Resistance to the 1892 Declaratory Act in Argyllshire
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Appendix a Short Course in Taylor Series
Appendix A Short Course in Taylor Series The Taylor series is mainly used for approximating functions when one can identify a small parameter. Expansion techniques are useful for many applications in physics, sometimes in unexpected ways. A.1 Taylor Series Expansions and Approximations In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It is named after the English mathematician Brook Taylor. If the series is centered at zero, the series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin. It is common practice to use a finite number of terms of the series to approximate a function. The Taylor series may be regarded as the limit of the Taylor polynomials. A.2 Definition A Taylor series is a series expansion of a function about a point. A one-dimensional Taylor series is an expansion of a real function f(x) about a point x ¼ a is given by; f 00ðÞa f 3ðÞa fxðÞ¼faðÞþf 0ðÞa ðÞþx À a ðÞx À a 2 þ ðÞx À a 3 þÁÁÁ 2! 3! f ðÞn ðÞa þ ðÞx À a n þÁÁÁ ðA:1Þ n! © Springer International Publishing Switzerland 2016 415 B. Zohuri, Directed Energy Weapons, DOI 10.1007/978-3-319-31289-7 416 Appendix A: Short Course in Taylor Series If a ¼ 0, the expansion is known as a Maclaurin Series. Equation A.1 can be written in the more compact sigma notation as follows: X1 f ðÞn ðÞa ðÞx À a n ðA:2Þ n! n¼0 where n ! is mathematical notation for factorial n and f(n)(a) denotes the n th derivation of function f evaluated at the point a. -
The Continuation, Breadth, and Impact of Evangelicalism in the Church of Scotland, 1843-1900
This thesis has been submitted in fulfilment of the requirements for a postgraduate degree (e.g. PhD, MPhil, DClinPsychol) at the University of Edinburgh. Please note the following terms and conditions of use: This work is protected by copyright and other intellectual property rights, which are retained by the thesis author, unless otherwise stated. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the author. The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the author. When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given. The Continuation, Breadth, and Impact of Evangelicalism in the Church of Scotland, 1843-1900 Andrew Michael Jones A Thesis Submitted to The University of Edinburgh, New College In Candidacy for the Degree of Doctor of Philosophy Edinburgh, United Kingdom 2018 ii Declaration This thesis has been composed by the candidate and is the candidate’s own work. Andrew M. Jones PhD Candidate iii Acknowledgements The research, composition, and completion of this thesis would have been impossible without the guidance and support of innumerable individuals, institutions, and communities. My primary supervisor, Professor Stewart J. Brown, provided expert historical knowledge, timely and lucid editorial insights, and warm encouragement from start to finish. My secondary supervisor, Dr. James Eglinton, enhanced my understanding of key cultural and theological ideas, offered wise counsel over endless cups of coffee, and reminded me to find joy and meaning in the Ph.D. -
At the University of Edinburgh
This thesis has been submitted in fulfilment of the requirements for a postgraduate degree (e.g. PhD, MPhil, DClinPsychol) at the University of Edinburgh. Please note the following terms and conditions of use: This work is protected by copyright and other intellectual property rights, which are retained by the thesis author, unless otherwise stated. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the author. The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the author. When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given. An Intellectual Biography of David Smith Cairns (1862-1946) Marlene Elizabeth Finlayson A thesis submitted for the Degree of Ph. D. University of Edinburgh 2014 An Intellectual Biography of David Smith Cairns (1862-1946) – Abstract This thesis explores the formative influences, development and impact of the theology of David Smith Cairns, Scottish minister, academic and writer, during the high point of British imperial expansion, and at a time of social tension caused by industrialisation. In particular, it describes and evaluates his role in the Church’s efforts to face major challenges relating to its relationships to the different world religions, its response to the First World War, and its attitude to the scientific disciplines that called into question some of its longstanding perceptions and suppositions. -
Ministers of ‘The Black Art’: the Engagement of British Clergy with Photography, 1839-1914
