Beta Function Research Articles

Alternating series in terms of Riemann zeta function and Dirichlet beta function

<abstract><p>By making use of the multisection series method, four classes of alternating infinite series are evaluated, in closed form, by the Riemann zeta function and the Dirichlet beta function.</p></abstract>

On Refined Netrusophic Fractional Calculus

Depending on the geometric isometry (AH-Isometry), it has been proven that every Neutrosophic real function is equivalent to three real functions. Then, the foundation of the Refined Netrusophic calculus was established, where new definitions of Refined Netrusophic integration and Refined Netrusophic differentiation were introduced, along with some illustrative examples. Following that, definitions for the Refined Netrusophic gamma function and Refined Netrusophic beta function were presented to pave the way towards achieving the desired goal, which is Refined Netrusophic Fractional calculus.

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

Extension of Beta Function and its Agricultural Applications

The first kind of Euler integral, known as the β-function, holds significant importance in various fields such as mathematics, engineering, statistics, and the chemical and physical sciences. Numerous extensions of the classical β-function can be found in the literature. In the realm of mathematics, the β-function plays a crucial role in addressing problems related to probability distribution, quantum mechanics, and fluid mechanics. This study aims to introduce new extensions of the β-function with a specific focus on their applications in agriculture. Additionally, the research delves into the applications of the β-function and explores various types of β-functions in conjunction with other special functions. The study further investigates additional extensions, properties, and mutual relationships, shedding light on recurrence relations and generating functions associated with the extended β-function.

ON SOME EXTENDED SPECIAL FUNCTIONS WITH THE BI-FOX-WRIGHT FUNCTION KERNEL

In the present article, our main goal is to study the new extended beta, Gauss and confluent hypergeometric functions defined by using the two Fox-Wright functions as their kernel. Some applications of the new extended beta function to distribution theory are also discussed.

New fractional integral inequalities via Euler's beta function

Abstract In this article, we present new fractional integral inequalities via Euler’s beta function in terms of s s -convex mappings. We develop some new generalizations of fractional trapezoid- and midpoint-type inequalities using the class of differentiable s s -convexity. The results obtained in this study extended other related results reported in the literature.

On Some Formulas for the Lauricella Function

Lauricella, G. in 1893 defined four multidimensional hypergeometric functions FA, FB, FC and FD. These functions depended on three variables but were later generalized to many variables. Lauricella’s functions are infinite sums of products of variables and corresponding parameters, each of them has its own parameters. In the present work for Lauricella’s function FA(n), the limit formulas are established, some expansion formulas are obtained that are used to write recurrence relations, and new integral representations and a number of differentiation formulas are obtained that are used to obtain the finite and infinite sums. In the presentation and proof of the obtained formulas, already known expansions and integral representations of the considered FA(n) function, definitions of gamma and beta functions, and the Gaussian hypergeometric function of one variable are used.

Estrogen receptor beta suppresses the androgen receptor oncogenic effects in triple-negative breast cancer.

Triple-negative breast cancer (TNBC) is an aggressive type of breast cancer associated with poor prognosis and limited treatment options. The androgen receptor (AR) has emerged as a potential therapeutic target for luminal androgen receptor (LAR) TNBC. However, multiple studies have claimed that anti-androgen therapy for AR-positive TNBC only has limited clinical benefits. This study aimed to investigate the role of AR in TNBC and its detailed mechanism. Immunohistochemistry and TNBC tissue sections were applied to investigate AR and nectin cell adhesion molecule 4 (NECTIN4) expression in TNBC tissues. Then, in vitro and in vivo assays were used to explore the function of AR and estrogen receptor beta (ERβ) in TNBC. Chromatin immunoprecipitation sequencing (ChIP-seq), co-immunoprecipitation (co-IP), molecular docking method, and luciferase reporter assay were performed to identify key molecules that affect the function of AR. Based on the TNBC tissue array analysis, we revealed that ERβ and AR were positive in 21.92% (32/146) and 24.66% (36/146) of 146 TNBC samples, respectively, and about 13.70% (20/146) of TNBC patients were ERβ positive and AR positive. We further demonstrated the pro-tumoral effects of AR on TNBC cells, however, the oncogenic biology was significantly suppressed when ERβ transfection in LAR TNBC cell lines but not in AR-negative TNBC. Mechanistically, we identified that NECTIN4 promoter -42bp to -28bp was an AR response element, and that ERβ interacted with AR thus impeding the AR-mediated NECTIN4 transcription which promoted epithelial-mesenchymal transition in tumor progression. This study suggests that ERβ functions as a suppressor mediating the effect of AR in TNBC prognosis and cell proliferation. Therefore, our current research facilitates a better understanding of the role and mechanisms of AR in TNBC carcinogenesis.

