Gaussian Frequency Distribution Research Articles

The Hubble tension survey: A statistical analysis of the 2012–2022 measurements

ABSTRACT In order to investigate the potential Hubble tension, we compile a catalogue of 216 measurements of the Hubble–Lemaître constant H0 between 2012 and 2022, which includes 109 model-independent measurements and 107 ΛCDM model-based measurements. Statistical analyses of these measurements show that the deviations of the results with respect to the average H0 are far larger than expected from their error bars if they follow a Gaussian distribution. We find that xσ deviation is indeed equivalent in a Gaussian distribution to xeqσ deviation in the frequency of values, where xeq = 0.72x0.88. Hence, a tension of 5σ, estimated between the Cepheid-calibrated type Ia supernovae and cosmic microwave background (CMB) data, is indeed a 3σ tension in equivalent terms of a Gaussian distribution of frequencies. However, this recalibration should be independent of the data whose tension we want to test. If we adopt the previous analysis of data of 1976–2019, the equivalent tension is reduced to 2.25σ. Covariance terms due to correlations of measurements do not significantly change the results. None the less, the separation of the data into two blocks with H0 < 71 and H0 ≥ 71 km s−1 Mpc−1 finds compatibility with a Gaussian distribution for each of them without removing any outlier. These statistical results indicate that the underestimation of error bars for H0 remain prevalent over the past decade, dominated by systematic errors in the methodologies of CMB and local distance ladder analyses.

Cellular automata to understand the prograding limit of deltaic tidal flat

Deltaic tidal flat is an important ecosystem that supports the livelihoods of millions of people globally. Due to the complexity and stochastic nature of tidal flat development, it is a challenging task to predict the tidal flat prograding limit accurately. In this study, we fill out this gap from the perspective of numerical simulation by focusing on the tidal flat extension limit of Nanhui Shoal (NHS), the largest marginal shoal of the Changjiang Estuary. Our results suggest that the tidal flat of NHS extended 29.16 m/y seaward in 1989–2002, had a rapid progradation of 158.08 m/y in 2002–2012 and a retreat rate of 26.88 m/y in 2012–2019, while the fluvial sediment supply decreased 56.78% from 1989–2002 to 2003–2012 and another 20.68% from 2003–2012 to 2013–2019. During the 1989–2002 period, the advancing rate of the −5 m isobaths followed a Gaussian frequency distribution, in agreement with simulation results of cellular automata. We suggest that in the absence of human perturbations, as the sediment input to NHS declines to the threshold of 0.01 × 108 t/y, the seaward advance of tidal flats may significantly slow down. However, human interferences generated large uncertainties on the development tendency of the NHS.

A Wavenumber–Frequency Spectrum Model for Sheared Convective Atmospheric Boundary Layer Flows

Abstract Parameterizing turbulence in the atmospheric boundary layer as a function of space and time is essential for weather and climate models. Here, we explore a model for wavenumber–frequency spectra based on a linear random advection approach to characterize sheared convective atmospheric boundary layer flows. Building on previous works, we obtain the wavenumber–frequency spectrum as a product of the wavenumber spectrum and a Gaussian frequency distribution, whose mean and variance are given by the mean advection and random sweeping velocities, respectively. The applicability of the model is tested with direct numerical simulation data in the mixed layer and the entrainment zone for the streamwise and vertical velocity components and buoyancy. To obtain a fully analytical model, we propose using a von Kármán wavenumber spectrum parameterized by the characteristic variances and integral length scales. These parameters are height dependent and vary considerably with the relative balance of buoyancy and shear forces. The introduced analytical model relies on fitting parameters obtained from numerical data in the relevant range of scales. The comparison of the von Kármán–based spectra for velocity and buoyancy to simulation results shows that the main features of the measured spectra are captured by the model.

