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For a fixed analytic function g on the unit disc, we consider the analytic paraproducts induced by g, which are formally defined by T g f (z) = ∫ 0 z f (ζ) g ′ (ζ) d ζ , S g f (z) = ∫ 0 z f ′ (ζ) g (ζ) d ζ , and M g f (z) = g (z) f (z). We are concerned with the study of the boundedness of operators in the algebra A g generated by the above operators acting on Hardy, or standard weighted Bergman spaces on the disc. The general question is certainly very challenging, since operators in A g are finite linear combinations of finite products (words) of T g , S g , M g which may involve a large amount of cancellations to be understood. The results in [1] show that boundedness of operators in a fairly large subclass of A g can be characterized by one of the conditions g ∈ H ∞ , or g n belongs to B M O A or the Bloch space, for some integer n > 0. However, it is also proved that there are many operators, even single words in A g whose boundedness cannot be described in terms of these conditions. The present paper provides a considerable progress in this direction. Our main result provides a complete quantitative characterization of the boundedness of an arbitrary word in A g in terms of a "fractional power" of the symbol g , that only depends on the number of appearances of each of the letters T g , S g , M g in the given word. [ABSTRACT FROM AUTHOR]

In many high-impact applications, it is important to ensure the quality of the output of a machine learning algorithm as well as its reliability in comparison to the complexity of the algorithm used. In this paper, we have initiated a mathematically rigorous theory to decide which models (algorithms applied on data sets) are close to each other in terms of certain metrics, such as performance and the complexity level of the algorithm. This involves creating a grid on the hypothetical spaces of data sets and algorithms so as to identify a finite set of probability distributions from which the data sets are sampled and a finite set of algorithms. A given threshold metric acting on this grid will express the nearness (or statistical distance) of each algorithm and data set of interest to any given application. A technically difficult part of this project is to estimate the so-called metric entropy of a compact subset of functions of infinitely many variables that arise in the definition of these spaces. [ABSTRACT FROM AUTHOR]

This paper deals with the error analysis of the trapezoidal rule for the computation of Fourier type integrals, based on two double exponential transformations. The theory allows to construct algorithms in which the steplength and the number of nodes can be a priori selected. The analysis is also used to design an automatic integrator that can be employed without any knowledge of the function involved in the problem. Several numerical examples, which confirm the reliability of this strategy, are reported. • Error analysis through the theory of analytic functions. • Error estimates for double exponential trapezoidal rules for Fourier type integrals. • Automatic integrator for Fourier type integrals. [ABSTRACT FROM AUTHOR]

In recent years, deep neural networks have been employed to approximate nonlinear continuous functionals F defined on L p ( [ − 1 , 1 ] s) for 1 ≤ p ≤ ∞. However, the existing theoretical analysis in the literature either is unsatisfactory due to the poor approximation results, or does not apply to the rectified linear unit (ReLU) activation function. This paper aims to investigate the approximation power of functional deep ReLU networks in two settings: F is continuous with restrictions on the modulus of continuity, and F has higher order Fréchet derivatives. A novel functional network structure is proposed to extract features of higher order smoothness harbored by the target functional F. Quantitative rates of approximation in terms of the depth, width and total number of weights of neural networks are derived for both settings. We give logarithmic rates when measuring the approximation error on the unit ball of a Hölder space. In addition, we establish nearly polynomial rates (i.e., rates of the form exp − a (log M) b with a > 0 , 0 < b < 1) when measuring the approximation error on a space of analytic functions. [ABSTRACT FROM AUTHOR]

In this paper we present an extension of our previous research, focusing on a method to numerically evaluate the error term in the Gaussian quadrature formula with the Legendre weight function, as discussed by Jandrlic et al. (2022). For an analytic integrand, the error term in Gaussian quadrature can be expressed as a contour integral. Consequently, determining the upper bound of the error term involves identifying the maximum value of the modulus of the kernel within the subintegral expression for the error along this contour. In our previous study, we investigated the position of this maximum point on the ellipse for Legendre polynomials. In this paper, we establish sufficient conditions for the maximum of the modulus of the kernel, which we derived analytically, to occur at one of the semi-axes for Gegenbauer polynomials. This result extends to a significantly broader case. We present an effective error estimation that we compare with the actual one. Some numerical results are presented. [ABSTRACT FROM AUTHOR]

