Terms Of Complex Variables Research Articles

A reciprocal-space formulation of mixed quantum-classical dynamics.

We derive a formulation of mixed quantum-classical dynamics for modeling electronic carriers interacting with phonons in reciprocal space. For dispersionless phonons, we start by expressing the real-space classical coordinates in terms of complex variables. Taking these variables as a Fourier series then yields the reciprocal-space coordinates. Evaluating the electron-phonon interaction term through Ehrenfest's theorem, we arrive at a reciprocal-space formalism that is equivalent to mean-field mixed quantum-classical dynamics in real space. This equivalence is numerically verified for the Holstein and Peierls models, for which we find the reciprocal-space Hellmann-Feynman forces to involve momentum-derivative contributions in addition to the position-derivative terms commonly seen in real space. To illustrate the advantage of the reciprocal-space formulation, we present a proof of concept for the inexpensive modeling of low-momentum carriers interacting with phonons using a truncated reciprocal-space basis, which is not possible within a real-space formulation.

Evaluation of singular integrals for anisotropic elastic boundary element analysis

To improve the numerical evaluation of weakly singular integrals appearing in the boundary element method, a logarithmic Gaussian quadrature formula is usually suggested in the literature. In this formula the singular function is expressed in terms of the distance between source point and field point, which is a real variable. When an anisotropic elastic solid is considered, most of the existing fundamental solutions are written in terms of complex variables. When the problems with holes, cracks, inclusions, or interfaces are considered, to suit for the shape of the boundaries usually a mapping function is introduced and then the solutions are expressed in terms of mapped complex variables. To deal with the trouble induced by the complex variables, in this study through proper change of variables we develop a simple way to improve the evaluation of weakly singular integrals, especially for the problems of anisotropic elastic solids containing holes, cracks, inclusions, or interfaces. By simple matrix expansion, the proposed method is extended to the problems with piezoelectric or magneto-electro-elastic solids. By using the dual reciprocity method, the proposed method employed for the elastostatic fundamental solution can also be applied to the elastodynamic analysis.

Dynamical analysis of a complex logistic-type map

A complex logistic-type map is considered in the present work. The dynamic behavior of the underlined map is discussed in two different cases: the first case is when the parameter of the map being real, and the second case is when the parameter being complex. Existence and stability of fixed points are derived. The conditions for existence of a pitchfork bifurcation, flip bifurcation and Neimark-Sacker bifurcation are derived by using the center manifold theorem and bifurcation theory. Numerical simulations including Lyapunov exponent, phase plane, bifurcation diagrams is carried out using matlab to ensure theoretical results and to reveal more complex dynamics of the map. The results show that expressing the logistic map in terms of complex variables leads to more distinguished behaviors, which could not be achieved in the logistic map with real variables. In addition, considering the control parameter as a complex parameter shows more interesting dynamics if compared to the case when considering it as a real parameter. Existence of a snapback repeller is proved in the sense of Marotto. Finally, chaos is controlled using the OGY feedback method.

Light-front and conformal field theories in two dimensions

Light-front (LF) quantization of massless fields in two spacetime dimensions is a long-standing and much debated problem. Even though the classical wave-equation is well-documented for almost two centuries, either as problems with initial values in spacetime variables or with initial data on characteristics in light-cone variables, the way to a consistent quantization in both types of frames is still a puzzle in many respects. This is in contrast to the most successful Conformal Field Theoretic (CFT) approach in terms of complex variables [Formula: see text], [Formula: see text] pioneered by Belavin, Polyakov and Zamolodchikov in the ’80s. It is shown here that the 2D-massless canonical quantization in both reference frames is completely consistent provided that quantum fields are treated as Operator-Valued Distributions (OPVD) with Partition of Unity (PU) test functions. We recall first that classical fields have to be considered as distributions (e.g. generalized functions in the Russian literature). Then, a necessary condition on the PU test function follows from the required matching of the classical solutions of the massless differential equations in both types of reference frame. Next we use a mathematical formulation for OPVD, developed in the recent past. Specifically, smooth [Formula: see text] fields are introduced through the convolution operation in the distributional context. Due to the specific behavior of the Fourier-transform of the initial test function, this convolution transform has a well-defined integral in the dual space, whatever the initial choice of the reference frame. The relation to the conformal fields method follows immediately from the transition to Euclidean time and leads directly to explicit calculations of a few correlation functions of the scalar field and its energy–momentum tensor. The LF derivation of the Virasoro algebra is then obtained from the [Formula: see text] and [Formula: see text] expansions of the canonical fields as distributional Laplace-transform in these variables. Finally, the popular and problematic Discretized Light Cone Quantization (DLCQ) method is scrutinized with respect to its zero mode and ultraviolet content as encompassed in the continuum OPVD formulation.

