2D calculation of parasitic fields in misaligned multipole electron-optical systems

The effects of geometrical imperfections in electron-optical components are usually evaluated in 3D simulations. These calculations inherently take a long time, require a large amount of memory, and do not directly produce the necessary axial field functions. We present a 2D perturbation method to calculate parasitic fields in misaligned multipole systems. Our method is based on finding an equivalent potential perturbation, similarly to Sturrock's method, but does not rely on the potential being differentiable. The method is directly applicable to both electrostatic and non-saturated magnetic problems. It does not require any 3D data and it is fully compatible with existing finite element method codes such as EOD. The proposed method produces axial field functions with an accuracy of units to a few tens of percents, depending on the number of unperturbed multipole field components used and the geometry. The results can then be used, for instance, to determine the parasitic imaging aberrations of the misaligned optical system using standard methods, in order to evaluate the effect of mechanical design tolerances.

A perturbative approach to the analysis of many-compartment models characterized by the presence of waning immunity

The waning of immunity after recovery or vaccination is a major factor accounting for the severity and prolonged duration of an array of epidemics, ranging from COVID-19 to diphtheria and pertussis. To study the effectiveness of different immunity level-based vaccination schemes in mitigating the impact of waning immunity, we construct epidemiological models that mimic the latter's effect. The total susceptible population is divided into an arbitrarily large number of discrete compartments with varying levels of disease immunity. We then vaccinate various compartments within this framework, comparing the value of [Formula: see text] and the equilibria locations for our systems to determine an optimal immunization scheme under natural constraints. Relying on perturbative analysis, we establish a number of results concerning the location, existence, and uniqueness of the system's endemic equilibria, as well as results on disease-free equilibria. We use numerical techniques to supplement our analytical ones, applying our model to waning immunity dynamics in pertussis, among other diseases. Our analytical results are applicable to a wide range of systems composed of arbitrarily many ODEs.

A Mathematical Modelling Study of Chemotactic Dynamics in Cell Cultures: The Impact of Spatio-temporal Heterogeneity

As motivated by studies of cellular motility driven by spatiotemporal chemotactic gradients in microdevices, we develop a framework for constructing approximate analytical solutions for the location, speed and cellular densities for cell chemotaxis waves in heterogeneous fields of chemoattractant from the underlying partial differential equation models. In particular, such chemotactic waves are not in general translationally invariant travelling waves, but possess a spatial variation that evolves in time, and may even oscillate back and forth in time, according to the details of the chemotactic gradients. The analytical framework exploits the observation that unbiased cellular diffusive flux is typically small compared to chemotactic fluxes and is first developed and validated for a range of exemplar scenarios. The framework is subsequently applied to more complex models considering the chemoattractant dynamics under more general settings, potentially including those of relevance for representing pathophysiology scenarios in microdevice studies. In particular, even though solutions cannot be constructed in all cases, a wide variety of scenarios can be considered analytically, firstly providing global insight into the important mechanisms and features of cell motility in complex spatiotemporal fields of chemoattractant. Such analytical solutions also provide a means of rapid evaluation of model predictions, with the prospect of application in computationally demanding investigations relating theoretical models and experimental observation, such as Bayesian parameter estimation.

Quantitative Comparison of Analytical Solution and Finite Element Method for Investigation of Near-infrared Light Propagation in Brain Tissue Model

Introduction

Functional near-infrared spectroscopy (fNIRS) is an imaging method in which a light source and detector are installed on the head; consequently, the re-emission of light from human skin contains information about cerebral hemodynamic alteration. The spatial probability distribution profile of photons penetrating tissue at a source spot, scattering into the tissue, and being released at an appropriate detector position, represents the spatial sensitivity.

Methods

Modeling light propagation in a human head is essential for quantitative near-infrared spectroscopy and optical imaging. The specific form of the distribution of light is obtained using the theory of perturbation. An analytical solution of the perturbative diffusion equation (DE) and finite element method (FEM) in a Slab media (similar to the human head) makes it possible to study light propagation due to absorption and scattering of brain tissue.

