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Two foremost barriers to the development of fast and efficient simulation algorithms for electromagnetic problems in half-space and planar media are efficient Sommerfeld integral evaluation techniques and fast solvers. In this paper, we focus on challenging these two difficulties in the impedance boundary half-space, which is a proper model for common air-lossy media half-space. Based on the Laplace transformation, we first propose an alternative exact image representation for the Sommerfeld integral in half-space Green’s function. In addition to its fast and absolute convergence property, this new exact image version does not contain any singularities and interprets the Sommerfeld integral as an integral over real image line. Furthermore, this representation allows a rigorous fast multipole expansion (FME), and thus it can be applied to the development of fast multipole solver with controllable precision. However, this exact image representation is not valid for the all Sommerfeld integrals in half-space Green’s function. Therefore, a more generalized single mirror image representation, also exact and computationally efficient, is then presented. The strength of the mirror image is obtained through Weyl’s method and steep descent path approach. Subsequently, an approximated FME scheme for this representation is proposed and studied via numerical examples. We find this approximated FME scheme has a higher accuracy than that of previous work. Numerical results show that the proposed work provides exact and efficient evaluations for Sommerfeld integrals in impedance boundary half-space Green’s function and valuable insights into the development of fast multipole solver for half-space electromagnetic problems. [ABSTRACT FROM AUTHOR]

Presents a comprehensive study of the temperature dependence of the Fowler-Nordheim (F-N) tunnel emission in a metal-oxide-semiconductor structure. Influence of the electron concentration of a degenerate semiconductor on the amplitude of F-N current; Use of Sommerfeld expansion; Importance of F-N tunnel emission phenomenon in metal-oxide-semiconductor structures.

It is shown that, above some critical value of Sommerfeld number, a condition of hydrodynamic lubrication may exist. An elementary theory is developed which is consistent with experimental results, the viscosity of typical lubricants and the pressure dependence of viscosity of the lubricant. [ABSTRACT FROM AUTHOR]

The realization problem of nonlinear time-varying input-output equations is considered. Differentials of the state coordinates, necessary for realization, are determined by the vector space of differential one-forms, spanned over the field of meromorphic functions. Formulas for computing the basis one-forms are given, based on the Euclidean division of noncommutative polynomials. Moreover, it is shown that in the case of a reducible system, the subspace admits a basis with certain structure, explicitly related to reduced input-output equations. [ABSTRACT FROM AUTHOR]

The one-particle Dirac Hamiltonian with Coulomb interaction is known to be realised, in a regime of large (critical) couplings, by an infinite multiplicity of distinct self-adjoint operators, including a distinguished physically most natural one. For the latter, Sommerfeld’s celebrated fine structure formula provides the well-known expression for the eigenvalues in the gap of the continuum spectrum. Exploiting our recent general classification of all other self-adjoint realisations, we generalise Sommerfeld’s formula so as to determine the discrete spectrum of all other self-adjoint versions of the Dirac-Coulomb Hamiltonian. Such discrete spectra display naturally a fibred structure, whose bundle covers the whole gap of the continuum spectrum. [ABSTRACT FROM AUTHOR]

Based on the generalized Rayleigh-Sommerfeld diffraction integral, the equations of nonparaxial propagation, far field propagation and paraxial propagation of a partially coherent four-petal Gaussian beam in free space are obtained. The results show that the far field propagation and paraxial propagation can be regarded as special cases of nonparaxial propagation. The normalized intensity and degree of coherence of a nonparaxial partially coherent four-petal Gaussian beam are illustrated and analyzed using numerical example. [ABSTRACT FROM AUTHOR]

We present analytical formulas for the Sommerfeld corrections to the annihilation of massive colored particles into quarks and gluons through the strong interaction. These corrections are essential to accurately compute the darkmatter relic density for coannihilationwith colored partners. Our formulas allow us to compute the Sommerfeld effect, not only for the lowest term in the angular momentum expansion of the amplitude, but for all orders in the partial wave expansion. In particular, we carefully account for the effects of the spin of the annihilating particle on the symmetry of the two-particle wave function. This work focuses on strongly interacting particles of arbitrary spin in the triplet, sextet and octet color representations. For typical velocities during freeze-out, we find that including Sommerfeld corrections on the next-to-leading order partial wave leads to modifications of up to 10 to 20 percent on the total annihilation cross section. Complementary to QCD, we generalize our results to particles charged under an arbitrary unbroken SU(N) gauge group, as encountered in dark glueball models. In connection with this paper a Mathematica notebook is provided to compute the Sommerfeld corrections for colored particles up to arbitrary order in the angular momentum expansion. [ABSTRACT FROM AUTHOR]

