Analytical Approximation Research Articles

Enhanced ClNO2 Formation at the Interface of Sea-Salt Aerosol.

The reactive uptake of N2O5 on sea-spray aerosol plays a key role in regulating the NOx concentration in the troposphere. Despite numerous field and laboratory studies, a microscopic understanding of its heterogeneous reactivity remains unclear. Here, we use molecular simulation and theory to elucidate the chlorination of N2O5 to form ClNO2, the primary reactive channel within sea-spray aerosol. We find that the formation of ClNO2 is markedly enhanced at the air-water interface due to the stabilization of the charge-delocalized transition state, as evident from the formulation of bimolecular rate theory in heterogeneous environments. We explore the consequences of the enhanced interfacial reactivity in the uptake of N2O5 using numerical solutions of molecular reaction-diffusion equations as well as their analytical approximations. Our results suggest that the current interpretation of aerosol branching ratios needs to be revisited.

Antiviral capacity of the early CD8 T-cell response is predictive of natural control of SIV infection: Learning in vivo dynamics using ex vivo data.

While most individuals suffer progressive disease following HIV infection, a small fraction spontaneously controls the infection. Although CD8 T-cells have been implicated in this natural control, their mechanistic roles are yet to be established. Here, we combined mathematical modeling and analysis of previously published data from 16 SIV-infected macaques, of which 12 were natural controllers, to elucidate the role of CD8 T-cells in natural control. For each macaque, we considered, in addition to the canonical in vivo plasma viral load and SIV DNA data, longitudinal ex vivo measurements of the virus suppressive capacity of CD8 T-cells. Available mathematical models do not allow analysis of such combined in vivo-ex vivo datasets. We explicitly modeled the ex vivo assay, derived analytical approximations that link the ex vivo measurements with the in vivo effector function of CD8-T cells, and integrated them with an in vivo model of virus dynamics, thus developing a new learning framework that enabled the analysis. Our model fit the data well and estimated the recruitment rate and/or maximal killing rate of CD8 T-cells to be up to 2-fold higher in controllers than non-controllers (p = 0.013). Importantly, the cumulative suppressive capacity of CD8 T-cells over the first 4-6 weeks of infection was associated with virus control (Spearman's ρ = -0.51; p = 0.05). Thus, our analysis identified the early cumulative suppressive capacity of CD8 T-cells as a predictor of natural control. Furthermore, simulating a large virtual population, our model quantified the minimum capacity of this early CD8 T-cell response necessary for long-term control. Our study presents new, quantitative insights into the role of CD8 T-cells in the natural control of HIV infection and has implications for remission strategies.

Analysis of Mechanical Properties and Parameter Dependency of Novel, Doubly Re-Entrant Auxetic Honeycomb Structures.

This study proposes a new, doubly re-entrant auxetic unit-cell design that is based on the widely used auxetic honeycomb structure. Our objective was to develop a structure that preserves and enhances the advantages of the auxetic honeycomb while eliminating all negative aspects. The doubly re-entrant geometry design aims to enhance the mechanical properties, while eliminating the buckling deformation characteristic of the re-entrant deformation mechanism. The effects of the geometric modification are described and evaluated using two parameters, offset and deg. A series of experiments were conducted on a wide range of parameters based on these two parameters. Specimens were printed via the vat photopolymerization process and were subjected to a compression test. Our aim was to investigate the mechanical properties (energy absorption and compressive force) and the deformation behaviour of these specimens in relation to the relevant parameters. The novel geometry achieved the intended properties, outperforming the original auxetic honeycomb structure. Increasing the offset and deg parameters results in increasing the energy absorption capability (up to 767%) and the maximum compressive force (up to 17 times). The right parameter choice eliminates buckling and results in continuous auxetic behaviour. Finally, the parameter dependency of the deformation behaviour was predicted by analytical approximation as well.

Channel‐Spanning Logjams and Reach‐Scale Hydraulic Resistance in Mountain Streams

AbstractLogjams create an upstream backwater of deepened, slower water, locally reducing bed shear stress. We compared hydraulic impact of logjam series across 37 geomorphically diverse reaches of mountain streams observed over 11 years in the US Southern Rockies. To enable reach‐scale comparison of logjam structure and spacing, we identified the modeled best‐fit effective resistance coefficient minimizing difference between outflow exiting a 1D channel with logjams present, and the same model channel with elevated channel resistance. Effective resistance increased with ratio of jam upstream depth to depth without a logjam, ratio of backwater length to average spacing, and decreased for randomly distributed jams due to close spacing, which reduced backwater impact. An analytic approximation and boundaries for region of relative spacing with steepest increase in effective resistance are provided. Our results can assist in targeting interventions to areas where hydraulic impact is greatest, providing value for money in nature‐based solution design.

