A History of Greek Mathematics
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CYP2D6 and CYP2C19 Genotyping in Psychiatry
CYP2D6 and CYP2C19 genotyping in psychiatry Citation for published version (APA): Koopmans, A. B. (2021). CYP2D6 and CYP2C19 genotyping in psychiatry: bridging the gap between practice and lab. ProefschriftMaken. https://doi.org/10.26481/dis.20210512ak Document status and date: Published: 01/01/2021 DOI: 10.26481/dis.20210512ak Document Version: Publisher's PDF, also known as Version of record Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. -
Squaring, Cubing, and Cube Rooting Arthur T
Squaring, Cubing, and Cube Rooting Arthur T. Benjamin Arthur T. Benjamin ([email protected]) has taught at Harvey Mudd College since 1989, after earning his Ph.D. from Johns Hopkins in Mathematical Sciences. He has served MAA as past editor of Math Horizons and the Spectrum Book Series, has written two books for MAA, and served as Polya Lecturer from 2006 to 2008. He frequently performs as a mathemagician, a term popularized by Martin Gardner. I still recall the thrill and simultaneous disappointment I felt when I first read Math- ematical Carnival [4] by Martin Gardner. I was thrilled because, as my high school teacher had told me, mathematics was presented there in a playful way that I had never seen before. I was disappointed because Gardner quoted a formula that I thought I had “invented” a few years earlier. I have always had a passion for mental calculation, and the formula (1) below appears in Gardner’s chapter on “Lightning Calculators.” It was used by the mathematician A. C. Aitken to square large numbers mentally. Squaring Aitken took advantage of the following algebraic identity. A2 (A d)(A d) d2. (1) = − + + Naturally, this formula works for any value of d, but we should choose d to be the distance to a number close to A that is easy to multiply. Examples. To square the number 23, we let d 3 to get = 232 20 26 32 520 9 529. = × + = + = To square 48, let d 2 to get = 482 50 46 22 2300 4 2304. = × + = + = With just a little practice, it’s possible to square any two-digit number in a matter of seconds. -
Lesson 3: Rectangles Inscribed in Circles
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M5 GEOMETRY Lesson 3: Rectangles Inscribed in Circles Student Outcomes . Inscribe a rectangle in a circle. Understand the symmetries of inscribed rectangles across a diameter. Lesson Notes Have students use a compass and straightedge to locate the center of the circle provided. If necessary, remind students of their work in Module 1 on constructing a perpendicular to a segment and of their work in Lesson 1 in this module on Thales’ theorem. Standards addressed with this lesson are G-C.A.2 and G-C.A.3. Students should be made aware that figures are not drawn to scale. Classwork Scaffolding: Opening Exercise (9 minutes) Display steps to construct a perpendicular line at a point. Students follow the steps provided and use a compass and straightedge to find the center of a circle. This exercise reminds students about constructions previously . Draw a segment through the studied that are needed in this lesson and later in this module. point, and, using a compass, mark a point equidistant on Opening Exercise each side of the point. Using only a compass and straightedge, find the location of the center of the circle below. Label the endpoints of the Follow the steps provided. segment 퐴 and 퐵. Draw chord 푨푩̅̅̅̅. Draw circle 퐴 with center 퐴 . Construct a chord perpendicular to 푨푩̅̅̅̅ at and radius ̅퐴퐵̅̅̅. endpoint 푩. Draw circle 퐵 with center 퐵 . Mark the point of intersection of the perpendicular chord and the circle as point and radius ̅퐵퐴̅̅̅. 푪. Label the points of intersection . -
Squaring the Circle a Case Study in the History of Mathematics the Problem
Squaring the Circle A Case Study in the History of Mathematics The Problem Using only a compass and straightedge, construct for any given circle, a square with the same area as the circle. The general problem of constructing a square with the same area as a given figure is known as the Quadrature of that figure. So, we seek a quadrature of the circle. The Answer It has been known since 1822 that the quadrature of a circle with straightedge and compass is impossible. Notes: First of all we are not saying that a square of equal area does not exist. If the circle has area A, then a square with side √A clearly has the same area. Secondly, we are not saying that a quadrature of a circle is impossible, since it is possible, but not under the restriction of using