DOCSLIB.ORG

The Binary Arithmetic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Binary Arithmetic Binary Arithmetic for Finite-Word-Length Linear Controllers : MEMS Applications Abdelkrim Kamel Oudjida To cite this version: Abdelkrim Kamel Oudjida. Binary Arithmetic for Finite-Word-Length Linear Controllers : MEMS Applications. Automatic Control Engineering. Université de Franche-Comté, 2014. English. NNT : 2014BESA2001. tel-01124332 HAL Id: tel-01124332 https://tel.archives-ouvertes.fr/tel-01124332 Submitted on 6 Mar 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Thèse de Doctorat école doctorale sciences pour l’ingénieur et microtechniques UNIVERSITÉ DE FRANCHE-COMTÉ Binary Arithmetic for Finite-Word-Length Linear Controllers: MEMS Applications nABDELKRIM KAMEL OUDJIDA THÈSE présentée par Abdelkrim Kamel OUDJIDA pour obtenir le Grade de Docteur de L'Université de Franche-Compté Spécialité: Automatique Binary Arithmetic for Finite-Word-Length Linear Controllers: MEMS Applications Unité de Recherche: FEMTO-ST, UMR CNRS 6174 Soutenue le 20 Janvier 2014 devant le jury: Amara AMARA Président Professeur des Universités, ISEP, Paris Adel BELOUCHRANI Rapporteur Professeur, ENP, Alger Maître de Conférences HDR, Chouki AKTOUF Rapporteur INP de Grenoble Maître de Conférences, Thibault HILAIRE Examinateur Université Paris 6 Directeur de Recherche, Brahim BOUZOUIA Examinateur CDTA, Alger Nicolas CHAILLET Directeur de Thèse Professeur des Universités, Université de Besançon Gödel's Incompleteness Theorems Theorem 1 − Blind spot: There is a sentence G such that if the formal system is consistent, G is not a theorem nor is it not a theorem. Theorem 2 − Consistency: No consistent theory, with a certain amount of arithmetic, can prove its own consistency. i Acknowledgments I wish to thank my supervisor, Prof. Nicolas Chaillet, for the support and trust he so generously provided me in the course of the accomplishment of my task. His continuous enthusiasm and encouragement constituted the driving force of this work. I am deeply grateful to Dr. Brahim Bouzouia, Director of CDTA, for his help and support during all the period of the thesis. I am so indebted to Prof. Noureddine Zerhouni, Mr. Karim Henda, and Prof. Yassine Haddab for their precious help. Without their effective contribution, this thesis would not have been concretized. My special thanks go to my two colleagues, Mohamed Lamine Berrandjia and Ahmed Liacha, for their technical help in a warm and joyful atmosphere. I am much thankful to my ex-student, Khaled Tahraoui, for having provided me with a tremendous number of papers that much contributed to enhance the quality of the work. I would like to acknowledge Dr. Joël Agnus, research engineer within AS2M department of Femto- St institute, for his valuable advices. Finally, I would like to thank my wife and sons for their great patience and support during my seemingly never-ending nights preparing the thesis. ii Abstract This thesis addresses the problem of optimal hardware-realization of finite-word-length (FWL) linear controllers dedicated to MEMS applications. The biggest challenge is to ensure satisfactory control performances with a minimal hardware. To come up, two distinct but complementary optimizations can be undertaken: in control theory and in binary arithmetic. Only the latter is involved in this work. Because MEMS applications are targeted, the binary arithmetic must be fast enough to cope with the rapid dynamic of MEMS; power-efficient for an embedded control; highly scalable for an easy adjustment of the control performances; and easily predictable to provide a precise idea on the required logic resources before the implementation. r The exploration of a number of binary arithmetics showed that radix-2 is the best candidate that fits the aforementioned requirements. It has been fully exploited to designing efficient multiplier cores, which are the real engine of the linear systems. r The radix-2 arithmetic was applied to the hardware integration of two FWL structures: a linear time variant PID controller and a linear time invariant LQG controller with a Kalman filter. Both controllers showed a clear superiority over their existing counterparts, or