This paper presents a new control methodology for a class of integrating systems. A PID controller augmented with a first order lead/lag filter is proposed for improved response. The polynomial approach is employed to derive the controller parameters. The novelty of the proposed method lies in the selection of pole locations. Multiple pole locations are considered where one of the poles is placed for cancelling the zero introduced by controller. The selection of tuning parameter is based on maximum sensitivity (MS). Set point weighting is employed to reduce the overshoot and settling time in the servo response. Various bench marking examples are adopted to evaluate the proposed method in terms of various performance indices. The results are superior to the recently proposed works in terms of both set point tracking and disturbance rejection. [ABSTRACT FROM AUTHOR]

Linear parameter varying (LPV) controller synthesis is a systematic approach for designing gain-scheduled controllers. The advances in LPV controller synthesis have spurred the development of system identification techniques that deliver the required models. The aim of this paper is to present an accurate and fast frequency response function (FRF) identification methodology for LPV systems. A local parametric modeling approach is developed that exploits smoothness over frequencies and scheduling parameters. By exploiting the smoothness over frequency as well as over the scheduling parameters, increased time efficiency in experimentation time and accuracy of the FRF is obtained. Traditional approaches, i.e., the local polynomial method/local rational method, are recovered as a special case of the proposed approach. The application potential is illustrated by a simulation example as well as real-life experiments on a medical X-ray system. [ABSTRACT FROM AUTHOR]

In a recent study, the classical problem of a large circular loop antenna carrying uniform current and situated at the Earth's surface has been revisited, with the scope to derive a totally analytical explicit expression for the radial distribution of the generated magnetic field. Yet, the solution arising from the study exhibits two major drawbacks. First, it describes the vertical magnetic field component only. Second, it is a valid subject to the quasi-static field assumption, which limits its applicability to the low-frequency range. The purpose of the present work is to provide the exact canonical solution to the problem, describing all the generated electromagnetic field components and valid in both the quasi-static and non-quasi-static frequency regions. These two features constitute an improvement with respect to the preceding solution. The canonical solution, which is obtained by reducing the field integrals to combinations of known Sommerfeld integrals, is seen to be also advantageous over the previous numerical and analytical-numerical approaches, since its usage takes negligible computation time. Numerical simulations are performed to show the accuracy of the obtained field expressions and to investigate the behavior of the above surface ground-and lateral-wave contributions to the fields in a wide frequency range. It is shown that in the near-zone the two waves do not predominate over each other, while the effect of the lateral wave becomes negligible only when the source-receiver distance is far greater than the skin depth in the Earth. [ABSTRACT FROM AUTHOR]

We study the quantum spectral curve of the Argyres-Douglas theories in the Nekrasov-Sahashvili limit of the Omega-background. Using the ODE/IM correspondence we investigate the quantum integrable model corresponding to the quantum spectral curve. We show that the models for the A -type theories are non-unitary coset models ( A ) × ( A ) /( A ) at the fractional level $$ L=\frac{2}{2N+1}-2 $$ , which appear in the study of the 4d/2d correspondence of $$ \mathcal{N} $$ = 2 superconformal field theories. Based on the WKB analysis, we clarify the relation between the Y-functions and the quantum periods and study the exact Bohr-Sommerfeld quantization condition for the quantum periods. We also discuss the quantum spectral curves for the D and E type theories. [ABSTRACT FROM AUTHOR]

We show that a set of n algebraic plane curves of constant maximum degree can be cut into O ( n 3 / 2 polylog n ) Jordan arcs, so that each pair of arcs intersect at most once, i.e., they form a collection of pseudo-segments . This extends a similar (and slightly better) bound for pseudo-circles due to Marcus and Tardos. Our result is based on a technique of Ellenberg, Solymosi and Zahl that transforms arrangements of plane curves into arrangements of space curves, so that lenses (pairs of subarcs of the curves that intersect at least twice) become vertical depth cycles. We then apply a variant of a technique of Aronov and Sharir to eliminate these depth cycles by making a small number of cuts, which corresponds to a small number of cuts to the original planar arrangement of curves. After these cuts have been performed, the resulting curves form a collection of pseudo-segments. Our cutting bound leads to new incidence bounds between points and constant-degree algebraic curves. The conditions for these incidence bounds are slightly stricter than those for the current best-known bound of Pach and Sharir; for our result to hold, the curves must be algebraic and of bounded maximum degree, while Pach and Sharir's bound only imposes weaker, purely topological constraints on the curves. However, when our conditions hold, the new bounds are superior for almost all ranges of parameters. We also obtain new bounds on the complexity of a single level in an arrangement of constant-degree algebraic curves, and a new bound on the complexity of many marked faces in an arrangement of such curves. [ABSTRACT FROM AUTHOR]

