Compact Formula Research Articles

Точное решение основного уравнения акустики для развивающейся по двум направлениям волны давления

The authors proceed from the hyperbolic equation for acoustic pressure. Using the integral Fourier transform along the axial coordinate, an equation in partial derivatives for the kernel of this transformation is found. This equation contains only one spatial coordinate and time. Applying the integral Laplace transform in time to the last equation, we obtain an ordinary differential equation with respect to the radial coordinate for the corresponding image. It turns out that the solution of the last equation is the well-known Macdonald function. For this function, it was possible to find the original image according to Laplace. All this made it possible to write an integral formula for the pressure in a sound wave. If the function of the initial pressure distribution along the pipe axis is taken in the form of a Gaussian impulse, then the integrals included in the representation of the desired solution are taken explicitly. As a result, we obtain an explicit compact formula for the acoustic pressure distribution in the axisymmetric case. It is convenient to use this formula to analyze the distribution of sound disturbances both along the pipe axis and in the radial direction. Therefore, the results are presented as isobars in the (z, r) plane corresponding to different times.

Non-stationary oscillations of an instantly loaded oscillator under conditions of nonlinear resistance

The motion of an oscillator instantaneously loaded with a constant force under conditions of nonlinear external resistance, the components of which are quadratic viscous resistance, dry and positional friction, are considered. Using the first integral of the equation of motion and the Lambert function, compact formulas for calculating the ranges of oscillations are derived. In order to simplify the search for the values of the Lambert function, asymptotic formulas are given that, with an error of less than one percent, express this special function in terms of elementary functions. It is shown that as a result of the action of the resistance force, including dry friction, the oscillation process has a finite number of cycles and is limited in time, since the oscillator enters the stagnation region, which is located in the vicinity of the static deviation of the oscillator caused by the applied external force. The system dynamic factor is less than two. Examples of calculations that illustrate the possibilities of the stated theory are considered. In addition to analytical research, numerical computer integration of the differential equation of motion was carried out. The complete convergence of the results obtained using the derived formulas and numerical integration is established, which confirms that using analytical solutions it is possible to determine the extreme displacements of the oscillator without numerical integration of the nonlinear differential equation. To simplify the calculations, the literature is also recommended, where tables of the Lambert function are printed, allowing you to find its value for interpolating tabular data. Under conditions of nonlinear external resistance, the components of which are quadratic viscous resistance, dry and positional friction, the process of oscillations of an instantly loaded oscillator has a limited number of cycles. The dependences obtained in this work using the Lambert function make it possible to determine the range of oscillations without numerical integration of the nonlinear differential equation of motion both for an oscillator with quadratic viscous resistance and dry friction, and for an oscillator with quadratic resistance and positional and dry friction.
 Keywords: nonlinear oscillator, instantaneous loading, quadratic viscous resistance, Lambert function, oscillation amplitude.

Asymptotics of $$\mathrm {SL}(2,{{\mathbb {C}}})$$ coherent invariant tensors

We study the semiclassical limit of a class of invariant tensors for infinite-dimensional unitary representations of $\mathrm{SL}(2,\mathbb{C})$ of the principal series, corresponding to generalized Clebsch-Gordan coefficients with $n\geq3$ legs. We find critical configurations of the quantum labels with a power-law decay of the invariants. They describe 3d polygons that can be deformed into one another via a Lorentz transformation. This is defined viewing the edge vectors of the polygons are the electric part of bivectors satisfying a (frame-dependent) relation between their electric and magnetic parts known as $\gamma$-simplicity in the loop quantum gravity literature. The frame depends on the SU(2) spin labelling the basis elements of the invariants. We compute a saddle point approximation using the critical points and provide a leading-order approximation of the invariants. The power-law is universal if the SU(2) spins have their lowest value, and $n$-dependent otherwise. As a side result, we provide a compact formula for $\gamma$-simplicity in arbitrary frames. The results have applications to the current EPRL model, but also to future research aiming at going beyond the use of fixed time gauge in spin foam models.

An exact and compact formula for the optical intersubband response of finite-barrier quantum wells, wires and dots

The optical response of quantum wells, wires and dots is frequently modelled without including continuum states. This is reasonable in a low-energy regime but fails to capture phenomena such as bound-to-continuum transitions. We show that, in fact, the full optical intersubband response of quantum confined structures can be expressed in an exceedingly simple exact formula for arbitrary barrier height, confinement dimension, and optical frequency. We illustrate the consequences of transitions reaching the continuum for several representative cases, including asymmetric quantum wells. Moreover, we derive a simple modification that may be added to bound-states-only simulation in order to include the continuum. For the asymmetric quantum well, both linear and second-order nonlinear response is obtained. Finally, excellent agreement with measured bound-to-continuum absorption in narrow quantum wells is demonstrated.