The Hybrid Event-B framework was introduced to add continuously varying behaviour to the discrete changes of state characteristic of the well established Event-B method. This is made necessary by the needs of verifying the hybrid and cyber-physical systems that are increasingly prevalent today. The semantic foundation of Hybrid Event-B rests on piecewise absolutely continuous functions of time. This enables unproblematic modelling of all classical physical phenomena, as well as the specification of conventional discrete changes of state, regardless of whether these arise in the physical arena or as abstractions of computational behaviour. In this paper, the large gap between arbitrary piecewise absolutely continuous functions, and what can be reasoned about mechanically/symbolically, is addressed. First, piecewise absolutely continuous real functions are restricted to piecewise complex analytic functions, real and without singularities on a semi-infinite portion of the real axis. This class has good properties with respect to symbolic manipulation and thus provides a good foundation for an approach to system verification that avoids dealing with the interleaved quantifiers of mathematical analysis, thus reducing the verification of the proof obligations of Hybrid Event-B to calculational checks. The individual proof obligations, whose discharge assures the correctness of a Hybrid Event-B machine, are examined, and results establishing sufficient conditions for their successful discharge via calculation are given. A small scale case study illustrates the verification process in this setting. • The paper considers in detail the reasoning processes that formal verification of Hybrid Event-B machines would require. • It presents a series of results concerning how such reasoning could be reduced to symbolic manipulation. • Thereby, individual Hybrid Event-B events can be proved correct, leading to correctness of the machine as a whole. [ABSTRACT FROM AUTHOR]

This paper investigates the approximation properties of deep neural networks with piecewise-polynomial activation functions. We derive the required depth, width, and sparsity of a deep neural network to approximate any Hölder smooth function up to a given approximation error in Hölder norms in such a way that all weights of this neural network are bounded by 1. The latter feature is essential to control generalization errors in many statistical and machine learning applications. • Rates and complexity for smooth function approximation in Hölder norms by ReQU neural networks. • Explicit and uniform bounds for weights of the approximating neural network. • Exponential convergence rates for analytic functions. [ABSTRACT FROM AUTHOR]

This paper investigates the fully complex mini-batch gradient algorithm for training complex-valued neural networks. Mini-batch gradient method has been widely used in neural network training, however, its convergence analysis is usually restricted to real-valued neural networks and of probability nature. By introducing a new Taylor mean value theorem for analytic functions, in this paper we establish deterministic convergence results for the fully complex mini-batch gradient algorithm under mild conditions. The deterministic convergence here means that the algorithm will deterministically converge, and both the weak convergence and strong convergence will be proved. Benefited from the newly introduced mean value theorem, our results are of global nature in that they are valid for arbitrarily given initial values of the weights. The theoretical findings are validated with a simulation example. [ABSTRACT FROM AUTHOR]

In this paper, for some α , ζ , α ∈ [ 0 , 1) , | ζ | ≤ 1 , we consider normalized analytic functions f such that (1 − ζ) z f ′ (z) / [ f (z) − f (ζ z) ] lies in half-plane { w : Re w > α } for all z , | z | < 1. The paper is devoted to the coefficient problems of such functions for real and for complex numbers ζ. [ABSTRACT FROM AUTHOR]

In this paper, we propose a consistent non-Gaussian Bayesian filter for which the system state is a continuous function. The distributions of the true system states, and those of the system and observation noises, are only assumed Lebesgue integrable with no prior constraints on what function classes they fall within. This type of filter has significant merits in both theory and practice, in that it is able to ameliorate the curse of dimensionality present in the particle filter, a popular non-Gaussian Bayesian filter for which the system state is parameterized by discrete particles and corresponding weights. We first propose a new type of statistics, called the generalized logarithmic moments. Together with the power moments, they are used to form a density surrogate, parameterized as an analytic function, to approximate the true system state. The map from the parameters of the proposed density surrogate to both the power moments and the generalized logarithmic moments is proved to be a diffeomorphism, establishing the fact that there exists a unique density surrogate which satisfies both moment conditions. This diffeomorphism also allows us to use gradient methods to treat the convex optimization problem in determining the parameters. Last but not least, simulation results reveal the advantage of using both sets of moments for estimating mixtures of complicated types of functions. A robot localization simulation is also given, as an engineering application to validate the proposed filtering scheme. [ABSTRACT FROM AUTHOR]

Littlewood's theorem is one of the pioneering results in random analytic functions over the open unit disk. In this paper, we prove some analogues of this theorem for Hardy spaces in infinitely many variables. Our results not only cover finite-variable setting, but also apply in cases of Dirichlet series. [ABSTRACT FROM AUTHOR]

Signed Distance Fields (SDFs) have found widespread utility in collision detection applications due to their superior query efficiency and ability to represent continuous geometries. However, little attention has been paid to calculating the intersection of two arbitrary SDFs. In this paper, we propose a novel, accurate, and real-time approach for SDF-based collision detection between two solids, both represented as SDFs. Our primary strategy entails using interval calculations and the SDF gradient to guide the search for intersection points within the geometry. For arbitrary objects, we take inspiration from existing collision detection pipelines and segment the two SDFs into multiple parts with bounding volumes. Once potential collisions between two parts are identified, our method quickly computes comprehensive intersection information such as penetration depth, contact points, and contact normals. Our method is general in that it accepts both continuous and discrete SDF representations. Experiment results show that our method can detect collisions in high-precision models in real time, highlighting its potential for a wide range of applications in computer graphics and virtual reality. • The first real-time and accurate general SDF-SDF collision detection method. • A novel method for testing the intersection of analytic distance functions. • An accurate method for estimating contact information for SDF-SDF collision response stages. [ABSTRACT FROM AUTHOR]

