Chemical Kinetics: Introduction with Mathcad/Maple/MCS
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Sagemath and Sagemathcloud
Viviane Pons Ma^ıtrede conf´erence,Universit´eParis-Sud Orsay [email protected] { @PyViv SageMath and SageMathCloud Introduction SageMath SageMath is a free open source mathematics software I Created in 2005 by William Stein. I http://www.sagemath.org/ I Mission: Creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab. Viviane Pons (U-PSud) SageMath and SageMathCloud October 19, 2016 2 / 7 SageMath Source and language I the main language of Sage is python (but there are many other source languages: cython, C, C++, fortran) I the source is distributed under the GPL licence. Viviane Pons (U-PSud) SageMath and SageMathCloud October 19, 2016 3 / 7 SageMath Sage and libraries One of the original purpose of Sage was to put together the many existent open source mathematics software programs: Atlas, GAP, GMP, Linbox, Maxima, MPFR, PARI/GP, NetworkX, NTL, Numpy/Scipy, Singular, Symmetrica,... Sage is all-inclusive: it installs all those libraries and gives you a common python-based interface to work on them. On top of it is the python / cython Sage library it-self. Viviane Pons (U-PSud) SageMath and SageMathCloud October 19, 2016 4 / 7 SageMath Sage and libraries I You can use a library explicitly: sage: n = gap(20062006) sage: type(n) <c l a s s 'sage. interfaces .gap.GapElement'> sage: n.Factors() [ 2, 17, 59, 73, 137 ] I But also, many of Sage computation are done through those libraries without necessarily telling you: sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]) sage : G . g a p () Group( [ (3,4), (1,2,3)(4,5) ] ) Viviane Pons (U-PSud) SageMath and SageMathCloud October 19, 2016 5 / 7 SageMath Development model Development model I Sage is developed by researchers for researchers: the original philosophy is to develop what you need for your research and share it with the community. -
Ordinary Differential Equations (ODE) Using Euler's Technique And
Mathematical Models and Methods in Modern Science Ordinary Differential Equations (ODE) using Euler’s Technique and SCILAB Programming ZULZAMRI SALLEH Applied Sciences and Advance Technology Universiti Kuala Lumpur Malaysian Institute of Marine Engineering Technology Bandar Teknologi Maritim Jalan Pantai Remis 32200 Lumut, Perak MALAYSIA [email protected] http://www.mimet.edu.my Abstract: - Mathematics is very important for the engineering and scientist but to make understand the mathematics is very difficult if without proper tools and suitable measurement. A numerical method is one of the algorithms which involved with computer programming. In this paper, Scilab is used to carter the problems related the mathematical models such as Matrices, operation with ODE’s and solving the Integration. It remains true that solutions of the vast majority of first order initial value problems cannot be found by analytical means. Therefore, it is important to be able to approach the problem in other ways. Today there are numerous methods that produce numerical approximations to solutions of differential equations. Here, we introduce the oldest and simplest such method, originated by Euler about 1768. It is called the tangent line method or the Euler method. It uses a fixed step size h and generates the approximate solution. The purpose of this paper is to show the details of implementing of Euler’s method and made comparison between modify Euler’s and exact value by integration solution, as well as solve the ODE’s use built-in functions available in Scilab programming. Key-Words: - Numerical Methods, Scilab programming, Euler methods, Ordinary Differential Equation. 1 Introduction availability of digital computers has led to a Numerical methods are techniques by which veritable explosion in the use and development of mathematical problems are formulated so that they numerical methods [1]. -
WEEK 10: FAREWELL to PYTHON... WELCOME to MAPLE! 1. a Review of PYTHON's Features During the Past Few Weeks, We Have Explored
