Beta Function Research Articles

Some generalized inequalities involving extended beta and gamma functions for several variables

AbstractRecently, extensions of the gamma and beta functions have been studied due to their appealing properties and wide range of applications in various scientific fields. This note aims to investigate generalized inequalities associated with the extended beta and gamma functions.

Optimal placement of renewable distributed generators and electric vehicles using multi-population evolution whale optimization algorithm

AbstractThis research takes on a crucial task- exploring the optimal placement of Renewable Distributed Generators such as Solar Photovoltaic, wind turbines and Electric Vehicles into the Radial Distribution System (RDS). This is a strategic move aimed at minimising power loss (PLoss) and improving the voltage profile and stability index. The RDGs are integrated into RDS with and without considering the uncertainty of the different load demands for 24 h. The probability function of Beta and Weibull distribution functions are employed to attain the solar irradiance and wind speed in a particular region. In addition, EVs are also integrated into RDS, employing meta-heuristic algorithms intended to reduce power loss (PLoss) and improve the voltage profile. The study uses an Indian 28-bus test system mimicking a balanced radial distribution network to integrate distributed generators (DGs) and EV charging stations. The simulated results demonstrate that integrating DGs into power systems has offered considerable benefits, including reduced PLoss, heightened efficiency, decreased dependency on centralised generation, and improved environmental sustainability. It is discovered that the Multi-population Evolution Whale Optimization Algorithm (MEWOA) produces better results than other methods in the literature and is valuable and practical for handling these nonlinear optimisation situations.

Regularized-renormalized-resummed loop corrected power spectrum of non-singular bounce with Primordial Black Hole formation

AbstractWe present a complete and consistent exposition of the regularization, renormalization, and resummation procedures in the setup of having a contraction and then non-singular bounce followed by inflation with a sharp transition from slow-roll (SR) to ultra-slow roll (USR) phase for generating primordial black holes (PBHs). We consider following an effective field theory (EFT) approach and study the quantum loop corrections to the power spectrum from each phase. We demonstrate the complete removal of quadratic UV divergences after renormalization and softened logarithmic IR divergences after resummation and illustrate the scheme-independent nature of our renormalization approach. We further show that the addition of a contracting and bouncing phase allows us to successfully generate PBHs of solar-mass order, $$M_\textrm{PBH}\sim \mathcal{O}(M_{\odot })$$ M PBH ∼ O ( M ⊙ ) , by achieving the minimum e-folds during inflation to be $$\Delta N_{\textrm{Total}}\sim \mathcal{O}(60)$$ Δ N Total ∼ O ( 60 ) and in this process successfully evading the strict no-go theorem. We notice that varying the effective sound speed between $$0.88\leqslant c_{s}\leqslant 1$$ 0.88 ⩽ c s ⩽ 1 , allows the peak spectrum amplitude to lie within $$10^{-3}\leqslant A \leqslant 10^{-2}$$ 10 - 3 ⩽ A ⩽ 10 - 2 , indicating that causality and unitarity remain protected in the theory. We analyse PBHs in the extremely small, $$M_{\textrm{PBH}}\sim \mathcal{O}(10^{-33}-10^{-27})M_{\odot }$$ M PBH ∼ O ( 10 - 33 - 10 - 27 ) M ⊙ , and the large, $$M_{\textrm{PBH}}\sim \mathcal{O}(10^{-6}-10^{-1})M_{\odot }$$ M PBH ∼ O ( 10 - 6 - 10 - 1 ) M ⊙ , mass limits and confront the PBH abundance results with the latest microlensing constraints. We also study the cosmological beta functions across all phases and find their interpretation consistent in the context of bouncing and inflationary scenarios while satisfying the pivot scale normalization requirement. Further, we estimate the spectral distortion effects and shed light on controlling PBH overproduction.

Different origins of running in the 2D CP(1) model

We revisit the computation of the beta function in the two-dimensional CP(1) sigma model. We show that in different schemes, different diagrams are responsible for the running, such as momentum-independent tadpoles or even UV-finite bubble diagrams. We also comment on the relation between the beta functions and the energy dependence of scattering amplitudes. Published by the American Physical Society 2024

Predicting Feynman periods in ϕ4-theory

We present efficient data-driven approaches to predict the value of subdivergence-free Feynman integrals (Feynman periods) in ϕ4-theory from properties of the underlying Feynman graphs, based on a statistical examination of almost 2 million graphs. We find that the numbers of cuts and cycles determines the period to better than 2% relative accuracy. Hepp bound and Martin invariant allow for even more accurate predictions. In most cases, the period is a multi-linear function of the properties in question. Furthermore, we investigate the usefulness of machine-learning algorithms to predict the period. When sufficiently many properties of the graph are used, the period can be predicted with better than 0.05% relative accuracy.We use one of the constructed prediction models for weighted Monte-Carlo sampling of Feynman graphs, and compute the primitive contribution to the beta function of ϕ4-theory at L ∈ {13, … , 17} loops. Our results confirm the previously known numerical estimates of the primitive beta function and improve their accuracy. Compared to uniform random sampling of graphs, our new algorithm is 1000-times faster to reach a desired accuracy, or reaches 32-fold higher accuracy in fixed runtime.The dataset of all periods computed for this work, combined with a previous dataset, is made publicly available. Besides the physical application, it could serve as a benchmark for graph-based machine learning algorithms.

