Wolfram Language Research Articles

FlexibleSUSY extended to automatically compute physical quantities in any beyond the standard model theory: Charged lepton flavor violation processes, Higgs decays, and user-defined observables

FlexibleSUSY is a framework for the automated computation of physical quantities (observables) in models beyond the Standard Model (BSM). This paper describes an extension of FlexibleSUSY which allows to define and add new observables that can be enabled and computed in applicable user-defined BSM models. The extension has already been used to include Charged Lepton Flavor Violation (CLFV) observables, but further observables can now be added straightforwardly. The paper is split into two parts. The first part is non-technical and describes from the user's perspective how to enable the calculation of predefined observables, in particular CLFV observables. The second part of the paper explains how to define new observables such that their automatic computation in any applicable BSM model becomes possible. A key ingredient is the new NPointFunctions extension which allows to use tree-level and loop calculations in the model-independent setup of observables. Three examples of increasing complexity are fully worked out. This illustrates the features and provides code snippets that may be used as a starting point for implementation of further observables. Program summaryProgram title:NPointFunctionsCPC Library link to program files:https://doi.org/10.17632/kf7m8gn8vp.2Developer's repository link:https://github.com/FlexibleSUSY/FlexibleSUSYLicensing provisions: GPLv3Programming language:C++, Wolfram Language, Fortran, Bourne shellJournal reference of previous version:: Comput. Phys. Commun. 230 (2018) 145–217; PoS CompTools2021 (2022) 036Does the new version supersede the previous version?: YesReasons for the new version: Program extension including new observables and file structuresNature of problem: Determining observables for an arbitrary extension of the Standard Model supported by FlexibleSUSY, input by the user.Solution method: Generation of the code from automated algebraic manipulations. Automatic filling and compiling of predefined template files.Additional comments including restrictions and unusual features: Vertices with a direct product of Lorentz and color structures are supported. Settings of the advanced NPointFunctions mode rely on explicit specification of topologies.

Some Finite Summation Identities Comprising Binomial Coefficients for Integrals of the Bernstein Polynomials and Their Applications

Certain finite sums, including the Catalan numbers, factorial functions, binomial coefficients, and their computational formulas are of indispensable importance both in probability and statistics applications and in other branches of science. The primary aim of this article is to give the integral representation of the finite sum containing the products of the Bernstein polynomials, given in our article, by applying the Beta function and the Euler gamma functions. Other aims of this paper are to bring to light novel finite sum formulae containing binomial coefficients by analyzing and unifying this integral representation. Finally, some relations among these sums, binomial coefficients, and the Catalan numbers are given. We also give the Wolfram language codes. By applying these codes to the finite sums, we give some numerical values.

A Mathematica-Based Interface for the Exploration of Inter- and Intra-Regional Financial Flows

This work surveys the use of directed weighted graphs in conducting comparative static analyses. The paper discusses the implementation of a computer-aided process for building spreadsheet-based graph models for inter- and intra-regional financial flows. The graph-theoretic techniques are programmed to enable the interactive visualization and analysis of financial data using Wolfram technologies (i.e., Mathematica software v. 11.3 or later, Wolfram player v. 12 or later). The paper describes the workflow for several interactive visualizations applicable to financial networks. The author provides four programs, written in the Wolfram language, that customize input–output financial models by combining the Manipulate command with built-in Mathematica functions and functions of the IGraph package (IGraph/M). The study is a tutorial article for the generation of a suite of visual schemes that provide patterns, practices, and roadmaps of the financial markets across the globe.

Modeling of heat transfer and fluid flow in the metal being welded during DC and HFPC TIG spot welding

A mathematical model of non-stationary electromagnetic, heat and mass transfer processes in the metal during high frequency pulsed current (HFPC) TIG spot welding has been proposed. A computational algorithm for the the numerical solution of the corresponding mathematical problems has been developed and implemented in Wolfram Language. A comparative analysis of the hydrodynamic and heat transfer processes in the specimen of steel S-235JR during HFPC TIG welding with 10 kHz square wave pulse current modulation and direct current (DC) TIG welding with current equal to the mean current of HFPC mode has been carried out. As simulations showed, the HFPC modulation results to appreciable weld penetration with minor influence on the radius of the fusion zone compared to the DC process.

