Cylindrical Functions Research Articles

Natural waves in a spatial viscoelastic cylinder with a radial crack

The investigation of the dispersion of waves in elastic bodies with different cutouts is of great interest in various scientific and technological fields. The objective of this research is to examine the propagation of free damped waves in elastic dissipative bodies with longitudinal notches. The dynamic behavior of cylindrical bodies with different cutouts is described by the equations of viscoelasticity. The solution of a system of differential equations is expressed by cylindrical Bessel and Hankel functions. The frequency equation is solved using the Muller, Gauss, and orthogonal run methods. It is found that with a decrease in the oscillation frequency in a cylinder of a given radius, the phase and group velocities of the longitudinal wave tend to the common limit - the phase velocity of the rod waves. It is found that as the oscillation frequency reduces in a cylinder of a given radius, the phase and group velocities of the longitudinal wave tend to converge towards the common limit. Additionally, it was found that for a specific value of the Poisson's ratio (0.2833), the frequencies of the shear and radial longitudinal resonances coincide in the cylinder.

Implementation of a cylindrical distribution function for the analysis of anisotropic molecular dynamics simulations.

The cylindrical distribution function (CDF) is a convenient anisotropic analogue of the radial distribution function, the difference being the use of cylindrical shells for binning. As such, CDF analysis can be a powerful tool for the analysis of positional correlations within anisotropic systems, such as liquid crystals. Here we describe a lightweight Python tool, cylindr, for the calculation of cylindrical distribution function, which is compatible with the output of a number of popular MD engines. We demonstrate the use of cylindr in computing the CDF of a number of exemplar materials: classical and ferroelectric nematics; lamellar and columnar liquid crystals.

Star product approach for loop quantum cosmology

Guided by recent developments toward the implementation of the deformation quantization program within the loop quantum cosmology (LQC) formalism, in this paper we address the introduction of both the integral and differential representation of the star product for LQC. To this end, we consider the Weyl quantization map for cylindrical functions defined on the Bohr compactification of the reals. The integral representation contains all of the common properties that characterize a star product which, in the case under study here, stands for a deformation of the usual pointwise product of cylindrical functions. Our construction also admits a direct comparison with the integral representation of the Moyal product which may be reproduced from our formulation by judiciously substituting the appropriate characters that identify such representation. Further, we introduce a suitable star commutator that correctly reproduces both the quantum representation of the holonomy-flux algebra for LQC and, in the proper limit, the holonomy-flux classical Poisson algebra emerging in the cosmological setup. Finally, we propose a natural way to obtain the quantum dynamical evolution in LQC in terms of this star commutator for cylindrical functions as well as a differential representation of the star product using discrete finite differences. We expect that our findings may contribute to a better understanding of certain issues arising within the LQC program, in particular, those related to the semiclassical limit and the dynamical evolution of quantum states.

Threshold conditions for transversal modes of tunable plasmonic nanolasers shaped as single and twin graphene-covered circular quantum wires

We implement the lasing eigenvalue problem (LEP) approach to study the electromagnetic field in the presence of a circular quantum wire (QW) made of a gain material and wrapped in graphene cover and a dimer of two identical graphene-covered QWs, at the threshold of stationary emission. LEP delivers the mode-specific eigenvalue pairs, namely the frequencies and the threshold values of the QW gain index for the plasmon and the wire modes of such nanolasers. In our analysis, we use quantum Kubo formalism for the graphene conductivity and classical Maxwell boundary-value problem for the field functions. The technique involves the resistive boundary conditions, the separation of variables in the local coordinates, and, for the dimer, the addition theorem for the cylindrical functions. For single-wire plasmonic laser, we derive approximate engineering expressions for the lasing frequencies and threshold values of the gain index that complement the full-wave computations. For the dimer, we derive separate determinantal equations for four different classes of symmetry of the lasing supermodes and solve them numerically. Our investigation of the mode frequencies and thresholds versus the graphene and QW parameters shows that plasmon modes or, for the dimer, plasmon supermodes have lower frequencies and thresholds than the wire modes provided that the QW radius is smaller than 10 μm, however in thicker wires they are comparable. Only the plasmon-mode characteristics are well-tunable using the graphene chemical potential. In the dimer, all lasing supermodes form closely located quartets, however, they quickly approach the single-wire case if the inter-wire separation becomes comparable to the radius. These results open a way for building essentially single-mode plasmonic nanolasers and their arrays and suggest certain engineering rules for their design.