Distinct functions for beta and alpha bursts in gating of human working memory

Multiple neural mechanisms underlying gating to working memory have been proposed with divergent results obtained in human and animal studies. Previous findings from non-human primates suggest prefrontal beta frequency bursts as a correlate of transient inhibition during selective encoding. Human studies instead suggest a similar role for sensory alpha power fluctuations. To cast light on these discrepancies we employed a sequential working memory task with distractors for human participants. In particular, we examined their whole-brain electrophysiological activity in both alpha and beta bands with the same single-trial burst analysis earlier performed on non-human primates. Our results reconcile earlier findings by demonstrating that both alpha and beta bursts in humans correlate with the filtering and control of memory items, but with region and task-specific differences between the two rhythms. Occipital beta burst patterns were selectively modulated during the transition from sensory processing to memory retention whereas prefrontal and parietal beta bursts tracked sequence order and were proactively upregulated prior to upcoming target encoding. Occipital alpha bursts instead increased during the actual presentation of unwanted sensory stimuli. Source reconstruction additionally suggested the involvement of striatal and thalamic alpha and beta. Thus, specific whole-brain burst patterns correlate with different aspects of working memory control.

Criterion of perturbativity with the mass-dependent beta function in extended Higgs models

In order to realize an electroweak first-order phase transition, a category of extended Higgs models with relatively large self-coupling constants is often considered. In such a scenario, the running coupling constants can blow up at an energy scale much below the Planck scale. To clarify the allowed parameter space of the model, it is important to evaluate the scale where perturbative calculation breaks down. In the renormalization group equation analysis using the mass-independent renormalization scheme, we need a matching condition to connect the low energy theory and the high energy theory. Consequently, the analysis depends on the details of the matching condition. On the other hand, an analysis with the mass-dependent beta function can be performed in a simpler way because the threshold effect is automatically included in the beta function. In this paper, using the mass-dependent beta function, we discuss the application limit of perturbative calculations at the one-loop level. First, we explain the essence of our method and compare our results with those based on the mass-independent beta function in a toy model with two complex scalar fields. We then apply our method to a more realistic model, i.e., the inert doublet model. We find that, in our method, in which the threshold effect is automatically included, the energy scale where the perturbative calculation breaks down can be higher than the one using the mass-independent beta function, especially when the blowup scale is relatively low. Published by the American Physical Society 2024

Enhancement Method Based on Multi-Strategy Improved Pelican Optimization Algorithm and Application to Low-Illumination Forest Canopy Images

Enhancement is a crucial step in the field of image processing, as it significantly improves image analysis and understanding. One of the most commonly used methods for image contrast enhancement is the incomplete beta function (IBF). However, the key challenge lies in determining the optimal parameters for the IBF. This paper introduces a multi-strategy improved pelican optimization algorithm (MIPOA) to address the low-illumination color image enhancement problem. The MIPOA algorithm utilizes a nonlinear decreasing coefficient to boost the exploration ability and convergence speed, whereas the Hardy–Weinberg principle compensates for the unsound exploitation mechanism. Additionally, the diversity variation operation improves the ability of the algorithm to escape local optimal solutions. The performance of the proposed MIPOA algorithm was evaluated using a benchmark function and was found to outperform five variant algorithms in extensive comparisons. To further harness the potential of the MIPOA algorithm, the authors propose a low-light forest canopy image enhancement method based on the MIPOA algorithm. The MIPOA algorithm searches for the optimal parameters of the IBF, leading to fast contrast enhancement of the image. The segmented gamma correction function is designed to enhance the brightness of the low-light forest canopy images. In determining the optimal parameters of IBF, the MIPOA algorithm demonstrates superior performance compared to other intelligent algorithms in the feature similarity index (FSIM), entropy, and contrast improvement index (CII) of 75%, 58.33%, and 75%, respectively. The proposed MIPOA-based enhancement method achieves a moderate pixel mean and surpasses the conventional enhancement method with an average gradient of 91.67%. The experimental results indicate that the MIPOA effectively addresses the limitations of low optimization accuracy in IBF parameters, and the enhancement method based on the MIPOA provides a more efficacious approach for enhancing low-light forest canopy images.

