Cauchy Distribution Research Articles

Calibration of the JAGB method for the Magellanic Clouds and Milky Way from Gaia DR3, considering the role of oxygen-rich AGB stars

The JAGB method is a new way of measuring distances in the Universe with the use of asymptotic giant branch (AGB) that are situated in a selected region in a $J$ versus $J- colour--magnitude diagram (CMD), and relying on the fact that the absolute $J$ magnitude is (almost) constant. It is implicitly assumed in the method that the selected stars are carbon-rich AGB stars (carbon stars). However, as the sample selected to determine J $ is purely colour based, there can also be contamination by oxygen-rich AGB stars in principle. As the ratio of carbon-rich to oxygen-rich stars is known to depend on metallicity and initial mass, the star formation history and age--metallicity relation in a galaxy should influence the value of $M_ J $. The aim of this paper is to look at mixed samples of oxygen-rich and carbon-rich stars for the Large Magellanic Cloud (LMC), Small Magellanic Cloud (SMC), and Milky way (MW) using the Gaia catalogue of long-period variables (LPVs) as a basis. The advantage of this catalogue is that it contains a classification of O- and C-stars based on the analysis of Gaia Rp spectra. The LPV catalogue is correlated with data from the Two Micron All Sky Survey (2MASS) and samples in the LMC, SMC, and the MW are retrieved. Following methods proposed in the literature, we report the mean and median magnitudes of the selected sample using different colour and magnitude cuts and the results of fitting Gaussian and Lorentzian profiles to the luminosity function (LF). For the SMC and LMC, we confirm previous results in the literature. The LFs of the SMC and LMC JAGB stars are clearly different, yet it can be argued that the mean magnitude inside a selection box agrees at the 0.021 mag level. The results of our analysis of the MW sample are less straightforward. The contamination by O-rich stars is substantial for a classical lower limit of $(J- 1.3$, and becomes less than 10<!PCT!> only for $(J- 1.5$. The sample of AGB stars is smaller than for the MCs for two reasons. Nearby AGB stars (with potentially the best determined parallax) tend to be absent as they saturate in the 2MASS catalogue, and the parallax errors of AGB stars tend to be larger compared to non-AGB stars. Several approaches have been taken to improve the situation but finally the JAGB LF for the MW contains about 130 stars and the fit of Gaussian and Lorentzian profiles is essentially meaningless. The mean and median magnitudes are fainter than for the MC samples by about 0.4 mag which is not predicted by theory. We do not confirm the claim in the literature that the absolute calibration of the JAGB method is independent of metallicity up to solar metallicity. A reliable calibration of the JAGB method at (near) solar metallicity should await further data releases, or should be carried out in another environment.

Development of craquelure patterns in paintings on canvas

Canvas paintings are layered structures composed of canvas support sized with animal glue, a preparatory layer of the ground, and paint and varnish layers on the top. Preventing or limiting humidity-induced stresses in these structures requires an understanding of the relevant processes and risks. A three-dimensional model of a canvas painting was used to analyse stresses and crack development in the two-layer structure comprised of a glue-sized canvas on a wooden stretcher with a layer of stiff chalk-glue ground representing a pictorial layer in historic canvas paintings. The model was subjected to a large relative humidity fall which induced shrinkage of the glue-sized canvas. The modelling revealed that when a stretcher with flexible wooden bars is considered, high tensile stresses arise in the ground layer at the corners of the painting, and cracks are formed in these areas in the direction perpendicular to the painting’s diagonal. Ratios of critical distances between cracks to the ground layer thickness for which stresses in the midpoints between the cracks dropped to below the level inducing fracture in the material were estimated for various magnitudes of the relative humidity drop and thicknesses of the ground layer. Increasing ground layer thickness limits the hygric response of the sized canvas and makes the paintings less vulnerable to humidity variations. The ratio of stress along the diagonal calculated for painting with one crack to the solution without cracks was described by the double Lorentz function. A simple procedure of calculating stress variations along the diagonal—using the function—on a sequential addition of cracks was developed. Cracks in central parts of canvas painting were found to be induced by permanent cumulative drying shrinkage of the oil-based paints and grounds due to the evolution of the molecular composition of the oil binder. The outcome of the modelling indicated that the risk of cracking of the pictorial layers in canvas paintings due to drops in ambient relative humidity was small.