Size effects on the dynamic indentation modulus of films

The dynamic mechanical characteristics of viscoelastic films play a significant role in the load-bearing functions of various engineering and natural composite materials systems. Identifying these film characteristics is a prime objective in both fundamental and applicative materials science fields. However, as the film thickness decreases, its analysis by dynamic nanoindentation methods becomes convoluted due to the emergence of mechanical coupling between the film and its underlying substrate. These yet unexplored size effects pose major limitations in approaching the elemental mechanical characteristics of thin films. Here, we analyze the film size effects on its dynamic indentation modulus by theoretical modeling and Finite-Element simulations. We develop compact analytical formulae that link the indentation modulus magnitude, loss coefficient, storage modulus, and loss modulus to those of the pristine film, draw functional insights into the energy-storage and energy-dissipation capabilities of the integrated film–substrate laminate, and outline an approach to back-calculate the elemental film characteristics via simple linear scaling of its dynamic indentation modulus measurements. Our analysis generally holds for synthetic and biological film materials—paving the way to identify the dynamic mechanical characteristics of various thin-film configurations, such as polymeric assemblies, functional surfaces, and biomechanical coatings.

The determinant representation of Darboux transformation for the Kulish–Sklyanin model and novel soliton solutions for [formula omitted

In this article, we formulate a determinant representation of the Darboux transformation for the Kulish–Sklyanin (KS) model, which can be viewed as a generalization of the nonlinear Schrödinger equation, and obtain a new compact formula of n-soliton solution for the KS system. As applications of the formula, the abundant dynamics of one-, two- and three-soliton solutions for the case m=2 are discussed. The different structures of soliton solitons are obtained by setting different parameters.

ABOUT THE ANALYTICAL SOLUTION OF THE EQUATION OF IMPACT FORCE OF TWO ELASTIC BODIES

A nonlinear differential equation of the force of direct central impact of elastic bodies of revolution, which have a singular point on the boundary contact surface, where its curvature is infinite, is compiled. To determine the coefficients of the equation and the order of its power nonlinearity, the well-known solution of the axisymmetric contact problem of the theory of elasticity, constructed by I. Ya. Shtaermann, is used. In the formulation of the dynamic problem, the classical assumptions of the theory of quasi-static impact proposed by H. Hertz were also used. The constituted equation of impact force is reduced to the Bernoulli equation and its closed analytical solution is constructed, which is expressed in terms of the Ateb-sine. Analytical time dependences of the impact force and the convergence of the centers of mass of elastic bodies are obtained. Compact formulas have been derived for calculating the maxima of these quantities, as well as the durations of the process of compression and impact of bodies. Compact approximations of Ateb-sine by elementary functions are proposed. Thanks to these approximations, it was possible to obtain a fairly simple analytical sweep in time of a fast-flowing mechanical process. Traditionally, in other works such a scan was obtained by numerical solution of the corresponding integral equations that determine the force of an impact. Examples of calculations are given in which the influence of various factors on the main characteristics of a body impact with a small initial velocity is investigated. The limitation on the collision rate is due to the elastic formulation of the problem, where the possibility of plastic deformations is excluded. As a result of this formulation, the need to determine the rate of recovery rate has disappeared, for it is equal to one. Comparison of numerical results is carried out, to which the obtained analytical solutions and the numerical integration of the impact force equation on a computer lead. Small divergences of the results confirmed the accuracy of the derived calculation formulas. Numerical results relate to the impact of a steel body on a fixed rubber half-space, the analogue of which is observed in practice when falling pieces of mineral raw materials on the rolls of a vibration classifier lined with rubber.

A MODIFIED BACKWARD EULER SCHEME FOR THE DIFFUSION EQUATION SUBJECT TO NONLINEAR NONLOCAL BOUNDARY CONDITIONS

In this article, Modified Backward Euler Scheme is developed to solve the diffusion equation subject to nonlinear nonlocal boundary conditions. The proposed scheme is derived by combining a fourth-order compact finite difference formula in space and a backward differ- entiation for the time derivative term. Nonlinear terms in boundary conditions are linearized by Taylor expansion. Numerical examples are provided to verify the accuracy and efficiency of our proposed method.

Fine-Structure Constant Connects Electronic Polarizability and Geometric van-der-Waals Radius of Atoms.

