Simple Analytical Formula Research Articles

Nonparametric estimation of variable productivity Hawkes processes

AbstractHawkes models are frequently used to describe point processes that are clustered spatial‐temporally, and have been used in numerous applications including the study of earthquakes, invasive species, and contagious diseases. An extension of the Hawkes model is considered where the productivity is variable. In particular, the case is explored where each point may have its own productivity and a simple analytic formula is derived for the maximum likelihood estimators of these productivities. This estimator is compared with an empirical estimator and ways are explored of stabilizing both estimators by lower truncating, smoothing, and rescaling the estimates. Properties of the estimators are explored in simulations, and the methods are applied to seismological and epidemic datasets to show and quantify substantial variation in productivity.

Lepton flavor violating decays l_jrightarrow {l_igamma } in the U(1)_XSSM model within the mass insertion approximation

Three singlet new Higgs superfields and right-handed neutrinos are added to MSSM to obtain U(1)_XSSM model. Its local gauge group is SU(3)_Ctimes SU(2)_L times U(1)_Y times U(1)_X. In the framework of U(1)_XSSM, we study muon anomalous magnetic moment and lepton flavor violating decays l_jrightarrow {l_igamma }(j=2,3;i=1,2) within the mass insertion approximation (MIA). Through the MIA method, we can find the parameters that directly affect the analytical result of the lepton flavor violating decays l_jrightarrow {l_igamma }, which make our work more convenient. We want to provide a set of simple analytic formulas for the form factors and the associated effective vertices, that may be very useful for future phenomenological studies of the lepton flavor violating decays. According to the accuracy of the numerical results which the influence of different sensitive parameters, we come to the conclusion that the non-diagonal elements which correspond to the generations of the initial lepton and final lepton are main sensitive parameters and lepton flavor violation (LFV) sources. This work can provide a clear signal of new physics (NP).

Asymmetric Reflection Spectrum of Fabry-Perot Interferometer and the Application in Pressure Sensing

The performance of a fiber optic Fabry-Perot (F-P) sensor largely depends on the spectral characteristics of F-P interference, nevertheless there is little research on the spectral characteristic, such as the asymmetry shapes which are often observed in practical experiments. Here, a simple analytical formula for simulating the reflected spectrum of a F-P resonator have been provided, based on which the asymmetry spectral shape can be diversified through adjusting the refined parameters in the analytical formula. In addition, the composite film of Pt/Ge with different thickness on fiber tip has been theoretically demonstrated to change the parameters, and the F-P interferometer composed by composite film of Pt/Ge on fiber tip and gold film has been simulated to exhibit variated asymmetric reflection spectra. Meanwhile, the reflection spectrum from fiber F-P interferometer based on the composite film deposited on the end face of optical fiber has been experimentally proven to change according to the thickness variation of Ge film. Finally, a fiber based miniature extrinsic Fabry-Perot interferometer (EFPI) showing a saw tooth wave shape spectrum has been fabricated to sense static pressure through a dual-wavelength based intensity demodulation scheme, which possesses the potential to overcome the offset between sensitive and dynamic range for a traditional sensor.

Inhomogeneous broadening, narrowing and shift of molecular lines under frequent velocity-changing collisions

We demonstrate that speed-dependent hard-collision profiles collapse into a simple Lorentz profile in the regime dominated by the velocity-changing collisions. We present simple analytical formulas for the effective width and shift of the Lorentzian, derived from the speed-dependent collisional broadening and shift. While the effective width increases with speed dependence of collisional shift, it is reduced by speed dependence of collisional width. On the other hand, the effective shift is modified by thermal average of speed dependence of the product of collisional width and shift. We validate these formula numerically applying the quadratic speed dependence of the collisional width and shift.

Simple Analytic Formula Relating the Mass and Spin of Accreting Compact Objects to Their Rapid X-Ray Variability

Abstract Following the previous research on epicyclic oscillations of accretion disks around black holes (BHs) and neutron stars (NSs), a new model of high-frequency quasiperiodic oscillations (QPOs) has been proposed, so-called cusp torus (CT) model, which deals with oscillations of fluid in marginally overflowing accretion tori (i.e., tori terminated by cusps). According to preliminary investigations, the model provides better fits of the NS QPO data compared to the relativistic precession (RP) model. It also implies a significantly higher upper limit on the Galactic microquasar BH spins. A short analytic formula has been noticed to well reproduce the model’s predictions on the QPO frequencies in Schwarzschild spacetimes. Here we derive an extended version of this formula that applies to rotating compact objects. We start with the consideration of Kerr spacetimes and derive a formula that is not restricted to a particular specific angular momentum distribution of the inner accretion flow, such as a Keplerian or constant one. Finally, we consider Hartle–Thorne spacetimes and include corrections implied by the NS oblateness. For a particular choice of a single parameter, our relation provides frequencies predicted by the CT model. For another value, it provides frequencies predicted by the RP model. We conclude that the formula is well applicable to rotating oblate NSs and both models. We briefly illustrate the application of our simple formula on several NS sources and confirm the expectation that the CT model is compatible with realistic values of the NS mass and provides better fits of data than the RP model.

