An Elementary Proof of the Mean Inequalities
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Gpindex: Generalized Price and Quantity Indexes
Package ‘gpindex’ August 4, 2021 Title Generalized Price and Quantity Indexes Version 0.3.4 Description A small package for calculating lots of different price indexes, and by extension quan- tity indexes. Provides tools to build and work with any type of bilateral generalized-mean in- dex (of which most price indexes are), along with a few important indexes that don't be- long to the generalized-mean family. Implements and extends many of the meth- ods in Balk (2008, ISBN:978-1-107-40496-0) and ILO, IMF, OECD, Euro- stat, UN, and World Bank (2020, ISBN:978-1-51354-298-0) for bilateral price indexes. Depends R (>= 3.5) Imports stats, utils License MIT + file LICENSE Encoding UTF-8 URL https://github.com/marberts/gpindex LazyData true Collate 'helper_functions.R' 'means.R' 'weights.R' 'price_indexes.R' 'operators.R' 'utilities.R' NeedsCompilation no Author Steve Martin [aut, cre, cph] Maintainer Steve Martin <[email protected]> Repository CRAN Date/Publication 2021-08-04 06:10:06 UTC R topics documented: gpindex-package . .2 contributions . .3 generalized_mean . .8 lehmer_mean . 12 logarithmic_means . 15 nested_mean . 18 offset_prices . 20 1 2 gpindex-package operators . 22 outliers . 23 price_data . 25 price_index . 26 transform_weights . 33 Index 37 gpindex-package Generalized Price and Quantity Indexes Description A small package for calculating lots of different price indexes, and by extension quantity indexes. Provides tools to build and work with any type of bilateral generalized-mean index (of which most price indexes are), along with a few important indexes that don’t belong to the generalized-mean family. Implements and extends many of the methods in Balk (2008, ISBN:978-1-107-40496-0) and ILO, IMF, OECD, Eurostat, UN, and World Bank (2020, ISBN:978-1-51354-298-0) for bilateral price indexes. -
Excursions in Classical Analysis Pathways to Advanced Problem Solving and Undergraduate Research
i i “master” — 2011/5/9 — 10:51 — page i — #1 i i Excursions in Classical Analysis Pathways to Advanced Problem Solving and Undergraduate Research i i i i i i “master” — 2011/5/9 — 10:51 — page ii — #2 i i Chapter 11 (Generating Functions for Powers of Fibonacci Numbers) is a reworked version of material from my same name paper in the International Journalof Mathematical Education in Science and Tech- nology, Vol. 38:4 (2007) pp. 531–537. www.informaworld.com Chapter 12 (Identities for the Fibonacci Powers) is a reworked version of material from my same name paper in the International Journal of Mathematical Education in Science and Technology, Vol. 39:4 (2008) pp. 534–541. www.informaworld.com Chapter 13 (Bernoulli Numbers via Determinants)is a reworked ver- sion of material from my same name paper in the International Jour- nal of Mathematical Education in Science and Technology, Vol. 34:2 (2003) pp. 291–297. www.informaworld.com Chapter 19 (Parametric Differentiation and Integration) is a reworked version of material from my same name paper in the Interna- tional Journal of Mathematical Education in Science and Technology, Vol. 40:4 (2009) pp. 559–570. www.informaworld.com c 2010 by the Mathematical Association of America, Inc. Library of Congress Catalog Card Number 2010924991 Print ISBN 978-0-88385-768-7 Electronic ISBN 978-0-88385-935-3 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “master” — 2011/5/9 — 10:51 — page iii — #3 i i Excursions in Classical Analysis Pathways to Advanced Problem Solving and Undergraduate Research Hongwei Chen Christopher Newport University Published and Distributed by The Mathematical Association of America i i i i i i “master” — 2011/5/9 — 10:51 — page iv — #4 i i Committee on Books Frank Farris, Chair Classroom Resource Materials Editorial Board Gerald M. -
AHM As a Measure of Central Tendency of Sex Ratio