Modeling of Tensile Stress Distribution Considering Anisotropy of Mechanical Properties of Thin-Walled AlSi10Mg Samples Obtained by Selective Laser Melting

The work that is being presented demonstrates that there is a critical point at which the engineering stress–strain diagram’s elastic–plastic region transitions to yield and fracture stresses. This transition is demonstrated using thin-walled specimens made using selective laser melting technology from high-strength aluminum alloys (AlSi10Mg) that have undergone preliminary heat treatment. It was discovered that the strain-hardening coefficient, which was determined in the section from yield strength to fracture strength, and the critical point have a highly statistically significant association (0.83 by Spearman and 0.93 by Pearson). It was possible to derive a regression equation that connected the strain-hardening coefficient with the crucial transition point. The type of stress distribution in the elastic–plastic region changes (the Weibull distribution changes to a normal distribution) as the plasticity of the thin-walled samples increases. Additionally, the contribution of the probability density of the stress distribution described by the Cauchy distribution increases in a mode near the point at which the probability density of the fracture increases.

Chaos in Bitcoin Cryptocurrency Metrics: Analysis and Forecasts

AbstractCryptocurrencies, particularly Bitcoin have attracted a lot of attention in the last decades of humanity. Analyzing cryptocurrencies algorithmic differences, chaotic behavior and self-similarity in cryptocurrency metrics might give significant insights for identifying risks and opportunities. Determining the degree of chaos in crypto metrics is critical for understanding complexity, improving prediction capabilities, and supporting decision-making. This study focuses on the analysis of chaos and self-similarity in Bitcoin dynamics for predictability perspective. Return, rate of return and volume quantities in different scales are analyzed with using rescaled range method to reveal the degree of self-similarity. Hurst parameter extracts a comprehensive summary providing information on how current values depend on previous ones to reveal any persistence in Bitcoin metrics. Daily rate of return and return give Hurst degree around 0.64 while they are in between 0.52–0.55 for minutely and hourly based prices. However, an increasing persistence is observed with the increasing time window. Although the largest Lyapunov exponents stay in the positive region for prices and returns of Bitcoin, they are approximately zero for inspected statistics. Periodic characteristics of Bitcoin are also investigated to reveal any dependencies on halving mechanism of Bitcoin. Detailed self-similarity analysis on specific periods shows that bull and bear market seasons don’t make any significant effect on the degree of Hurst parameter. Due to nonlinear and unpredictable characteristics of Bitcoin metrics, distribution fittings are applied to characterize BTC return and rate of return. While Wakeby distribution gives best fitting for daily return, Cauchy distribution gives best for hourly returns.

Seismic scattering inversion for multiple parameters of overburden-stressed isotropic media

Subsurface reservoirs are compressed by overburden stress resulting from the gravity of overlying masses, and the resulting stress changes significantly affect the seismic reflection responses generated at the reservoir interfaces. Several exact reflection coefficient equations have been well established to delineate the role that in situ stress plays in altering the energy transition and amplitude of seismic reflection responses. These exact equations, however, cannot be effectively used in practice due to their intricate formulations and the difficult geophysical estimation for the third-order elastic constants (3oECs) embedded in reflection coefficients. Based on the theories of nonlinear elasticity and elastic wave inverse scattering, we derive an approximate seismic reflection coefficient equation for overburden-stressed isotropic media in terms of the P-wave modulus, shear modulus, density, and a defined stress-related parameter (SRP). The SRP is a combined quantity of elastic moduli, 3oECs, and overburden stress, which can be naturally treated as a dimensionless stress-induced anisotropy parameter. Its inclusion effectively eliminates the need for 3oECs information when using our equation to estimate the desired reservoir properties from seismic observations. By comparing our equation to the exact one, we confirm its validity within the moderate-stress range. Then, we introduce a Bayesian inversion approach incorporating the new reflection coefficient equation to estimate four model parameters. In our approach, the Cauchy and Gaussian distribution functions are used for the a priori probability and the likelihood distributions, respectively. The synthetic tests from two well-log data sets and a field example demonstrate that four parameters can be reasonably inverted using our approach with rather smooth initial models, which illustrates the feasibility of our inversion approach.

Exploring mixture estimators in stratified random sampling.