Spectral approach with iterative clarification of a radiation boundary conditions for modeling of quasimodes of a gyrotrons open cavities

Purpose. The article presents a new method for numerical simulation of quasi-eigenmode oscillations in open resonators of gyrotrons — powerful vacuum generators of electromagnetic waves in the millimeter and submillimeter ranges. The gyrotron cavity has the shape of a weakly inhomogeneous hollow circular metal waveguide. Methods. The proposed approach uses the inhomogeneous string equation with radiation boundary conditions to formulate a nonlinear spectral boundary value problem describing oscillations in a resonator, neglecting the couplings of waves with different radial indices. By linearizing with respect to frequency the radiation boundary conditions, the boundary value problem is reduced to a linear boundary value problem. To discretize this boundary value problem, the finite difference method is used and a linear generalized matrix eigenvalue problem is formulated. This problem is solved by the Arnoldi method with eigenvalues calculation in a shift-invert mode. An iterative algorithm is proposed that makes it possible to sequentially calculate a given number of frequencies and quality factors of quasi-eigenmodes of oscillations. Results. The computer program was developed written in the Wolfram Language and Fortran using the algorithms proposed in the work. The results of test calculations for real gyrotron resonators are presented, which demonstrate the high accuracy of the obtained values of frequencies, quality factors and field distributions of quasi-eigenmode oscillations in the studied resonators. Conclusion. The methods, algorithms and created program proposed in the article can significantly facilitate the process of developing gyrotrons intended for various practical applications and operating in new frequency ranges. The method of iterative refinement of boundary conditions can be generalized to the case of equations of the linear theory of a gyrotron and used to develop new methods for analyzing the starting conditions for the soft self-excitation in gyrotrons — generators.

Numerical computation and prediction analysis on multilayer flow through porous medium saturated with mono nanofluid

Many scientific and technological disciplines in the specialization of chemical and petroleum sectors are related to multilayer fluid flow systems. In light of this perspective, the current article focuses on the steady, laminar flow of mono nanofluid saturated with porous media sandwiched between a viscous fluid filled in a vertical channel to develop a new model. The main subject of this scientifical research is to explore the prediction analysis through machine learning algorithm for the physical quantities across the entire region of the proposed fluid. This present physical phenomenon is witnessed in the mathematical form of nonlinear PDEs through the conservation principles and Darcy’s law. The system of governing PDEs are nondimensionalized by treating the suitable fundamental variables to facilitate the simulation process. After that, the dimension-free form of momentum and temperature equations is numerically simulated by utilizing the Wolfram language algorithm. In addition, the comparison analysis is accounted to find the higher heat transfer performance for the particles After ensuring the numerical computation for the specific flow scenarios, an innovative machine learning-based multiple linear regression technique is employed to anticipate the flow and heat transfer rate for the inclusion of Ag nanoparticle. The validation of the present study is highlighted by comparing with the existing literature. The current findings exhibit that the region of single-phase nanosized particles suspended in water affects the momentum and thermal behavior of the clear viscous fluid regions. Silver Ag nanosized particles exhibit better performance in heat transfer than the MgO and Also, the MLR approach offers a new viewpoint on flow control with higher accuracy 99.59% and lower error The outcomes of the suggested techniques enrich the application of the multilayer fluid flow model, specifically fire suppression systems, petroleum sectors, agricultural sprays, and oil recovery.

