Complex Variable Theory Research Articles

On elementary complex variable theory applied to sinc and related integrals**

An elementary application of complex variable theory demonstrates that some properties of sinc and related integrals, which have been variously described as “tricky,” “intriguing” and “deluding intuition,” arise as simple consequences of Jordan’s lemma. This approach, in offering an interesting alternative to the established treatment based upon Fourier transform theory, may thus be recommended in addition for its pedagogic value. The present treatment extends the earlier results significantly in applications to both divergent and convergent sequences, by introducing the useful concepts of “turning points” and “ranks” of the various contributions to summations over complex path integrals. Finally, a result stated as a student problem in the well-known treatise of Whittaker and Watson (1927) is similarly extended in its application to sequences of both types.

Analytical solution of displacement and stress induced by the sequential excavation of noncircular tunnels in viscoelastic rock

The sequential excavation of a tunnel cross-section is usually employed in large scale tunnel construction to reduce the excavation impact on surrounding rock. Time-dependent sequential excavation combined with rock rheology makes the mechanical response of the surrounding rock a complex function of time, and it is possibly closely related to the excavation path. This study presents new analytical solutions that account for an arbitrary sequential excavation of the tunnel's cross-sections, the ground's viscoelastic properties, arbitrary tunnel shapes and pressures exerted at the internal tunnel boundaries due to liners or water pressures. The time-dependent analytical solutions for the stresses and displacements in each excavation step are derived by complex variable theory, conformal mapping and the Laplace transform technique. The solutions agree very well with the FEM numerical results for models that are completely consistent, and they are qualitatively consistent with the field data. In addition, the mechanical mechanism of the time-dependency of the displacement is analyzed in detail for two types of viscoelastic models. Then parametric analyses are conducted for the sequential excavation of a rectangular tunnel, a semi-arched tunnel with a vertical wall and a horseshoe tunnel to investigate the influence of the excavation sequences, excavation time and the supporting pressures on the ground displacements and stresses. The proposed analytical solutions can help reveal the particular mechanical mechanism of the time-dependent ground responses due to sequential excavations combined with rock rheology, as well as provide an alternative method in the preliminary designs of future tunnels.

Analytical solutions for the displacement and stress of lined circular tunnel subjected to surcharge loadings in semi-infinite ground

For tunnels located at shallow depths, the adjacent surcharge loadings caused by new building constructions or overloading on the ground surface significantly influence the stability of the tunnel and lining. This study presents a new rigorous analytical solution for the stress and displacement around lined circular tunnels at a shallow depth, fully considering the interaction between the ground and rigid lining, and any distributed surcharge loadings.A new analytical approach is provided, where the original problem is divided into two problems: Problem 1 is a semi-infinite plane subjected to surcharge loads without any holes, and Problem 2 is a tunnel contained in a semi-infinite plane loaded by the prescribed displacements along the tunnel boundary. The solutions, which strictly satisfy all the boundary and compatibility conditions of the original problem, are finally addressed by the superposition of the solutions of Problems 1 and 2 respectively obtained by the modified Flamant's solution and complex variable theory. The good agreement between the analytical and numerical results under the same assumptions verifies the correctness of the proposed analytical solutions. Furthermore, the possible deviation due to the simplification of the analytical solution is also discussed.Based on the obtained analytical solution, a parametric study is carried out to investigate the influence of the key parameters on the stress and displacement of the ground and lining.

Mathematical Tools for the Internet of Things Analysis

An overview of recent publications on the use of mathematical methods and models for the analysis of the Internet of Things is given. It is shown that IoT modeling uses such sections of mathematics as game theory, probability theory, theory of random processes, Boolean and matrix algebra, graph theory, number theory, complex variable theory, measure theory, optimization theory, simulation modeling, cluster analysis, and numerical and mathematical analysis.