A complex-valued first integral of Navier-Stokes equations: Unsteady Couette flow in a corrugated channel system

For a two-dimensional incompressible viscous flow, a first integral of the governing equations of motion is constructed based on a reformulation of the unsteady Navier-Stokes equations in terms of complex variables and the subsequent introduction of a complex potential field; complementary solid and free surface boundary conditions are formulated. The methodology is used to solve the challenging problem of unsteady Couette flow between two sinusoidally varying corrugated rigid surfaces utilising two modelling approaches to highlight the versatility of the first integral. In the Stokes flow limit, the results obtained in the case of steady flow are found to be in excellent agreement with corresponding investigations in the open literature. Similarly, for unsteady flow, the results are in accord with related investigations, exploring material transfer between trapped eddies and the associated bulk flow, and vice versa. It is shown how the work relates to the classical complex variable method for solving the biharmonic problem and perspectives are provided as to how the first integral may be further utilised to investigate other fluid flow features.

Plane elastostatic stress analysis in complex variables: A wavelet processing perspective

The well-known Kolosov–Muskhelishvili (KM) representation of the Airy function for 2D stress analysis in complex variable terms is enhanced by combining it with Walsh wavelets decomposition. It allows us to perform general analytical derivations up to the maximum extent possible which, in turn, provides a basis for developing a new stress computation algorithm readily incorporated into the routine single scale KM scheme. The mathematical treatment of the wavelet application is supported by a number of examples where non-trivial closed-form solutions are known and serve as a benchmark for numerical simulations. The comparison shows that the proposed framework has better performance than the conventional Fourier transform, especially when it comes to non-smooth stress distributions.

Some Exact Solutions to Equations of Motion of an Incompressible Second Grade Fluid

In this paper, we derive some exact solutions of the equations governing the steady plane motions of an incompressible second grade fluid. For this purpose, the vorticity and stream functions both are expressed in terms of complex variables and complex functions. The derived solutions represent the flows having streamlines as a family of ellipses, parabolas, concentric circles, and rectangular hyperbolas. Some physical features of the derived solutions are also illustrated by their contour plots.

A planar grained structure with a multiphase nested inclusion in a periodic cell: Elastostatic solution and the equi-stressness

By using the specially constructed Kolosov–Muskhelishvili potentials a concise formulation of 2D elastostatic problems for general regular structures with multiphase nested inclusions is obtained in complex-variable terms. Analytical averaging of the stress/strain fields over the representative cell of the structure gives its effective moduli in the perturbation-like form that was known thus far only for less complicated phase arrangements. These derivations are further extended to prove the existence of the equi-stress nested inclusions under the square symmetry of the structure. In sharp contrast to the one-phase (homogeneous) inclusion, they no longer saturate the attainable Hashin–Shtrikman bounds on the effective bulk modulus, but continue to be a subject by themselves.

Rotating frame analysis of rigid body dynamics in space phasor variables

We propose a new method of attacking problems in rigid body rotation, focusing on the heavy symmetric top. The technique is a direct extension of the method traditionally applied to the free symmetric top. We write Euler's equations in a frame which is attached to the top and thus shares its entire angular velocity. The structure of the resulting equations is such that it is advantageous to cast and solve them in terms of complex variables (space phasors). Through this formalism, we obtain a direct link between the initial conditions at the time of launch and the subsequent behavior of the top. The insertion of a damping term allows us to further explain the behavior of a top where the pivot is non-ideal and has friction. Finally, we make some suggestions regarding experimental verification of our results.

