962 research outputs found
Big Data Analysis and Programming for Engineers
This text serves to cover critical programming, data analysis, statistical analysis, and mathematical skills for engineers. In particular fundamental programming skills are demonstrated using Mathematica specifically the importing of data sets, loop structures, plotting and statistically analyzing data, image analysis, and machine learning. Critical engineering topics such as solid mechanics, vibrations, and engineering problems which require solving ODEs and PDEs are covered.https://scholarlycommons.pacific.edu/open-textbooks/1026/thumbnail.jp
Symbolic computation of solitary wave solutions and solitons through homogenization of degree
A simplified version of Hirota's method for the computation of solitary waves
and solitons of nonlinear PDEs is presented. A change of dependent variable
transforms the PDE into an equation that is homogeneous of degree. Solitons are
then computed using a perturbation-like scheme involving linear and nonlinear
operators in a finite number of steps.
The method is applied to a class of fifth-order KdV equations due to Lax,
Sawada-Kotera, and Kaup-Kupershmidt. The method works for non-quadratic
homogeneous equations for which the bilinear form might not be known.
Furthermore, homogenization of degree allows one to compute solitary wave
solutions of nonlinear PDEs that do not have solitons. Examples include the
Fisher and FitzHugh-Nagumo equations, and a combined KdV-Burgers equation. When
applied to a wave equation with a cubic source term, one gets a bi-soliton
solution describing the coalescence of two wavefronts. The method is largely
algorithmic and is implemented in Mathematica.Comment: Proceedings Conference on Nonlinear and Modern Mathematical Physics
(NMMP-2022) Springer Proceedings in Mathematics and Statistics, 60pp,
Springer-Verlag, New York, 202
Making Presentation Math Computable
This Open-Access-book addresses the issue of translating mathematical expressions from LaTeX to the syntax of Computer Algebra Systems (CAS). Over the past decades, especially in the domain of Sciences, Technology, Engineering, and Mathematics (STEM), LaTeX has become the de-facto standard to typeset mathematical formulae in publications. Since scientists are generally required to publish their work, LaTeX has become an integral part of today's publishing workflow. On the other hand, modern research increasingly relies on CAS to simplify, manipulate, compute, and visualize mathematics. However, existing LaTeX import functions in CAS are limited to simple arithmetic expressions and are, therefore, insufficient for most use cases. Consequently, the workflow of experimenting and publishing in the Sciences often includes time-consuming and error-prone manual conversions between presentational LaTeX and computational CAS formats. To address the lack of a reliable and comprehensive translation tool between LaTeX and CAS, this thesis makes the following three contributions. First, it provides an approach to semantically enhance LaTeX expressions with sufficient semantic information for translations into CAS syntaxes. Second, it demonstrates the first context-aware LaTeX to CAS translation framework LaCASt. Third, the thesis provides a novel approach to evaluate the performance for LaTeX to CAS translations on large-scaled datasets with an automatic verification of equations in digital mathematical libraries. This is an open access book
Routines and Applications of Symbolic Algebra Software
Computing has become an essential resource in modern research and has found application
across a wide range of scientific disciplines. Developments in symbolic algebra tools have been
particularly valuable in physics where calculations in fields such as general relativity, quantum
field theory and physics beyond the standard model are becoming increasing complex and
unpractical to work with by hand. The computer algebra system Cadabra is a tensor-first
approach to symbolic algebra based on the programming language Python which has been used
extensively in research in these fields while also having a shallow learning curve making it an
excellent way to introduce students to methods in computer algebra.
The work in this thesis has been concentrated on developing Cadabra, which has involved
looking at two different elements which make up a computer algebra program. Firstly, the
implementation of algebraic routines is discussed. This has primarily been focused on the
introduction of an algorithm for detecting the equivalence of tensorial expressions related by
index permutation symmetries. The method employed differs considerably from traditional
canonicalisation routines which are commonly used for this purpose by using Young projection
operators to make such symmetries manifest.
The other element of writing a computer algebra program which is covered is the infrastruc-
ture and environment. The importance of this aspect of software design is often overlooked by
funding committees and academic software users resulting in an anti-pattern of code not being
shared and contributed to in the way in which research itself is published and promulgated.
