Analytic Functions Research Articles

A Composition Formula for the Modified Analytic Function Space Fourier–Feynman Transform

Composition formula is one of the most important research topics in functional analysis theory. Various relationships can be obtained using the composition formula. Since the generalized Brownian motion process used in this paper has a non-zero mean function, there are many restrictions on obtaining a composition formula for the modified analytic function space Fourier–Feynman transform. This paper contains an idea of how the composition Formula (9) below is established for the modified analytic function space Fourier–Feynman transform on function space. Using this idea, we are able to solve a problem that had never been solved before.

Analytical potential energy functions for CO in its ground and excited electronic states.

Accurate functions to analytically represent the potential energy interactions of CO diatomic system in , , and electronic states are proposed. The new functions depend upon only four parameters directly obtained from experimental data, without any fitting procedure. These functions have been developed from the modified generalized potential proposed by Araújo and Ballester. The function for the electronic state represents a significant improvement to the previously proposed model. To quantify the accuracy of the potential energy functions, the Lippincont test is used. The novel potential was also compared with the classical Morse potential and with the recently proposed Improved Generalized Pöschl-Teller potential. Furthermore, the main spectroscopic constants and vibrational energy levels are calculated and compared for all potentials. The present results agree excellently with the experiment Rydberg-Klein-Rees (RKR) potentials. The rovibrational energy levels of the proposed diatomic potentials were asserted by solving radial the Schrödinger equation of the nuclear motion with the aid of the LEVEL program.

On the two nonzero boundary problems of the AB system with multiple poles

The Riemann–Hilbert method is used to study AB systems with two nonzero boundary conditions. In order to reconstruct the potentials, spectral analysis of Lax pair and the corresponding Riemann–Hilbert problem are discussed. For these two nonzero boundary problems, we consider the cases where the sectional analytic functions have double and triple poles, respectively. The derived scattering data has multiple zeros or pairs of multiple zeros, from which the relationship between the eigenfunctions at the corresponding poles can be obtained. By solving algebraic equations, we can derive the formulas of N-order solutions in different cases. As an application, we obtain different types of solitons and breathers in the reflectionless case and the dynamic analysis of these solutions is revealed.

Symbolic-numeric algorithm for parameter estimation in discrete-time models with exp

Dynamic models describe phenomena across scientific disciplines, yet to make these models useful in application the unknown parameter values of the models must be determined. Discrete-time dynamic models are widely used to model biological processes, but it is often difficult to determine these parameters. In this paper, we propose a symbolic-numeric approach for parameter estimation in discrete-time models that involve univariate non-algebraic (locally) analytic functions such as exp. We illustrate the performance (precision) of our approach by applying our approach to two archetypal discrete-time models in biology (the flour beetle ‘LPA’ model and discrete Lotka-Volterra competition model). Unlike optimization-based methods, our algorithm guarantees to find all solutions of the parameter values up to a specified precision given time-series data for the measured variables provided that there are finitely many parameter values that fit the data and that the used polynomial system solver can find all roots of the associated polynomial system with interval coefficients.