in comparison to their initial forms. iii Glossary Abstraction Simplification of details, approximation of complex problems ADC Analog to Digital Converter AFM Atomic Force Microscopy ALU Arithmetic and Logic Unit ASIC Application Specific Integration Circuit Ath Adder Depth, the maximum number of serial adder-operations from input to output Avg Average number of additions BHM Bull Horrocks modified, an existing heuristic MCM algorithm BIGE Bounded Inverse Graph Enumeration, an optimal SCM algorithm BMA Booth Multiplication Algorithm, an existing algorithm for signed multiplication CAD Computer-aided design, tools for design automation CDE Common Digit Elimination CLB Configurable Logic Bloc CMOS Complementary Metal Oxide Semiconductor COTS Commercial Off-The-Shelf CSD Canonical Signed Digit, the SD form with no adjacent nonzero digits and the minimum number of nonzero digits CSE Common Subexpression Elimination, a framework for solving SCM and MCM DAC Digital to Analog Converter DAG Directed Acyclic Graphs, a framework for solving SCM and MCM DBNS Double Base Number System, an existing number representation system DFS Dynamic Frequency Scaling Digit Clashing The CSE problem of disappearing patterns due to colliding digits DMAC Double Multiply-And-Accumulate DSP Digital-Signal-Processor/Processing FF Flip-Flop FPGA Field Programmable Gate Array FPR Fixed-Point Representation FWL Finite Word Length H(k) Heuristic with k extra nonzero digits, an existing heuristic SCM algorithm H(k)+ODP The H(k) algorithm with ODPs, a proposed heuristic SCM algorithm Hcub Cumulative Benefit Heuristic, an existing MCM algorithm iv HDL Hardware Description Language Heuristic An effective but potentially suboptimal method for solving a problem HIS Host Side Interface HPM High Performance Multiplication, an existing adder reduction technique IOB Input Output Block Logic resources An abstraction of the amount of silicon required to implement a logic function LQG Linear Quadratic Gaussian LTI Linear Time Invariant LTV Linear Time Variant MAC Multiply-And-Accumulate MAG Minimized Adder Graph, an existing optimal SCM algorithm MBMA Modified Booth Multiplication Algorithm, , an existing algorithm for signed multiplication MCM Multiple Constant Multiplication, find a low-cost add-shift-subtract realization of multiplication by each of the given constants MEMS Micro-Electro-Mechanical Structure MM Matrix Multiplication MPC Model Predictive Control MPM Multi-Precision Multiplication MSD Minimal Signed Digit, any SD representation with the minimum number of nonzero digits MV Multiplication by a Variable NEMS Nano-Electro-Mechanical Structure NP-hard Non-deterministic Polynomial-time hard ODMAC Optimized Double Multiply-And-Accumulate ODP Overlapping Digit Pattern, a proposed technique for partially resolving the CSE digit clashing problem OS Operating System Pattern (CSE) A collection of signed digits that define how existing terms are added-operated together PC Personal Computer PID Proportional Integral Derivative, an existing control law PLD Programmable Logic Device PPG Partial Product Generator RAG-n n-dimensional Reduced Adder Graph, an existing heuristic MCM algorithm RMRMA Recursive-Multibit-Recoding Multiplication Algorithm, a new algorithm RNS Residue Number System, an existing number representation system RTL Register Transfer Level RTOS Real Time Operating System SCM Single Constant Multiplication, same as MCM but for a single constant SD Signed Digit, a recoding used in the CSE framework v Search space The set of all solutions that can be found by an algorithm, this is smaller than the solution space for heuristic algorithms SEM Scanning Electron Microscopy SiGe Silicon Germanium SM Sign and Magnitude SOC System On Chip Solution space The set of all feasible solutions SRAM Static Random Access Memory TC True and Complement TSMC Taiwan Semiconductor Manufacturing Company Upb Upper bound, maximum number of additions VHDL Very high speed integrated circuit Hardware Description Language VLIW Very Large Instruction Word vi Contents 1 General Introduction . 1 1.1 Motivation and Problem Statement . 1 1.2 Objective of the Thesis. 1 1.3 Requirements and Specifications. 2 1.4 Contribution of the Thesis. 