Dynamics of topological defects in the noisy Kuramoto model in two dimensions

We consider the two-dimensional (2D) noisy Kuramoto model of synchronization with short-range coupling and a Gaussian distribution of intrinsic frequencies, and investigate its ordering dynamics following a quench. We consider both underdamped (inertial) and over-damped dynamics, and show that the long-term properties of this intrinsically out-of-equilibrium system do not depend on the inertia of individual oscillators. The model does not exhibit any phase transition as its correlation length remains finite, scaling as the inverse of the standard deviation of the distribution of intrinsic frequencies. The quench dynamics proceeds via domain growth, with a characteristic length that initially follows the growth law of the 2D XY model, although is not given by the mean separation between defects. Topological defects are generically free, breaking the Berezinskii-Kosterlitz-Thouless scenario of the 2D XY model. Vortices perform a random walk reminiscent of the self-avoiding random walk, advected by the dynamic network of boundaries between synchronised domains; featuring long-time super-diffusion, with the anomalous exponentα= 3/2.

The study of the dynamics of the order parameter of coupled oscillators in the Ott-Antonsen scheme for generic frequency distributions.

The Ott-Antonsen ansatz shows that, for certain classes of distribution of the natural frequencies in systems of N globally coupled Kuramoto oscillators, the dynamics of the order parameter, in the limit N → ∞, evolves, under suitable initial conditions, in a manifold of low dimension. This is not possible when the frequency distribution, continued in the complex plane, has an essential singularity at infinity; this is the case, for example, of a Gaussian distribution. In this work, we propose a simple approximation scheme that allows one to extend also to this case the representation of the dynamics of the order parameter in a low dimensional manifold. Using the Gaussian frequency distribution as a working example, we compare the dynamical evolution of the order parameter of the system of oscillators, obtained by the numerical integration of the N equations of motion, with the analogous dynamics in the low dimensional manifold obtained with the application of the approximation scheme. The results confirm the validity of the approximation. The method could be employed for general frequency distributions, allowing the determination of the corresponding phase diagram of the oscillator system.

Synchronization in the Kuramoto model in presence of stochastic resetting.

What happens when the paradigmatic Kuramoto model involving interacting oscillators of distributed natural frequencies and showing spontaneous collective synchronization in the stationary state is subject to random and repeated interruptions of its dynamics with a reset to the initial condition? While resetting to a synchronized state, it may happen between two successive resets that the system desynchronizes, which depends on the duration of the random time interval between the two resets. Here, we unveil how such a protocol of stochastic resetting dramatically modifies the phase diagram of the bare model, allowing, in particular, for the emergence of a synchronized phase even in parameter regimes for which the bare model does not support such a phase. Our results are based on an exact analysis invoking the celebrated Ott-Antonsen ansatz for the case of the Lorentzian distribution of natural frequencies and numerical results for Gaussian frequency distribution. Our work provides a simple protocol to induce global synchrony in the system through stochastic resetting.

Contour Maps Description of Natural Radiation in The Soil of Najaf Province

The soil is one of the most important natural sources of ionizing radiation to which humans are exposed. Naturally, radionuclides such as 238U series,232Th series and potassium 40K radioisotope are widespread in the soil and are considered the main sources the terrestrial natural radiation. The study under investigation is regarded as one of the rare studies that deals with the effect of the contents of the radionuclides on levels of natural radiation, besides being helpful in drawing of radiological maps for the area under study. The gamma radiation has been measured to determine 238U, 232Th and 40K natural radioactivity in the soil of selected area in Najaf city, south western part of Iraq. Google Earth program was used to get an upper view picture or photo of the area of study. In addition to this photo, our study involved maps that supported by the center of engineering affairs in the University of Kufa to get a geographical location of area under the study. As soil surface samples, 30 soil samples were obtained in a methodical manner. These samples were investigated using gamma-ray spectrometry, which included a high-counting-efficiency NaI (Tl) detector with a 2"×2" crystal dimension linked to the amplifier via a unit of preamplifier, an analog to digital converter (ADC), and a multichannel analyzer (MCA) with 4096 Cassy type channels. The findings of our research were shown in contoured radiological maps and analyzed using the Gaussian frequency distribution. Finally, all radiation results in the area of study were lower than limits of the global average, and they were also compared to the values recommended by international authorities and 40determined to be within permissible limits

