Analytic Functions Research Articles

Data-driven evaluation of the Paris’ law parameters in polyethylene pipe grades — Increasing the precision of fracture mechanical lifetime estimation

The Paris’ Law parameters A and m are a necessity for predicting lifetimes of structural components under static or fatigue loading that fail due to crack initiation and propagation. Conventional methods require measurements of crack growth kinetics that involve direct or indirect monitoring of physical crack extension during long-term experiments. Usually, measurement series also involve multiple specimens in order to obtain a crack growth controlled failure diagram of an investigated material under relevant load conditions. In this contribution a combination of simple numerical, statistical and analytical approaches is presented to obtain A and m without the need to measure actual crack growth. This is accomplished by reformulating the Paris’ Law to express A as a function of m. The parameter m is varied within a reasonable range to generate an analytical function for A that solves the equation of the Paris’ Law based lifetime for a single specimen. A subsequent superposition of all available specimens reveals an intersection of all A functions at the technically relevant pair of A and m values that are capable of describing the lifetime of all specimens with a minimum error. The obtained best-fitting A and m are in good agreement with literature and are able to predict the lifetime of previously published sample data based upon cyclic Cracked Round Bar test results with an average error of 3.30 ± 2.67%.

Finite Sum of Integral Operators from the Fractional Cauchy Spaces to Bloch-Type and Zygmund-Type Spaces

The family of fractional Cauchy transforms, defined on the open unit disc in the complex plane, is of classical and modern interest. Membership of an analytic function in the family is determined by the requirement that the function can be expressed as an integral of a certain kernel against a complex Borel measure on the disc. Such an integral representation imposes a growth condition on the function and its derivatives. This exposes a connection between the families of Cauchy transforms and familiar spaces of analytic functions, such as the Bloch spaces and the Zygmund space. The notion of a composition operator has been a fruitful area of study. More generally, many authors have studied weighted composition operators, the differentiation operator, integral-type operators, and various products of such operators, acting from one normed linear space of analytic functions to another such space. A common theme of such works is to characterize the operator-theoretic notions of boundedness and compactness in terms of the inducing symbols of the operator. We extend these studies to a specific linear transformation which will be defined as the sum of finitely many integral operators. Our conclusions include a complete characterization of boundedness and compactness of the integral sum, acting from the fractional Cauchy spaces to the Bloch-type and Zygmund-type spaces.

Complex Generalized Stancu-Schurer Operators

Abstract In the present study, we determine a Schurer type of complex generalized Stancu operators and examine its some approximation properties. Firstly, we obtain upper quantitative estimates for the convergence and then we get lower estimates from a qualitative Voronovskaja-type result for these specified operators attached to the analytic functions in an open disk centered at the origin. Hence, we present the exact order of simultaneous approximation by using obtained upper and lower estimates.

A semi-analytical transient undisturbed velocity correction scheme for wall-bounded two-way coupled Euler-Lagrange simulations

Force closure models employed in Euler-Lagrange (EL) point-particle simulations rely on the accurate estimation of the undisturbed fluid velocity at the particle center to evaluate the fluid forces on each particle. Due to the self-induced velocity disturbance of the particle in the fluid, two-way coupled EL simulations only have access to the disturbed velocity. The undisturbed velocity can be recovered if the particle-generated disturbance is estimated. In the present paper, we model the velocity disturbance generated by a regularized forcing near a planar wall, which, along with the temporal nature of the forcing, provides an estimate of the unsteady velocity disturbance of the particle near a planar wall. We use the analytical solution for a singular in-time transient Stokeslet near a planar wall (Felderhof [1]) and derive the corresponding time-persistent Stokeslets. The velocity disturbance due to a regularized forcing is then obtained numerically via a discrete convolution with the regularization kernel. The resulting Green's functions for parallel and perpendicular regularized forcing to the wall are stored as pre-computed temporal correction maps. By storing the time-dependent particle force on the fluid as fictitious particles, we estimate the unsteady velocity disturbance generated by the particle as a scalar product between the stored forces and the pre-computed Green's functions. Since the model depends on the analytical Green's function solution of the singular Stokeslet near a planar wall, the obtained velocity disturbance exactly satisfies the no-slip condition and does not require any fitted parameters to account for the rapid decay of the disturbance near the wall. The numerical evaluation of the convolution integral makes the present method suitable for arbitrary regularization kernels. Additionally, the generation of parallel and perpendicular correction maps enables to estimate the velocity disturbance due to particle motion in arbitrary directions relative to the local flow. The convergence of the method is studied on a fixed particle near a planar wall, and verification tests are performed in the Stokes regime on a settling particle parallel to a wall and a free-falling particle perpendicular to the wall.