PRIME: A CyberGIS Platform for Resilience Inference Measurement and Enhancement

In an era of increased climatic disasters, there is an urgent need to develop reliable frameworks and tools for evaluating and improving community resilience to climatic hazards at multiple geographical and temporal scales. Defining and quantifying resilience in the social domain is relatively subjective due to the intricate interplay of socioeconomic factors with disaster resilience. To broaden upon it, the choice of indicators and their subsequent ranking for the aggregation into an index is subjective in nature. This aggregation is not empirically validated and is prone to omit the nuances of localized resilience changes and causal factors affecting it, while leading to oversimplified conclusions. Meanwhile, there is a lack of scientifically and computationally rigorous, user-friendly tools that can support customized resilience assessment with consideration of local conditions. This study addresses these gaps through the power of CyberGIS with three objectives: 1) To develop an empirically validated disaster resilience model - Customizable Resilience Inference Measurement (RIM), designed for multi-scale community resilience assessment and influential socioeconomic factors identification; 2) To implement a Platform for Resilience Inference Measurement and Enhancement (PRIME) module in the CyberGISX platform backed by high-performance computing, enabling users to apply and customize RIM to compute and visualize disaster resilience; 3) To demonstrate the utility of PRIME through a representative study to understand the geographical disparities of county-level community resilience to natural hazards in the United States and identifying the driving factors of resilience in the social domain. Customizable RIM generates vulnerability, adaptability, and overall resilience scores derived from empirical parameters—hazard threat, damage, and recovery. Computationally intensive Machine Learning (ML) methods are employed to explain the intricate relationships between these scores and socioeconomic driving factors. PRIME provides a web-based notebook interface guiding users to select study areas, configure parameters, calculate and geo-visualize resilience scores, and interpret socioeconomic factors shaping resilience capacities. A representative study showcases the efficiency of the platform while explaining how the visual results obtained may be interpreted. The essence of this work lies in its comprehensive architecture that encapsulates the requisite data, analytical and geo-visualization functions, and ML models for resilience assessment. This setup provides a foundation for assessing resilience and strategizing enhancement interventions.

Toeplitz Matrices whose Elements are Coefficients of new Subclasses of Analytical Functions

Toeplitz Matrices whose Elements are Coefficients of new Subclasses of Analytical Functions

FUNCTIONS WHOSE DERIVATIVE HAS POSITIVE REAL PART

It is well-known that a normalized analytic function for which the quantity $1+zf''(z)/f'(z)$ lies in the half-plane $\real w>-1/2$ is close-to-convex and hence univalent. In this paper, we show that the derivative of the function $f$ has positive real part if the quantity $1+\alpha zf''(z)/f'(z)$ with $\alpha>0$ lies in the sector $|\arg w| <\arctan( \alpha )$.

Chebyshev–Padé approximants for multivalued functions

The paper discusses the connection between the linear Chebyshev–Padé approximants for an analytic function f f and diagonal type I Hermite–Padé polynomials for the set of functions [ 1 , f 1 , f 2 ] [1, f_1, f_2] , where the pair of functions f 1 , f 2 f_1, f_2 forms a Nikishin system. Both problems can ultimately be reduced to certain convergence problems for multipoint Padé approximants. On the other hand, the denominators of multipoint Padé approximants are non-Hermitian orthogonal polynomials with analytical weights. Thus, to study all the above problems, the general method created by Herbert Stahl can be applied. Stahl’s method is not yet sufficiently developed to obtain general results on these problems. In particular, many key convergence problems for Chebyshev–Padé approximants for functions with arbitrary configurations of branch points remain open. In this paper, we consider several important general and particular results related to this case, some already well known, and also formulate two general hypotheses in the indicated direction.

The Gonchar–Chudnovskies conjecture and a functional analogue of the Thue–Siegel–Roth theorem

This article examines the Gonchar–Chudnovskies conjecture about the limited size of blocks of diagonal Padé approximants of algebraic functions. The statement of this conjecture is a functional analogue of the famous Thue–Siegel–Roth theorem. For algebraic functions with branch points in general position, we will show the validity of this conjecture as a consequence of recent results on the uniform convergence of the continued fraction for an analytic function with branch points. We will also discuss related problems on estimating the number of “spurious” (“wandering”) poles for rational approximations (Stahl’s conjecture), and on the appearance and disappearance of defects (Froissart doublets).

On functions of finite analytical complexity

We construct examples of polynomials and analytic functions of any predetermined finite analytical complexity n n . We obtain an estimate of the order of derivative of the differential-algebraic criteria for membership in the class C l n Cl_{n} of functions of analytical complexity not higher than n n . We find uniform estimates for finite values d n d_{n} of the analytic spectrum { d n } \{d_{n}\} for systems of differential-algebraic equations of fixed order of derivative δ \delta .