Glucotoxicity suppresses function of pancreatic beta and duct cells via miR-335-targeted Runx2 and insulin-mediated mechanism.

Pancreatic cell dynamics have important contributions to the development of type 2 diabetes and related diseases such as nonalcoholic fatty pancreas disease. The aim of this study was to investigate the effects of prolonged excessive glucose exposure on the functions of pancreatic beta cells and duct cells in single and co-culture conditions. In this study, we focused on the effects of glucotoxicity on insulin secretion which is the main function of beta cells and on progenitor functions of duct cells. Rat primary INS1 beta cells and ARIP duct cells were exposed to glucose (25mM) for 72h under single or indirect co-culture conditions. Glucotoxicity stimuli increased insulin secretion and decreased insulin expression in single beta cells while stimulating beta-cell differentiation and adipogenesis in single duct cells. On the other hand, glucotoxicity caused functional loss and increased proliferation and apoptosis in beta cells while increasing proliferation but suppressed beta-cell differentiation and adipogenesis in duct cells under co-culture conditions. The expression level of miR-335, a microRNA known to be upregulated by leptin and target Runx2, was measured. As a result, unlike single-cell culture, glucotoxicity upregulated miR-335, downregulated Runx2, and decreased insulin signaling in beta cells while downregulating miR-335 and upregulating Runx2, and decreased insulin signaling in duct cells under co-culture conditions. When the results of single and co-culture experiments are compared, insulin and miR-335 may be seen as important mediators for setting up the relation between beta and duct cells. Our findings are important for preventing the development of type 2 diabetes and nonalcoholic fatty pancreas disease, even developing new diagnosis and treatment strategies.

Analysis of the beta-logarithmic function and its properties

Abstract In this article, we aim to present a new extension of the beta-logarithmic function with the help of an extended beta function and logarithmic mean. Then we introduce its statistical properties and investigate some basic important formulas of extended beta-logarithmic functions, such as integral representation, summation formula and Mellin transform. Further, we consider a new type of hypergeometric and confluent hypergeometric functions using this extended beta-logarithmic function and discuss their various result.

An accurate wavelets‐collocation technique for neutral delay distributed‐order fractional optimal control problems

AbstractIn this study, a new class of optimal control problems called neutral delay distributed‐order fractional optimal control problems is introduced, this problem is solved based on an efficient computational scheme. To solve the problem, we derive an exact formula for the Riemann–Liouville fractional integral operator of Genocchi wavelets based on beta functions for the first time. By taking into account this operator, collocation method, and Gauss–Legendre integration formula, the solution of fractional optimal control problems (FOCPs) under consideration is converted to a nonlinear programming one to which existing well‐developed algorithms may be applied. The mentioned scheme is applied to both FOCPs with or without delay. Error analysis associated with the proposed idea is also investigated under several mild conditions. The effectiveness of the strategy is showed by several illustrative examples, furthermore, a comparison with the previous methods highlights the preference of this scheme.

Inequalities for Basic Special Functions Using Hölder Inequality

Let p,q≥1 be two real numbers such that 1p+1q=1, and let a,b∈R be two parameters defined on the domain of a function, for example, f. Based on the well known Hölder inequality, we propose a generic inequality of the form |f(ap+bq)|≤|f(a)|1p|f(b)|1q, and show that many basic special functions, such as the gamma and polygamma functions, Riemann zeta function, beta function and Gauss and confluent hypergeometric functions, satisfy this type of inequality. In this sense, we also present some particular inequalities for the Gauss and confluent hypergeometric functions to confirm the main obtained inequalities.

Special functions with general kernel: Properties and applications to fractional partial differential equations

Abstract In this paper, we reconstruct the gamma and beta functions using a general kernel function in their integral representations. We also reconstruct the Gauss and confluent hypergeometric functions using the beta function with general kernel in their series representations. The general kernel function we use here can be chosen as any special function such as exponential function, Mittag-Leffler function, Wright function, Fox-Wright function, Kummer function or M-series. Using different choices of this general kernel function, various of the generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions in the literature can be obtained. In this paper, we first obtain the integral representations, functional relations, summation, derivative and transformation formulas and double Laplace transforms of the special functions we construct. Furthermore, we compute the solutions of some fractional partial differential equations involving special functions with general kernel via the double Laplace transform and graph some of the solutions for specific values. Finally, we obtain the incomplete beta function with general kernel by defining the beta distribution with general kernel.