Precise analytical approximations of the eigenvalues of the decatic anharmonic potential

Abstract Precise analytical approximations have been determined for the eigenvalues of the ground state of the decatic anharmonic potential x 2 + λx 10 in the one-dimensional Schrödinger equation. The results have been found using the technique multipoint quasi-rational approximation (MPQA). With the new method, power and asymptotic expansions have been determined. The analytic function here obtained is derived connecting both expansions. The maximum relative error of the best analytical approximation here determined is 0.04. However, most of the relative errors for other values of λ, are smaller than 1% (less than 0.01).

Analytical approximate solutions of some fractional nonlinear evolution equations through AFVI method

In this article, we construct and investigate a new fractional variational iteration technique which is named as AFVI method. After that, we formulate some IVPs corresponding to the fractional nonlinear evolution equations NLTFFWE, mNLTFFWE, TFmCHE and TFmDPE. The Caputo fractional order derivative has been used to fractionalize the considered NLEEs. Then, we solve the considered IVPs by applying the formulated AFVI method. Lastly, we check the accuracy of the attained AASs by making some graphical and numerical comparisons with their corresponding exact solutions and other existing equivalent AASs. The result of this article confirm that the efficiency, appropriateness and time spent capability of the newly constructed AFVI method are better than that of the other existing analogous fractional analytical approximation methods. Here, we apply the Maple 2021 programming software to obtain the AASs of the formulated IVPs and drawing the 3D graphs of the attained AASs. Finally, in this article we validate the applicability of the Caputo fractional order derivative to form a novel analytical approximation method as well as to fractionalize some essential NLEEs of Mathematical physics.

Effect of MHD and radiation on biviscous Bingham fluid flow on Marangoni boundary for heat source/sink with chemical reaction

A significance of Marangoni-driven convective magnetohydrodynamics (MHD) was investigated in a planned study. The present study examines the impact of radiation and chemical reactions on the two dimensional boundary layer flow of bi-viscous Bingham fluid on a thermosolutal Marangoni boundary with magnetic field and heat source/sink. The physical flow problem is mathematically modelled into Navier–Stokes equations. These nonlinear partial differential equations are then mapped into a set of nonlinear ordinary differential equations using similarity transformation. The elegant analytical method is used to obtain analytic approximations for the resulting system of nonlinear differential equations. The analytical solution for the temperature and concentration profiles is expressed in terms of Kummer's confluent hypergeometric function. The features of flow characteristics such as velocity, temperature, and concentration profiles in response to the variations of the emerging parameters are simulated and examined with a physical explanation through graphs. Surface tension is considered to be corresponding to heat and mass. The results reveal that the velocity profile decreases with increasing the magnetic field, while the Marangoni number value increases with increasing the velocity profile. Moreover, the temperature profile rises with rising the radiation parameter and whereas the value chemical reaction parameter upsurges with decline concentration profile. The present problem has great interest due to its applications in industrial applications such as drying of silicon wafers, thin layers of paint, glues, in heat exchangers, crystal growth in space, biological fluids, blood, synovial liquids, plasma, lubricants, many pharmaceutical products, proteins, sewage sludge, china clay, and many more.

Метод обработки сигналов токов статора асинхронного двигателя для диагностики обрыва стержня ротора

The study is relevant due to high requirements to the reliability of the operation of auxiliary mechanisms of thermal power plants (TPP). The main source of mechanical power of auxiliary mechanisms of TPPs is high-voltage induction motor with squirrel cage rotor (IMSC). One of the most difficult causes of emergency shutdown to detect early is a broken rotor bar. The breakage of IMSC rotor bar is of latent character. It can be for a long time without considerable influence on the machine operation mode. Nevertheless, the fact of the breakage can be considered as an emergency condition of IMSC, and the consequences of leaving the groove of the magnetic core are irreversible. Scientifically, early diagnostics of IMSC rotor bars breakage of TPP auxiliary needs is both a difficult and extremely urgent task. The existing methods of diagnostics of such damage are based on measurement and analysis of vibration and acoustic physical quantities. The most common electrical parameter, used as a source of information, as a rule, is current consumption. And most of methods are based on Fourier analysis. The purpose of the study is to develop a mathematical algorithm to process digitized curves of stator phase currents to diagnose high-voltage IMSC. The theory of electric machines, the method of analytical approximation of experimental data, the method of regression analysis have been used to solve the problem. Experimental data is obtained, and testing of the algorithm is conducted on an experimental installation with a 0,4 kV induction motor with a modified design of the active part of the rotor developed for physical simulation of IMSC broken rotor bar. This paper proposes a new method of digital processing of stator current signals based on the regression analysis method for diagnostics of IMSC rotor winding. The authors have developed an algorithm for mathematical processing of digitized stator current curves based on regression analysis in the cosine-sine basis to identify the diagnostic sign of failure of one rotor bar of IMSC rotor. The authors have determined the number of harmonic components that best describe the initial signal from the point of view of the balance between saving computational resources and accuracy of approximation. A mathematical algorithm to detect the rotor bar breakage of an induction motor based on regression analysis of stator currents has been obtained and tested. It allows to detect rotor circuit damage with 97 % accuracy. At the same time the diagnostic sign changes its level when one bar breaks by 5 %. It is recommended to adapt the algorithm to the programming languages of microprocessor relay protection units.