Effect of a Boundary Layer on Cavity Flow

Cavity flow past an obstacle in the presence of an inflow vorticity is considered. The proposed approach to the solution of the problem is based on replacing the continuous vorticity with its discrete form in which the vorticity is concentrated along vortex lines coinciding with the streamlines. The flow between the streamlines is vortex free. The problem is to determine the shape of the streamlines and cavity boundary. The pressure on the cavity boundary is constant and equal to the vapour pressure of the liquid. The pressure is continuous across the streamlines. The theory of complex variables is used to determine the flow in the following subregions coupled via their boundary conditions: a flow in channels with curved walls, a cavity flow in a jet and an infinite flow along a curved wall. The numerical approach is based on the method of successive approximations. The numerical procedure is verified considering a body with a sharp edge, for which the point of cavity detachment is fixed. For smooth bodies, the cavity detachment is determined based on Brillouin’s criterion. It is found that the inflow vorticity delays the cavity detachment and reduces the cavity length. The results obtained are compared with experimental data.

Exact Solutions for Interaction of Parallel Screw Dislocations with a Wedge Crack in One-Dimensional Hexagonal Quasicrystal with Piezoelectric Effects

The purpose of this paper is to consider the interaction between many parallel dislocations and a wedge-shaped crack and their collective response to the external applied generalized stress in one-dimensional hexagonal piezoelectric quasicrystal, employing the complex variable function theory and the conformal transformation method; the problem for the crack is reduced to the solution of singular integral equations, which can be further reduced to solving Riemann–Hilbert boundary value problems. The analytical solutions of the generalized stress field are obtained. The dislocations are subjected to the phonon field line force, phason field line force, and line charge at the core. The positions of the dislocations are arbitrary, but the dislocation distribution is additive. The dislocation is not only subjected to the external stress and the internal stress generated by the crack, but also to the force exerted on it by other dislocations. The closed-form solutions are obtained for field intensity factors and the image force on a screw dislocation in the presence of a wedge-shaped crack and a collection of other dislocations. Numerical examples are provided to show the effects of wedge angle, dislocation position, dislocation distribution containing symmetric configurations and dislocation quantities on the field intensity factors, energy release rate, and image force acting on the dislocation. The principal new physical results obtained here are (1) the phonon stress, phason stress, and electric displacement singularity occur at the crack tip and dislocations cores, (2) the increasing number of dislocations always accelerates the crack propagation, (3) the effect of wedge angle on crack propagation is related to the distribution of dislocations, and (4) the results of the image force on the dislocation indicate that the dislocations can either be attracted or rejected and reach stable positions eventually.

Perturbations of Christoffel–Darboux Kernels: Detection of Outliers

Two central objects in constructive approximation, the Christoffel–Darboux kernel and the Christoffel function, encode ample information about the associated moment data and ultimately about the possible generating measures. We develop a multivariate theory of the Christoffel–Darboux kernel in $$\mathbb {C}^d$$ , with emphasis on the perturbation of Christoffel functions and their level sets with respect to perturbations of small norm or low rank. The statistical notion of leverage score provides a quantitative criterion for the detection of outliers in large data. Using the refined theory of Bergman orthogonal polynomials, we illustrate the main results, including some numerical simulations, in the case of finite atomic perturbations of area measure of a 2D region. Methods of function theory of a complex variable and (pluri) potential theory are widely used in the derivation of our perturbation formulas.

Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces

Abstract After the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂ ¯ {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are L p ⁢ ( ℝ n ) {L^{p}({\mathbb{R}^{n}})} bounded for 1 < p < ∞ {1<p<\infty} , but only bounded on local Hardy spaces h p ⁢ ( ℝ n ) {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for 0 < p ≤ 1 {0<p\leq 1} . Though much work has been done on the L p ⁢ ( ℝ n 1 × ℝ n 2 ) {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 1 < p < ∞ {1<p<\infty} and Hardy H p ⁢ ( ℝ n 1 × ℝ n 2 ) {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for 0 < p ≤ 1 {0<p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of 0 < p ≤ 1 {0<p\leq 1} . The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces h p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for 0 < p ≤ 1 {0<p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

