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Characterizations of Several Split Regular Functions on Split Quaternion in Clifford Analysis
East Asian Math. J. Vol. 33 (2017), No. 3, pp. 309{315 http://dx.doi.org/10.7858/eamj.2017.023 CHARACTERIZATIONS OF SEVERAL SPLIT REGULAR FUNCTIONS ON SPLIT QUATERNION IN CLIFFORD ANALYSIS Han Ul Kang, Jeong Young Cho, and Kwang Ho Shon* Abstract. In this paper, we investigate the regularities of the hyper- complex valued functions of the split quaternion variables. We define several differential operators for the split qunaternionic function. We re- search several left split regular functions for each differential operators. We also investigate split harmonic functions. And we find the correspond- ing Cauchy-Riemann system and the corresponding Cauchy theorem for each regular functions on the split quaternion field. 1. Introduction The non-commutative four dimensional real field of the hypercomplex num- bers with some properties is called a split quaternion (skew) field S. Naser [12] described the notation and the properties of regular functions by using the differential operator D in the hypercomplex number system. And Naser [12] investigated conjugate harmonic functions of quaternion variables. In 2011, Koriyama et al. [9] researched properties of regular functions in quater- nion field. In 2013, Jung et al. [1] have studied the hyperholomorphic functions of dual quaternion variables. And Jung and Shon [2] have shown hyperholo- morphy of hypercomplex functions on dual ternary number system. Kim et al. [8] have investigated regularities of ternary number valued functions in Clif- ford analysis. Kang and Shon [3] have developed several differential operators for quaternionic functions. And Kang et al. [4] researched some properties of quaternionic regular functions. Kim and Shon [5, 6] obtained properties of hyperholomorphic functions and hypermeromorphic functions in each hyper- compelx number system. -
Math/CS 467 (Lesieutre) Homework 2 September 9, 2020 Submission
Math/CS 467 (Lesieutre) Homework 2 September 9, 2020 Submission instructions: Upload your solutions, minus the code, to Gradescope, like you did last week. Email your code to me as an attachment at [email protected], including the string HW2 in the subject line so I can filter them. Problem 1. Both prime numbers and perfect squares get more and more uncommon among larger and larger numbers. But just how uncommon are they? a) Roughly how many perfect squares are there less than or equal to N? p 2 For a positivep integer k, notice that k is less than N if and only if k ≤ N. So there are about N possibilities for k. b) Are there likely to be more prime numbers or perfect squares less than 10100? Give an estimate of the number of each. p There are about 1010 = 1050 perfect squares. According to the prime number theorem, there are about 10100 10100 π(10100) ≈ = = 1098 · log 10 = 2:3 · 1098: log(10100) 100 log 10 That’s a lot more primes than squares. (Note that the “log” in the prime number theorem is the natural log. If you use log10, you won’t get the right answer.) Problem 2. Compute g = gcd(1661; 231). Find integers a and b so that 1661a + 231b = g. (You can do this by hand or on the computer; either submit the code or show your work.) We do this using the Euclidean algorithm. At each step, we keep track of how to write ri as a combination of a and b. -
The Enigmatic Number E: a History in Verse and Its Uses in the Mathematics Classroom
To appear in MAA Loci: Convergence The Enigmatic Number e: A History in Verse and Its Uses in the Mathematics Classroom Sarah Glaz Department of Mathematics University of Connecticut Storrs, CT 06269 [email protected] Introduction In this article we present a history of e in verse—an annotated poem: The Enigmatic Number e . The annotation consists of hyperlinks leading to biographies of the mathematicians appearing in the poem, and to explanations of the mathematical notions and ideas presented in the poem. The intention is to celebrate the history of this venerable number in verse, and to put the mathematical ideas connected with it in historical and artistic context. The poem may also be used by educators in any mathematics course in which the number e appears, and those are as varied as e's multifaceted history. The sections following the poem provide suggestions and resources for the use of the poem as a pedagogical tool