Turing Patterns in Networks

The first pattern formation model was proposed by the mathematician Alan M. Turing. This model consists of a system of reaction-diffusion equations that produces stationary patterns by means of the so-called “Turing instability.” In this paper, we found the conditions that the network and the parameters need to fulfill in order to achieve the Turing instability in a particular reaction-diffusion system called the Mimura–Murray model on different network topologies, including some simulations on an innovative kind of network, based on the Wolfram model, that evolves over time, generating interesting topologies that exhibit lattice-like topology. In addition, the equations are solved and simulated in Wolfram Language, and some examples of applications in biology and sociology are presented.

A novel algorithm for seismic events multiplets search

A common practice of seismology is to analyze earthquake occurrence in terms of events catalogues, with the aim to either find useful correlations between internal mechanisms under study and their outcome in the spatial/temporal series of the events or, more directly, to assess some statistical rules from observations. With this approach, catalogues are often searched for some recognizable patterns or behaviors: in this work we present a software tool created to reveal a particular kind of events sequences.The idea follows from the concept of multiplets, a well known events pattern often found in seismic series. A multiplet is defined as a sequence of events, all near in space and time and exhibiting similar magnitudes. The amount of multiplets in seismic series is related, as it is for other clustering mechanisms, to underlying correlations in the physics of the events.The software, built from scratch, scans seismic catalogues in search of events clustered as “multiplets”: this is done through the thorough application of comparison tests whose parameters thresholds are both user defined and semi-automated. The tool is however more “general” in the sense that by varying values of the filtering parameters it can reveal other kind of patterns too.While we think that this tool can be thought as a general purpose space–time series analyzer, we have found it particularly useful when applied to the results of a seismic simulator with the purpose of assessing their adherence with the observed seismicity. It can be used as a sort of metric to quantify the simulation predictions effectiveness in terms of presence of similar multiplets distributions in simulated vs. real catalogues.The software has been entirely developed in the Wolfram Language (Mathematica), a commercial powerful environment for scientific calculus and results report, but the main computational routine has been also ported to python for open-source, copyleft usage.

MHD boundary layer flow due to an exponentially stretching surface through porous medium with radiation effect

The purpose of this article is to study the boundary layer flow and heat transfer of the MHD second-grade fluid. By utilizing similarity transformations, the governing equations are transformed into a set of non-linear ordinary differential equations. To get semi-analytical formulations of velocity, temperature, and other variables, we use the homotopy analysis technique (HAM). Then, we employ the Wolfram Language function NSolve to get the solutions. The main finding of the present work is that the flow variables have been influenced by the magnetic field parameter, the porous parameter, and the radiation parameter.

Investigating hygrothermal bending behavior of FG nanobeams via local/nonlocal stress gradient theory of elasticity with general boundary conditions

The bending response of functionally graded (FG) nanobeams under hygrothermal loading was investigated to emphasize the different scenarios that arise when using simplified and original boundary conditions. The governing equations were derived by using the principle of virtual work on the basis of the local/nonlocal stress gradient theory of elasticity. A Wolfram language code in Mathematica was written by the authors to develop a numerical investigation for different values of the material gradient index, the gradient length parameter, the nonlocal parameter, and considering two distinct types of thermal loading, that is, uniform temperature rise and heat conduction across the thickness of FG nanobeam cross-section.

Impact on Casson Nanofluid Flow Across an Inclined, Slanted Surface by Radiation, Energy and Mass Transfer

Casson nanofluid flow across a porous, slanted surface is investigated. In addition to the impact of a heat source, thermal energy, and a chemical reaction also are considered. Applying similarity transformations, the controlling nonlinear PDEs are converted to ODEs. To resolve the ordinary differential equations (ODEs), we used the universal numerical differential equation solver Wolfram Language function NDSolve. The distributions under heat source are affected by chemical processes and other variables are explored. While liquid velocity reduced due to inclination and the Casson factor, mass and energy transit rates rose. The findings are represented graphically and are consistent with past research. The findings would provide valuable insights into the flow patterns, temperature distribution, and concentration profiles within the system. This information can be crucial for designing and optimizing heat transfer systems, energy-efficient processes, and catalytic reactors involving Casson nanofluids and porous media.