Impact of charge noise on electron exchange interactions in semiconductors

The electron exchange interaction is a promising medium for the entanglement of single-spin qubits in semiconductors as it results in high-speed two-qubit gates. The quality of such entangling gates is reduced by the presence of noise caused by nearby defects acting as two-level fluctuators. To date, the effect of charge noise has been calculated assuming a Gaussian distribution of exchange interaction frequencies between the qubits equivalent to a linear coupling of charge noise with the exchange interaction. In reality the coupling can differ significantly from this linear-coupling approximation depending on the inter-qubit tunnel coupling, detuning of the qubit system, and the magnitude of charge noise. We derive analytical expressions for the frequency spectra of exchange oscillations that encompasses both linear and non-linear coupling to charge-noise. The resulting decoherence times and decay profiles of the exchange oscillations vary considerably. When compared with recent experiments our model shows that non-linear charge-noise coupling is significant and requires consideration to characterise and optimise exchange-based entangling gates.

Positive and negative linear chirping of Gaussian pulses in non-dispersive dielectric layers

In this work, the first and second order effects for reflected and transmitted pulses with a Gaussian distribution of frequencies are analytically developed and interpreted. They show some peculiar characteristics that bear certain parallelism with those of monochromatic beams with a Gaussian distribution of amplitudes. However, when the former are considered, the results of the first and second order effects obtained analytically are more difficult to interpret and are clearly limited by the principle of causality. We show that pulses show some peculiar characteristics when they reflect and transmit through linear media: time delay, shifting of the mean frequency and modification of the pulse width. In addition to these effects we show and analyze the unexpected presence of chirping when a simple structure of non-dispersive materials is considered.

Theoretical Analysis of the EPR Spectrum Saturation Effect with Account for Spectral Diffusion in a System with a Gaussian Distribution of Spin Resonance Frequencies

We have analyzed the effect of saturation of the EPR spectrum for the model situation when a set of resonance frequencies of spin packets is described by a Gaussian distribution and spectral diffusion, which is a random process without correlation, exists. We have obtained the form of the spectrum for arbitrary values of magnetic–resonant parameters: relaxation times for the longitudinal and transverse components of the magnetization vector of the system, the spectral diffusion rate, the dispersion of the Gaussian distribution of resonance frequency, and the magnetic induction of the microwave field. We have derived an analytic expression for the saturation curve of the EPR spectrum. Detailed analysis of the dependence of the spectrum saturation curve on the spin system parameters is performed for the optimal case when the microwave field frequency coincides with the average frequency of the Gaussian distribution for resonance frequencies. The value of the microwave field induction at which the peak of the saturation curve is attained has been determined for arbitrary values of the paramagnetic relaxation time, spectral diffusion rate, and dispersion of the Gaussian spin frequency distribution. We have formulated the algorithm for determining the spin–lattice relaxation time using the experimentally determined microwave field induction for which the saturation curve passes through the peak. It should be noted that numerical calculations have been performed for parameters typical of EPR spectroscopy. The general analytic expression derived for the shape of the spectrum in the saturation conditions and qualitative conclusions concerning the effect of spectral diffusion on the saturation of stationary spectra obtained in this study can also be used in other fields of spectroscopy.

Role of disorder on the signal amplification in an array of unidirectionally coupled MEMS

In a recent work, it was shown that sinusoidal signals are amplified in an array of unidirectionnally coupled MEMS and when the number of MEMS units was large enough, an instability leading to chaotic behavior was observed [M.V. Tchakui et al., Eur. Phys. J. B 89, 22 (2016)]. In this work, we explore the role of disorder in natural frequencies on the propagation and amplification of a signal in such an array. Considering a random Gaussian distribution of natural frequencies, it is found that for small disorder strength, the propagation takes place leading to oscillations in the array. Along the network, a decrease of oscillations amplitude is observed in proportion as the disorder strength grows up. With large coupling strength, the second transition to chaotic behavior is further postponed to a large MEMS rank up to its disappearance as the disorder strength increases in a broaden frequency range. For very large disorder strength the signal amplitude is damped to zero and there appears oscillations death.