in a variety of mathematics courses. They also place these suggestions in the context of other efforts made by educators in this direction by briefly outlining the uses of historical mathematical poems for teaching mathematics at high-school and college level. Historical Background The number e is a newcomer to the mathematical pantheon of numbers denoted by letters: it made several indirect appearances in the 17 th and 18 th centuries, and acquired its letter designation only in 1731. Our history of e starts with John Napier (1550-1617) who defined logarithms through a process called dynamical analogy [1]. Napier aimed to simplify multiplication (and in the same time also simplify division and exponentiation), by finding a model which transforms multiplication into addition. -
An Amazing Prime Heuristic.Pdf
This document has been moved to https://arxiv.org/abs/2103.04483 Please use that version instead. AN AMAZING PRIME HEURISTIC CHRIS K. CALDWELL 1. Introduction The record for the largest known twin prime is constantly changing. For example, in October of 2000, David Underbakke found the record primes: 83475759 264955 1: · The very next day Giovanni La Barbera found the new record primes: 1693965 266443 1: · The fact that the size of these records are close is no coincidence! Before we seek a record like this, we usually try to estimate how long the search might take, and use this information to determine our search parameters. To do this we need to know how common twin primes are. It has been conjectured that the number of twin primes less than or equal to N is asymptotic to N dx 2C2N 2C2 2 2 Z2 (log x) ∼ (log N) where C2, called the twin prime constant, is approximately 0:6601618. Using this we can estimate how many numbers we will need to try before we find a prime. In the case of Underbakke and La Barbera, they were both using the same sieving software (NewPGen1 by Paul Jobling) and the same primality proving software (Proth.exe2 by Yves Gallot) on similar hardware{so of course they choose similar ranges to search. But where does this conjecture come from? In this chapter we will discuss a general method to form conjectures similar to the twin prime conjecture above. We will then apply it to a number of different forms of primes such as Sophie Germain primes, primes in arithmetic progressions, primorial primes and even the Goldbach conjecture. -
6Th Online Learning #2 MATH Subject: Mathematics State: Ohio
6th Online Learning #2 MATH Subject: Mathematics State: Ohio Student Name: Teacher Name: School Name: 1 Yari was doing the long division problem shown below. When she finishes, her teacher tells her she made a mistake. Find Yari's mistake. Explain it to her using at least 2 complete sentences. Then, re-do the long division problem correctly. 2 Use the computation shown below to find the products. Explain how you found your answers. (a) 189 × 16 (b) 80 × 16 (c) 9 × 16 3 Solve. 34,992 ÷ 81 = ? 4 The total amount of money collected by a store for sweatshirt sales was $10,000. Each sweatshirt sold for $40. What was the total number of sweatshirts sold by the store? (A) 100 (B) 220 (C) 250 (D) 400 5 Justin divided 403 by a number and got a quotient of 26 with a remainder of 13. What was the number Justin divided by? (A) 13 (B) 14 (C) 15 (D) 16 6 What is the quotient of 13,632 ÷ 48? (A) 262 R36 (B) 272 (C) 284 (D) 325 R32 7 What is the result when 75,069 is divided by 45? 8 What is the value of 63,106 ÷ 72? Write your answer below. 9 Divide. 21,900 ÷ 25 Write the exact quotient below. 10 The manager of a bookstore ordered 480 copies of a book. The manager paid a total of $7,440 for the books. The books arrived at the store in 5 cartons. Each carton contained the same number of books. A worker unpacked books at a rate of 48 books every 2 minutes. -
Lara Mantovan Nominal Modification in Italian Sign Language Sign Languages and Deaf Communities
