A Collection of Matlab Functions for the Computation of Elliptical Integrals and Jacobian Elliptic Functions of Real Arguments
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
The Cool Package∗
The cool package∗ nsetzer December 30, 2006 This is the cool package: a COntent Oriented LATEX package. That is, it is designed to give LATEX commands the ability to contain the mathematical meaning while retaining the typesetting versatility. Please note that there are examples of use of each of the defined commands at the location where they are defined. This package requires the following, non-standard LATEX packages (all of which are available on www.ctan.org): coolstr, coollist, forloop 1 Implementation 1 \newcounter{COOL@ct} %just a general counter 2 \newcounter{COOL@ct@}%just a general counter 1.1 Parenthesis 3 \newcommand{\inp}[2][0cm]{\mathopen{}\left(#2\parbox[h][#1]{0cm}{}\right)} 4 % in parentheses () 5 \newcommand{\inb}[2][0cm]{\mathopen{}\left[#2\parbox[h][#1]{0cm}{}\right]} 6 % in brackets [] 7 \newcommand{\inbr}[2][0cm]{\mathopen{}\left\{#2\parbox[h][#1]{0cm}{}\right\}} 8 % in braces {} 9 \newcommand{\inap}[2][0cm]{\mathopen{}\left<{#2}\parbox[h][#1]{0cm}{}\right>} 10 % in angular parentheses <> 11 \newcommand{\nop}[1]{\mathopen{}\left.{#1}\right.} 12 % no parentheses \COOL@decide@paren \COOL@decide@paren[hparenthesis typei]{hfunction namei}{hcontained texti}. Since the handling of parentheses is something that will be common to many elements this function will take care of it. If the optional argument is given, \COOL@notation@hfunction nameiParen is ignored and hparenthesis typei is used hparenthesis typei and \COOL@notation@hfunction nameiParen must be one of none, p for (), b for [], br for {}, ap for hi, inv for \left.\right. 13 \let\COOL@decide@paren@no@type=\relax 14 \newcommand{\COOL@decide@paren}[3][\COOL@decide@paren@no@type]{% 15 \ifthenelse{ \equal{#1}{\COOL@decide@paren@no@type} }% 16 {% 17 \def\COOL@decide@paren@type{\csname COOL@notation@#2Paren\endcsname}% 18 }% ∗This document corresponds to cool v1.35, dated 2006/12/29. -
An Introduction to the SAS System
An Introduction to the SAS System Dileep K. Panda Directorate of Water Management Bhubaneswar-751023 [email protected] Introduction The SAS – Statistical Analysis System (erstwhile expansion of SAS) - is the one of the most widely used Statistical Software package by the academic circles and Industry. The SAS software was developed in late 1960s at North Carolina State University and in 1976 SAS Institute was formed. The SAS system is a collection of products, available from the SAS Institute in North Carolina. SAS software is a combination of a statistical package, a data – base management system, and a high level programming language. The SAS is an integrated system of software solutions that performs the following tasks: Data entry, retrieval, and management Report writing and graphics design Statistical and mathematical analysis Business forecasting and decision support Operations research and project management Applications development At the core of the SAS System is the Base SAS software. The Base SAS software includes a fourth-generation programming language and ready-to-use programs called procedures. These integrated procedures handle data manipulation, information storage and retrieval, statistical analysis, and report writing. Additional components offer capabilities for data entry, retrieval, and management; report writing and graphics; statistical and mathematical analysis; business planning, forecasting, and decision support; operations research and project management; quality improvement; and applications development. In general, the Base SAS software has the following capabilities A data management facility A programming language Data analysis and reporting utilities Learning to use Base SAS enables you to work with these features of SAS. It also prepares you to learn other SAS products, because all SAS products follow the same basic rules. -
Information Technology Laboratory Technical Accomplishments
CONTENTS Director’s Foreword 1 ITL at a Glance 4 ITL Research Blueprint 6 Accomplishments of our Research Program 7 Foundation Research Areas 8 Selected Cross-Cutting Themes 26 Industry and International Interactions 36 Publications 44 NISTIR 7169 Conferences 47 February 2005 Staff Recognition 50 U.S. DEPARTMENT OF COMMERCE Carlos M. Gutierrez, Secretary Technology Administration Phillip J. Bond Under Secretary of Commerce for Technology National Institute of Standards and Technology Hratch G. Semerjian, Jr., Acting Director About ITL For more information about ITL, contact: Information Technology Laboratory National Institute of Standards and Technology 100 Bureau Drive, Stop 8900 Gaithersburg, MD 20899-8900 Telephone: (301) 975-2900 Facsimile: (301) 840-1357 E-mail: [email protected] Website: http://www.itl.nist.gov INFORMATION TECHNOLOGY LABORATORY D IRECTOR’S F OREWORD n today’s complex technology-driven world, the Information Technology Laboratory (ITL) at the National Institute of Standards and Technology has the broad mission of supporting U.S. industry, government, and Iacademia with measurements and standards that enable new computational methods for scientific inquiry, assure IT innovations for maintaining global leadership, and re-engineer complex societal systems and processes through insertion of advanced information technology. Through its efforts, ITL seeks to enhance productivity and public safety, facilitate trade, and improve the Dr. Shashi Phoha, quality of life. ITL achieves these goals in areas of Director, Information national priority by drawing on its core capabilities in Technology Laboratory cyber security, software quality assurance, advanced networking, information access, mathematical and computational sciences, and statistical engineering. utilizing existing and emerging IT to meet national Information technology is the acknowledged engine for priorities that reflect the country’s broad based social, national and regional economic growth. -