3 1.5 Organization of the Thesis . 4 2 Problem Background . 6 2.1 Application Context . .6 2.2 Micromanipulation as a MEMS application . .8 2.3 Embedded Control-Electronics for MEMS . 10 2.4 Review of the Basic Digital Solutions for Embedded Control . 12 2.4.1 Commercial Off-The-Shelf (COTS) Electronics Components . 12 2.4.2 Digital Signal Processors (DSPs) . 12 2.4.3 Field Programmable Gate Arrays (FPGA) . .13 2.4.4 Application Specific Integrated Circuits (ASIC) . 14 2.5 Overview of Finite-Word-Length (FWL) Controller Optimizations . 16 2.5.1 Definition of the FWL Effect. 16 2.5.2 Control-Theory Based Optimization. 17 2.5.3 Binary-Arithmetic Based Optimization . 19 2.5.3.1 Multiplication by a Constant . 19 2.5.3.2 Multiplication by a Variable . .21 2.5.3.3 Multi-Precision Multiplication. 22 2.6 Conclusion . 23 3 The Binary Arithmetic. 28 3.1 Introduction to the Binary Arithmetic . 28 3.2 Number Representation Formats. 29 3.2.1 Fixed-Point Format. 30 3.2.1.1 Representation of Nonnegative Integers. 30 3.2.1.2 Representation of Signed Integers . 32 3.2.1.3 Fixed-Point Arithmetic of Two's Complement Numbers . 33 3.2.1.4 Multiplication. 34 3.2.1.5 Addition. 35 3.2.1.6 Overflow Detection . 35 3.2.2 Floating-Point Format. 35 vii 3.2.2.1 Dynamic Range. 36 3.2.2.2 Precision . .36 3.3 Number Representation Systems. 38 3.3.1 Canonical Signed Digit (CSD) . 39 3.3.2 Double Base Number System (DBNS) .
Load more

Recommended publications
  • Dynamical Directions in Numeration Tome 56, No 7 (2006), P
    R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Guy BARAT, Valérie BERTHÉ, Pierre LIARDET & Jörg THUSWALDNER Dynamical directions in numeration Tome 56, no 7 (2006), p. 1987-2092. <http://aif.cedram.org/item?id=AIF_2006__56_7_1987_0> © Association des Annales de l’institut Fourier, 2006, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 56, 7 (2006) 1987-2092 DYNAMICAL DIRECTIONS IN NUMERATION by Guy BARAT, Valérie BERTHÉ, Pierre LIARDET & Jörg THUSWALDNER (*) Abstract. — This survey aims at giving a consistent presentation of numer- ation from a dynamical viewpoint: we focus on numeration systems, their asso- ciated compactification, and dynamical systems that can be naturally defined on them. The exposition is unified by the fibred numeration system concept. Many examples are discussed. Various numerations on rational integers, real or complex numbers are presented with special attention paid to β-numeration and its gener- alisations, abstract numeration systems and shift radix systems, as well as G-scales and odometers.
    [Show full text]
  • Basic Computer Arithmetic
    BASIC COMPUTER ARITHMETIC TSOGTGEREL GANTUMUR Abstract. First, we consider how integers and fractional numbers are represented and manipulated internally on a computer. The focus is on the principles behind the algorithms, rather than on implementation details. Then we develop a basic theoretical framework for analyzing algorithms involving inexact arithmetic. Contents 1. Introduction 1 2. Integers 2 3. Simple division algorithms 6 4. Long division 11 5. The SRT division algorithm 16 6. Floating point numbers 18 7. Floating point arithmetic 21 8. Propagation of error 24 9. Summation and product 29 1. Introduction There is no way to encode all real numbers by using finite length words, even if we use an alphabet with countably many characters, because the set of finite sequences of integers is countable. Fortunately, the real numbers support many countable dense subsets, and hence the encoding problem of real numbers may be replaced by the question of choosing a suitable countable dense subset. Let us look at some practical examples of real number encodings. ● Decimal notation. Examples: 36000, 2:35(75), −0:000072. ● Scientific notation. Examples: 3:6⋅104, −72⋅10−6. The general form is m⋅10e. In order to have a unique (or near unique) representation of each number, one can impose a normalization, such as requiring 1 ≤ ∣m∣ < 10. 2 −1 ● System with base/radix β. Example: m2m1m0:m−1 = m2β +m1β +m0 +m−1β . The dot separating the integer and fractional parts is called the radix point. ● Binary (β = 2), octal (β = 8), and hexadecimal (β = 16) numbers. ● Babylonian hexagesimal (β = 60) numbers.