On the domain of non-symmetric and, possibly, degenerate Ornstein–Uhlenbeck operators in separable Banach spaces

Let X be a separable Banach space and let X^* be its topological dual. Let Q:X^*\rightarrow X be a linear, bounded, non-negative, and symmetric operator and let A:D(A)\subseteq X\rightarrow X be the infinitesimal generator of a strongly continuous semigroup of contractions on X . We consider the abstract Wiener space (X,\mu_\infty,H_\infty) , where \mu_\infty is a centered non-degenerate Gaussian measure on X with covariance operator defined, at least formally, as Q_\infty=\int_0^{+\infty} e^{sA}Qe^{sA^*}\,ds, and H_\infty is the Cameron–Martin space associated to \mu_\infty . Let H be the reproducing kernel Hilbert space associated with Q with inner product [\cdot,\cdot]_H . We assume that the operator Q_\infty A^*:D(A^*)\subseteq X^*\rightarrow X extends to a bounded linear operator B\in\mathcal{L}(H) which satisfies B+B^*=-\mathrm{Id}_H , where \mathrm{Id}_H denotes the identity operator on H . Let D and D^2 be the first and second order Fréchet derivative operators. We denote by D_H and (D_H,D^2_H) the closure in L^2(X,\mu_\infty) of the operators QD and (QD,QD^2) , respectively, defined on smooth cylindrical functions, and by W^{1,2}_H(X,\mu_\infty) and W^{2,2}_H(X,\mu_\infty) , respectively, their domains in L^2(X,\mu_\infty) . Furthermore, we denote by D_{A_\infty} the closure of the operator Q_\infty A^*D in L^2(X,\mu_\infty) defined on smooth cylindrical functions, and by W^{1,2}_{A_\infty}(X,\mu_\infty) the domain of D_{A_\infty} in L^2(X,\mu_\infty) . We characterize the domain of the operator L , associated to the bilinear form (u,v)\mapsto-\int_{X}[BD_Hu,D_Hv]_H\,d\mu_\infty, \quad u,v\in W^{1,2}_H(X,\mu_\infty), in L^2(X,\mu_\infty) . More precisely, we prove that D(L) coincides, up to an equivalent renorming, with a subspace of W^{2,2}_H(X,\mu_\infty)\cap W^{1,2}_{A_\infty}(X,\mu_\infty) . We stress that we are able to treat the case when L is degenerate and non-symmetric.

Electromagnetic wave focusing with 2-D gridded parabolic reflector made of parallel circular PEC wires at oblique incidence

We consider the two-dimensional (2-D) scattering of the TE- and TM-polarized plane electromagnetic waves from a discrete grid-like parabolic reflector consisting of parallel circular perfectly electrically conducting (PEC) wires at oblique incidence. Our analysis is performed in the exact formulation, fully taking into account the wave interactions between the wires. We use the method of separation of variables in the local cylindrical coordinate systems in combination with Graf’s addition formulas for cylindrical functions. The near field in the vicinity of the reflector is calculated at different angles of incidence. The dependences of the maximum field amplitude in the focal area and the position of this maximum on the angle of incidence are plotted. The far field scattered patterns, total scattering cross-sections and backward scattering cross-sections are calculated and discussed.

A Robust Method for Solving the Modal Fields in Radially Unbounded Cylindrical Waveguides With Two Layers Under Extreme Conductive Conditions

In this article, we present a numerically stable formulation for the electromagnetic analysis of lossy circular waveguides. The associated boundary value problem is solved analytically via an appropriate set of scaled cylindrical functions. A new numerically stable characteristic equation that simplifies the application of Cauchy’s argument principle is introduced. Numerical results are presented and show that the proposed approach can solve modal fields even in extreme lossy scenarios.

Исследование математических подходов к определению частотно-зависимого внутреннего сопротивления провода воздушной линии электропередачи

