Exact solutions for nonlinear partial differential equations via a fusion of classical methods and innovative approaches

This paper presents a new approach for finding exact solutions to certain classes of nonlinear partial differential equations (NLPDEs) by combining the variation of parameters method with classical techniques such as the method of characteristics. Our primary focus is on NLPDEs of the formu tt + a ( x , t ) u xt + b ( t ) u t = α ( x , t ) + G ( u ) ( u t + a ( x , t ) u x ) e - ∫ b ( t ) d t andu t m ( u tt + a ( x , t ) u xt ) + b ( t ) u t m + 1 = e - ( m + 1 ) ∫ b ( t ) d t ( u t + a ( x , t ) u x ) F ( u , u t e ∫ b ( t ) d t ) . We provide numerical validation through several examples to ensure accuracy and reliability. Our approach enhances the applicability of analytical solution methods for a broader range of NLPDEs.

Fifth-order asymptotic geometric aberrations of electron lenses

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.

Theory and calculation of abelian and non-abelian geometric phase factors with SpinDynamica

Cyclic quantum evolution is accompanied by a systematic change in the phase of the initial state vector. This change only depends upon the path traced out by the system itself. Such effects are collectively known as geometric phase factors. The geometric foundations of these phase factors are most elegantly formulated in terms of fibre bundle theory and differential forms, both of which can represent a significant hurdle to master. We present a derivation of the abelian and non-abelian Berry phase in terms of embedded manifolds of linear vector spaces. Embedded manifolds offer the advantage of being less abstract than fibre bundles, and are well-suited for explicit calculations. Essential features of the derivation reduce to matrix-vector manipulations. We further discuss a numerical strategy for the calculation of abelian and non-abelian phase factors. Our approach is based upon Hungarian method and the polar decomposition, and is made freely available as a SpinDynamica addon. Additionally, all derivations and analytic calculations are supported by Mathematica notebooks.

X-ray powder diffraction in education. Part II. Intensity of a powder pattern

This article is the second part of a series dealing with the description and visualization of mathematical functions used to describe a powder diffraction pattern for teaching and education purposes. The first part dealt with the instrumental and sample contributions to the profile of a Bragg peak [Dinnebier & Scardi (2021 ▸). J. Appl. Cryst. 54, 1811-1831]. The second part, here, deals with the mathematics and physics of the intensity in X-ray powder diffraction. Scholarly scripts are again provided using the Wolfram language in Mathematica.

X-ray powder diffraction in education. Part I. Bragg peak profiles

A collection of scholarly scripts dealing with the mathematics and physics of peak profile functions in X-ray powder diffraction has been written using the Wolfram language in Mathematica. Common distribution functions, the concept of convolution in real and Fourier space, instrumental aberrations, and microstructural effects are visualized in an interactive manner and explained in detail. This paper is the first part of a series dealing with the mathematical description of powder diffraction patterns for teaching and education purposes.

A series acceleration algorithm for the gamma-Pareto (type I) convolution and related functions of interest for pharmacokinetics

Abstract

The gamma-Pareto type I convolution (GPC type I) distribution, which has a power function tail, was recently shown to describe the disposition kinetics of metformin in dogs precisely and better than sums of exponentials. However, this had very long run times and lost precision for its functional values at long times following intravenous injection. An accelerated algorithm and its computer code is now presented comprising two separate routines for short and long times and which, when applied to the dog data, completes in approximately 3 min per case. The new algorithm is a more practical research tool. Potential pharmacokinetic applications are discussed.

Graphic abstract

Supplementary Information

The online version contains supplementary material available at 10.1007/s10928-021-09779-4.

Variation in Methylmercury Metabolism and Elimination in Humans: Physiological Pharmacokinetic Modeling Highlights the Role of Gut Biotransformation, Skeletal Muscle, and Hair