Effective detection of early warning signal with power spectrum in climate change system

The development of early warning signals (EWS) before abrupt changes can help prevent system collapse. Current EWS, such as increasing autocorrelation (AC) and variance, provide general indicators of impending tipping points by detecting the slowing down of dynamics near transitions. However, these conventional EWS often fail to distinguish between oscillatory behavior (e.g., Hopf bifurcation) and shifts to a distant attractor (e.g., Fold bifurcation). Additionally, traditional EWS are less reliable in systems affected by density-dependent noise. To address these limitations, alternative EWS based on power spectrum analysis, known as spectral EWS, have been proposed. In this study, we apply analytical approximations for EWS as systems approach different types of local bifurcations. This novel method allows us to use spectral EWS that offer enhanced sensitivity to approaching transitions and increased robustness to density-dependent noise. We demonstrate the application of spectral EWS as robust indicators across three general models with different bifurcations. Our analysis also reveals distinct signals preceding transitions in data from sea ice loss. This combined approach underscores the advantages of incorporating spectral EWS into existing methodologies, providing a more comprehensive toolkit for anticipating critical transitions in real-world systems.

Temporal variability in transmission spectra of H2-dominated exoplanets: The influence of thermal evolution and stellar irradiation on atmospheric composition

Planets and their host stars undergo evolutionary changes over time, resulting in variations in internal temperature and incoming radiation, which significantly impact the temperature structure and composition of their atmospheres. These evolving conditions give rise to distinctive features in planetary spectra that are observable only during specific stages of planetary evolution. We aim to understand how the composition of planets with H2-dominated atmospheres changes over longer timescales due to their thermal evolution. We also investigate time-dependent features in the transmission spectra. These features could provide insights in both the formation and evolution of these gaseous planets, as well as the timescales of these changes, enabling us to study the potential variability of exoplanets over time. We evolve a ∼0.04 MJup and ∼0.45 MJup planet around a 1.0 M⊙ and 1.3 M⊙ star respectively for 109.5 years. In both systems, the planets are considered at semi-major axes of 0.1 AU and 1.0 AU. The star-planet systems are evolved by making use of Modules for Experiments in Stellar Astrophysics (MESA). The temperature–pressure profiles are obtained at selected time-steps using an analytical approximation based on the internal and irradiation temperature of the planet at each time step. We then use VULCAN, a photochemical kinetics code, to see how the composition changes with time in the atmosphere due to the thermal evolution of the planets. By making use of the radiative transfer code petitRADTrans, we also simulate the evolution of the transmission spectra of the planets to find potential time-dependent spectral features. Our findings show a prominent change in the CO2 feature at ∼4.3μm. For the 0.45 MJup case, this feature is visible in the pre-main-sequence phase of the host star, regardless of orbital distance from the host star. In the case of the ∼0.04 MJup planet, this CO2 feature is visible until t ≤ 106 years, and then it reappears after t ≥ 108 years when the planet is 0.1 AU away the host star. The CH4 features at ∼3.3μm and ∼7.5μm are only time-dependent when the planet is located at 0.1 AU from the host star and experiences high irradiation since the high temperature at early stages favoured CO2 over CH4. When the planet is 1.0 AU away from its host star, the CH4 features are always visible regardless of the mass and hence internal temperature of the planet.