The focus in this area has been on implementing a packaging system for Cadabra which allows
the writing of generic libraries which can be shared by the community, and interfacing with
other scientific computing packages to increase the capabilities of Cadabra
Π’Π΅Ρ Π½ΠΎΠ»ΠΎΠ³ΠΈΡ ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ½ΠΎΠΉ ΠΏΠΎΠ΄Π΄Π΅ΡΠΆΠΊΠΈ ΠΆΠΈΠ·Π½Π΅Π½Π½ΠΎΠ³ΠΎ ΡΠΈΠΊΠ»Π° ΡΠ΅ΠΌΠ°Π½ΡΠΈΡΠ΅ΡΠΊΠΈ ΡΠΎΠ²ΠΌΠ΅ΡΡΠΈΠΌΡΡ ΠΈΠ½ΡΠ΅Π»Π»Π΅ΠΊΡΡΠ°Π»ΡΠ½ΡΡ ΠΊΠΎΠΌΠΏΡΡΡΠ΅ΡΠ½ΡΡ ΡΠΈΡΡΠ΅ΠΌ Π½ΠΎΠ²ΠΎΠ³ΠΎ ΠΏΠΎΠΊΠΎΠ»Π΅Π½ΠΈΡ
Π ΠΈΠ·Π΄Π°Π½ΠΈΠΈ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΎ ΠΎΠΏΠΈΡΠ°Π½ΠΈΠ΅ ΡΠ΅ΠΊΡΡΠ΅ΠΉ Π²Π΅ΡΡΠΈΠΈ ΠΎΡΠΊΡΡΡΠΎΠΉ ΡΠ΅Ρ
Π½ΠΎΠ»ΠΎΠ³ΠΈΠΈ ΠΎΠ½ΡΠΎΠ»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΠΏΡΠΎΠ΅ΠΊΡΠΈΡΠΎΠ²Π°Π½ΠΈΡ, ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ΄ΡΡΠ²Π° ΠΈ ΡΠΊΡΠΏΠ»ΡΠ°ΡΠ°ΡΠΈΠΈ ΡΠ΅ΠΌΠ°Π½ΡΠΈΡΠ΅ΡΠΊΠΈ ΡΠΎΠ²ΠΌΠ΅ΡΡΠΈΠΌΡΡ
Π³ΠΈΠ±ΡΠΈΠ΄Π½ΡΡ
ΠΈΠ½ΡΠ΅Π»Π»Π΅ΠΊΡΡΠ°Π»ΡΠ½ΡΡ
ΠΊΠΎΠΌΠΏΡΡΡΠ΅ΡΠ½ΡΡ
ΡΠΈΡΡΠ΅ΠΌ (Π’Π΅Ρ
Π½ΠΎΠ»ΠΎΠ³ΠΈΠΈ OSTIS). ΠΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½Π° ΡΡΠ°Π½Π΄Π°ΡΡΠΈΠ·Π°ΡΠΈΡ ΠΈΠ½ΡΠ΅Π»Π»Π΅ΠΊΡΡΠ°Π»ΡΠ½ΡΡ
ΠΊΠΎΠΌΠΏΡΡΡΠ΅ΡΠ½ΡΡ
ΡΠΈΡΡΠ΅ΠΌ, Π° ΡΠ°ΠΊΠΆΠ΅ ΡΡΠ°Π½Π΄Π°ΡΡΠΈΠ·Π°ΡΠΈΡ ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ² ΠΈ
ΡΡΠ΅Π΄ΡΡΠ² ΠΈΡ
ΠΏΡΠΎΠ΅ΠΊΡΠΈΡΠΎΠ²Π°Π½ΠΈΡ, ΡΡΠΎ ΡΠ²Π»ΡΠ΅ΡΡΡ Π²Π°ΠΆΠ½Π΅ΠΉΡΠΈΠΌ ΡΠ°ΠΊΡΠΎΡΠΎΠΌ, ΠΎΠ±Π΅ΡΠΏΠ΅ΡΠΈΠ²Π°ΡΡΠΈΠΌ ΡΠ΅ΠΌΠ°Π½ΡΠΈΡΠ΅ΡΠΊΡΡ ΡΠΎΠ²ΠΌΠ΅ΡΡΠΈΠΌΠΎΡΡΡ ΠΈΠ½ΡΠ΅Π»Π»Π΅ΠΊΡΡΠ°Π»ΡΠ½ΡΡ
ΠΊΠΎΠΌΠΏΡΡΡΠ΅ΡΠ½ΡΡ
ΡΠΈΡΡΠ΅ΠΌ ΠΈ ΠΈΡ
ΠΊΠΎΠΌΠΏΠΎΠ½Π΅Π½ΡΠΎΠ², ΡΡΠΎ
ΡΡΡΠ΅ΡΡΠ²Π΅Π½Π½ΠΎΠ΅ ΡΠ½ΠΈΠΆΠ΅Π½ΠΈΠ΅ ΡΡΡΠ΄ΠΎΠ΅ΠΌΠΊΠΎΡΡΠΈ ΡΠ°Π·ΡΠ°Π±ΠΎΡΠΊΠΈ ΡΠ°ΠΊΠΈΡ
ΡΠΈΡΡΠ΅ΠΌ.