The paper deals with the asymptotic description of a diffraction pattern similar to the classical Weyl-Van der Pol phenomenon (the Weyl-Van der Pol formula). The latter arises in the problem of diffraction of waves generated by a source located near an impedance plane. An incident wave illuminates an impedance wedge or cone. The singular points of the wedge's (the edge points) or cone's (the vertex of the cone) boundary play the role of an imaginary source, giving rise to a specific boundary layer in some neighborhood of the corresponding impedance surface, provided that the surface impedance is relatively small. From the mathematical point of view, the description of the phenomenon is given by means of the far field asymptotics for the Sommerfeld integral representations of the scattered field. For small impedance of the scattering surface, the singularities describing the surface wave, which propagates from the edge (or from the vertex) along the impedance surface, may be located in a neighborhood of saddle points. The latter are responsible for a cylindrical wave from the edge of the wedge (or for a spherical wave from the vertex of the cone). As a result, the asymptotics of the Sommerfeld integral are uniformly represented by a Fresnel type integral for the wedge problem or by a parabolic cylinder type function for the cone problem. [ABSTRACT FROM AUTHOR]

To investigate the effects of ionospheric single thin layer height on GPS based total electron content (TEC) derivation by a single station, the GPS dual frequency pseudoranges and carrier phases received in Beijing at solar minimum and maximum are used for derivation at different heights through grid method, Kalman filtering method and polynomial method respectively. Detailed analyses proceed to be conducted on derived vertical TEC (VTEC) and estimated instrumental biases as well as residual errors of slant TEC (STEC) and errors of satellite biases, during which derived VTEC is compared with VTEC offered by CODE and IRI2012. VTEC at receiver zenith derived through three methods tends to increase with the height, whereas that through grid method at relatively large height may be unsolvable. Biases estimated via Kalman filtering method and polynomial method are less affected by the height. Those via grid method at height below 400 km are similar to those via other two methods, but the errors rise sharply above that value. Under the research condition the optimum height for Kalman filtering method as well as polynomial method at solar minimum is 380 km, and 310 km at solar maximum, while for grid method the optimum height ranges from 280 km to 350 km. [ABSTRACT FROM AUTHOR]

In order to better study the trajectory of robot motion, a motion trajectory planning and simulation based on 6-DOF manipulator arm robot is designed. Three and five polynomial methods are adopted to plan motion trajectories. The advantages and disadvantages of the three and five polynomial trajectory planning are summarized. The trajectory planning in the right angle space can make the robot arm work according to the prescribed movement path, and realize the continuous and controllable movement path. Thus, the study of six DOF robot arm and mobile robot has broad application fields. The experimental results show that the intelligent robot is the highest performance of mechatronics, and it continues to be active in the field of scientific and technological development. Based on the above finding, it is concluded that the five -polynomial trajectory planning can guarantee the continuous angular acceleration of the joint, make the steering gear run smoothly and reduce the mechanical wear of the motor. [ABSTRACT FROM AUTHOR]

Great attention has been paid to the propagation of electromagnetic (EM) waves across the sea surface due to its important applications. Most of the previous research, however, focus on the half-space model illustrating the deep sea environment. In this communication, by taking into account the presence of a seafloor, the EM fields generated above shallow sea by a submerged horizontal electric dipole have been investigated theoretically and experimentally with a three-layer model. A set of formulas for EM fields in air expressed by Sommerfeld integrals are derived with recursive propagation approach, and the fields are computed using a complementary numerical integration technique. The effects of frequency, sea depth, seafloor conductivity, and receiver height on the fields are discussed. An experiment was conducted on shallow sea, and the test results agree well with theoretical predication in the quasi-near range. A transmission distance over 3 km above the sea surface was realized with reasonable transmitting power, which shows the potential of our theoretical model for applications in shallow sea environment. [ABSTRACT FROM PUBLISHER]