MATLAB implementation of the complex power flow

PurposeThe purpose of this work is that of providing the guidelines of an efficient implementation of power flow computations using the MATLAB computation environment.Design/methodology/approachThe goal of obtaining high efficiency from MATLAB programs often proves elusive unless special care is taken in exploiting the vectorising capability of MATLAB programming. In the present paper the implementation of Newton‐Raphson power flow in MATLAB is examined with particular emphasis on the way of obtaining a vectorisable code capable of achieving effective numerical performance by exploiting its formulation in terms of complex variables.FindingsTests on actual networks with up to 1,300 buses are presented. They show that the complex power flow is as efficient as the best implementations of the Newton Raphson power flow using real variables, as long as the operations involved are reordered with the aim of exploiting the vectorisation capabilities of the MATLAB environment.Originality/valueIt is shown that improved numerical efficiency in the MATLAB can be obtained through its formulation in terms of complex variables. The complex Newton‐Raphson load flow, not very common in practical uses, is shown to have many desirable qualities from the point of view of MATLAB programming and is presented in detail.

Effects of Surface Tension and General Rotation on the Rayleigh-Taylor Instability of Three-Layer Flow

The stability of two interfaces separating three fluids, where the fluids are assumed to be incompressible, inviscid, and of constant density, has been investigated for a system that is acted upon by a general rotation. The effect of surface tension at the two interfaces is taken into account. A general dispersion relation for the system is obtained analytically by formulating the problem in terms of complex variables. Numerical calculations were performed for a hexane-NaCl-CCl4 system to investigate stable case, and special cases that isolate the effect of various parameters on the growth rate of the Rayleigh-Taylor instability are discussed. It is found that the two cutoff wave numbers for the system with surface tension are unchanged by the addition of a general rotation, and that for the system considered, all growth rates are reduced in the presence of a general rotation.

Electromechanical fields around a screw dislocation in a cubic piezoelectric crystal

Explicit and closed-form expressions for the electromechanical fields around a screw dislocation in a cubic piezoelectric crystal are presented. Governing equations for anti-plane elastic deformation coupled with in-plane electric field are derived and solutions in terms of complex variables are obtained. The effect of coordinate rotation on the solution is studied. The results are expressed in both rectangular and cylindrical coordinates. Also determined is the electric enthalpy associated with the dislocation. The interaction of two parallel dislocations is discussed with reference to the Peach–Koehler force. A crack problem is solved as an illustration of the application of present solutions.

A generalization of the equi-stress principle in optimizing the mechanical performance of two-dimensional grained composites

A new optimization criterion of minimizing the variations of the boundary tractions in 2D bi-material elastostatic problems is proposed as a relaxation of the well-known equi-stress principle far beyond its primary application. This integral-type assessment of the local stresses offers significant numerical advantages over their direct minimization. In particular, it allows us to obtain reliable results at moderate computational cost through the same flexible scheme as in the author’s previous research on optimizing perforated plates. The scheme combines a genetic algorithm optimization with an efficient direct solver and with an economic shape parametrization, both formulated in the complex variable terms. After an extended analysis of the criterion, its effectiveness is illustrated, as before, by applying to a checkerboard grained plate where the equi-stress inclusions cease to exist at not-too-small volume fractions. The results obtained permit us to make some empirical conclusions which may stimulate further studies in both theoretical and practical directions.

Stress-smoothing holes in an elastic plate: The triangular checkerboard lattice of low symmetry

The shape optimization problem for a linearly elastic perforated plate with the triangular checkerboard of identical holes is considered with respect to the recently advanced criterion of minimizing the hoop stress variation. The paper continues the author’s previous research on optimizing the square-symmetric checkerboard and extends the results obtained to the non-symmetric lattice of the holes which are varied not only in their shapes, but also in their location. This novelty is resolved by enhancing the same three-component algorithm as before. It combines a genetic algorithm (GA) optimization with an efficient direct solver and with an economic shape parametrization, both formulated in the complex variable terms. The averaging nature of the stress variation criterion permits its effective numerical implementation for a wide range of the holes volume fraction at reasonable computational efforts. For an engineering audience, the elastic behaviour of the optimal structures is detailed in a graphics set.

A first integral of Navier–Stokes equations and its applications

Although it is well known that Bernoulli's equation is obtained as the first integral of Euler's equations in the absence of vorticity and that in the case of non-vanishing vorticity a first integral of them can be found using the Clebsch transformation for inviscid flow, generalization of the procedure for viscous flow has remained elusive. Accordingly, in this paper, a first integral of the Navier–Stokes equations for steady flow is constructed. In the case of a two-dimensional flow, this is made possible by formulating the governing equations in terms of complex variables and introducing a new scalar potential. Associated boundary conditions are considered, and an extension of the theory to three dimensions is proposed. The capabilities of the new approach are demonstrated by calculating a Reynolds number correction to the laminar shear flow generated in the narrow gap between a flat moving and a stationary wavy wall, as is often encountered in lubrication problems. It highlights the first integral as a suitable tool for the development of new analytical and numerical methods in fluid dynamics.