Mathematical modeling of a power transmission line (PTL) is an important issue for a wide range of tasks in the electric power industry. Mathematical modeling is necessary when studying the transient processes in electric power systems. Analytical expressions that allow modeling power transmission lines in a wide frequency range, including low frequencies and direct current, make it possible to study the operation of relay protection and automation devices in various modes and improve the accuracy of devices of fault location. It is important not only to obtain analytical expressions to determine the frequency dependences of the resistance, but also to verify these expressions in comparison with more accurate methods based on the finite elements method (FEM). The study has been carried out using a mathematical tools based on cylindrical Bessel functions. To develop formulas, it is necessary to determine the constants of integration based on boundary conditions. To verify the obtained expressions, modeling has been performed in the COMSOL Multiphysics software package, which is based on the FEM. The article presents a study of the internal resistance of the wires of power transmission line using the example of AC 185/24 wire. An analytical expression has been obtained to determine the internal complex resistance of a bimetallic wire. The reliability of the obtained expressions is confirmed by the convergence of the simulation results in comparison with the results of simulation modeling in Comsol software and mathematical modeling using known analytical expressions. The proposed approach to determine the internal resistance of a wire makes it possible to get more accurate analytic definition of characteristics of an overhead power transmission line. And thus, to design more qualitative models to analyze transient processes in power transmission lines and investigate the operation of relay protection devices. The wire models developed in Comsol software can be considered as more accurate in a wide range of frequencies.

A coupled mix-mode cohesive law based on a cylindrical potential function

A novel mixed-mode cohesive law derived from a potential function is presented. The potential function is formulated using physical parameters that can be extracted from any fracture mechanics test capable of providing R-curves in terms of the J-integral as a function of the normal and tangential end-openings. The proposed cohesive law is able to describe the fracture behaviour of composites with large fracture process zones, including fibre bridging. As such, it is capable of describing both the crack tip, as well as the bridging region. An important aspect of the formulation is that the shape of the mixed-mode cohesive laws are derived and not assumed. The mixed-mode cohesive law was tested using synthetic data emulating mixed-mode fracture mechanics tests. The cohesive tractions extracted from the method exhibited characteristics which were not seeded on the model such as negative normal tractions under pure mode shear loading and non-zero shear loading under pure normal mode loading.

Light scattering by an infinitely long circular cylinder above a plane interface

An exact analytical method is presented to calculate light scattering by an infinitely long circular cylinder above a plane interface. The transformation relations of rectangular and cylindrical vector wave functions are used to deal with the coupling scattering between the circular cylinder and plane interface. By applying the boundary conditions over the cylindrical surface, the unknown expansion coefficients of the scattered field by the circular cylinder in terms of cylindrical vector wave functions are determined. Mueller matrix elements are computed, and also compared with those measured for a plane wave with normal incidence.

On the sine polarity and the Lp-sine Blaschke-Santaló inequality

This paper is dedicated to study the sine version of polar bodies and establish the Lp-sine Blaschke-Santaló inequality for the Lp-sine centroid body.The Lp-sine centroid body ΛpK for a star body K⊂Rn is a convex body based on the Lp-sine transform, and its associated Blaschke-Santaló inequality provides an upper bound for the volume of Λp∘K, the polar body of ΛpK, in terms of the volume of K. Thus, this inequality can be viewed as the “sine cousin” of the Lp Blaschke-Santaló inequality established by Lutwak and Zhang. As p→∞, the limit of Λp∘K becomes the sine polar body K⋄ and hence the Lp-sine Blaschke-Santaló inequality reduces to the sine Blaschke-Santaló inequality for the sine polar body. The sine polarity naturally leads to a new class of convex bodies Cen, which consists of all origin-symmetric convex bodies generated by the intersection of origin-symmetric closed solid cylinders. Many notions in Cen are developed, including the cylindrical support function, the supporting cylinder, the cylindrical Gauss image, and the cylindrical hull. Based on these newly introduced notions, the equality conditions of the sine Blaschke-Santaló inequality are settled.

Transmission characteristics of terahertz Bessel vortex beams through a multi-layered anisotropic magnetized plasma slab

The transmission of terahertz (THz) Bessel vortex beams through a multi-layered anisotropic magnetized plasma slab is investigated by using a hybrid method of cylindrical vector wave functions (CVWFs) and Fourier transform. On the basis of the electromagnetic boundary conditions on each interface, a cascade form of expansion coefficients of the reflected and transmitted fields is obtained. Taking a double Gaussian distribution of the plasma density as an example, the influences of the applied magnetic field, the incident angle and polarization mode of the incident beams on the magnitude, OAM mode and polarization of the transmitted beams are analyzed in detail. The results indicate that the applied magnetic field has a major effect upon the polarization state of the transmitted fields but not upon the transmitted OAM spectrum. The incident angle has a powerful influence upon both the amplitude profile and the OAM spectrum of the transmitted beam. Furthermore, for multiple coaxial vortex beams, an increase of the maximum value of the plasma density causes more remarkable distortion of both the profile and OAM spectrum of the transmitted beam. This research makes a stable foundation for the THz OAM multiplexing/demultiplexing technology in a magnetized plasma environment.