This paper presents the fuzzy power series method to solve the second-order differential equation under generalized Hukuhara differentiability with fuzzy and real coefficients. For this purpose, different types of fuzzy analytic functions by attention to generalized Hukuhara differentiability, ordinary and singular points are introduced. Since in the discussion of fuzzy power series, the concept of the fuzzy convergence radius is one of the most essential and fundamental concepts, the fuzzy convergence radius under the generalized division is defined. Fundamental theorems, such as fuzzy ratio tests, the convergence of the fuzzy geometric series, are expressed and proven. In addition, it has been shown that the fuzzy convergence radius of fuzzy power series does not change concerning derivatives under operators such as derivatives and integrals. It has been shown that the fuzzy analytic functions are still fuzzy analytic functions concerning derivatives under fuzzy operators, summation, multiplication, and generalized division. Then, the uniqueness of the solution of the second-order fuzzy differential equations with fuzzy and real coefficients in the form of a fuzzy power series by attention to the type of generalized differentiability is shown. Finally, there are examples to demonstrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

• An analytical solution for a finite thickness soil layer is presented. • The interaction between soil layer and strip footing is considered. • Both rigid footings and deformable footings are reserched. • The complex variable method is applied to the mixed boundary problem. • The mechanical influence of bedrock buried depth on soil layer is studied. Consider an elastic soil layer with a shallow-buried strip footing at the upper boundary and bedrock at the lower boundary, the displacement along the contact surface between the soil layer and the bedrock can be considered as zero when the bedrock is sufficiently rigid relative to the soil layer. The stress and displacement fields in the soil layer are solved on the premise that the displacement on the contact surface between the soil layer and the strip footing is given. The soil layer with the finite thickness on the physical plane is mapped onto the inner domain of the unit circle on the image plane based on the conformal transformation. Then, the complex variable method is applied to give the stress and displacement fields in the soil layer by two analytic functions, the coefficients of which can be determined by the stress and displacement boundary conditions on the upper and lower boundaries of the soil layer. In calculation, the power series method is introduced to establish a set of linear equations containing the analytic function coefficients, and then the coefficients are determined by solving the linear equations. This paper presents the analytical solutions of stress and displacement in the soil layer under two different displacement boundary conditions on the contact surface between the soil layer and the footing. The obtained results are compared with the results of the numerical method based on ANSYS. Finally, the influence of different displacement boundary conditions and soil thickness on the stresses on the surface and internal region of the soil layer is analyzed. [ABSTRACT FROM AUTHOR]

Off-axis incidence is inherent to sun tracking mirrors, named heliostats. Moreover, heliostat defocus takes place, provided that the focal length of the mirrors usually differs from their distance to the target. Defocus, along with high incidence angles, lead to astigmatic aberrations, that are overlooked by analytic functions based on convolution. This paper presents an analytic function on the image plane that reproduces these aberrations. As the reflection of the heliostat axes is non-orthogonal with defocus, the identification of the reflected axes plays a key role in the resulting model. In comparison with Monte Carlo Ray-Tracing, the model shows correlation coefficients above 90%, except at the tangential and sagittal foci, where the analytic function is indeterminate. An astigmatic parameter is provided to assess the model validity, which is complementary to UNIZAR at high incidence angles. • Analytic function that reproduces astigmatic aberrations in defocused heliostats. • Accounts for non-orthogonality of heliostat reflected axes under defocus. • Flux spot lines up along the sagittal axis at the tangential focus, and viceversa. • Indeterminate analytic function at the astigmatic foci. • Appropriate for 1 − cos ω L H / f > 0.005, complementing UNIZAR. [ABSTRACT FROM AUTHOR]

The linear Mohr–Coulomb (M-C) failure criterion is widely used to describe the behaviour of materials with compression/tension asymmetry, such as geological materials. Difference in compressive and tensile fracture/yielding stress is so-called strength difference or SD effect. A modified M-C criterion was proposed previously to describe the failure of ductile materials and to better fit the experimental data [ 1 ]. A circular failure envelope, tangent to the Mohr's circles corresponding to the limit state of stress for uniaxial tension, compression and pure shear, was considered. In this paper, the above circular M-C criterion is extended to brittle materials (with SD effect). The new criterion has been compared with fundamental criteria in plasticity theory and experimental results taken from literature. In the space σ 1 -σ 3 it is represented by two segments of hyperbola and allows a smooth transition from linear M-C to Rankine criteria. It proves to be adaptable to a wide range of materials and this is a major advantage. Moreover, the paper demonstrates also that all yield criteria for standard plastic materials, without SD effect, under biaxial state of stress, are insensitive to hydrostatic pressure. [ABSTRACT FROM AUTHOR]

