Simple Analytical Formula Research Articles

The role of losses in determining hyperbolic material figures of merit

Uniaxial materials have achieved new prominence in photonics because they can have hyperbolic spectral regions with metallic (ε<0) and dielectric (ε>0) permittivities along different crystal axes. In the lossless case, this results in an open hyperboloid dispersion relation, allowing materials to support highly confined modes with extremely large wavevectors. However, even small losses change the character of the hyperbolic dispersion from open hyperboloids to closed surfaces with finite maximum k, significantly limiting the extent to which highly-confined modes can be achieved. Here, we derive a simple analytic formula for the dispersion relation in the presence of loss and show that for some typical materials the maximum wavevector in hyperbolic materials is roughly ten times the free-space. The scaling of the maximum wavevector is derived, and it is shown that there is a universal scaling relation between the propagation length and the wavelength, which implies that the shortest wavelengths in any hyperbolic material are strongly attenuated.

Direct Identification of the Continuous Relaxation Time and Frequency Spectra of Viscoelastic Materials

Relaxation time and frequency spectra are not directly available by measurement. To determine them, an ill-posed inverse problem must be solved based on relaxation stress or oscillatory shear relaxation data. Therefore, the quality of spectra models has only been assessed indirectly by examining the fit of the experiment data to the relaxation modulus or dynamic moduli models. As the measures of data fitting, the mean sum of the moduli square errors were usually used, the minimization of which was an essential step of the identification algorithms. The aim of this paper was to determine a relaxation spectrum model that best approximates the real unknown spectrum in a direct manner. It was assumed that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. A modified relaxation frequency spectrum was defined as a quotient of the real relaxation spectrum and relaxation frequency and expanded into a series of linearly independent exponential functions that are known to constitute a basis of the space of square-integrable functions. The spectrum model, given by a finite series of these basis functions, was assumed. An integral-square error between the real unknown modified spectrum and the spectrum model was taken as a measure of the model quality. This index was proved to be expressed in terms of the measurable relaxation modulus at uniquely defined sampling instants. Next, an empirical identification index was introduced in which the values of the real relaxation modulus are replaced by their noisy measurements. The identification consists of determining the spectrum model that minimizes this empirical index. Tikhonov regularization was applied to guarantee model smoothness and noise robustness. A simple analytical formula was derived to calculate the optimal model parameters and expressed in terms of the singular value decomposition. A complete identification algorithm was developed. The analysis of the model smoothness and model accuracy for noisy measurements was carried out. The equivalence of the direct identification of the relaxation frequency and time spectra has been demonstrated when the time spectrum is modeled by a series of functions given by the product of the relaxation frequency and its exponential function. The direct identification concept can be applied to both viscoelastic fluids and solids; however, some limitations to its applicability have been pointed out. Numerical studies have shown that the proposed identification algorithm can be successfully used to identify Gaussian-like and Kohlrausch–Williams–Watt relaxation spectra. The applicability of this approach to determining other commonly used classes of relaxation spectra was also examined.

A simplified analytical formula for the coefficient of 3rd-order Hermite moment model and its application

Based on the 3rd-order softened Hermite nonlinear equations, a new simplified formula of the Hermite moment model is proposed. According to the wind tunnel pressure test data of a flat roof, the new and existing simplified formulas are used to calculate the Hermite parameters and peak factors respectively. The calculation accuracies of different simplified formulas are compared and analyzed to verify the applicability of the new simplified formula. The results show that the new simplified formula has higher accuracy for the peak factor calculation of high-order softened time histories, and the relative error is kept within 15%. And the applicability for mild non-Gaussian time histories is relatively better. For low-order softened time histories, the calculation errors of the new simplified formula are relatively large for the Hermite parameters, but the calculation accuracy of the peak factor is kept within the applicable range. Specifically, the calculation errors of the negative peak factors distributed only in the 1st-order softened applicable range are less than 20%, and the calculation errors of the negative peak factors distributed only in the 2nd-order softened applicable range are less than 15%. Finally, the new simplified formula shows good applicability to the extreme value analysis of the high-order and low-order softened wind pressure time histories.