    [Show full text]
  • The Mayan Long Count Calendar Thomas Chanier
    The Mayan Long Count Calendar Thomas Chanier To cite this version: Thomas Chanier. The Mayan Long Count Calendar. 2015. hal-00750006v11 HAL Id: hal-00750006 https://hal.archives-ouvertes.fr/hal-00750006v11 Preprint submitted on 8 Dec 2015 (v11), last revised 16 Dec 2015 (v12) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Mayan Long Count Calendar Thomas Chanier∗1 1 Universit´ede Brest, 6 avenue Victor le Gorgeu, F-29285 Brest Cedex, France The Mayan Codices, bark-paper books from the Late Postclassic period (1300 to 1521 CE) contain many astronomical tables correlated to ritual cycles, evidence of the achievement of Mayan naked- eye astronomy and mathematics in connection to religion. In this study, a calendar supernumber is calculated by computing the least common multiple of 8 canonical astronomical periods. The three major calendar cycles, the Calendar Round, the Kawil and the Long Count Calendar are shown to derive from this supernumber. The 360-day Tun, the 365-day civil year Haab’ and the 3276-day Kawil-direction-color cycle are determined from the prime factorization of the 8 canonical astronomical input parameters.
    [Show full text]
  • Ed 040 737 Institution Available from Edrs Price
    DOCUMENT RESUME ED 040 737 LI 002 060 TITLE Automatic Data Processing Glossary. INSTITUTION Bureau of the Budget, Washington, D.C. NOTE 65p. AVAILABLE FROM Reprinted and distributed by Datamation Magazine, 35 Mason St., Greenwich, Conn. 06830 ($1.00) EDRS PRICE EDRS Price MF-$0.50 HC-$3.35 DESCRIPTORS *Electronic Data Processing, *Glossaries, *Word Lists ABSTRACT The technology of the automatic information processing field has progressed dramatically in the past few years and has created a problem in common term usage. As a solution, "Datamation" Magazine offers this glossary which was compiled by the U.S. Bureau of the Budget as an official reference. The terms appear in a single alphabetic sequence, ignoring commas or hyphens. Definitions are given only under "key word" entries. Modifiers consisting of more than one word are listed in the normally used sequence (record, fixed length). In cases where two or more terms have the same meaning, only the preferred term is defined, all synonylious terms are given at the end of the definition.Other relationships between terms are shown by descriptive referencing expressions. Hyphens are used sparingly to avoid ambiguity. The derivation of an acronym is shown by underscoring the appropriate letters in the words from which the acronym is formed. Although this glossary is several years old, it is still considered the best one available. (NH) Ns U.S DEPARTMENT OF HEALTH, EDUCATION & WELFARE OFFICE OF EDUCATION THIS DOCUMENT HAS BEEN REPRODUCED EXACTLY AS RECEIVED FROM THE PERSON OR ORGANIZATION ORIGINATING IT POINTS OF VIEW OR OPINIONS STATED DO NOT NECES- SARILY REPRESENT OFFICIAL OFFICE OF EDU- CATION POSITION OR POLICY automatic data processing GLOSSA 1 I.R DATAMATION Magazine reprints this Glossary of Terms as a service to the data processing field.