The biological half-life (t1/2) of methylmercury (MeHg) shows considerable individual variability (t1/2 < 30 to > 120 days), highlighting the importance of mechanisms controlling MeHg metabolism and elimination. Building on a prior physiologically based pharmacokinetic (PBPK) model, we elucidate parameters that have the greatest influence on variability of MeHg t1/2 in the human body. Employing a dataset of parameters for mean organ volumes and blood flow rates appropriate for man and woman (25-35 years) and child (4 - 6 years), we demonstrate model fitness by simulating data from our prior controlled study of MeHg elimination in people. Model predictions give MeHg t1/2 of 46.9, 38.9, and 31.5 days and steady-state blood MeHg of 2.6, 2.6, and 2.3 µg/l in man, woman, and child, respectively, subsequent to a weekly dose of 0.7 µg/kg body weight. The major routes of elimination are biotransformation to inorganic Hg in the gut lumen (73% in adults, 61% in child) and loss of MeHg via excretion within growing hair (13% in adults, 24% in child). Local and global sensitivity analyses of model parameters reveal that variation in biotransformation rate in the gut lumen, and rates of transport between gut lumen and gut tissue, have the greatest influence on MeHg t1/2. Volume and partition coefficients for skeletal muscle (SM) and gut tissue also show significant sensitivity affecting model output of MeHg t1/2. Our results emphasize the role of gut microbiota in MeHg biotransformation, transport kinetics at the level of the gut, and SM mass as moderators of MeHg kinetics in the human body.

Insights into Gene Therapy for Urea Cycle Defects by Mathematical Modeling

Metabolic liver diseases are attractive gene therapy targets that necessitate reconstitution of enzymatic activity in functionally complex biochemical pathways. The levels of enzyme activity required in individual hepatocytes and the proportion of the hepatic cell mass that must be gene corrected for therapeutic benefit vary in a disease-dependent manner that is difficult to predict. While empirical evaluation is inevitably required, useful insights can nevertheless be gained from knowledge of disease pathophysiology and theoretical approaches such as mathematical modeling. Urea cycle defects provide an excellent example. Building on a previously described one-compartment model of the urea cycle, we have constructed a two-compartment model that can simulate liver-targeted gene therapy interventions using the computational program Mathematica. The model predicts that therapeutically effective reconstitution of ureagenesis will correlate most strongly with the proportion of the hepatic cell mass transduced rather than the level of enzyme-encoding transgene expression achieved in individual hepatocytes. Importantly, these predictions are supported by experimental data in mice and human genotype/phenotype correlations. The most notable example of the latter is ornithine transcarbamylase deficiency (X-linked) where impairment of ureagenesis in male and female patients is closely simulated by the one- and two-compartment models, respectively. Collectively, these observations support the practical value of mathematical modeling in evaluation of the disease-specific gene transfer challenges posed by complex metabolic phenotypes.

Classification of Rota-Baxter operators on semigroup algebras of order two and three

In this paper, we determine all the Rota-Baxter operators of weight zero on semigroup algebras of order two and three with the help of computer algebra. We determine the matrices for these Rota-Baxter operators by directly solving the defining equations of the operators. We also produce a Mathematica procedure to predict and verify these solutions.

SpinDynamica: Symbolic and numerical magnetic resonance in a Mathematica environment

SpinDynamica is a set of Mathematica packages for performing numerical and symbolic analysis of a wide range of magnetic resonance experiments and phenomena. An overview of the SpinDynamica architecture and functionality is given, with some simple representative examples.

Third-rank chromatic aberrations of electron lenses

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.

Mathematical Development and Computational Analysis of Harmonic Phase-Magnetic Resonance Imaging (HARP-MRI) Based on Bloch Nuclear Magnetic Resonance (NMR) Diffusion Model for Myocardial Motion

Harmonic Phase-Magnetic Resonance Imaging (HARP-MRI) is a tagged image analysis method that can measure myocardial motion and strain in near real-time and is considered a potential candidate to make magnetic resonance tagging clinically viable. However, analytical expressions of radially tagged transverse magnetization in polar coordinates (which is required to appropriately describe the shape of the heart) have not been explored because the physics required to directly connect myocardial deformation of tagged Nuclear Magnetic Resonance (NMR) transverse magnetization in polar geometry and the appropriate harmonic phase parameters are not yet available. The analytical solution of Bloch NMR diffusion equation in spherical geometry with appropriate spherical wave tagging function is important for proper analysis and monitoring of heart systolic and diastolic deformation with relevant boundary conditions. In this study, we applied Harmonic Phase MRI method to compute the difference between tagged and untagged NMR transverse magnetization based on the Bloch NMR diffusion equation and obtained radial wave tagging function for analysis of myocardial motion. The analytical solution of the Bloch NMR equations and the computational simulation of myocardial motion as developed in this study are intended to significantly improve healthcare for accurate diagnosis, prognosis and treatment of cardiovascular related deceases at the lowest cost because MRI scan is still one of the most expensive anywhere. The analysis is fundamental and significant because all Magnetic Resonance Imaging techniques are based on the Bloch NMR flow equations.

rdml: A Mathematica package for parsing and importing Real-Time qPCR data

OBJECTIVE

The purpose and objective of the research presented is to provide a package for easy importing of Real-Time PCR data markup language (RDML) data to Mathematica.