The impact of constrained interacting dark energy on the bound-zone velocity profile

We numerically study the effects of constrained interacting dark energy (CIDER) on the bound-zone velocity profiles around massive dark matter halos. Analyzing the CIDER simulations performed by ref. [1] for three different cases of dark sector coupling (β = 0.03, 0.05 and 0.08) as well as for the standard ΛCDM cosmology (β = 0), we determine the mean peculiar velocity profiles in the bound zones around the friends-of-friends halos with masses larger than M cut = 3 × 1013 h-1 M ⊙ at three redshifts, z = 0, 0.5 and 1. It is found that the universal power-law formula proposed by ref. [2] originally for the ΛCDM case still describes well the bound-zone velocity profiles, V(r), even in the CIDER models. The slope of V(r), turns out to be significantly affected by the CIDER, progressively decreasing as β increases. Meanwhile, the amplitude of V(r) exhibits little dependence on β, which is ascribed to the identical Hubble parameters shared by the ΛCDM and CIDER models in the entire redshift range. Our results imply that the bound-zone velocity slope can break a degeneracy even between the ΛCDM and CIDER models with β ≤ 0.03, which the standard cosmological diagnostics fail to distinguish. We devise a simple analytic formula for the bound-zone slope as a function of β, and prove its validity at all of the three redshifts. It is concluded that the slope of the mean bound-zone peculiar velocity profile should be in principle a powerful probe of dark sector interaction.

Estimate for the Bulk Viscosity of Strongly Coupled Quark Matter Using Perturbative QCD and Holography.

Modern hydrodynamic simulations of core-collapse supernovae and neutron-star mergers require knowledge not only of the equilibrium properties of strongly interacting matter, but also of the system's response to perturbations, encoded in various transport coefficients. Using perturbative and holographic tools, we derive here an improved weak-coupling and a new strong-coupling result for the most important transport coefficient of unpaired quark matter, its bulk viscosity. These results are combined in a simple analytic pocket formula for the quantity that is rooted in perturbative quantum chromodynamics at high densities but takes into account nonperturbative holographic input at neutron-star densities, where the system is strongly coupled. This expression can be used in the modeling of unpaired quark matter at astrophysically relevant temperatures and densities.

Performance Prediction of GFRP-Reinforced Concrete Deep Beams Containing a Web Opening in the Shear Span

This study aimed to investigate the nonlinear structural behavior of concrete deep beams internally reinforced with glass fiber-reinforced polymer (GFRP) reinforcing bars and containing a web opening of various sizes and locations within the shear span. Three-dimensional (3D) numerical simulation models were developed for large-scale GFRP-reinforced concrete deep beams (300 mm × 1200 mm × 5000 mm) with a shear span-to-depth ratio (a/h) of 1.04. Predictions of the numerical models were validated against published experimental data. A parametric study was conducted to examine the effect of varying the opening size and location on the shear response. Results of the numerical analysis indicated that the strength of the deep beam models with an opening in the middle of the shear span decreased with an increase in either the opening width or height. The rate of the strength reduction caused by increasing the opening height was, however, more significant than that produced by increasing the opening width. Placing a web opening in the compression zone close to the load plate was very detrimental to the beam strength. Conversely, a negligible strength reduction was recorded when the web opening was placed in the tension side above the flexural reinforcement and away from the natural load path. Data of the parametric study were utilized to introduce simplified analytical formulas capable of predicting the shear capacity of GFRP-reinforced concrete deep beams with a web opening in the shear span.

