Sum to Product and Product to Sum Formulas

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1 Customer satisfaction: review of literature and application to the product-service systems Final report to the Society for Non-Traditional Technology, Japan Oksana Mont Andrius Plepys Research Associates International Institute for Industrial Environmental Economics at Lund University <http://www.iiiee.lu.se/> P. O. Box 196 Tegnersplatsen 4 SE- 221 00 Lund Sweden Phone: +46 46 222 0200 Fax: +46 46 222 0230 oksana.mont@iiiee.lu.se andrius.plepys@iiiee.lu.se Lund, February 28 2003 2 Acknowledgments The authors would like to thank the National Institute for Advanced Industrial Science and Technology in Japan and the Ministry of Economy, Trade and Industry (METI) of Japan for financially supporting this study and for useful comments on the drafts. We would like to thank our supervisor, Prof. Thomas Lindhqvist for valuable guidance and challenging comments. 3 Executive summary This feasibility study commissioned by the National Institute for Advanced Industrial Science and Technology in Japan (AIST) and supported by the Sustainable Consumption Unit (UNEP) provided an overview of approaches used in different disciplines for evaluating consumer behaviour. The study analysed the applicability of existing research concepts, theories, and tools for evaluating consumer satisfaction with product-service systems (PSS). It included a discussion of their strengths/weaknesses. BACKGROUND It has been recognised that eco-efficiency improvements at production and product design level can be significantly reduced or totally negated by rebound effect from increased consumption levels. In line with this problem factor 10 to 20 material and energy efficiency improvements have been suggested (Factor 10 Club 1994; Schmidt-Bleek 1996; Bolund, Johansson et al. 1998; Ryan 1998). The improvements, however, if not carefully done, may still lead to rebound effects through changes in resource prices. As a potential solution to the factor 10/20 vision, system level improvements have to be made, contrary to redesigning individual products or processes (Weterings and Opschoor 1992; Vergragt and Jansen 1993; von Weizsäcker, Lovins et al. 1997; Ryan 1998; Manzini 1999; Brezet, Bijma et al. 2001; Ehrenfeld and Brezet 2001). The product service system (PSS) concept has been suggested as a way to contribute to this system level improvement (Goedkoop, van Halen et al. 1999; Mont 2000). Here the environmental impacts of products and associated services could be addressed already at the product and service design stage. Special focus should be given to the use phase by providing alternative system solutions to owning products. A number of examples in the business-to-business (B2B) area exist that confirm the potential of PSS for reducing life cycle environmental impact. It is, however, increasingly evident that business examples are difficult to directly apply to the private consumer market. Private consumers, contrary to businesses, prefer product ownership to service substitutes (Schrader 1996; Littig 1998). Even if accepted, the environmental impacts of “servicised products” offers depend to a large extent on consumer behaviour. To address this problem, either behavioural or service system design changes are needed. Changing human behaviour and existing lifestyles contributes to the vision of sustainable development, but at the same time, it is an extremely difficult and time-consuming process. A Sum-to-Product and Product-to-Sum Formulas Sum-to-Product and Product-to-Sum Formulas By: OpenStaxCollege The UCLA marching band (credit: Eric Chan, Flickr) A band marches down the field creating an amazing sound that bolsters the crowd That sound travels as a wave that can be interpreted using trigonometric functions For example, [link] represents a sound wave for the musical note A In this section, we will 1/16 Sum-to-Product and Product-to-Sum Formulas investigate trigonometric identities that are the foundation of everyday phenomena such as sound waves Expressing Products as Sums We have already learned a number of formulas useful for expanding or simplifying trigonometric expressions, but sometimes we may need to express the product of cosine and sine as a sum We can use the product-to-sum formulas, which express products of trigonometric functions as sums Let’s investigate the cosine identity first and then the sine identity Expressing Products as Sums for Cosine We can derive the product-to-sum formula from the sum and difference identities for