Ministers of ‘the Black Art’: the engagement of British clergy with photography, 1839-1914 Submitted by James Downs to the University of Exeter as a thesis for the degree of Doctor of Philosophy in English in March 2018 This thesis is available for Library use on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. I certify that all material in this thesis which is not my own work has been identified and that no material has previously been submitted and approved for the award of a degree by this or any other University. Signature: ………………………………………………………….. Abstract 1 Ministers of ‘the Black Art’: the engagement of British clergy with photography, 1839- 1914 This thesis examines the work of ordained clergymen, of all denominations, who were active photographers between 1839 and the beginning of World War One: its primary aim is to investigate the extent to which a relationship existed between the religious culture of the individual clergyman and the nature of his photographic activities. Ministers of ‘the Black Art’ makes a significant intervention in the study of the history of photography by addressing a major weakness in existing work. Taking an interdisciplinary approach, the research draws on a wide range of primary and secondary sources such as printed books, sermons, religious pamphlets, parish and missionary newsletters, manuscript diaries, correspondence, notebooks, biographies and works of church history, as well as visual materials including original glass plate negatives, paper prints and lantern slides held in archival collections, postcards, camera catalogues, photographic ephemera and photographically-illustrated books. -
Evangelicals and the Synoptic Problem
EVANGELICALS AND THE SYNOPTIC PROBLEM by Michael Strickland A thesis submitted to the University of Birmingham for the degree of DOCTOR OF PHILOSOPHY Department of Theology and Religion School of Philosophy, Theology and Religion University of Birmingham January 2011 University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder. Dedication To Mary: Amor Fidelis. In Memoriam: Charles Irwin Strickland My father (1947-2006) Through many delays, occasioned by a variety of hindrances, the detail of which would be useless to the Reader, I have at length brought this part of my work to its conclusion; and now send it to the Public, not without a measure of anxiety; for though perfectly satisfied with the purity of my motives, and the simplicity of my intention, 1 am far from being pleased with the work itself. The wise and the learned will no doubt find many things defective, and perhaps some incorrect. Defects necessarily attach themselves to my plan: the perpetual endeavour to be as concise as possible, has, no doubt, in several cases produced obscurity. Whatever errors may be observed, must be attributed to my scantiness of knowledge, when compared with the learning and information necessary for the tolerable perfection of such a work. -
MATHEMATICAL INDUCTION SEQUENCES and SERIES
MISS MATHEMATICAL INDUCTION SEQUENCES and SERIES John J O'Connor 2009/10 Contents This booklet contains eleven lectures on the topics: Mathematical Induction 2 Sequences 9 Series 13 Power Series 22 Taylor Series 24 Summary 29 Mathematician's pictures 30 Exercises on these topics are on the following pages: Mathematical Induction 8 Sequences 13 Series 21 Power Series 24 Taylor Series 28 Solutions to the exercises in this booklet are available at the Web-site: www-history.mcs.st-andrews.ac.uk/~john/MISS_solns/ 1 Mathematical induction This is a method of "pulling oneself up by one's bootstraps" and is regarded with suspicion by non-mathematicians. Example Suppose we want to sum an Arithmetic Progression: 1+ 2 + 3 +...+ n = 1 n(n +1). 2 Engineers' induction € Check it for (say) the first few values and then for one larger value — if it works for those it's bound to be OK. Mathematicians are scornful of an argument like this — though notice that if it fails for some value there is no point in going any further. Doing it more carefully: We define a sequence of "propositions" P(1), P(2), ... where P(n) is "1+ 2 + 3 +...+ n = 1 n(n +1)" 2 First we'll prove P(1); this is called "anchoring the induction". Then we will prove that if P(k) is true for some value of k, then so is P(k + 1) ; this is€ c alled "the inductive step". Proof of the method If P(1) is OK, then we can use this to deduce that P(2) is true and then use this to show that P(3) is true and so on. -
The Trisectrix of Maclaurin, That Provides One Way of Geometrically Trisecting an Angle Exactly