The Spectra of Multiplication Operators and Weighted Composition Operators on Iterated Weighted-Type Banach Spaces of Analytic Functions

This research aims to analyze the spectral of multiplication operators acting on weighted Banach spaces of analytic functions defined on the unit disk. These spaces, denoted by Sn : n ∈ N, include well-known cases such as the Bloch space and the Zygmund space for n = 1 and n = 2, respectively. Additionally, we offer a description of the spectra of weighted composition operators within these spaces. The outline of this paper provides a structured framework for organizing the research, starting from the introduction to the conclusion and references. The main section is to investigate the spectrum of multiplication operators on Sn spaces, and it followed by a section that is designed to build upon the previous one, leading to characterize the spectrum of weighted composition operators on the same spaces.

A note on transcendental analytic functions with rational coefficients mapping $\mathbf{Q}$ into itself

A note on transcendental analytic functions with rational coefficients mapping $\mathbf{Q}$ into itself

On Subclass of Analytic Functions Defined by q-Analogue of Modified Tremblay Fractional Derivative Operator

In this research, by using the principle of quantum calculus, we introduce a modified fractional derivative operator $\mathcal{T}^{\xi,\digamma}_{q,\varsigma}$ of the analytic functions in the open unit disc $\diamondsuit=\{\varsigma:\varsigma\in\mathbb{C},|\varsigma|

Electrochemical molecularly imprinted polymer sensor for simple and fast analysis of tetrodotoxin in seafood

Tetrodotoxin (TTX) is a marine biotoxin whose biosynthesis is associated with the pufferfish. Its distribution is primarily focused in Asian and tropical marine areas. Currently, this group of toxins is classified as emerging in Europe, and its presence could be related to climate change. This incidence has prompted the European Union, with the European Food Safety Authority, to establish control and monitoring mechanisms for TTX in marine products in Europe. In this context, the development of analytical tools capable of ensuring the safety of food products, especially seafood and fish, is a crucial task. This study describes the development of a molecularly imprinted polymer (MIP) based electrochemical sensor for the analysis of TTX. The MIP was synthesized through the electropolymerization of a functional monomer, ortho-phenylenediamine in the presence of a dummy template, voglibose. The MIP sensor was constructed on a screen-printed gold electrode and characterized by cyclic voltammetry. Differential pulse voltammetry, using a redox probe ([Fe(CN)6]3-/4-), was used in the analysis protocol. The developed sensor exhibited a linear response between 5.0 μg mL−1 and 25.0 μg mL−1, with a limit of detection of 1.14 μg mL−1. Its high imprinting efficiency conferred outstanding selectivity towards TTX. The sensor's applicability was confirmed through recovery assays on spiked mussel samples, achieving recoveries of 81.0 %, 110.2 %, and 102.5 % for external standard addition at 30.0, 44.0, and 60.0 μg kg−1, respectively, with relative standard deviations below 15 %. These results are comparable to those obtained using Hydrophilic Interaction Liquid Chromatography coupled with Tandem Mass Spectrometry, a validated method carried out by the European Reference Laboratory for Marine Biotoxins. Thus, the MIP sensor represents a portable, simple, and fast tool with essential analytical functionalities for the sampling phase and pre-selection of laboratory samples for analysis.

New Results on Differential Subordination and Superordination for Multivalent Functions Involving New Symmetric Operator

This article aims to significantly advance geometric function theory by providing a valuable contribution to analytic and multivalent functions. It focuses on differential subordination and superordination, which characterize the interactions between analytic functions. To achieve our goal, we employ a method that relies on the characteristics of differential subordination and superordination. As one of the latest advancements in this field, this technique is able to derive several results about differential subordination and superordination for multivalent functions defined by the new operator within the open unit disk . Additionally, by employing the technique, the differential sandwich outcome is achieved. Therefore, this work presents crucial exceptional instances that follow the results. The findings of this paper can be applied to a wide range of mathematical and engineering problems, including system identification, orthogonal polynomials, fluid dynamics, signal processing, antenna technology, and approximation theory. Furthermore, this work significantly advances the knowledge and understanding of the analytical functions of the unit and its interactive higher relations. The characteristics and consequences of differential subordination theory are symmetric to those of differential superordination theory. By combining them, sandwich-type theorems can be derived.