This paper is concerned with the fractional nonlinear Schrödinger equation with periodic boundary conditions i u t + − Δ s 0 u + − Δ r 0 f ( | u | 2) u = 0 , x ∈ T ≔ R / 2 π Z , where s 0 ∈ (1 / 2 , 1) , r 0 ∈ [ 0 , s 0 − 1 / 2 ] and f is a real analytic function in some neighborhood of the origin in ℂ. The above equation (0.1) is reversible with respect to the involution G : (u , u ̄) ↦ (u ̄ , u). By the KAM (Kolmogorov–Arnold–Moser) theorem for an infinite dimensional reversible system with unbounded perturbation, we obtain that there exists Cantor families of quasi-periodic solutions of small amplitude to above equation. • The quasi-periodic solutions of fractional nonlinear Schrödinger equation. • The corresponding system is a reversible system. • The nonlinearity with fractional derivative which is unbounded. • Fractional calculus is used in fluid dynamics and quantum mechanics, etc. [ABSTRACT FROM AUTHOR]

In control engineering, control processes are regulated to prevent transient shocks to test rigs. As an important part of the control loop, the reference may be continuous and bounded to avoid the transient impulse generated by the controller. Therefore, a high-order bounded reference (HOBR) is proposed for the trajectory generation. This paper studies the smooth control process and constrained reference generation for large testing platforms. Utilizing the Hermite interpolation technique, a planning method for the HOBR is proposed to ensure a smooth and bounder generated trajectory, which derives an analytic function for the desired reference. In addition, an observer-based state feedback controller is presented and applied to enable the system output to converge asymptotically to the desired reference. The theoretical analysis shows that the HOBR satisfies continuous constraints and boundary constraints at the same time. Finally, the effectiveness of the proposed HOBR in two constraints is demonstrated by both simulations and experiments. • The proposed HOBR reference and its derivatives are continuous and bounded. • The proposed HOBR technique is robust to various periodical reference scenarios. • The proposed technique generates analytic solutions in polynomial time complexity. • The proposed controller utilizes the reference information to achieve convergence. [ABSTRACT FROM AUTHOR]

• Properties of multiple discrete-continuous (MDC) choice models with linear utility profile for outside good (L γ-profile MDC choice models). • Suitability of linear outside good utility (vis-à-vis non-linear utility) functions for modeling MDC choices. • Derivation of optimal demand distributions resulting from L γ-profile MDC choice models with large budgets. • Relationship between Tobit models and L γ-profile MDC choice models with large budgets. • Empirical application to the analysis of expenditures on domestic tourism trips in India. In this paper, we enhance the current understanding of the properties of multiple discrete-continuous (MDC) choice models with additively separable (AS), independent and identically distributed (IID) utility functions, and linear utility form on the essential outside good. First, we highlight that the prior implementations of this model in the literature ignore primal feasibility conditions related to the budget constraint and the essential nature of outside good in formulating the model likelihood function. Second, we evaluate the suitability and performance of the model for alternative consumption patterns relative to the budget. In addition, we provide a systematic comparison of the performance of MDC choice models with the linear outside good utility form (i.e., the L γ-profile model) and those with the non-linear outside good utility form (i.e., the NL γ-profile model). Third, for the L γ-profile model with infinite budgets (i.e., when a very small proportion of the budget is allocated to inside goods), we derive the distributions of the resulting optimal demand functions and analytic expressions for the corresponding first and second moments, and identify a property that makes it easy to estimate the utility function parameters of an inside alternative even when consumption data is not available for other alternatives. In addition, perhaps for the first time in the literature, we show how an independent system of Tobit models can be derived as a restricted version of the utility-theoretic L γ-profile MDC model structure. Finally, we apply the model for an empirical analysis of expenditure patterns of leisure trips from a domestic tourism survey sample of households in India. [ABSTRACT FROM AUTHOR]

The main concern of this paper is the three-variable Karcher (alternatively, Riemannian or Cartan) matrix mean for 2 × 2 linearly independent triples of positive definite matrices. We show that such triples form an analytic manifold and obtain three analytic functions from (1 , ∞) 2 × (0 , π / 2) to (0 , 1) realizing the Karcher mean as a linear combination of variables. [ABSTRACT FROM AUTHOR]

