Beta Function Research Articles

Osteopontin: A novel marker of pre-symptomatic sporadic Alzheimer's disease.

We investigate the role of osteopontin (OPN) in participants with Pre-symptomatic Alzheimer's disease (AD), mild cognitive impairment (MCI), and in AD brains. Cerebrospinal fluid (CSF) OPN, AD, and synaptic biomarker levels were measured in 109 cognitively unimpaired (CU), parental-history positive Pre-symptomatic Evaluation of Experimental or Novel Treatments for Alzheimer's Disease (PREVENT-AD) participants, and in 167 CU and 399 participants with MCI from the Alzheimer's Disease Neuroimaging Initiative (ADNI) cohort. OPN levels were examined as a function of amyloid beta (Aβ) and tau positivity. Survival analyses investigated the link between OPN and rate of conversion to AD. In PREVENT-AD, CSF OPN was positively correlated with synaptic biomarkers. In PREVENT-AD and ADNI, OPN was elevated in CSF Aβ42/40(+)/total tau(+) and CSF Aβ42/40(+)/phosphorylated tau181(+) individuals. In ADNI, OPN was increased in Aβ(+) positron emission tomography (PET) and tau(+) PET individuals, and associated with an accelerated rate of conversion to AD. OPN was elevated in autopsy-confirmed AD brains. Strong associations between CSF OPN and key markers of AD pathophysiology suggest a significant role for OPN in tau neurobiology, particularly in the early stages of the disease. In the Pre-symptomatic Evaluation of Experimental or Novel Treatments for Alzheimer's Disease cohort, we discovered that cerebrospinal fluid (CSF) osteopontin (OPN) levels can indicate early synaptic dysfunction, tau deposition, and neuronal loss in cognitively unimpaired elderly with a parental history. CSF OPN is elevated in amyloid beta(+) positron emission tomography (PET) and tau(+) PET individuals. Elevated CSF OPN is associated with an accelerated rate of conversion to Alzheimer's disease (AD). Elevated CSF OPN is associated with an accelerated rate of cognitive decline on the Alzheimer's Disease Assessment Scale-Cognitive subscale 13, Montreal Cognitive Assessment, Mini-Mental State Examination, and Clinical Dementia Rating Scale Sum of Boxes. OPN mRNA and protein levels are significantly upregulated in the frontal cortex of autopsy-confirmed AD brains.

Coupled twiss parameters estimation from turn-by-turn data

Linear optics parameters are often estimated using turn-by-turn (TbT) data. In-plane beta functions, which are the most common measurement objectives, can be estimated from the amplitudes or phases of TbT data, providing an overall characterization of the linear lattice. In addition to estimating uncoupled Twiss parameters, TbT data can also be utilized for characterizing coupled linear motion. We investigate several methods for constructing a full normalization matrix at each beam position monitor (BPM). BPMs provide information on beam centroid transverse coordinates, but direct observation of transverse momenta is not available. To estimate momenta, one can use a pair of BPMs or fit data obtained from several BPMs. By using both coordinates and momenta, it is possible to fit the one-turn matrix (or its power) at each BPM, allowing for the computation of coupled Twiss parameters. Another approach involves the minimization of side peak amplitudes in the spectrum of normalized complex coordinates. Similarly, the normalization matrix can be estimated by fitting linear coupled invariants. In this study, we derive and utilize a special form of the normalization matrix, which remains non-singular in the zero coupling limit. Coupled Twiss parameters can be obtained from the normalization matrix. The paper presents the results of applying and comparing these methods to both modeled and measured VEPP-4M TbT data, demonstrating effective estimation of coupled Twiss parameters.

First-order formalism for β functions in bosonic sigma models from supersymmetry breaking

We consider the renormalization group flow equation for the two-dimensional sigma models with the Kähler target space. The first-order formulation allows us to treat perturbations in these models as current-current deformations. We demonstrate, however, that the conventional first-order formalism misses certain anomalies in the measure, and should be amended. We reconcile beta functions obtained within the conformal perturbation theory for the current-current deformations with traditional “geometric” results obtained in the background field methods, in this way resolving the peculiarities pointed out in O. Gamayun [Peculiarities of beta functions in sigma models, ]. The result is achieved by the supersymmetric completion of the first-order sigma model. Published by the American Physical Society 2024

