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Is there any proof that Queen Jane I of England actually existed?
Proof? Here you go - have a nice day out in the archives. All credit to A Primary Source Bibliography on the subject of Lady Jane Grey Dudley, England's Nine Days Queen of July 1553.Manuscript Primary Sources:British Library, LondonAdditional 10617 – Seventeenth-century transcriptions of National Archives Public Record Office, Kings Bench 8/22–25Additional 12065 – Sixteenth-century transcript of John Dudley’s Scaffold SpeechAdditional 14024 – Seventeenth- or eighteenth-century transcriptions of Privy Council Register from reign of Edward VIAdditional 15215 – Anonymous sixteenth-century chronicle of Wyatt’s RebellionAdditional 18738 – Letter from Privy Council dated 15 July 1553Additional 22563 – 1833 transcript of letter from Privy Council to the Sheriff of Wiltshire in the name of Queen Jane, 7 July 1553Additional 26748 – Minutes of the Privy Council for late-1553Additional 33230 – Letter from officials in Kent noting they had proclaimed Mary as Queen, 19 July 1553Additional 34152 – Letters from persons in Kent related to Wyatt’s RebellionAdditional 48036 – Seventeenth-century transcript of Acts of Privy Council for Mary and PhilipCotton Caligula B IX/1 – Notes on matters of the succession in England, 1556–1570Cotton Galba B XII – Originals of letters from Edward and the Privy Council to the English Ambassadors in BrusselsCotton Julius B IX – Register of Grants, etc. under the Great Seal, 1551–1553Cotton Julius F VI – Sixteenth-century transcription of a fifteenth-century treatise against accession to the Crown by a woman and of proclamation of accession of Queen JaneCotton Titus B II – Letter from Mary to Earl of Sussex to raise forces in her defense, 29 July 1553Cotton Vespasian F III – Mary’s proclamation against Sir Thomas WyattCotton Vitellius F XII – St Edmund’s Chronicle of London, 1189–1555Egerton 2009–2012 – Four-volume set of madrigal, motets, etc. in four voices, including a ballad about Queen JaneEgerton 2148 – Thomas Hoby’s ‘Booke of the Trauaile and lief of me’Egerton 2642 – A sixteenth-century almanac, includes chronicle of events of 1553Egerton 2986 – Letter from Francis Hastings reporting capture of Henry Grey, February 1554Harley 284 – Contemporary transcript of John Dudley’s scaffold speechHarley 353 – Contemporary copies of sixteenth-century letters and papersHarley 416 – Papers of John Foxe, accessible only via British Library microfilmHarley 422 – Papers of John Foxe, accessible only via British Library microfilmHarley 424 – Papers of John Foxe, accessible only via British Library microfilmHarley 425 – Papers of John Foxe, accessible only via British Library microfilmHarley 523 – Contemporary copies of papers of the Privy Council for reign of Edward VIHarley 2194 – Late-seventeenth-century transcript of records of Lords High Steward of EnglandHarley 2342 – Lady Jane Grey’s prayer-book. Accessible only via British Library microfilm. Sometimes on display in the British Library‘s ‘Treasures’ exhibition.Harley 2370 – De Arte Moriendi, containing copy of Jane Grey’s letter to her sister, February 1554Lansdowne 2 – William Cecil’s copy of Jane Grey’s letter to Thomas HardingLansdowne 3 – Original letters to and from the Privy Council, July 1553Lansdowne103 – Papers of William Cecil, Lord BurghleyLansdowne 104 – Papers of William Cecil, Lord BurghleyLansdowne 198 – Royal Proclamations 1504–1642Lansdowne 1236 – Original letters, including several under autograph signature of Queen Jane, 11 July 1553Royal 12A XXVI – Late-seventeenth-century transcript of John Dudley’s speech from the scaffoldRoyal 18C XXIV – Original register of all bills signed by King and Privy Council from 19 October 1551Sloane 1086 – Book of heraldic emblems, including Grey family bannerSloane 3011 – Regole et Institutioni della Lingua Thoscana, by Michelangelo Florio, inscribed to