ΠΠ½ΠΈΠ³Π° ΠΏΡΠ΅Π΄Π½Π°Π·Π½Π°ΡΠ΅Π½Π° Π²ΡΠ΅ΠΌ, ΠΊΡΠΎ ΠΈΠ½ΡΠ΅ΡΠ΅ΡΡΠ΅ΡΡΡ ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ°ΠΌΠΈ ΠΈΡΠΊΡΡΡΡΠ²Π΅Π½Π½ΠΎΠ³ΠΎ ΠΈΠ½ΡΠ΅Π»Π»Π΅ΠΊΡΠ°, Π° ΡΠ°ΠΊΠΆΠ΅ ΡΠΏΠ΅ΡΠΈΠ°Π»ΠΈΡΡΠ°ΠΌ Π² ΠΎΠ±Π»Π°ΡΡΠΈ ΠΈΠ½ΡΠ΅Π»Π»Π΅ΠΊΡΡΠ°Π»ΡΠ½ΡΡ
ΠΊΠΎΠΌΠΏΡΡΡΠ΅ΡΠ½ΡΡ
ΡΠΈΡΡΠ΅ΠΌ ΠΈ ΠΈΠ½ΠΆΠ΅Π½Π΅ΡΠΈΠΈ Π·Π½Π°Π½ΠΈΠΉ. ΠΠΎΠΆΠ΅Ρ Π±ΡΡΡ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½Π° ΡΡΡΠ΄Π΅Π½ΡΠ°ΠΌΠΈ, ΠΌΠ°Π³ΠΈΡΡΡΠ°Π½ΡΠ°ΠΌΠΈ ΠΈ Π°ΡΠΏΠΈΡΠ°Π½ΡΠ°ΠΌΠΈ ΡΠΏΠ΅ΡΠΈΠ°Π»ΡΠ½ΠΎΡΡΠΈ Β«ΠΡΠΊΡΡΡΡΠ²Π΅Π½Π½ΡΠΉ ΠΈΠ½ΡΠ΅Π»Π»Π΅ΠΊΡΒ».
Π’Π°Π±Π». 8. ΠΠ». 223. ΠΠΈΠ±Π»ΠΈΠΎΠ³Ρ.: 665 Π½Π°Π·Π²
ΠΠΎΠ΄Π΅Π»ΡΠ²Π°Π½Π½Ρ ΠΊΠΎΠΌΠΏ'ΡΡΠ΅ΡΠ½ΠΎ-ΡΠ½ΡΠ΅Π³ΡΠΎΠ²Π°Π½ΠΈΡ ΡΠΈΠ»ΠΎΠ²ΠΈΡ Π΅Π½Π΅ΡΠ³Π΅ΡΠΈΡΠ½ΠΈΡ
Π ΠΎΠ·Π³Π»ΡΠ½ΡΡΠΎ Π°Π΄Π°ΠΏΡΠ°ΡΡΠΉΠ½Ρ ΠΌΠ΅ΡΠΎΠ΄ΠΈ ΡΠ° Π·Π°ΡΠΎΠ±ΠΈ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ½ΠΎΠ³ΠΎ ΠΌΠΎΠ΄Π΅Π»ΡΠ²Π°Π½Π½Ρ ΠΏΡΠΎΡΠ΅ΡΡΠ² ΡΡΠ½ΠΊΡΠΎΠ½ΡΠ²Π°Π½Π½Ρ ΠΊΠΎΠΌΠΏ'ΡΡΠ΅ΡΠ½ΠΎ-ΡΠ½ΡΠ΅Π³ΡΠΎΠ²Π°Π½ΠΈΡ
ΡΠΈΡΡΠ΅ΠΌ (ΡΡΠΎΡΠΎΠ²Π½ΠΎ Π΄ΠΎ ΡΠΈΠ»ΠΎΠ²ΠΈΡ
Π΅Π½Π΅ΡΠ³Π΅ΡΠΈΡΠ½ΠΈΡ
ΡΡΡΠ°Π½ΠΎΠ²ΠΎΠΊ). ΠΠ°Π²ΠΎΠ΄ΡΡΡΡΡ ΠΎΠ±ΡΠΈΡΠ»ΡΠ²Π°Π»ΡΠ½Ρ Π°Π»Π³ΠΎΡΠΈΡΠΌΠΈ ΡΠ° ΠΏΡΠΈΠΊΠ»Π°Π΄ΠΈ ΡΠΎΠ·'ΡΠ·ΡΠ²Π°Π½Π½Ρ ΠΏΡΠΈΠΊΠ»Π°Π΄Π½ΠΈΡ
Π·Π°Π΄Π°Ρ. ΠΠ»Ρ Π½Π°ΡΠΊΠΎΠ²ΡΡΠ², Π½Π°ΡΠΊΠΎΠ²ΠΎ-ΠΏΠ΅Π΄Π°Π³ΠΎΠ³ΡΡΠ½ΠΈΡ
ΠΏΡΠ°ΡΡΠ²Π½ΠΈΠΊΡΠ² ΡΠ° ΡΠ½ΠΆΠ΅Π½Π΅ΡΡΠ², ΡΠΊΡ Π·Π°ΠΉΠΌΠ°ΡΡΡΡΡ ΠΌΠΎΠ΄Π΅Π»ΡΠ²Π°Π½Π½ΡΠΌ Π΅Π½Π΅ΡΠ³Π΅ΡΠΈΡΠ½ΠΈΡ
ΡΡΡΡΠ΅ΠΌ, Π° ΡΠ°ΠΊΠΎΠΆ Π°ΡΠΏΡΡΠ°Π½ΡΡΠ² ΡΠ° ΡΡΡΠ΄Π΅Π½ΡΡΠ² Π²ΡΠ΄ΠΏΠΎΠ²ΡΠ΄Π½ΠΈΡ
ΡΠ΅Ρ
Π½ΡΡΠ½ΠΈΡ
ΡΠΏΠ΅ΡΡΠ°Π»ΡΠ½ΠΎΡΡΠ΅ΠΉ
Finite element modeling of lead acid batteries
This thesis investigates the finite element method with regard to the macrohomogeneous theory for flooded porous electrochemical cells, more specifically lead-acid cells. One- and two-dimensional finite element models are developed for flooded porous electrochemical lead acid cells. Chapter One introduces the background of the technology of lead-acid batteries, theory fundamentals, previous mathematical models for lead acid batteries, and the reason for the work. Chapter Two develops Newmanβs macrohomogeneous equations for flooded porous electrodes. Chapter Three details the finite element theory, and how it is used to solve time dependent coupled non-linear partial differential equations. Chapter Four applies finite element theory to one-dimensional macrohomogeneous equations that describe lead-acid batteries. The results of the model are compared to previously published papers utilising finite difference methods. In Chapter Five, the technique is extend to two-dimensions and is validated with previously published papers of models on lead-acid batteries
Review on Fixed and Floating Offshore Structures. Part II: Sustainable Design Approaches and Project Management
Offshore structures exist in a variety of forms, and they are used for a variety of functions in varied sea depths. These structures are tailored for certain environments and sea depths. Different actions for suitable equipment selection, platform type design, and drilling/production processes are required for the applications of these offshore structures, as given in Part I. This paper is the second part, which outlines various processes, loads, design approaches and project management of offshore platforms. To achieve these, proper planning must be conducted for lifting, transportation, installation, design, fabrication, and