Distortions of Gaussian pulses transmitted through a transparent isotropic layer

Los láseres pueden operar en dos regímenes: modo continuo ó pulsado. En los casos más simples, el modo continuo se corresponde a haces monocromáticos con distribución de amplitudes gaussiana (es un haz limitado en el espacio); mientras que el modo pulsado corresponde a haces policromáticos con distribución gaussiana de frecuencias (pulso limitado en el tiempo).Cuando los haces pulsados se reflejan y refractan en diferentes tipos de interfaces, sufren distorsiones peculiares que tienen cierto paralelismo con los encontrados para haces limitados en el espacio. Estos efectos, como se muestra en un trabajo anterior, corresponden al retardo de tiempo (primer orden) y al cambio de ancho de pulso (segundo orden).Las distorsiones están claramente limitadas por el principio de causalidad y su interpretación, aunque no es directa, emerge claramente cuando los campos asociados se expresan en magnitud y fase. Pero como las expresiones analíticas no son simples (incluso en el caso de que el pulso se transmita a través de una capa única de material lineal, homogéneo, isotrópo y transparente) se hace difícil resolver el problema inverso.En este trabajo, presentamos un desarrollo analítico alternativo que hace posible determinar explícitamente estos efectos de distorsión cuando un pulso incide normalmente en una capa isotrópica transparente sumergida en un medio de características análogas.

Effect of the Polymer Chain Arrangement on Exciton and Polaron Dynamics in P3HT and P3HT:PCBM Films

Arrangement of polymer chains at the interface with an acceptor is one of the key issues which influences the interfacial charge transfer. Films of P3HT and P3HT:PCBM blends with the different polymer chain arrangement have been prepared from one-component and binary solvents representing a mixture of chloroform and different poor solvents with high boiling temperature. Electronic absorption and photoluminescence spectra evidenced in favor of reduced disorder of P3HT films as a result of use of poor solvents; the ordering was displayed through structuring and narrowing the spectral bands, indicative of decreasing width of Gaussian distribution of molecular transition frequencies. At the same time, transient absorption of singlet excitons showed that exciton decay in highly ordered P3HT films slows down as compared to the disordered film, and this effect was reproduced for the different poor solvents used. A more pronounced effect was revealed in P3HT:PCBM blends where much faster decay of excitons was found in disordered as-prepared P3HT:PCBM film from chloroform as compared to the annealed film or films prepared from the binary solvents. A complex behavior of polarons in the different regions of the blend film, i.e., within crystalline domains of P3HT and at the interface of P3HT/PCBM, was observed. The conclusion was drawn that chain disorder induces easier exciton dissociation and charge transfer at the P3HT/PCBM interface.

The persistence of tuberculosis in the United States: Spatial analysis and predictive modeling in the move toward elimination of tuberculosis (TB)

Though tuberculosis (TB) prevalence has decreased dramatically in the United States, its continual presence remains a threat to those whose needs are often overlooked. Those already impacted by poverty are the most vulnerable to TB, and stand to bear the worst health impacts, should they contract this disease. Mathematical modeling and spatial analysis have become invaluable tools in TB surveillance monitoring and elimination efforts. In this contribution, we demonstrate the capability of employing a time series, interpolative, vulnerability model to forecasted, state-level TB prevalence in the United States by determining areas influenced by poverty, as well as existing TB data acquired from the Center of Disease Control (CDC). The random effects term in this orthogonal eigenvector spatial filter model was comprised of spatially structured and stochastic effects (that is, spatially unstructured) terms, which were substituted for diagnostic, remote, and clinical covariates in our model. It was assumed that random effects terms in the TB risk model had followed a Gaussian frequency distribution with a mean of zero. The estimate equations were as follows: and . The resulting estimated number of cases for a given state and year was . The Moran coefficient (MC) was 0.66, and its Geary Ratio (GR) was 0.35. The spatially unstructured random effects terms have only trace levels of spatial autocorrelation, with MC = 0.02, and Gr = 0.89. Thus, the assumption of non-zero spatial autocorrelation was violated. The forecast revealed possible hyperendemic transmission of TB in non-coastal, Northwestern states, as well as in some Northeastern states. As such, more intervention efforts should be directed towards these areas. Key words: Tuberculosis (TB), poverty, center of disease control (CDC).