Non-perturbative SQED beta function using the functional renormalization group approach and the NSVZ exact beta function

Abstract The renormalization group equations of massive $\mathcal {N}=1$ supersymmetric quantum electrodynamics are studied using the functional renormalization group approach. A non-perturbative form of the beta function has been computed via a derivative expansion of the effective action. In the local potential approximation, the functional form of the non-perturbative beta function is closely related to the form of the Novikov–Shifman–Vainshtein–Zakharov (NSVZ) exact beta function; this relationship is exact if an effective fine-structure constant is defined. The non-massive limit of the same is also analyzed. Furthermore, the calculation of the beta function has been improved by incorporating the influence of momentum modes on the propagation of the superfields in the non-perturbative running of the electric charge, applying a second-order truncation for the derivative expansion, which we use to find the momentum contributions to the β function. Again, we find the NSVZ relation for an effective fine-structure constant. It is with sadness that I say goodbye to my professor, Iván Schmidt Andrade, who left us during the course of this work. His passion for research and his special vision of physics work will remain with us. Thank you for everything.

A flexible-predefined-time convergence and noise-suppression ZNN for solving time-variant Sylvester equation and its application to robotic arm

Although the zeroing neural network (ZNN) has strong competitiveness in solving the time-varying Sylvester equation (TVSE), its convergence and robustness as two key indicators still need to be further improved. To address these problems, we propose a flexible-predefined-time convergence and noise-suppression ZNN (FPCNS-ZNN). Different from most existing ZNN variants, the proposed FPCNS-ZNN provides the following three new improvements: (1) In FPCNS-ZNN, a new specially designed time-variant-gain (SD-TVG) is proposed to better improve the convergence speed at each stage of iterative computation. (2) In FPCNS-ZNN, a new flexible nonlinear activation function (FN-AF) is proposed to flexibly control the convergence speed and predefine convergence time with certain robustness. (3) SD-TVG and FN-AF are dexterously combined to construct the new FPCNS-ZNN, which can obtain faster convergence speed and stronger robustness. Through detailed theoretical analysis and mathematical derivation, the proposed FPCNS-ZNN model has been confirmed to have promising stability, flexible-predefined-time, and strong robustness. It should be noted that the predefined-time upper bound of the proposed FPCNS-ZNN is accurately calculated by dexterously using the Beta function. Computer simulations and comparative experiments indicate that the proposed FPCNS-ZNN has faster convergence speed and stronger robustness compared with three state-of-the-art ZNN variants. Finally, the proposed FPCNS-ZNN is used to solve the motion planning problem of the robotic arm, and the experiment results illustrate the utility of the model.

Notes on a surface defect in the O(N) model

We study a surface defect in the free and critical O(N) vector models, defined by adding a quadratic perturbation localized on a two-dimensional subspace of the d-dimensional CFT. We compute the beta function for the corresponding defect renormalization group (RG) flow, and provide evidence that at long distances the system flows to a nontrivial defect conformal field theory (DCFT). We use epsilon and large N expansions to compute several physical quantities in the DCFT, finding agreement across different expansion methods. We also compute the defect free energy, and check consistency with the so-called b-theorem for RG flows on surface defects.