• An analytical method for solving the gravity-induced stresses in slope is presented. • The usable expression of mapping function is given. • The method doesn't involve complicated analytic continuation and Cauchy integral. • The complex potentials are directly solved through the boundary condition. A new analytical approach is presented for gravity-induced stresses in elastic half plane with a slope. The half plane is mapped onto the unit circle in ζ plane by conformal transformations. The mapping function proposed by Schwarz-Christoffel is irrational and difficult to be applied to the problem in the paper. Therefore, the explicit expression of the mapping function, which is easy to use and has high precision, is proposed through power series approximation. Based on the complex potentials with body force, a simple method for solving gravity-induced stresses which do not involve analytic continuation and Cauchy integral is established. The analytic functions are expressed as power series. Through the stress boundary condition on the ground surface, a set of linear equations with the coefficients of the power series can be directly constructed and solved. The stress results obtained by the presented analytical method agree well with the numerical solution. The stress distributions under different Poisson's ratio and slope angle are studied. [ABSTRACT FROM AUTHOR]

The direct application of the classical method of fundamental solutions (MFS) is restricted to homogeneous linear partial differential equations (PDEs). The use of fundamental solutions with different frequencies allowed the extension of the MFS to non-homogeneous PDEs, in particular, for Poisson or Helmholtz equations and for elastostatic or elastodynamic problems. This method has been called method of fundamental solutions for domains (MFS-D), but it faces an approximation problem when the non-homogeneous term presents discontinuities, because the fundamental solutions are analytic functions outside the source point set. In this paper we analyze two domain decomposition techniques for overcoming this approximation problem. The problem is set in the context of the modified Helmholtz equation, and we also establish the missing density results that justify both the MFS and the MFS-D approximations. Numerical results are presented comparing a direct and an iterative domain decomposition technique, with simulations in non-trivial domains. [ABSTRACT FROM AUTHOR]

In this paper we consider weighted integrals with respect to a modification of the generalized Laguerre weight functions w α (x) = x α e − x , α > − 1 , on (0 , + ∞) by the oscillatory factor exp (i ζ x) , where ζ > 0. For calculation of such integrals for analytic real-valued functions in D = { z ∈ ℂ ∣ Re z ≥ 0 , Im z ≥ 0 } we construct polynomials orthogonal with respect to the linear functional L : P → ℂ , given by L [ p ] = ∫ 0 + ∞ p (x) w α (x) exp (i ζ x) d x , where P is a linear space of all algebraic polynomials, as well as the corresponding quadrature rules of Gaussian type. The existence of such orthogonal polynomials and the corresponding three-term recurrence relations are proved. A few numerical examples are presented, including computation of the Cauchy principal value integrals. [ABSTRACT FROM AUTHOR]

In this paper, we delve into the b -family of equations and explore regularity properties of its global solutions. Our findings reveal that, irrespective of the real choice of the constitutive parameter, when the initial datum is confined to an analytic Gevrey function the resulting global solution is analytic in both temporal and spatial variables. [ABSTRACT FROM AUTHOR]

The paper proposes a new approach for function approximation by ‘estimating’ the second derivative, using Taylor series, where the samples of the function and its first derivatives at the sample points are known. Without such ‘estimation’, these initial data, can approximate the function traditionally using the first order Taylor approximation, but with more error. If we desire to improve the approximation via second order Taylor series, then we can estimate the ‘pseudo’ second derivatives of the function in three different ways. All these three ways are investigated in this paper. The pseudo second derivatives help in computing many more sample points of the function within each sampling interval. Thus, the approach acts like a mathematical ‘magnifying glass’. Two examples are treated to compare the efficiencies of the methods. Also, a qualitative study for upper bound of error of the approximations is studied in detail. [ABSTRACT FROM AUTHOR]

This paper introduces two algorithms of the phase retrieval, Gerchberg–Saxton algorithm and gradient search algorithm. We respectively get the function of the object of gradient search algorithm about the generalized pupil, wavefront and the zernike coefficients of the partial derivatives when double-frame images and their defocus as the input. The relationship between GS algorithm and the gradient search algorithm are revealed. This paper designs the simulation experiment with GS algorithm and gradient search algorithm when single-frame images and double-frame images are used as input. The experiment results show that the gradient search algorithm is superior to GS algorithm for a single-frame image as input. Both GS algorithm and gradient search algorithm can primely work out wavefront for double-frame images of different defocus as the input, but the convergence rate of gradient search algorithm is evidently better than GS algorithm. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give some conditions for univalence of analytic functions in the unit disc | z | < 1. We consider starlike functions such that z f ′ (z) ∕ f (z) lies in a vertical strip. In the second part we present some conditions on z f ′ ′ (z) ∕ f ′ (z) for f to be univalent. [ABSTRACT FROM AUTHOR]