A new approach to MacMahon’s Partition Analysis and associated applications

We introduce a new elimination procedure for Partition Analysis to obtain generating functions free of Omega variables starting with a crude generating function containing or not factored polynomials. In contrast to Han’s approach (Ann Comb. 2003;7(4):467–480), we work with factored polynomials by judiciously constructing the quotient of a series modulo a certain prime ideal. Our results include diverse areas such as an Omega-based approach to compute the Hadamard product of power series and certain trigonometric integrals generalizing the main result of Kim (J Integer Seq. 2009;12:09.7.4) and recovering an integral identity of Boros and Moll (J Comput Appl Math. 1999;106(2):361–368), respectively. Furthermore, we construct a new Omega representation of analytic matrix functions which requires no spectral information and uses the matrix rank to simplify the calculations. By specializing to tridiagonal matrices, the main result of Kiliç (Appl Math Comput. 2008;197(1):345–357) is obtained.

Formation and conversion of high-peak-power structured beams by an Nd:YVO4/Cr4+:YAG laser subject to tight-focusing off-axis pumping

A compact Nd:YVO4/Cr4+:YAG passively Q-switched laser in a near-hemispherical resonator is exploited to realize high-peak-power pulsed beams with high spatial degrees of freedom. Beneficial from the advantages of strong intracavity beam focusing as well as the point-like excitation condition for the proposed cavity design, various high-order structured pulses as coherent superpositions of multiple degenerate eigenmodes are stably generated under different off-axis pump schemes. Besides, by employing external-cavity astigmatic mode conversion (AMC), the oval-shaped and chessboard-like structured pulses under on-axis and 1D off-axis pumping are transformed into exotic modes with polygonal and figure-eight-shaped envelopes to further enrich the spatial complexity of the generated fields. With well-defined beam structures that are reconstructed using the analytical resonant wave functions of the resonator, the phase structures of AMC pulsed fields are numerically resolved to present a variety of singularity arrays. Experimental results reveal that the overall peak power of the on-axis and off-axis generated structured pulses respectively exceeds 600 W and 1 kW while maintaining good pulse train stability with peak-to-peak amplitude fluctuation to be less than 10% and 15%.

Starlikeness, Convexity, Close-to-Convexity, and Quasi-Convexity for Functions with Fixed Initial Coefficients

In this paper, we employ the theory of differential subordination to establish a theorem that delineates certain sufficient conditions for starlikeness, convexity, close-to-convexity, and quasi-convexity in relation to functions with fixed initial coefficients. Furthermore, we introduce some results derived from these conditions. Building upon this framework, we derive an extension of Nunokawa’s lemma for analytic functions with fixed initial coefficients.

Multi-Objective optimization for stable and efficient cargo transportation of partial space elevator

This paper proposed a new libration decoupling analytical speed function (LD-ASF) in lieu of the classic analytical speed function to control the climber’s speed along a partial space elevator to improve libration stability in cargo transportation. The LD-ASF is further optimized for payload transportation efficiency by a novel coordinate game theory to balance competing control objectives among payload transport speed, stable end body’s libration, and overall control input via model predictive control. The transfer period is divided into several sections to reduce computational burden. The validity and efficacy of the proposed LD-ASF and coordinate game-based model predictive control are demonstrated by computer simulation. Numerical results reveal that the optimized LD-ASF results in higher transportation speed, stable end body’s libration, lower thrust fuel consumption, and more flexible optimization space than the classic analytical speed function.

Some Coefficient Problems for Subclasses of Holomorphic Functions in Complex Order Associating Sălăgeăn q-Differential Operator

A function with complex values and at every point of the specific domain contains a derivative is commonly known as analytic functions which is also referred as holomorphic functions. We begin by interpreting \(A\) as the class for all holomorphic functions \(L(\xi)\) with a Taylor series expansion written in the form: \[L(\xi) = \xi + \sum_{i=2}^{\infty} x_i \xi^i\] where \(x_i \in \mathbb{C}\) and \(\xi \in D\). \(D\) is the open unit disk where \(D = \{\xi : \xi \in \mathbb{C}, |\xi| < 1 \}\). Furthermore, we suggest the subclass of \(A\) that is univalent in \(D\) represented as \(S\). It is commonly known that the main subclasses of class \(S\) are the class of starlike functions and the class of convex functions. To develop and analyze the Fekete-Szegö problems, the theory of geometric function contributes significantly to this. Moreover, the frequent use of \(q\)-calculus as a general direction of research among mathematicians has caught our attention. In this research, we attained the initial coefficients, \(x_2\) and \(x_3\), and the upper bound for the functional \(|x_3 - \nu x_2^2|\) of functions \(L\) in the two new subclasses that are introduced by involving the Sălăgeăn \(q\)-differential operator, \(M_q^\eta L(\xi)\) and the definition of subordination.