Modular Curves and the Refined Distance Conjecture
MITP/21-034 Modular Curves and the Refined Distance Conjecture Daniel Kl¨awer PRISMA+Cluster of Excellence and Mainz Institute for Theoretical Physics, Johannes Gutenberg-Universit¨at,55099 Mainz, Germany Abstract We test the refined distance conjecture in the vector multiplet moduli space of 4D N = 2 compactifications of the type IIA string that admit a dual heterotic description. In the weakly coupled regime of the heterotic string, the moduli space geometry is governed by the perturbative heterotic dualities, which allows for exact computations. This is reflected in the type IIA frame through the existence of a K3 fibration. We identify the degree d = 2N of the K3 fiber as a parameter that could potentially lead to large distances, which is substantiated by studying several explicit models. The moduli space geometry degenerates into the modular curve for the congruence subgroup + Γ0(N) . In order to probe the large N regime, we initiate the study of Calabi-Yau threefolds fibered by general degree d > 8 K3 surfaces by suggesting a construction as complete intersections in Grassmann bundles. arXiv:2108.00021v1 [hep-th] 30 Jul 2021 Contents 1 Introduction2 2 Refined Distance Conjecture for Simple K3 Fibrations5 3 2.1 Fibration by P1113[6] - SL(2; Z).........................6 3 + 2.2 Fibration by P [4] - Γ0(2) ........................... 10 4 + 2.3 Fibration by P [2; 3] - Γ0(3) .......................... 13 5 + 2.4 Fibration by P [2; 2; 2] - Γ0(4) ......................... 14 3 RDC for CY Threefolds Fibered by Degree 2N K3 Surfaces 15 3.1 K3 Fibrations with h11 = 2: Generalities . 16 3.2 Violating the Refined Distance Conjecture? . -
The Unity of Mathematics As a Method of Discovery
The unity of mathematics as a method of discovery (long version of a talk given at the 7th French Philosophy of Mathematics Workshop 5-7 November 2015, University of Paris-Diderot) Jean Petitot CAMS (EHESS), Paris J. Petitot The unity of mathematics Introduction 1. Kant used to claim that \philosophical knowledge is rational knowledge from concepts, mathematical knowledge is rational knowledge from the construction of concepts" (A713/ B741). As I am rather Kantian, I will consider here that philosophy of mathematics has to do with \rational knowledge from concepts" in mathematics. J. Petitot The unity of mathematics 2. But \concept" in what sense? Well, in the sense introduced by Galois and deeply developped through Hilbert to Bourbaki. Galois said: \Il existe pour ces sortes d'´equationsun certain ordre de consid´erationsm´etaphysiquesqui planent sur les calculs et qui souvent les rendent inutiles." \Sauter `apieds joints sur les calculs, grouper les op´erations,les classer suivant leur difficult´eet non suivant leur forme, telle est selon moi la mission des g´eom`etresfuturs." So, I use \concept" in the structural sense. In this perspective, philosophy of mathematics has to do with the dialectic between, on the one hand, logic and computations, and, on the other hand, structural concepts. J. Petitot The unity of mathematics 3. In mathematics, the context of justification is proof. It has been tremendously investigated. But the context of discovery remains mysterious and is very poorly understood. I think that structural concepts play a crucial role in it. 4. In this general perspective, my purpose is to investigate what could mean \complex" in a conceptually complex proof. -
Paquetes Estadísticos Con Licencia Libre (I) Free Statistical
2013, Vol. 18 No 2, pp. 12-33 http://www.uni oviedo.es/reunido /index.php/Rema Paquetes estadísticos con licencia libre (I) Free statistical software (I) Carlos Carleos Artime y Norberto Corral Blanco Departamento de Estadística e I. O. y D.M.. Universidad de Oviedo RESUMEN El presente artículo, el primero de una serie de trabajos, presenta un panorama de los principales paquetes estadísticos disponibles en la actualidad con licencia libre, entendiendo este concepto a partir de los grandes proyectos informáticos así autodefinidos que han cristalizado en los últimos treinta años. En esta primera entrega se presta atención fundamentalmente al programa R. Palabras clave : paquetes estadísticos, software libre, R. ABSTRACT This article, the first in a series of works, presents an overview of the major statistical packages available today with free license, understanding this concept from the large computer and self-defined projects that have crystallized in the last thirty years. In this first paper we pay attention mainly to the program R. Keywords : statistical software, free software, R. Contacto: Norberto Corral Blanco Email: [email protected] . 12 Revista Electrónica de Metodología Aplicada (2013), Vol. 18 No 2, pp. 12-33 1.- Introducción La palabra “libre” es usada, y no siempre de manera adecuada, en múltiples campos de conocimiento así que la informática no iba a ser una excepción. La palabra es especialmente problemática porque su traducción al inglés, “free”, es aun más polisémica, e incluye el concepto “gratis”. En 1985 se fundó la Free Software Foundation (FSF) para defender una definición rigurosa del sófguar libre. La propia palabra “sófguar” (del inglés “software”, por oposición a “hardware”, quincalla, aplicado al soporte físico de la informática) es en sí misma problemática. -