    [Show full text]
  • Non-Power Positional Number Representation Systems, Bijective Numeration, and the Mesoamerican Discovery of Zero
    Non-Power Positional Number Representation Systems, Bijective Numeration, and the Mesoamerican Discovery of Zero Berenice Rojo-Garibaldia, Costanza Rangonib, Diego L. Gonz´alezb;c, and Julyan H. E. Cartwrightd;e a Posgrado en Ciencias del Mar y Limnolog´ıa, Universidad Nacional Aut´onomade M´exico, Av. Universidad 3000, Col. Copilco, Del. Coyoac´an,Cd.Mx. 04510, M´exico b Istituto per la Microelettronica e i Microsistemi, Area della Ricerca CNR di Bologna, 40129 Bologna, Italy c Dipartimento di Scienze Statistiche \Paolo Fortunati", Universit`adi Bologna, 40126 Bologna, Italy d Instituto Andaluz de Ciencias de la Tierra, CSIC{Universidad de Granada, 18100 Armilla, Granada, Spain e Instituto Carlos I de F´ısicaTe´oricay Computacional, Universidad de Granada, 18071 Granada, Spain Keywords: Zero | Maya | Pre-Columbian Mesoamerica | Number rep- resentation systems | Bijective numeration Abstract Pre-Columbian Mesoamerica was a fertile crescent for the development of number systems. A form of vigesimal system seems to have been present from the first Olmec civilization onwards, to which succeeding peoples made contributions. We discuss the Maya use of the representational redundancy present in their Long Count calendar, a non-power positional number representation system with multipliers 1, 20, 18× 20, :::, 18× arXiv:2005.10207v2 [math.HO] 23 Mar 2021 20n. We demonstrate that the Mesoamericans did not need to invent positional notation and discover zero at the same time because they were not afraid of using a number system in which the same number can be written in different ways. A Long Count number system with digits from 0 to 20 is seen later to pass to one using digits 0 to 19, which leads us to propose that even earlier there may have been an initial zeroless bijective numeration system whose digits ran from 1 to 20.
    [Show full text]
  • Parallel Branch-And-Bound Revisited for Solving Permutation Combinatorial Optimization Problems on Multi-Core Processors and Coprocessors Rudi Leroy
    Parallel Branch-and-Bound revisited for solving permutation combinatorial optimization problems on multi-core processors and coprocessors Rudi Leroy To cite this version: Rudi Leroy. Parallel Branch-and-Bound revisited for solving permutation combinatorial optimization problems on multi-core processors and coprocessors. Distributed, Parallel, and Cluster Computing [cs.DC]. Université Lille 1, 2015. English. tel-01248563 HAL Id: tel-01248563 https://hal.inria.fr/tel-01248563 Submitted on 27 Dec 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Ecole Doctorale Sciences Pour l’Ingénieur Université Lille 1 Nord-de-France Centre de Recherche en Informatique, Signal et Automatique de Lille (UMR CNRS 9189) Centre de Recherche INRIA Lille Nord Europe Maison de la Simulation Thèse présentée pour obtenir le grade de docteur Discipline : Informatique Parallel Branch-and-Bound revisited for solving permutation combinatorial optimization problems on multi-core processors and coprocessors. Défendue par : Rudi Leroy Novembre 2012 - Novembre 2015 Devant le jury
    [Show full text]
  • DYNAMICAL DIRECTIONS in NUMERATION Contents 1
    DYNAMICAL DIRECTIONS IN NUMERATION GUY BARAT, VALERIE´ BERTHE,´ PIERRE LIARDET, AND JORG¨ THUSWALDNER Abstract. This survey aims at giving a consistent presentation of numer- ation from a dynamical viewpoint: we focus on numeration systems, their associated compactification, and dynamical systems that can be naturally defined on them. The exposition is unified by the fibred numeration system concept. Many examples are discussed. Various numerations on rational integers, real or complex numbers are presented with special attention paid to β-numeration and its generalisations, abstract numeration systems and shift radix systems, as well as G-scales and odometers. A section of appli- cations ends the paper. Le but de ce survol est d’aborder d´efinitions et propri´et´es concernant la num´eration d’un point de vue dynamique : nous nous concentrons sur les syst`emes de num´eration, leur compactification, et les syst`emes dynamiques d´efinis sur ces espaces. La notion de syst`eme de num´eration fibr´eunifie la pr´esentation. De nombreux exemples sont ´etudi´es. Plusieurs num´erations sur les entiers naturels, relatifs, les nombres r´eels ou complexes sont pr´esent´ees. Nous portons une attention sp´eciale `ala β-num´eration ainsi qu’`ases g´en´erali- sations, aux syst`emes de num´eration abstraits, aux syst`emes dits “shift radix”, de mˆeme qu’aux G-´echelles et aux odom`etres. Un paragraphe d’appli- cations conclut ce survol. Contents 1. Introduction 2 1.1. Origins 2 1.2. What this survey is (not) about 4 2. Fibred numeration systems 7 2.1. Numeration systems 7 2.2.