WEEK 10: FAREWELL TO PYTHON... WELCOME TO MAPLE! 1. A review of PYTHON’s features During the past few weeks, we have explored the following aspects of PYTHON: • Built-in types: integer, long integer, float, complex, boolean, list, se- quence, string etc. • Operations on built-in types: +, −, ∗, abs, len, append, etc. • Loops: N = 1000 i=1 q = [] while i <= N q.append(i*i) i += 1 • Conditional statements: if x < y: min = x else: min = y • Functions: def fac(n): if n==0 then: return 1 else: return n*fac(n-1) • Modules: math, turtle, etc. • Classes: Rational, Vector, Polynomial, Polygon, etc. With these features, PYTHON is a powerful tool for doing mathematics on a computer. In particular, the use of modules makes it straightforward to add greater functionality, e.g. GL (3D graphics), NumPy (numerical algorithms). 2. What about MAPLE? MAPLE is really designed to be an interactive tool. Nevertheless, it can also be used as a programming language. It has some of the basic facilities available in PYTHON: Loops; conditional statements; functions (in the form of “procedures”); good graphics and some very useful additional modules called “packages”: • Built-in types: long integer, float (arbitrary precision), complex, rational, boolean, array, sequence, string etc. • Operations on built-in types: +, −, ∗, mathematical functions, etc. • Loops: 1 2 WEEK 10: FAREWELL TO PYTHON... WELCOME TO MAPLE! Table 1. Some of the most useful MAPLE packages Name Description Example DEtools Tools for differential equations exactsol(ode,y) linalg Linear Algebra gausselim(A) plots Graphics package polygonplot(p,axes=none) stats Statistics package random[uniform](10) N := 1000 : q := array(1..N) : for i from 1 to N do q[i] := i*i : od : • Conditional statements: if x < y then min := x else min := y fi: • Functions: fac := proc(n) if n=0 then return 1 else return n*fac(n-1) fi : end proc : • Packages: See Table 1. -
Practical Estimation of High Dimensional Stochastic Differential Mixed-Effects Models
Practical Estimation of High Dimensional Stochastic Differential Mixed-Effects Models Umberto Picchinia,b,∗, Susanne Ditlevsenb aDepartment of Mathematical Sciences, University of Durham, South Road, DH1 3LE Durham, England. Phone: +44 (0)191 334 4164; Fax: +44 (0)191 334 3051 bDepartment of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark Abstract Stochastic differential equations (SDEs) are established tools to model physical phenomena whose dynamics are affected by random noise. By estimating parameters of an SDE intrin- sic randomness of a system around its drift can be identified and separated from the drift itself. When it is of interest to model dynamics within a given population, i.e. to model simultaneously the performance of several experiments or subjects, mixed-effects mod- elling allows for the distinction of between and within experiment variability. A framework to model dynamics within a population using SDEs is proposed, representing simultane- ously several sources of variation: variability between experiments using a mixed-effects approach and stochasticity in the individual dynamics using SDEs. These stochastic differ- ential mixed-effects models have applications in e.g. pharmacokinetics/pharmacodynamics and biomedical modelling. A parameter estimation method is proposed and computational guidelines for an efficient implementation are given. Finally the method is evaluated using simulations from standard models like the two-dimensional Ornstein-Uhlenbeck (OU) and the square root models. Keywords: automatic differentiation, closed-form transition density expansion, maximum likelihood estimation, population estimation, stochastic differential equation, Cox-Ingersoll-Ross process arXiv:1004.3871v2 [stat.CO] 3 Oct 2010 1. INTRODUCTION Models defined through stochastic differential equations (SDEs) allow for the represen- tation of random variability in dynamical systems. -
Numerical Methods for Ordinary Differential Equations
