Concise Analytical Form Research Articles

Wave propagation in stress-driven nonlocal Rayleigh beam lattices

This paper focuses on small-size planar beam lattices, where size effects are modelled by the stress-driven nonlocal elasticity theory in conjunction with the Rayleigh beam theory. The purpose is to propose two novel computational approaches for elastic wave propagation analysis. In a first dynamic-stiffness approach, every lattice member is modelled by a unique two-node beam element, the exact dynamic-stiffness matrix of which is built solving, in concise analytical form, the stress-driven differential equations of motion. In a second finite-element approach, every lattice member is discretized by an increasingly refined mesh of two-node beam elements; in this case, the stiffness and mass matrices of the lattice member are obtained from shape functions built based on the exact solutions of the stress-driven differential equations for static equilibrium. Advantages of the two approaches are compared and discussed. Dispersion curves are calculated for a typical planar lattice, highlighting the role of nonlocality.

Frequency-dependent Factor Expression of the GTD Scattering Center Model for the Arbitrary Multiple Scattering Mechanism

This paper presents a derivation of a formula with a concise and uniform analytic form by the Stationary Phase Method (SPM) plus Geometrical Optics (GO), the Physical Theory of Diffraction (PTD), and Geometrical Theory of Diffraction (GTD) to calculate the frequency-dependent factor for the arbitrary multiple scattering mechanism, validated by the simulated and measured data of a series of canonical ensembles, validated by the simulated and measured data of a series of canonical ensembles. Although the GTD model, a scattering center model, can accurately describe the frequency-dependent characteristic of several main scattering mechanisms of the radar target, no explicit and general expression relates the frequency-dependent factor to the type of scattering mechanism. The derived formula relates the scattering center’s frequency-dependent factor with bounce times, dimensions of all the encountered geometrical elements, and a caustic type of ray contributing to the scattering center and can be applied to determine the parameter value of frequency-dependent factor of the GTD model and its derived versions in the forward parametric modeling.

A simplified model for fast analysis of the deployment dynamics of tethered-net in space

The tethered-net is one of the most promising flexible space debris cleaning technologies to be developed in recent years. Momentum and geometric analysis were used to select four characteristic parameters of the tethered-net ejection system—the radius of the circumcircle of the net, the ratio of the bullet mass to the net mass, the ejection velocity of the bullet, and the ejection angle—to establish a set of formulas that describe the kinematics of the net deployment. The net deployment performance evaluation criteria were obtained with a concise analytical form. The calculation results of this set of analytical formulas are in good agreement with the numerical simulation results of the mass-spring model. Furthermore, this set of analytical formulas reveals some intrinsic characteristics of the net deployment dynamics. For example, the flying distance of the net from the start of ejection to the largest area is mainly determined by the radius of the circumcircle of the net and ejection angle. It is believed that the analytic formulas proposed are suitable for the preliminary design of a tethered-net system, especially when numerical simulation is time consuming.

Thermodynamics of two aligned Kerr-Newman black holes

Using the notion of thermodynamic length, the first law of thermodynamics is consistently derived for two binary configurations of equal Kerr-Newman black holes separated by a massless strut. Like in the electrostatic systems of two Reissner-Nordstr\"om black holes and stationary vacuum systems of two Kerr black holes considered earlier, the thermodynamic length $\ell$ turns out to be defined by the same simple formula $\ell=L\exp(\gamma_0)$, $L$ being the coordinate length of the strut and $\gamma_0$ the value of the metric function $\gamma$ on the strut, which permits the elaboration of $\ell$ in a concise analytic form. The expression of the free energy in the case of two generic Kerr-Newman black holes is also proposed.

Abstract representations of events arise from mental errors in learning and memory

Humans are adept at uncovering abstract associations in the world around them, yet the underlying mechanisms remain poorly understood. Intuitively, learning the higher-order structure of statistical relationships should involve complex mental processes. Here we propose an alternative perspective: that higher-order associations instead arise from natural errors in learning and memory. Using the free energy principle, which bridges information theory and Bayesian inference, we derive a maximum entropy model of people’s internal representations of the transitions between stimuli. Importantly, our model (i) affords a concise analytic form, (ii) qualitatively explains the effects of transition network structure on human expectations, and (iii) quantitatively predicts human reaction times in probabilistic sequential motor tasks. Together, these results suggest that mental errors influence our abstract representations of the world in significant and predictable ways, with direct implications for the study and design of optimally learnable information sources.