A p–y curve model for laterally loaded XCC pile in soft clay

This paper presents a p–y curve model for laterally loaded X-sectional cast-in-place concrete (XCC) piles, which are a type of non-circular cross-sectional pile, in soft clay. The recently developed empirical solutions for the elastic stiffness and the ultimate lateral resistance of the two-dimensional laterally loaded X-shaped cross-sectional pile segment–soil system by the authors using the complex variable elasticity theory and finite element limit analysis are used as the two key parameters in constructing the p–y curve. An elastic–perfectly plastic p–y curve is immediately obtained with the linear elastic state described by the elastic stiffness of the two-dimensional system, and the perfectly plastic state is controlled by the ultimate lateral resistance. Furthermore, to consider the nonlinear behaviour, a more robust model using the hyperbolic function is selected for producing the p–y curve with the elastic stiffness and the ultimate lateral resistance incorporated. Subsequently, the p–y model is introduced to the Winkler model to generate solutions for the response of laterally loaded XCC piles. The proposed model considers the non-circular cross-sectional effect of the XCC pile and can also be regarded as a simple method for constructing the p–y curve for other non-circular cross-sectional piles.

Slice regular functions of several octonionic variables

A slice theory of several octonionic variables is introduced in the article as a generalization of the holomorphic theory of several complex variables. Our new trick is to focus our attentions on some subset of with the same complex structures. In our setting, the Bochner‐Martinelli formula and the Hartogs extension theorems are established for slice functions of several octonionic variables.

The Tangential $$\varvec{k}$$-Cauchy–Fueter Operator and $$\varvec{k}$$-CF Functions Over the Heisenberg Group

In this paper, we investigate quaternionic analysis on the $$(4n+1)$$-dimensional Heisenberg group. The tangential k-Cauchy–Fueter operator and k-CF functions are counterparts of the tangential Cauchy–Riemann operator and CR functions on the Heisenberg group in the theory of several complex variables, respectively. We give the Penrose integral formula for k-CF functions and establish the Bochner–Martinelli formula for the tangential k-Cauchy–Fueter operator. We also construct the tangential k-Cauchy–Fueter complex.

Numerical determination of coefficients of multi-parameter asymptotic expansion of the crack-tip stress field using FEM

In this study coefficients of the multi-parameter Williams series expansion for the stress field in the vicinity of the central crack in the rectangular plate and are obtained by the use of finite element analysis. The higher-order terms in the Williams asymptotic expansion are retained. It allows us to give more accurate estimation of stress, strain and displacement fields and to extend the domain of validity for the Williams series expansion. The higher-order coefficients are extracted from finite element method calculations implemented in multi-purpose software package Simulia Abaqus. The plate with a small central crack has been considered either. This kind of the cracked specimen has been utilized for comparison of coefficients of the Williams series expansion obtained from finite element analysis with the coefficients known from the theoretical solution based on the complex variable theory in plane elasticity. It is shown that the coefficients of the Williams series expansion match with good accuracy.

An Analytical Solution for the Frost Heaving Force considering the Freeze‐Thaw Damage and Transversely Isotropic Characteristics of the Surrounding Rock in Cold‐Region Tunnels

Frost damage is a frequent occurrence in cold regions and can threaten the normal use and structural stability of tunnel engineering projects. To accurately determine the frost heaving force and effectively evaluate the frost damage in cold‐region tunnels, an analytical solution for the frost heaving force considering the freeze‐thaw (F‐T) damage and transversely isotropic characteristics of surrounding rock is presented based on complex variable theory and the power series method. The calculation results indicate that the frost heaving force acts on the lining considering that the transversely isotropic characteristics of surrounding rock are significantly greater than those when assuming the surrounding rock is homogeneous isotropic media. This demonstrates that the transversely isotropic characteristics of surrounding rock have a considerable impact on the frost heaving force and should be considered. The frost heaving force continuously increases as the bedding angle increases from 0° to 90°, and the maximum frost heaving force in the Guanjiao tunnel (the rock mass bedding angle is 30°) of the Xining‐Geermu Railway in China is approximately 1.04 MPa. In addition, the influence of F‐T cycles on the frost heaving force in cold‐region tunnels is investigated based on the analytical solution of the frost heaving force proposed in this paper. The frost heaving force acting on the lining decreases with an increasing number of F‐T cycles due to the deterioration of the mechanical parameters of the surrounding rock.

