Modified Beta Function Research Articles

Modified special functions: properties, integral transforms and applications to fractional differential equations

In this paper, we defined the modified gamma and beta functions, which involving the generalized M-series at their kernels, and then defined the modified Gauss and confluent hypergeometric functions by using the modified beta function. Furthermore, we presented some of their properties such that, integral representations, summation formulas and derivative formulas. Also, we applied beta, Mellin, Laplace, Sumudu, Elzaki and general integral transforms to newly defined the modified special functions. Then, we obtained solution of fractional differential equations involving the modified special functions, as examples. Finally, we gave the relationships between the modified functions with some of the generalized special functions, which can be found in the literature.

M-Srivastava hypergeometric functions: integral representations and solution of fractional differential equations

In this paper, we introduce a new extensions of Srivastava’s triple hypergeometric functions HA, HB and HC, by using an modified beta function, which given with a generalized M-series in its kernel. We also introduce a new extension of Appell’s hypergeometric function of the first kind by using the same modified beta function. Furthermore, we give some integral representations of the new extensions of Srivastava’s triple hypergeometric functions. Finally, we obtain solutions of fractional differential equations involving new extensions of Srivastava’s triple hypergeometric functions.

ANALISIS UMUR REPRODUKSI PALING AWAL DAN PALING AKHIR SERTA PUNCAK FERTILITAS PADA WANITA DI SETIAP KABUPATEN DAN KOTA DI PROVINSI BANTEN

The birth rate (fertility) is one of the factors that influences the increase in population. Banten Province has 8 districts and cities with different fertility rates. One of the fertility indicators presented in the Long Form SP2020 data is Birth Rate data by Age Group / Age Specific Fertility Rate (ASFR). The aim of this research is to determine the earliest, latest reproductive age and the age of women who have the highest fertility rates in each district and city in Banten Province. The skewness (measure of slope) and kurtosis (measure of sharpness) values of the continuous curve formed will also be studied. ASFR data presented by the Central Statistics Agency (BPS) is discrete data. Fertility rates will be approached using the MBF (Modified Beta Function) continuous fertility model. In this research, information was obtained that the earliest reproductive age in Banten Province was 15.64 years. This happened to women in Lebak Regency. Meanwhile, the latest reproductive age in Banten Province is 49.85 years. This happened to women in Pandeglang Regency. It can also be seen that the highest peak fertility rate for the population of Banten Province occurred in women aged 27.54 years in the Cilegon City area, with a peak fertility value of 0.13055. Apart from being based on calculations, this is in accordance with the curve visualization carried out using Mathematica software. The highest skewness value occurs in the continuous fertility curve for the Lebak Regency area. This means that Lebak Regency has the most left-leaning curve. All continuous fertility curves in districts and cities in Banten Province have a platykurtic distribution (flat curve) because the kurtosis value is smaller than 0.263

M-Lauricella hypergeometric functions: integral representations and solutions of fractional differential equations

In this paper, using the modified beta function involving the generalized M-series in its kernel, we described new extensions for the Lauricella hypergeometric functions $F_{A}^{(r)}$, $F_{B}^{(r)}$, $F_{C}^{(r)}$ and $F_{D}^{(r)}$. Furthermore, we obtained various integral representations for the newly defined extended Lauricella hypergeometric functions. Then, we obtained solution of fractional differential equations involving new extensions of Lauricella hypergeometric functions, as examples.