cosine If we add the two equations, we get: cos α cos β + sin α sin β = cos(α − β) + cos α cos β − sin α sin β = cos(α + β) cos α cos β = cos(α − β) + cos(α + β) Then, we divide by to isolate the product of cosines: cos α cos β = [cos(α − β) + cos(α + β)] How To Given a product of cosines, express as a sum Write the formula for the product of cosines Substitute the given angles into the formula Simplify Writing the Product as a Sum Using the Product-to-Sum Formula for Cosine 2/16 Sum-to-Product and Product-to-Sum Formulas Write the following product of cosines as a sum: cos ( 7x2 ) cos 3x2 We begin by writing the formula for the product of cosines: cos α cos β = [cos(α − β) + cos(α + β)] We can then substitute the given angles into the formula and simplify cos ( 7x2 )cos( 3x2 ) = (2)( 12 )[cos( 7x2 − 3x2 ) + cos( 7x2 + 3x2 )] 4x 10x = cos( ) + cos( [ 2 )] = cos 2x + cos 5x Try It Use the product-to-sum formula to write the product as a sum or difference: cos(2θ)cos(4θ) (cos6θ + cos2θ) Expressing the Product of Sine and Cosine as a Sum Next, we will derive the product-to-sum formula for sine and cosine from the sum and difference formulas for sine If we add the sum and difference identities, we get: sin(α + β) = sin α cos β + cos α sin β + sin(α − β) = sin α cos β − cos α sin β _ sin(α + β) + sin(α − β) = sin α cos β Then, we divide by to isolate the product of cosine and sine: sin(α + β) + sin(α − β)] 2[ Writing the Product as a Sum Containing only Sine or Cosine sin α cos β = Express the following product as a sum containing only sine or cosine and no products: sin(4θ)cos(2θ) 3/16 Sum-to-Product and Product-to-Sum Formulas Write the formula for the product of sine and cosine Then substitute the given values into the formula and simplify sin α cos β = sin(4θ)cos(2θ) = sin(α + β) + sin(α − β)] 2[ sin(4θ + 2θ) + sin(4θ − 2θ)] 2[ = sin(6θ) + sin(2θ)] 2[ Try It Use the product-to-sum formula to write the product as a sum: sin(x + y)cos(x − y) (sin2x + sin2y) Expressing Products of Sines in Terms of Cosine Expressing the product of sines in terms of cosine is also derived from the sum and difference identities for cosine In this case, we will first subtract the two cosine formulas: cos(α − β) = cos α cos β + sin α sin β cos(α + β) = − (cos α cos β − sin α sin β) − cos(α − β) − cos(α + β) = sin α sin β Then, we divide by to isolate the product of sines: sin α sin β = cos(α − β) − cos(α + β)] 2[ Similarly we could express the product of cosines in terms of sine or derive other product-to-sum formulas A General Note The Product-to-Sum Formulas The product-to-sum formulas are as follows: cos α cos β = [cos(α − β) + cos(α + β)] 4/16 Sum-to-Product and Product-to-Sum Formulas sin(α + β) + sin(α − β)] 2[ sin α sin β = [cos(α − β) − cos(α + β)] cos α sin β = [sin(α + β) − sin(α − β)] Express the Product as a Sum or Difference sin α cos β = Write cos(3θ) cos(5θ) as a sum or difference We have the product of cosines, so we begin by writing the related formula Then we substitute the given angles and simplify cos α cos β = [cos(α − β) + cos(α + β)] cos(3θ)cos(5θ) = [cos(3θ − 5θ) + cos(3θ + 5θ)] = [cos(2θ) + cos(8θ)] Use even-odd identity Try It Use the product-to-sum formula to evaluate cos 11π 12 cos π 12 − − √3 Expressing Sums as Products Some problems require the reverse of the process we just used The sum-to-product formulas allow us to express sums of sine or cosine as products These formulas can be derived from the product-to-sum identities For example, with a few substitutions, we u+v u−v can derive the sum-to-product identity for sine Let = α and = β Then, 5/16 Sum-to-Product and Product-to-Sum Formulas α+β= = u+v + u−v 2u =u α−β= = u+v − u−v 2v =v Thus, replacing α and β in the product-to-sum formula with the substitute expressions, we have sin α cos β = [sin(α + β) + sin(α − β)] ( ) ( ) u+v u−v sin( ... The Sensitivity of Bank Net Interest Margins and Profitability to Credit, Interest-Rate, and Term-Structure Shocks Across Bank Product Specializations Gerald Hanweck Professor of Finance School of Management George Mason University Fairfax, VA 22030 ghanweck@gmu.edu and Visiting Scholar Division of Insurance and Research FDIC Lisa Ryu Senior Financial Economist Division of Insurance and Research FDIC lryu@fdic.gov January 2005 Working Paper 2005-02 The authors wish to thank participants at the FDIC’s Analyst/Economists Conference, October 7–9, 2003, and at the Research Seminar at the School of Management, George Mason University, for helpful comments and suggestions. The authors would also like to thank Richard Austin, Mark Flannery, and FDIC Working Paper Series reviewers for their comments and suggestions. All errors and omissions remain the responsibility of the authors. The opinions expressed in this paper are those of the authors and do not necessarily reflect those of the FDIC or its staff. 