Introduction to Monte-Carlo Methods for Transport and Diffusion Equations

This paper develops and validates through a series of presentable examples, a comprehensive high-precision, and ultrafast computing concept for solving nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN). The core of this concept is a straightforward scheme that we call "nonlinear adaptive optimization (NAOP),” which is used for a precise template calculation for solving nonlinear ODEs and PDEs through CNN processors. One of the key contributions of this work is to demonstrate the possibility of transforming different types of nonlinearities displayed by various classical and well-known nonlinear equations (e.g., van der Pol-, Rayleigh-, Duffing-, Rössler-, Lorenz-, and Jerk-equations, just to name a few) unto first-order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN templates. Furthermore, in the case of PDE solving, the same concept also allows a mapping unto first-order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDE in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultrafast solver of nonlinear ODEs and/or PDEs. This clearly enables a CNN-based, real-time, ultraprecise, and low-cost computational engineering. As proof of concept, two examples of well-known ODEs are considered namely a second-order linear ODE and a second order nonlinear ODE of the van der Pol type. For each of these ODEs, the corresponding precise CNN templates are derived and are used to deduce the expected solutions. An implementation of the concept developed is possible even on embedded digital platforms (e.g., field programmable gate array (FPGA), digital signal processor (DSP), graphics processing unit (GPU), etc.). This opens a broad range of applications. Ongoing works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting ODEs and PDEs equation models such as Lorenz-, Rössler-, Navier Stokes-, Schrödinger-, Maxwell-, etc.

A topic of interest in financial mathematics is the Black-Scholes model. However, the underlying asset price in the stock market may not be satisfied by this linear model, which was developed under a number of assumptions, including liquidity and the absence of transaction costs. The linear model has restricted its precision in actual market conditions. We study the transaction cost model for modelling illiquid markets from the extended nonlinear Black-Scholes model. Using a semi-discretization finite difference approach, the nonlinear partial differential equation is transformed into a nonlinear ordinary differential equation. Deep Learning (DL) is an advanced technique of machine learning solves the converted ordinary differential equation by fully connected neural network (FCNN). By modelling the complex and nonlinear relationships among market variables, DL models can generate option pricing forecasts that are more dependable and precise, not just for continuous data but also for discontinuous data (at jump point). Introduction: Nonlinear partial differential equation plays a crucial role in financial modelling, especially in the pricing of derivatives like options. The Black- Scholes model, introduced in 1973, continues to be one of the most widely used frameworks for pricing European options. However, this model is a linear one and offers an analytical solution, yet it is not appropriate for the complexities of real market assumptions that exhibit nonlinear effects. We examine the transaction cost model for modelling illiquid markets from the extended nonlinear Black-Scholes model created by Seelama et al. (2021). First using a semi-discretization finite difference approach, the nonlinear partial differential equation is transformed into a nonlinear ordinary differential equation. Solves the converted ordinary differential equation by Deep Learning (DL) based fully connected neural network (FCNN) algorithm. This algorithm is capable of handling the nonlinear behaviour of model and produce more accurate option value for European call. Objectives: Find the solution of more realistic nonlinear model of Black-Scholes equation include transaction cost in illiquid market with deep learning algorithm for a European call option. Methods: From extended nonlinear Black-Scholes model, the nonlinear model of transaction cost in illiquid market is considered for study. The nonlinear partial differential equation is converted into a nonlinear ordinary differential equation by semi-discretization finite difference method. DL is a sophisticated machine learning technique that solves transformed ordinary differential equations using fully connected neural network (FCNN) algorithm. DL algorithm uses a Python program. Results: For European call option, option values are predicted for different number of neurons with different loss functions like MSE and MAE at the time of maturity. Graphical representation shows the accuracy of the algorithm at continuous as well as at jump point (strike price). Conclusions: The method of solving a complex nonlinear partial differential equation by transforming it into a nonlinear ordinary differential equation is valuable. More precise pricing estimates for option values with nonlinear effects in financial data could be improved by deep learning.