    [Show full text]
  • Glossary of Data , Processing and Communications Terms ."!
    GLOSSARY OF DATA , PROCESSING AND COMMUNICATIONS TERMS ."! ... FIRST EDITION First Printing, August, 1964 Second Printing, October, 1964 SECOND EDITION First Printing, January, 1965 Second Printing, April, 1965 THIRD EDITION First Printing, April, 1966 Honeywell Inc. Electronic Data Processing Division Wellesley Hills, Massachusetts 02181 Printed in U.S.A. When ordering this publication please specify 143.0000.0000.0·m WP·8784 title and underscored portion of file number. 10466 I I ,..- ~ { t ,,- GLOSSARY OF DATA , PROCESSING AND COMMUNICATIONS TERMS Honeywell ELECTRONIC DATA PROCESSING ! r FOREWORD A glossary is indispensable to both newcomers and veterans in the data processing field. The newcomer finds a glossary helpful in finding his bearings as quickly and as surely as possible. He marks his way with definitions, and relationships eventually become evident and whole patterns take form. Even veterans find that they have all too often been incorrect in their assumptions as to the meaning of a term. Or, they will feel the needofa glossary when precision is criticalin communi­ cating with others. Whichever the case, a glossary, tobe useful, mustbe comprehensive and its definitions lucid. We believe that the Honeywell Glossary is one of the most comprehensive data processing glossaries ever published by a computer manufacturer. In addition to containing the standard EDP terms, it keeps pace with the fast-rising data transmission field by the inclusion of several hundred communications terms that will soon be common data processing language. It is also one of the most current glossaries available, inasmuch as it includes hundreds of the latest ASA'I.< definitions. This book is largely based on definitions found in the Bureau of the Budget's Automatic Data Processing Glossary.
    [Show full text]
  • Reciprocal Divisor Method 2007
    1 The Reciprocal Divisor Method for Abbreviation of Multiplication Tables 0 9 1 8 2 7 3 6 4 5 2 This booklet is dedicated to my wife, Laura Ann. First Edition, 3 October 2007 This booklet and all its art are copyright 2007 by author Michael Thomas De Vlieger, all rights reserved. No presentation or reproduction of any portion of this booklet is to be made without express permission from the author. This first edition of “Reciprocal Divisor Method” is a draft of the work and is presented exclusively at the 6 October 2007 meeting of the Dozenal Society of America on Long Island, New York. Michael Thomas De Vlieger Vinci LLC 5750 Delor Street 5106 Hampton Avenue Suite 205 Saint Louis, Missouri 63109-2830 Saint Louis, Missouri 63109-3315 314-351-7456 [email protected] www.vincico.com 3 0 7 1 6 2 5 3 4 The Reciprocal Divisor Method for Abbreviation of Multiplication Tables w x 0 1 2 0 u v 3 4 t 5 9 1 s 6 r 7 q 8 8 2 p 9 o A 7 3 n B m C 6 4 l D 5 k E j F 0 i G Ç 1 h H ç 2 g I f J e K 9 3 d L c M 8 4 b N a O 7 5 Z P 6 Y X Q W V U T S R F 0 1 E 2 D 3 C 4 B 5 A 6 9 8 7 Michael De Vlieger • 3 October 2007 • Saint Louis, Missouri 4 Contents Contents .............................................................................................................................
