Functions Of Complex Variables Research Articles

Analytical solution of surrounding rock stress and plastic zone of rectangular roadway under non-uniform stress field

The shape and extent of the plastic zone can effectively reflect the degree of deformation and failure of the surrounding rock. By applying complex variable functions and the theory of elasticity, a mechanical model for the surrounding rock of a rectangular roadway is established. The analytical solution for the stress in the surrounding rock of a rectangular roadway under a non-uniform stress field is derived, and the spatial stress distribution pattern in the surrounding rock of the rectangular roadway is obtained. Furthermore, using the Mohr-Coulomb criterion, the boundary equation of the plastic zone in the surrounding rock of a rectangular roadway is derived, and the morphological characteristics of the plastic zone are obtained. The results indicate that the plastic zones on both sides of the rectangular roadway are arc-shaped. The asymmetry of the stress environment has a significant impact on the plastic zones in the roof and floor of the rectangular roadway, causing the shape of the plastic zones to resemble a “petal” pattern. The lateral pressure coefficient and the height-to-width ratio have a significant impact on the stress distribution and the morphological characteristics of the plastic zone in the surrounding rock of the rectangular roadway. By validating the computational accuracy of the analytical solution through numerical simulations and conducting an engineering applicability analysis based on a coal mine roadway, it has been demonstrated that the analytical calculation method is both rational and possesses good engineering applicability. This provides reliable theoretical guidance for the design calculations and construction planning of similar roadway projects.

Study on the effective drainage range of long horizontal borehole for goaf gas drainage

AbstractLaying long horizontal borehole (LHB) in mining‐induced strata fractures for coal‐bed gas drainage in the goaf is a pivotal method for eliminating safety hazards, preventing atmospheric pollution, and realizing gas utilization. A crucial premise of this method is to determine the effective drainage range of the borehole, a key parameter directly affecting boreholes spacing and drainage efficiency. In this study, a model of gas seepage during LHB drainage within a circular equipotential boundary was presented, and the complex variable function theory and the mirror principle were adopted for deriving the complex potential function of gas flow under the condition of LHB drainage and the drainage volume function of a single center borehole. Besides, the main indicator to determinate the effective drainage range of LHB was given. Specifically, the indicator refers to the distance between the turning point (the point at which the change rate of the gas flow velocity is −0.002 around the borehole) and the center of the borehole. Subsequently, numerical simulation was conducted to obtain the effective drainage range of LHB within mining‐induced fractures of overlying strata, that is, 4 m < r < 7 m. Furthermore, discussion of the effective drainage range of LHB was carried out by analyzing gas flow velocity distributions and LHB drainage volumes. A case study demonstrates that a double‐borehole layout with a 10 m spacing can effectively control pressure‐relief gas in the goaf. Meanwhile, such a layout can achieve well‐balanced drainage efficiency of each borehole with a minimal difference of 0.31 m3 min−1, which is only about 7.9% of the average drainage volume of the two boreholes. The above results verify that the determined effective drainage range of LHB is reasonable. The research results can provide references for determining the drainage range of LHB in multi‐borehole layout of coal mines, and then determining the spacing of LHBs arrangement, thereby improving the efficiency of gas extraction.

Stress Characteristics and Mechanical Behavior of Rock Masses with an Opening under Complex Deep Underground Stress Conditions

In this study, the impact of principal stress states on the stress characteristics and initial failure of the rock mass surrounding a three-center arch opening was investigated using complex variable function methods and Discrete Element Method (DEM) numerical modeling. First, the mapping function of the opening was determined using the trigonometric interpolation method, and the influence of the number of terms in the mapping function on its accuracy was revealed. Based on this, the far-field stress state of the underground rock mass was characterized by the ratio of the minimum to maximum principal stress (λ) and the angle (β) between the principal stress and the vertical direction. This stress state was then converted into normal and shear stresses. Using complex variable function theory, the stress characteristics at the boundary of the opening under different stress states were analyzed. Finally, DEM numerical modeling was employed to study the initial failure characteristics at the boundary of the opening and its relationship with the stress distribution. The results indicate that the lateral pressure coefficient significantly affects the stability of the opening by influencing stress concentration around the surrounding rock. Low lateral pressure coefficients lead to tensile stress concentration at the boundary perpendicular to the maximum principal stress. As the coefficient increases, tensile stress decreases, and compressive stress areas expand. While the principal stress direction has a minor effect on stress concentration, it notably impacts stress distribution at the boundary. When λ < 1.0 and β = 45°, stress distribution asymmetry is most pronounced, with the highest compressive stress. The early failure distribution aligns with stress concentration areas, validating the use of stress analysis in predicting opening stability and failure characteristics.