• Analytic function of the steady-state water flow in the transitional Reynolds area. • Expected to be adaptable to all temperatures above freezing point and pipe dimensions. • Satisfies gradient at laminar flow boundary Re = 2000 and the gradient at turbulent boundary Re = 4000. • Developed for the calibration of roughness values in water networks. • Incorporates Colebrook–White's friction factor in Darcy–Weisbach's pressure drop relation. In this paper we derive a mathematical description of the steady-state water flow in the transitional Reynolds area, i.e. between Reynolds numbers 2000 and 4000. Specifying the flow as two dimensional function of the pressure drop and the roughness of a conduit pipe, a description is obtained which not only satisfies the boundary conditions, but also the gradient on the laminar-transitional as well as the transitional-turbulent boundary to a sufficient degree of accuracy. This is motivated by the need to identify individual friction parameters per pipe in a water supply network, a necessity for being able to detect and localise faults reliably. It occurs that although some flows have never been in the turbulent regime, one often yet mistakenly tries to find roughness values for corresponding pipes during the network's calibration. In order to let the calibration algorithm itself decide if an appropriate pipe flow has been laminar or turbulent, a continuous and smooth description in between is needed. Effectively, this work lays part of the foundation for water network calibration algorithms, which fully account for the different flow regimes using pressure sensors primarily. [ABSTRACT FROM AUTHOR]

Let f = h + g ‾ be a normalized and sense-preserving convex harmonic mapping in the unit disk D. In a recent paper, Ponnusamy and Sairam Kaliraj conjectured that there is a θ ∈ [ 0 , 2 π) such that the function h + e i θ g is convex in D. In this article, we first disprove a more flexible conjecture: " Let f = h + g ‾ be a convex harmonic mapping in the disk D. Then there is a θ ∈ [ 0 , 2 π) such that the function h + e i θ g is starlike in D ". In addition, we present an example to show that there exists a harmonic automorphism f = h + g ‾ of a disk such that the function h + e i θ g is convex in only one direction for θ ≠ 0 , and that the analytic function h + g is not starlike therein. The article concludes with a new conjecture. [ABSTRACT FROM AUTHOR]

The widely linear estimator offers theoretical and practical advantages over the strictly linear counterpart for the processing of the generality of complex signals. When applying the widely linear technique to a nonlinear estimator of learning systems, the resulting augmented nonlinear estimator also empirically produces performance advantages over the standard (strictly) nonlinear estimator. However, the theoretical justification for this observation is still lacking in the literature. To this end, we establish performance bounds of the augmented nonlinear estimator over the strictly nonlinear one in this paper. By expanding the analytic nonlinear function with the first-order Taylor formula, we obtain the closed form of the MSE difference △ MSE and that of the complementary MSE difference △ CMSE for the two estimators. By jointly considering the MSE and the complementary MSE, it is theoretically established that △ MSE > 0 always holds for noncircular input signals, and MSE performance advantages always exist in both the real and imaginary channels over its strictly nonlinear counterpart except for the jointly circular case. The simulation results validate our theoretical analysis. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to investigate some subordination implications for a class of nonlinear integral operators associated with multivalent functions in the open unit disk. Some known results associated with a class of nonlinear averaging integral operators are also pointed out as special cases of the main findings which are presented here. [ABSTRACT FROM AUTHOR]

With the prevalence of implicit shape processing and reconstruction, extracting a polygonal mesh of an isosurface from volume data, which plays an important role in these tasks, is receiving more and more attention. GradNormal, a recently proposed marching tetrahedra method, can effectively extract high-quality meshes from analytic functional shapes but suffers from detail loss and computational inefficiency issues. In this paper, we improve GradNormal from four aspects. First, we extend GradNormal from an analytic function to an arbitrary geometric domain equipped with the projection operation. Second, we select a seed tetrahedron and find only the tetrahedra intersecting the implicit surface, in a region-growing style, which helps save memory and accelerate calculation. Third, we invent a hierarchical tiling mechanism to enhance the recovery accuracy of the resulting mesh, unlike the uniform tiling used in GradNormal. Finally, we propose to accurately predict how the underlying surface goes through a tetrahedral element so that complicated topological structures such as thin plates and gaps can be well captured. Extensive experimental results on challenging shapes show that the improved GradNormal is able to quickly produce a feature-adapted triangle mesh that is more topologically and geometrically accurate than the state-of-the-art. [Display omitted] • Propose an improved GradNormal method to extract an accurate mesh from volume data. • Extend GradNormal to an arbitrary geometric domain equipped with projection operation. • Search the tetrahedra intersecting the implicit surface in a region-growing style. • Invent a hierarchical tiling mechanism to enhance the recovery mesh accuracy. • Accurately predict how the underlying surface passes a tetrahedral element. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper, the authors derive several sharp coefficient estimates for the function class consisting of the inverses of functions in a certain class of spirallike functions in the open unit disk , which was introduced by Libera. They also obtain a number of sharp coefficient bounds of functions in the more general classes and , each of which is introduced here. Some of the results derived in this paper would generalize those in a recent work of Kapoor and Mishra. [Copyright &y& Elsevier]

