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Genetic algorithm (GA) and particle swarm optimisation (PSO) techniques have been integrated with the differential algebra (DA) method in charged particle optics to optimise an Einzel lens. The DA method is a robust and efficient tool for the calculation of aberration coefficients of electrostatic lenses, which makes use of nonstandard analysis for ray tracing a particle as it is subjected to the field generated by a lens. In this study, initial populations of lenses with random geometrical configurations are generated. These initial populations are then subjected to GA and PSO algorithms to alter the geometry of each lens for a set number of iterations. The lens performance is evaluated by calculating the spot size using the aberrations coefficients up to third-order generated by the DA method. Moreover, a focusing column comprising two lenses and a Wien filter was optimised using GA method., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2024 The Author(s). Published by Elsevier B.V. All rights reserved.)

In this article the analytical expressions of the fifth-order asymptotic geometric aberrations of electron lenses are derived by Mathematica. The process of the derivation is analogous to the method described in "Principles of Electron Optics" by P.W. Hawkes and E. Kasper. All the analytical formulae for asymptotic aberration coefficients in polynomials in the reciprocal magnification are numerically cross-validated with the differential algebraic (DA) method. The results indicate that the derived formulae are doubtless correct., Competing Interests: Declaration of Competing Interest The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: none, (Copyright © 2023. Published by Elsevier B.V.)

The successful application of epidemic models hinges on our ability to estimate model parameters from limited observations reliably. An often-overlooked step before estimating model parameters consists of ensuring that the model parameters are structurally identifiable from the observed states of the system. In this tutorial-based primer, intended for a diverse audience, including students training in dynamic systems, we review and provide detailed guidance for conducting structural identifiability analysis of differential equation epidemic models based on a differential algebra approach using differential algebra for identifiability of systems (DAISY) and Mathematica (Wolfram Research). This approach aims to uncover any existing parameter correlations that preclude their estimation from the observed variables. We demonstrate this approach through examples, including tutorial videos of compartmental epidemic models previously employed to study transmission dynamics and control. We show that the lack of structural identifiability may be remedied by incorporating additional observations from different model states, assuming that the system's initial conditions are known, using prior information to fix some parameters involved in parameter correlations, or modifying the model based on existing parameter correlations. We also underscore how the results of structural identifiability analysis can help enrich compartmental diagrams of differential-equation models by indicating the observed state variables and the results of the structural identifiability analysis., (© 2023. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.)

Characterisation of the electron beams trajectory in an electron microscope is possible in a few select commercial software packages, but these tools and their source code are not available in a free and accessible manner. This paper introduces the free and open-source software TEMGYM Advanced, which implements ray tracing methods that calculate the path of electrons through a magnetic or electrostatic lens and allow evaluation of the first-order properties and third-order geometric aberrations. Validation of the aberration coefficient calculations is performed by implementing two independent methods - the aberration integral and differential algebra (DA) methods and by comparing the results of each. This paper also demonstrates parallelised electron ray tracing through a series of magnetic components, which enables near real-time generation of a physically accurate beam-spot including aberrations and brings closer the realisation of a digital twin of an electron microscope. TEMGYM Advanced represents a valuable resource for the electron microscopy community, providing an accessible and open source means of characterising electron lenses. This software utilises the Python programming language to complement the growing ecosystem of free and open-source software within the electron microscopy community, and to facilitate the application of machine learning to an electron microscope digital twin for instrument automation. The software is available under GNU Public License number Three (GPL 3)., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2023 The Author(s). Published by Elsevier B.V. All rights reserved.)

A complete analysis including finite element method (FEM) calculation, focal length properties, and thirdorder geometric aberrations of the open-source electrostatic lens from the NanoMi project is presented. The analysis is carried out by the software TEMGYM Advanced, a free package developed to carry out ray-tracing and lens characterisation in Python. Previously TEMGYM Advanced has shown how to analyse the aberrations of analytical lens fields; this paper expands upon this work to demonstrate how to apply a suitable fitting method to discrete lens fields obtained via FEM methods so that the aberrations of real lens designs can be calculated. Each software platform used in this paper is freely available in the community and creates a free and viable alternative to commercial lens design packages., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2023. Published by Elsevier Ltd.)