Biometrics & Biostatistics International Journal Research Article Open Access AHM as a measure of central tendency of sex ratio Abstract Volume 10 Issue 2 - 2021 In some recent studies, four formulations of average namely Arithmetic-Geometric Mean Dhritikesh Chakrabarty (abbreviated as AGM), Arithmetic-Harmonic Mean (abbreviated as AHM), Geometric- Department of Statistics, Handique Girls’ College, Gauhati Harmonic Mean (abbreviated as GHM) and Arithmetic-Geometric-Harmonic Mean University, India (abbreviated as AGHM) have recently been derived from the three Pythagorean means namely Arithmetic Mean (AM), Geometric Mean (GM) and Harmonic Mean (HM). Each Correspondence: Dhritikesh Chakrabarty, Department of of these four formulations has been found to be a measure of central tendency of data. in Statistics, Handique Girls’ College, Gauhati University, India, addition to the existing measures of central tendency namely AM, GM & HM. This paper Email focuses on the suitability of AHM as a measure of central tendency of numerical data of ratio type along with the evaluation of central tendency of sex ratio namely male-female ratio and female-male ratio of the states in India. Received: May 03, 2021 | Published: May 25, 2021 Keywords: AHM, sex ratio, central tendency, measure Introduction however, involve huge computational tasks. Moreover, these methods may not be able to yield the appropriate value of the parameter if Several research had already been done on developing definitions/ observed data used are of relatively small size (and/or of moderately 1,2 formulations of average, a basic concept used in developing most large size too) In reality, of course, the appropriate value of the 3 of the measures used in analysis of data. -
Stabilizability of the Stolarsky Mean and Its Approximation in Terms of the Power Binomial Mean
Int. Journal of Math. Analysis, Vol. 6, 2012, no. 18, 871 - 881 Stabilizability of the Stolarsky Mean and its Approximation in Terms of the Power Binomial Mean Mustapha Ra¨ıssouli Taibah University, Faculty of Science, Department of Mathematics Al Madinah Al Munawwarah, P.O. Box 30097, Zip Code 41477 Saudi Arabia raissouli [email protected] Abstract We prove that the Stolarsky mean Ep,q of order (p, q)is(Bq−p,Bp)- stabilizable, where Bp denotes the power binomial mean. This allows us to approximate the nonstable Stolarsky mean by an iterative algorithm involving the stable power binomial mean. Mathematics Subject Classification: 26E60 Keywords: Means, Stable means, Stabilizable means, Stolarsky mean, Convergence of iterative algorithms 1. Introduction Throughout this paper, we understand by binary mean a map m between two positive real numbers satisfying the following statements. (i) m(a, a)=a, for all a>0; (ii) m(a, b)=m(b, a), for all a, b > 0; (iii) m(ta, tb)=tm(a, b), for all a, b, t > 0; (iv) m(a, b) is an increasing function in a (and in b); (v) m(a, b) is a continuous function of a and b. A binary mean is also called mean with two variables. Henceforth, we shortly call mean instead of binary mean. The definition of mean with three or more variables can be stated in a similar manner. The set of all (binary) means can be equipped with a partial ordering, called point-wise order, defined by: m1 ≤ m2 if and only if m1(a, b) ≤ m2(a, b) for every a, b > 0. -
Pappus of Alexandria: Book 4 of the Collection
Pappus of Alexandria: Book 4 of the Collection For other titles published in this series, go to http://www.springer.com/series/4142 Sources and Studies in the History of Mathematics and Physical Sciences Managing Editor J.Z. Buchwald Associate Editors J.L. Berggren and J. Lützen Advisory Board C. Fraser, T. Sauer, A. Shapiro Pappus of Alexandria: Book 4 of the Collection Edited With Translation and Commentary by Heike Sefrin-Weis Heike Sefrin-Weis Department of Philosophy University of South Carolina Columbia SC USA [email protected] Sources Managing Editor: Jed Z. Buchwald California Institute of Technology Division of the Humanities and Social Sciences MC 101–40 Pasadena, CA 91125 USA Associate Editors: J.L. Berggren Jesper Lützen Simon Fraser University University of Copenhagen Department of Mathematics Institute of Mathematics University Drive 8888 Universitetsparken 5 V5A 1S6 Burnaby, BC 2100 Koebenhaven Canada Denmark ISBN 978-1-84996-004-5 e-ISBN 978-1-84996-005-2 DOI 10.1007/978-1-84996-005-2 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2009942260 Mathematics Classification Number (2010) 00A05, 00A30, 03A05, 01A05, 01A20, 01A85, 03-03, 51-03 and 97-03 © Springer-Verlag London Limited 2010 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. -