A low-frequency approximation of the discrete Sommerfeld diffraction problems, involving the scattering of a time harmonic lattice wave incident on square lattice by a discrete Dirichlet or a discrete Neumann half-plane, is investigated. It is established that the exact solution of the discrete model converges to the solution of the continuum model, i.e., the continuous Sommerfeld problem, in the discrete Sobolev space defined by Hackbusch. A proof of convergence has been provided for both types of boundary conditions when the imaginary part of incident wavenumber is positive. [ABSTRACT FROM AUTHOR]

We show a method for constructing a polynomial interpolating roots' multiplicities of another polynomial, that does not use companion matrices. This leads to a modification to Guersenzvaig-Szechtman square-free decomposition algorithm that is more efficient both in theory and in practice. [ABSTRACT FROM AUTHOR]

Essentially non-oscillatory (ENO) and weighted ENO (WENO) methods are efficient high order numerical methods for solving hyperbolic conservation laws designed to reduce the Gibbs oscillations. The original ENO and WENO methods are based on the polynomial interpolation and the overall convergence rate is uniquely determined by the total number of interpolation points involved for the approximation. In this paper, we propose non-polynomial ENO and WENO finite volume methods in order to enhance the local accuracy and convergence. The infinitely smooth radial basis functions (RBFs) are adopted as a non-polynomial interpolation basis. Particularly we use the multi-quadratic and Gaussian RBFs. The non-polynomial interpolation such as the RBF interpolation offers the flexibility to control the local error by optimizing the free parameter. Then we show that the non-polynomial interpolation can be represented as a perturbation of the polynomial interpolation. To guarantee the essentially non-oscillatory property, the monotone polynomial interpolation method is introduced as a switching method to the polynomial reconstruction adaptively near the non-smooth area. The numerical results show that the developed non-polynomial ENO and WENO methods with the monotone polynomial interpolation method enhance the local accuracy and give sharper solution profile than the ENO/WENO methods based on the polynomial interpolation. [ABSTRACT FROM AUTHOR]

Motivated by the Sommerfeld diffraction problem, we consider interface problems for the n-dimensional Helmholtz equation in [ABSTRACT FROM AUTHOR]

We define the bisector energy $$\mathcal E(\mathcal P)$$ of a set $$\mathcal P$$ in $$\mathbb R^2$$ to be the number of quadruples $$(a,b,c,d)\in \mathcal P^4$$ such that a, b determine the same perpendicular bisector as c, d. Equivalently, $$\mathcal E(\mathcal P)$$ is the number of isosceles trapezoids determined by $$\mathcal P$$ . We prove that for any $$\varepsilon >0$$ , if an n-point set $$\mathcal P$$ has no M( n) points on a line or circle, then we have We derive the lower bound $$\mathcal E(\mathcal P)=\Omega (M(n)n^2)$$ , matching our upper bound when M( n) is large. We use our upper bound on $$\mathcal E(\mathcal P)$$ to obtain two rather different results: [ABSTRACT FROM AUTHOR]

In this paper, the classic problem of determining the fields of a vertically oriented current-carrying small circular-loop antenna located near a plane air-earth boundary is revisited. It is well known that the Sommerfeld integrals describing the field components generated by the loop source cannot be exactly evaluated and that their numerical integration is made difficult by the presence of highly oscillatory terms in the integrands. Here, for both source and observation points on the interface, a rigorous method is developed that allows analytical integration of the Sommerfeld integrals, providing fast-convergent series representations for the surface fields. The obtained formulas permit to overcome all the drawbacks of the previously published approximate solutions to this problem, and to avoid the necessity of using purely numerical approaches when high computational accuracy is required. At the same time, the proposed series representations constitute an analytical benchmark for numerical procedures employed to solve electromagnetic boundary value problems, with applications in antenna design and close-to-the-surface communication. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Considerable efforts have been made to investigate the electromagnetic (EM) field in air produced by a magnetic dipole in sea. The conventional methods for this investigation mostly involve complicated analysis of Sommerfeld-type integrals. To conveniently provide a meaningful physical explanation of the EM field radiation in a two-layered media, this study considers a current sheet at the sea-to-air interface created by an underwater magnetic dipole. This current behaves like an antenna located on the surface and creates a wave that propagates through the atmosphere. Analysis of this current sheet can take advantage of the Huygens-Fresnel principle. The results using this approach can be verified by previous calculation and experiments on the sea, and show good agreement with experimental data. The present approach, which involves simple algebra, can easily explain the physical aspect of the EM wave propagation across the sea-to-air interface. [ABSTRACT FROM AUTHOR]