Bounds for the Second Hankel Determinant of a General Subclass of Bi-Univalent Functions

The Hankel determinant, which plays a significant role in the theory of univalent functions, is investigated here in the context of bi-univalent analytic functions. Specifically, this paper is dedicated to deriving an upper-bound estimate for the second-order Hankel determinant for a general subclass of bi-univalent analytic functions that incorporate Gegenbauer polynomials within the unit disk. Through the careful specialization of parameters in our main result, we unveil several novel findings.

Design of coma-free Hettrick-Underwood type spectrometers for high energy resolution RIXS experimental station.

The Hettrick-Underwood (HU) type design, consisting of a pre-focusing mirror and a varied-line-spacing planar grating (VLS-PG), for the high resolution resonant inelastic x-ray scattering spectrometer is considered. The light path of the HU design is analyzed, and the analytical functions determining the groove parameters of VLS-grating are given. Efforts are made to compare the different choices of the pre-focusing mirrors, circular cylindrical (HU-CM) and elliptical cylindrical (HU-EM). Although the HU-EM type offers great tuning properties with variable-included-angle, it imposes high requirement on the optical element manufacturing accuracy, while the HU-CM type design couples the coma contributions from the pre-mirror and the grating, allowing for mutual compensation of the manufacturing errors of the two optical elements.

A Fully Analytical Approach for the Real-Time Dynamic Reliability Evaluation of Composite Power Systems with Renewable Energy Sources

Renewable energy sources (RES) have strong uncertainties, which significantly increase the risks of power imbalance and load shedding in composite power systems. It is thus necessary to evaluate the operational reliability for guiding economic dispatch and reducing the risks. Current methods cannot meet the requirement for the operational timeliness of reliability evaluations due to the high computational complexity of the optimal power flow (OPF) calculations of massive contingencies. This paper proposes a fully analytical approach to construct fast-to-run analytical functions of reliability indices and avoid reassessments when the load and RES change. The approach consists of uniform design (UD)-based contingency screening and a modified stochastic response surface method (mSRSM). The contingency screening method is used to select critical contingencies while considering the uncertainties. The mSRSM is used to construct the analytical functions of the load shedding to the load and RES generation for the selected contingencies. An analytical function of a smooth virtual variable that maps to the load shedding is established in such a way that, when the load and RES vary, the reliability can be assessed within a very short time rather than using laborious OPF calculations. Case studies illustrate the excellent performance of the proposed method for real-time reliability evaluation.

Optimal domain of generalized Volterra operators

For g in BMOA, we consider the generalized Volterra operator Tg acting on Hardy spaces Hp. This article aims to study the largest space of analytic functions, which is mapped by Tg into the Hardy space Hp. We call this space the optimal domain of Tg and we describe its structural properties. Motivation for this comes from the work of G. Curbera and W. Ricker [7] who studied the optimal domain of the classical Cesáro operator.

Electric transport and topological properties of binary heterostructures in topological insulators

The design of devices with coupled hybrid structures offers an approach to creating synthetic topological materials. This work discusses the topological and transport properties of low-dimensional binary heterostructures of topological and trivial materials. By adjusting the parameters of each component, we control the global topological properties to enhance tunneling and optimize the transmission of the topological edge states (TES). Considering a one-dimensional tight-binding model, we build heterostructures of coupled chains employing Green’s functions (GF) formalism. We determine the topological characteristics of chains and couple them together, applying Dyson’s equation to generate the heterostructure. The intensity and decay length of the TES vary depending on the coupling parameters and the size of each chain. We investigate the topological diagrams phase using the energy bands of the periodic system and calculating the invariant from the Zak phase. Using cross-band condition, we derive analytical functions of the parameter space to get the phase topological diagram, which can be compared with the LDOS maps at zero energy. Finally, we calculate the differential conductance with the Keldysh GF technique to demonstrate the tunneling of the TES at the zero bias voltage and discuss potential design and experimental applications.

Intractability results for integration in tensor product spaces

We prove lower bounds on the worst-case error of numerical integration in tensor product spaces. The information complexity is the minimal number N of function evaluations that is necessary such that the N-th minimal error is less than a factor ε times the initial error, i.e., the error for N=0, where ε belongs to (0,1). We are interested to which extent the information complexity depends on the number d of variables of the integrands. If the information complexity grows exponentially fast in d, then the integration problem is said to suffer from the curse of dimensionality.Under the assumption of the existence of a worst-case function for the uni-variate problem, we present two methods for providing lower bounds on the information complexity. The first method is based on a suitable decomposition of the worst-case function and can be seen as a generalization of the method of decomposable reproducing kernels. The second method, although only applicable for positive quadrature rules, does not require a suitable decomposition of the worst-case function. Rather, it is based on a spline approximation of the worst-case function and can be used for analytic functions. Several applications of both methods are presented.