1 The Sensitivity of Bank Net Interest Margins and Profitability to Credit, Interest-Rate, and Term-Structure Shocks Across Bank Product Specializations Abstract This paper presents a dynamic model of bank behavior that explains net interest margin changes for different groups of banks in response to credit, interest-rate, and term-structure shocks. Using quarterly data from 1986 to 2003, we find that banks with different product-line specializations and asset sizes respond in predictable yet fundamentally dissimilar ways to these shocks. Banks in most bank groups are sensitive in varying degrees to credit, interest-rate, and term-structure shocks. Large and more diversified banks seem to be less sensitive to interest-rate and term-structure shocks, but more sensitive to credit shocks. We also find that the composition of assets and liabilities, in terms of their repricing frequencies, helps amplify or moderate the effects of changes and volatility in short-term interest rates on bank net interest margins, depending on the direction of the repricing mismatch. We also analyze subsample periods that represent different legislative, regulatory, and economic environments and find that most banks continue to be sensitive to credit, interest-rate, and term-structure shocks. However, the sensitivity to term-structure shocks seems to have lessened over time for certain groups of banks, although the results are not universal. In addition, our results show that banks in general are not able to hedge fully against interest-rate volatility. The sensitivity of net interest margins to interest-rate volatility for different groups of banks varies across subsample periods; this varying sensitivity could reflect interest-rate regime shifts as well as the degree of hedging activities and market competition. Finally, by investigating the sensitivity of ROA to interest-rate and credit shocks, we have some evidence that banks of different specializations were able Annals of Mathematics The Erd˝os-Szemer´edi problem on sum set and product set By Mei-Chu Chang* Annals of Mathematics, 157 (2003), 939–957 The Erd˝os-Szemer´edi problem on sum set and product set By Mei-Chu Chang* Summary The basic theme of this paper is the fact that if A is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erd˝os-Szemer´edi [E-S]. (see also [El], [T], and [K-T] for related aspects.) Only much weaker results or very special cases of this conjecture are presently known. One approach consists of assuming the sum set A + A small and then deriving that the product set AA is large (using Freiman’s structure theorem) (cf. [N-T], [Na3]). We follow the reverse route and prove that if |AA| <c|A|, then |A + A| >c  |A| 2 (see Theorem 1). A quantitative version of this phenomenon combined with the Pl¨unnecke type of inequality (due to Ruzsa) permit us to settle completely a related conjecture in [E-S] on the growth in k.If g(k) ≡ min{|A[1]| + |A{1}|} over all sets A ⊂ of cardinality |A| = k and where A[1] (respectively, A{1}) refers to the simple sum (resp., product) of elements of A. (See (0.6), (0.7).) It was conjectured in [E-S] that g(k) grows faster than any power of k for k →∞. We will prove here that ln g(k) ∼ (ln k) 2 ln ln k (see Theorem 2) which is the main result of this paper. Introduction Let A, B be finite sets of an abelian group. The sum set of A, B is (0.1) A + B ≡{a + b | a ∈ A, b ∈ B}. We denote by (0.2) hA ≡ A + ···+ A (h fold) the h-fold sum of A. *Partially supported by NSA. 940 MEI-CHU CHANG Similarly we can define the product set of A, B and h-fold product of A. AB ≡{ab | a ∈ A, b ∈ B},(0.3) A h ≡ A ···A (h fold).(0.4) If B = {b},asingleton, we denote AB by b · A. In 1983, Erd˝os and Szemer´edi [E-S] conjectured that for subsets of integers, the sum set and the product set cannot both be small. Precisely, they made the following conjecture. Conjecture 1 (Erd˝os-Szemer´edi). For any ε>0 and any h ∈ there is k 0 = k 0 (ε) such that for any A ⊂ with |A|≥k 0 , (0.5) |hA ∪ A h ||A| h−ε . We note that there is an obvious upper bound |hA∪A h |≤2  |A| + h −1 h  . Another related conjecture requires the following notation of simple sum and simple product. A[1] ≡  k  i=1 ε i a i | a i ∈ A, ε i =0 or 1  ,(0.6) A{1}≡  k  i=1 a ε i i | a i ∈ A, ε i =0 or 1  .