    [Show full text]
  • SEL Glossary DP Tech Terms.Pdf
    GLOSSARY OF DA TA PROCESSING TECHNICAL TERMS s E L ■ V ■ TEMB ENGINEERING LABORATORIES, INCORPORATED P . 0. BOX 9148 • FORT LAUDERDALE. FLA. 33310 • AREA CODE 305 587-2900 INTRODUCTION The terms and expressions listed on the fol~owing pages were compiled from The Proposed American Standard Dictionary~:, . of American Standard Association Terminology, and other reliable sources. These terms are those whose individual mefnings are different in the information processing field than from their meanings in the general technical vocabulary. ,:, Copyright 1964, Commerce Clearing House, Inc. -A- ADDRESS PART A part of an inatruction word that specifies the address ABSOLUTE ADDRESS of an operand. ( 1) An a ddr,'ss th;it is p,•r manently assigned by the machine dPsig,wr to" stor,ig,• location. (2)A pattern of characters ADDRESS REGISTER that idt>ntifies a uniqun storage location without further modification. ·Synonymous with machine addresa. A register that stores an addr~ss. ABSOLUTE ERROR ADP. The value of the error without regard to its algebraic sign. Automatic data processing per~aining to automatic data processing equipment such as EAM and EDP equipment. ACCESS TIME ALGOL. ( 1) The time interval between the instant at which data are called for from a storage device and the instant deliverr Algorithmec oriented language. An international pro­ is completed, i.e., the read time. (2) The time interval cedure -oriented language, between the instant at which data are to be stored and the instant at which storage is completed, i.e. the write time. ALGORITHM ACCUMULATOR A prescribed set of well defined rules or processes for the solution of a problem in a finite number of steps,e.g., A register in which the result of an arithmetic or logic a full statement of an arithmetic procedure for evaluating operation is formed.
    [Show full text]
  • Proposal for the Universal Unit System
    Proposal for the Universal Unit System Takashi Suga, Ph.D. May 1X;(22.), 1203;(2019.) 1 Abstract We define the Universal Unit System based on the combination of fundamental physical constants and propose the Harmonic System as its practical implementation.2 1. The Universal Unit System 1.1. Before the Universal Unit System A unit of measure is “a quantity that is used as the basis for expressing a given quantity and is of the same type as the quantity that is to be expressed”. A unit that is used in exchanges between people must be guaranteed to have a constant magnitude within the scope of that exchange. Quantities that can, by consensus, serve as common standards over a broad scope were sought and selected to serve as units. The ultimate example of such a quantity is an entity common to all of the humankind, the Earth itself, which was selected as the foundation for the metric system. 1.2. The next stage? Of course, we can consider going beyond the framework of the Earth and defining units with concepts for which agreement can be reached within a broader scope. The quantities that then become available to serve as the standards for defining units include the quantities of the fundamental physical constants, quantities such as the speed of light in a vacuum (c0), the quantum of action (ħ), the Boltzmann constant (kB), and so on. These quantities are believed to have values that remain constant everywhere in the universe. When trying to construct a coherent unit system, however, it is not possible to use all of the fundamental physical constants in the definitions of units.
    [Show full text]
  • Pre-Publication Accepted Manuscript
    Liam Solus Simplices for numeral systems Transactions of the American Mathematical Society DOI: 10.1090/tran/7424 Accepted Manuscript This is a preliminary PDF of the author-produced manuscript that has been peer-reviewed and accepted for publication. It has not been copyedited, proofread, or finalized by AMS Production staff. Once the accepted manuscript has been copyedited, proofread, and finalized by AMS Production staff, the article will be published in electronic form as a \Recently Published Article" before being placed in an issue. That electronically published article will become the Version of Record. This preliminary version is available to AMS members prior to publication of the Version of Record, and in limited cases it is also made accessible to everyone one year after the publication date of the Version of Record. The Version of Record is accessible to everyone five years after publication in an issue. This is a pre-publication version of this article, which may differ from the final published version. Copyright restrictions may apply. SIMPLICES FOR NUMERAL SYSTEMS LIAM SOLUS n Abstract. The family of lattice simplices in R formed by the convex hull of the standard basis vectors together with a weakly decreasing vector of negative integers include simplices that play a central role in problems in enumerative algebraic geometry and mirror symmetry. From this perspective, it is use- ful to have formulae for their discrete volumes via Ehrhart h∗-polynomials. Here we show, via an association with numeral systems, that such simplices yield h∗-polynomials with properties that are also desirable from a combinato- rial perspective.
    [Show full text]