only a straightedge and compass. Precursors It has been written, in many places, that the quadrature problem appears in one of the earliest extant mathematical sources, the Rhind Papyrus (~ 1650 B.C.). This is not really an accurate statement. If one means by the “quadrature of the circle” simply a quadrature by any means, then one is just asking for the determination of the area of a circle. This problem does appear in the Rhind Papyrus, but I consider it as just a precursor to the construction problem we are examining. The Rhind Papyrus The papyrus was found in Thebes (Luxor) in the ruins of a small building near the Ramesseum.1 It was purchased in 1858 in Egypt by the Scottish Egyptologist A. -
Mathematicians
MATHEMATICIANS [MATHEMATICIANS] Authors: Oliver Knill: 2000 Literature: Started from a list of names with birthdates grabbed from mactutor in 2000. Abbe [Abbe] Abbe Ernst (1840-1909) Abel [Abel] Abel Niels Henrik (1802-1829) Norwegian mathematician. Significant contributions to algebra and anal- ysis, in particular the study of groups and series. Famous for proving the insolubility of the quintic equation at the age of 19. AbrahamMax [AbrahamMax] Abraham Max (1875-1922) Ackermann [Ackermann] Ackermann Wilhelm (1896-1962) AdamsFrank [AdamsFrank] Adams J Frank (1930-1989) Adams [Adams] Adams John Couch (1819-1892) Adelard [Adelard] Adelard of Bath (1075-1160) Adler [Adler] Adler August (1863-1923) Adrain [Adrain] Adrain Robert (1775-1843) Aepinus [Aepinus] Aepinus Franz (1724-1802) Agnesi [Agnesi] Agnesi Maria (1718-1799) Ahlfors [Ahlfors] Ahlfors Lars (1907-1996) Finnish mathematician working in complex analysis, was also professor at Harvard from 1946, retiring in 1977. Ahlfors won both the Fields medal in 1936 and the Wolf prize in 1981. Ahmes [Ahmes] Ahmes (1680BC-1620BC) Aida [Aida] Aida Yasuaki (1747-1817) Aiken [Aiken] Aiken Howard (1900-1973) Airy [Airy] Airy George (1801-1892) Aitken [Aitken] Aitken Alec (1895-1967) Ajima [Ajima] Ajima Naonobu (1732-1798) Akhiezer [Akhiezer] Akhiezer Naum Ilich (1901-1980) Albanese [Albanese] Albanese Giacomo (1890-1948) Albert [Albert] Albert of Saxony (1316-1390) AlbertAbraham [AlbertAbraham] Albert A Adrian (1905-1972) Alberti [Alberti] Alberti Leone (1404-1472) Albertus [Albertus] Albertus Magnus -
Divisibility Rules by James D
DIVISIBILITY RULES BY JAMES D. NICKEL Up to this point in our study, we have explored addition, subtraction, and multiplication of whole num- bers. Recall that, on average, out of every 100 arithmetical problems you encounter 70 will require addition, 20 will require multiplication, 5 will require subtraction, and 5 will require division. Although least used, di- vision is a part of everyday life. Parents, in providing inheritances for their children, generally seek to divide the estate equally between their children. When four people play with a deck of 52 cards, they seek to divide the cards equally to each player. We are now ready to explore the nature of division and, to speak truthfully, division is one of the hard- est processes for a student to master because it is the most complicated of all the arithmetical operations. Note what was said about it in 1600: Division is esteemed one of the busiest operations of Arithmetick, and such as requireth a mynde not wandering, or setled uppon other matters.1 So, we shall tread this division ground lightly before we start digging. We shall provide a few lessons of “interlude” first that I think you will find very enjoyable. The ancient Greeks of the Classical era (ca. 600–300 BC) were fascinated with the structure of number, especially the natural numbers. For the = 2 = 1 philosophers of this period, the study of the structure of natural numbers = 4 = 3 was a favorite “thought” hobby. We have already seen one example of this in the classification of even and odd numbers. -
A History of Cynicism
A HISTORY OF CYNICISM Downloaded from https://www.holybooks.com Downloaded from https://www.holybooks.com A HISTORY OF CYNICISM From Diogenes to the 6th Century A.D. by DONALD R. DUDLEY F,llow of St. John's College, Cambrid1e Htmy Fellow at Yale University firl mll METHUEN & CO. LTD. LONDON 36 Essex Street, Strand, W.C.2 Downloaded from https://www.holybooks.com First published in 1937 PRINTED IN GREAT BRITAIN Downloaded from https://www.holybooks.com PREFACE THE research of which this book is the outcome was mainly carried out at St. John's College, Cambridge, Yale University, and Edinburgh University. In the help so generously given to my work I have been no less fortunate than in the scenes in which it