Lara Mantovan Nominal Modification in Italian Sign Language Sign Languages and Deaf Communities Editors Annika Herrmann, Markus Steinbach, Ulrike Zeshan Editorial board Carlo Geraci, Rachel McKee, Victoria Nyst, Sibaji Panda, Marianne Rossi Stumpf, Felix Sze, Sandra Wood Volume 8 Lara Mantovan Nominal Modification in Italian Sign Language ISHARA PRESS ISBN 978-1-5015-1343-5 e-ISBN (PDF) 978-1-5015-0485-3 e-ISBN (EPUB) 978-1-5015-0481-5 ISSN 2192-516X e-ISSN 2192-5178 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2017 Walter de Gruyter Inc., Boston/Berlin and Ishara Press, Preston, UK Printing and binding: CPI books GmbH, Leck ♾ Printed on acid-free paper Printed in Germany www.degruyter.com E Pluribus Unum (uncertain origin, attributed to Virgilio, Moretum, v. 103) Acknowledgements This book is a revised version of my 2015 dissertation which was approved for the PhD degree in Linguistics at Ca’ Foscari University of Venice. When I first plunged into the world of academic research, almost five years ago, I would never have imagined it was possible to achieve such an important milestone. Being so close to finalizing this book, I would like to look back briefly and remember and thank all the people who showed me the way, supported me, and encouraged me to grow both academically and personally. -
Euler Number in Vedic Physics
The Purpose of the Euler Number in Vedic Physics By John Frederic Sweeney Abstract The Euler Number, or e logarithm, arises naturally from our combinatorial universe, the result of the interactive combination of three distinct states of matter. This paper provides the details about how this combinatorial process takes place and why the range of visible matter extends from Pi to the Euler Number. 1 Table of Contents Introduction 3 Wikipedia on the Euler Number 5 Vedic Physics on the Euler Number 6 Conclusion 8 Bibliography 9 2 Introduction The author posted a few papers on Vixra during 2013 which made statements about Pi and the Euler Number that lacked support. This paper offers the support for those statements. Since the explanation is mired, twisted and difficult to follow, the author wished to make other aspects of Vedic Particle Physics clear. This paper begins with Wikipedia on Euler, then to the Vedic Explanation. This paper, along with Why Pi? Which explains the reason why Pi carries such importance in a combinatorial world, should make it easier for readers to grasp the fundamental concepts of Vedic Particle Physics. Essentially, three states of matter exist in a combinatorial world, the axes of the states run from Pi to the Euler number. The simple formula of n + 1 suggests Pascal’s Triangle or Mount Meru, and the author has published a paper on this theme about Clifford Algebras, suggesting them as a most useful tool in a combinatorial universe. Someone once said that only those with clear understanding can explain things clearly and simply to others. -
On Hypercomplex Number Systems*
ON HYPERCOMPLEX NUMBER SYSTEMS* (FIRST PAPER) BY HENRY TABER Introduction. The method invented by Benjamin Peirce t for treating the problem to determine all hypereomplex number systems (or algebras) in a given number of units depends chiefly, first, upon the classification of hypereomplex number sys- tems into idempotent number systems, containing one or more idempotent num- bers, \ and non-idempotent number systems containing no idempotent number ; and, second, upon the regularizaron of idempotent number systems, that is, the classification of each of the units of such a system with respect to one of the idempotent numbers of the system. For the purpose of such classification and regularizaron the following theorems are required : Theorem I.§ In any given hypereomplex number system there is an idem- potent number (that is, a number I + 0 such that I2 = I), or every number of the system is nilpotent. || Theorem H.*rj In any hypereomplex number system containing an idem- potent number I, the units ex, e2, ■■ -, en can be so selected that, with reference ♦Presented to the Society February 27, 1904. Received for publication February 27, 1904, and September 6, 1904. t American Journal of Mathematics, vol. 4 (1881), p. 97. This work, entitled Linear Associative Algebra, was published in lithograph in 1870. For an estimate of Peirce's work and for its relation to the work of Study, Scheffers, and others, see articles by H. E. Hawkes in the American Journal of Mathematics, vol. 24 (1902), p. 87, and these Transactions, vol. 3 (1902), p. 312. The latter paper is referred to below when reference is made to Hawkes' work. -
Microfilmed 1996 Information to Users
UMI MICROFILMED 1996 INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 NonhZeeb Road, Ann Arbor MI 48106-1346 USA 313/761-4700 800/521-0600 Interpoint Distance Methods for the Analysis of High Dimensional Data DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Graduate School of The Ohio State University By John Lawrence, B. -