On the Applicability of Kramers–Kronig Dispersion Relations to Guided and Surface Waves

In unbounded media, the acoustic attenuation as function of frequency is related to the frequency-dependent sound velocity (dispersion) via Kramers–Kronig dispersion relations. These relations are fundamentally important for better understanding of the nature of attenuation and dispersion and as a tool in physical acoustics measurements, where they can be used for control purposes. However, physical acoustic measurements are frequently carried out not in unbounded media but in acoustic waveguides, e.g., inside liquid-filled pipes. Surface acoustic waves are also often used for physical acoustics measurements. In the present work, the applicability of Kramers–Kronig relations to guided and surface waves is investigated using the approach based on the theory of functions of complex variables. It is demonstrated that Kramers–Kronig relations have limited applicability to guided and surface waves. In particular, they are not applicable to waves propagating in waveguides characterised by the possibility of wave energy leakage from the waveguides into the surrounding medium. For waveguides without leakages, e.g., those formed by rigid walls, Kramers–Kronig relations remain valid for both ideal and viscous liquids. Examples of numerical calculations of wave dispersion and attenuation using Kramers–Kronig relations, where applicable, are presented for unbounded media and for waveguides formed by two rigid walls.

Study on the directional extension law of hydraulic fractures induced by pre-cast slot under tru-triaxial

Directional hydraulic fracturing technology serves as an effective means to overcome the stress constraints in deep coal mines and achieve targeted depressurization and permeability enhancement in specific areas. This study conducted directional hydraulic fracturing tests on samples with pre-cast slots under true triaxial conditions. Considering the azimuth angle (i.e., the angle between the long axis of the pre-cast slot and the maximum principal stress) and long axis length, the stress distribution along the fracture edge was determined using the method of complex variable function, and the evolution mechanism of the stress distribution along the fracture edge was investigated. The main conclusions are as follows: Experimental results indicate that the initiation pressure and directional extension effectiveness show a relative growth pattern of “slow-fast-slow” in the azimuth angle range of 30° to 90°, with the most significant decrease in initiation pressure occurring around 45°, and the enhancement trend of directional extension is most pronounced in the range of 45° to 60°. When the ratio of the long axis to the short axis of the pre-cast slot changes from 2:1 to 1:1 and 3:1, the initiation pressure increases by −30.44% and 28.08% respectively, while the amplification of directional extension effect is −74.6% and 166.8%. Considering the azimuth angle, an expression for the circumferential stress at the edge of the pre-cast slot was derived, and a criterion for determining the initiation of the pre-cast slot edge was proposed. As the azimuth angle increases from 0° to 90°, the circumferential stress at the edge of the pre-cast slot fluctuates, which is consistent with the trend of initiation pressure observed in the experimental results. As the long axis length increases, the initiation pressure decreases, and the initiation position moves closer to the tip.

On solutions of certain compatible systems of quadratic trinomial Partial differential-difference equations

This paper has involved the use of a variety of variations of the Fermat-type equation $f^n(z)+g^n(z)=1$, where $n(\geq 2)\in\mathbb{N}$. Many researchers have demonstrated a keen interest to investigate the Fermat-type equations for entire and meromorphic solutions of several complex variables over the past two decades. Researchers utilize the Nevanlinna theory as the key tool for their investigations. Throughout the paper, we call the pair $(f,g)$ as a finite order entire solution for the Fermat-type compatible system $\begin{cases} f^{m_1}+g^{n_1}=1;\\ f^{m_2}+g^{n_2}=1,\end{cases}$\!\! if $f$, $g$ are finite order entire functions satisfying the system, where $m_1,m_2,n_1,n_2\in\mathbb{N}\setminus\{1\} .$\ Taking into the account the idea of the quadratic trinomial equations, a new system of quadratic trinomial equations has been constructed as follows: $\begin{cases} f^{m_1}+2\alpha f g+g^{n_1}=1;\\ f^{m_2}+2\alpha f g+g^{n_2}=1,\end{cases}$ \!\! where $\alpha\in\mathbb{C}\setminus\{0,\pm1\}.$ In this paper, we consider some earlier systems of certain Fermat-type partial differential-difference equations on $\mathbb{C}^2$, especially, those of Xu {\it{et al.}} (Entire solutions for several systems of nonlinear difference and partial differential-difference equations of Fermat-type, J. Math. Anal. Appl. 483(2), 2020) and then construct some systems of certain quadratic trinomial partial differential-difference equations with arbitrary coefficients. Our objective is to investigate the forms of the finite order transcendental entire functions of several complex variables satisfying the systems of certain quadratic trinomial partial differential-difference equations on $\mathbb{C}^n$. These results will extend the further study of this direction.