Multiple interacting circular nano-inhomogeneities with surface/interface effects

A two-dimensional problem of multiple interacting circular nano-inhomogeneities or/and nano-pores is considered. The analysis is based on the Gurtin and Murdoch model [Gurtin, M.E., Murdoch, A.I., 1975. A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323.] in which the interfaces between the nano-inhomogeneities and the matrix are regarded as material surfaces that possess their own mechanical properties and surface tension. The precise component forms of Gurtin and Murdoch's three-dimensional equations are derived for interfaces of arbitrary shape to provide a basis for critical review of various modifications used in the literature. The two-dimensional specification of these equations is considered and their representation in terms of complex variables is provided. A semi-analytical method is proposed to solve the problem. Solutions to several example problems are presented to: (i) examine the difference between the results obtained with the original and modified Gurtin and Murdoch's equations, (ii) compare the results obtained using Gurtin and Murdoch's model and those for a problem of nano-inhomogeneities with thin membrane-type interphase layers, and (iii) demonstrate the effectiveness of the approach in solving problems with multiple nano-inhomogeneities.

ACTIVE CONTROL AND GLOBAL SYNCHRONIZATION OF THE COMPLEX CHEN AND LÜ SYSTEMS

Chaos synchronization is a very important nonlinear phenomenon, which has been studied to date extensively on dynamical systems described by real variables. There also exist, however, interesting cases of dynamical systems, where the main variables participating in the dynamics are complex, for example, when amplitudes of electromagnetic fields are involved. Another example is when chaos synchronization is used for communications, where doubling the number of variables may be used to increase the content and security of the transmitted information. It is also well-known that similar generalization of the Lorenz system to one with complex ODEs has been introduced to describe and simulate the physics of a detuned laser and thermal convection of liquid flows. In this paper, we study chaos synchronization by applying active control and Lyapunov function analysis to two such systems introduced by Chen and Lü. First we show that, written in terms of complex variables, these systems can have chaotic dynamics and exhibit strange attractors. We calculate numerically the values of the parameters at which these attractors exist. Active control and global synchronization techniques are then applied to study the phenomenon of chaos synchronization. Analytical criteria concerning the stability of these techniques are implemented and excellent agreement is found upon comparison with numerical experiments. In particular, studying the time evolution of "errors" (or differences between drive and control dynamics), we show that both techniques are very effective for controlling the behavior of these systems, even in regimes of very strong chaos.

Analytic Functions, Ideal Fluid Flow, and Bernoulli’s Equation

It is shown that Bernoulli’s equation is an immediate consequence of describing the kinematical and dynamical equations for the steady two-dimensional flow of an ideal incompressible, irrotational fluid in integral form in terms of complex variables and then evoking the theorems of Morera and Cauchy–Goursat. Thus, the traditional derivation of Bernoulli’s equation via differential equations is bypassed and the connection between fluid flow and complex variables demonstrated at a very basic level.

Low-speed two-dimensional free-surface flow past a body

This paper defines the linearized problem of free-surface flow past a shallow two-dimensional body in terms of complex variables and provides an asymptotic solution for the Froude number tending to zero. The simple case of flow past a rectangular box with a triangular extension in front is worked out as an example. The low-Froude-number solution for ‘double-body’ flow in a fully infinite fluid past the same box with its reflection in the upper half-plane is also calculated. The upstream free-surface height for the linear solution is compared to the equivalent free-surface height for the double-body flow. Experimental results for this body are also presented and compared.

A note on the Darwin-Prins rocking curve for perfect crystals

For the purpose of calculating the reflectivity of perfect crystals in the Bragg case, taking into account both normal and Borrmann absorption, it is suggested that its original form is used expressed in terms of complex variables. Care is required regarding the phase angle for the algebraic expression of the square root. Two expressions which are directly usable for programming are presented. One of them is equivalent to the traditional one given by Zachariasen [Theory of X-ray Diffraction in Crystals (1945). New York: Wiley]. The other can be used without any problems of infinite values whatever values the Fourier coefficients of the polarizability takes. Some numerical examples of rocking curves are demonstrated for very special values of the Fourier coefficients.