Theory on Linear L-Fractional Differential Equations and a New Mittag–Leffler-Type Function

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We develop a theory of this fractional derivative as follows. We prove a fundamental theorem of calculus. We deal with linear systems of autonomous homogeneous parts, which correspond to Caputo linear equations of non-autonomous homogeneous parts. The associated L-fractional integral operator, which is closely related to the beta function and the beta probability distribution, and the estimates for its norm in the Banach space of continuous functions play a key role in the development. The explicit solution is built by means of Picard’s iterations from a Mittag–Leffler-type function that mimics the standard exponential function. In the second part of the paper, we address autonomous linear equations of sequential type. We start with sequential order two and then move to arbitrary order by dealing with a power series. The classical theory of linear ordinary differential equations with constant coefficients is generalized, and we establish an analog of the method of undetermined coefficients. The last part of the paper is concerned with sequential linear equations of analytic coefficients and order two.

Renormalization group analysis of the Anderson model on random regular graphs

We present a renormalization group (RG) analysis of the problem of Anderson localization on a random regular graph (RRG) which generalizes the RG of Abrahams, Anderson, Licciardello, and Ramakrishnan to infinite-dimensional graphs. The RG equations necessarily involve two parameters (one being the changing connectivity of subtrees), but we show that the one-parameter scaling hypothesis is recovered for sufficiently large system sizes for both eigenstates and spectrum observables. We also explain the nonmonotonic behavior of dynamical and spectral quantities as a function of the system size for values of disorder close to the transition, by identifying two terms in the beta function of the running fractal dimension of different signs and functional dependence. Our theory provides a simple and coherent explanation for the unusual scaling behavior observed in numerical data of the Anderson model on RRG and of many-body localization.

An observation on the beta functions in quadratic gravity

We study the beta functions for the dimensionless couplings in quadratic curvature gravity, and find that there is a simple argument to restrict the possible form of the beta functions as derived from the counterterms at an arbitrary loop. The relation to the recent different results on beta functions is also commented on.

Physical Running of Couplings in Quadratic Gravity.

We argue that the well-known beta functions of quadratic gravity do not correspond to the physical dependence of scattering amplitudes on external momenta, and derive the correct physical beta functions. Asymptotic freedom turns out to be compatible with the absence of tachyons.

Some New Estimations of Ostrowski-Type Inequalities for Harmonic Fuzzy Number Convexity via Gamma, Beta and Hypergeometric Functions

This paper demonstrates several of Ostrowski-type inequalities for fuzzy number functions and investigates their connections with other inequalities. Specifically, employing the Aumann integral and the Kulisch–Miranker order, as well as the inclusion order on the space of real and compact intervals, we establish various Ostrowski-type inequalities for fuzzy-valued mappings (F·V·Ms). Furthermore, by employing diverse orders, we establish connections with the classical versions of Ostrowski-type inequalities. Additionally, we explore new ideas and results rooted in submodular measures, accompanied by examples and applications to illustrate our findings. Moreover, by using special functions, we have provided some applications of Ostrowski-type inequalities.

Momentum-dependent field redefinitions in asymptotic safety

We discuss general momentum-dependent field redefinitions in the context of quantum-gravitational scattering amplitudes in general, and asymptotic safety in particular. Implementing such redefinitions at the lowest curvature order, we can bring the graviton propagator into tree-level form, avoiding issues of fiducial ghost poles and their associated violations of unitarity. We compute the beta function for Newton’s constant, and find an asymptotically safe fixed point whose critical exponent changes by 0.4% compared to not resolving the momentum-dependent field redefinition. This provides a strong indication that this fixed point does not feature extra degrees of freedom related to ghostly modes, and has a good chance of being related to a unitary theory. Published by the American Physical Society 2024

The Effect of Interferon Beta and Natalizumab on miR-20b Expression in Patients with Relapsing-Remitting Multiple Sclerosis is Potentially Mediated by Modulation of the Jak-STAT Signaling Pathway: A Case-control Study.