Advancements in sensor technology have brought a revolution in data generation. Therefore, the study variable and several linearly related auxiliary variables are recorded due to cost-effectiveness and ease of recording. These auxiliary variables are commonly observed as quantitative and qualitative (attributes) variables and are jointly used to estimate the study variable's population mean using a mixture estimator. For this purpose, this work proposes a family of generalized mixture estimators under stratified sampling to increase efficiency under symmetrical and asymmetrical distributions and study the estimator's behavior for different sample sizes for its convergence to the Normal distribution. It is found that the proposed estimator estimates the population mean of the study variable with more precision than the competitor estimators under Normal, Uniform, Weibull, and Gamma distributions. It is also revealed that the proposed estimator follows the Cauchy distribution when the sample size is less than 35; otherwise, it converges to normality. Furthermore, the implementation of two real-life datasets related to the health and finance sectors is also presented to support the proposed estimator's significance.

Intellectual fitting of hard X-ray resonant reflectance maps obtained for low-contrast oxide multilayers across L absorption edges of Eu and Gd

The present work describes an intellectual computational approach to X-ray resonant reflectometry – a synchrotron-based method aimed at non-destructive characterization of epitaxial nanoscale multilayers with low optical contrast between the sublayer materials. Resonant reflectance from specially designed bilayer structures was measured as a function of grazing angle and photon energy across the L absorption edges of rare earth elements and then modeled with the aid of a specially developed software utilizing OpenCL high speed computations performed on a graphical processing unit. The map fitting was carried out in the blind mode in which the spectral shapes of the optical constants of the sublayers were reconstructed by the fitting routine without initial knowledge of the chemical composition and without using any reference spectra. The model uses stepped Lorentzian line shape for the imaginary part of the scattering length density and the Kramers-Kronig consistent line shape for the corresponding real part. Epitaxially grown Eu2O3 / Gd3Ga5O12 bilayers were used as a test bench system to demonstrate the benefits of the proposed 2D mapping technique over conventional X-ray reflectometry. To increase reliability, the map fitting was carried out simultaneously at the absorption edges of two different chemical elements. The derived shapes and positions of the absorption peaks were used to uniquely identify rare earth elements within the corresponding sublayers and provide information on their oxidation states. The method was experimentally tested on the bilayers having different material order to show that not only the chemical composition but also the layer sequence can be effectively reconstructed in the automatic way from the 2D resonant reflectance maps.

Investigating an Asymmetric Ratio Cosine Distribution

Most real-world data have asymmetric features, slight or pronounced, that cannot be analyzed in depth using classical symmetric distributions. This is especially true when the underlying phenomena take their values in bounded support. Examples include data with support (−1, 1), which may correspond to daily temperature anomalies, stock market returns and satisfaction ratings, among others. Motivated by this last point, this article introduces a novel asymmetric cosine distribution of the ratio type. It aims to extend the functionalities of the classical cosine distribution by incorporating two adjustable parameters that allow for flexible levels of asymmetry. The main functions and theoretical properties such as moments, quantiles and distributional properties are studied. Some derived distributions are also established, extending the scope of the Cauchy and half-Cauchy distributions. Applications to simulated data in hypothetical environmental, financial and sentiment analysis scenarios demonstrate the practical utility of the asymmetric cosine distribution in capturing nuanced behavior.

Effective phase diffusion for spin phase evolution under random nonlinear magnetic field.

The general theoretical description of spin self-diffusion under a nonlinear gradient magnetic field is proposed, which extends the effective phase diffusion method for a linear gradient field. Based on the phase diffusion, the proposed method reveals the general features of phase evolutions in nonlinear gradient fields. There are three types of phase evolutions: phase diffusion, float phase evolution, and shift evolution based on the starting position. For spin diffusion near the origin of the nonlinear field, these three phase evolutions significantly affect the nuclear magnetic resonance (NMR) signal. The traditional methods have difficulties in handling these three-phase evolutions. Notably, the phase from float phase evolution is missed or misplaced in traditional methods, which leads to incorrect NMR signal attenuation or phase shift. The method here shows that the diffusing and float phase evolutions come from the first and second derivatives of the gradient field. Based on these three phase evolutions, the phase variance and corresponding NMR signal attenuation are obtained, as demonstrated by calculating the phase diffusions under both parabolic and cubic fields. The results indicate that signal attenuation obeys Gaussian attenuation for a short time, then changes to follow Lorentzian or Mittag-Leffler function attenuations as time increases, significantly different from Gaussian attenuation. For spins starting diffusion far away from the origin of the field gradient, the signal attenuation is Gaussian, but the float phase still has an important effect on the total phase shift of even-order gradient fields, which could be used to measure the diffusion coefficient directly. Random walk simulations were performed, which support the obtained theoretical results. General theoretical expressions are obtained, which can handle random order nonlinear gradient fields. The results could help develop advanced experimental techniques based on a nonlinear gradient field in NMR and magnetic resonance imaging.