Результаты проведённых опросов и различных тестов по россиеведению в Китае показывают, что сегодня студенты, изучающие русский как иностранный (продвинутого уровня), ошибочно воспринимают русскую интеллигенцию как русскую элиту по принципу: элитарность начинается с интеллигентности. Благодаря устоявшимся стереотипам о великом культурном наследии России, студенты даже выпускных курсов не разделяют эти понятияи ассоциативно уравновешивают представителей русской интеллигенции с элитой. В статье дается краткий лингвокультурологический анализ преемственности русской культурной традиции по курсам обучения русскому языку как иностранному. Представителем русской интеллектуальной элиты и безупречным интеллигентом считается Д.С. Лихачев. И сегодня его личность задает очень высокую планку, поскольку ученый полностью посвятил себя служению науке, родине и делу гуманизма в ХХ веке. Формирование современной китайской интеллигенции является сегодня приоритетом в образовательной стратегии страны. Границы этического понятия интеллигентность коррелируются разными учебными курсами на протяжении всего процесса обучения. Темпы образовательной гонки влияют на увеличение числа интеллигенции в Китае (Цань Сюе, Мо Янь, Лао Ше, Лунь Интай и мн. др.), чему посвящается немало современных исследований. Поэтому научная деятельность Д.С. Лихачева привлекает внимание тем, что формирует представление о духовно-нравственной сути русской интеллигентности и духовной мощи интеллигенции, которая черпается из любви к России. Преданность и служение родине усваивается китайской молодежью как основа основ и руководство к действию. Особого внимания заслуживает тот факт, что из числа многих выдающихся деятелей науки и культуры России лишь немногие удостоены того, чтобы их имена проходили красной нитью на всех курсах межпредметного обучения русскому языку. Мы остановились на самых известных представителях русской интеллигенции, без которых померкла бы и русская литература, и русская наука.

It is a pleasure to present the citation for this year's Maurice Ewing medalist, John Ewing.John was born in Texas and completed his early education there. During or shortly after high school, he spent 1 year working on a farm. Working on a farm included tinkering with tractors, automobiles, and various other machines. John has told me that this one year of fooling around with machines gave him better training in instrumentation than he subsequently obtained at more hallowed institutions, such as Harvard, where he obtained a B.S. degree in physics. John subsequently took some graduate courses at Columbia. At Columbia a course in mathematics for physics students (which was a disguised course in applied quantum mechanics) had to compete for John's attention with adventures at sea with Maurice Ewing. Adventures at sea won out. Joining John in his march out of graduate school were a number of others who went on to become illustrious scientists.

Purpose: While hearing care professions recently submitted commentary about proposed regulatory guidelines for over-the-counter (OTC) hearing aids, educational considerations were not highlighted much. This article examines audiology education regarding OTC devices, specifically addressing how graduate curricula may incorporate the topic and examining graduate audiology students' perspectives on proposed OTC regulations. Graduate audiology curricula include a wide breadth of core classes and several specialized courses. The topic of OTC devices may fit well within graduate coursework focused on Professional Issues, Public Health, or Practice Management. This article presents the curricular option implemented at a mid-Atlantic university, including a brief course description, module overview for the OTC topic, and two related assignments. Graduate audiology training with a public health focus will generally address the core functions, assessment, policy, and assurance, which facilitates a topic such as OTC devices. This educational framework also optimized the examination of graduate students' views by qualitative analysis of assignment responses ( n = 21) from audiology students. Thematic analysis of data revealed four themes with respective subthemes and categories. Major themes are examined in a discussion highlighting the perspectives of audiology graduate students. Conclusions: Regular teaching course audits facilitate curricular changes and may incorporate professional topics that are new, controversial, and/or in the development phase to optimize audiology curricular transformation. Analyzing student perspectives on current topics, such as OTC devices, provided an avenue for immersing public health audiology concepts into graduate training with an overall goal to improve equity and inclusion in clinical practice. The topic of OTC devices was appropriate for educating future audiologists.

A simple numerical approach for assessing coupled transport processes in partitioning systems

An in-depth study of the concept of a consequence relation, culminating in the concept of a Lindenbaum-Tarski algebra, intended for advanced undergraduate and graduate students in mathematics and graduate students in philosophy, as well as researchers in the field of mathematical and philosophical logic. The book is aimed to fill the niche left after the publication of Rasiowa and Sikorski’s The Mathematics of Metamathematics (1970), Rasiowa’s An Algebraic Approach to Non-Classical Logics (1974), and Wójcicki’s Theory of Logical Calculi (1988). The problem of obtaining confirmation that a statement is a consequence of a set of statements as prerequisites, on the one hand, and the problem of demonstrating that such confirmation does not exist in the structure under consideration, on the other hand, are considered in a balanced way, using predominantly algebraic methods, as one two-sided problem. Namely, for the second part of this problem, the concept of the Lindenbaum-Tarski algebra plays a key role, which becomes even more important when the considered consequence relation is placed in the context of decidability. This role is traced in the book for various formal objective languages. The included exercises are an essential component of the reader’s assimilation of the book’s material.