Abstract: For analytic functions the remainder term of quadrature formulae can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points and a sum of semi-axes , for Gauss–Radau quadrature formula with Chebyshev weight function of the third kind. Starting from the explicit expression of the corresponding kernel, derived by Gautschi, we determine the locations on the ellipses where maximum modulus of the kernel is attained. The obtained values confirm the corresponding conjectured values given by Gautschi in his paper [W. Gautschi, On the remainder term for analytic functions of Gauss–Lobatto and Gauss–Radau quadratures, Rocky Mounatin J. Math. 21 (1991) 209-206]. In this way the last unproved conjecture from the mentioned paper is now verified. [Copyright &y& Elsevier]

Abstract: In a recent paper, Hernández and Salanova proposed an interesting Newton-like method for the solution of nonlinear Fredholm integral equations. The aim of the present paper is to carry over this method to nonlinear operator equations of similar type. By using a functional analytic approach, the presentation becomes much clearer and more general. Apart from this, some new results in the Banach space setting as well as in the applications are obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce and investigate an interesting subclass of analytic and bi-univalent functions in the open unit disk . For functions belonging to the class , we obtain estimates on the first two Taylor–Maclaurin coefficients and . The results presented in this paper would generalize and improve some recent work of Srivastava et al. [H.M. Srivastava, A.K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23 (2010) 1188–1192]. [Copyright &y& Elsevier]

Abstract: For functions belonging to the class and , of analytic functions in , which are investigated in this paper, the author derives several interesting strong differential subordination results. These strong differential subordinations are established by means of a special case of the extended multiplier transformations namely where , and , A number of interesting consequences of some of these strong subordination results are discussed. Relevant connections of some of the new results obtained in this paper with those in earlier works are also provided. [Copyright &y& Elsevier]

• The stress function of the wedge-splitting test on a specimen was obtained by the boundary collocation method. • The wedge-splitting test on a compact tension specimen was carried out, the cohesive zone length and the crack opening displacement (COD) in the cohesive zone were obtained by using the digital image correlation method (DICM). • Based on the stress function and test data, the cohesive zone model analytic solutions that satisfied the conditions of COD in the cohesive zone was determined by the weighted integral method. • The tensile strain softening curve and the fracture energy were obtained, and the fracture energy was very close to the fracture energy determined using softening curve forms those are frequently used, and the calculated fracture energy was very close to the measured fracture energy. • The method of determining cohesive zone model analytic solutions that satisfied the conditions of COD in the cohesive zone is applicability, revealing a new method for solving the tensile strain softening curve. In this paper, the cohesive zone of concrete is investigated using wedge-splitting test on a compact tension specimen. First, the stress function of the wedge-splitting test on a specimen was obtained by the boundary collocation method. Followed by the actual the wedge-splitting test on a compact tension specimen, the cohesive zone length and the crack opening displacement (COD) in the cohesive zone were obtained by using the digital image correlation method (DICM). Based on the stress function and test data, the weight function that satisfied the conditions of COD in the cohesive zone was determined by superposition of analytic functions with the same crack length but different cohesive zone lengths. Next, the cohesive zone model analytic solutions was determined by the weighted integral method, and the tensile strain softening curve and the fracture energy were obtained. Finally, by comparing the results, it was found that the COD obtained from the cohesive zone model analytic solution was close to the experimental COD, and in addition, the fracture energy was very close to the fracture energy determined using softening curve forms those are frequently used, and the calculated fracture energy was very close to the measured fracture energy. The work shows that the method of determining cohesive zone model analytic solutions using the COD works, revealing a new method for solving the tensile strain softening curve. [ABSTRACT FROM AUTHOR]

Abstract Error estimation for the pointwise error of partial expansion of the Legendre series of a given function is a classical problem. Asymptotic error estimates exist expressed by the modulus of continuity of the given function, for functions of bounded variation, and for both analytic and piecewise analytic functions. These estimates are all asymptotic in the sense that the order of the partial expansion tends to infinity. It is interesting that in spite of many attempts the estimates are not optimal. This fact is demonstrated here, and the optimal estimate for the piecewise analytic functions is formulated as a hypothesis, which in turn is validated computationally. Further, in most applications the polynomial orders are relatively small compared to those in the asymptotic range. The existing theory does not address the behaviour of the preasymptotic error. Inherent difficulties addressing this question are discussed and illustrated here. Highlights • This paper extends the theory of pointwise Legendre expansions. • The new theory sharpens the existing results. • We show the existence of preasymptotic and asymptotic ranges, and underline the difficulty of identifying these ranges a priori. [ABSTRACT FROM AUTHOR]

Abstract A numerical method for solving polynomial partial differential equations in two independent variables, defined in the paper, is presented. The technique is based on polynomial approximation. Properties and the operational matrices for partial derivatives for a polynomial in two variables are presented first. These properties are then used to reduce the solution of partial differential equations in two independent variables to a system of algebraic equations. Five illustrative examples are presented to prove the effectiveness of the present method. Results show that the numerical scheme is very convenient for solving polynomial partial differential equations. [ABSTRACT FROM AUTHOR]

Abstract: For functions belonging to the class and , of normalized analytic functions in the open unit disc , which is investigated in this paper, the author derives several interesting differential subordination results. These subordinations are established by means of a special case of the multiplier transformations namely where , , and , A number of interesting consequences of some of these subordination results are discussed. Relevant connections of some of the new results obtained in this paper with those in earlier works are also provided. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to generalize the conic domain defined by Kanas and Wisniowska, and define the class of functions which map the open unit disk onto this generalized conic domain. A brief comparison between these conic domains is the main motivation of this paper. A correction is made in selecting the range interval of order of conic domain. [Copyright &y& Elsevier]

Abstract: Results about the study of nonanalytic systems’ center-focus and bifurcations of limit cycles are hardly seen in published references up till now. In this paper, we investigated the problems of determining center or focus and bifurcations for a class of planar quasi cubic analytic systems. The recursive formula to figure out generalized focal values is given, ulteriorly the conditions for four limit cycles from the origin or the point at infinity are obtained and center problems are considered. What is worth pointing out is that we offer a kind of interesting phenomenon that the exponent parameter λ control the non-analyticity of studied system in this paper. In terms of nonanalytic differential systems, our work is new. [ABSTRACT FROM AUTHOR]

Abstract A numerical model to predict mechanical properties of Ni-base superalloys for turbine disk applications is developed by modifying and combining previous models that consider various microstructural factors. Modifications are made so that as many necessary model parameters as possible, whose values have been obtained from mechanical measurements and microstructural observation, can be determined using a thermodynamic calculation and a precipitation kinetics simulation. The model's predictions of tensile properties, creep resistance and fatigue damage tolerance have good correlation with experimental data. The applicability and limits of the present model as an alloy design tool for new Ni-base disk superalloys are discussed in this paper. Highlights • We propose a numerical model to predict tensile properties, creep resistance and fatigue damage tolerance. • The model has been achieved by combining earlier empirical models with modifications. • The model considers various microstructural factors based on deformation mechanisms. • The advantage of our model is to enable to calculate all model parameters from computational approaches. • The model enables to predict mechanical properties without any further experimental measurement. [ABSTRACT FROM AUTHOR]

In this paper, we propose an inexact Newton-like conditional gradient method for solving constrained systems of nonlinear equations. The local convergence of the new method as well as results on its rate are established by using a general majorant condition. Two applications of such condition are provided: one is for functions whose derivatives satisfy a Hölder-like condition and the other is for functions that satisfy a Smale condition, which includes a substantial class of analytic functions. Some preliminary numerical experiments illustrating the applicability of the proposed method are also presented. [ABSTRACT FROM AUTHOR]

The increasing application of plasma based technologies over a wide range of fields has led to the necessity of an optimal determination of the characteristic parameters of the plasma systems. Optical Emission Spectroscopy techniques have been proved as an excellent tool to that end. These techniques are based on the collection and analysis of the radiation coming from the plasma, being classified between molecular and atomic emission. Among atomic emission lines, Hydrogen Balmer series lines are the most measured. The H β line is usually the most employed one for plasma diagnosis but under some conditions, the H β line is not always detectable and only the H α line can be detected. For these lines, it is well known that their shapes are the result of the convolution of three profile types: Lorentzians, Gaussians and Starks. But while the first two types present analytical functions, models explaining the Stark profile do not offer these. So, in this paper we propose an accurate analytical function for the Stark profile of the H α line allowing improving the determination of characteristic parameters of plasma with respect to the methods traditionally used. [ABSTRACT FROM AUTHOR]

Mohr-Coulomb (M-C) criterion is widely used for isotropic brittle materials, although it has some disadvantages which limit its wider application. One important disadvantage of this theory is that it ignores the effect of the intermediate principal stress, although it has an important influence on materials behaviour. Therefore, Mohr-Coulomb theory is actually used only for materials in biaxial state of stress. In this paper, a modified form of Mohr 's criterion has been suggested, in order to overcome this limitation. The modified criterion takes into account the influence of intermediate principal stress in the case of materials under triaxial loading conditions. On the basis of two new hypotheses, a real triaxial state of stress is replaced by a virtual biaxial state of stress. The above mentioned stress states are considered to be equivalent for a material when the failure is imminent in both cases. Both Mohr 's theory and the concept of equivalence of stress states are combined in order to obtain the virtual biaxial state of stress, which is determined using all three principal stresses. The modified Mohr 's theory can be used only for triaxial state of stress. For biaxial and uniaxial state of stress it is reduced to the classical one. The proposed hypotheses have an important advantage because does not require additional material coefficients. The calculated values using the virtual biaxial state of stress and the linear M-C criterion are in good agreement with experimental data for grey cast iron. The article also presents a compact statistical analysis for both stresses and errors, in order to objectively assess the effectiveness of the formulated hypotheses. [ABSTRACT FROM AUTHOR]