was pursued. I am much indebted for criticism and advice to Professor M. Rostovtseff and Professor E. R. Goodonough of Yale, to Professor A. E. Taylor of Edinburgh, to Professor F. M. Cornford of Cambridge, to Professor J. L. Stocks of Liverpool, and to Dr. W. H. Semple of Reading. I should also like to thank the electors of the Henry Fund for enabling me to visit the United States, and the College Council of St. John's for electing me to a Research Fellowship. Finally, to• the unfailing interest, advice and encouragement of Mr. M. P. Charlesworth of St. John's I owe an especial debt which I can hardly hope to repay. These acknowledgements do not exhaust the list of my obligations ; but I hope that other kindnesses have been acknowledged either in the text or privately. -
Meditations the Philosophy Classic
MEDITATIONS THE PHILOSOPHY CLASSIC MEDITATIONS THE PHILOSOPHY CLASSIC THE INTERNATIONAL BESTSELLER MARCUS AURELIUS WITH AN INTRODUCTION BY DONALD ROBERTSON MEDITATIONS Also available in the same series: Beyond Good and Evil: The Philosophy Classic by Friedrich Nietzsche (ISBN: 978-0-857-08848-2) On the Origin of Species: The Science Classic by Charles Darwin (ISBN: 978-0-857-08847-5) Tao Te Ching: The Ancient Classic by Lao Tzu (ISBN: 978-0-857-08311-1) The Art of War: The Ancient Classic by Sun Tzu (ISBN: 978-0-857-08009-7) The Game of Life and How to Play It: The Self-Help Classic by Florence Scovel Shinn (ISBN: 978-0-857-08840-6) The Interpretation of Dreams: The Psychology Classic by Sigmund Freud (ISBN: 978-0-857-08844-4) The Prince: The Original Classic by Niccolo Machiavelli (ISBN: 978-0-857-08078-3) The Prophet: The Spirituality Classic by Kahlil Gibran (ISBN: 978-0-857-08855-0) The Republic: The Influential Classic by Plato (ISBN: 978-0-857-08313-5) The Science of Getting Rich: The Original Classic by Wallace Wattles (ISBN: 978-0-857-08008-0) The Wealth of Nations: The Economics Classic by Adam Smith (ISBN: 978-0-857-08077-6) Think and Grow Rich: The Original Classic by Napoleon Hill (ISBN: 978-1-906-46559-9) MEDITATIONS The Philosophy Classic MARCUS AURELIUS With an Introduction by DONALD ROBERTSON This edition first published 2020 Introduction copyright © Donald Robertson, 2020 The material for Meditations is based on The Thoughts of the Emperor M. Aurelius Antoninus, translated by George Long, published by Bell & Daldy, London 1862, and is now in the public domain. -
Plato As "Architectof Science"
Plato as "Architectof Science" LEONID ZHMUD ABSTRACT The figureof the cordialhost of the Academy,who invitedthe mostgifted math- ematiciansand cultivatedpure research, whose keen intellectwas able if not to solve the particularproblem then at least to show the methodfor its solution: this figureis quite familiarto studentsof Greekscience. But was the Academy as such a centerof scientificresearch, and did Plato really set for mathemati- cians and astronomersthe problemsthey shouldstudy and methodsthey should use? Oursources tell aboutPlato's friendship or at leastacquaintance with many brilliantmathematicians of his day (Theodorus,Archytas, Theaetetus), but they were neverhis pupils,rather vice versa- he learnedmuch from them and actively used this knowledgein developinghis philosophy.There is no reliableevidence that Eudoxus,Menaechmus, Dinostratus, Theudius, and others, whom many scholarsunite into the groupof so-called"Academic mathematicians," ever were his pupilsor close associates.Our analysis of therelevant passages (Eratosthenes' Platonicus, Sosigenes ap. Simplicius, Proclus' Catalogue of geometers, and Philodemus'History of the Academy,etc.) shows thatthe very tendencyof por- trayingPlato as the architectof sciencegoes back to the earlyAcademy and is bornout of interpretationsof his dialogues. I Plato's relationship to the exact sciences used to be one of the traditional problems in the history of ancient Greek science and philosophy.' From the nineteenth century on it was examined in various aspects, the most significant of which were the historical, philosophical and methodological. In the last century and at the beginning of this century attention was paid peredominantly, although not exclusively, to the first of these aspects, especially to the questions how great Plato's contribution to specific math- ematical research really was, and how reliable our sources are in ascrib- ing to him particular scientific discoveries. -
Cool Math Essay May 2013: DIVISIBILITY