Stress partitioning and effective behavior of N-phase laminates in anisotropic elasticity from a fast explicit method

In this work, a fast explicit method, easy to implement numerically, is proposed in order to compute the effective behavior and the distribution of stresses in a general N-phase laminate made of parallel, planar and perfectly bonded interfaces. The solutions are exact for a homogeneous far-field loading and work for an arbitrary number of phases, a general linear anisotropic elasticity, as well as different uniform thermal and plastic strains in the phases. A simple direct analytical formula is also derived to compute the stress in a given phase once the effective behavior of the laminate is known. Moreover, the correctness of the proposed method is checked by comparisons with finite element simulation results on a same boundary value problem, showing excellent agreements. An application of the method is performed for a near-β titanium alloy with elongated grains, by comparing the level of internal stresses for different elastic loadings within a N-phase laminate made of 100,000 orientations and a 2-phase laminate of equal volume fraction with maximal elastic contrast. Interestingly, the maximum von Mises stress of the 2-phase laminate is always the lowest, which is explained by a volume fraction effect. Finally, comparisons with elastic self-consistent models considering oblate spheroidal grains of different aspect ratios are performed.

Vertical Heterostructures between Transition‐Metal Dichalcogenides—A Theoretical Analysis of the NbS2/WSe2 Junction

AbstractLow‐dimensional metal‐semiconductor vertical heterostructures (VH) are promising candidates in the search of electronic devices at the extreme limits of miniaturization. Within this line of research, here a theoretical/computational study of the NbS2/WSe2 metal‐semiconductor vertical hetero‐junction using density functional theory (DFT) and conductance simulations is presented. First atomistic models of the NbS2/WSe2 VH considering all the five possible stacking orientations at the interface are constructed, and DFT and quantum‐mechanical (QM) scattering simulations are conducted to obtain information on band structure and transmission coefficients. Then an analysis of the QM results in terms of electrostatic potential, fragment decomposition, and band alignment is carried out. The behavior of transmission expected from this analysis is in excellent agreement with, and thus fully rationalizes, the DFT results, and the peculiar double‐peak profile of transmission. Finally, maximally localized Wannier functions, projected density of states, and a simple analytic formula to predict and explain quantitatively the differences in transport in the case of epitaxial misorientation are used. Within the class of Transition‐Metal Dichalcogenide systems, the NbS2/WSe2 VH exhibits a wide interval of finite transmission and a double‐peak profile, features that can be exploited in applications.

Kapitza pendulum effects in a Josephson junction coupled to a nanomagnet under external periodic drive

We investigate reorientation effects under external periodic drive in the nanomagnet dynamics coupled to a Josephson junction. The Kapitsa pendulum is introduced as a mechanical analog to this system and we demonstrate the reorientation of the easy axis of the nanomagnet. The magnetic field generated by the Josephson junction and external drive plays the role of the oscillating force of the suspension point in the Kapitsa pendulum. The high frequency oscillations change the orientation of the magnetic moment. The magnetic field of the quasiparticle current determines the frequency dependence of the magnetic moment's orientation. We obtain simple analytical formulas for the stable position of the magnetic moment, both under the external periodic drive and without it. The influence of external periodic drive on the voltage of complete reorientation have been demonstrated.

Useful zero friction simulations for assessing MBS codes Pascal's formula giving wheelsets frequency for zero wheel-rail friction

Present study provides a simple analytical formula, the “Klingel-like formula” or “Pascal's Formula” that can be used as a reference to test some results of existing railway codes and specifically those using rigid contact. It develops properly the 3D Newton-Euler equations governing the 6 degrees of freedom (DoF) of unsuspended loaded wheelsets in case of zero wheel-rail friction and constant conicity. Thus, by solving numerically these equations, we got pendulum like harmonic oscillations of which the calculated angular frequency is used for assessing the accuracy of the proposed formula so that it can in turn be used as a fast practical target for testing multi-body system (MBS) railway codes. Due to the harmonic property of these pendulum-like oscillations, the square ω2 of their angular frequency can be made in the form of a ratio K/M where K depends on the wheelset geometry and load and M on its inertia. Information on K and M are useful to understand wheelsets behavior. The analytical formula is derived from the first order writing of full trigonometric Newton-Euler equations by setting zero elastic wheel-rail penetration and by assuming small displacements. Full trigonometric equations are numerically solved to assess that the formula provides ω2 inside a 1% accuracy for usual wheelsets dimensions. By decreasing the conicity down to 1 × 10−4 rad, the relative formula accuracy is under 3 × 10−5. In order to test the formula reliability for rigid contact formulations, the stiffness of elastic contacts can be increased up to practical rigidity (Hertz stiffness × 1000).