CHAPTER 1 Numerical Methods for Ordinary Differential Equations In this chapter we discuss numerical method for ODE . We will discuss the two basic methods, Euler’s Method and Runge-Kutta Method. 1. Numerical Algorithm and Programming in Mathcad 1.1. Numerical Algorithm. If you look at dictionary, you will the following definition for algorithm, 1. a set of rules for solving a problem in a finite number of steps; 2. a sequence of steps designed for programming a computer to solve a specific problem. A numerical algorithm is a set of rules for solving a problem in finite number of steps that can be easily implemented in computer using any programming language. The following is an algorithm for compute the root of f(x) = 0; Input f, a, N and tol . Output: the approximate solution to f(x) = 0 with initial guess a or failure message. ² Step One: Set x = a ² Step Two: For i=0 to N do Step Three - Four f(x) Step Three: Compute x = x ¡ f 0(x) Step Four: If f(x) · tol return x ² Step Five return ”failure”. In analogy, a numerical algorithm is like a cook recipe that specify the input — cooking material, the output—the cooking product, and steps of carrying computation — cooking steps. In an algorithm, you will see loops (for, while), decision making statements(if, then, else(otherwise)) and return statements. ² for loop: Typically used when specific number of steps need to be carried out. You can break a for loop with return or break statement. 1 2 1. NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS ² while loop: Typically used when unknown number of steps need to be carried out. -
Getting Started with Euler
Getting started with Euler Samuel Fux High Performance Computing Group, Scientific IT Services, ETH Zurich ETH Zürich | Scientific IT Services | HPC Group Samuel Fux | 19.02.2020 | 1 Outlook . Introduction . Accessing the cluster . Data management . Environment/LMOD modules . Using the batch system . Applications . Getting help ETH Zürich | Scientific IT Services | HPC Group Samuel Fux | 19.02.2020 | 2 Outlook . Introduction . Accessing the cluster . Data management . Environment/LMOD modules . Using the batch system . Applications . Getting help ETH Zürich | Scientific IT Services | HPC Group Samuel Fux | 19.02.2020 | 3 Intro > What is EULER? . EULER stands for . Erweiterbarer, Umweltfreundlicher, Leistungsfähiger ETH Rechner . It is the 5th central (shared) cluster of ETH . 1999–2007 Asgard ➔ decommissioned . 2004–2008 Hreidar ➔ integrated into Brutus . 2005–2008 Gonzales ➔ integrated into Brutus . 2007–2016 Brutus . 2014–2020+ Euler . It benefits from the 15 years of experience gained with those previous large clusters ETH Zürich | Scientific IT Services | HPC Group Samuel Fux | 19.02.2020 | 4 Intro > Shareholder model . Like its predecessors, Euler has been financed (for the most part) by its users . So far, over 100 (!) research groups from almost all departments of ETH have invested in Euler . These so-called “shareholders” receive a share of the cluster’s resources (processors, memory, storage) proportional to their investment . The small share of Euler financed by IT Services is open to all members of ETH . The only requirement -
Freemat V3.6 Documentation
FreeMat v3.6 Documentation Samit Basu November 16, 2008 2 Contents 1 Introduction and Getting Started 5 1.1 INSTALL Installing FreeMat . 5 1.1.1 General Instructions . 5 1.1.2 Linux . 5 1.1.3 Windows . 6 1.1.4 Mac OS X . 6 1.1.5 Source Code . 6 2 Variables and Arrays 7 2.1 CELL Cell Array Definitions . 7 2.1.1 Usage . 7 2.1.2 Examples . 7 2.2 Function Handles . 8 2.2.1 Usage . 8 2.3 GLOBAL Global Variables . 8 2.3.1 Usage . 8 2.3.2 Example . 9 2.4 INDEXING Indexing Expressions . 9 2.4.1 Usage . 9 2.4.2 Array Indexing . 9 2.4.3 Cell Indexing . 13 2.4.4 Structure Indexing . 14 2.4.5 Complex Indexing . 16 2.5 MATRIX Matrix Definitions . 17 2.5.1 Usage . 17 2.5.2 Examples . 17 2.6 PERSISTENT Persistent Variables . 19 2.6.1 Usage . 19 2.6.2 Example . 19 2.7 STRING String Arrays . 20 2.7.1 Usage . 20 2.8 STRUCT Structure Array Constructor . 22 2.8.1 Usage . 22 2.8.2 Example . 22 3 4 CONTENTS 3 Functions and Scripts 25 3.1 ANONYMOUS Anonymous Functions . 25 3.1.1 Usage . 25 3.1.2 Examples . 25 3.2 FUNCTION Function Declarations . 26 3.2.1 Usage . 26 3.2.2 Examples . 28 3.3 KEYWORDS Function Keywords . 30 3.3.1 Usage . 30 3.3.2 Example . 31 3.4 NARGIN Number of Input Arguments . 32 3.4.1 Usage . -
Towards a Fully Automated Extraction and Interpretation of Tabular Data Using Machine Learning