The mechanisms of the function of interferon beta (IFN-β) and natalizumab (NTZ) in multiple sclerosis (MS) patients have not yet been fully understood. Over the past decades, many studies have been conducted to evaluate gene expression changes especially regulatory non-coding RNAs such as microRNAs (miRNAs) following therapy in MS patients. To assess the changes in the expression of miR-20b in MS patients treated with IFN-β or NTZ. Sixty patients with relapsing-remitting MS (RRMS) and 30 healthy controls (HCs) were enrolled. The patients were categorized as untreated (N=20), IFN-β-treated (N=20), and NTZtreated (N=20). For the expression analysis, real-time PCR was performed on the whole blood. The bioinformatic tools were applied for signaling pathways enrichment analysis of miR-20b targetome. The relative expression of miR-20b was significantly downregulated in the untreated patients compared with the HCs (-1.726-fold, p<0.001), while IFN-β-treated and NTZ-treated patients showed no statistical difference compared with the HCs (0.733-fold, p=0.99 for IFN-β and 1.025-fold, p=0.18 for NTZ). This indicates the restoration of miR-20b expression to normal level in the treated patients. Additionally, in silico analysis demonstrated that the Jak-STAT signaling pathway is enriched with miR-20b targets (p<0.0001). Our findings suggest that the positive effects of IFN-β and NTZ in the RRMS patients could be potentially mediated by returning miR-20b expression to baseline.

Holographic entanglement renormalisation for fermionic quantum matter

We demonstrate the emergence of a holographic dimension in a system of 2D non-interacting Dirac fermions placed on a torus, by studying the scaling of multipartite entanglement measures under a sequence of renormalisation group (RG) transformations applied in momentum space. Geometric measures defined in this emergent space can be related to the RG beta function of the spectral gap, hence establishing a holographic connection between the spatial geometry of the emergent spatial dimension and the entanglement properties of the boundary quantum theory. We prove, analytically, that changing the boundedness of the holographic space involves a topological transition accompanied by a critical Fermi surface in the boundary theory. We go on to show that this results in the formation of a quantum wormhole geometry that connects the UV and the IR of the emergent dimension. The additional conformal symmetry at the transition also supports a relation between the emergent metric and the stress-energy tensor. In the presence of an Aharonov–Bohm flux, the entanglement gains a geometry-independent piece which is shown to be topological, sensitive to changes in boundary conditions, and related to the Luttinger volume of the system. Upon the insertion of a strong transverse magnetic field, we show that the Luttinger volume is linked to the Chern number of the occupied single-particle Landau levels.

Pointwise and weighted estimates for Bernstein-Kantorovich type operators including beta function

Pointwise and weighted estimates for Bernstein-Kantorovich type operators including beta function

Bounds for novel extended beta and hypergeometric functions and related results

We introduce a new unified extension of the integral form of Euler’s beta function with a MacDonald function in the integrand and establish functional upper bounds for it. We use this definition to extend as well the Gaussian and Kummer’s confluent hypergeometric functions, for which we provide bounding inequalities. Moreover, we use our extension of the beta function to define a new probability distribution, for which we establish raw moments and moment inequalities and, as by-products, Turán inequalities for the initially defined extended beta function.

Fractional calculus of modified special functions involving the generalized M-series in their kernels and illustrative examples

In this paper we apply the Riemann–Liouville, Erdelyi–Kober and Caputo fractional operators to the modified beta, modified Gauss hypergeometric and modified confluent hypergeometric functions in which the generalized M-series are included in their kernels. Furthermore, as examples, we obtain solutions of some fractional differential equations involving the above modified special functions.

On New Representations of the Values of the Dirichlet Beta Function at Even Points

On New Representations of the Values of the Dirichlet Beta Function at Even Points

An Exploration of Some Special Functions and their Applications

Special functions are a class of mathematical functions that fall outside of ele-mentary functions. They have distinct properties and wide-ranging applications in areas such as engineering, applied mathematics, physics, statistics, economics, and finance. In this research paper, I study two popular special functions: the Gamma function Γ(m) and the Beta function β(m, n). I develop some of their important properties and results, prove these properties, and demonstrate some of their practical applications. Understanding the properties and applications of these functions is vital for solving mathematical problems that arise in practical modeling. This study sheds new light on the significance and versatility of Γ(m) and β(m, n), their applications, and mathematical properties.

Field Theory Expansions of String Theory Amplitudes.