Analysis of two-dimensional planar gratings and metasurfaces in the quasi-static approximation

Currently, composite materials based on metal gratings with a period D much smaller than the wavelength l are widely used. With the use of such materials, it is possible to control the amplitude, phase and polarization of an electromagnetic wave, which opens wide possibilities for the creation of new types of antennas, radio-absorbing coatings, etc. This paper considers gratings of metallic strip or patches on the surface of a thin metal-backed dielectric slab to control the phase of the reflected wave. In the quasi-static approximation, the input impedance of such materials can be calculated using simple analytical formulas. This paper investigates the accuracy of the quasi-static approximation at different angles of incidence of the electromagnetic wave on two-dimensional gratings. Using the equivalent circuits method, analytical expressions for the input impedance of metasurfaces are obtained. The dispersion properties of composite materials “inductive or capacitive grating on the surface of a thin metal-backed dielectric slab” are investigated, and a technique for determining the resonant frequency, which corresponds to the maximum of the input impedance of the metasurface, is presented. The obtained results show that at any angle of incidence, it is possible to evaluate the condition of the quasi-static approximation applicability for two-dimensional gratings and for metasurfaces based on such gratings. If this condition is satisfied, that simple analytical expressions can be used to calculate the input impedance. The use of these expressions allows us to determine the frequency at which the input impedance reaches its maximum. It is established that for inductive and capacitive gratings the impedance boundary condition of M.A. Leontovich is equivalent to the averaged boundary condition of M.I. Kontorovich. It is obtained that for grating in free space, quasi-static approximation corresponds to the M.A. Leontovich impedance condition. The dispersion properties of composite materials “inductive or capacitive grating on a thin metal-backed dielectric slab” are investigated. It is shown that the phase transition of the reflection coefficient through zero occurs at maximum values of the input impedance. Using the considered method of equivalent circuits, the presented results can be easily generalized to any type of multilayer metasurfaces.

Easy bootstrap for the 3D Ising model: a hybrid approach of the lightcone bootstrap and error minimization methods

As a simple lattice model that exhibits a phase transition, the Ising model plays a fundamental role in statistical and condensed matter physics. The Ising transition is realized by physical systems, such as the liquid-vapor transition. Its continuum limit also furnishes a basic example of interacting quantum field theories and universality classes. Motivated by a recent hybrid bootstrap study of the quantum quartic oscillator, we revisit the conformal bootstrap approach to the 3D Ising model at criticality, without resorting to positivity constraints. We use at most 10 nonperturbative crossing constraints at low derivatives from the Taylor expansion around a crossing symmetric point. The high-lying contributions are approximated by simple analytic formulae deduced from the lightcone singularity structure. Surprisingly, the low-lying properties are determined to good accuracy by this computationally very cheap approach. For instance, the results for the two relevant scaling dimensions (∆σ, ∆ϵ) ≈ (0.518153, 1.41278) are close to the most precise rigorous bounds obtained at a much higher computational cost.

Analytical perturbations of relativistic images in Kerr space-time

Light rays passing very close to black holes may wind several times before escaping. For any given electromagnetic source around the black hole, a distant observer would thus observe two infinite sequences of images on either side of the black hole. These images are generated by light rays performing an increasing numbers of loops. The strong deflection limit provides a simple analytic formalism to describe such higher order images for spherically symmetric metrics, while for axially symmetric black holes one typically resorts to numerical approaches. Here we present the leading order perturbation to higher order images when the black hole spin is turned on. We show that the images slide around the black hole shadow as an effect of space-time dragging. We derive analytical formulae for their shifts and the perturbation of their time delays. We also discuss how such simple analytical formulae for images by Kerr black holes can be of great help in many applications.