Results

The simulation result indicates that sensitivity is slowly decreasing in the deep area, and the sensitivity below the source and detector is the highest. The depth sensitivity and computation time of both analytical and FEM methods are compared. The simulation time of the analytical approach is four times larger than the FEM.

Conclusion

In this paper, an analytical solution and the performance of FEM methods when applied to the diffusion equation for heterogeneous media with a single spherical defect are compared. The depth sensitivity along with the computation time of simulation has been investigated for both methods. For simple and Slab modes of the human brain, the analytical solution is the right candidate. Whenever the brain model is sophisticated, it is possible to use FEM methods, but it costs a higher computation time.

Highlights

Analytical and finite element method (FEM) depth sensitivity are almost the same.FEM requires more computation time, but can handle complicated head models.The analytical method is proposed for the first step and simple head models.

Plain Language Summary

The functional near-infrared spectroscopy (fNIRS) is a type of neuromonitoring that uses near-infrared light to measure brain activity indirectly and is similar to electroencephalography (EEG). A single-channel fNIRS system contains a near-infrared light source, which emits near-infrared light (NIR), and a detector is placed near the source. A light intensity change received by detectors indicates brain activity when NIR light penetrates into the gray matter. It is necessary to have a prior understanding of light penetration depth in order to measure brain activity more accurately. fNIRS can be better understood, optimized, and investigated through modeling light propagation in brain tissue. In order to study light in tissues, analytical and numerical methods can be used. In this work, we compared these two approaches quantitatively in a simple slab medium. We concluded that the numerical method takes too much time to calculate the results, but it can be applied to complicated head models. The results of these studies provide researchers with new insights into the modeling and simulation of fNIRS and diffuse optical tomography.

Derivation of Bessel function closed-form solutions in zero dimensional φ4-field theory

The integral ∫ - ∞ ∞ e - x 2 - g x 4 d x is used as an introductory learning tool in the study of Quantum Field Theory and path integrals. Typically, it is analyzed via perturbation theory. Closed-form solutions have been quoted for which I could not find any derivation. Using a simple and elegant transformation, the close form solutions for the integral and its even positive integer moments can be obtained in terms of Bessel functions.

Long-range seed dispersal enables almost stationary patterns in a model for dryland vegetation

Spatiotemporal patterns of vegetation are a ubiquitous feature of semi-arid ecosystems. On sloped terrain, vegetation patterns occur as stripes perpendicular to the contours. Field studies report contrasting long-term dynamics between different observation sites; some observe slow uphill migration of vegetation bands while some report stationary patterns. In this paper, we show that long-range seed dispersal provides a mechanism that enables the occurrence of both migrating and stationary patterns. We utilise a nonlocal PDE model in which seed dispersal is accounted for by a convolution term. The model represents vegetation patterns as periodic travelling waves and numerical continuation shows that both migrating and almost stationary patterns are stable if seed dispersal distances are sufficiently large. We use a perturbation theory approach to obtain analytical confirmation of the existence of almost stationary patterned solutions and provide a biological interpretation of the phenomenon.

Enhanced stark effect in Dirac materials

The Stark effect in confined geometries is sensitive to boundary conditions. The vanishing wave function required on the boundary of nanostructures described by the infinite-barrier Schrödinger equation means that such states are only weakly polarizable. In contrast, materials described by the Dirac equation are characterized by much less restrictive boundary conditions. Focusing on honeycomb-lattice armchair nanoribbons, we demonstrate an enhancement by more than an order of magnitude. This result follows from an exact Dirac polarizability valid for arbitrary mass, momentum and ribbon width. Moreover, an exact expression for the frequency-dependent dynamic polarizability has been derived. Our analytic Dirac results have been validated by comparison to numerical results from atomistic models.