Motivated by quantum field theory (QFT) considerations, we present new representations of the Euler-Beta function and tree-level string theory amplitudes using a new two-channel, local, crossing symmetric dispersion relation. Unlike standard series representations, the new ones are analytic everywhere except at the poles, sum over poles in all channels, and include contact interactions, in the spirit of QFT. This enables us to consider mass-level truncation, which preserves all the features of the original amplitudes. By starting with such expansions for generalized Euler-Beta functions and demanding QFT-like features, we single out the open superstring amplitude. We demonstrate the difficulty in deforming away from the string amplitude and show that a class of such deformations can be potentially interesting when there is level truncation. Our considerations also lead to new QFT-inspired, parametric representations of the Zeta function and π, which show fast convergence.

Exploring the Extended Beta-Logarithmic Function: Matrix Arguments and Properties

The beta-logarithmic function substantially generalizes the standard beta function, which is widely recognized for its significance in many applications. This article is devoted to the study of a generalization of the classical beta-logarithmic function in a matrix setting called the extended beta-logarithmic matrix function. The proofs of some essential properties of this extension, such as convergence, partial derivative formulas, functional relations, integral representations, inequalities, and finite and infinite sums, are established. Moreover, an application of the extended beta-logarithmic function in matrix arguments is proposed in probability theory. Further, numerical examples and graphical presentations of the new generalization are obtained.

A new generalized beta function associated with statistical distribution and fractional kinetic equation

Several authors have extensively investigated beta function, hypergeometric function and confluent hypergeometric function, their extensions and generalizations due to their several application in many areas of engineering, probability theory and science. The main purpose of this paper is to present a new generalization of extended beta function, hypergeometric function and confluent hypergeometric function with the help of m-parameter Mittag-Leffler function, as well as examine some important properties like integral representations, differential formula and summation formulas. We also examine generalized Caputo fractional derivative operator with the help of m-parameter Mittag-Leffler function with associated properties using the generalized beta function. We define a new beta distribution involving the new generalized beta function. The mean, variance, coefficient of variance, moment generating function and characteristic function and cumulative distribution are derived. Further, we derive the solution of fractional Kinetic equation involving generalized hypergeometric function.

Software Implementation of the Epps-Pulley Criterion in Matlab Modeling Environment

Purpose. Modeling systems and programming platforms provide ample opportunities for the use of statistical tools in research activities. Since the normal distribution is one of the most common distribution laws, the criterion for checking the sample for normality is in high demand among statistical assessment tools, among which the Epps-Pulley test has the status as one of the most powerful tests to check the deviation of the distribution from the normal one. There are a number of implementations of this test in the R and Python languages. However, this test is not implemented in one of the most popular Matlab modeling software. Thus, the purpose of this study is to develop a software implementation of the Epps-Pulley criterion in the Matlab environment and verify the correctness of the performed calculations.Materials and Methods. We implemented the calculation of Epps-Pulley statistics by two methods – classical, using cycles, and matrix-vector, using linear algebra operations. The classical method requires calculating the intermediate values necessary to obtain the criterion statistics using two independent cycles, the second cycle being a double one, in which one cycle is nested into the other. The matrix-vector method requires fewer code by performing calculations using linear algebra operations on matrices and vectors. We obtained critical statistical values for the sample size from 8 to 1000 observations with two-dimensional linear interpolation of tabular values. We used an approximation by a beta function of the third kind for a sample of over 1000 elements.Results. An assessment of the computational efficiency of the methods showed that the cyclic approach is about three times higher than the matrix-vector approach in terms of consumed time, which is presumably due to the processing of insignificant elements in triangular matrices when performing component-by-component operations. The correctness of the software implementation of the Epps-Pulley criterion was tested on several examples, which confirmed the compliance of the calculated values of the criterion statistics, as well as the critical values of statistics, with known data. We carried out a criterion statistical evaluation based on the empirical values of the error of the first kind. We obtained the error values correspondence to the specified significance levels. We performed comparative estimates of the Epps-Pulley test with the Anders-Darling and Shapiro-Wilk tests in terms of the criterion empirical power and tabulated the evaluation results. We published the software implementation of the Epps-Pulley test on the MATLAB Central Internet resource and for free use.Conclusion. We developed software implementation of the Epps-Pulley criterion as a new research tool that was previously unavailable in the Matlab modeling environment. We used the time spent on calculations to make a reasonable choice of the calculation algorithm for the criterion statistics. We confirmed correctness of the calculation algorithms by a set of selective checks and statistical estimates that showed the compliance with well-known theoretical provisions.