Stochastic Differential Equations in Finance

Harnessing the power of Mathematica and the Wolfram language for stochastic or probabilistic models

On low rank fusion rings

We present a method to generate all fusion rings of a specific rank and multiplicity. This method generated exhaustive lists of fusion rings up to order 9 for several multiplicities. We introduce a class of non-commutative fusion rings based on a group with transitive action on a set. This construction generalises the Tambara–Yamagami (TY) and Haagerup-Izumi (HI) fusion rings. We give an example of two such rings that are categorifiable and not of TY or HI type. The structure of non-commutative fusion rings with a subgroup is reviewed, and the one- and two-particle extensions of groups are classified. A website containing data on fusion rings is introduced, and an overview of a Wolfram Language package for working with these rings is given. We also applied several categorifiability criteria to the fusion rings. We provide a table of all multiplicity-free fusion rings up to rank 9 with info on the categorifiability of each ring.

Ternary Hybrid Nanofluid Flow Emerging on a Symmetrically Stretching Sheet Optimization with Machine Learning Prediction Scheme

Nanofluids holding three distinct sorts of nanosized particles suspended in base fluid possess excellent thermal performance. In light of this novel use in coolant applications, the current work dealt with the optimal design and performance estimation of a ternary hybrid nanofluid, based on a modern machine learning prediction technique. The synthesis of (Cu), (TiO2), and (SiO2) ternary hybrid nanoparticles suspended in water over a symmetrically stretching sheet was scrutinized. The flow over a stretching sheet is the most noteworthy symmetry analysis for momentum and thermal boundary layers, due to the implications of heat transfer, and is applied in various industries and technological fields. The governing equations were transformed to a dimension-free series of ODEs, by handling similarity transformable with symmetry variables, after which, the series of ODEs were treated scientifically, with the help of the Wolfram Language tool. The precision of the current estimates was assessed by comparison to existing research. Moreover, the natures of the physical phenomena were forecast by designing a support vector machine algorithm with an emphasis on machine learning, which delivers a robust and efficient structure for every fluid application that infers physical influences. To validate the proposed research, some of the statistical metrics were taken for error assessment between true and anticipated values. It was revealed that the presented approach is the best strategy for predicting physical quantities. This investigation established that ternary hybrid nanofluid possesses excellent thermal performance, greater than that of hybrid nanofluid. The current optimization process delivers a new beneficial viewpoint on the production of polymer sheets, glass fiber, petroleum, plastic films, heat exchangers, and electronic devices. Hence, the obtained results are recommended for the development of industrial devices setups.

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.

Graphical User Interface for Design, Analysis, Validation, and Reporting of Continuous-Time Systems Using Wolfram Language

A graphical user interface presented is intended for fast design, symbolic analysis, accurate simulation, exact verification, and test report preparation. It helps to skip the gap between theory and practice in electrical engineering because the numeric analysis is usually approximate, and the power of symbolic systems has insufficient speed even, for simple engineering problems. The software is written using a computer algebra system that is free on small computers. The mathematical representation of the system can be obtained automatically from the schematic description. Further automated symbolic manipulations are possible according to the user’s aspirations.

IGraph/M: graph theory and network analysis for Mathematica

IGraph/M is an efficient general purpose graph theory and network analysis package for Mathematica. IGraph/M serves as the Wolfram Language interfaces to the igraph C library, and also provides several unique pieces of functionality not yet present in igraph, but made possible by combining its capabilities with Mathematica's. The package is designed to support both graph theoretical research as well as the analysis of large-scale empirical networks.

Fixed Income and Debt Instruments: Annuities, Bonds, Cashflows and Mortgages

This is the first in a sequence of articles on finance using the Wolfram language. While this essay covers fixed-income instruments, future essays will cover financial derivatives, time series analysis, machine learning in finance, financial visualization and data services.