Quantifying the quadrupolar interaction by 45Sc-NMR spectroscopy of single crystals

Single crystals of the compound [ [Formula: see text] ]Cl₄ ⋅ 2H₂O were studied by ⁴⁵Sc-NMR, with the effect of the quadrupolar coupling interaction on the spectra of the spin-7/2 nucleus analysed in the hierarchical framework of perturbation theory. Orientation-dependent spectra acquired at B₀ = 17.6 T showed strong second-order effects due to the comparatively large coupling constant of χ = |14.613 ± 0.006| MHz, with an associated asymmetry parameter of η_Q = 0.540 9 ± 0.000 4. By analysing the splittings of the ±3/2 satellites, which in good approximation are subjected to first-order effects only, the full quadrupolar coupling tensor could be determined. The second-order effects caused by this tensor were calculated according to theoretical predictions for all orientations, and subtracted from both the centres of gravity of the satellites, and the central transitions. This allowed extraction of the full chemical shift tensor, with the eigenvalues being δ₁₁ = (5.6 ± 0.9) ppm, δ₂₂ = (12.4 ± 0.9) ppm, and δ₃₃ = (38.5 ± 0.9) ppm. In spectra acquired at a lower magnetic field of B₀ = 9.4 T, third-order effects could be detected, and similarly quantified using analytical expressions.

Machine Learning and Perturbation Theory Machine Learning (PTML) in Medicinal Chemistry, Biotechnology, and Nanotechnology

Recently, different authors have reported Perturbation Theory (PT) methods combined with machine learning (ML) to obtain PTML (PT + ML) models. They have applied PTML models to the study of different biological systems. Here we present one state-of-art review about the different applications of PTML models in Organic Synthesis, Medicinal Chemistry, Protein Research, and Technology. The aim of the models is to find relations between the molecular descriptors and the biological characteristics to predict key properties of new compounds. An area where the ML has been very useful is the drug discovery process. The entire process of drug discovery leads to the generation of lots of data, and it is also a costly and time-consuming process. ML comes with the opportunity of analyzing significant amounts of chemical data obtaining outcomes to find potential drug candidates.

Using the heteronuclear Bloch-Siegert shift of protons for B1 calibration of insensitive nuclei not present in the sample

Indirect rf field calibration using the heteronuclear Bloch-Siegert shift is presented. This method is useful for calibrating ω₁ = -γB₁ for the rf channels of small volume fast-spinning probes on which direct rf calibration is practically inconvenient or difficult for insensitive low-γ nuclei. Proton signals are observed for the rf calibration of the insensitive nuclei without requiring their presence in the sample. For a linearly modulated rf field, the heteronuclear Bloch-Siegert shift is given by, Δω_BS=ω₀ω₁²/ω₀²-ω_irr², where ω₀ and ω_irr are the Larmor and irradiation frequencies, respectively. A short protocol using full-echo acquisition of protons is described for measurement of the phase change induced by the Bloch-Siegert shift. The calibration procedure is validated by a comparison with direct ¹³C calibration and demonstrated for ¹⁴N rf field measurement of a 0.75 mm 100 kHz triple-resonance magic-angle spinning probe.

On the validity of several previously published perturbation formulas for the acoustoelastic effect on Rayleigh waves

This article revisits the evaluation by a perturbation theory of the modification of the Rayleigh wave velocity under a static loading varying with depth. Two derivations, that have been exposed in the past and presented as comparable, are questioned. A new derivation of the perturbation formula is given by adapting Auld's approach. Validation with exact calculations is provided. The examples cover depth-varying static stress as well as depth-varying third order elastic properties.