On the Normality of Numbers
ON THE NORMALITY OF NUMBERS Adrian Belshaw B. Sc., University of British Columbia, 1973 M. A., Princeton University, 1976 A THESIS SUBMITTED 'IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Mathematics @ Adrian Belshaw 2005 SIMON FRASER UNIVERSITY Fall 2005 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. APPROVAL Name: Adrian Belshaw Degree: Master of Science Title of Thesis: On the Normality of Numbers Examining Committee: Dr. Ladislav Stacho Chair Dr. Peter Borwein Senior Supervisor Professor of Mathematics Simon Fraser University Dr. Stephen Choi Supervisor Assistant Professor of Mathematics Simon Fraser University Dr. Jason Bell Internal Examiner Assistant Professor of Mathematics Simon Fraser University Date Approved: December 5. 2005 SIMON FRASER ' u~~~snrllbrary DECLARATION OF PARTIAL COPYRIGHT LICENCE The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. The author has further granted permission to Simon Fraser University to keep or make a digital copy for use in its circulating collection, and, without changing the content, to translate the thesislproject or extended essays, if technically possible, to any medium or format for the purpose of preservation of the digital work. -
On Moduli for Which the Fibonacci Sequence Contains a Complete System of Residues S
ON MODULI FOR WHICH THE FIBONACCI SEQUENCE CONTAINS A COMPLETE SYSTEM OF RESIDUES S. A. BURR Belt Telephone Laboratories, Inc., Whippany, New Jersey Shah [1] and Bruckner [2] have considered the problem of determining which moduli m have the property that the Fibonacci sequence {u }, de- fined in the usual way9 contains a complete system of residues modulo m, Following Shah we say that m is defective if m does not have this property, The results proved in [1] includes (I) If m is defective, so is any multiple of m; in particular, 8n is always defective. (II) if p is a prime not 2 or 5, p is defective unless p = 3 or 7 (mod 20). (in) If p is a prime = 3 or 7 (mod 20) and is not defective^ thenthe set {0, ±1, =tu3, ±u49 ±u5, • • • , ±u }9 where h = (p + l)/2 ? is a complete system of residues modulo p„ In [2], Bruckner settles the case of prime moduli by showing that all primes are defective except 2, 3S 5, and 7. In this paper we complete the work of Shah and Bruckner by proving the following re suit f which completely characterizes all defective and nondefective moduli. Theorem. A number m is not defective if and only if m has one of the following forms: k k k 5 , 2 * 5 f 4»5 , 3 k k 3 *5 5 6«5 , k k 7-5 , 14* 5* , where k ^ 0S j — 1. Thus almost all numbers are defective. We will prove a series of lem- mas, from which the theorem will follow directly, We first make some use- ful definitions. -
Multistage Hybrid Arabic/Indian Numeral OCR System
(IJCSIS) International Journal of Computer Science and Information Security, Vol. 8, No. 1, 2010 Multistage Hybrid Arabic/Indian Numeral OCR System Yasser M. Alginaih, Ph.D., P.Eng. IEEE Member Abdul Ahad Siddiqi, Ph.D., Member IEEE & PEC Dept. of Computer Science Dept. of Computer Science Taibah University Taibah University Madinah, Kingdom of Saudi Arabia Madinah, Kingdom of Saudi Arabia [email protected] [email protected] Abstract— The use of OCR in postal services is not yet numeral OCR systems for Postal services have been used universal and there are still many countries that process in some countries, but still there are problems in such mail sorting manually. Automated Arabic/Indian numeral systems, stemming from the fact that machines are unable Optical Character Recognition (OCR) systems for Postal to read the crucial information needed to distribute the services are being used in some countries, but still there are mail efficiently. Historically, most civilizations have errors during the mail sorting process, thus causing a different symbols that convey numerical values, but the reduction in efficiency. The need to investigate fast and Arabic version is the simplest and most widely efficient recognition algorithms/systems is important so as to correctly read the postal codes from mail addresses and to acceptable. In most Middle Eastern countries both the eliminate any errors during the mail sorting stage. The Arabic (0,1,2,3,4,5,6,7,8,9) and Indian ۷,۸,۹) numerals are used. The objective of,٤,٥,٦,objective of this study is to recognize printed numerical (۰,۱,۲,۳ postal codes from mail addresses.