UPTEC F 19050 Examensarbete 30 hp August 2019 Towards a fully automated extraction and interpretation of tabular data using machine learning Per Hedbrant Per Hedbrant Master Thesis in Engineering Physics Department of Engineering Sciences Uppsala University Sweden Abstract Towards a fully automated extraction and interpretation of tabular data using machine learning Per Hedbrant Teknisk- naturvetenskaplig fakultet UTH-enheten Motivation A challenge for researchers at CBCS is the ability to efficiently manage the Besöksadress: different data formats that frequently are changed. Significant amount of time is Ångströmlaboratoriet Lägerhyddsvägen 1 spent on manual pre-processing, converting from one format to another. There are Hus 4, Plan 0 currently no solutions that uses pattern recognition to locate and automatically recognise data structures in a spreadsheet. Postadress: Box 536 751 21 Uppsala Problem Definition The desired solution is to build a self-learning Software as-a-Service (SaaS) for Telefon: automated recognition and loading of data stored in arbitrary formats. The aim of 018 – 471 30 03 this study is three-folded: A) Investigate if unsupervised machine learning Telefax: methods can be used to label different types of cells in spreadsheets. B) 018 – 471 30 00 Investigate if a hypothesis-generating algorithm can be used to label different types of cells in spreadsheets. C) Advise on choices of architecture and Hemsida: technologies for the SaaS solution. http://www.teknat.uu.se/student Method A pre-processing framework is built that can read and pre-process any type of spreadsheet into a feature matrix. Different datasets are read and clustered. An investigation on the usefulness of reducing the dimensionality is also done. -
A Case Study in Fitting Area-Proportional Euler Diagrams with Ellipses Using Eulerr
A Case Study in Fitting Area-Proportional Euler Diagrams with Ellipses using eulerr Johan Larsson and Peter Gustafsson Department of Statistics, School of Economics and Management, Lund University, Lund, Sweden [email protected] [email protected] Abstract. Euler diagrams are common and user-friendly visualizations for set relationships. Most Euler diagrams use circles, but circles do not always yield accurate diagrams. A promising alternative is ellipses, which, in theory, enable accurate diagrams for a wider range of input. Elliptical diagrams, however, have not yet been implemented for more than three sets or three-set diagrams where there are disjoint or subset relationships. The aim of this paper is to present eulerr: a software package for elliptical Euler diagrams for, in theory, any number of sets. It fits Euler diagrams using numerical optimization and exact-area algorithms through a two-step procedure, first generating an initial layout using pairwise relationships and then finalizing this layout using all set relationships. 1 Background The Euler diagram, first described by Leonard Euler in 1802 [1], is a generalization of the popular Venn diagram. Venn and Euler diagrams both visualize set relationships by mapping areas in the diagram to relationships in the data. They differ, however, in that Venn diagrams require all intersections to be present| even if they are empty|whilst Euler diagrams do not, which means that Euler diagrams lend themselves well to be area-proportional. Euler diagrams may be fashioned out of any closed shape, and have been implemented for triangles [2], rectangles [2], ellipses [3], smooth curves [4], poly- gons [2], and circles [5, 2]. -
An Introduction to Maple
A little bit of help with Maple School of Mathematics and Applied Statistics University of Wollongong 2008 A little bit of help with Maple Welcome! The aim of this manual is to provide you with a little bit of help with Maple. Inside you will find brief descriptions on a lot of useful commands and packages that are commonly needed when using Maple. Full worked examples are presented to show you how to use the Maple commands, with different options given, where the aim is to teach you Maple by example – not by showing the full technical detail. If you want the full detail, you are encouraged to look at the Maple Help menu. To use this manual, a basic understanding of mathematics and how to use a computer is assumed. While this manual is based on Version 10.01 of Maple on the PC platform on Win- dows XP, most of it should be applicable to other versions, platforms and operating systems. This handbook is built upon the original version of this handbook by Maureen Edwards and Alex Antic. Further information has been gained from the Help menu in Maple itself. If you have any suggestions or comments, please email them to Grant Cox ([email protected]). Table of contents : Index 1/81 Contents 1 Introduction 4 1.1 What is Maple? ................................... 