On some new inequalities and fractional kinetic equations associated with extended gauss hypergeometric and confluent hypergeometric function

Fractional kinetic equations are of immense importance in describing and solving numerous intriguing problems of physics and astrophysics. Inequalities are important topics in special functions. In this paper, we studied the monotonicity of the extended Gauss and confluent hypergeometric function that are derived by using the inequalities on generalized beta function involving Appell series and Lauricella function. We also establish generalized fractional kinetic equation involving extended hypergeometric and confluent hypergeometric functions. The solutions of generalized fractional kinetic equation is derived and studied as an application of extended hypergeometric and confluent hypergeometric function using the General integral transform. The results obtained here are general and can be used to derive many new solutions of fractional kinetic equations involving various types of special functions.

Statistics of Feynman amplitudes in ϕ4-theory

The amplitude of subdivergence-free logarithmically divergent Feynman graphs in ϕ4-theory in 4 spacetime dimensions is given by a single number, the Feynman period. We numerically compute the periods of 1.3 million completed graphs, this represents more than 33 million graphs contributing to the beta function. Our data set includes all primitive graphs up to 13 loops, and non-complete samples up to 18 loops, with an accuracy of ca. 4 significant digits.We implement all known symmetries of the period in a new computer program and count them up to 14 loops. Combining the symmetries, we discover relations between periods that had been overlooked earlier. All expected symmetries are respected by the numerical values of periods.We examine the distribution of the numerically computed Feynman periods. We confirm the leading asymptotic growth of the average period with growing loop order, up to a factor of 2. At high loop order, a limiting distribution is reached for the amplitudes near the mean. A small class of graphs, most notably the zigzags, grows significantly faster than the mean and causes the limiting distribution to have divergent moments even when normalized to unit mean. We examine the relation between the period and various properties of the underlying graphs. We confirm the strong correlation with the Hepp bound, the Martin invariant, and the number of 6-edge cuts. We find that, on average, the amplitude of planar graphs is significantly larger than that of non-planar graphs, irrespective of O(N) symmetry.We estimate the primitive contribution to the 18-loop beta function of the O(N)-symmetric theory. We find that primitive graphs constitute a large part of the beta function in MS for L → ∞ loops. The relative contribution of planar graphs increases with growing N and decreases with growing loop order L.

Discrete phase space-continuous time relativistic Klein–Gordon and Dirac equations, and a new non-singular Yukawa potential

This paper deals with the second quantization of interacting relativistic Fermionic and Bosonic fields in the arena of discrete phase space and continuous time. The mathematical formulation involves partial difference equations. The corresponding Feynman diagrams and a new S^{#}-matrix theory is developed. In the special case of proton-proton Møller scattering via an exchange of a neutral meson, the explicit second order element langle f | S^{#}_{(2)} |i rangle is deduced. In the approximation of very low external three-momenta, a new Yukawa potential is explicitly derived from langle f | S^{#}_{(2)} |i rangle . Moreover, it is rigorously proved that this new Yukawa potential is divergence-free. The mass parameter of the exchanged meson may be set to zero to obtain a type of scalar Boson exchange between hypothetical Fermions. This provides a limiting case of a new Coulomb type potential directly from the new singularity free Yukawa potential. A divergence-free Coulomb potential between two Fermions at two discrete points is shown to be proportional to the Euler beta function. Within this relativistic discrete phase space continuous time, a single quanta is shown to occupy the hyper-tori S^{1}_{n^1} times S^{1}_{n^3} times S^{1}_{n^3} where S^{1}_{n} is a circle of radius sqrt{2n+1}.

Certain results concerning (p, q)-parameterized beta logarithmic function and their properties

The primary object of this article is to introduce (p, q)-beta logarithmic function with extended beta function by using the logarithmic mean. We evaluate different properties and representations of beta logarithmic function. Further, it is evaluated logarithmic distribution, hypergeometric and confluent hypergeometric functions via logarithmic mean are evaluated and their essential properties are studied. Numerous formulas of (p, q)-beta logarithmic functions such as integral formula, derivative formula, transformation formula and generating function are analyzed.