Evolution of Cosmological Parameters and Fundamental Constants in a Flat Quintessence Cosmology: A Dynamical Alternative to ΛCDM

The primary purpose of this work is the provision of accurate, analytic, evolutionary templates for cosmological parameters and fundamental constants in a dynamical cosmology. A flat quintessence cosmology with a dark energy potential that has the mathematical form of the Higgs potential is the specific cosmology and potential addressed in this work. These templates, based on the physics of the cosmology and potential, are intended to replace those parameterizations currently used to determine the likelihoods of dynamical cosmologies. Acknowledging that, unlike ΛCDM, the evolutions are dependent on both the specific cosmology and the dark energy potential, the templates are referred to as specific cosmology and potential (SCP) templates. The requirements set for the SCP templates are that they must be accurate, analytic functions of an observable, such as the scale factor or redshift. This is achieved through the utilization of a modified beta function formalism that is based on a physically motivated dark energy potential to calculate the beta function. The methodology developed here is designed to be adaptable to other cosmologies and dark energy potentials. The SCP templates are essential tools in determining the relative likelihoods of a range of dynamical cosmologies and potentials. The ultimate purpose is the determination of whether dark energy is dynamical or static in a quantitative manner. It is suggested that the SCP templates calculated in this work can serve as fiducial dynamical templates in the same manner as ΛCDM serves for static dark energy.

Temperature Effects on the Shoot and Root Growth, Development, and Biomass Accumulation of Corn (Zea mays L.)

Temperature is a critical environmental factor regulating plant growth and yield. Corn is a major agronomic crop produced globally over a vast geographic region, and highly variable climatic conditions occur spatially and temporally throughout these regions. Current literature lacks a comprehensive study comparing the effects of temperature on above versus below-ground growth and development and biomass partitioning of corn measured over time. An experiment was conducted to quantify the impact of temperature on corn’s early vegetative growth and development. Cardinal temperatures (Tmin, Topt, and Tmax) were estimated for different aspects of above- and below-ground growth processes. Plants were subjected to five differing day/night temperature treatments of 20/12, 25/17, 30/22, 35/27, and 40/32 °C using sun-lit controlled environment growth chambers for four weeks post-emergence. Corn plant height, leaves, leaf area, root length, surface area, volume, numbers of tips and forks, and plant component part dry weights were measured weekly. Cardinal temperatures were estimated, and the relationships between parameters and temperature within these cardinal limits were estimated using a modified beta function model. Cardinal temperature limits for whole plant dry weight production were 13.5 °C (Tmin), 30.5 °C (Topt), and 38 °C (Tmax). Biomass resources were prioritized to the root system at low temperatures and leaves at high temperatures. Root growth displayed the lowest optimum temperature compared to root development, shoot growth, and shoot development. The estimated cardinal temperatures and functional algorithms produced in this study, which include both above and below-ground aspects of plant growth, could be helpful to update crop models and could be beneficial to estimate corn growth under varying temperature conditions. These results could also be applicable when considering management decisions for maximizing field production and implementing emerging precision agriculture technology.

Testing the suitability of thermal time models for forecasting spring wheat phenological development in western Canada

Predicting crop development stages is fundamental to many aspects of agronomy (e.g., pesticides and fertilizer applications). Temperature is the main factor affecting plant development and its impact on crop development is often measured using thermal-time. We compared different thermal-time models to identify the best model for simulating spring wheat development in western Canada. Models compared include (i) North-Dakota growing-degree-day (NDGDD), (ii) growing-degree-day base-temperature zero (GDD0), (iii) growing-degree-day base-temperature five (GDD5), (iv) beta-function (BF), and (v) modified-beta-function (MBF). We utilised agro-meteorological data collected across western Canada from 2009–2011. Results showed that accumulated heat units/daily growth rates from the different models correlated well with spring wheat phenology with R2 ≥ 0.91 and P < 0.001. However, when the developed models were used to predict time (calendar-days) from planting to anthesis for cultivar AC-Barrie, the BF and MBF models performed poorly. Average predicted times from planting to anthesis by NDGDD, GDD0, GDD5, BF, and MBF models were 63, 63, 62, 65, and 64 d, respectively; while the actual observed time was 60 d. Root-mean-square error (RMSE) for NDGDD was 4 d, 5 d for GDD0 and GDD5, and 6 d for BF and MBF. These findings suggest that simple GDD-based models performed better than more complex BF-based models.