A New Class of Modular Equation for Weber Functions
A NEW CLASS OF MODULAR EQUATION FOR WEBER FUNCTIONS WILLIAM B. HART Abstract. We describe the construction of a new type of modular equation for Weber functions. These bear some relationship to Weber’s modular equa- tions of irrational kind. Numerous examples of these equations are explicitly computed. We also obtain some modular equations of irrational kind which Weber was not able to compute. Introduction A modular equation, in its simplest form, is a polynomial identity relating a modular function f(τ) of some level n with the function f(mτ) for some m ∈ N (most commonly with (m, n) = 1). This m is called the degree of the modular equation. The basic example of such an identity comes from the Klein j-function, where for each m ∈ N one has a symmetric polynomial Dm(X, Y ) ∈ Z[X, Y ], such that Dm(j(τ), j(mτ)) = 0 is an identity for all τ in the complex upper half plane (see [6] for details). A second example is given by Schl¨aflimodular equations. These provide identities for the Weber functions − πi τ+1 τ e 24 η η √ η (2τ) (1) f(τ) = 2 , f (τ) = 2 , f (τ) = 2 , η(τ) 1 η(τ) 2 η(τ) where η(τ) is the well known Dedekind eta function. We let u(τ) be any one of these functions and define v(τ) = u(mτ) for any m with (m, 2) = 1. Then if we let P (τ) = u(τ)v(τ) and Q(τ) = u(τ)/v(τ) be the product and quotient of these functions, a Schl¨aflimodular equation is a polynomial identity relating 2 k 1 l A = P k + and B = Ql ± , P Q where the sign in B and the powers k, l ∈ N depend on the degree m. -
The Statistics Tutor's Pocket Book Guide
statstutor community project encouraging academics to share statistics support resources All stcp resources are released under a Creative Commons licence Stcp-marshall_owen-pocket The Statistics Tutor’s Pocket Book Guide to Statistics Resources Version 1.0 Dated 25/08/2016 Please note that if printed, this guide has been designed to be printed as an A5 booklet double sided on A4 paper. © Alun Owen (University of Worcester), Ellen Marshall (University of Sheffield) www.statstutor.ac.uk and Jonathan Gillard (Cardiff University). Reviewer: Mark Johnston (University of Worcester) Page 2 of 51 Contents INTRODUCTION ................................................................................................................................................................................. 4 SECTION 1 MOST POPULAR RESOURCES .................................................................................................................................... 7 THE MOST RECOMMENDED STATISTICS BOOKS ......................................................................................................................................... 8 THE MOST RECOMMENDED ONLINE STATISTICS RESOURCES ........................................................................................................................ 9 SECTION 2 DESIGNING A STUDY AND CHOOSING A TEST ......................................................................................................... 11 DESIGNING AN EXPERIMENT OR SURVEY AND CHOOSING A TEST ............................................................................................................... -
Eta Products, BPS States and K3 Surfaces
Eta Products, BPS States and K3 Surfaces Yang-Hui He1 & John McKay2 1 Department of Mathematics, City University, London, EC1V 0HB, UK and School of Physics, NanKai University, Tianjin, 300071, P.R. China and Merton College, University of Oxford, OX14JD, UK [email protected] 2 Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec, H3G 1M8, Canada [email protected] Abstract Inspired by the multiplicative nature of the Ramanujan modular discriminant, ∆, we consider physical realizations of certain multiplicative products over the Dedekind eta-function in two parallel directions: the generating function of BPS states in certain heterotic orbifolds and elliptic K3 surfaces associated to congruence subgroups of the modular group. We show that they are, after string duality to type II, the same K3 arXiv:1308.5233v3 [hep-th] 8 Jan 2014 surfaces admitting Nikulin automorphisms. In due course, we will present identities arising from q-expansions as well as relations to the sporadic Mathieu group M24. 1 Contents 1 Introduction and Motivation 3 1.1 Nomenclature . .5 2 Eta Products and Partition Functions 7 2.1 Bosonic String Oscillators . .7 2.2 Eta Products . .9 2.3 Partition Functions and K3 Surfaces . 10 2.4 Counting 1/2-BPS States . 11 3 K3 Surfaces and Congruence Groups 13 3.1 Extremal K3 Surfaces . 13 3.2 Modular Subgroups and Coset Graphs . 14 3.3 Summary . 17 3.4 Beyond Extremality . 20 3.5 Elliptic Curves . 21 4 Monsieur Mathieu 24 5 A Plethystic Outlook 28 6 Conclusions and Prospects 30 A Further Salient Features of Eta 33 2 A.1 Modularity . -