4 1.2 Getting started ................................... 4 1.3 Operators ...................................... 7 1.3.1 The Ditto Operator ............................. 9 1.4 Assign and unassign ................................ 9 1.5 Special constants .................................. 11 1.6 Evaluation ...................................... 11 1.6.1 Digits ................................... -
The Python Interpreter
The Python interpreter Remi Lehe Lawrence Berkeley National Laboratory (LBNL) US Particle Accelerator School (USPAS) Summer Session Self-Consistent Simulations of Beam and Plasma Systems S. M. Lund, J.-L. Vay, R. Lehe & D. Winklehner Colorado State U, Ft. Collins, CO, 13-17 June, 2016 Python interpreter: Outline 1 Overview of the Python language 2 Python, numpy and matplotlib 3 Reusing code: functions, modules, classes 4 Faster computation: Forthon Overview Scientific Python Reusing code Forthon Overview of the Python programming language Interpreted language (i.e. not compiled) ! Interactive, but not optimal for computational speed Readable and non-verbose No need to declare variables Indentation is enforced Free and open-source + Large community of open-souce packages Well adapted for scientific and data analysis applications Many excellent packages, esp. numerical computation (numpy), scientific applications (scipy), plotting (matplotlib), data analysis (pandas, scikit-learn) 3 Overview Scientific Python Reusing code Forthon Interfaces to the Python language Scripting Interactive shell Code written in a file, with a Obtained by typing python or text editor (gedit, vi, emacs) (better) ipython Execution via command line Commands are typed in and (python + filename) executed one by one Convenient for exploratory work, Convenient for long-term code debugging, rapid feedback, etc... 4 Overview Scientific Python Reusing code Forthon Interfaces to the Python language IPython (a.k.a Jupyter) notebook Notebook interface, similar to Mathematica. Intermediate between scripting and interactive shell, through reusable cells Obtained by typing jupyter notebook, opens in your web browser Convenient for exploratory work, scientific analysis and reports 5 Overview Scientific Python Reusing code Forthon Overview of the Python language This lecture Reminder of the main points of the Scipy lecture notes through an example problem. -
Sage Tutorial (Pdf)
Sage Tutorial Release 9.4 The Sage Development Team Aug 24, 2021 CONTENTS 1 Introduction 3 1.1 Installation................................................4 1.2 Ways to Use Sage.............................................4 1.3 Longterm Goals for Sage.........................................5 2 A Guided Tour 7 2.1 Assignment, Equality, and Arithmetic..................................7 2.2 Getting Help...............................................9 2.3 Functions, Indentation, and Counting.................................. 10 2.4 Basic Algebra and Calculus....................................... 14 2.5 Plotting.................................................. 20 2.6 Some Common Issues with Functions.................................. 23 2.7 Basic Rings................................................ 26 2.8 Linear Algebra.............................................. 28 2.9 Polynomials............................................... 32 2.10 Parents, Conversion and Coercion.................................... 36 2.11 Finite Groups, Abelian Groups...................................... 42 2.12 Number Theory............................................. 43 2.13 Some More Advanced Mathematics................................... 46 3 The Interactive Shell 55 3.1 Your Sage Session............................................ 55 3.2 Logging Input and Output........................................ 57 3.3 Paste Ignores Prompts.......................................... 58 3.4 Timing Commands............................................ 58 3.5 Other IPython