(0.7) For the rest of the introduction, we only consider A ⊂ . Conjecture 2 (Erd˝os-Szemer´edi). Let g(k) ≡ min |A|=k {|A[1]|+|A{1}|}. Then for any t, there is k 0 = k 0 (t) such that for any k ≥ k 0 ,g(k) >k t . Toward Conjecture 1, all work has been done so far, are for the case h =2. Erd˝os and Szemer´edi [E-S] got the first bound: Theorem (Erd˝os-Szemer´edi). Let f(k) ≡ min |A|=k |2A ∪A 2 |. Then there are constants c 1 ,c 2 , such that (0.8) k 1+c 1 <f(k) <k 2 e −c 2 ln k ln ln k . Nathanson showed that f(k) >ck 32 31 , with c =0.00028 . At this point, the best bound is (0.9) |2A ∪ A 2 | >c|A| 5/4 obtained by Elekes [El] using the Szemer´edi-Trotter theorem on line-incidences in the plane (see [S-T]). THE ERD ˝ OS-SZEMER ´ EDI PROBLEM 941 On the other hand, Nathanson and Tenenbaum [N-T] concluded some- thing stronger by assuming the sum set is small. They showed Theorem (Nathanson-Tenenbaum). If (0.10) |2A|≤3|A|−4, then (0.11) |A 2 |  |A| ln |A|  2 . Very recently, Elekes and Ruzsa [El-R] again using the Szemer´edi-Trotter theorem, established the following general inequality. Theorem (Elekes-Ruzsa). If A ⊂ is a finite set, then (0.12) |A + A| 4 |AA|ln|A| > |A| 6 . In particular, their result implies Product Innovation Toolbox Beckley_ffirs.indd iBeckley_ffirs.indd i 2/4/2012 1:02:09 AM2/4/2012 1:02:09 AM Product Innovation Toolbox A Field Guide to Consumer Understanding and Research Edited by Jacqueline Beckley The Understanding & Insight Group LLC Denville, New Jersey USA Dulce Paredes, Ph.D. Takasago International Corporation (USA) Rockleigh, New Jersey USA Kannapon Lopetcharat, Ph.D. NuvoCentric Bangkok Thailand A John Wiley & Sons, Ltd., Publication Beckley_ffirs.indd iiiBeckley_ffirs.indd iii 2/4/2012 1:02:09 AM2/4/2012 1:02:09 AM This edition first published 2012 © 2012 by John Wiley & Sons, Inc Wiley-Blackwell is an imprint of John Wiley & Sons, formed by the merger of Wiley’s global Scientific, Technical and Medical business with Blackwell Publishing. Editorial Offices: 2121 State Avenue, Ames, Iowa 50014-8300, USA The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, UK 9600 Garsington Road, Oxford OX4 2DQ, UK For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com/wiley-blackwell. Authorization to photocopy items for internal or personal use, or the internal or personal use of specific clients, is granted by Blackwell Publishing, provided that the base fee is paid directly to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923. For those organizations that have been granted a photocopy license by CCC, a separate system of payments has been arranged. The fee codes for users of the Transactional Reporting Service are ISBN-13: 978-0-8138-2397-3/2012. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Product innovation toolbox : a field guide to consumer understanding and research / edited by Jacqueline Beckley, Dulce Paredes, Kannapon Lopetcharat. p. cm. Includes bibliographical references and index. ISBN 978-0-8138-2397-3 (hard cover : alk. paper) 1. New products. 2. Consumer behavior. 3. Marketing research. I. Beckley, Jacqueline H. II. Paredes, Dulce III. Lopetcharat, Kannapon. TS170.P758 2012 658.8′3–dc23 2011037446 A catalogue record for this book is available from the British Library. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Set in 9/12pt Interstate Light by SPi Publisher Services, Pondicherry, India 1 2012 Beckley_ffirs.indd ivBeckley_ffirs.indd iv 2/4/2012 1:02:09 AM2/4/2012 1:02:09 AM v Contents Contributors xiv Acknowledgments xvi Introduction: From Pixel to Picture xvii Jacqueline Beckley, Dulce Paredes and Kannapon Lopetcharat Scoping the innovation landscape xix How this book is organized xix Part I xx Part II xxi Part III xxiii References xxiv PART I STARTING THE JOURNEY AS A CONSUMER EXPLORER 1 1 Setting the Direction: First, Know Where You Are 4 Howard Moskowitz and Jacqueline Beckley 1.1 Roles in the corporation – the dance of the ... 8/16 Sum- to- Product and Product -to -Sum Formulas Media Access these online resources for additional instruction and practice with the productto -sum and sum- to- product identities • Sum to Product. .. Sum- to- product Formulas Key Concepts • From the sum and difference identities, we can derive the product -to -sum formulas and the sum- to- product formulas for sine and cosine • We can use the product -to -sum. .. into the formula Simplify Writing the Product as a Sum Using the Product -to -Sum Formula for Cosine 2/16 Sum- to- Product and Product -to -Sum Formulas Write the following product of cosines as a sum: