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The complex variable finite element method (ZFEM) is a numerical technique which aims to find the partial derivatives of the independent variables with respect to variation in dependent parameters declared in the physics. This is done by combining the complex Taylor series expansion within the weak formulation of the governing equation in a coupled system of linear equations forming a complex valued block matrix given by the Cauchy–Riemann matrix representation. In this work, two‐dimensional linear first‐order elements have been implemented in ZFEM to predict the design derivatives of the mechanical displacement field and the voltage potential field for a layered piezoelectric system in a steady‐state study with Dirichlet boundary condition applied at the top and bottom edges of the geometry. This approach allows the standard FEM solution to quantify the sensitivity of the mechanical displacement and voltage potential fields with respect to small variations in the material properties through the information obtained from the computation of the derivatives. The domain is formed by a layered body with PZT‐4 and PZT‐5 stacked together. For result verification, the numerical solution obtained with ZFEM was compared to results from a commercial FEM package and the solution from the imaginary part was compared to the exact solution of a well‐known benchmark problem. Comparison of the results showed good agreement for both the real and imaginary parts of the solution and the largest sensitivities were found in PZT‐5 specifically in C13, C33, and ε33. [ABSTRACT FROM AUTHOR]

• Robust Complex-Valued Levenberg-Marquardt algorithm. • Bi-quadratic convergence property. • Rectangular coordinates and quadratic mismatch functions. • Easier computer implementation. This paper deals with a robust complex-valued Levenberg-Marquardt algorithm specially developed for solving ill-conditioned power flow problems. Moreover, it can also be a useful tool for voltage instability and voltage collapse studies. Because power flow models are nonlinear, the Wirtinger calculus is applied to develop iterative algorithms based on Taylor series expansions of nonlinear functions of complex variables and their complex conjugates. Our proposal in complex plane is straightforward derived in rectangular coordinates. Consequently, its performance is compared to the well-known optimized multiplier based load flow method. Aiming this purpose, we show that few changes in the codes are required to transform the complex-valued Newton-Raphson power flow algorithm into the complex-valued Levenberg-Marquardt power flow one. Furthermore, we show that the latter lends itself well to modeling new smart grid technologies while exhibiting a bi-quadratic convergence rate and superior performance as compared to the former procedure. The performance of our proposal is demonstrated and analyzed on well-conditioned IEEE-14, −30, −57 and −118 bus systems and the Brazilian Southern-equivalent system termed SIN-1916 bus. Furthermore, its performance is also demonstrated on the ill-conditioned IEEE-11, −43 bus systems besides the SIN-1916 under stressed operating conditions or higher R / X ratios of transmission lines. [ABSTRACT FROM AUTHOR]

The three-body potential for the ground state of the helium trimer is determined by an extended geminal model. The basis set for the calculation is an uncontracted (19s,7p,6d,5f,4g,2h) set of Gaussian-type functions. Three different types of configurations were considered: (i) equilateral triangles, (ii) linear configurations with R12=R23, and (iii) a set of pseudorandom configurations. The interatomic distances were selected within the interval [3.0,9.0] bohrs. The computed points have been fitted to global potential functions. The fit is characterized by a maximum absolute error equal to 0.69 μEh and a mean error equal to -0.018 μEh. [ABSTRACT FROM AUTHOR]

This paper presents new methodologies and modeling aimed at studying interwoven AC and DC subsystems of a power system in the complex plane. It is well known that the development of any former power flow application in real domain addressed to embedding FACTS devices is preceded by an arduous algebra task. To overcome this difficulty, a novel power flow solution method is proposed in this work. Thanks to the Wirtinger calculus, a Newton–Raphson method based on Taylor series expansions of nonlinear functions of complex variables and their complex conjugates is developed. Thus, the voltage-source-converter multi-terminal direct current is formulated in complex plane without any loss of accurateness and is termed as VSC-MTDC. Its performance is assessed through IEEE-test systems interconnected across a DC network prone to several scenarios, e.g., topology, loading magnitude and interchanging of active power. • Complex valued data processing is no longer computational restricted. • Flexible and direct VSC modeling in the complex plane. • Power flow analysis of VSC MTDC system using NR method in the complex plane. • Modulation of new technologies for power flow analysis with complex values. • Sequential hybrid AC/DC power flow algorithms. [ABSTRACT FROM AUTHOR]