Modeling of scattering properties of mineral aerosols using modified beta function

[1] In the combined experiment from Amsterdam and modeling study, the spheroid is applied to represent the real mineral aerosols, and the single scattering properties of spheroids are computed using the T matrix combined with the improved geometric optical method in the wider size range and aspect ratio range. After that a modeling method for the scattering matrices of the feldspar, quartz, and red clay is proposed. In this approach, the ensemble-average scattering matrix is calculated by using the number density distribution deduced from the fit of the measured normalized projected surface area distribution with the modified beta function and the shape distribution obtained by the least squares fit of all the six measured scattering elements. On the other hand, the models of scattering matrices using the lognormal function to fit the measured normalized number distribution and fit the measured normalized projected surface area distribution are also conducted as comparison. Furthermore, the asymmetry parameter is calculated in order to further verify the reliability of the proposed method. Simulation experiments indicate that the simulated scattering matrices with the proposed method are rather close to the measured scattering matrices for the three mineral aerosols, and this proposed numerical model can provide a simple, reliable, and efficient method to reproduce the scattering properties of the mineral aerosols based on the Amsterdam database.

Early-stage evolution of particle size distribution with Johnson's SB function due to Brownian coagulation

The moment method can be used to determine the time evolution of particle size distribution due to Brownian coagulation based on the general dynamic equation (GDE). But the function form of the initial particle size distribution must be determined beforehand for the moment method. If the assumed function type of the initial particle size distribution has an obvious deviation from the true particle population, the evolution of particle size distribution may be different from the real evolution tendency. Thus, a simple and general method is proposed based on the moment method. In this method, the Johnson's SB function is chosen as a general distribution function to fit the initial distributions including the log normal (L-N), Rosin–Rammler (R-R), normal (N-N) and gamma distribution functions, respectively. Meanwhile, using the modified beta function to fit the L-N, R-R, N-N and gamma functions is also conducted as a comparison in order to present the advantage of the Johnson's SB function as the general distribution function. And then, the time evolution of particle size distributions using the Johnson's SB function as the initial distribution can be obtained by several lower order moment equations of the Johnson's SB function in conjunction with the GDE during the Brownian coagulation process. Simulation experiments indicate that fairly reasonable results of the time evolution of particle size distribution can be obtained with this proposed method in the free molecule regime, transition regime and continuum plus near continuum regime, respectively, at the early time stage of evolution. The Johnson's SB function has the ability of describing the early time evolution of different initial particle size distributions.

On a modified beta function and some applications

Abstract In this work, we study the modified beta function B 1 2 ( x ; a , b ) = ∫ 1 / 2 x ( 1 - v ) a - 1 v b - 1 dv , x ∈ ( 0 , 1 ) that appeared in Marus˘ic and Bajzer’s solution [M. Marus˘ic, Z˘. Bajzer, Generalized two-parameter equation of growth, J. Math. Anal. Appl. 179 (1993) 446–462] of the generalized two-parameter equation of tumor growth y′ = ayμ + byν. In particular, we provide methods for computing this function for special values of the function’s parameters. Some analytic solutions of the generalized two-parameter growth equation are obtained.

Inversion of particle size distribution from spectral extinction data using the bimodal Johnson's SB function

In spectral extinction particle sizing technique, an inversion method in the dependent model is proposed, in which the commonly used bimodal, mix-bimodal, and monomodal particle size distributions are retrieved using the bimodal Johnson's S B function as the assumed type of distribution function. The feasibility and reliability of this inversion method are investigated, and the inversion using the bimodal modified beta function as the assumed type of distribution function is also discussed for comparison. Numerical simulations and experimental results indicate that the commonly used particle size distributions can be retrieved well using the bimodal Johnson's S B function as a versatile distribution function in the dependent model with the spectral extinction data. This inversion method not only overcomes the difficulty caused by the inability to determine the type of specified distribution function accurately beforehand, but also allows the description of different types of distributions by a single model. In spectral extinction technique, the commonly used particle size distributions are retrieved using the bimodal Johnson's S B function as the assumed type of distribution function. This inversion method overcomes the difficulty caused by the inability to determine the type of specified distribution function accurately beforehand. Moreover, it allows the description of different types of distributions by a single model.