Resistive accretion flows around a Kerr black hole

AbstractIn this paper, we present the stationary axisymmetric configuration of a resistive magnetised thick accretion disc in the vicinity of external gravity and intrinsic dipolar magnetic field of a slowly rotating black hole. The plasma is described by the equations of fully general relativistic magnetohydrodynamics (MHD) along with the Ohm’s law and in the absence of the effects of radiation fields. We try to solve these two-dimensional MHD equations analytically as much as possible. However, we sometimes inevitably refer to numerical methods as well. To fully understand the relativistic geometrically thick accretion disc structure, we consider all three components of the fluid velocity to be non-zero. This implies that the magnetofluid can flow in all three directions surrounding the central black hole. As we get radially closer to the hole, the fluid flows faster in all those directions. However, as we move towards the equator along the meridional direction, the radial inflow becomes stronger from both the speed and the mass accretion rate points of view. Nonetheless, the vertical (meridional) speed and the rotation of the plasma disc become slower in that direction. Due to the presence of pressure gradient forces, a sub-Keplerian angular momentum distribution throughout the thick disc is expected as well. To get a concise analytical form of the rate of accretion, we assume that the radial dependency of radial and meridional fluid velocities is the same. This simplifying assumption leads to radial independency of mass accretion rate. The motion of the accreting plasma produces an azimuthal current whose strength is specified based on the strength of the external dipolar magnetic field. This current generates a poloidal magnetic field in the disc which is continuous across the disc boundary surface due to the presence of the finite resistivity for the plasma. The gas in the disc is vertically supported not only by the gas pressure but also by the magnetic pressure.

Periodic Green functions for 2D magneto-electric quadrupolar arrays: explicitly satisfying the optical theorem

We present all the periodic Green function dyadics that enter a description of a 2D array of emitters at the level that includes the electric dipole, magnetic dipole and electric quadrupole moment of each emitter. We find a concise analytic form for the radiative contributions to the periodic Green function dyadics that give rise to radiation reaction fields, and so our description of the scattered light explicitly satisfies the optical theorem; we give the non-radiative contributions that do not affect energy balance in the form of rapidly converging series. Finally, we present an approximation scheme for evaluating periodic Green function dyadics at long wavelengths that rigorously respects energy conservation. The scheme extends the range of validity of the usual static approximation by the inclusion of a simple dynamic correction.

Group V sensitive vapor–liquid–solid growth of Au-catalyzed and self-catalyzed III–V nanowires

We present a new theoretical model that treats the group V sensitive growth rates and structures of Au-catalyzed and self-catalyzed III–V nanowires within a single kinetic picture. It is shown that Au-catalyzed III–V nanowires can grow with a time-independent radius within a wide range of parameters. At high V/III flux ratios, the vapor–liquid–solid growth of Au catalyzed III–V nanowires is controlled by surface diffusion of the group III adatoms, while at low V/III flux ratios it becomes nucleation-limited. Conversely, self-catalyzed III–V nanowires cannot grow with a time-independent droplet size and instead such nanowires may either swell or shrink or converge to a certain stationary radius depending on the V/III flux ratio. Quite importantly, the results are presented in a concise analytical form which is convenient for comparison with experimental data or prior theoretical works. We demonstrate how the model fits the data obtained previously for Au- and Ga-catalyzed GaAs nanowires.

Metric of two balancing Kerr particles in physical parametrization

The present paper aims at elaborating a completely physical representation for the general 4-parameter family of the extended double-Kerr spacetimes describing two spinning sources in gravitational equilibrium. This involved problem is solved in a concise analytical form by using the individual Komar masses and angular momenta as arbitrary parameters, and the simplest equatorially symmetric specialization of the general expressions obtained by us yields the physical representation for the well-known Dietz-Hoenselaers superextreme case of two balancing identical Kerr constituents. The existence of the physically meaningful "black hole-superextreme object" equilibrium configurations permitted by the general solution may be considered as a clear indication that the spin-spin repulsion force might actually be by far stronger than expected earlier, when only the balance between two superextreme Kerr sources was thought possible. We also present the explicit analytical formulas relating the equilibrium states in the double-Kerr and double-Reissner-Nordstr\"om configurations.

Zernike expansion of derivatives and Laplacians of the Zernike circle polynomials.

The partial derivatives and Laplacians of the Zernike circle polynomials occur in various places in the literature on computational optics. In a number of cases, the expansion of these derivatives and Laplacians in the circle polynomials are required. For the first-order partial derivatives, analytic results are scattered in the literature. Results start as early as 1942 in Nijboer's thesis and continue until present day, with some emphasis on recursive computation schemes. A brief historic account of these results is given in the present paper. By choosing the unnormalized version of the circle polynomials, with exponential rather than trigonometric azimuthal dependence, and by a proper combination of the two partial derivatives, a concise form of the expressions emerges. This form is appropriate for the formulation and solution of a model wavefront sensing problem of reconstructing a wavefront on the level of its expansion coefficients from (measurements of the expansion coefficients of) the partial derivatives. It turns out that the least-squares estimation problem arising here decouples per azimuthal order m, and per m the generalized inverse solution assumes a concise analytic form so that singular value decompositions are avoided. The preferred version of the circle polynomials, with proper combination of the partial derivatives, also leads to a concise analytic result for the Zernike expansion of the Laplacian of the circle polynomials. From these expansions, the properties of the Laplacian as a mapping from the space of circle polynomials of maximal degree N, as required in the study of the Neumann problem associated with the transport-of-intensity equation, can be read off within a single glance. Furthermore, the inverse of the Laplacian on this space is shown to have a concise analytic form.