The complex-valued finite element method, ZFEM, is proposed as a new virtual crack extension method to compute the energy release rate. The energy release rate is computed as a numerical derivative of the strain energy with respect to a crack extension using the complex Taylor series expansion method (CTSE). This is accomplished using a finite element method with complex nodal coordinates and extending the crack by a very small quantity along the imaginary directions of the complex nodal coordinates. The resulting finite element formulation becomes complex valued with the imaginary component of the strain energy containing the energy release rate. This method retains the conceptual simplicity of numerical differentiation but eliminates numerical issues regarding perturbation of the crack size. Both two- and three-dimensional examples are provided. Numerical examples indicate that the energy release rate computed using ZFEM is of the same accuracy as the domain-based J integral formulation. The method is very general and can be applied to self-similar or non-self-similar crack extensions. [ABSTRACT FROM AUTHOR]

Abstract: We consider β-expansions of formal Laurent series over finite fields. If the base β is a Pisot or Salem series, we prove that the β-expansion of a Laurent series α is automatic if and only if α is algebraic. [Copyright &y& Elsevier]

In the present investigation, by taking ϕ(z) as an analytic function, sharp upper bounds of the Fekete-Szegö functional ∣a3 - μa22 ∣ for functions belonging to the class Mαg,h(ϕ) are obtained. A few applications of our main result are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, authors introduce a new class R(β,α,n) of Salageantype analytic functions. We obtain extreme points of R(β,α,n) and some sharp bounds for Re{Dnf(z)/z} and Re{Dn-1f(z)/z}. Relevant connections of the results presented here with various known results are briefly indicated. [ABSTRACT FROM AUTHOR]

In this paper we consider a new integral operator I (f1, ..., fn; g1, ..., gn) (z) for analytic functions fi(z), gi(z) in the open unit disk U. The main object of the present paper is to study the order of convexity for this integral operator. [ABSTRACT FROM AUTHOR]

The article discusses the best linear methods and width values for some analytic function classes in the Bergman weight space. It presents results associated to the exact value calculation in analytical function classes on unit disk and the best linear method construction. It presents the interpretation of the results of the problems provided as assertions on the functions' coding and optimal reconstruction. It also explains the calculation of Taylor coefficient absolute values' least upper bounds.

Abstract: The sensitivity of the computed cycles-to-failure and other lifing estimates to the various input parameters is a valuable, yet largely unexploited, aspect of a fatigue lifing analysis. Two complex variable sensitivity methods, complex Taylor series expansion (CTSE) and Fourier differentiation (FD), are adapted and applied to fatigue analysis through the development of a complex variable fatigue analysis software (CVGROW). The software computes the cycles-to-failure and the sensitivities of the computed cycles-to-failure with respect to parameters of interest such as the initial crack size, material properties, geometry, and loading. The complex variable methods are shown to have advantages over traditional numerical differentiation in that more accurate and stable first and second order derivatives are obtained using CTSE and more accurate and stable higher order derivatives are obtained using FD. [Copyright &y& Elsevier]

We consider Ruscheweyh derivative Dnf(z) and Sălăgean derivative dnf(z), for n ∈ {0, 1, 2, …}, on a class [Multiple line equation(s) cannot be represented in ASCII text] is analytic in ιzι < 1}. On this paper we study relations between two subclasses T R(n,m;α) and [Multiple line equation(s) cannot be represented in ASCII text] [ABSTRACT FROM AUTHOR]

In the present paper, by means of Carlson-Shaffer operator and a multiplier transformation, we consider a new class of analytic functions BL(m, l, a, c, μ,α, λ). A sufficient condition for functions to be in this class and the angular estimates are provided. [ABSTRACT FROM AUTHOR]

Let A be the class of all analytic functions in the open unit disc D = {zι ιzι < 1} of the form f(z) = z + a2z2 + a3z3 + …. Let g(z) be an element of A satisfying the condition [Multiple line equation(s) cannot be represented in ASCII text] where ιαι [Multiple line equation(s) cannot be represented in ASCII text] is analytic in D and satisfies the conditions ɸ(0) = 0, ιɸ(z)ι < 1 for every z ∈ D. Then g(z) is called Janowski αspirallike functions in the unit disc. The class of such functions is denoted by S*α(A,B). The aim of this paper is to give growth and distortion theorems for the class S*α(A,B). [ABSTRACT FROM AUTHOR]