Interface and mixing zone between soil waters arising from upward and downward seepage - Part I: Homogeneous total density

Thin water lenses floating on top of the main groundwater body are important for many natural and agricultural systems, owing to their different properties in terms of chemical composition or density compared to the surrounding groundwater. In settings with upward seeping groundwater, lenses may form that have thicknesses ranging from tens of centimeters to a few meters, making them prone to changing conditions in the short (seasonal) or long term (climate change). Knowing their thickness, shape, movement and mixing zone width may help in managing these lenses.In a series of two papers, we present a mathematical description of the flow of water and transport of solute in a 2D cross-section between two parallel outflow faces and compare a simplified model to a complete model as described by the numerical code SUTRA. In this first paper of the series, we consider situations with a homogeneous density distribution. In the simplified model we employ the sharp interface approximation to obtain an expression for the stream function, the interface between the two types of water and the corresponding maximum lens thickness in steady state in the domain considered. This steady state description is used for travel time analyses and forms the basis for the transient analyses. For a typical example of oscillatory (e.g. seasonal) fluctuations in boundary conditions, we obtain expressions of the movement of the interface midway between two outflow faces by separating the problem into two timescales using the interface motion equation. This analysis provides insight into the importance of parameters on the vulnerability of water lenses under changing conditions, and may easily be extended to situations with abrupt or gradual changes in boundary conditions reflecting changes in land use or climate, respectively. Finally, we derive an analytical approximation of the mixing zone midway between the drains for steady state solutions, stepping away from the sharp interface approach. For a variety of examples, we validate the obtained expressions of the simplified mathematical model against the numerical model code SUTRA, which solves the fluid and solute mass balances explicitly.

Analysis of Entropy Generation via Non-Similar Numerical Approach for Magnetohydrodynamics Casson Fluid Flow with Joule Heating.

This analysis emphasizes the significance of radiation and chemical reaction effects on the boundary layer flow (BLF) of Casson liquid over a linearly elongating surface, as well as the properties of momentum, entropy production, species, and thermal dispersion. The mass diffusion coefficient and temperature-dependent models of thermal conductivity and species are used to provide thermal transportation. Nonlinear partial differential equations (NPDEs) that go against the conservation laws of mass, momentum, heat, and species transportation are the form arising problems take on. A set of coupled dimensionless partial differential equations (PDEs) are obtained from a set of convective differential equations by applying the proper non-similar transformations. Local non-similarity approaches provide an analytical approximation of the dimensionless non-similar system up to two degrees of truncations. The built-in Matlab (Version: 7.10.0.499 (R2010a)) solver bvp4c is used to perform numerical simulations of the local non-similar (LNS) truncations.

Primordial black holes from supercooled phase transitions

Cosmological first-order phase transitions (1stOPTs) are said to be strongly supercooled when the nucleation temperature is much smaller than the critical temperature. These are often encountered in theories that admit a nearly scale-invariant potential, for which the bounce action decreases only logarithmically with temperature. During supercooled 1stOPTs the equation of state of the universe undergoes a rapid and drastic change, transitioning from vacuum domination to radiation domination. The statistical variations in bubble nucleation histories imply that distinct causal patches percolate at slightly different times. Patches which percolate the latest undergo the longest vacuum-domination stage and as a consequence develop large overdensities triggering their collapse into primordial black holes (PBHs). We derive an analytical approximation for the probability of a patch to collapse into a PBH as a function of the 1stOPT duration, β−1, and deduce the expected PBH abundance. We find that 1stOPTs which take more than 15% of a Hubble time to complete (β/H≲7) produce observable PBHs. Their abundance is independent of the duration of the supercooling phase, in agreement with the de Sitter no hair conjecture. Published by the American Physical Society 2024

A Variant of the Local Similarity Theory and Approximations of Vertical Profiles of Turbulent Moments of the Atmospheric Convective Boundary Layer

The approximation of the turbulent moments of the atmospheric convective layer is based on a variant of the local similarity theory using the concepts of the semi-empirical theory of Prandtl turbulence. In the proposed variant of the local similarity theory, the second moment of vertical velocity and the “spectral” Prandtl mixing length are used as basic parameters. This approach allows us to extend Prandtl’s theory to turbulent moments of vertical velocity and buoyancy and additionally offer more than ten new approximations. The comparison of the proposed approximation with other variants of the theory of local similarity is considered. It is shown that the selected basic parameters significantly improve the agreement between the local similarity approximations and experimental data. The approximations are consistent with observations in the turbulent convective layer of the atmosphere, the upper boundary of which nearly corresponds to the lower boundary of the temperature inversion. Analytical approximations of local similarity can find applications in the construction of high-order moment closures in the vortex of resolving numerical turbulence models, as well as in the construction of “mass-flux” parametrization.