A Differential algebraic method is of an effective technique in computer numerical analysis. It implements conveniently differentiation up to arbitrary high order, based on the nonstandard analysis. In this paper, the differential algebra (DA) method has been employed to compute the high order aberrations up to the fifth order of electron lenses with quadrupole and octupole corrector whose electric/magnetic fields are in the forms of discrete arrays, for example, the files computed by FEM or FDM method. The quadrupole and octupole electro-magnetic fields of arbitrary point are obtained by local analytic expressions, and then field potentials are transformed into new forms which can be operated in the DA calculation. The program has been developed and tested as well. The geometric and chromatic aberrations coefficients up to fifth order of electron lenses with quadrupole and octupole corrector are calculated by the developed program., (Copyright © 2018. Published by Elsevier B.V.)

In this paper the third-rank chromatic aberration coefficients of round electron lenses are analytically derived and numerically calculated by Mathematica. Furthermore, the numerical results are cross-checked by the differential algebraic (DA) method, which verifies that all the formulas for the third-rank chromatic aberration coefficients are completely correct. It is hoped that this work would be helpful for further chromatic aberration correction in electron microscopy., (Copyright © 2017 Elsevier B.V. All rights reserved.)

Multistage clonal expansion (MSCE) models of carcinogenesis are continuous-time Markov process models often used to relate cancer incidence to biological mechanism. Identifiability analysis determines what model parameter combinations can, theoretically, be estimated from given data. We use a systematic approach, based on differential algebra methods traditionally used for deterministic ordinary differential equation (ODE) models, to determine identifiable combinations for a generalized subclass of MSCE models with any number of preinitation stages and one clonal expansion. Additionally, we determine the identifiable combinations of the generalized MSCE model with up to four clonal expansion stages, and conjecture the results for any number of clonal expansion stages. The results improve upon previous work in a number of ways and provide a framework to find the identifiable combinations for further variations on the MSCE models. Finally, our approach, which takes advantage of the Kolmogorov backward equations for the probability generating functions of the Markov process, demonstrates that identifiability methods used in engineering and mathematics for systems of ODEs can be applied to continuous-time Markov processes., (© 2016 Society for Risk Analysis.)

Identifiability concerns finding which unknown parameters of a model can be estimated, uniquely or otherwise, from given input-output data. If some subset of the parameters of a model cannot be determined given input-output data, then we say the model is unidentifiable. In this work, we study linear compartment models, which are a class of biological models commonly used in pharmacokinetics, physiology, and ecology. In past work, we used commutative algebra and graph theory to identify a class of linear compartment models that we call identifiable cycle models, which are unidentifiable but have the simplest possible identifiable functions (so-called monomial cycles). Here we show how to modify identifiable cycle models by adding inputs, adding outputs, or removing leaks, in such a way that we obtain an identifiable model. We also prove a constructive result on how to combine identifiable models, each corresponding to strongly connected graphs, into a larger identifiable model. We apply these theoretical results to several real-world biological models from physiology, cell biology, and ecology.

Mathematical models of physical systems often have parameters that must be identified from physical data. This makes the analysis of the parameter identifiability of the given model system an essential prerequisite. Thus far, several methods have been proposed for analyzing the parameter identifiability of ordinary differential equation (ODE) systems. But, to the best of our knowledge, the parameter identifiability of differential algebraic equation (DAE) systems has scarcely been analyzed as a specific topic. Traditional differential algebraic (DA) methods developed for ODE systems are often applied directly on DAE systems. These methods, however, are not always applicable, e.g., when the prime ideal condition is not satisfied by a DAE system. In this paper, we propose a novel method to analyze the identifiability of DAE systems, based on the concept of space extension, through which the algebraic and differential variables can be decoupled. Furthermore, an inherent, low-dimensional, regular ODE system can be obtained, which is the external equivalent of the original DAE system. Subsequently, the differential algebraic (DA) method can then be used to analyze the identifiability of the low-dimension ODE system. Theoretical analysis is also presented for the proposed method. Two examples, including a simplified interaction model and an isothermal reactor system, are presented to illustrate the detailed steps and effectiveness of the proposed method., (Copyright © 2013 ISA. Published by Elsevier Ltd. All rights reserved.)