Abstract: Weight functions are a critical component of a damage tolerance fracture control plan in that they allow the stress intensity factor to be computed quickly from the stress along the uncracked crack line. The traditional method to compute weight functions is to use several (2–4) reference stress solutions or auxiliary conditions to develop the coefficients in a series solution. While this method has been shown to provide good results in many scenarios, the truncated series provides a source of error that is difficult to quantify and the method requires multiple high-quality reference solutions or other auxiliary information. In contrast, the WCTSE method presented here, provides a method to accurately and efficiently develop the weight function for an arbitrary geometry and loading scenario from a single complex variable finite element solution without other reference solutions or auxiliary information. The complex Taylor series expansion method is used within the finite element formulation to obtain the derivatives of the crack opening displacements with respect to crack length directly from the finite element analysis. These derivatives allow the direct evaluation of the weight function. The method requires a small perturbation of the crack length along the imaginary axis; the real coordinate mesh is unaltered. Since the real coordinate mesh is unaltered, standard finite element meshes and meshing algorithms can be used. Given that the error in the weight function is controlled by the accuracy of the mesh, typical convergence tests can be used to obtain high confidence in the weight functions. Several numerical examples are computed and compared to other well known published weight function solutions or finite element (J integral) or boundary element solutions. [Copyright &y& Elsevier]

In this paper, we introduce a new class of analytic functions in the unit disk defined by a generalized linear operator referring to Cătaş and Lupasş. We study the sufficient conditions for a function f to belong to this new considered class. [ABSTRACT FROM AUTHOR]

In this paper integral properties of a certain class of analytic functions with negative coefficients are studied. This class is defined using a generalized Sal agean operator. The obtained results are sharp and they improve known results. [ABSTRACT FROM AUTHOR]

Abstract: Shape sensitivity analysis of finite element models is useful for structural optimization and design modifications. Complex variable methods for shape sensitivity analysis have some potential advantages over other methods. In particular, for first order sensitivities using the complex Taylor series expansion method (CTSE), the implementation is straightforward, only requiring a perturbation of the finite element mesh along the imaginary axis. That is, the real valued coordinates of the mesh are unaltered and no other modifications to the software are required. Fourier differentiation (FD) provides higher order sensitivities by conducting an FFT analysis of multiple complex variable analyses around a sampling radius in the complex plane. Implementation of complex variable sensitivity methods requires complex variable finite element software such that complex nodal coordinates can be used to implement a perturbation in the shape of interest in the complex domain. All resulting finite element outputs such as displacements, strains and stresses become complex and accurate derivatives of all finite element outputs with respect to the shape parameter of interest are available. The methodologies are demonstrated using two-dimensional finite element models of linear elasticity problems with known analytical solutions. It is found that the error in the sensitivities is primarily defined by the error in the finite element solution not the error in the sensitivity method. Hence, more accurate sensitivities can be obtained through mesh refinement. [Copyright &y& Elsevier]

The additive Schwarz method with harmonic extension (ASH) was introduced by Cai and Sarkis (1999) as an efficient variant of the additive Schwarz method that converges faster and requires less communication. We show that ASH can also be used with optimized transmission conditions to obtain faster convergence. We show that when the decomposition into subdomains contains no cross points, optimized ASH can be reformulated as an iteration that is closely related to the optimized Schwarz method at the continuous level. In fact, the iterates of ASH are identical to the iterates of the discretized parallel Schwarz method outside the overlap, whereas inside the overlap they are linear Combinations of previous Schwarz iterates. Thus, one method converges if and only if the other one does, and they do so at the same asymptotic rate, unlike additive Schwarz, which fails to converge inside the overlap. However, when cross points are present, then ASH and the Schwarz methods are incomparable, i.e.. there are cases where one method converges and the other diverges, and vice versa. [ABSTRACT FROM AUTHOR]

In this paper we obtain sufficient conditions for the analyticity and the univalence of the functions defined by an integral operator. [ABSTRACT FROM AUTHOR]

In this paper, we drive several interesting subordination results of certain classes of analytic functions defined by convolution with complex order. [ABSTRACT FROM AUTHOR]

Let A denote the class of analytic functions f(z) defined in the unit disc U = {z : ∣z∣ < 1} and satisfying the conditions f(0) = f′(0) - 1 = 0. We introduce some new subclasses of strongly starlike functions defined by the Sălăgean integral operator and study their properties. [ABSTRACT FROM AUTHOR]