Characterizing polarization switching kinetics of ferroelectric Hf0.5Zr0.5O2 at cryogenic temperature

We have characterized polarization switching kinetics of Hf0.5Zr0.5O2 (HZO) at cryogenic temperature (T) down to 100 K. By the nucleation-limited-switching model with Lorentzian distribution fittings, the dependences of the average switching time (log τ1) and switching time distribution (ω) on T are extracted. As T decreases from 300 to 100 K, log τ1 first rapidly decreases and then gradually stabilizes. Meanwhile, ω exhibits different trends under different external electric fields (Eext), decreasing under low Eext while increasing under high Eext, eventually stabilizing at non-zero constants. With the further decrease in T (<100 K), log τ1 and ω exhibited by HZO are almost unchanged, indicating the intrinsic properties of ferroelectric multiple domains, which are different from the prediction in previous literature studies that log τ1 and ω will linearly decrease as T decreases.

Exact stark analytical function for Hα line based on the FFM model related with plasma parameters

Optical Emission Spectroscopy is a widely used technique for plasma diagnosis, with particular interest in hydrogen atomic emission due to its prevalence in plasmas. However, accurately determining plasma parameters like electron density, electron temperature, and gas temperature starting from the experimental profiles remains a challenge. This paper introduces a comprehensive model for Stark broadening of the Hα line in a wide range of plasma conditions, addressing the limitations of existing analytical expressions for line shapes. The proposed model encompasses the full splitting of the transition into fifteen Lorentzian profiles and electric micro-field fluctuations surrounding the emitting atoms due to collisions with charged particles. Starting from accurate spectral data obtained from realistic computer simulations, fitting parameters of the model, have been obtained by using an optimization method based on a genetic algorithm. The set of parameters of the model are reported for a wide range of plasma conditions. The behavior of these parameters is analyzed to understand their dependence in terms of the electron density and temperature and gas density of the plasma. The model parameters here obtained constitute a useful tool in plasma diagnosis to obtaining the values of the physical parameters of the plasma starting from the experimental profiles.

Cauchy noise removal using high-order total variation and overlapping group sparsity

Image denoising presents a significant challenge in computer vision, aiming to eliminate unwanted noise from images and restore their original quality. Cauchy noise, characterized by its heavy-tailed distribution, poses a unique hurdle among various types of noise. While several denoising models have been proposed, Total Variation (TV)-based methods, such as the Remove Cauchy Noise model, are widely used for their effectiveness. However, these approaches often suffer from issues such as oversmoothing, staircase artifacts, and the introduction of false details. To address these limitations, recent research has explored the combination of high-order TV with Overlapping Group Sparsity (OGS), showing promising results in noise removal. Inspired by this, our article introduces a novel Cauchy denoising model. Our approach leverages OGS and directional higher-order TV to effectively remove Cauchy noise while preserving image details and minimizing aliasing and smoothing artifacts. The Chambolle-Pock algorithm efficiently solves the underlying optimization problem. Through qualitative and quantitative evaluations, including visualization and parameter measurements, we demonstrate the competitiveness of our model compared to existing methods.

Nonlinear Vlasov and Fokker-Planck Dynamics in Confined Systems with Drag: A Numerical Study

Abstract We explore numerically the behavior of a one-dimensional many-body system consisting of particles that interact through short range repulsive forces, and are also under the effects of drag forces, and of an external confining potential. The statistical dynamics of systems of this kind exhibits interesting links with the thermostatistical formalism based on the Sq non-additive entropies. In the regime of overdamped motion, these systems admit an effective description in terms of a non-linear Fokker-Planck equation. When the overdamped condition is relaxed, and inertial effects are explicitly taken into account, the system can be described by a Vlasov-like effective mean field dynamics. The Vlasov-like description of this type of systems has been recently investigated in the literature from an analytical point of view. In the present contribution we explore the behaviour of these system numerically, through direct molecular dynamics simulations. We consider examples of systems with four different short range repulsive forces, with a dependence on distance given by exponential, Heaviside, Lorentzian, and Bessel functions. The results of our numerical simulations are fully consistent with the predictions derived from the Vlasov-like mean field description. In particular, we verify that, in the asymptotic limit of large times, the system evolves towards a state exhibiting a spatial distribution of particles that coincides with the stationary solution of an appropriate nonlinear Fokker-Planck equation. This limit spatial distribution has the form of a q-Gaussian.