Weber Equation Model of Flow Through a Variable-Permeability Porous Core Bounded by Fluid Layers

Abstract Flow-through and over a Brinkman porous layer of variable permeability, immersed in a fluid-filled channel, is modeled. The model governing equation through the porous layer inevitably gives rise to an inhomogeneous Weber's differential equation, solved in this work and solutions expressed in terms of parabolic cylindrical functions. Using state-of-the-art computational techniques and a body of knowledge, the parabolic cylindrical functions are evaluated for a range of flow and medium parameters in order to illustrate intrinsic characteristics of the flow quantities. The approach followed in this work is novel and sets precedent in the study of flow through general porous media configurations and flow domains with variable permeability.

Mitigation of railway-induced vibrations by using periodic wave impeding barriers

Railway-induced vibrations can cause significant environmental issues. This paper proposes an efficient analytical method to investigate the mitigation of railway-induced vibrations by using periodic barriers in a layered half-space. The general solutions for layered ground and multiple inclusions are derived by using the potential decomposition and multiple scattering theory. The conversion equation between cylindrical and exponential functions and the addition theorem are introduced to achieve the transformation between plane and cylindrical wave functions and the translation between cylindrical wave functions. Combined with the transfer matrix method, the fundamental solution for the soil-inclusion dynamic interaction in a layered half-space is derived. The railway train and track are subsequently coupled to the ground-inclusion system. Numerical results demonstrate that the phononic crystal effect induced by the periodic distribution of barriers improves the mitigation efficiency at high frequencies. The increase in the number, size, and stiffness of barriers can give a higher mitigation efficiency in a wider frequency range. The mitigation efficiency of periodic barriers can be guaranteed when their depth is shorter than half the Rayleigh wavelength in the considered frequency range. Owing to the scattering of waves at layer interfaces, the periodic barriers beneath the track have a higher efficiency than those located next to the track, which does not appear in the homogeneous half-space. The performance of periodic barriers is significantly affected by the soil stiffness of the upper shallow layer, while it is less affected by the soil stiffness of the bottom stiffer layer.

Pengaruh Proprioceptive Neuromuscular Facilitation terhadap Kekuatan Fungsi Prehension Cylindrical Grip pada Pasien Stroke Non-Hemoragik

Stroke is a disease that is most commonly found today, stroke has a high mortality and disability rate. Stroke can cause physical disability in productive age and old age. This study aims to determine the effect of Proprioceptive Neuromuscular Facilitation (PNF) on the strength of the prehension Cylindrical grip function in non-hemorrhagic stroke patients. This research is a quantitative study with a quasi-experimental design, one group pretest and posttest, the samples were taken by purposive sampling, namely non-hemorrhagic stroke patients who visited the physiotherapy poly at Grandmed Lubuk Pakam Hospital. The results showed a p-value of 0.001 which means that there is an effect of PNF on the strength of the prehension Cylindrical grip function in non-hemorrhagic stroke patients.

НАПРЯЖЕННО-ДЕФОРМИРОВАННОЕ СОСТОЯНИЕ ПОДКРЕПЛЕННЫХ СЛОИСТЫМИ ОБДЕЛКАМИ ТОННЕЛЕЙ МЕЛКОГО ЗАЛОЖЕНИЯ ПРИ ДЕЙСТВИИ ТРАНСПОРТНЫХ НАГРУЗОК

Based on the solution of the problem of the action of a transport loads moving at a constant speed on a three-layer or two-layer shells in an elastic half-space, a comparative analysis was performed for the rock mass in the vicinity of a shallows circular tunnel with various designs of their lining (three-layer and two-layer), along the inner surface of which an axisymmetric normal load moves at a constant speed. The inner layer of the three-layer lining is a thick-walled concrete shell, and the outer layers are thin-walled steel shells of equal thickness. The two-layer lining is structurally different from the considered three-layer lining by the absence of the inner layer of the lining. Dynamic equations of elasticity theory in Lame potentials are used to describe the motion of the half-space and the inner layer of the shell. The vibrations of the outer layers of the shell are described by the classical equations of the theory of thin shells. The equations are represented in a movable coordinate system. The contact between the shell and the array is assumed to be either rigid or sliding. The contact between the layers of the shell is supposed to be rigid. To solve the problem, the method of incomplete separation of variables is used. The solution for potentials is presented in the form of a superposition of Fourier-Bessel series and contour integrals of the Fourier type. Next, the method of decomposition of potentials into plane waves and re-decomposition of plane waves into series according to cylindrical functions is used. The solution is obtained for the case when the velocity of the load is less than its critical velocities. The calculation results are presented in the form of graphs and are analyzed in detail. From the analysis of the calculation results, it follows that the use of a three-layer lining as a building envelope is more efficient than a two-layer lining, since in this case the dynamic effect of the moving loads on the rock mass is significantly less.