WILD COOL MATH! CURIOUS MATHEMATICS FOR FUN AND JOY MAY 2013 PROMOTIONAL CORNER: PUZZLER: Here’s a delight for dividing Have you an event, a workshop, a website, by nine. To compute 12401÷ 9 , for some materials you would like to share with example, just read form left to right and the world? Let me know! If the work is about record partial sums. The partial sums give deep and joyous and real mathematical the quotient and the remainder! doing I would be delighted to mention it here. *** Are you in the DC/Maryland are looking for an exciting, rich and challenging summer- course experience for students? The Institute for Academic Challenge is offering a really cool course on The puzzle is: Why does this work? Combinatorics, August 12-23. Check out www.academicchallenge.org/classes.html . COMMENT: If we didn’t carry digits in our arithmetic system, this method would always hold. For instance, 3282÷ 9 = 3|5|13remainder 15 is mathematically correct! © James Tanton 2013 26+ 78 = 104! That is, DIVIDING BY 97 6 AND OTHER SUCH QUANTITIES 1378÷ 98 = 14 . 98 Quick! What’s 602÷ 97 ? Qu ery : Look at the picture on the left. Can Well… you use it to see that 602÷ 103 equals 6 602 is basically 600 , with a remainder of −16 , a negative 97 is essentially 100 , and −16 remainder? So 602÷ 103 = 6 . What is 600÷ 100 = 6 . 103 approximate decimal value of this? So 602÷ 97 is basically 6 . How far off is this answer? What is 598÷ 103 ? A picture of 602 dots shows that each COMMENT : If x= y and z= y , then group of 100 is off from being a group of surely it is true that x= z . -
Unit 3, Lesson 1: How Well Can You Measure?
GRADE 7 MATHEMATICS NAME DATE PERIOD Unit 3, Lesson 1: How Well Can You Measure? 1. Estimate the side length of a square that has a 9 cm long diagonal. 2. Select all quantities that are proportional to the diagonal length of a square. A. Area of a square B. Perimeter of a square C. Side length of a square 3. Diego made a graph of two quantities that he measured and said, “The points all lie on a line except one, which is a little bit above the line. This means that the quantities can’t be proportional.” Do you agree with Diego? Explain. 4. The graph shows that while it was being filled, the amount of water in gallons in a swimming pool was approximately proportional to the time that has passed in minutes. a. About how much water was in the pool after 25 minutes? b. Approximately when were there 500 gallons of water in the pool? c. Estimate the constant of proportionality for the number of gallons of water per minute going into the pool. Unit 3: Measuring Circles Lesson 1: How Well Can You Measure? 1 GRADE 7 MATHEMATICS NAME DATE PERIOD Unit 3: Measuring Circles Lesson 1: How Well Can You Measure? 2 GRADE 7 MATHEMATICS NAME DATE PERIOD Unit 3, Lesson 2: Exploring Circles 1. Use a geometric tool to draw a circle. Draw and measure a radius and a diameter of the circle. 2. Here is a circle with center and some line segments and curves joining points on the circle. Identify examples of the following. -
On the Standard Lengths of Angle Bisectors and the Angle Bisector Theorem
Global Journal of Advanced Research on Classical and Modern Geometries ISSN: 2284-5569, pp.15-27 ON THE STANDARD LENGTHS OF ANGLE BISECTORS AND THE ANGLE BISECTOR THEOREM G.W INDIKA SHAMEERA AMARASINGHE ABSTRACT. In this paper the author unveils several alternative proofs for the standard lengths of Angle Bisectors and Angle Bisector Theorem in any triangle, along with some new useful derivatives of them. 2010 Mathematical Subject Classification: 97G40 Keywords and phrases: Angle Bisector theorem, Parallel lines, Pythagoras Theorem, Similar triangles. 1. INTRODUCTION In this paper the author introduces alternative proofs for the standard length of An- gle Bisectors and the Angle Bisector Theorem in classical Euclidean Plane Geometry, on a concise elementary format while promoting the significance of them by acquainting some prominent generalized side length ratios within any two distinct triangles existed with some certain correlations of their corresponding angles, as new lemmas. Within this paper 8 new alternative proofs are exposed by the author on the angle bisection, 3 new proofs each for the lengths of the Angle Bisectors by various perspectives with also 5 new proofs for the Angle Bisector Theorem. 1.1. The Standard Length of the Angle Bisector Date: 1 February 2012 . 15 G.W Indika Shameera Amarasinghe The length of the angle bisector of a standard triangle such as AD in figure 1.1 is AD2 = AB · AC − BD · DC, or AD2 = bc 1 − (a2/(b + c)2) according to the standard notation of a triangle as it was initially proved by an extension of the angle bisector up to the circumcircle of the triangle.