This textbook provides a broad introduction to continuous and discrete dynamical systems. With its hands-on approach, the text leads the reader from basic theory to recently published research material in nonlinear ordinary differential equations, nonlinear optics, multifractals, neural networks, and binary oscillator computing. Dynamical Systems with Applications Using Python takes advantage of Python’s extensive visualization, simulation, and algorithmic tools to study those topics in nonlinear dynamical systems through numerical algorithms and generated diagrams. After a tutorial introduction to Python, the first part of the book deals with continuous systems using differential equations, including both ordinary and delay differential equations. The second part of the book deals with discrete dynamical systems and progresses to the study of both continuous and discrete systems in contexts like chaos control and synchronization, neural networks, and binary oscillator computing. These later sections are useful reference material for undergraduate student projects. The book is rounded off with example coursework to challenge students’ programming abilities and Python-based exam questions. This book will appeal to advanced undergraduate and graduate students, applied mathematicians, engineers, and researchers in a range of disciplines, such as biology, chemistry, computing, economics, and physics. Since it provides a survey of dynamical systems, a familiarity with linear algebra, real and complex analysis, calculus, and ordinary differential equations is necessary, and knowledge of a programming language like C or Java is beneficial but not essential.

Materials that simultaneously possess ultralow thermal conductivity, high stiffness, and damage tolerance are highly desirable for engineering applications. However, this combination of properties has never been demonstrated in a single material because thermal and mechanical properties are coupled in most fully dense and porous solids. A new class of lattice materials with nanoscale features, called nanolattices, can fill this void in the material property space by virtue of their architecture and nanoscale dimensions. Extensive work on nanolattice mechanical properties report their excellent stiffness-to-density ratio and recoverability from large compressive strains. In contrast, the framework for studying their thermal properties has not been established. Our work develops the computational and experimental tools necessary to study heat conduction in nanoarchitected materials and applies those tools to prove the viability of octet-truss nanolattices as multifunctional thermal insulators. We implement significant improvements to a phonon Monte Carlo method to solve the Boltzmann transport equation (BTE) in highly complex geometries like the octet-truss. No prior works solve the BTE in a domain as intricate as a nanolattice, so we create a geometry representation scheme that can model any arbitrary 3-D body. Our enhanced variance-reduced Monte Carlo code incorporates this scheme, allowing us to predict the thermal conductivity of nanolattices and analyze the phonon transport behavior in them. Results suggest that hollow-beam silicon nanolattices indeed reach ultralow thermal conductivities. Based on Monte Carlo and finite element simulations, we develop a predictive thermal conductivity model that accounts for both diffusive and radiative phonon transport in nanolattices. We also devise custom modifications to the 3ω method to experimentally measure the thermal conductivity of additively manufactured nanolattices. Since the serial fabrication process of nanolattices makes it costly to cover large areas, we design a specialized 3ω sample that minimizes the required structure size while maintaining good experimental sensitivity. We derive a new thermal model to account for conductive losses through the heater line in our novel sample geometry. 3ω measurements and compression tests of hollow-beam alumina nanolattices show that they combine ultralow thermal conductivity with excellent mechanical stiffness and resilience, which proves that nanolattices occupy a previously unreachable region in material property space. Our work provides motivation to further investigate and improve the thermal properties of architected materials.

We have studied quantum-mechanical transport equations for nondegenerate electrons in semiconductors in high electric fields. Our calculations use Kadanoff and Baym's formalism based on Green functions, treat only fields that are constant in space and time, and are restricted to weak scattering. First, we derive an approximate solution to the equation of motion of the retarded Green function in an electric field, with a careful check of its validity. This is used in deducing the conditions under which the quantum-mechanical transport equation reduces to the Boltzmann equation. We find, in agreement with some previous studies, that the electric field causes a ``broadening'' of the \ensuremath{\delta} function in semiclassical transition rates, a result of the ``intracollisional field effect.'' The Boltzmann equation fails when this broadening exceeds some characteristic energy scale (usually ${k}_{B}$T), which occurs at fields of a few MV ${m}^{\mathrm{\ensuremath{-}}1}$ in conventional semiconductors. These results are strongly dependent on the ansatz used to reduce the Green function to a distribution function. The scattering-out term is usually much less sensitive to the electric field than the scattering-in term. We exploit this to construct an integral transport equation, valid in high electric fields, which differs from the Boltzmann equation only in having a broadened function replacing the \ensuremath{\delta} function in the scattering-in rates. It should be possible to solve this equation using standard numerical techniques and gain quantitative information on the intracollisional field effect.