The Pythagorean Theorem Crown Jewel of Mathematics
The Pythagorean Theorem Crown Jewel of Mathematics 5 3 4 John C. Sparks The Pythagorean Theorem Crown Jewel of Mathematics By John C. Sparks The Pythagorean Theorem Crown Jewel of Mathematics Copyright © 2008 John C. Sparks All rights reserved. No part of this book may be reproduced in any form—except for the inclusion of brief quotations in a review—without permission in writing from the author or publisher. Front cover, Pythagorean Dreams, a composite mosaic of historical Pythagorean proofs. Back cover photo by Curtis Sparks ISBN: XXXXXXXXX First Published by Author House XXXXX Library of Congress Control Number XXXXXXXX Published by AuthorHouse 1663 Liberty Drive, Suite 200 Bloomington, Indiana 47403 (800)839-8640 www.authorhouse.com Produced by Sparrow-Hawke †reasures Xenia, Ohio 45385 Printed in the United States of America 2 Dedication I would like to dedicate The Pythagorean Theorem to: Carolyn Sparks, my wife, best friend, and life partner for 40 years; our two grown sons, Robert and Curtis; My father, Roscoe C. Sparks (1910-1994). From Earth with Love Do you remember, as do I, When Neil walked, as so did we, On a calm and sun-lit sea One July, Tranquillity, Filled with dreams and futures? For in that month of long ago, Lofty visions raptured all Moonstruck with that starry call From life beyond this earthen ball... Not wedded to its surface. But marriage is of dust to dust Where seasoned limbs reclaim the ground Though passing thoughts still fly around Supernal realms never found On the planet of our birth. And I, a man, love you true, Love as God had made it so, Not angel rust when then aglow, But coupled here, now rib to soul, Dear Carolyn of mine. -
ΜΕΣΌΤΗΣ in PLATO and ARISTOTLE1 Roberto Grasso
ΜΕΣΌΤΗΣ IN PLATO AND ARISTOTLE1 Roberto Grasso Universidade Estadual de Campinas Resumo: Neste artigo, proponho uma revisão da lexicografia comumemente aceita sobre o termo mesotēs, restrita ao uso da palavra por Platão e Aristóteles. Nas obras dos dois filósofos, mesotēs nunca indica simplesmente “algo que está no meio”, e, em vez disso, sugere algo que medeia entre dois extremos com base em uma razão bem determinada, estabelecendo uma relação parecida com a de ‘analogia’ (ἀναλογία). Particular atenção é dada a algumas ocorrências controversas da palavra mesotēs em Aristóteles que estão ligadas às doutrinas éticas e perceptivas do médio, expostas no livro II da Ética Nicomachea e em De Anima II.12. Nessas passagens, a habitual suposição de que mesotēs deve indicar um estado intermediário faz com que os argumentos de Aristóteles sejam controversos, se não incoerentes. Ao destacar essas ocorrências problemáticas de ‘mesotēs’, o artigo pretende ser aporético: seu objetivo não consiste na elaboração de uma solução, mas sim na individuação das limitações da nossa atual compreensão dessa importante noção em Platão e Aristóteles. No entanto, no que diz respeito à Ética Nicomachea (1106b27-28) sugere-se que a conjectura de que mesotēs pode se referir à atividade de ‘encontrar a média’ pode, pelo menos, salvar o argumento da acusação de ser um non-sequitur. Palavras-chave: Platão, Aristóteles, meio, ética, percepção. Abstract: I propose a revision of the received lexicography of μεσότης with regard to Plato’s and Aristotle’s use of the word. In their works, μεσότης never indicates something that merely ‘lies in the middle’, and rather hints at what establishes a reason-grounded, ἀναλογία-like relationship between two extremes. -
Logarithmic Mean for Several Arguments