The fine-structure constant (FSC) measures the coupling strength between photons and charged particles and is more strongly associated with quantum electrodynamics than with atomic and molecular physics. Here we present an elementary derivation that accurately predicts the electronic polarizability of atoms from their geometric van-der-Waals (vdW) radius RvdW and the FSC α through the compact formula = (4πε0/a04) × α4/3RvdW7, where ε0 is the permittivity of free space and a0 is the Bohr radius. The validity of this formula is empirically confirmed by estimating the value of α from nonrelativistic quantum calculations of atomic polarizabilities and atomic vdW radii obtained from both theory and experiment. Our heuristic derivation based on empirical data extends the influence of FSC from quantum electrodynamics and specific materials properties such as the visual transparency of graphene to atomic electronic properties throughout the periodic table of elements.

Superradiance and anomalous hyperfine splitting in inhomogeneous ensembles

Collective effects in the interaction of light with ensembles of identical scatterers play an important role in many fields of physics. However, often the term ``identical'' is not accurate due to the presence of hyperfine fields which induce inhomogeneous transition shifts and splittings. Here we develop a formalism based on the Green's-function method to model the linear response of such inhomogeneous ensembles in one-dimensional waveguides. We obtain a compact formula for the collective spectrum, which exhibits deviations from the uniform frequency shift and broadening expected of two-level systems. In particular, if the coherent contribution to the collective coupling is large, the effect of inhomogeneous broadening can be suppressed, with the linewidth approaching that of the superradiant value. We apply this formalism to describe collective effects in x-ray scattering off thin-film waveguides for inhomogeneous hyperfine parameters.

Quantum-mechanical force balance between multipolar dispersion and Pauli repulsion in atomic van der Waals dimers

The structure and stability of atomic and molecular systems with van der Waals (vdW) bonding are often determined by the interplay between attractive dispersion interactions and repulsive interactions caused by electron confinement. Arising due to different mechanisms -- electron correlation for dispersion and the Pauli exclusion principle for exchange-repulsion -- these interactions do not appear to have a straightforward connection. In this paper, we use a coarse-grained approach for evaluating the exchange energy for two coupled quantum Drude oscillators and investigate the mutual compensation of the attractive and repulsive forces at the equilibrium distance within the multipole expansion of the Coulomb potential. This compensation yields a compact formula relating the vdW radius of an atom to its multipole polarizabilities, $R_{\rm vdW} = A_l^{\,}\, \alpha_l^{{2}/{7(l+1)}}$, where $l$ is the multipole rank and $A_l$ is a conversion factor. Such a relation is compelling because it connects an electronic property of an isolated atom (atomic polarizability) with an equilibrium distance in a dimer composed of two closed-shell atoms. We assess the accuracy of the revealed formula for noble-gas, alkaline-earth, and alkali atoms and show that the $A_l$ can be assumed to be universal constants. Besides a seamless definition of vdW radii, the proposed relation can also be used for the efficient determination of atomic multipole polarizabilities solely based on the corresponding dipole polarizability and the vdW radius. Finally, our work provides a basis for the construction of efficient and minimally-empirical interatomic potentials by combining multipolar interatomic exchange and dispersion forces on an equal footing.

Dynamic fracture analysis of the linearly uncoupled and coupled physical phenomena by the variable-node multiscale XFEM

This paper aims to develop a variable-node multiscale extended finite element method (V-XFEM) for dynamic fracture analysis of the linearly uncoupled and coupled physical phenomena in a compact formula. The general governing equations for the linearly uncoupled and coupled physical phenomena are presented in a compact form. The local mesh refinement technique for modeling cracks is used to improve the accuracy and efficiency, in which variable-node elements without modifying the system matrix or impose additional boundary conditions are taken to connect/link different scale elements. In addition, the time-dependent equations are solved by the unconditionally stable implicit Newmark time integration method, and the dynamic intensity factors (DIFs) are derived from the domain forms of the interaction integrals. Numerical results of the linearly elastic and piezoelectric problems show that V-XFEM is an efficient numerical approach to simulate the dynamic fracture problems of the linearly uncoupled and coupled physical phenomena.

Hamiltonian truncation in Anti-de Sitter spacetime

Quantum Field Theories (QFTs) in Anti-de Sitter (AdS) spacetime are often strongly coupled when the radius of AdS is large, and few methods are available to study them. In this work, we develop a Hamiltonian truncation method to compute the energy spectrum of QFTs in two-dimensional AdS. The infinite volume of constant timeslices of AdS leads to divergences in the energy levels. We propose a simple prescription to obtain finite physical energies and test it with numerical diagonalization in several models: the free massive scalar field, ϕ4 theory, Lee-Yang and Ising field theory. Along the way, we discuss spontaneous symmetry breaking in AdS and derive a compact formula for perturbation theory in quantum mechanics at arbitrary order. Our results suggest that all conformal boundary conditions for a given theory are connected via bulk renormalization group flows in AdS.

Scalar product for the XXZ spin chain with general integrable boundaries * *Dedicated to the memory of Omar Foda.