From Generalized Dirac Equations to a Candidate for Dark Energy

We consider extensions of the Dirac equation with mass terms m1+iγ5m2 and im1+γ5m2. The corresponding Hamiltonians are Hermitian and pseudo-Hermitian (γ5 Hermitian), respectively. The fundamental spinor solutions for all generalized Dirac equations are found in the helicity basis and brought into concise analytic form. We postulate that the time-ordered product of field operators should yield the Feynman propagator (iϵ prescription), and we also postulate that the tardyonic as well as tachyonic Dirac equations should have a smooth massless limit. These postulates lead to sum rules that connect the form of the fundamental field anticommutators with the tensor sums of the fundamental plane-wave eigenspinors and the projectors over positive-energy and negative-energy states. In the massless case, the sum rules are fulfilled by two egregiously simple, distinguished functional forms. The first sum rule remains valid in the case of a tardyonic theory and leads to the canonical massive Dirac field. The second sum rule is valid for a tachyonic mass term and leads to a natural suppression of the right-handed helicity states for tachyonic particles and left-handed helicity states for tachyonic spin-1/2 antiparticles. When applied to neutrinos, the theory contains a free tachyonic mass parameter. Tachyons are known to be repulsed by gravity. We discuss a possible role of a tachyonic neutrino as a contribution to the accelerated expansion of the Universe “dark energy.”

A note on two-loop superloop

We explore the duality between supersymmetric Wilson loop on null polygonal contours in maximally supersymmetric Yang–Mills theory and next-to-maximal helicity violating (NMHV) scattering amplitudes. Earlier analyses demonstrated that the use of a dimensional regulator for ultraviolet divergences, induced due to presence of the cusps on the loop, yields anomalies that break both conformal symmetry and supersymmetry. At one-loop order, these are present only in Grassmann components localized in the vicinity of a single cusp and result in a universal function for any number of sites of the polygon that can be subtracted away in a systematic manner leaving a well-defined supersymmetric remainder dual to corresponding components of the superamplitude. The question remains though whether components which were free from the aforementioned supersymmetric anomaly at leading order of perturbation theory remain so once computed at higher orders. Presently we verify this fact by calculating a particular component of the null polygonal super Wilson loop at two loops restricting the contour kinematics to a two-dimensional subspace. This allows one to perform all computations in a concise analytical form and trace the pattern of cancellations between individual Feynman graphs in a transparent fashion. As a consequence of our consideration we obtain a dual conformally invariant result for the remainder function in agreement with one-loop NMHV amplitudes.

Unified theory of linear instability of anisotropic surfaces and interfaces under capillary, electrostatic, and elastostatic forces: The regrowth of epitaxial amorphous silicon

The first-order unified linear instability analysis (LISA) of the governing equation for the evolution of surfaces and interfaces under capillary, electromigration (EM), and elastostatic forces is developed. A formal treatment of the thermomigration (Soret effect) driven by the nonuniform temperature distribution caused by exothermic phase transformation (growth) at the surface and interfacial layers is presented and its apparent influence on the capillary force in connection with the stability is also established in a concise analytical form. This unified approach, which relies on a rigorous theory of irreversible thermodynamics of surfaces and interfaces, seriously considers the anisotropies associated with the generalized growth mobility, the interfacial specific Gibbs free energy (i.e., the surface stiffness), and the surface diffusivity in thin solid films. The singularity in the surface stiffness at the cusp regions of the Wulff construction of the surface Gibbs free energy is fully elaborated by using a modified cycloid-curtate function as a basis for generating the Dirac $\ensuremath{\delta}$ distribution, which shows an unusually strong anomalous effect on the surface morphological instability even in the absence of EM forces, as illustrated clearly by the graphical representation of the EM-induced instability threshold level as a function of tilt angle and wave number, in a three-dimensional plot for various intrinsic and normalized system parameters. In the development of LISA theory special attention is paid to the origin of the elastostatic forces, which include not only the elastic strain energy density, but also the elastic dipole tensor interaction between mobile atomic species and the applied stress field. The profound influence of the anomalous surface stiffness anisotropy on the surface morphological evolution under the applied stress system is demonstrated by three-dimensional computer graphics applied for copper and silicon thin single-crystal solid films having, respectively, sixfold {111}- and fourfold {100}-symmetric singular (vicinal) planes as the top surfaces, which reveal the fine features of the theory and give insight into some controversial issues related to LISA in the literature. This unified approach also considers the stress dependence of the generalized growth mobility and its profound influence on the stability of the interface displacement and roughening in thin solid films. As a special application of the theory, the effects of uniaxial and biaxial applied stresses on the recrystallization and the interfacial morphological evolution of amorphous Si deposited on silicon substrates are thoroughly analyzed and excellent quantitative agreement is found with the published experimental data in the literature.