The Maclaurin Trisectrix Justin Seago December 8, 2008 Colin Maclaurin (1698-1746) • Scottish mathematician • Entered University of Glasgow at the age of 11 and received his M.A. at 14. [2] • Elected professor of mathematics at the University of Aberdeen at the age of 19, making him the youngest professor in history until March 2008. [4] • Made use of the form of a Taylor Series that bears his name in A Treatise on Fluxions (1742) wherein he defended Newton’s calculus. [2] [3] Geometry: Trisecting an Angle Imagine two lines separated by an arbitrary angle (θ). Now, our objective is to precisely trisect this angle; θ that is, geometrically divide the angle into three equal angles (θ/3). If we made a line spanning the angle and divided it a into three equal lengths, would there be three equal angles? θ/3? θ/3? a θ/3? a Trisection of General Angles Here GeoGebra is employed to model the attempted trisection. If θ is taken to be the angle we wish to trisect, and θ = 96.29°, then θ/3 should be approximately 32.01° As you can see in the screen capture from GeoGebra at right, trisecting a line spanning the arc of θ gives at best a rough approximation of θ/3. Trisection of General Angles Here is a second attempt. Although this greatly improves our approximation from the previous method, drawing this accurately is difficult and the resulting angles are still not exactly θ/3. Maclaurin’s Solution In 1742 Colin Maclaurin discovered a curve, now called the Maclaurin Trisectrix or The Trisectrix of Maclaurin, that provides one way of geometrically trisecting an angle exactly. -
The Supremacy of God in Preaching – Part 5
The Supremacy of God in Preaching – Part 5 2 TIMOTHY 3:16-2 TIMOTHY 4:2 TEXT, EXPOSITION AND PRACTICAL HELPS West Los Angeles Living Word Christian Center Centro Cristiano Palabra Viva Dr. Eddie Ildefonso, Senior Pastor 6520 Arizona Ave. Los Ángeles, CA 90045 Email: [email protected] Website: WWW.WLALWCC.ORG Phillips Brooks From Wikipedia, the free encyclopedia Jump to: navigation, search For other persons named Phillip Brooks, see Phil Brooks. Phillips Brooks. Anglicanism portal Phillips Brooks (December 13, 1835 – January 23, 1893) was a noted American clergyman and author, who briefly served as Bishop of Massachusetts in the Episcopal Church during the early 1890s. In the Episcopalian liturgical calendar he is remembered on January 23. 156 Contents [hide] • 1 Background o 1.1 Early life and education o 1.2 Pastoral career • 2 Influence and Legacy o 2.1 Publications o 2.2 Awards and Historical Monuments • 3 External links • 4 References [edit] Background [edit] Early life and education Brooks was born in Boston, Massachusetts in 1835. Through his father, William Gray Brooks, he was descended from the Rev. John Cotton; through his mother, Mary Ann Phillips, a very devout woman, he was a great-grandson of Samuel Phillips, Jr., the founder of Phillips Academy, Andover, Massachusetts. Four of the couple's six sons -- Phillips, Frederic, Arthur and John Cotton -- were ordained in the Episcopal Church. Phillips Brooks prepared for college at the Boston Latin School and graduated from Harvard University in 1855 at the age of 20, where he was elected to the A.D. Club. After a brief period as a teacher at Boston Latin , he began in 1856 to study for ordination in the Episcopal Church in the Virginia Theological Seminary at Alexandria, Virginia. -
Who, Where and When: the History & Constitution of the University of Glasgow
Who, Where and When: The History & Constitution of the University of Glasgow Compiled by Michael Moss, Moira Rankin and Lesley Richmond © University of Glasgow, Michael Moss, Moira Rankin and Lesley Richmond, 2001 Published by University of Glasgow, G12 8QQ Typeset by Media Services, University of Glasgow Printed by 21 Colour, Queenslie Industrial Estate, Glasgow, G33 4DB CIP Data for this book is available from the British Library ISBN: 0 85261 734 8 All rights reserved. Contents Introduction 7 A Brief History 9 The University of Glasgow 9 Predecessor Institutions 12 Anderson’s College of Medicine 12 Glasgow Dental Hospital and School 13 Glasgow Veterinary College 13 Queen Margaret College 14 Royal Scottish Academy of Music and Drama 15 St Andrew’s College of Education 16 St Mungo’s College of Medicine 16 Trinity College 17 The Constitution 19 The Papal Bull 19 The Coat of Arms 22 Management 25 Chancellor 25 Rector 26 Principal and Vice-Chancellor 29 Vice-Principals 31 Dean of Faculties 32 University Court 34 Senatus Academicus 35 Management Group 37 General Council 38 Students’ Representative Council 40 Faculties 43 Arts 43 Biomedical and Life Sciences 44 Computing Science, Mathematics and Statistics 45 Divinity 45 Education 46 Engineering 47 Law and Financial Studies 48 Medicine 49 Physical Sciences 51 Science (1893-2000) 51 Social Sciences 52 Veterinary Medicine 53 History and Constitution Administration 55 Archive Services 55 Bedellus 57 Chaplaincies 58 Hunterian Museum and Art Gallery 60 Library 66 Registry 69 Affiliated Institutions -
Section 9.10 Taylor and Maclaurin Series • Find a Taylor Or Maclaurin Series for a Function