Equations for efficient cycle-by-cycle computation of fatigue crack retardation and acceleration due to amplitude changes

The paper presents an efficient methodology for computation of the evolution of the fatigue crack retardation and acceleration effects due to crack closure during variable amplitude loading. The presented approach can lead to a very effective tool for estimation of residual fatigue life of engineering structures. Conventionally, consideration of variable-amplitude loading is either computationally too demanding, such as in finite element modelling, or challenging to unify the strip-yield model with another computational code. The simple analytical formulae available in literature are based on residual stress, which does not describe correctly the delayed retardation effect and in some cases it can be non-conservative. This work presents simple analytical equations describing the development of the crack closure based on the results of the strip-yield model for amplitude changes at the load ratio R = 0.1. The parabolic function enables simple computation of the maximum crack closure occurring after an overload without running any simulation. The equations also enable computation of crack closure behaviour following amplitude changes with the consideration of various cyclic material properties reflected in the tuneable ratio between the monotonic and cyclic plastic zone sizes.

Massive Debris Disks May Hinder Secular Stirring by Planetary Companions: An Analytic Proof of Concept

Debris disks or exo-Kuiper belts, detected through their thermal or scattered emission from their dusty components, are ubiquitous around main-sequence stars. Since dust grains are short-lived, their sustained presence is thought to require dynamical excitation, i.e., “stirring,” of a massive reservoir of large planetesimals, such that mutual collisions are violent enough to continually supply fresh dust. Several mechanisms have been proposed to explain debris disk stirring, with the commonly accepted being long-term, secular planet–debris disk interactions. However, while effective, existing planet-stirring models are rudimentary; namely, they ignore the (self-)gravity of the disk, treating it as a massless reservoir of planetesimals. Here, using a simple analytical model, we investigate the secular interactions between eccentric planets and massive, external debris disks. We demonstrate that the disk gravity drives fast apsidal precession of both planetesimal and planetary orbits, which, depending on the system parameters, may well exceed the planet-induced precession rate of planetesimals. This results in strong suppression of planetesimal eccentricities and thus relative collisional velocities throughout the disk, often by more than an order of magnitude when compared to massless disk models. We thus show that massive debris disks may hinder secular stirring by eccentric planets orbiting near, e.g., the disk’s inner edge, provided the disk is more massive than the planet. We provide simple analytic formulae to describe these effects. Finally, we show that these findings have important implications for planet inferences in debris-bearing systems, as well as for constraining the total masses of debris disks (as done for β Pic).

Comparison of reduced model predictions for divertor detachment onset and reattachment timescales in ASDEX Upgrade and JET experiments

Building on prior analysis of ASDEX Upgrade (AUG) experiments (Henderson et al 2023 Nucl. Fusion 63 086024), this study compares simple analytical formula predictions for divertor detachment onset and reattachment timescales in JET experiments. Detachment onset primarily scales with divertor neutral pressure, impurity concentration, power directed to the targets, machine size, and integral perpendicular power decay length. JET experiments, focusing on seeding mixtures of Ne and Ar, align with the detachment onset predictions. Radiation efficiencies among the impurities show good agreement with the model predictions, contrasting with AUG observations which suggested higher efficiency for Ar and lower efficiency for Ne. The time taken to re-ionise the neutral volume in front of the outer target in fully detached divertor conditions was measured following both abrupt increases in injected neutral beam power and, separately, cutting of the impurity gas flow. Re-ionisation of the neutrals occurs within approximately 1 s on JET, which aligns with the simple model prediction derived from AUG data. While the AUG results are not new, their comparison with the JET results enhances understanding, reinforcing confidence in using simple models to predict future reactor scenarios.

How mutation accumulation depends on the structure of the cell lineage tree.

All the cells of a multicellular organism are the product of cell divisions that trace out a single binary tree, the so-called cell lineage tree. Because cell divisions are accompanied by replication errors, the shape of the cell lineage tree is a key determinant of how somatic evolution, which can potentially lead to cancer, proceeds. Carcinogenesis requires the accumulation of a certain number of driver mutations. By mapping the accumulation of mutations into a graph theoretical problem, we present an exact numerical method to calculate the probability of collecting a given number of mutations and show that for low mutation rates it can be approximated with a simple analytical formula, which depends only on the distribution of the lineage lengths, and is dominated by the longest lineages. Our results are crucial in understanding how natural selection can shape the cell lineage trees of multicellular organisms and curtail somatic evolution.