Black Hole Magnetospheres around Thin Disks Driving Inward and Outward Winds

We construct a simple model for stationary, axisymmetric black hole magnetospheres, in which the poloidal magnetic field is generated by a toroidal electric current in a thin disk with an inner edge, by solving the vacuum Maxwell equations in the Schwarzschild background. In this work, to obtain a concise analytical form of the magnetic stream function, we use the approximation that the inner edge is far distant from the event horizon. The global magnetospheric structure with a closed-loop and open field lines threading the inner and outer parts of the disk is explicitly shown, claiming that the model is useful as a starting point to study astrophysical problems involving inward disk-driven winds to a black hole and outward ones to infinity. The asymptotic shape of the field lines at the event horizon becomes nearly cylindrical, while at infinity it becomes conical. The magnetic spot in the disk connected with the black hole through the loop field lines occupies a very narrow region with the ring area roughly equal to the horizon area. By taking account of the existence of a uniform (external) magnetic field, we also obtain a model for collimated open field lines. Then, it is found that the magnetic connection between the black hole and the disk breaks down if the uniform field is strong enough. Considering slow rotation of the magnetosphere and angular momentum transfer by inward winds from the disk, the final discussion is devoted to gradual disruption of the closed loops due to radial accretion of disk plasma toward the black hole.

Nine-parameter electrovac metric involving rational functions

An analytically extended nine-parameter family of the electrovac rational function solutions of the Einstein-Maxwell equations generalizing the Chen-Guo-Ernst class of hyperextreme spacetimes is presented. The general four-soliton asymptotically flat solution possessing the equatorial symmetry and involving five independent real parameters is derived in a concise analytical form and its relevance to the equilibrium problem of two extreme particles is discussed.

Transition probability of lithium atom and lithiumlike ions with weakest bound electron wave functions and coupled equations

One of us has presented a new theory called the Weakest Bound Electron Potential Model theory (WBEPM theory) to determine the exact potential field in atoms. A concise analytic form of the potential of the weakest-bound electron is proposed. In the new potential and the wave function derived afterward, the integral atomic number Z and quantum numbers n and l are replaced by nonintegral Z*, n*, and l*, respectively. In this article, we employed the WBEPM theory to calculate the transition probability for the lithium atom and lithiumlike ions and obtained very good results compared with the accepted values and other results. © 1999 John Wiley & Sons, Inc. Int J Quant Chem 76: 51–61, 2000

Arbitrarily deformed Kerr-Newman black hole in an external gravitational field

An exact axisymmetric solution of the Einstein-Maxwell equations possessing two infinite sets of arbitrary real parameters and able to describe a deformed Kerr-Newman black hole in an external gravitational field is presented in a concise analytic form. The validity of Smarr's mass formula is demonstrated for a Kerr-Newman black hole surrounded by an external static gravitational field.

A Kerr black hole in the external gravitational field

The full metric describing a Kerr black hole in an arbitrary static and axisymmetric gravitational field is presented in a concise analytical form which allows a straightforward verification of the mass formula for black holes. A sufficient condition of the regularity of the metric in the region exterior to the black hole horizon is formulated.

Scattering of light by light in the electroweak Standard Model

We calculate the invariant amplitudes for the process γ→γ in the full electroweak standard model and present them in concise analytical form. The differential polarized and total unpolarized cross sections are discussed in the MeV region and in the GeV region. The cross section is completely dominated by the fermion contributions at low energies whereas it is determined by the W-boson diagrams at high energies\(\left( {\sqrt s \gtrsim 1TeV} \right)\).

Generalizations of the Kerr and Kerr-Newman metrics possessing an arbitrary set of mass-multipole moments

The authors present in a concise analytical form two asymptotically flat metrics describing the superposition of the Kerr solution with an arbitrary static vacuum Weyl field which differ in their angular momentum distributions. They are then used for the construction of two asymptotically flat generalizations of the Kerr-Newman spacetime possessing the full set of mass-multipole moments able to describe the exterior gravitational field of a charged rotating arbitrary axisymmetric mass.