We calculate the scalar product of Bethe states of the XXZ spin- chain with general integrable boundary conditions. The off-shell equations satisfied by the transfer matrix and the off-shell Bethe vectors allow one to derive a linear system for the scalar product of off-shell and on-shell Bethe states. We show that this linear system can be solved in terms of a compact determinant formula that involves the Jacobian of the transfer matrix eigenvalue and certain q-Pochhammer polynomials of the boundary couplings.

Dressed Dirac propagator from a locally supersymmetric [formula omitted] spinning particle

We study the Dirac propagator dressed by an arbitrary number N of photons by means of a worldline approach, which makes use of a supersymmetric N=1 spinning particle model on the line, coupled to an external Abelian vector field. We obtain a compact off-shell master formula for the tree level scattering amplitudes associated to the dressed Dirac propagator. In particular, unlike in other approaches, we express the particle fermionic degrees of freedom using a coherent state basis, and consider the gauging of the supersymmetry, which ultimately amounts to integrating over a worldline gravitino modulus, other than the usual worldline einbein modulus which corresponds to the Schwinger time integral. The path integral over the gravitino reproduces the numerator of the dressed Dirac propagator.

A new approach for computing the inverse of confluent Vandermonde matrices via Taylor's expansion

In this paper, a novel method is presented to compute an explicit formula for the inverse of the confluent Vandermonde matrices. The proposed results may have many interesting perspectives in diverse areas of mathematics and natural sciences, notably on the situations where the Vandermonde matrices have acquired much usefulness. The method presented here is direct and straightforward, it gives explicit and compact formulas. Several examples are presented to highlight the results. The main tools are some elementary basic linear algebra.

Analytic results for scalar-mediated Higgs boson production in association with two jets

We present compact analytic formulae for all one-loop amplitudes representing the production of a Higgs boson in association with two jets, mediated by a colour triplet scalar particle. Many of the integral coefficients present for scalar mediators are identical to the case when a massive fermion circulates in the loop, reflecting a close relationship between the two theories. The calculation is used to study Higgs boson production in association with two jets in a simplified supersymmetry (SUSY) scenario in which the dominant additional contributions arise from loops of top squarks. The results presented here facilitate an indirect search for top squarks in this channel, by a precision measurement of the corresponding cross section. However, we find that the potential for improved discrimination between the Standard Model and SUSY cases suggested by the pattern of results in the one- and two-jet samples is unlikely to be realized due to the loss in statistical power compared to an inclusive analysis.

Notes on gravity multiplet correlators in AdS3 \xd7 S3

We present a compact formula in Mellin space for the four-point tree-level holographic correlators of chiral primary operators of arbitrary conformal weights in (2, 0) supergravity on AdS3× S3, with two operators in tensor multiplet and the other two in gravity multiplet. This is achieved by solving the recursion relation arising from a hidden six-dimensional conformal symmetry. We note the compact expression is obtained after carefully analysing the analytic structures of the correlators. Various limits of the correlators are studied, including the maximally R-symmetry violating limit and flat-space limit.

Power-Flow and Mechanical Efficiency Computation in Two-Degrees-of-Freedom Planetary Gear Units: New Compact Formulas

The mechanical efficiency is a computed value for comparing the performance of the multi degrees-of-freedom geared transmissions of hybrid vehicles. Most of the current methods for estimating gear trains mechanical efficiency require the decomposition of gear transmissions in basic structural elements or planetary gear units (PGU). These are two degrees-of-freedom components whose mechanical efficiency has a deep influence on the overall device. The authors (E.L.E., E.P.) already evidenced that, under certain kinematic conditions, the classic Radzimovsky’s formulas, widely accepted for computing the mechanical efficiency of PGUs, are not adequate. In this paper, more general and reliable formulas for computing the mechanical efficiency are deduced. The proposed formulas herein, exploiting the concept of potential or virtual power, evidence the dependency between kinematics and efficiency. A numerical example compares our results with previous work on the subject.

Buildup time of Autler-Townes splitting in attosecond transient absorption spectroscopy

Although the time-dependent buildup processes of Autler-Townes splitting in attosecond transient absorption spectra have been observed in plenty of experiments, there still exists an open question of when the splitting is maximized, which tells us how fast the system responds to the pump field and how well the adiabatic following is in a slowly varying few-cycle laser pulse. In this Letter we work out a compact formula to quantify the buildup time of the Autler-Townes splitting, which is related to multiple timescales of the laser-atom system, e.g., the Rabi period, the laser cycle, and the pulse duration, according to a universal power law. This scaling law applies not only to singly excited states, but also to doubly excited states in which complex electron-electron interaction gets involved. Our findings have potential applications in calibrating the zero delay in a transient absorption spectroscopy.