Nonlinear effects of micro-cracks on long-wavelength symmetric Lamb waves

For an elastic medium containing a homogeneous distribution of micro-cracks, an effective one-dimensional stress-strain relation has been determined with finite element simulations. In addition to flat micro-cracks, voids were considered that contain a Hertzian contact, which represents an example for micro-cracks with internal structure. The orientation of both types of micro-cracks was fully aligned or, for flat micro-cracks, totally random. For micro-cracks with Hertzian contacts, the case of random orientation was treated in an approximate way. The two types of defects were found to give rise to different degrees of non-analytic behavior of the effective stress-strain relation, which governs the nonlinear propagation of symmetric (S0) Lamb waves in the long-wavelength limit. The presence of flat micro-cracks causes even harmonics to grow linearly with propagation distance with amplitudes proportional to the amplitude of the fundamental wave, and gives rise to a static strain. The presence of the second type of defects leads to a linear growth of all harmonics with amplitudes proportional to the power 3/2 of the fundamental amplitude, and to a strain-dependent velocity shift. Simple expressions are given for the growth rates of higher harmonics of S0 Lamb waves in terms of the parameters occurring in the effective stress-strain relation. They have partly been determined quantitatively with the help of the FEM results for different micro-crack concentrations.

A perturbative approach for enhancing the performance of time series forecasting

This paper proposes a method to perform time series prediction based on perturbation theory. The approach is based on continuously adjusting an initial forecasting model to asymptotically approximate a desired time series model. First, a predictive model generates an initial forecasting for a time series. Second, a residual time series is calculated as the difference between the original time series and the initial forecasting. If that residual series is not white noise, then it can be used to improve the accuracy of the initial model and a new predictive model is adjusted using residual series. The whole process is repeated until convergence or the residual series becomes white noise. The output of the method is then given by summing up the outputs of all trained predictive models in a perturbative sense. To test the method, an experimental investigation was conducted on six real world time series. A comparison was made with six other methods experimented and ten other results found in the literature. Results show that not only the performance of the initial model is significantly improved but also the proposed method outperforms the other results previously published.

Perturbative method for maximum likelihood estimation of the Weibull distribution parameters

The two-parameter Weibull distribution is the predominant distribution in reliability and lifetime data analysis. The classical approach for estimating the scale \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha )$$\end{document}(α) and shape \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\beta )$$\end{document}(β) parameters employs the maximum likelihood estimation (MLE) method. However, most MLE based-methods resort to numerical or graphical techniques due to the lack of closed-form expressions for the Weibull \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}β parameter. A Weibull \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}β parameter estimator based on perturbation theory is proposed in this work. An explicit expression for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document}β is obtained, making the estimation of both parameters straightforward. Several right-censored lifetime data sets with different sample sizes and censoring percentages were analyzed in order to assess the performance of the proposed estimator. Study case results show that our parameter estimator provides solutions of high accuracy, overpassing limitations of other parameter estimators.

Planar aromaticity of D N h -symmetrical systems as a perturbed two-dimensional (2D) rigid rotor

Energy levels for D _{N h} -symmetrical cyclic polyenes with planar aromaticity were obtained using two-dimensional (2D) rigid rotor as the zeroth approximation. The addition of nuclei was simulated by the cosine-type potential and treated at the first-order perturbation theory. The result is qualitatively equivalent to that obtained using Hückel's molecular orbital theory. As an addition, the energetic shift between σ and π orbitals is obtained using the model of electron oscillating around the center of a positively charged ring.

General continuous-time Markov model of sequence evolution via insertions/deletions: local alignment probability computation

Background

Insertions and deletions (indels) account for more nucleotide differences between two related DNA sequences than substitutions do, and thus it is imperative to develop a method to reliably calculate the occurrence probabilities of sequence alignments via evolutionary processes on an entire sequence. Previously, we presented a perturbative formulation that facilitates the ab initio calculation of alignment probabilities under a continuous-time Markov model, which describes the stochastic evolution of an entire sequence via indels with quite general rate parameters. And we demonstrated that, under some conditions, the ab initio probability of an alignment can be factorized into the product of an overall factor and contributions from regions (or local alignments) delimited by gapless columns.