332460_0910.qxd 11/4/04 3:12 PM Page 676 676 CHAPTER 9 Infinite Series Section 9.10 Taylor and Maclaurin Series • Find a Taylor or Maclaurin series for a function. • Find a binomial series. • Use a basic list of Taylor series to find other Taylor series. Taylor Series and Maclaurin Series In Section 9.9, you derived power series for several functions using geometric series with term-by-term differentiation or integration. In this section you will study a general procedure for deriving the power series for a function that has derivatives of all orders. The following theorem gives the form that every convergent power series must take. Bettmann/Corbis THEOREM 9.22 The Form of a Convergent Power Series COLIN MACLAURIN (1698–1746) ͑ ͒ ϭ ͚ ͑ Ϫ ͒n If f is represented by a power series f x an x c for all x in an open The development of power series to represent interval I containing c, then a ϭ f ͑n͒͑c͒͞n! and functions is credited to the combined work of n many seventeenth and eighteenth century fЉ͑c͒ f ͑n͒͑c͒ f͑x͒ ϭ f͑c͒ ϩ fЈ͑c͒͑x Ϫ c͒ ϩ ͑x Ϫ c͒2 ϩ . ϩ ͑x Ϫ c͒n ϩ . mathematicians. Gregory, Newton, John and 2! n! James Bernoulli, Leibniz, Euler, Lagrange, Wallis, and Fourier all contributed to this work. However, the two names that are most Proof Suppose the power series a ͑x Ϫ c͒n has a radius of convergence R. Then, commonly associated with power series are n by Theorem 9.21, you know that the nth derivative of f exists for Խx Ϫ cԽ < R, and Brook Taylor (1685–1731) and Colin Maclaurin. -
Taylor Polynomials and Infinite Series a Chapter for a First-Year Calculus Course
Taylor Polynomials and Infinite Series A chapter for a first-year calculus course By Benjamin Goldstein Table of Contents Preface...................................................................................................................................... iii Section 1 – Review of Sequences and Series ...............................................................................1 Section 2 – An Introduction to Taylor Polynomials ...................................................................18 Section 3 – A Systematic Approach to Taylor Polynomials .......................................................34 Section 4 – Lagrange Remainder...............................................................................................48 Section 5 – Another Look at Taylor Polynomials (Optional) .....................................................62 Section 6 – Power Series and the Ratio Test..............................................................................66 Section 7 – Positive-Term Series...............................................................................................81 Section 8 – Varying-Sign Series................................................................................................96 Section 9 – Conditional Convergence (Optional).....................................................................112 Section 10 – Taylor Series.......................................................................................................116 © 2013 Benjamin Goldstein Preface The subject of infinite -
Lecture 1 Overview (MA2730,2812,2815) Lecture 1
Overview (MA2730,2812,2815) lecture 1 Overview (MA2730,2812,2815) lecture 1 Contents of the teaching and assessment blocks MA2730: Analysis I Lecture slides for MA2730 Analysis I Analysis — taming infinity Maclaurin and Taylor series. Simon Shaw Sequences. people.brunel.ac.uk/~icsrsss Improper Integrals. [email protected] Series. College of Engineering, Design and Physical Sciences Convergence. bicom & Materials and Manufacturing Research Institute Brunel University LATEX 2ε assignment in December. Question(s) in January class test. Question(s) in end of year exam. September 21, 2015 Web Page: http://people.brunel.ac.uk/~icsrsss/teaching/ma2730 Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel MA2730, Analysis I, 2015-16 MA2730, Analysis I, 2015-16 Overview (MA2730,2812,2815) lecture 1 Overview (MA2730,2812,2815) lecture 1 Lecture 1 Lecture 1 MA2730: topics for Lecture 1 Functions — Level 1 revision What is a function? How do we define a function? What ingredients do we need? A function is a triple: Lecture 1 A rule, for example f(x)= x2 + sin(√x). A domain, for example. Revision of functions and factorials (from Level 1) D = [0, ) R. ∞ ⊆ Maclaurin polynomials — some intuition A range, for example. Taylor polynomials R = [ 1, ) R − ∞ ⊆ Definitions and formality We then write f : D R with rule f(x)= x2 + sin(√x). → Examples and exercises Often the range is replaced with the co-domain. The co-domain is The introductory material covered in this lecture can be found in usually R. Usually the domain is taken to be the largest subset of The Handbook in Chapter 1 and Section 1.1.