Augmentation of magnetohydrodynamic nanofluid flow through a permeable stretching sheet employing Machine learning algorithm

An incompressible MHD nanofluid boundary layer flow over a vertical stretching permeable surface employing Buongiorno’s design investigated by considering the convective states. The Brownian motion and thermophoresis effects are used to implement the nanofluid model. Operating the similarity transmutations, to transform the governing partial differential equations into ordinary differential equations consisting of the momentum, energy, and concentration fields and later worked by using a program written together with the stiffness shifting in Wolfram Language. The consequences of various physical parameters on the velocity, temperature, and concentration fields are analyzed, such as magnetic parameter M, Brownian motion parameter Nb, thermophoresis parameter Nt, Lewis number Le, temperature Biot number Biθ, concentration Biot number Biϕ, and suction parameter fw. Furthermore, the Skin friction coefficient, local Nusselt, and local Sherwood numbers concerning magnetic parameter for various values of physical parameters (i.e. fw, Nb) are obtained graphically, then the outcome is validated with other recent works. Finally, introduced a new environment to employ machine learning by performing the sensitivity analysis based on the iterative method for predicting the Skin friction coefficient, reduced Nusselt number, and Sherwood number with respect to magnetic parameter for suction parameter and Brownian motion parameter. Machine learning algorithms provide a strong and quick data processing structure to enhance the actual research procedures and industrial application of fluid mechanics. These techniques have been upgraded and organized for fluid flow characteristics. The present optimization process has the potential for a new perspective on the metallurgical process, heat exchangers in electronics, and some medicinal applications.

FlexibleDecay: An automated calculator of scalar decay widths

We present FlexibleDecay, a tool to calculate decays of scalars in a broad class of BSM models. The tool aims for high precision particularly in the case of Higgs boson decays. In the case of scalar and pseudoscalar Higgs boson decays the known higher order SM QED, QCD and EW effects are taken into account where possible. The program works in a modified MS‾ scheme that exhibits a decoupling property with respect to heavy BSM physics, with BSM parameters themselves treated in the MS‾/DR‾-scheme allowing for an easy connection to high scale tests for, e.g., perturbativity and vacuum stability, and the many observable calculations readily available in MS‾/DR‾ programs. Pure BSM effects are taken into account at the leading order, including all one-loop contributions to loop-induced processes. The program is implemented as an extension to FlexibleSUSY, which determines the mass spectrum for arbitrary BSM models, and does not require any extra configuration from the user. We compare our predictions for Higgs decays in the SM, singlet extended SM, type II THDM, CMSSM and MRSSM, as well as for squark decays in the CMSSM against a selection of publicly available tools. The numerical differences between our and other programs are explained. The release of FlexibleDecay officially deprecates the old effective couplings routines in FlexibleSUSY. Program summaryProgram Title:FlexibleDecayCPC Library link to program files:https://doi.org/10.17632/2c38hcpp5n.1Developer's repository link:https://github.com/FlexibleSUSY/FlexibleSUSYLicensing provisions: GPLv3Programming language: C++, Wolfram Language, Fortran, Bourne shellNature of problem: Calculation of decay widths of scalar bosons in extensions of the Standard Model of particle physics. For the calculation of Higgs boson decay widths known higher-order contributions are taken into account to achieve a high precision.Solution method: The decay widths of the scalar bosons are expressed in terms of Feynman diagrams in a general renormalizable quantum field theory and specialized to the considered model. The resulting expressions are numerically evaluated using special functions and numerical integration.Additional comments including restrictions and unusual features:FlexibleDecay is an addon to FlexibleSUSY [1, 2]. The decay widths can only be calculated in models that can be treated with FlexibleSUSY. Decay widths of fermions or vector bosons are not calculated. For non-Higgs scalar bosons the calculation is restricted to the leading order in the perturbation series.