We generalize the Szemerédi-Trotter incidence theorem, to bound the number of complete flags in higher dimensions. Specifically, for each $$i=0,1,\ldots ,d-1$$ , we are given a finite set $$S_i$$ of i-flats in $${\mathbb {R}}^d$$ or in $${\mathbb {C}}^d$$ , and a (complete) flag is a tuple $$(f_0,f_1,\ldots ,f_{d-1})$$ , where $$f_i\in S_i$$ for each i and $$f_i\subset f_{i+1}$$ for each $$i=0,1,\ldots ,d-2$$ . Our main result is an upper bound on the number of flags which is tight in the worst case. We also study several other kinds of incidence problems, including (i) incidences between points and lines in $${\mathbb {R}}^3$$ such that among the lines incident to a point, at most O(1) of them can be coplanar, (ii) incidences with Legendrian lines in $${\mathbb {R}}^3$$ , a special class of lines that arise when considering flags that are defined in terms of other groups, and (iii) flags in $${\mathbb {R}}^3$$ (involving points, lines, and planes), where no given line can contain too many points or lie on too many planes. The bound that we obtain in (iii) is nearly tight in the worst case. Finally, we explore a group theoretic interpretation of flags, a generalized version of which leads us to new incidence problems. [ABSTRACT FROM AUTHOR]

An extended version of the Bogomolny-Prasad-Sommerfeld (BPS) Skyrme model that admits time-dependent solutions is discussed. Initially, by introducing a power law at the original potential term of the BPS Skyrme model, the existence, stability, and structure of the corresponding solutions are investigated. Then, the frequency and half-lifes of the radial oscillations of the constructed time-dependent solutions are determined. [ABSTRACT FROM AUTHOR]

An exact solution is provided for a discrete analog of each of the two Sommerfeld diffraction problems using a triangular lattice model, with nearest neighbor interactions, deforming in the anti-plane direction. The discrete Helmholtz equation with time-harmonic data prescribed on semi-infinite row(s) of lattice sites is solved using the discrete Wiener–Hopf method. An asymptotic expression of the exact solution, given in the form of a contour integral, has been obtained in far field using the standard approximation of the diffraction integrals based on the method of stationary phase. In case of the rigid constraint diffraction, the displacement of particles on the semi-infinite row complementing the constrained lattice sites, as well as on the adjacent row, is presented in closed form as a discrete convolution. For the crack diffraction problem, the length of both types of slant bonds on the semi-infinite row complementing the crack, as well as the crack opening displacement, is given in a similar form but in terms of the Fourier coefficients, of the associated Wiener–Hopf kernel, which are not available in closed form. The displacement field associated with the scattered waves at sites far from the defect tip, as well as few sites near the tip, is compared graphically with that of a numerical solution on a finite grid. Both discrete Sommerfeld problems are naturally relevant to their continuous counterparts, involving the traditional Helmholtz equation, that model the diffraction of electromagnetic and acoustic waves by a semi-infinite screen, based on 7-point discretization on a triangular grid. [ABSTRACT FROM AUTHOR]

The Sommerfeld halfspace problem is revisited allowing for multilayer and plasmonic media, with a focus on the surface field computation. Alternative field representations are discussed and an improved Sommerfeld–Norton groundwave formula is developed. The role of the Zenneck wave and surface plasmon polariton in the surface field and their relationship is elucidated. The theoretical development is illustrated and validated by representative numerical results. [ABSTRACT FROM PUBLISHER]

Rotor dynamic systems are often analyzed with ideal drive assumption. However, all drives are essentially non-ideal, i.e., they can only provide a limited amount of power. One basic fact often ignored in rotor dynamics studies is that the drive dynamics has complex coupling with the dynamics of the driven system. Increase in drive power input near resonance may contribute to increasing the transverse vibrations rather than increasing the rotor spin, which is referred to as the Sommerfeld effect. In this article, we generate the rotor response with finite element (FE) model by assuming an ideal drive. Thereafter, the rotor system’s response with ideal drive is used in a power balance equation to theoretically predict the amplitude and speed characteristics of the same rotor system when it is driven through a non-ideal drive. The integrated system with drive-rotor interaction is modeled in bond graph (BG) form and the transient analysis from the BG model is used to validate the theoretical results. The results are important from the point of actuator sizing for rotor dynamic systems. [ABSTRACT FROM AUTHOR]