Experimental evaluation of coefficients of multi-parameter asymptotic expansion of the crack-tip stress field using digital photoelasticity

In this study coefficients of the multi-parameter Williams series expansion for the stress field in the vicinity of the central crack in the rectangular plate are obtained by the use of the digital photoelasticity method and finite element analysis. The higher-order terms in the Williams asymptotic expansion are retained. It allows us to give more accurate estimation of stress, strain and displacement fields and to extend the domain of validity for the Williams series expansion. The program is specially developed for the interpretation and processing of experimental data from the phototelasticity experiments. The developed tool allows us to find points that belong to isochromatic fringes with minimal light intensity. The digital image processing with the aid of the developed tool is performed. The points determined with the adopted tool are used further for the calculations of stress intensity factor, T-stresses and coefficients of higher-order terms in the Williams series expansion. Numerical simulations of the same cracked specimen as in the experimental photoelasticity method are performed. The higher-order coefficients are extracted from finite element method calculations implemented in Simulia Abaqus software package and the outcomes are compared to experimental values. The plate with a small central crack has been considered either. This kind of the cracked specimen has been utilized for comparison of coefficients of the Williams series expansion obtained from finite element analysis with the coefficients known from the theoretical solution based on the complex variable theory in plane elasticity. It is shown that the coefficients of the Williams series expansion match with good accuracy.

Математические модели насыщенной асинхронной машины в полярных координатах

This study is aimed at filling a certain gap in the mathematical modeling of steady-state and transient processes in induction machines. A set of mathematical models written in polar coordinates that take into account the machine main magnetic circuit saturation is proposed. For solving the formulated problems, methods of the complex variable theory were used. The proposed mathematical models were implemented by means of software, and the processes simulated using them were investigated in the MATLAB computation environment (using the SIMULINKpackage of application computer programs). Two versions of induction machine mathematical models written in polar coordinates are presented, which take into account saturation of the main magnetic circuit and differ from each other in the set of machine state variables. The specific features of implementing such models by means of software are determined. It has been found that these models describe the processes in an induction machine in the working part of its mechanical characteristic with sufficient accuracy. It is shown that the mathematical models written in polar coordinates simulate the processes in an induction machine on the whole with the same accuracy as the widely used similar models written in the Cartesian coordinates. At the same time, by using the models written in polar coordinates, it is possible to observe induction machine state variables that cannot be observed in the case of using the mathematical models written in the Cartesian coordinates.

Analytical study on interaction between existing and new tunnels parallel excavated in semi-infinite viscoelastic ground

Abstract A new tunnel is usually excavated in close proximity to existing ones, which leads to significant negative influence on the existing tunnels. Moreover, the rheology of the ground probably induces quite different time-dependent ground deformation if the new tunnel is excavated at different time. This study focuses on the interaction between the existing and the new tunnels which excavated in rheological ground at a shallow depth. Through the strict derivation, a new analytical solution is proposed for ground stress and displacement induced by the interaction of new and existing tunnels. The ground rheology, excavation delay of the new tunnel, tunnel size and various tunnel arrangement, are all taken into account. The complex variable theory combined with the extension of corresponding principle are employed in the derivation. By deriving the potentials in complex variable theory for the problems in all the excavation stages, the time-dependent stresses and displacements are finally addressed for the whole excavation process, where the ground is simulated by any linear viscoelastic models. To verify and validate the analytical solutions, the analytical solution is compared with numerical results under simplified and complex ground conditions, which shows good consistency except the solution for Case 1 (considering gravity gradient). A parametric study is finally preformed to find the influence of excavation time and location of the new tunnel, the tunnel spacing and relative size of the new tunnel, on stresses/displacements around tunnels and surface settlements. The results show that the excavation time of the new tunnel (t2) significantly influence the additional displacements around the existing tunnel which is a decrease exponential function of t2; when the distance from center to center is larger than 2.5R1 (2.0R1), the interaction between two tunnels can be neglected from perspective of displacement (stress).