Results

Here, using our formulation, we attempt to approximately calculate the probabilities of local alignments under space-homogeneous cases. First, for each of all types of local pairwise alignments (PWAs) and some typical types of local multiple sequence alignments (MSAs), we numerically computed the total contribution from all parsimonious indel histories and that from all next-parsimonious histories, and compared them. Second, for some common types of local PWAs, we derived two integral equation systems that can be numerically solved to give practically exact solutions. We compared the total parsimonious contribution with the practically exact solution for each such local PWA. Third, we developed an algorithm that calculates the first-approximate MSA probability by multiplying total parsimonious contributions from all local MSAs. Then we compared the first-approximate probability of each local MSA with its absolute frequency in the MSAs created via a genuine sequence evolution simulator, Dawg. In all these analyses, the total parsimonious contributions approximated the multiplication factors fairly well, as long as gap sizes and branch lengths are at most moderate. Examination of the accuracy of another indel probabilistic model in the light of our formulation indicated some modifications necessary for the model’s accuracy improvement.

Conclusions

At least under moderate conditions, the approximate methods can quite accurately calculate ab initio alignment probabilities under biologically more realistic models than before. Thus, our formulation will provide other indel probabilistic models with a sound reference point.

Electronic supplementary material

The online version of this article (doi:10.1186/s12859-016-1167-6) contains supplementary material, which is available to authorized users.

PeTTSy: a computational tool for perturbation analysis of complex systems biology models

Background

Over the last decade sensitivity analysis techniques have been shown to be very useful to analyse complex and high dimensional Systems Biology models. However, many of the currently available toolboxes have either used parameter sampling, been focused on a restricted set of model observables of interest, studied optimisation of a objective function, or have not dealt with multiple simultaneous model parameter changes where the changes can be permanent or temporary.

Results

Here we introduce our new, freely downloadable toolbox, PeTTSy (Perturbation Theory Toolbox for Systems). PeTTSy is a package for MATLAB which implements a wide array of techniques for the perturbation theory and sensitivity analysis of large and complex ordinary differential equation (ODE) based models. PeTTSy is a comprehensive modelling framework that introduces a number of new approaches and that fully addresses analysis of oscillatory systems. It examines sensitivity analysis of the models to perturbations of parameters, where the perturbation timing, strength, length and overall shape can be controlled by the user. This can be done in a system-global setting, namely, the user can determine how many parameters to perturb, by how much and for how long. PeTTSy also offers the user the ability to explore the effect of the parameter perturbations on many different types of outputs: period, phase (timing of peak) and model solutions. PeTTSy can be employed on a wide range of mathematical models including free-running and forced oscillators and signalling systems.

To enable experimental optimisation using the Fisher Information Matrix it efficiently allows one to combine multiple variants of a model (i.e. a model with multiple experimental conditions) in order to determine the value of new experiments. It is especially useful in the analysis of large and complex models involving many variables and parameters.

Conclusions

PeTTSy is a comprehensive tool for analysing large and complex models of regulatory and signalling systems. It allows for simulation and analysis of models under a variety of environmental conditions and for experimental optimisation of complex combined experiments. With its unique set of tools it makes a valuable addition to the current library of sensitivity analysis toolboxes. We believe that this software will be of great use to the wider biological, systems biology and modelling communities.

Electronic supplementary material

The online version of this article (doi:10.1186/s12859-016-0972-2) contains supplementary material, which is available to authorized users.