Differential geometry and general relativity with algebraifolds

It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential geometry which eliminate the need for an underlying manifold. While the literature contains various independent approaches to this, we focus on one particular approach that we argue to be the most natural one based on the definition of algebraifold, by which we mean a commutative algebra A for which the module of derivations of A is finitely generated projective. Over R as the base ring, this class of algebras includes the algebra C∞(M) of smooth functions on a manifold M, and similarly for analytic functions. An importantly different example is the Colombeau algebra of generalized functions on M, which makes distributional differential geometry an instance of our formalism. Another instance is a fibred version of smooth differential geometry, since any smooth submersion M→N makes C∞(M) into an algebraifold with C∞(N) as the base ring. Over any field k of characteristic zero, examples include the algebra of regular functions on a smooth affine variety as well as any function field.Our development of differential geometry in terms of algebraifolds comprises tensors, connections, curvature, geodesics and we briefly consider general relativity.

Numerical conformal bootstrap with analytic functionals and outer approximation

This paper explores the numerical conformal bootstrap in general spacetime dimensions through the lens of a distinct category of analytic functionals, previously employed in two-dimensional studies. We extend the application of these functionals to a more comprehensive backdrop, demonstrating their adaptability and efficacy in general spacetime dimensions above two. The bootstrap is implemented using the outer approximation methodology, with computations conducted in double precision. The crux of our study lies in comparing the performance of this category of analytic functionals with conventional derivatives at crossing symmetric points. It is worth highlighting that in our study, we identified some novel kinks in the scalar channel during the maximization of the gap in two-dimensional conformal field theory. Our numerical analysis indicates that these analytic functionals offer a superior performance, thereby revealing a potential alternative paradigm in the application of conformal bootstrap.

Differential Subordination and Coefficient Functionals of Univalent Functions Related to $\cos z$

Differential subordination in the complex plane is the generalization of a differential inequality on the real line. In this paper, we consider two subclasses of univalent functions associated with the trigonometric function $\cos z$. Using some properties of the hypergeometric functions, we determine the sharp estimate on the parameter $\beta$ such that the analytic function $p(z)$ satisfying $p(0)=1$, is subordinate to $\cos z$ when the differential expression $p(z)+\beta z (dp(z)/dz)$ is subordinate to the Janowski function. We compute sharp bounds on coefficient functional Hermitian--Toeplitz determinants of the third and the fourth order with an invariance property for such functions. In addition, we estimate bound on Hankel determinants of the second and the third order.

Certain theorems involving differential superordination and sandwich-type results

Abstract. To obtain the main result of the present paper, we use the technique of differential superordination. As special cases of our main result, we obtain sufficient conditions for f ∈ A to be φ−like, parabolic φ−like, starlike, parabolic starlike, close-to-convex and uniform close-to-convex. We also obtain sandwich-type results regarding these functions. For demonstration of the results, we have plotted the images of open unit disk under certain functions using Mathematica 7.0. Mathematics Subject Classification (2010): 30C80, 30C45. Keywords: Analytic function, differential superordination, φ−like function, starlike function, close-to-convex function.

On P-Stable Functions

Let h_1, h_2 be two analytic functions defined in the open unit disc Δ:={z∈C:|z|<1} which are normalized by the condition h_1 (0)=1=h_2 (0). Then h_1 is P-stable with respect to h_2, whenever (P_n (h_1,z))/(h_1 (z))≺1/(h_2 (z)) (z∈Δ),holds for all n∈N. Here ’≺’ stands for subordination and P_n (h,z)=P_n (z)*h(z) where P_n (z) denote the n-degree polynomial induced by the (n+1)th row entities in an admissible lower triangular matrix. The main purpose of this article is to prove that the function ((Az+1)/(Bz+1))^δ is P-stable with respect to (Bz+1)^(-δ), for δ∈(0,1] and -1≤B<A≤0 but not P-stable with respect to itself, when -1≤B<A<0 and δ∈(0,1]. As an application, considered different admissible lower triangular matrices to derive various results related on stability.