LOGARITHMIC MEAN FOR SEVERAL ARGUMENTS SEPPO MUSTONEN Abstract. The logarithmic mean is generalized for n positive arguments x1,...,xn by examining series expansions of typical mean numbers in case n = 2. The generalized logarithmic mean defined as a series expansion can then be presented also in closed form which proves to be the (n−1)th divided differ- u ence (multiplied by (n−1)!) of values f(u1),...,f(un) where f(ui)=e i = xi, i =1,...,n. Various properties of this generalization are studied and an effi- cient recursive algorithm for computing it is presented. 1. Introduction Some statisticians and mathematicians have proposed generalizations of the log- arithmic mean for n arguments (n>2), see E.L.Dodd [3] and A.O.Pittenger [11]. The generalization presented in this paper differs from the earlier suggestions and has its origin in an unpublished manuscript of the author [6]. This manuscript based on a research made in early 70’s is referred to in the paper of L.T¨ornqvist, P.Vartia, Y.O.Vartia [13]. It essentially described a generalization in cases n =3, 4 and provided a suggestion for a general form which will be derived in this paper. The logarithmic mean L(x1,x2) for two arguments x1 > 0, x2 > 0 is defined by x1 − x2 (1) L(x1,x2)= for x1 =6 x2 and L(x1,x1)=x1. log (x1/x2) Obviously Leo T¨ornqvistwas the first to advance the ”log-mean” concept in his fundamental research work related to price indexes [12]. Yrj¨oVartia then imple- mented the logarithmic mean in his log-change index numbers [14]. -
Janusz Matkowski GENERALIZATIONS OF
DEMONSTRATIO MATHEMATICA Vol. XLIII No 4 2010 Janusz Matkowski GENERALIZATIONS OF LAGRANGE AND CAUCHY MEAN-VALUE THEOREMS Abstract. Some generalizations of the Lagrange Mean-Value Theorem and Cauchy Mean-Value Theorem are proved and the extensions of the corresponding classes of means are presented. 1. Introduction Recall that a function M : I2 —> I is called a mean in a nontrivial interval / C R I if it is internal, that is if min(x,y) < M(x,y) < max(i,y) for all x,y € I. The mean M is called strict if these inequalities are strict for all x,y € /, x ^ y, and symmetric if M(x,y) = M(y,x) for all x,y € I. The Lagrange Mean-Value Theorem can be formulated in the following way. If a function f : I —> R is differentiable, then there exists a strict symmetric mean L : I2 —> I such that, for all x,y € /,x ^ y, (i) Mzx-y M = meii)). If f is one-to-one then, obviously, L^ := L is uniquely determined and is called a Lagrange mean generated by /. Note that formula (1) can be written in the form x x iim-H -¥), n -¥)-m\ fl(T( „ cT , 21 + £±g-t, )=/(L(^))' x,y€l,x^y, \ x 2 2 y ' 2000 Mathematics Subject Classification: Primary 26A24. Key words and phrases: mean, mean-value theorem, Lagrange theorem, Cauchy the- orem, generalization, Lagrange mean, Cauchy mean, logrithmic mean, Stolarsky mean. 766 J. Matkowski or, setting A(x,y) := in the form f(x)-f(A(x,y)) f(A(x,y)) - f(y)\ (2) A = f'(L(x,y)), x — A(x, y) ' A(x, y) — y ) x,y€ I, x^y, which shows a relationship of the mean-value theorem and the arithmetic mean. -
New Variants of Newton's Method for Nonlinear Unconstrained
Intelligent Information Management, 2010, 2, 40-45 doi:10.4236/iim.2010.21005 Published Online January 2010 (http://www.scirp.org/journal/iim) New Variants of Newton’s Method for Nonlinear Unconstrained Optimization Problems V. K A N WA R 1, Kapil K. SHARMA2, Ramandeep BEHL2 1University Institute of Engineering and Technology, Panjab University, Chandigarh160 014, India 2Department of Mathematics, Panjab University, Chandigarh160 014, India Email: [email protected], [email protected], [email protected] Abstract In this paper, we propose new variants of Newton’s method based on quadrature formula and power mean for solving nonlinear unconstrained optimization problems. It is proved that the order of convergence of the proposed family is three. Numerical comparisons are made to show the performance of the presented meth- ods. Furthermore, numerical experiments demonstrate that the logarithmic mean Newton’s method outper- form the classical Newton’s and other variants of Newton’s method. MSC: 