RESULTS

Real-Time qPCR is the most widely used experimental method for the accurate quantification of gene expression. To enable the straightforward archiving and sharing of qPCR data and its associated experimental information, an XML-based data standard was developed-the Real-Time PCR data markup language (RDML)-devised by the RDML consortium. Here, we present rdml, a package to parse and import RDML data into Mathematica, allowing the quick loading and extraction of relevant data, thus promoting the re-analysis, meta-analysis or experimental re-validation of gene expression data deposited in RDML format.

MathIOmica-MSViewer: a dynamic viewer for mass spectrometry files for Mathematica

MathIOmica-MSViewer is an add-on graphical user interface utility for the Mathematica software system which facilitates the visualization and exploration of spectra from open format mass spectrometry files (mzXML and mzML standard community formats). The viewer was designed for simplicity and handling of large mass spectrometry data files. To facilitate searches, users may use search filters for the spectra based on mass to charge ratios and retention times, and visualize precursor spectra associated to a parent spectrum.

AVAILABILITY

The viewer is available as a Mathematica notebook (MathIOmica-MSViewer.nb) at https://doi.org/10.5281/zenodo.321385. The software is provided under an MIT License. © 2017 The Authors. Journal of Mass Spectrometry published by John Wiley & Sons, Ltd.

Novel scripts for improved annotation and selection of variants from whole exome sequencing in cancer research

Sequencing the exome is quickly becoming the preferred method for discovering disease-inducing mutations. While obtaining data sets is a straightforward procedure, the subsequent analysis and interpretation of the data is a limiting step for clinical applications. Thus, while the initial mutation and variant calling can be performed by a bioinformatician or trained researcher, the output from robust packages such as MuTect and GATK is not directly informative for the general life scientists. In attempt to obviate this problem we have created complementary Wolfram scripts, which enable easy downstream annotation and selection, presented here in the perspective of hematological relevance. It also provides the researcher with the opportunity to extend the analysis by having a full-fledged programming and analysis environment of Mathematica at hand. In brief, post-processing is performed by:

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Mapping of germ line and somatic variants to coding regions, and defining variant sets within Mathematica.

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Processing of variants in variant effect predictor.

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Extended annotation, relevance scoring and defining focus areas through the provided functions.

MESAFace, a graphical interface to analyze the MESA output

MESA (Modules for Experiments in Stellar Astrophysics) has become very popular among astrophysicists as a powerful and reliable code to simulate stellar evolution. Analyzing the output data thoroughly may, however, present some challenges and be rather time-consuming. Here we describe MESAFace, a graphical and dynamical interface which provides an intuitive, efficient and quick way to analyze the MESA output.

NATURE OF PROBLEM

Find a way to quickly and thoroughly analyze the output of a MESA run, including all the profiles, and have an efficient method to produce graphical representations of the data.

SOLUTION METHOD

We created two scripts (to be run consecutively). The first one downloads all the data from a MESA run and organizes the profiles in order of age. All the files are saved as tables or arrays of tables which can then be accessed very quickly by Mathematica. The second script uses the Manipulate function to create a graphical interface which allows the user to choose what to plot from a set of menus and buttons. The information shown is updated in real time. The user can access very quickly all the data from the run under examination and visualize it with plots and tables.

UNUSUAL FEATURES

Moving the slides in certain regions may cause an error message. This happens when Mathematica is asked to read nonexistent data. The error message, however, disappears when the slides are moved back. This issue does not preclude the good functioning of the interface.

ADDITIONAL COMMENTS

The program uses the dynamical capabilities of Mathematica. When the program is opened, Mathematica prompts the user to "Enable Dynamics". It is necessary to accept before proceeding.

RUNNING TIME

Depends on the size of the data downloaded, on where the data are stored (hard-drive or web), and on the speed of the computer or network connection. In general, downloading the data may take from a minute to several minutes. Loading directly from the web is slower. For example, downloading a 200MB data folder (a total of 102 files) with a dual-core Intel laptop, P8700, 2 GB of RAM, at 2.53 GHz took about a minute from the hard-drive and about 23 minutes from the web (with a basic home wireless connection).