On the Containment of the Unit Disc Image by Analytical Functions in the Lemniscate and Nephroid Domains

Suppose that A1 is a class of analytic functions f:D={z∈C:|z|<1}→C with normalization f(0)=1. Consider two functions Pl(z)=1+z and ΦNe(z)=1+z−z3/3, which map the boundary of D to a cusp of lemniscate and to a twi-cusped kidney-shaped nephroid curve in the right half plane, respectively. In this article, we aim to construct functions f∈A0 for which (i) f(D)⊂Pl(D)∩ΦNe(D) (ii) f(D)⊂Pl(D), but f(D)⊄ΦNe(D) (iii) f(D)⊂ΦNe(D), but f(D)⊄Pl(D). We validate the results graphically and analytically. To prove the results analytically, we use the concept of subordination. In this process, we establish the connection lemniscate (and nephroid) domain and functions, including gα(z):=1+αz2, |α|≤1, the polynomial gα,β(z):=1+αz+βz3, α,β∈R, as well as Lerch’s transcendent function, Incomplete gamma function, Bessel and Modified Bessel functions, and confluent and generalized hypergeometric functions.

Applications of Generalized Hypergeometric Distribution on Comprehensive Families of Analytic Functions

A sequence of n trials from a finite population with no replacement is described by the hypergeometric distribution as the number of successes. Calculating the likelihood that factory-produced items would be defective is one of the most popular uses of the hypergeometric distribution in industrial quality control. Very recently, several researchers have applied this distribution on certain families of analytic functions. In this study, we provide certain adequate criteria for the generalized hypergeometric distribution series to be in two families of analytic functions defined in the open unit disk. Furthermore, we consider an integral operator for the hypergeometric distribution. Some corollaries will be implied from our main results.

Formation of S2 species in different redox states by radiative association in atomic and ionic collisions

Radiative associations for formations of the S2, S2+ and S2- molecular species during atomic collisions S(3Pu) + S(3Pu), S(3Pu) + S+(4Su) and S(3Pu) + S-(2Pu) are investigated. The adiabatic potential energy curves (PECs) and spin-allowed transition dipole moments (TDMs) are obtained by the internally contracted multireference configuration interaction method with the Davidson correction (icMRCI+Q). A number of PECs and TDMs are chosen to calculate the corresponding cross-sections and rate coefficients of radiative associations. The calculated rate coefficients are valid for the temperatures from 100 to 16000 K and fitted to the analytical function according to the three-parameter Arrhenius–Kooij formula. These results indicate that transitions originating in the ΔΛ=0 selection rule are the main contributors for the radiative association process. The present study can elucidate the further understanding the radiative association, which plays an important role in the formation and evolution of the S2, S2+ and S2- molecules.

Popularity-Aware Service Provisioning Framework in Cloud Environment

To balance the tradeoff between quality of service (QoS) and operating expenditure (OPEX), the service provider should request the appropriate amount of resources to the cloud operator based on the estimated variation of service requests. This paper proposes a popularity-aware service provisioning framework (PASPF), which leverages the network data analytics function (NWDAF) to obtain analytics on service popularity variations. These analytics estimate the congestion level and list of top services contributing most of the traffic change. Based on the analytics, PASPF enables the service provider to request the appropriate amount of resources for each service for the next time period to the cloud operator. To minimize the OPEX of the service provider while keeping the average response time of the services below their requirements, we formulate a constrained Markov decision process (CMDP) problem. The optimal stochastic policy can be obtained by converting the CMDP model into a linear programming (LP) model. Evaluation results demonstrate that the PASPF can achieve less than 50% OPEX of the service provider compared to a popularity-non-aware scheme while keeping the average response time of the services below the requirement.

Linear equations with infinitely many derivatives and explicit solutions to zeta nonlocal equations

We summarize our theory on existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives of the formf(∂t)ϕ=J(t),t≥0, where f is an analytic function such as the (analytic continuation of the) Riemann zeta function. We explain how to analyse initial value problems for these equations, and we prove rigorously that the functionϕ(t)=∑n≥1μ(n)nhJ(t−ln⁡(n)), in which μ is the Möbius function and J satisfies some technical conditions to be specified in Section 4, is the solution to the zeta nonlocal equationζ(∂t+h)ϕ=J(t),t≥0, in which ζ is the Riemann zeta function and h>1. We also present explicit examples of solutions to initial value problems for this equation. Our constructions can be interpreted as highlighting the importance of the cosmological daemon functions considered by Aref'eva and Volovich (2011) [1]. Our main technical tool is the Laplace transform as a unitary operator between the Lebesgue space L2 and the Hardy space H2.