Computational ecotoxicology: simultaneous prediction of ecotoxic effects of nanoparticles under different experimental conditions

Nanotechnology has brought great advances to many fields of modern science. A manifold of applications of nanoparticles have been found due to their interesting optical, electrical, and biological/chemical properties. However, the potential toxic effects of nanoparticles to different ecosystems are of special concern nowadays. Despite the efforts of the scientific community, the mechanisms of toxicity of nanoparticles are still poorly understood. Quantitative-structure activity/toxicity relationships (QSAR/QSTR) models have just started being useful computational tools for the assessment of toxic effects of nanomaterials. But most QSAR/QSTR models have been applied so far to predict ecotoxicity against only one organism/bio-indicator such as Daphnia magna. This prevents having a deeper knowledge about the real ecotoxic effects of nanoparticles, and consequently, there is no possibility to establish an efficient risk assessment of nanomaterials in the environment. In this work, a perturbation model for nano-QSAR problems is introduced with the aim of simultaneously predicting the ecotoxicity of different nanoparticles against several assay organisms (bio-indicators), by considering also multiple measures of ecotoxicity, as well as the chemical compositions, sizes, conditions under which the sizes were measured, shapes, and the time during which the diverse assay organisms were exposed to nanoparticles. The QSAR-perturbation model was derived from a database containing 5520 cases (nanoparticle-nanoparticle pairs), and it was shown to exhibit accuracies of ca. 99% in both training and prediction sets. In order to demonstrate the practical applicability of our model, three different nickel-based nanoparticles (Ni) with experimental values reported in the literature were predicted. The predictions were found to be in very good agreement with the experimental evidences, confirming that Ni-nanoparticles are not ecotoxic when compared with other nanoparticles. The results of this study thus provide a single valuable tool toward an efficient prediction of the ecotoxicity of nanoparticles under multiple experimental conditions.

A perturbation theory study of electron vortices in electromagnetic fields: the case of infinitely long line charge and magnetic dipole

The novel discovery of electron vortices carrying quantized orbital angular momentum motivated intensive research of their basic properties as well as applications, e.g. structural characterization of magnetic materials. In this paper, the fundamental interactions of electron vortices within infinitely long atomic-column-like electromagnetic fields are studied based on the relativistically corrected Pauli-Schrödinger equation and the perturbation theory. The relative strengths of three fundamental interactions, i.e. the electron-electric potential interaction, the electron-magnetic potential/field interaction and the spin-orbit coupling are discussed. The results suggest that the perturbation energies of the last two interactions are in an order of 10(3)-10(4) smaller than that of the first one for electron vortices. In addition, it is also found that the strengths of these interactions are strongly dependant on the spatial distributions of the electromagnetic field as well as the electron vortices.

A path integral formulation of the Wright-Fisher process with genic selection

The Wright-Fisher process with selection is an important tool in population genetics theory. Traditional analysis of this process relies on the diffusion approximation. The diffusion approximation is usually studied in a partial differential equations framework. In this paper, I introduce a path integral formalism to study the Wright-Fisher process with selection and use that formalism to obtain a simple perturbation series to approximate the transition density. The perturbation series can be understood in terms of Feynman diagrams, which have a simple probabilistic interpretation in terms of selective events. The perturbation series proves to be an accurate approximation of the transition density for weak selection and is shown to be arbitrarily accurate for any selection coefficient.

Vibrational energy levels of difluorodioxirane computed with variational and perturbative methods from a hybrid force field

We have computed vibrational energy levels of difluorodioxirane (CF2O2). For the potential, a Taylor expansion in normal coordinates is used. The CCSD(T) and MP2 methods and correlation consistent basis sets of quadruple-zeta quality are used to determine the force constants. The vibrational Schrödinger equation was solved using both a variational method and second order perturbation theory. The Watson kinetic energy operator and a discrete variable representation were used with the DEWE (E. Mátyus, G. Czakó, B.T. Sutcliffe and A.G. Császár, J. Chem. Phys. 127 (2007) 084102) computer program to do the variational calculations. For the variational calculations, the average absolute deviation of fundamentals, with respect to experimental values, is less than 3 cm(-1). Perturbative results are almost as good. About 300 vibrational levels were computed. (16)O→(18)O isotopic shifts have also been calculated variationally for the lowest 75 vibrational energy levels and are compared to experimental results.