Theoretical realization of tunable hollow beams using a periodical ring structure with a complex phase

Hollow beam is a peculiar structure beam, which has been widely used in various areas. Here, we propose a novel diffraction optical element to generate tunable hollow beams. This element is composed of periodic concentric rings. The phase of each ring is periodically distributed between −π and π and satisfies a complex variable function. By tuning the parameters of the structure, we can flexibly manipulate the size and length of the hollow beam. The length of the beam can be increased from 98 λ to 248 λ, and the full width at half maximum varies from 0.43 λ to 0.61 λ. Moreover, the light intensity and side lobe of the hollow beam can also be regulated using the designed diffraction optical element. The potential applications of this highly tunable hollow beam include optical nanomanipulation, microscopic imaging, and nanolithography.

Selected trace element uptake by rice grain as affected by soil arsenic, water management and cultivar -a field investigation

Accumulation of arsenic (As) in rice grain was reported in many regions of the world, including the United States, which has been a threat to human health. This field research investigated the grain As accumulation and its relationship with the uptake of selenium (Se), molybdenum (Mo), and cadmium (Cd) in soils with and without monosodium methanearsonate (MSMA) amended, as effects of selected rice cultivars and water management. Results indicated that MSMA increased the accumulation of As and Se but decreased Mo for all six cultivars under four irrigation management. MSMA also increased grain-Cd in some cultivars. In no MSMA-amended soil (Native soil), intermittent flooding decreased grain-As by 66%, grain-Se by 21%, and grain-Mo by 63%, but increased grain-Cd by 64% in Zhe 733, a straighthead resistant cultivar, while in MSMA-amended soil, intermittent flooding decreased grain-As by 63% and grain-Mo by 44% but increased grain-Se by 68% and grain-Cd by three times. For all other five cultivars, intermittent flooding generally decreased grain-As and grain-Mo but increased grain-Se and grain-Cd. Zhe 733 cultivar resulted in the lowest grain concentrations of all trace elements in all water treatments. A negative grain As-Se correlation and a positive grain As-Mo correlation were significant but not the As-Cd correlation. This research showed that the uptake of As, Se, Mo, and Cd by rice grain occurred as a complex function of multiple variables, including cultivar type and soil chemistry. As a result, accumulation of As and other trace elements in rice grain may be controlled by selecting appropriate cultivars and adopting appropriate water management practices.

Dynamic inverse plane characteristics of circular holes in a strip-shaped medium under line source excitation

Based on the complex variable function method and the wave function expansion method, the dynamic anti-plane characteristics of circular holes in the strip domain under line source excitation were studied. The total wave field in the strip domain is constructed using the concept of infinite mirroring, which eliminates the mutual impact of the wave field under the action of the upper and lower limits of the strip domain. To determine the overall wave field, solve the wave field's unknown coefficients in accordance with the boundary conditions. The right number of mirror images is found using the dynamic stress distribution of a circular hole in the band-shaped domain as a specific calculation example. The effect of altering the incident frequency and band-shaped domain width on the dynamic stress concentration coefficient surrounding the circular hole is also explored. As a consequence of the findings, the dynamic stress concentration coefficient around the circular hole in the zone domain at low frequencies is substantially bigger than it is at medium and high frequencies. Under identical conditions, the dynamic stress concentration coefficient will exhibit periodic extreme values and converge as the distance increases between the lower boundary and the circular hole’s center.

Applications of Residue Theorem to Some Selected Integrals

A complex variable is a quantity that contains both a real part and an imaginary part. Complex analysis is an application of mathematics that analyzes the features of functions of complex variables. Physics, engineering, and computer science are just a few of the scientific disciplines that benefit from complex analysis. In order to tackle issues that are challenging or impossible to resolve using only real variables, complex analysis is crucial. This paper introduces an important theorem in complex analysis, which is Cauchy’s residue theorem. A relative general expression for a complex integral along a simple closed contour is provided by Cauchy's residue theorem. Cauchy's residue theorem can be used to select an acceptable closed contour for the calculation of some unusual definite integrals that may be highly complex and challenging to solve using traditional methods. The residue at the function's isolated singularities is then established. In the formula that is subtracted in Cauchy's residue theorem, the values of the residues are extracted. The integral along a simple closed contour can then be represented in two pieces, in which one along the real axis and the other along the circle.