Effect of a perfectly conducting corner space on the PAFs for an absorptive dielectric circular cylinder

The photophoretic asymmetry factors (PAFs) for an absorptive dielectric circular cylinder, located near a perfectly conducting and totally reflecting corner space are derived and computed. The method used in this analysis relies on the modal expansion method in cylindrical coordinates, the classical method of images, and the translational addition theorem of cylindrical wave functions. Initially, the components of the internal electric field vector are obtained stemming from an analysis of the scattering. Subsequently, the solution is used to integrate to the normalized intensity function over the cylinder’s volume to obtain the longitudinal (L) and transverse (T) PAFs. Both TM- and TE-polarized plane progressive waves with arbitrary incidence (in the polar plane) are considered. Attention is given to varying the dimensionless size parameter of the cylinder, the angle of incidence of the incoming waves, and the dimensionless distance parameters from the corner space. Numerical examples illustrate the analysis and demonstrate the net effect of the totally reflecting corner space on the L- and T-PAFs, where negative, positive, and neutral values have been predicted. The results are relevant in applications related to the emergence of the photophoretic force and torque on an absorptive particle located near surfaces and topics in electromagnetic/optical scattering, particle manipulation and assembly, optically bound matter, light–matter interactions, and photopheresis.

Illuminating finite-amplitude wave propagation: Hargrove’s use of recurrence relations

A classic problem in the theory of finite-amplitude wave propagation is that of the spectral evolution of a plane wave. An initial sinusoidal wave becomes triangular, with a vertical face at the shock distance. The wave in the pre-shock regime was represented by a Fourier series in the seminal work by Fubini, i.e., Fubini Ghiron [Alta Frequenza, 4, 530 (1935)]. Independently, Hargrove performed a similar derivation, also invoking recurrence relations for cylindrical Bessel functions, but presenting this with exceptional heuristic clarity and brevity [J. Acoust. Soc. Am. 32, 511 (1960)]. This was coincident with a third paper, by Keck and Beyer [Phys. Fluids 3, 346 (1960)], which knowingly followed Fubini. All of this work has facilitated understanding, providing cogent reminders of the power of recurrence relations when dealing with special functions.

Analytical solution of problem of shielding low-frequency magnetic field by thin-walled cylindrical screen in presence of cylinder

The analytical solution of boundary value problem describing the process of penetration of low-frequency magnetic field through thin-walled cylindrical screen with cylindrical inclusion is constructed by use of approximate boundary conditions. The source of the field is a thin thread of infinitely small length with an infinitely small cross-section where current circulates. Thread is located in a plane which is perpendicular to axis of cylindrical screen, in outer region with respect to a screen. Initially the potential of initial magnetic field is represented as spherical harmonic functions, then using addition theorems connecting spherical and cylindrical harmonic functions, it became as cylindrical harmonic functions superposition. Secondary potentials of magnetic field are also presented as superposition of cylindrical harmonic functions in three-dimensional space. It is shown that the solution of formulated boundary value problem is reduced to the solution of linear algebraic equations system for coefficients included in the representation of secondary fields. The influence of some aspects of the problem on the value of the screening coefficient of an external magnetic field when passing through a cylindrical copper screen in the presence of a cylindrical inclusion is studied numerically. Calculation results are presented in graphs form. Obtained results can be used to shield technical devices and biological objects against the effects of magnetic fields to provide ecological surrounding of operating electrical installations and devices.

Dynamical collapse of cylindrical symmetric dipolar Bose\u2013Einstein condensates

We study the formation of singularities for cylindrical symmetric solutions to the Gross–Pitaevskii equation describing a dipolar Bose–Einstein condensate. We prove that solutions arising from initial data with energy below the energy of the Ground State and that do not scatter collapse in finite time. The main tools to prove our result are the variational characterization of the Ground State energy, suitable localized virial identities for cylindrical symmetric functions, and general integral and pointwise estimates for operators involving powers of the Riesz transform.