Reducing bend-induced loss and crosstalk in a two-mode ridge waveguide by steplike thickness structuring.

This study demonstrated that two-step structuring of the waveguide thickness in a bent section significantly reduces bend-related inter-mode crosstalk, excess loss, and pure bending loss. A simple analytical formula was derived linking the thickness change required to compensate for bend-induced effects with geometrical parameters of the waveguide and bending radius. The effectiveness of the proposed approach was verified through rigorous numerical simulations of the loss and inter-mode crosstalk based on transformation optics formalism. The numerical results obtained for a TiO2:SiO2 medium index contrast waveguide exhibited significant improvement of all relevant parameters for TE00 and TE01 modes.

Parameterization on formation free energy of dislocation loops up to 1100 K in bcc iron

This work provides a simple analytical formula with a set of best-fit coefficients for evaluating formation free energies of prismatic dislocation loops as a function of temperature in body-centered-cubic iron. The parameterization is made based on the anisotropic elastic self-energies and corresponding non-linear core-energies of edge dislocations. The variation of these two energy terms with temperature is estimated by including temperature-dependent elastic stiffness tensors in the anisotropic elasticity calculation, and based on the stability of C15 clusters and dislocation loops, respectively. The new analytical formula explicitly contains temperature and is valid for temperatures up to 1100 K.

Energies of Doubly Excited <sup>1,3</sup>P° Resonances in He-like Systems Below the N = 2–14 Hydrogenic Threshold

In this present work, resonance energies and excitation energies of doubly ^1,3P° excited states of the helium isoelectronic sequence are calculated. The doubly excited states investigated are labelled in the (K, T, A) classification scheme. The energies are calculated in the framework of the variationnal procedure of the Screening constant by unit nuclear charge (SCUNC) formalism. The results obtained compared very well with theoretical and experimental literature values. The possibility to use the SCUNC formalism report rapidly with an excellent accuracy the position of the excitation resonances as well as their width within simple analytical formulae is demonstrated. It is demonstrated that the SCUNC-method can be used to assist fruitfully experiments for identifying narrow resonance energies. Thus, our results can be used as reference data for the interpretation of atomic spectra for the diagnosis of astrophysical and laboratory plasma. In the present work, a new correlated wave function is presented to express analytically the resonance energies and excitation energies of doubly ^1,3P° excited states in the He-like systems.

A note on parametric resonance induced by a singular parameter modulation

We investigate the classical problem of motion of a mathematical pendulum with an oscillating pivot. This simple mechanical setting is frequently used as the prime example of a system exhibiting the parametric resonance phenomenon, which manifests itself by surprising stabilisation/destabilisation effects. In the classical case the pivot oscillations are described by a cosine wave, and the corresponding stability analysis requires one to investigate the behaviour of solutions to the Mathieu equation. This is not a straightforward procedure, and it does not lead to exact and simple analytical results expressed in terms of elementary functions. Consequently, the explanation of the parametric resonance phenomenon can be in this case obscured by the relatively involved technical calculations. We show that the stability analysis is much easier if one considers the pivot motion described by a non-smooth function—a triangular or a nearly rectangular wave. The non-smooth pivot motion leads to the presence of singularities (Dirac distributions) in the corresponding Mathieu type equation, which seemingly further complicates the analysis. Fortunately, this is only a minor technical difficulty. Once the mathematical setting for the non-smooth forcing is settled down, the corresponding stability diagram is indeed straightforward to obtain, and the stability boundaries are, unlike in the classical case, given in terms of simple analytical formulae involving only elementary functions.

Quantitative Assessment and Measurement of X-ray Detector Performance and Solid Angle in the Analytical Electron Microscope.

A wide range of X-ray detectors and geometries are available today on transmission/scanning transmission analytical electron microscopes. While there have been numerous reports of their individual performance, no single experimentally reproducible metric has been proposed as a basis of comparison between the systems. In this paper, we detail modeling, experimental procedures, measurements, and specimens which can be used to provide a manufacturer-independent assessment of the performance of an analytical system. Using these protocols, the geometrical collection efficiency, system peaks, and minimum detection limits can be independently assessed and can be used to determine the best conditions to conduct modern hyperspectral and/or spectrally resolved tomographic analyses for an individual instrument. A simple analytical formula and specimen is presented which after suitable system calibrations can be used to experimentally determine the X-ray detector solid angle.

Analytical solution to dielectric droplet diffusiophoresis under Debye-Hückel approximation.