An Exploration of Cultivating Students' Creative Thinking in Teaching Functions of Complex Variables

As an important branch of higher mathematics, complex function occupies a core position in many fields such as mathematics, physics, and engineering, which not only involves rich theoretical knowledge but also shows great value in practical applications. Therefore, the teaching of complex functions is extremely important for students of mathematics and related majors. However, traditional teaching methods often emphasize theory over practice, lack effective ways to stimulate students' innovative thinking and limit the overall development of students' abilities. Based on this, this study will explore how to cultivate students' innovative thinking more effectively through the teaching of functions of a complex variable, aiming at finding and evaluating different teaching strategies and methods to better stimulate students' interest, improve their innovative and practical abilities, and promote the overall development of students' comprehensive abilities.

Numerical stability of the branched continued fraction expansion of Horn's hypergeometric function $H_4$

In this paper, we consider some numerical aspects of branched continued fractions as special families of functions to represent and expand analytical functions of several complex variables, including generalizations of hypergeometric functions. The backward recurrence algorithm is one of the basic tools of computation approximants of branched continued fractions. Like most recursive processes, it is susceptible to error growth. Each cycle of the recursive process not only generates its own rounding errors but also inherits the rounding errors committed in all the previous cycles. On the other hand, in general, branched continued fractions are a non-linear object of study (the sum of two fractional-linear mappings is not always a fractional-linear mapping). In this work, we are dealing with a confluent branched continued fraction, which is a continued fraction in its form. The essential difference here is that the approximants of the continued fraction are the so-called figure approximants of the branched continued fraction. An estimate of the relative rounding error, produced by the backward recurrence algorithm in calculating an nth approximant of the branched continued fraction expansion of Horn’s hypergeometric function H4, is established. The derivation uses the methods of the theory of branched continued fractions, which are essential in developing convergence criteria. The numerical examples illustrate the numerical stability of the backward recurrence algorithm.

Еdgе fiеld of dеflеctorplаtes with expanding screens

Deflector plates consist of two parallel conductive plates that create a deflecting electric field. They can be used to control the flow of charged particles ― electrons or ions. The effect of the еdgе fiеld of deflector plates leads to a change in the velocity of charged particles in the longitudinal and transverse directions, consequently of which their real trajectories change, deviating from ideal ones, which violates the space-time resolution of corpuscular optical devices in which they are used. Apart from that, the electric field at the input to the plates of deflector can vary over time, which must also be taken into account when the deflector diverts the beam of charged particles. Thus, in many cases, the use of deflector plates with open ends is inappropriate, since uncontrolled scattering fields are formed. In this article, we can consider the field of deflector plates with expanded screens at the output, so that deflecting beams of charged particles can be used at the output of deflector plates. Using the methods of the theory of complex variable functions, analytical expressions for the edge field of deflector plates with grounded screens were obtained. Firstly, by grounding the screens and shielding the plates from the deflection field, we can localize the edge electric field and reduce the uncontrolled scattering fields, and secondly, such a field can be accurately calculated analytically.

Dynamic anti-planar characteristics of circular holes in laminated structures under the action of line source

Based on the complex variable function method and the wave function expansion method, the dynamic anti-plane characteristics of the circular hole in the laminated structure under line source excitation were studied. To deal with the interface problem within the laminated structure, the idea of “fit” is adopted to construct continuity conditions that satisfy displacement and stress. The circular hole boundary stress-free condition is used, the wave field is constructed using the mirror method, and the unknown coefficients of the scattered wave field are solved through Fourier series expansion, and finally the total wave field in the laminated structure is obtained. Taking the dynamic stress distribution of a circular hole in a laminated structure as a specific calculation example, the effects of changes in line source position, layer thickness, and medium hard-soft ratio on the dynamic stress concentration coefficient around the circular hole are discussed. The results show that when the line source load is applied directly above the circular hole and the lower layer is narrow, the dynamic stress concentration coefficient will have a sudden change compared with other situations. When other conditions are the same, under low, medium, and high-frequency wave fields, the dynamic stress concentration caused by different soft-to-hard ratios shows completely different rules. These rules can be used to better detect defects in laminated structures.