Inversion of particle size distribution from spectral extinction data using the modified beta function

In spectral extinction particle sizing technique, an inversion method in the dependent model is proposed, in which the commonly used particle size distribution functions are retrieved using the modified beta function as the assumed type of distribution function. The feasibility and reliability of this inversion method are investigated, and the inversion using the Johnson's S B function as the assumed type of distribution function is also discussed for comparison. Simulation experiments indicate that the commonly used monomodal particle size distribution functions can be retrieved well using the modified beta function as a versatile function in the dependent model with the spectral extinction data. This inversion method not only overcomes the difficulty caused by the inability to determine the type of specified distribution function accurately beforehand, but also allows the description of different types of distributions by a single model.

Inclusion of Yang–Mills fields in string corrected supergravity

We consistently incorporate Yang–Mills matter fields into string corrected (deformed) D=10, N=1 supergravity. We solve the Bianchi identities within the framework of the modified beta function favored constraints to second order in the string slope parameter γ also including the Yang–Mills fields. In the torsion, curvature and H sectors we find that a consistent solution is readily obtained with a Yang–Mills modified supercurrent Aabc. We find a solution in the F sector following our previously developed method.

Implications of a light radion on the RG evolution of Higgs self coupling in the Randall–Sundrum model

In this Letter we determine how the beta function of the Higgs self coupling λ at one loop order is modified by a light stabilized radion in the Randall–Sundrum model. We then use the modified beta function to derive a lower bound on the radion vev 〈φ〉, both for perturbative and non-perturbative values of λ at the ultra violet cut off Λ. The lower bound on 〈φ〉 is obtained by demanding that the renormalized coupling λ(μ) at μ=114 GeV be consistent with the present experimental bound of 114 GeV on the Higgs mass from LEPII searches. We also show that important bounds on 〈φ〉 can be derived from the sign of β(λ) in the infrared regime.

An exact solution to the O(26) sigma model coupled to 2D quantum gravity.

By a mapping to the bosonic string theory, we present an exact solution to the O(26) sigma model coupled to 2-D quantum gravity. In particular, we obtain the exact gravitational dressing to the various matter operators classified by the irreducible representations of O(26). We also derive the exact form of the gravitationally modified beta function for the original coupling constant $e^2$. The relation between our exact solution and the asymptotic solution given in ref[3] is discussed in various aspects.

The modified beta density function as a model for synthetic aperture radar clutter statistics

The authors show that the modified beta distribution function is an adequate model for the underlying distribution function of the random variable used to model synthetic-aperture radar (SAR) image data. The model represents a range of SAR returns from different sea ice types by using a simple change in its parameter. The ability is explained by describing the distribution functions in, width. modified skewness space where the modified beta function covers a region while the other, more common, distribution functions cover only a curve. A procedure for comparing simple distribution functions with analytical functions specifically for digitized SAR data is presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Description of normal, log—normal and Rosin—Rammler particle populations by a modified version of the beta distribution function

Data generated by the normal, log—normal and Rosin—Rammler distribution functions were normalized and fitted with a slightly modified version of the beta distribution function. As long as the frequency function had a zero or practically zero value at the two ends of a finite size range, the fitted curves were, for all practical purposes, indistinguishable from the normal and Rosin—Rammler distributions. The fit of the modified beta function to narrow log—normal distributions was also excellent but it declined significantly as the distribution spread increased. It appears, though, that for real particle populations, having a finite size range, and not necessarily a perfectly smooth size distribution, the modified beta function can replace all three functions, thus providing a way to present and compare the different size distribution patterns in terms of a single mathematical expression.