Optically Tunable Many-Body Exciton-Phonon Quantum Interference.

This study introduces a novel paradigm for achieving widely tunable many-body Fano quantum interference in low-dimensional semiconducting nanostructures, beyond the conventional requirement of closely matched energy levels between discrete and continuum states observed in atomic Fano systems. Leveraging Floquet engineering, the remarkable tunability of Fano lineshapes is demonstrated, even when the original discrete and continuum states are separated by over 1eV. Specifically, by controlling the quantum pathways of discrete phonon Raman scattering using femtosecond laser pulses, the Raman intermediate states across the excitonic Floquet band are tuned. This manipulation yields continuous transitions of Fano lineshapes from antiresonance to dispersive and to symmetric Lorentzian profiles, accompanied by significant variations in Fano parameter q and Raman intensity spanning 2 orders of magnitude. A subtle shift in the excitonic Floquet resonance is further shown, achieved by controlling the intensity of the femtosecond laser, which profoundly modifies quantum interference strength from destructive to constructive interference. The study reveals the crucial roles of Floquet engineering in coherent light-matter interactions and opens up new avenues for coherent control of Fano quantum interference over a broad energy spectrum in low-dimensional semiconducting nanostructures.

Some non parametric tests for umbrella alternatives in experimental design: A simulation study

Testing the null hypothesis against the umbrella alternative arises in various practical situations. This article has studied some non parametric tests that may be used for umbrella alternatives and their performance in different symmetric and asymmetric distributions, including long-tailed, short-tailed, and a mixture of two distributions. We have considered viz., Mack and Wolfe test, Hettmansperger and Norton test, Shi test, SU test, and Modified Mack and Wolfe test. The simulation approach assesses the empirical level and power of test statistics. The study shows that the SU and Shi tests have performed almost the same in Normal, Cauchy, Contaminated normal, and mixture distributions. However, the overall performance of the SU test was found to be better than that of other selected tests.

Early emission lines in SN 2024ggi revealed by high-resolution spectroscopy

Context. We present an analysis of very early high-resolution spectroscopic observations of the Type II supernova (SN) 2024ggi, a nearby SN that occurred in the galaxy NGC 3621 at a distance of 7.24 Mpc (z ≈ 0.002435). These observations represent the earliest high-resolution spectra of a Type II SN ever made. Aims. We analyzed the very early-phase spectroscopic evolution of SN 2024ggi obtained in a short interval at 20.6 and 27.8 h after its discovery, or 26.6 and 33.8 h after the SN first light. Observations were obtained with the high-resolution spectrograph MIKE (R ≈ 22 600 − 28 000) at the 6.5 m Magellan Clay Telescope, located at the Las Campanas Observatory, on the night of April 12, 2024 UT. Methods. The emission lines were identified and studied in detail during the first hours of SN 2024ggi. We analyzed the evolution of ions of H I, He I, He II, N III, C III, Si IV, N IV, and C IV detected across the spectra. We modeled these features with multiple Gaussian and Lorentzian profiles, and estimated their velocities and full widths at half maximum (FWHMs). Results. The spectra show asymmetric emission lines of H I, He II, C IV, and N IV that can be described by narrow Gaussian cores (FWHM ≤ 200 km s−1) with broader Lorentzian wings, and symmetric narrow emission lines of He I, N III, and C III. The emission lines of He I are detected only in the first spectrum, indicating the rapid ionization of He I to He II. The narrow components of the emission lines show a systematic blueshift relative to their zero-velocity position, with an increase of ∼18 km s−1 in the average velocity between the two epochs. The broad Lorentzian components show a blueshift in velocity relative to the narrow components, and a significant increase in the average velocity of ∼103 km s−1. Such a rapid evolution and significant ionization changes in a short period of time were never observed before, and are probably a consequence of the radiative acceleration generated in the SN explosion.