Totally asymmetric simple exclusion process on a dynamic lattice with local inhomogeneity

In this manuscript, we introduce a totally asymmetric simple exclusion process on a dynamic lattice with inhomogeneity to model motor-based long-range transport along microtubules (MTs) which exhibit cycles of growth, shortening and regrowth at their growing tips. By mean-field approximation and numerical simulations, we explore phase diagrams for the particle density near the dynamic end of the lattice as well as over the entire lattice. In particular, we find seven different phases for the density over the entire lattice and explore the corresponding phase diagram by analytical analysis. Interestingly, we find that the density may have a linear profile in part of the lattice. When the lattice has a defect, we separate the lattice into two subsystems: a right subsystem with fixed length and a defect in the middle and a left subsystem with dynamic length. When the right subsystem is in maximum-current phase, we calculate an analytical approximation of the density away from the boundaries and the defect site and the density is in association with the reduced hopping rate at the defect site; the density at the left side of the defect site serves as the effective entry rate of the left subsystem. We find that with defects, the linear profiles no longer appear at the left end of the left subsystem and only five possible phases could occur. This is a step-forward to a more comprehensive mathematical description of intracellular long-range transport of organelles along MTs with instability.

Virial relations for elongated plasmas in tokamaks: Analytical approximations and numerical calculations

Virial relations for elongated plasmas in tokamaks: Analytical approximations and numerical calculations

Average Entropy of Gaussian Mixtures.

We calculate the average differential entropy of a q-component Gaussian mixture in Rn. For simplicity, all components have covariance matrix σ21, while the means {Wi}i=1q are i.i.d. Gaussian vectors with zero mean and covariance s21. We obtain a series expansion in μ=s2/σ2 for the average differential entropy up to order O(μ2), and we provide a recipe to calculate higher-order terms. Our result provides an analytic approximation with a quantifiable order of magnitude for the error, which is not achieved in previous literature.

Scaling law for a buckled elastic filament in a shear flow.

We analyze the three-dimensional (3D) buckling of an elastic filament in a shear flow of a viscous fluid at low Reynolds number and high Péclet number. We apply the Euler-Bernoulli beam (elastica) theoretical model. We show the universal character of the full 3D spectral problem for a small perturbation of a thin filament from a straight position of arbitrary orientation. We use the eigenvalues and eigenfunctions for the linearized elastica equationin the shear plane, found earlier by Liu etal. [Phys. Rev. Fluids 9, 014101 (2024)2469-990X10.1103/PhysRevFluids.9.014101] with the Chebyshev spectral collocation method, to solve the full 3D eigenproblem. We provide a simple analytic approximation of the eigenfunctions, represented as Gaussian wave packets. As the main result of the paper, we derive the square-root dependence of the eigenfunction wave number on the parameter χ[over ̃]=-ηsin2ϕsin^{2}θ, where η is the elastoviscous number and the filament orientation is determined by the zenith angle θ with respect to the vorticity direction and the azimuthal angle ϕ relative to the flow direction. We also compare the eigenfunctions with shapes of slightly buckled elastic filaments with a non-negligible thickness with the same Young's modulus, using the bead model and performing numerical simulations with the precise hydromultipole numerical codes.

Analytical approximations of laminar flow states above a curved wall heated from below

An analytical framework is developed for estimating wall-bounded plane laminar flow states above curved walls subject to external shear and heating. The framework is applicable to arbitrary lower-wall geometries with arbitrary temperature prescriptions. The study relies on perturbation expansions of flow states with respect to the inverse of the dimensionless channel wavelength. Exact closed-form solutions are obtained for systems of perturbation equations up to the third order, which have the freedom to allow for any flow constraint. As an application of the analytical framework, we examine flow states above two specific instances of wall geometries with long channel wavelengths. The approximated flow states shed insights into morphologies of laminar convection states at low shear rates (i.e. low Reynolds numbers).

Analytical proxy to families of numerical solutions: the case study of spherical mini-boson stars

The Einstein field equations, or generalizations thereof, are difficult to solve analytically. On the other hand, numerical solutions of the same equations have become increasingly common, in particular concerning compact objects. Whereas analytic approximations to each individual solution within a numerical family have been proposed, proxies for whole families are missing, which can facilitate studying properties across the parameter space, data compression and a wider usage of such solutions. In this work we tackle this need, proposing a simple strategy based on two differentexpansions of the unknown functions in an appropriately chosen basis, to build such proxy. We use as an exploratory case-study spherical, fundamental mini-boson stars, to illustrate the feasibility of such an approach, emphasise its advantage in reducing the data size, and the challenges, say, in covering large parameter spaces.