Using Monte Carlo Simulation to Forecast the Scientific Utility of Psychological App Studies: A Tutorial

Mobile applications offer a wide range of opportunities for psychological data collection, such as increased ecological validity and greater acceptance by participants compared to traditional laboratory studies. However, app-based psychological data also pose data-analytic challenges because of the complexities introduced by missingness and interdependence of observations. Consequently, researchers must weigh the advantages and disadvantages of app-based data collection to decide on the scientific utility of their proposed app study. For instance, some studies might only be worthwhile if they provide adequate statistical power. However, the complexity of app data forestalls the use of simple analytic formulas to estimate properties such as power. In this paper, we demonstrate how Monte Carlo simulations can be used to investigate the impact of app usage behavior on the utility of app-based psychological data. We introduce a set of questions to guide simulation implementation and showcase how we answered them for the simulation in the context of the guessing game app Who Knows (Rau et al., 2023). Finally, we give a brief overview of the simulation results and the conclusions we have drawn from them for real-world data generation. Our results can serve as an example of how to use a simulation approach for planning real-world app-based data collection.

Fate of false vacuum in non-perturbative regimes

We use some exact results in scalar field theory to revise the analysis by Coleman and Callan about false vacuum decay and propose a simple non-perturbative formalism. We introduce an exact Green’s function which incorporates non-perturbative corrections in the strong coupling regimes of the theory. The solution of the scalar field theory involves the Jacobi elliptical function and has been used to calculate the effective potential for any arbitrary coupling values. We demonstrate the use of this formalism in a simple λ ϕ 4 theory, and show that the effective potential exhibits a false minimum at the origin. We then calculate the false vacuum decay rate in the thin wall approximation, and suggest simple analytic formulae that may be useful for the analysis for the first-order phase transition beyond the perturbative regime. In our methodology, we show that the standard results obtained in perturbation theory are reproduced by making the coupling values very small.

Mathematical formulae for neutron self-shielding properties of media in an isotropic neutron field

In the current study, an ab initio derivation of the neutron self-shielding factor to solve the complex neutron transport problem of the decrease of the neutron flux as it penetrates into a material placed in an isotropic neutron field having equal flux in all directions. The theory of steady-state neutron transport was employed, starting from Stuart’s formula, to derive simple analytical formulae based on the integral cross-section parameters. The formulae could be adopted by the user according to various variables, such as the neutron flux distribution and geometry of the simulation at hand. The concluded formulae of the self-shielding factors comprise an inverted sigmoid function normalized with a weight representing the ratio between the macroscopic total and scattering cross-sections of the medium. The general convex volume geometries are reduced to a set of chord lengths, while the neutron interaction probabilities within the volume are parameterized to the epithermal and thermal neutron energies. The arguments of the inverted-sigmoid function were derived from a simplified version of neutron transport formulation. The derived analytic formulae agreed greatly with the experimental observations for different elements and geometries.

Magnetic-field-induced splitting of Rydberg Electromagnetically Induced Transparency and Autler-Townes spectra in 87Rb vapor cell.

We theoretically and experimentally investigate the Rydberg electromagnetically induced transparency (EIT) and Autler-Townes (AT) splitting of 87Rb vapor under the combined influence of a magnetic field and a microwave field. In the presence of static magnetic field, the effect of the microwave field leads to the dressing and splitting of each mF state, resulting in multiple spectral peaks in the EIT-AT spectrum. A simplified analytical formula was developed to explain the EIT-AT spectrum in a static magnetic field, and the theoretical calculations agree qualitatively with experimental results. The Rydberg atom microwave electric field sensor performance was enhanced by making use of the splitting interval between the two maximum absolute mF states separated by the static magnetic field, which was attributed to the stronger Clebsch-Gordon coefficients between the extreme mF states and the frequency detuning of the microwave electric field under the static magnetic field. The traceable measurement limit of weak electric field by EIT-AT splitting method was extended by an order of magnitude, which is promising for precise microwave electric field measurement.