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Lattice vibrations are the microscopic mechanism responsible for a large, if not dominant, contribution to heat transport in crystalline insulators. These vibrations are described in terms of phonons, collective excitations (or quasiparticles) in the form of waves of atomic displacements inside a crystal. Phonons are traditionally considered to be the quasiparticles responsible for carrying heat through the material. Heat transport is considered as a flux of a phonon gas, diffusing from hot areas (high phonon densities) to cold areas (low phonon densities) in an attempt to reestablish equilibrium, with phonon collisions being the source of heat flux dissipation. However, as dimensionality is reduced, the motion of phonons stimulated by temperature perturbations becomes correlated and this gas-like picture of thermal transport in terms of phonons becomes invalid. In this Thesis, we lay out an interpretation of thermal transport in 2D materials based on the Boltzmann transport equation in the form of collective excitations of phonons. These collective phonon excitations give raise to complex phenomena, such as high thermal conductivities, that are otherwise unexplained. As another example, collective excitations, at variance with conventional diffusive transport, can induce wave-like heat propagation, or second sound. This had been found only in a few exotic materials at cryogenic temperatures, but is present instead routinely in 2D materials at room temperature. The correlated-phonon description of heat transfer can be rationalised by introducing a new collective excitation, called 'relaxon', which is defined as the eigenstate of the collision operator. Whereas only oversimplifying assumptions endow phonons with well defined relaxation times (the average interval of time between collisions), relaxons have always well defined relaxation times and permit an exact description of thermal transport. The complex dynamic of heat transport in 2D is thus greatly simplified and a kinetic gas theory of thermal transport still applies, provided that the gas is not constituted by phonons, but by relaxons. Our work on thermal transport is part of a larger effort, aiming at the creation of a database of numerically computed properties of materials. The high-throughput production of simulated properties is a challenging task, since it necessitates the understanding of a physical model, but it also needs to face a myriad of technicalities and problems that hinder the execution of a large number of calculations. In order to allow the creation of computational materials databases, we developed AiiDA, an open-source automated interactive infrastructure and database for computational science. This platform tackles the problems of creation, management, analysis and sharing of data and simulations, summarized in the pillars of Automation, Data, Environment and Sharing. Automation is achieved by management of remote computational resources and the encoding of workflows, that allows the execution of complex sequences of calculations. The tight coupling between automation and data storage, handled by the platform, enables full reproducibility of the results and a suitable database design allows for efficient data analysis tools. Sharing of scientific knowledge is addressed by providing tools for distribution of data and of the underlying workflows that generated them, creating an ecosystem for computational materials science.

We present BTE-Barna (Boltzmann Transport Equation - Beyond the RTA for NAnosystems), a software package that extends the Monte Carlo (MC) module of the almaBTE solver of the Peierls-Boltzmann transport equation for phonons (PBTE) to work with nanosystems based on 2D materials with complex geometries. To properly capture how the phonon occupations evolve in momentum space as a result of scattering, we have supplemented the relaxation-time approximation with an implementation of the propagator for the full linearized version of the PBTE. The code can now find solutions for finite and extended devices under the effect of a thermal gradient, with isothermal reservoirs or with an arbitrary initial temperature distribution in space and time, writing out the temperature and heat flux distributions as well as their spectral decompositions. Besides the full deviational MC solver, a number of useful approximations for highly symmetric devices are also included. Program summaryProgram Title::BTE-BarnaCPC Library link to program files:https://doi.org/10.17632/t58z289tpc.1Developer's repository link:https://github.com/sousaw/BTE-BarnaCode Ocean capsule:https://codeocean.com/capsule/4171806Licensing provisions: Apache-2.0Programming language: C++Nature of problem: Calculation of temperature profiles, thermal flux and/or effective thermal conductivities for nanosystems based on 2D materials.Solution method: For highly symmetric systems confined along some direction(s), the linearized phonon Boltzmann transport equation is solved iteratively by partially suppressing the phonon lifetimes due to boundaries. For more complex geometries, an energy-based deviational Monte Carlo method including off-diagonal terms in the linearized scattering operator arising from the phonon-phonon interaction is used.Additional comments including restrictions and unusual features: Depends on the following external libraries: boost, Eigen, HDF5, MessagePack, MPI, oneTBB, spglib, and almaBTE.

This bibliography is offered as a practical guide to published papers, conference proceedings papers and theses/dissertations on the finite element (FE) and boundary element (BE) applications in different fields of biomechanics between 1976 and 1991. The aim of this paper is to help the users of FE and BE techniques to get better value from a large collection of papers on the subjects. Categories in biomechanics included in this survey are: orthopaedic mechanics, dental mechanics, cardiovascular mechanics, soft tissue mechanics, biological flow, impact injury, and other fields of applications. More than 900 references are listed.

Computer applications in medicine: By Edward E. Mason, M.D., Ph.D., Professor, Department of Surgery, Ohio State University College of Medicine; and William G. Bulgren, M.S., Graduate Student in Mathematics, Ohio State University. Springfield, Ill., 1963, Charles C Thomas, Publisher, 171 pages. Price $6.75