Аппроксимация статистических данных заболеваемости коронавирусной инфекцией с учетом расслоения по сопутствующим диагнозам

The article considers the stratification of concomitant diagnoses of Covid-19 recovery statistics for the city of Irkutsk for 2020-2021. The previous study was conducted without taking into account such stratification. Various options for approximating real statistics by Gaussian and Lorentz functions, gamma distribution, and Johnson curves are considered. It is shown that the stratification of recovery statistics improves the approximation of Gaussian and Lorentz functions in comparison with integral statistics, and the construction of an approximation based on the Lorentz function always describes the real statistics better. Estimates of mathematical expectation and variance based on statistical data are consistent with estimates of these values based on the Gaussian approximation of statistics by the least squares method, i.e. the approaches are equivalent. At the same time, calculations of the Pearson Chi-squared criterion reject the hypothesis that empirical data correspond to the assumed theoretical distribution. Therefore, we cannot talk about finding the distribution function, but only about approximating statistics by certain types of curves. The fitting of empirical data by Gaussian and Lorentz functions was carried out using the least squares method. In general, the approximation error due to the stratification of statistics on concomitant diagnoses decreases from 6% to 3%.

Noncommutative wormhole in de Rham-Gabadadze-Tolley like massive gravity

The wormhole solution in dRGT massive gravity is examined in this paper in the background of non-commutative geometry. In order to derive the wormhole model, along with the zero tidal force, we assume that the matter distribution is given by the Gaussian and Lorentzian distributions. The shape function in both models involves the massive gravity parameters m2c1 and m2c2. But the spacetime loses its asymptotic flatness due to the action of the massive gravity parameter. It is noticed that the asymptotic flatness is affected by the repulsive effect induced in the massive gravitons that push the spacetime geometry very strongly. We observed that each model violates the null energy criteria, indicating the presence of exotic matter which is necessary to sustain the wormholes. The exotic matter is measured using the volume integral quantifier. Moreover, it is discovered that the model is stable under the hydrostatic equilibrium condition by utilizing the TOV equation. Finally, our research encompassed an exploration of the repulsive influence exerted by gravity. Our findings demonstrated that the presence of repulsive gravity results in a negative deflection angle for photons following null geodesics. Remarkably, we consistently observed negative values for the deflection angle across all values of r0 in the two scenarios examined. This consistent negativity unequivocally signifies the manifestation of the repulsive gravity effect.

Inhomogeneous broadening effects on bistable behavior in a hybrid optomechanical system

Optical bistability is investigated for optomechanical system composed of an ensemble of inhomogeneously broadened two-level atoms embedded between the two mirrors of the optical cavity, that is subjected to two input (strong and probe) fields. We consider two types of inhomogeneous broadened distributions, namely, Lorentzian and Gaussian line shapes. The first order harmonic components of the output fields (due to the presence of probe field) show reversed and knotted bistability shapes simultaneously with the standard bistable shape of the fundamental component of the field at frequency ωc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega _{c}$$\\end{document}. The inhomogeneous broadened line width for both Lorentzian and Gaussian distribution affects the dispersive switching of the system, as well as the bistable behavior. Locations of the maxima of photon numbers associated with higher harmonics of the output fields are identified with respect to the probe field frequency.

Measurement of the Dispersion of χ(3)$\chi ^{(3)}$ of SiO2${\rm SiO}_2$ and SiN Across the THz and Far‐Infrared Frequency Bands

AbstractTerahertz (THz) radiation sources based on two‐color femtosecond plasmas in air are becoming a mature technology for coherent spectroscopy and strong‐field physics across the extended THz range to several tens of THz. The field‐resolved detection of such THz transients relies on the third‐order nonlinearity of the detection medium. Here, a comparative measurement is demonstrated with air‐biased coherent detection (ABCD) and solid‐state biased detection (SSBCD) as a novel method to measure the dispersion of the magnitude and phase of the relevant third‐order nonlinearity for fused silica () and silicon nitride (SiN). Based on the development of the ultrabroadband SSBCD device with a detection bandwidth exceeding 30 THz, measurements are obtained across the 1–28 THz frequency range, hence covering the THz and far‐infrared. It is shown that the vibrational modes in and SiN in the THz range lead to strong resonant enhancement and dispersion of the nonlinearity. The SSBCD devices operate down to nanojoule (nJ) probe energy, and their is demonstrated by measuring the dielectric function of the Lorentzian line profile of transverse‐optical (TO) phonon mode at 9 THz in single‐crystal gallium arsenide (GaAs) and observing the weak phonon combination bands near the TO phonon.