On a conjectural series of Sun for the mathematical constant β(4)

Let β ( z ) be the Dirichlet beta function. Series for β ( 4 ) are very rare in the historical development of mathematics. With the help of the operator method and a transformation formula for hypergeometric series, we prove a surprising conjectural series of Z.-W Sun for the mathematical constant β ( 4 ) . Furthermore, we find five new series for the same constant in this paper.

Eigenvalues of truncated unitary matrices: disk counting statistics

Let T be an n×n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\ imes n$$\\end{document} truncation of an (n+α)×(n+α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n+\\alpha )\ imes (n+\\alpha )$$\\end{document} Haar distributed unitary matrix. We consider the disk counting statistics of the eigenvalues of T. We prove that as n→+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\rightarrow + \\infty $$\\end{document} with α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} fixed, the associated moment generating function enjoys asymptotics of the form exp(C1n+C2+o(1)),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\exp \\big ( C_{1} n + C_{2} + o(1) \\big ), \\end{aligned}$$\\end{document}where the constants C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1}$$\\end{document} and C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{2}$$\\end{document} are given in terms of the incomplete Gamma function. Our proof uses the uniform asymptotics of the incomplete Beta function.

Fractional-order hybrid functions combining simulated annealing algorithm for solving fractional pantograph differential equations

A new and novel numerical method has been developed based on the fractional-order hybrid functions combining simulated annealing (SA) algorithm for the solutions of fractional pantograph delay differential equations. First of all, a fractional-order hybrid of block-pulse functions and Chebyshev polynomials (FOHBPCs) is defined. With the aid of regularized beta function, the exact formulas of FOHBPCs are derived under the definition of Riemann–Liouville fractional integral. And then, by the properties of FOHBPCs and the exact formulas together with the collocation method, the problem under consideration is simplified into algebraic equations, which are solved by Gaussian elimination method and Picard iteration method for linear and nonlinear cases, respectively. The error analysis of the proposed method is investigated. Due to the importance of the parameter α in the fractional-order hybrid functions method, SA algorithm is considered to find the optimal parameter α. Finally, the effectiveness and applicability of the suggested method are verified through some numerical examples.

Notes on Various Transforms Identified by Some Special Functions with Complex (or Real) Parameters and Some of Related Implications

The fundamental aim of this special research is first to introduce certain essential information in regards to some special functions, which are the Gamma function and the Beta function and play a big role in both (applied) mathematics and most engineering sciences, and then to present both a number of their familiar properties and several relationships between them. Afterward, various possible-undeniable effects of those special functions in the transformation theory, their special implications, and suggestions for the relevant researchers will be also considered as special information.

Peculiarities of beta functions in sigma models

In this paper we consider perturbation theory in generic two-dimensional sigma models in the so-called first-order formalism, using the coordinate regularization approach. Our goal is to analyze the first-order formalism in application to β functions and compare its results with the standard geometric calculations. Already in the second loop, we observe deviations from the geometric results that cannot be explained by the regularization/renormalization scheme choices. Moreover, in certain cases the first-order calculations produce results that are not symmetric under the classical diffeomorphisms of the target space. Although we could not present the full solution to this remarkable phenomenon, we found some indirect arguments indicating that an anomaly similar to that established in supersymmetric Yang-Mills theory might manifest itself starting from the second loop. We discuss why the difference between two answers might be an infrared effect, similar to that in β functions in supersymmetric Yang-Mills theories.In addition to the generic Kähler target spaces we discuss in detail the so-called Lie-algebraic sigma models. In particular, this is the case when the perturbed field Gij¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {G}^{i\\overline{j}} $$\\end{document} is a product of the holomorphic and antiholomorphic currents satisfying two-dimensional current algebra.