A simple formula for coherent specular reflection loss of underwater sound from rough ocean surfaces

The interaction of underwater sound with rough sea surfaces affects its propagation. The interaction may involve several coupled mechanisms: (1) a rough air-sea interface scatters and spreads the incident sound energy over a range of angles, (2) entrapped sub-surface air bubbles absorb and scatter sound energy, and (3) medium of bubble clouds refract sound energy upwards. In this paper, we ignore the effect of bubbles and consider the effect of rough air-sea interface only. We further limit our consideration to the coherent component in the specular direction. Two simple closed-form formulas are often used to model the coherent reflection loss from rough ocean surfaces. One is derived from perturbation approximation and suitable for smaller grazing angles. The other is from Kirchhoff approximation and suitable for larger grazing angles. More accurate approximations over a wide-range of grazing angles require numerical evaluation of integrals or complex special functions. In this work, we simplify and adapt earlier results by other authors to obtain a simple expression that transits from the perturbation to the Kirchhoff formulas as grazing angle increases. The new simple analytical formula is easy to use and compares well with results from complex numerical methods.

Thermodynamics of the electron–positron plasma at very high temperatures and sources of annihilation γ-quanta in our Galaxy

Thermodynamic properties and equations of state of the electron–positron plasma (or gas) at high and very high temperatures (T≥170 keV) are derived and investigated. We have derived a number of simple analytical formulas for the Fermi–Dirac distribution functions, which can be applied to various Fermi gases and plasmas in different cases. Almost all these formulas are represented in the form of series expansions. The coefficients in these expansions are relatively simple functions of the μT ratio, where T is the temperature and μ is the chemical potential of this Fermi system. Our approach works well for high-temperature electron–positron plasmas, which are in thermal equilibrium with the photon gas of annihilation γ-quanta, and for the model gas of fermions, where there is no radiation at all. The first case corresponds to the ultra-relativistic limit for high-temperature electron–positron plasma, while the second case represents a model Fermi gas of particles, which has some non-zero chemical potential. By investigating sources of annihilation γ-quanta in our Galaxy, we have arrived to a remarkable conclusion about the high-temperature limit in our regular (photon) optics.

The gravitational force field of proto-pancakes

It is well known that the first structures that form from small fluctuations in a self-gravitating, collisionless, and initially smooth cold dark matter (CDM) fluid are pancakes. We studied the gravitational force generated by such pancakes just after shell crossing and have found a simple analytical formula for the force along the collapse direction, which can be applied to both the single- and multi-stream regimes. We tested the formula on the early growth of CDM proto-haloes seeded by two or three crossed sine waves. Adopting the high-order Lagrangian perturbation theory (LPT) solution as a proxy for the dynamics, we confirm that our analytical prediction agrees well with the exact solution computed via a direct resolution of the Poisson equation, as long as the local caustic structure remains sufficiently one-dimensional. These results are further confirmed by comparisons of the LPT predictions performed this way to measurements in Vlasov simulations performed with the public code ColDICE. We also show that the component of the force orthogonal to the collapse direction preserves its single-stream nature – it does not change qualitatively before or after the collapse – allowing sufficiently high-order LPT acceleration to be used to approximate it accurately as long as the LPT series converges. As expected, solving the Poisson equation on the density field generated with LPT displacement provides a more accurate force than the LPT acceleration itself, as a direct consequence of the faster convergence of the LPT series for the positions than for the accelerations. This may provide a clue as to how we can improve standard LPT predictions. Our investigations represent a very needed first step in the study of gravitational dynamics in the multi-stream regime analytically: we estimate, at the leading order in time and space, the proper backreaction on the gravitational field inside the pancakes.