Modeling microwave heating of frozen mashed potato in a domestic oven incorporating electromagnetic frequency spectrum

Domestic microwave oven magnetrons produce microwaves in a frequency range of 2.45 ± 0.05 GHz. Most microwave heat transfer simulations simplify that the magnetron produces a monochromatic electromagnetic wave of frequency of 2.45 GHz to reduce the computational complexity. This study assumes that the magnetron produces a frequency spectrum defined by a Gaussian distribution of frequencies with a central frequency of 2.45 GHz and investigates the effect of Gaussian distribution variance of (0.05 GHz)2, (0.025 GHz)2, (0.017 GHz)2 on prediction accuracy when compared to using monochromatic frequency of 2.45 GHz. A three-dimensional finite element model coupling electromagnetic and heat transfer physics was developed to simulate heating of 550 g of frozen mashed potato for 6 min. The model was validated in a 1250 W rated microwave oven with the mashed potato tray placed at the center of the stationary turntable. The electromagnetic power densities were determined separately at five different frequencies equidistant between 2.4 and 2.5 GHz. They were then weighted averaged, based on the selected Gaussian distribution. Simulated temperature profiles of the models using the monochromatic frequency of 2.45 GHz and Gaussian frequency spectrum with different variances were compared with experimental temperature profiles obtained using a thermal imaging camera at the end of cooking and five fiber-optic thermocouples during cooking. The model results showed that predicted spatial surface temperature pattern by the model using frequency spectrum with the largest variance (0.05 GHz)2 had better agreement with the experimental temperature pattern when compared to that using a monochromatic frequency of 2.45 GHz. In the transient temperature profile measurement, the average RMSE value of five locations was 7.5 and 13.1 °C for simulations using frequency spectrum and monochromatic frequency of 2.45 GHz, respectively. When compared to using the monochromatic frequency of 2.45 GHz, the frequency spectrum with an assumption of having a Gaussian distribution with mean of 2.45 GHz and variance of (0.05 GHz)2 improved the accuracy of temperature field pattern and transient temperature profile.

Coherent potential approximation for the absorption spectra and the densities of states of cubic Frenkel exciton systems with Gaussian diagonal disorder

Using the coherent potential approximation, we investigate the optical absorption and the density of states of Frenkel exciton systems on simple, body centered, and face centered cubic lattices with nearest-neighbor interactions and a Gaussian distribution of transition frequencies (i.e. Gaussian diagonal disorder). The analysis is based on an elliptic integral approach with a variety of variances. The results for the simple cubic lattice are in good agreement with the finite array calculations of Schreiber and Toyozawa. Our findings suggest that the coherent potential approximation can be useful in interpreting the optical properties of cubic crystals where the optically excited states are Frenkel excitons with the dominant interactions limited to nearest-neighbors.

The Principles of Fractal Geometry and Its Applications for Pulp & Paper Industry