In this paper the generalized form of a non-ideal system which contains a pure nonlinear oscillator and a non-ideal energy source is studied. In the non-ideal oscillator-motor system there is an interaction between the motions of the oscillator and those of the motor, as the motor has an influence on the oscillator and vice versa. The mathematical model of the system is represented with two coupled nonlinear differential equations. The averaging method for solving these differential equations is based on the application of the Ateb function, which is the exact solution of the pure nonlinear oscillator. Using the obtained approximate solution, the resonant motion of the system is considered. Significant attention is paid to the steady-state motion and to the Sommerfeld effect. The influence of the order of nonlinearity on the dynamics of the nonideal system is evident. In the paper the procedure for determination of the parameters for suppression of the Sommerfeld effect of the non-ideal system is also given. [ABSTRACT FROM PUBLISHER]

Discussed are some classical and quantization problems of test affinely-rigid bodies moving in Riemannian spaces. We investigate the systems with potential energies for which the variables can be separated. The special case of constant curvature two-dimensional spaces is discussed. Some explicit solutions are found using the Sommerfeld polynomial method. [ABSTRACT FROM AUTHOR]

Intervalley charged-impurity scattering processes are examined. It is found that the scattering probability is enhanced due to the Coulomb interaction with the impurity by the Sommerfeld factor $$F_z \propto \varepsilon ^{2\sqrt {1 - 4g^2 } - 2}$$; where ∈ is the electron energy and g is the dimensionless constant of the Coulomb interaction. [ABSTRACT FROM AUTHOR]

We give a shorter proof of a slightly weaker version of a theorem from Guth and Katz (Ann Math 181:155-190, ): we prove that if $$\mathfrak {L}$$ is a set of $$L$$ lines in $$\mathbb {R}^3$$ with at most $$L^{1/2}$$ lines in any low degree algebraic surface, then the number of $$r$$ -rich points of $$\mathfrak {L}$$ is $$\lesssim L^{(3/2) + \varepsilon } r^{-2}$$ . This result is one of the main ingredients in the proof of the distinct distance estimate in Guth and Katz (). With our slightly weaker theorem, we get a slightly weaker distinct distance estimate: any set of $$N$$ points in $$\mathbb {R}^2$$ determines at least $$c_{\varepsilon } N^{1 -{\varepsilon }}$$ distinct distances. [ABSTRACT FROM AUTHOR]

The problem considered is a vertical electrically small wire antenna located underground, which transmits electromagnetic signals to the ground. Getting Green’s function of the vertical dipole underground was the first step to calculate this issue. A quasistatic situation was considered to make an approximation on Sommerfeld integral for easy solution. The method of moments was used to solve the current distribution on the antenna surface at different frequencies, which laid a good foundation for obtaining the electric field of the antenna. Then the axial and radial components of the electric field with the radial distance on the ground were investigated, as well as the voltage received on the ground. Furthermore, the influence of the frequency and stratum parameters on current and electric field was studied to understand the variation clearly. [ABSTRACT FROM AUTHOR]

The paper arose from the fact that the smallest element of the set of Salem numbers is not known. Indeed, it is not even known whether this set has a smallest element. The aim of this paper is to prove that the minimal polynomial of the smallest Salem series of degree n in the field of formal power series over a finite field is given by P(Y ) = Y n - XY n-1 - Y + X - 1, where we suppose that 1 is the least element of the finite field F-1 q (as a finite total ordered set). Consequently, we are led to deduce that F-1((X-1)) has no smallest Salem series. Moreover, we will prove that the root of P(Y ) of degree n = 2s + 1 in F-1m((X-1)) is well approximable. [ABSTRACT FROM AUTHOR]

The paper arose from the fact that the smallest element of the set of Salem numbers is not known. Indeed, it is not even known whether this set has a smallest element. The aim of this paper is to prove that the minimal polynomial of the smallest Salem series of degree n in the field of formal power series over a finite field is given by P(Y ) = Y n - XY n-1 - Y + X - 1, where we suppose that 1 is the least element of the finite field F-1 q (as a finite total ordered set). Consequently, we are led to deduce that F-1((X-1)) has no smallest Salem series. Moreover, we will prove that the root of P(Y ) of degree n = 2s + 1 in F-1m((X-1)) is well approximable. [ABSTRACT FROM AUTHOR]