A simple analytical formula is obtained for the diffusiophoresis of a dielectric fluid droplet in symmetric binary electrolyte solutions under Debye-Hückel approximation valid for weakly charged droplets. The chemiphoresis is found to yield negative mobilities most of the time for droplets of constant surface charge density, which implies that the droplets tend to move away from the source releasing ionic chemicals. This is undesirable in some practical applications like drug delivery with liposomes in terms of conveying the drug-carrying liposomes to the desired area in the human body releasing specific ionic chemicals utilizing the self-guiding nature of diffusiophoresis. The further involvement of the electrophoresis component, however, may change the scenario via the oriented electric field generated by the induced diffusion potential. The lesson here is that while the impact of the chemiphoresis component is determined by nature and uncontrollable, the electrophoresis component serves as an artificially adjustable factor via choosing droplets with the surface charge of appropriate sign in practical applications. The results here have potential use in practical applications such as drug delivery. The portable simple analytical formula is a powerful asset to experimental researchers and design engineers in colloid science and technology to facilitate their works.

Revisiting transverse momentum broadening in dense QCD media

We reconsider the problem of transverse momentum broadening of a highly-energetic parton suffering multiple scatterings in dense colored media, such as the thermal Quark-Gluon plasma or large nuclei. In the framework of Moli\`ere's theory of multiple scattering we re-derive a simple analytic formula, to be used in jet quenching phenomenology, that accounts for both the multiple soft and hard Rutherford scattering regimes. Further, we discuss the sensitivity of momentum broadening to modeling of the non-perturbative infrared sector by presenting a detailed analytic and numerical comparison between the two widely used models in phenomenology: the Hard Thermal Loop and the Gyulassy-Wang potentials. We show that for the relevant values of the parameters the non-universal, model dependent contributions are negligible, at LHC, RHIC and EIC energies thus consolidating the predictive power of jet quenching theory.

Closed-form solutions for the piezoresistivity properties of short-fiber reinforced composites with percolation-type behavior

A new analytical formulation for the modeling of piezoresistive fiber-reinforced composites with percolation-type behavior is presented in this work. Firstly, we develop a closed-form solution of the electrical conductivity of oriented short-fiber reinforced composites by using generalized spherical harmonics series expansions of a Mori-Tanaka (MT) model. Piezoresistive effects are accounted for by means of three distinct mechanisms, namely filler reorientation, volume expansion, and breakage/formation of conductive paths. Then, this solution is used to derive simple analytical formulas to estimate the linear piezoresistivity coefficients. To illustrate the potentials of the proposed formulation, numerical results and discussion are presented on its application to the modeling of the piezoresistive composites doped with carbon nanotubes (CNTs). The presented formulation is also inlaid in a standard 3D finite element code to simulate the electromechanical response of full-scale CNT-based structural elements. The reported results demonstrate the capabilities of the proposed formulation to link the microstructural properties of short-fiber composites with the macroscopic response of structural systems with extraordinarily fast computation times and accuracy.

Assortativity and bidegree distributions on Bernoulli random graph superpositions

AbstractA probabilistic generative network model with$n$nodes and$m$overlapping layers is obtained as a superposition of$m$mutually independent Bernoulli random graphs of varying size and strength. When$n$and$m$are large and of the same order of magnitude, the model admits a sparse limiting regime with a tunable power-law degree distribution and nonvanishing clustering coefficient. In this article, we prove an asymptotic formula for the joint degree distribution of adjacent nodes. This yields a simple analytical formula for the model assortativity and opens up ways to analyze rank correlation coefficients suitable for random graphs with heavy-tailed degree distributions. We also study the effects of power laws on the asymptotic joint degree distributions.

Prediction of adhesion between randomly rough surfaces by order statistics

Understanding the adhesion between rough surfaces has practical significance. We derive a simple analytical formula on the basis of the classic order statistics to predict the interfacial binding energy between rough surfaces. It is found that the strong length scale dependence of adhesion ranging from the nominal size scale down to any artificially defined cutoff length scale in constructing a rough profile can be considered as a purely statistic performance resulted from different samplings and can be further described by a single parameter called sampling number. We compare the formula predictions with the experimental results and demonstrate that our simple formula holds its accuracy especially for the Derjaguin–Muller–Toporov adhesion case.

Optical Stark shift to control the dark exciton occupation of a quantum dot in a tilted magnetic field

When a detuned and strong laser pulse acts on an optical transition, a Stark shift of the corresponding energies occurs. We analyze how this optical Stark effect can be used to prepare and control the dark exciton occupation in a semiconductor quantum dot. The coupling between the bright and dark exciton states is facilitated by an external magnetic field. Using sequences of laser pulses, we show how the dark exciton and different superposition states can be prepared. We give simple analytic formulas, which yield a good estimate for optimal preparation parameters. The preparation scheme is quite robust against the influence of acoustic phonons. We further discuss the experimental feasibility of the used Stark pulses. Giving a clear physical picture our results will stimulate the usage of dark excitons in schemes to generate photons from quantum dots.