Until Mandelbrot introduced the concept of fractal geometry and fractal dimension in early 1970s, it has been generally considered that the geometry of nature should be too complex and irregular to describe analytically or mathematically. Here fractal dimension indicates a non-integer number such as 0.5, 1.5, or 2.5 instead of only integers used in the traditional Euclidean geometry, i.e., 0 for point, 1 for line, 2 for area, and 3 for volume. Since his pioneering work on fractal geometry, the geometry of nature has been found fractal. Mandelbrot introduced the concept of fractal geometry. For example, fractal geometry has been found in mountains, coastlines, clouds, lightning, earthquakes, turbulence, trees and plants. Even human organs are found to be fractal. This suggests that the fractal geometry should be the law for Nature rather than the exception. Fractal geometry has a hierarchical structure consisting of the elements having the same shape, but the different sizes from the largest to the smallest. Thus, fractal geometry can be characterized by the similarity and hierarchical structure. A process requires driving energy to proceed. Otherwise, the process would stop. A hierarchical structure is considered ideal to generate such driving force. This explains why natural process or phenomena such as lightning, thunderstorm, earth quakes, and turbulence has fractal geometry. It would not be surprising to find that even the human organs such as the brain, the lung, and the circulatory system have fractal geometry. Until now, a normal frequency distribution (or Gaussian frequency distribution) has been commonly used to describe frequencies of an object. However, a log-normal frequency distribution has been most frequently found in natural phenomena and chemical processes such as corrosion and coagulation. It can be mathematically shown that if an object has a log-normal frequency distribution, it has fractal geometry. In other words, these two go hand in hand. Lastly, applying fractal principles is discussed, focusing on pulp and paper industry. The principles should be applicable to characterizing surface roughness, particle size distributions, and formation. They should be also applicable to wet-end chemistry for ideal mixing, felt and fabric design for papermaking process, dewatering, drying, creping, and post-converting such as laminating, embossing, and printing.

A new robust statistical model for interpretation of differences in serial test results from an individual.

Population-based reference intervals have very limited value for the interpretation of laboratory results when analytes display high biological individuality. In these cases, the longitudinal evaluation of individual results using the reference change value (RCV) is the recommended approach. However, the traditional model for RCV calculation requires a Gaussian frequency distribution of data and risks to overestimate the parameter if a correlation between within-subject serial measurements is present. We propose and validate an alternative non-parametric statistical model for interpretation of differences in serial results from an individual, overcoming data distribution and correlation issues. After describing the traditional and newly proposed statistical models, we compared them with each other using a simulation on three specific analytes displaying different concentration distributions in biological setting. We demonstrated that when analyte concentrations followed a Gaussian frequency distribution, as in the case of glycated hemoglobin, both methods can be used equally. On the contrary, if analyte concentrations present a bimodal (e.g., chromogranin A) or skewed (e.g., C-reactive protein) distribution, the information obtained by two statistical methods is different. The proposed statistical approach may be more appropriate in assessing difference between serial measurements when individual data are not normally distributed.

Coherent potential approximation for the absorption spectra of a two-dimensional Frenkel exciton system with Gaussian diagonal disorder

We investigate the optical absorption and the density of states of a Frenkel exciton system on a square lattice with nearest-neighbor interactions and a Gaussian distribution of transition frequencies (i.e. Gaussian diagonal disorder). Results are presented for the absorption and the density of states of direct and indirect edge systems for a range of variances. There is reasonable agreement with the corresponding finite array calculations of Schreiber and Toyozawa. The existence of an Urbach tail is also investigated.

On the lorentzian versus Gaussian character of time-domain spin-echo signals from the brain as sampled by means of gradient-echoes: Implications for quantitative transverse relaxation studies.

To determine whether Lorentzian or Gaussian intra-voxel frequency distributions are better suited for modeling data acquired with gradient-echo sampling of single spin-echoes for the simultaneous characterization of irreversible and reversible relaxation rates. Clinical studies (e.g., of brain iron deposition) using such acquisition schemes have typically assumed Lorentzian distributions. Theoretical expressions of the time-domain spin-echo signal for intra-voxel Lorentzian and Gaussian distributions were used to fit data from a human brain scanned at both 1.5 Tesla (T) and 3T, resulting in maps of irreversible and reversible relaxation rates for each model. The relative merits of the Lorentzian versus Gaussian model were compared by means of quality of fit considerations. Lorentzian fits were equivalent to Gaussian fits primarily in regions of the brain where irreversible relaxation dominated. In the multiple brain regions where reversible relaxation effects become prominent, however, Gaussian fits were clearly superior. The widespread assumption that a Lorentzian distribution is suitable for quantitative transverse relaxation studies of the brain should be reconsidered, particularly at 3T and higher field strengths as reversible relaxation effects become more prominent. Gaussian distributions offer alternate fits of experimental data that should prove quite useful in general. Magn Reson Med 74:51-62, 2015. © 2014 Wiley Periodicals, Inc.