Theoretical modeling and dynamic analysis of designable grillages for repairing material surface defect

Designing structural grillages is an innovative solution to repair material surface defects. The grillages have advantages as lightweight and directionally enhanced, but their stability and stress concentration under dynamic loads may pose significant risks. Three analytical models of grillages above a semicircular defect are constructed. The analytical solution of the frequency-domain wave field around the semicircular defect under SH wave incidence is solved. The complex variable function method is introduced to simplify the wave field expression. The accuracy is verified by comparing it with classic examples. Applying the Modified Time-Frequency Transform to obtain the time-domain wave field. Based on the Multipoint Transient Input Method (MTIM), the transient surface displacement response of the grillage above the defect is derived, mainly discussing the superposition behavior of input waves in the grillage. MTIM is based on the analytical solution of local time-domain wave field and takes the time histories of the surface characteristic points as the wave input. It has been validated as an effective method for studying the dynamic response of structures. The numerical results show that grillage form, incident angle, and wave velocity ratio are important factors affecting the structural responses of grillage above the defect. This research aims to reveal the propagation characteristics of waves in grillages in filling material defects, providing theoretical guidance for the design and application of grillages in this field.

Stress Intensity Factors for Thermoelectric Bonded Materials Weakened by an Inclined Crack

Thermoelectric Bonded Materials (TEBM) weakened by an Inclined Crack Problems (ICP) subjected to remote stress was presented in this study. The problems are addressed by employing the Modified Complex Variable Function (MCVF) method, which incorporates the Continuity Conditions (CC) for the Resultant Electric Force (REF) and Displacement Electric Function (DEF) to formulate the Hypersingular Integral Equations (HSIEs) associated with these problems. By applying the curved length coordinate method, the unknown functions of Crack Opening Displacement (COD), Electric Current Density (ECD), and Energy Flux Load (EFL) are mapped onto the square root singularity function. The resulting equations are then numerically solved using appropriate quadrature formulas, with the traction along the crack utilized as the right-hand term. The obtained COD, ECD and EFL functions is then used to compute the dimensionless Stress Intensity Factors (SIFs) in order to determine the stability behavior of TEBM weakened by an ICP. The numerical results provided demonstrate the dimensionless SIFs at the crack tips. These results exhibit excellent agreement with previous studies conducted on the subject. Additionally, it is observed that the dimensionless SIFs at the crack tips are influenced by factors such as the ratio of Elastic Constants (ECR), the geometry of the cracks, and the coefficients associated with the Electric Current Density (ECD).

Recent progress in the theory of functions of several complex variables and complex geometry

Recent progress in the theory of functions of several complex variables and complex geometry

On Grothendieck-type duality for spaces of holomorphic functions of several variables

We describe the strong dual space $({\mathcal O} (D))^*$ of the space ${\mathcal O} (D)$ of holomorphic functions of several complex variables in a bounded domain $D$ with Lipschitz boundary and connected complement (as usual, ${\mathcal O} (D)$ is endowed with the topology of local uniform convergence in $D$). We identify the dual space with the closed subspace of the space of harmonic functions on the closed set ${\mathbb C}^n\setminus D$, $n>1$, whose elements vanish at the point at infinity and satisfy the Cauchy-Riemann tangential conditions on $\partial D$. In particular, we generalize classical Grothendieck-Köthe-Sebastião e Silva duality for holomorphic functions of one variable to the multivariate situation. We prove that the duality we produce holds if and only if the space ${\mathcal O} (D)\cap H^1 (D)$ of Sobolev-class holomorphic functions in $D$ is dense in ${\mathcal O} (D)$. Bibliography: 35 titles.

Recent progress in the theory of functions of several complex variables and complex geometry

Дан обзор недавних новых результатов в области обратных теорем о $L^2$-существовании и $L^2$-продолжении, составляющих две основные части $L^2$-теории. Эти результаты используются для получения критерия положительности по Гриффитсу и условий положительности по Накано для (сингулярных) эрмитовых метрик голоморфных векторных расслоений, а также для доказательства сильной открытости и устойчивости пучков мультипликативных подмодулей, связанных с сингулярными неотрицательными по Накано эрмитовыми метриками на голоморфных векторных расслоениях.

Inequalities for meromorphic functions with prescribed poles

The extremal problems for functions of complex variables, as well as approaches for obtaining classical inequalities on the base of various methods of the geometric function theory, are known for various norms and for many classes of functions such as rational functions with various constraints and for various domains in the complex plane. It is important to mention that different types of Bernstein-type inequalities appeared in the literature in more generalized forms in which the underlying polynomial was replaced by a more general class of functions. One such generalization is the passage from polynomials to rational functions. In this paper, we prove some inequalities for meromorphic functions with prescribed poles and restricted zeros. These results not only generalize some Bernstein-type inequalities for rational functions, but also improve and generalize some known polynomial inequalities. These inequalities have their own importance in the approximation theory.