In this paper, we generalize certain integral operators given by Pescar [8] and determine conditions for univalence of these general integral operators. [ABSTRACT FROM AUTHOR]

new method of establishing the completeness of systems of analytic functions in the space A( D) is considered. We indicate some applications of the results obtained to the case of the principle of doubly symmetric Kaz'min sets, to the Abel-Goncharov problem (the uniqueness and construction problem), and to some other cases. [ABSTRACT FROM AUTHOR]

Wiman-Valiron theory and results of Macintyre about 'flat regions' describe the asymptotic behaviour of entire functions in certain discs around points of maximum modulus. We estimate the size of these discs for Macintyre's theory from above and below. [ABSTRACT FROM AUTHOR]

A four-phase confocal elliptical cylinder model is proposed from which a generalised self-consistent method is developed for predicting the thermal conductivity of coated fibre reinforced composites. The method can account for the influence of the fibre section shape ratio on conductivity, and the physical reasonableness of the model is demonstrated by using the fibre distribution function. An exact solution is obtained for thermal conductivity by applying conformal mapping and Laurent series expansion techniques of the analytic function. The solution to the three-phase confocal elliptical model, which simulates composites with idealised fibre-matrix interfaces, is arrived at as the degenerated case. A comparison with other available micromechanics methods, Hashin and Shtrikman's bounds and experimental data shows that the present method provides convergent and reasonable results for a full range of variations in fibre section shapes and for a complete spectrum of the fibre volume fraction. Numerical results show the dependence of the effective conductivities of composites on the aspect ratio of coated fibres and demonstrate that a coating is effective in enhancing the thermal transport property of a composite. The present solutions are helpful to analysis and design of composites. [ABSTRACT FROM AUTHOR]

recursive operator method was used to obtain a general solution of the system of linearized differential equations of heat and mass transfer in hardening concrete in a heating chamber. The solution contains arbitrary analytic functions determined from the boundary and initial conditions. [ABSTRACT FROM AUTHOR]

We consider locally defined operators of the form Dn ○ K where D is the operator of differentiation and K maps the space of continuous functions into the space of n-times differentiable functions. As a corollary we obtain a characterization of the Volterra operator. Locally defined operators acting in the space of analytic functions are also discussed. [ABSTRACT FROM AUTHOR]

For a positive integer n, applying Schwarz's lemma related to analytic functions w(z) = cnzn + · · · in the open unit disk ..., some assertion in a certain lemma which is well-known as Jack's lemma proven by Miller and Mocanu [J. Math. Anal. Appl. 65 (1978), 289-305] is given. Further, by using a certain method of the proof of subordination relation which was discussed by Suffridge [Duke Math. J. 37 (1970), 775-777] and MacGregor [J. London Math. Soc. (2) 9 (1975), 530-536], some differential subordination property concerning with the subordination p(z) ... q(zn) (z ∈ ...) for functions p(z) = a+anzn + · · · and q(z) = a+bnz + · · · which are analytic in ... is deduced, and an extension of some subordination relation is given. [ABSTRACT FROM AUTHOR]

We develop a power series representation and estimates for an effective action of the form: ln[∫ef(α1,...,αs;z*,z)dμ(z*,z)/∫ef(0,...,0;z*,z)dμ(z*,z)]. Here, f(α1,...,αs;z*,z) is an analytic function of the complex fields α1(x),...,αs(x), z*(x), z(x) indexed by x in a finite set X, and dμ(z*,z) is a compactly supported product measure. Such effective actions occur in the small field region for a renormalization group analysis. Using methods similar to a polymer expansion, we estimate the power series of the effective action. [ABSTRACT FROM AUTHOR]

In a previous paper, we developed a power series representation and estimates for an effective action of the form ln[∫ef(α1,...,αs;z*,z)dμ(z*,z)/∫ef(0,...,0;z*,z)dμ(z*,z)]. Here, f(α1,...,αs;z*,z) is an analytic function of the complex fields α1(x),...,αs(x),z*(x),z(x) indexed by x in a finite set X and dμ(z*,z) is a compactly supported product measure. Such effective actions occur in the small field region for a renormalization group analysis. We illustrate the technique by a model renormalization group flow motivated by the ultraviolet regime in many boson systems. [ABSTRACT FROM AUTHOR]

