. Financial data in high-volatility markets may be better modelled, in many cases, by the Linnik distribution in comparison to the stable distribution. This highly flexible family of distributions is better capable of modelling the inflection points and tail behaviour compared to the other existing models. The estimation of the tail thickness of heavy-tailed financial data is important in the context of modelling. However, the new probability distributions do not admit any closed analytical form of representation. Thus, novel methods need to be developed, as only a few can be found in the literature. Here, we recall a recent novel method, developed by the authors, based on a trigonometric moment estimator using circular distributions. The linear data may be transformed to yield circular data. This transformation is solely for yielding a suitable estimator. Our aim in this paper is to provide a review of the few existing methods, discuss some of their drawbacks, and also provide a universal (∀α∈(0,2]), efficient, and easily implementable estimator of α based on the transformation mentioned above. Novel, circular-statistics-based tests for the index parameter α of the stable and Linnik distributions are introduced and also exemplified with real-life financial data. Two real-life data sets are analysed to exemplify the methods recommended and enhanced by the authors.

Noninvasive Characterization of Metabolic Changes in Ischemic Stroke Using Z-spectrum-fitted Multiparametric Chemical Exchange Saturation Transfer-weighted Magnetic Resonance Imaging.

This study aimed to noninvasively characterize the metabolic alterations in ischemic brain tissues using Z-spectrum-fitted multiparametric chemical exchange saturation transfer-weighted magnetic resonance imaging (CEST-MRI). Three sets of Z-spectrum data with saturation power (B1) values of 1.5, 2.5, and 3.5 µT, respectively, were acquired from 17 patients with ischemic stroke. Multiple contrasts contributing to the Z-spectrum, including fitted amide proton transfer (APTfitted), +2 ppm peak (CEST@2ppm), concomitantly fitted APTfitted and CEST@2ppm (APT&CEST@2ppm), semisolid magnetization transfer contrast (MT), aliphatic nuclear Overhauser effect (NOE), and direct saturation of water (DSW), were fitted with 4 and 5 Lorentzian functions, respectively. The CEST metrics were compared between ischemic lesions and contralateral normal white matter (CNWM), and the correlation between the CEST metrics and the apparent diffusion coefficient (ADC) was assessed. The differences in the Z-spectrum metrics under varied B1 values were also investigated. Ischemic lesions showed increased APTfitted, CEST@2ppm, APT&CEST@2ppm, NOE, and DSW as well as decreased MT. APT&CEST@2ppm, MT, and DSW showed a significant correlation with ADC [APT&CEST@2ppm at the 3 B1 values: R=0.584/0.467/0.551; MT at the 3 B1 values: R=-0.717/-0.695/-0.762 (4-parameter fitting), R=-0.734/-0.711/-0.785 (5-parameter fitting); DSW of 4-/5-parameter fitting: R=0.794/0.811 (2.5 µT), R=0.800/0.790 (3.5 µT)]. However, the asymmetric analysis of amide proton transfer (APTasym) could not differentiate the lesions from CNWM and showed no correlation with ADC. Furthermore, the Z-spectrum contrasts varied with B1. The Z-spectrum-fitted multiparametric CEST-MRI can comprehensively detect metabolic alterations in ischemic brain tissues.

Identifying Probable Submarine Hydrothermal Spots in North Santorini Caldera Using the Generalized Moments Method

The presence of active hydrothermal vent fields near residential areas and their possible link to volcanic activity poses a potential hazard to the environment, society, and the economy. By capitalizing on Autonomous Underwater Vehicle sampling methodologies and applying the Generalized Moments Method model for geological and physical processes in these environments, we shed light on the underlying dynamics shaping the physicochemical characteristics of the vents. In this study, we focus on the Northern Caldera of Santorini and, more specifically, on the recorded CTD data (Conductivity, Temperature, Depth). The data sets were collected in 2017 in Santorini using an Autonomous Underwater Vehicle during the GEOMAR POS510 mission. Our research shows that the active vent field within the caldera probably follows a multifractal behavior and exhibits a weak memory effect. Depth Profiles and Time Series show similar behavior among conductivity and temperature. The variance and moments of both parameters underline the existence of two different mechanisms governing the behavior of the vent field. Finally, the structure function shows that changes in the time series are described by a Cauchy–Lorentz distribution.

Bayesian inversion with α-stable priors

We propose using Lévy α-stable distributions to construct priors for Bayesian inverse problems. The construction is based on Markov fields with stable-distributed increments. Special cases include the Cauchy and Gaussian distributions, with stability indices α = 1, and α = 2, respectively. Our target is to show that these priors provide a rich class of priors for modeling rough features. The main technical issue is that the α-stable probability density functions lack closed-form expressions, and this limits their applicability. For practical purposes, we need to approximate probability density functions through numerical integration or series expansions. For Bayesian inversion, the currently available approximation methods are either too time-consuming or do not function within the range of stability and radius arguments. To address the issue, we propose a new hybrid approximation method for symmetric univariate and bivariate α-stable distributions that is both fast to evaluate and accurate enough from a practical viewpoint. In the numerical implementation of α-stable random field priors, we use the constructed approximation method. We show how the constructed priors can be used to solve specific Bayesian inverse problems, such as the deconvolution problem and the inversion of a function governed by an elliptic partial differential equation. We also demonstrate hierarchical α-stable priors in the one-dimensional deconvolution problem. For all numerical examples, we use maximum a posteriori estimation. To that end, we exploit the limited-memory BFGS and its bounded variant for the estimator.