Surface Effects on the Scattering of SH-Wave Around an Arbitrary Shaped Nano-Cavity

By using the complex variable function theory and the conformal mapping method, the scattering of plane shear wave (SH-wave) around an arbitrary shaped nano-cavity is studied. Surface effects at the nanoscale are explained based on the surface elasticity theory. According to the generalized Yong–Laplace equations, the boundary conditions are given, and the infinite algebraic equations for solving the unknown coefficients of the scattered wave solutions are established. The numerical solutions of the stress field can be obtained by using the orthogonality of trigonometric functions. Lastly, the numerical results of dynamic stress concentration factor around a circular hole, an elliptic hole and a square hole as the special cases are discussed. The numerical results show that the surface effect and wave number have a significant effect on the dynamic stress concentration, and prove that our results from theoretical derivation are correct.

Mechanical properties and fracture characteristics of pre-holed rocks subjected to uniaxial loading: A comparative analysis of five hole shapes

This study systematically investigates the influence of hole shape on mechanical properties and fracture characteristics of rocks containing a hole under uniaxial loading. First, combined with digital image correlation (DIC) and acoustic emission (AE) equipment, quantities of uniaxial compression tests on brittle prismatic sandstone samples containing a circular, inverted U-shaped, trapezoid, rectangular or square hole were conducted to study the strength, deformation and fracture characteristics. After that, the analytical solutions of stress for the five types of holes were derived using complex variable function theory. The experimental results suggest that the mechanical properties of the samples are greatly weakened by the existence of the holes, and the degradation degree depends on the hole shape. The stability order of these holes is ranked as: circle > inverted U-shape > trapezoidal > square > rectangle. According to the formation mechanism and sequence, four types of cracks, namely, primary tensile cracks, slabbing fractures, remote cracks and shear cracks, are formed around all the holes, but the crack sequence, initiation location and propagation characteristics of each hole are slightly different. The variation of AE signals matches well with the fracture evolution. The shear failure mode results from the coalescence of the shear cracks and V-shaped notches. Theoretical analysis shows that crack development mechanism can be well interpreted by the stress distributions around the holes. Moreover, the length of the primary tensile cracks is theoretically solved and compared, which is agreeable with the measured result.

The Improved Complex Variable Element-Free Galerkin Method for Bending Problem of Thin Plate on Elastic Foundations

In this paper, the improved complex variable element-free Galerkin (ICVEFG) method is proposed for solving the bending problem of thin plate on elastic foundations. In the ICVEFG method, the approximation function regarding the deflection of thin plate is formed with the improved complex moving least-squares (ICVMLS) approximation, the discrete equation is obtained from Galerkin weak form of bending problem of thin plate on different elastic foundations, and essential boundary conditions are considered based on penalty method. As the ICVMLS approximation is based on the complex variable theory, it can obtain the shape function quickly with high precision. Three sample problems are used to discuss the advantages of the ICVEFG method, and the numerical results show that the ICVEFG method presented in this paper has a fast convergence speed and great computational accuracy.

Reflection and transmission of light by a half-space with a nonlocal response: A complex variable approach

A solution of the electromagnetic reflection and transmission problem by a half-space of a simple material with a nonlocal response to external electric fields is investigated with the aid of several mathematical results of the complex variable theory, the Sokhotski relations, and the Hilbert transform. We found that the problem of reflection and transmission of electromagnetic radiation in the Fourier space is equivalent to an integral equation arising in the Riemann boundary problem. The classical results of Fresnel’s formulas for S polarization are recovered, and the special case of the reflection and transmission of light by a half-space of small spheres in the diluted regime of particle concentration is found to also follow Fresnel’s formula with effective optical properties.