For functions belonging to the class Sm (δ, α), δ ϵ [0, 1), α ≥ 0 and in ϵ N, of normalized analytic functions in the open unit disc U, which are 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 Ip (m, λ, l) ƒ(z) namely >>>>>Ip (m, λ, l) ƒ(z) ≔ zp + Σj=p+n∞ (p + λ(j - 1) + l/p + 1)m ajzj, where p, n ϵ N, m ϵ NU {0}, λ, l ≥ 0 and ƒ ϵ A(p,n), A (p,n) = {ƒ ϵ H(U) : ƒ(z) = zp + Σj=p+n∞ ajzj, z ϵ U). 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. [ABSTRACT FROM AUTHOR]

It is shown that, for analytic functions f, systems of the form $\ddot {z}=f(z,\dot {z})$ and $\ddot {z}=f(z)$ cannot produce chaos; and that systems of the form $\ddot {z}=f(z^\ast , \dot {z}^\ast )$ and $\ddot {z}=f(z,z^\ast )$ are conservative. Eight simple chaotic systems of the form $\ddot {z}=f(z, z^\ast )$ with quadratic and cubic polynomial f(z, z*) are given. Lyapunov spectra are calculated, and the systems' phase space trajectories are displayed. For each system, a Hamiltonian is given, if one exists. [ABSTRACT FROM AUTHOR]

Let A be the class of analytic functions in the unit disk that are normalized with f(0) = f′(0) - 1 = 0: In this paper we study the class Gλ = { f ϵ A : ∣1+z f″(z)/f′(z)/zf′(z)/f(z)=1∣<λ, z ϵ u}, and give sharp sufficient conditions that embed it into the class S*[A, B, b] = {f ϵ A : +1/b (z f′(z)/f(z)-1) ≺ 1+Az/1+Bz}where -1 ≤ B < A ≤ 1; b ≠ 0 and " ≺ " denotes the usual subordination. [ABSTRACT FROM AUTHOR]

Consider a family of infinite tri-diagonal matrices of the form L + zB, where the matrix L is diagonal with entries L kk = k2, and the matrix B is off-diagonal, with nonzero entries B k, k+1 = B k+1, k = k α, 0 ≤ α < 2. The spectrum of L + zB is discrete. For small | z| the nth eigenvalue E n ( z), E n (0) = n2, is a well-defined analytic function. Let R n be the convergence radius of its Taylor’s series about z = 0. It is proved that . [ABSTRACT FROM AUTHOR]

Considering the class {ie48-01} of analytic functions {ie48-02} in the unit disk with a m,n ε ℝ and the nonvanishing nth divided difference [ F( z); z0, ⋯, z n] for all z0, ℝ, zn ∈ E we establish that {ie48-03}, where {ie48-04}. If n is an odd number then {ie48-05}. [ABSTRACT FROM AUTHOR]

Certain classical results on cluster sets and boundary cluster sets of analytic functions, due to Iversen, Lindelof, Noshiro, Tsuji, Ohtsuka, Pommerenke, Carmona, Cufi and others, are extended to n-dimensional quasiconformal mappings. Unlike what is usually the case in the context of analytic functions, our considerations are not restricted to mappings of a disk or ball only. It is shown, for instance, that quasiconformal cluster sets and boundary cluster sets, taken at a non-isolated boundary point of an arbitrary domain, coincide. More refined versions are established in the special case where the domain is the open unit ball. These include cluster set considerations of the induced radial boundary extension and results where certain exceptional sets on the boundary are allowed for. [ABSTRACT FROM AUTHOR]

Let T be the class of analytic functions with real coefficients in the open unit disk 핌. For f(z) belonging to the class T some sufficient conditions for starlikeness and convexity are discussed. [ABSTRACT FROM AUTHOR]

In the present paper, the structure of solutions of some important classes of singular elliptic systems on the plane are investigated. In particular, it is proved that the solutions of such systems have in principle nonanalytic behavior in the neighborhood of fixed singular points. These results make it possible to state correctly the boundary value problems and give their complete analysis. [ABSTRACT FROM AUTHOR]

This paper deals with boundary value problems of linear conjugation with shift for analytic functions in the case of piecewise continuous coefficients. Int main goal is the construction of a canonical matrix for these problems. Boundary value problems with shift for generalized analytic functions and vectors as well as differential boundary value problems are studied. [ABSTRACT FROM AUTHOR]