The authors present a formulation of the Sommerfeld integration method to calculate antenna radiation in an arbitrary stack of dielectric slabs. Historically, Sommerfeld integration, in the method of moments, has been used only for two semi-infinite layers (the lower layer often referred to as a ground plane). Here, the authors have returned to Sommerfeld's original formulation to extend the method for an arbitrary number of layers. The authors show that this algorithm gives accurate results as proved by comparison to other simulations and approximate analytical formulas. The authors motivation is to calculate antenna coupling in a seawater environment at ultra-low frequencies, taking account of seawaves at the seabed and water surface. This is an important problem in marine engineering which cannot be addressed by finite element methods owing to the high aspect ratios and 106-order jumps in complex permittivity across the different layers. As well as validating the algorithm for multiple layers the authors show results from a realistic air/sea/seabed setup. The authors demonstrate that, if the antenna is close to the seabed, surface waves can form on the sea/seabed interface as well as the air/sea interface potentially improving reception. [ABSTRACT FROM AUTHOR]

The objective of this presentation is to illustrate the accuracy of the Schelkunoff formulation over the Sommerfeld solution for a vertical electric dipole radiating over an imperfect ground. In an earlier paper, the alternate form of the Sommerfeld Green's function developed by Schelkunoff was presented (Schelkunoff, 1943 and Dyab, 2013). Here we demonstrate the application of this new methodology for two classes of problems. First, the problem of predicting the propagation path loss in a wireless communication environment is illustrated. The second application problem described in this paper deals with the verification of experimental data related to propagation over an Aluminum sheet at THz frequencies. It is seen that the main contribution of the reflected field is due to a specular image point as expected for a metal and the presence of surface waves in the total reflected field is absent, even though the permittivity of the metal is negative at these frequencies. Both theoretical predictions and experimental data demonstrate that there is little contribution to the reflected field due to a surface wave. Also, a clear definition is made to characterize surface waves as there is confusion as to what a surface wave really is. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper we develop numerical approximations of the wave equation in mixed form supplemented with non-reflecting boundary conditions (NRBCs) of Sommerfeld-type on artificial boundaries for truncated domains. We consider three different variational forms for this problem, depending on the functional space for the solution, in particular, in what refers to the regularity required on artificial boundaries. Then, stabilized finite element methods that can mimic these three functional settings are described. Stability and convergence analyses of these stabilized formulations including the NRBC are presented. Additionally, numerical convergence test are evaluated for various polynomial interpolations, stabilization methods and variational forms. Finally, several benchmark problems are solved to determine the accuracy of these methods in 2D and 3D. [Copyright &y& Elsevier]

Abstract: For the generalized Fermi–Dirac integrals, , of orders k =−1/2, 1/2, 3/2, and 5/2, we explicitly obtained the first 11 terms of their Sommerfeld expansions. The main terms of the last three orders are rewritten so as to avoid the cancelation problem. If is not so small, say not less than 13.5, 12.0, 10.9, and 9.9 when k =−1/2, 1/2, 3/2, and 5/2, respectively, the first 8 terms of the expansion assure the single precision accuracy for arbitrary value of . Similarly, the 15-digits accuracy is achieved by the 11 terms expansion if is greater than 36.8, 31.6, 30.7, and 26.6 when k =−1/2, 1/2, 3/2, and 5/2, respectively. Since the truncated expansions are analytically given in a closed form, their computational time is sufficiently small, say at most 4.9 and 6.7 times that of the integrand evaluation for the 8- and 11-terms expansions, respectively. When is larger than a certain threshold value as indicated, these appropriately-truncated Sommerfeld expansions provide a factor of 10–80 acceleration of the computation of the generalized Fermi–Dirac integrals when compared with the direct numerical quadrature. [Copyright &y& Elsevier]

In this article, in a simple way, simple relations are derived between the electric field components of an electrically uniaxial medium and those of an isotropic medium. The permittivity of the isotropic medium is the same as the permittivity of the uniaxial medium that is common to the axes transverse to the optic axis. Using the spectral representation, the vector wave equation for the electric field intensity vector of the uniaxial medium is solved for the x directed, y directed and z directed point sources. For the x directed and y directed point sources, the electric field components transverse to the optic axis are written in terms of the corresponding components of the isotropic medium plus some other terms. Part of these terms are closed forms expressions, and the rest are Sommerfeld type integrals. Elements of each group are related to each other by coordinate transformations. The electric field components parallel to the optic axis are shown to be obtained from the isotropic medium components using coordinate transformations. The relations between the uniaxial medium and isotropic medium field components are verified by comparing the results of a previous study in the literature to the results obtained using the relations in this study. Good agreement is achieved between these results. [ABSTRACT FROM AUTHOR]