The R Project for Statistical Computing a Free Software Environment For
The R Project for Statistical Computing A free software environment for statistical computing and graphics that runs on a wide variety of UNIX platforms, Windows and MacOS OpenStat OpenStat is a general-purpose statistics package that you can download and install for free. It was originally written as an aid in the teaching of statistics to students enrolled in a social science program. It has been expanded to provide procedures useful in a wide variety of disciplines. It has a similar interface to SPSS SOFA A basic, user-friendly, open-source statistics, analysis, and reporting package PSPP PSPP is a program for statistical analysis of sampled data. It is a free replacement for the proprietary program SPSS, and appears very similar to it with a few exceptions TANAGRA A free, open-source, easy to use data-mining package PAST PAST is a package created with the palaeontologist in mind but has been adopted by users in other disciplines. It’s easy to use and includes a large selection of common statistical, plotting and modelling functions AnSWR AnSWR is a software system for coordinating and conducting large-scale, team-based analysis projects that integrate qualitative and quantitative techniques MIX An Excel-based tool for meta-analysis Free Statistical Software This page links to free software packages that you can download and install on your computer from StatPages.org Free Statistical Software This page links to free software packages that you can download and install on your computer from freestatistics.info Free Software Information and links from the Resources for Methods in Evaluation and Social Research site You can sort the table below by clicking on the column names. -
Cumulation of Poverty Measures: the Theory Beyond It, Possible Applications and Software Developed
Cumulation of Poverty measures: the theory beyond it, possible applications and software developed (Francesca Gagliardi and Giulio Tarditi) Siena, October 6th , 2010 1 Context and scope Reliable indicators of poverty and social exclusion are an essential monitoring tool. In the EU-wide context, these indicators are most useful when they are comparable across countries and over time. Furthermore, policy research and application require statistics disaggregated to increasingly lower levels and smaller subpopulations. Direct, one-time estimates from surveys designed primarily to meet national needs tend to be insufficiently precise for meeting these new policy needs. This is particularly true in the domain of poverty and social exclusion, the monitoring of which requires complex distributional statistics – statistics necessarily based on intensive and relatively small- scale surveys of households and persons. This work addresses some statistical aspects relating to improving the sampling precision of such indicators in EU countries, in particular through the cumulation of data over rounds of regularly repeated national surveys. 2 EU-SILC The reference data for this purpose are EU Statistics on Income and Living Conditions, the major source of comparative statistics on income and living conditions in Europe. A standard integrated design has been adopted by nearly all EU countries. It involves a rotational panel, with a new sample of households and persons introduced each year to replace one-fourth of the existing sample. Persons enumerated in each new sample are followed-up in the survey for four years. The design yields each year a cross- sectional sample, as well as longitudinal samples of 2, 3 and 4 year duration. -
Comparison of Three Common Statistical Programs Available to Washington State County Assessors: SAS, SPSS and NCSS
Washington State Department of Revenue Comparison of Three Common Statistical Programs Available to Washington State County Assessors: SAS, SPSS and NCSS February 2008 Abstract: This summary compares three common statistical software packages available to county assessors in Washington State. This includes SAS, SPSS and NCSS. The majority of the summary is formatted in tables which allow the reader to more easily make comparisons. Information was collected from various sources and in some cases includes opinions on software performance, features and their strengths and weaknesses. This summary was written for Department of Revenue employees and county assessors to compare statistical software packages used by county assessors and as a result should not be used as a general comparison of the software packages. Information not included in this summary includes the support infrastructure, some detailed features (that are not described in this summary) and types of users. Contents General software information.............................................................................................. 3 Statistics and Procedures Components................................................................................ 8 Cost Estimates ................................................................................................................... 10 Other Statistics Software ................................................................................................... 13 General software information Information in this section was