On a new extension of extended (p,q) Beta function

The main objective of this paper is to introduce a further extension of extended (p, q)-Beta function by considering product of two Mittag-Leffler functions in the kernel. We investigate various properties of this newly defined Beta function such as integral representations, summation formulas and Mellin transform. Further, some known and new relations for various forms of extended Beta functions are obtained as special cases of the main results.

Detachment of leading-edge vortex enhances wake capture force production

During stroke reversals, insect wings interact with their own wake flow from the preceding half-stroke, resulting in an unsteady aerodynamic mechanism known as ‘wing–wake interaction’ or ‘wake capture’. To better elucidate this mechanism, we numerically solved the incompressible Navier–Stokes equations at Reynolds numbers $10^2$ and $10^3$ . Simulations were conducted for wing planforms defined using the beta function distribution with varying aspect ratios ( $AR=2\unicode{x2013}6$ ) and radial centroid locations ( $\hat {r}_1=0.4\unicode{x2013}0.6$ ), whilst employing representative normal hovering kinematics. The wake development from the considered flapping wing planforms was investigated, and the wake capture contribution to aerodynamic force production was quantified by comparing the force generation between the fifth and first stroke cycles at multiple sections along the wingspan. Our results revealed that on the inboard wing region experiencing an attached leading-edge vortex (LEV) structure, wing–wake interaction is dominated by an unsteady downwash effect, resulting in a reduction in local force production. However, in regions closer to the wingtip experiencing detachment of the LEV, wing–wake interaction is dominated by an unsteady upwash effect, leading to an increase in local force production. Consequently, the global wake capture force production is controlled by the extent of LEV detachment, which primarily increases with the increase of wing aspect ratio. This suggests that for normal hovering flapping wings, the typical loss in translational force production due to wingtip stall is partially mitigated by wake capture effects.

Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions

Abstract Three classes of trigonometric integrals involving an integer parameter are evaluated by the contour integration and the residue theorem. The resulting formulae are expressed in terms of Riemann zeta function and Dirichlet beta function. Several remarkable integral identities are presented.

A numerical algorithm for the computation of the noncentral beta distribution function

AbstractThe noncentral beta distribution function is a generalization of the central beta distribution (the regularized incomplete beta function) that includes a noncentrality parameter. This paper describes an algorithm and provides a Matlab implementation for calculating the noncentral beta distribution function. Through a series of numerical tests, we demonstrate that the algorithm is accurate and efficient across a wide range of parameters.

Gravitational corrections to the two-loop beta function in a non-Abelian gauge theory

This paper investigates the coupling of massive fermions to gravity within the context of a non-Abelian gauge theory, utilizing the effective field theory framework for quantum gravity. Specifically, we calculate the two-loop beta function of the gauge coupling constant in a non-Abelian gauge theory, employing the one-graviton exchange approximation. Our findings reveal that gravitational corrections may lead to a nontrivial UV fixed point in the beta function of the gauge coupling constant, contingent upon the specific gauge group and the quantity of fermions involved. Published by the American Physical Society 2024

On Extended Beta Function and Related Inequalities

In this article, we consider a unified generalized version of extended Euler’s Beta function’s integral form a involving Macdonald function in the kernel. Moreover, we establish functional upper and lower bounds for this extended Beta function. Here, we consider the most general case of the four-parameter Macdonald function Kν+12pt−λ+q(1−t)−μ when λ≠μ in the argument of the kernel. We prove related bounding inequalities, simultaneously complementing the recent results by Parmar and Pogány in which the extended Beta function case λ=μ is resolved. The main mathematical tools are integral representations and fixed-point iterations that are used for obtaining the stationary points of the argument of the Macdonald kernel function Kν+12.

The Exact Density of the Eigenvalues of the Wishart and Matrix-Variate Gamma and Beta Random Variables

The determination of the distributions of the eigenvalues associated with matrix-variate gamma and beta random variables of either type proves to be a challenging problem. Several of the approaches utilized so far yield unwieldy representations that, for instance, are expressed in terms of multiple integrals, functions of skew symmetric matrices, ratios of determinants, solutions of differential equations, zonal polynomials, and products of incomplete gamma or beta functions. In the present paper, representations of the density functions of the smallest, largest and jth largest eigenvalues of matrix-variate gamma and each type of beta random variables are explicitly provided as finite sums when certain parameters are integers and, as explicit series, in the general situations. In each instance, both the real and complex cases are considered. The derivations initially involve an orthonormal or unitary transformation whereby the wedge products of the differential elements of the eigenvalues can be worked out from those of the original matrix-variate random variables. Some of these results also address the distribution of the eigenvalues of a central Wishart matrix as well as eigenvalue problems arising in connection with the analysis of variance procedure and certain tests of hypotheses in multivariate analysis. Additionally, three numerical examples are provided for illustration purposes.