commissioning of these offshore platforms. Some historical developments of some offshore structures are presented, and some project planning routines are undertaken in this research. The ultimate goal is to provide a general overview of the many processes of offshore platform design, construction, loadout, transportation, and installation. Some discussions on the design parameters such as water depth and environmental conditions were presented. It also lists various software programs used in engineering designs covering software programs for structural analysis, 3D rendering, computer-aided design (CAD), hydrodynamic design, oceanic flow analysis, offshore structures analysis, mathematical modelling, coding/algorithm development software, and programming software to aid analytical calculations. The review also includes information on cutting-edge offshore platforms and industry advancements. Ultimately, for long-term operations, various types of offshore platforms for specific seawater depths are available
Methods of symmetry reduction and their application
In this thesis methods of symmetry reduction are applied to several physically relevant partial differential equations.
The first chapter serves to acquaint the reader with the symmetry methods used in this thesis. In particular the classical method of Lie, an extension of it by Bluman and Cole [1969], known as the nonclassical method, and the direct method of Clarkson and Kruskal [1989] are described. Other known extensions of these methods are outlined, including potential symmetries, introduced by Bluman, Kumei and Reid [1988]. Also described are the tools used in practice to perform the calculations. The remainder of the thesis is split into two parts.
In Part One the classical and nonclassical methods are applied to three classes of scalar equation: a generalised Boussinesq equation, a class of third order equations and a class of fourth order equations. Many symmetry reductions and exact solutions are found.
In Part Two each of the classical, nonclassical and direct methods are applied to various systems of partial differential equations. These include shallow water wave systems, six representations of the Boussinesq equation and a reaction-diffusion equation written as a system. In Chapters Five and Six both the actual application of these methods and their results is compared and contrasted. In such applications, remarkable phenomena can occur, in both the nonclassical and direct methods. In particular it is shown that the application of the direct method to systems of equations is not as conceptually straightforward as previously thought, and a way of completing the calculations of the nonclassical method via hodograph transformations is introduced. In Chapter Seven it is shown how more symmetry reductions may be found via nonclassical potential symmetries, which are a new extension on the idea of potential symmetries.
In the final chapter the relationship between the nonclassical and direct methods is investigated in the light of the previous chapters. The thesis is concluded with some general remarks on its findings and on possible future work
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