65H05. Keywords: Unconstrained Optimization, Newton’s Method, Order of Convergence, Power Means, Initial Guess 1. Introduction the one dimensional methods are most indispensable and the efficiency of any method partly depends on them The celebrated Newton’s method [10]. The purpose of this work is to provide some alterna- f xn (1) tive derivations through power mean and to revisit some xxnn1 f xn well-known methods for solving nonlinear unconstrained used to approximate the optimum of a function is one of optimization problems. the most fundamental tools in computational mathemat- ics, operation research, optimization and control theory. 2. Review of Definition of Various Means It has many applications in management science, indus- trial and financial research, chaos and fractals, dynamical For a given finite real number , the th power systems, stability analysis, variational inequalities and mean m of positive scalars a and b , is defined as even to equilibrium type problems. -
Pappus of Alexandria: Book 4 of the Collection
Pappus of Alexandria: Book 4 of the Collection For other titles published in this series, go to http://www.springer.com/series/4142 Sources and Studies in the History of Mathematics and Physical Sciences Managing Editor J.Z. Buchwald Associate Editors J.L. Berggren and J. Lützen Advisory Board C. Fraser, T. Sauer, A. Shapiro Pappus of Alexandria: Book 4 of the Collection Edited With Translation and Commentary by Heike Sefrin-Weis Heike Sefrin-Weis Department of Philosophy University of South Carolina Columbia SC USA [email protected] Sources Managing Editor: Jed Z. Buchwald California Institute of Technology Division of the Humanities and Social Sciences MC 101–40 Pasadena, CA 91125 USA Associate Editors: J.L. Berggren Jesper Lützen Simon Fraser University University of Copenhagen Department of Mathematics Institute of Mathematics University Drive 8888 Universitetsparken 5 V5A 1S6 Burnaby, BC 2100 Koebenhaven Canada Denmark ISBN 978-1-84996-004-5 e-ISBN 978-1-84996-005-2 DOI 10.1007/978-1-84996-005-2 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2009942260 Mathematics Classification Number (2010) 00A05, 00A30, 03A05, 01A05, 01A20, 01A85, 03-03, 51-03 and 97-03 © Springer-Verlag London Limited 2010 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. -
A Note on Stolarsky, Arithmetic and Logarithmic Means
A note on Stolarsky, arithmetic and logarithmic means Vania Mascioni Department of Mathematical Sciences Ball State University Muncie, IN 47306-0490, USA [email protected] Subject Classifications: Primary: 26D20, Secondary: 26E60 Keywords: Stolarsky mean, arithmetic mean, logarithmic mean Abstract We present a way to study differences of some Stolarsky means as a way to discover new inequalities, or place known inequalities in a wider context. In particular, as an application we prove a very sharp upper bound for the difference between the arithmetic and the logarithmic means of two positive numbers. 1 Introduction and main results Alomari, Darus and Dragomir [1], as an application of Hadamard-Hermite type inequalities, give some estimates for the difference between some generalized logarithmic means and power means (see Propositions 1,2, and 3 in [1]). For example, they prove bn+1 − an+1 an + bn n(n − 1) − ≤ (b − a)2 maxfjajn−2; jbjn−2g; (n + 1)(b − a) 2 12 where a < b are real and n ≥ 2 is an integer. This inequality is very sharp, though the authors do not specify if constants such as 12 are best possible. In our Theorem 2 below (see part (c) in reference to the example we just quoted) we extend these inequalities to the full range of powers p 2 R, thus showing interesting reversals and changes that occur when p falls in certain intervals, besides putting this kind of result in a broader, perhaps more natural context (our proofs also show that the inequalities given here are optimal within their natural category). The broader context we suggest is the one provided by the large family of Stolarsky means (or difference means, as they are sometimes called | see 1 below for the definitions).