The d-dimensional Hamming torus is the graph whose vertices are all of the integer points inside an a1n × a2n × ⋅⋅⋅ × adn box in $\mathbb{R}^d$ (for constants a1, . . ., ad > 0), and whose edges connect all vertices within Hamming distance one. We study the size of the largest connected component of the subgraph generated by independently removing each vertex of the Hamming torus with probability 1 − p. We show that if p = λ/n, then there exists λc > 0, which is the positive root of a degree d polynomial whose coefficients depend on a1, . . ., ad, such that for λ < λc the largest component has O(log n) vertices (w.h.p. as n → ∞), and for λ > λc the largest component has $(1-q) \lambda \bigl(\prod_i a_i \bigr) n^{d-1} + o (n^{d-1})$ vertices and the second largest component has O(log n) vertices w.h.p. An implicit formula for q < 1 is also given. The value of λc that we find is distinct from the critical value for the emergence of a giant component in bond percolation on the Hamming torus. [ABSTRACT FROM AUTHOR]

A constitutive integral equation (CIE) formulation is proposed to model the lightning return stroke. From the CIE, the electromagnetic field radiated by the return stroke considering the lossy ground is determined by using the moment method (MM). To implement a fast numerical and rigorous approach, an exact image theory is first employed to accelerate the computation of Sommerfeld integral, and a translational mesh is then constructed under the thin-wire approximation. Such grid representation and the convolution properties of the integral kernel equip the fast Fourier transform to have the ability to efficiently solve the MM matrix equations. The feasibility and merits of the proposed method are positively confirmed by the comparison of its performances with those of an existing one on a case study. [ABSTRACT FROM AUTHOR]

We consider statements of the elasticity problem about the wave propagation in an elastic cylindrical waveguide under different radiation conditions at infinity. For the energy radiation conditions we develop the mathematical theory based on the Mandelstam principle and the Umov–Poynting vector, as well as the technique of weighted spaces with detached asymptotics and the energy flux (symplectic) form. As is shown by examples, the Sommerfeld radiation conditions and the limiting absorption principle contradict the Mandelstam radiation conditions in certain situations. Bibliography: 64 titles. Illustrations: 8 figures. [ABSTRACT FROM AUTHOR]

We analyze the description of quantum many-body mixed states using matrix product states and operators. We consider two such descriptions: (i) as a matrix product density operator of bond dimension D; and (ii) as a purification that is written as a matrix product state of bond dimension D′. We show that these descriptions are inequivalent in the sense that D′ cannot be upper bounded by D only. Then we provide two constructive methods to obtain (ii) out of (i). The sum of squares (sos) polynomial method scales exponentially in the number of different eigenvalues, and its approximate version is formulated as a semidefinite program, which gives efficient approximate purifications whose D′ only depends on D. The eigenbasis method scales quadratically in the number of eigenvalues, and its approximate version is very efficient for rapidly decaying distributions of eigenvalues. Our results imply that a description of mixed states which is both efficient and locally positive semidefinite does not exist, but that good approximations do. [ABSTRACT FROM AUTHOR]

In this paper, we study the 3D Helmholtz equation in perturbed stratified media, allowing the presence of guided waves. Our assumptions on the perturbed and source terms are too few. On the basis of the knowledge of the Green's function for the 3D homogeneous Helmholtz equation in unperturbed stratified media, we introduce a generalized (out-going) Sommerfeld-Rellich radiation condition, and then we prove the uniqueness and existence of solutions for the studied 3D Helmholtz equation satisfying our radiation condition. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this article I shall present a brief review of the hundred-year young Bohr model of the atom. In particular, I will first introduce the Thomson and the Rutherford models of atoms, their shortcomings and then discuss in some detail the development of the atomic model by Niels Bohr. Further, I will mention its refinements at the hand of Sommerfeld and also its shortcomings. Finally, I will discuss the implication of this model in the development of quantum mechanics. [ABSTRACT FROM AUTHOR]