Lady Jane GreyCambridge University LibraryDd. XI.46, Regole de la lingua Thoscana by Michelangelo FlorioCorporation of London Record OfficeThe Repertories of the Court of Aldermen, 1485–1835Folger Shakespeare Library, Washington , D.C.Loseley L.b.24 – Letter from Queen Jane to Thomas Cawarden, 16 July 1553Loseley L.b.341 – Letter from Queen Mary to Thomas Cawarden, January 1554Loseley L.b.342 – Letters Patent of Edward VI granting offices to Thomas CawardenLoseley L.b.503 – Letter from Privy Council to Thomas Cawarden, 9 July 1553Loseley L.b.504 – Letter from Henry Grey to Thomas Cawarden, 16 July 1553National Archives at Kew, LondonC 1/1289 – Records of lawsuit, related to Chancery 1/1301C 1/1301 – Lawsuit catalogued as ‘time of Queen Jane’E 101/631 – Inventory of John Dudley’s properties seized in August 1553E 154/2 – Inventory of John Dudley’s properties seized in August 1553E 154/6 – Inventory of John Dudley’s properties seized in August 1553KB 8/22–25 – Documents related to trial of Jane and Guildford Dudley et al., 13 November 1553LR 2/118 – Inventory of John Dudley’s properties seized in August 1553LR 2/119 – Inventory of John Dudley’s properties seized in August 1553LR 2/120 – Inventory of John Dudley’s properties seized in August 1553SP 10/5/4–5, Extract 94/2/1–2 – Jane Grey’s letter to Thomas Seymour, 1548SP 10/15/79 – Letters and papers of the reign of Edward VIHatfield House (UK)Salisbury 1 – Drafts of diplomatic letters, July 1553Salisbury 133 – Documents related to trial of Thomas Seymour, 1549Salisbury 138 – Documents related to trial of Thomas Seymour, 1549Salisbury 150 – Documents related to trial of Thomas Seymour, 1549Salisbury 198 – Diplomatic letters, June and July 1553All accessible only via microfilm at Folger Shakespeare Library (Washington, DC) as Folger Shakespeare Library microfilms number 164, reels 1, 30, 39, and 52, or in the same format at the British Library (London) or via The Cecil Papers subscription-only online database.Surrey History Centre, Woking (UK)Loseley COR 3 – Original letters from Queen Jane and Privy Council to officials in SurreyLoseley 2014 – Original letters from Queen Jane and Privy Council to official in SurreyLoseley 6729/3 – Original letters under autograph signature of Queen JaneUnclassified 1922 – Original copy of proclamation of accession of Queen JanePrinted Primary Sources:Acts of the Privy Council of England, New Series, Vol. III . A.D. 1550–1552. Edited by John Roche Dasent. London: HMSO, 1891.Anon. The life, death and actions of the most chast, learned, and religious lady, the Lady Iane Gray, daughter to the Duke of Suffolke Containing foure principall discourses written with her owne hands. The first an admonition to such as are weake in faith: the second a catechisme: the third an exhortation to her sister: and the last her words at her death. London: Printed by G. Eld, for John Wright, 1615. Accessible via Early English Books Online (subscription only).Ascham, Roger. The Scholemaster, or plaine and perfite way of teachyng children, to understand, write, and speake, the latin tong, but specially purposed for the private brynging up of youth in Jentlemen and Noble mens houses.... London: John Daye, 1570. Accessible via Early English Books Online.Ascham, Roger. The Whole Works of Roger Ascham. Edited by J. A. Giles. New York: AMS Press, 1965; reprinted from edition of London: John Russell Smith, 1864.Calendar of Letters and Papers, Foreign and Domestic of the Reign of Henry VIII, 1509–1545, Preserved in the Public Record Office, the British Museum, and Elsewhere in England. Edited by James Gairdner. London: Published for Her Majesty's Stationery Office by Longman, l826 and Vaduz: Kraus Reprint, 1965.Calendar of Letters and State Papers Relating to English Affairs, preserved principally in the Archives of Simancas, Vol. 1: Elizabeth I: 1558–1567. Edited by Martin A. Hume. Nendeln, Liechtestein: Kraus Reprints, 1971.Calendar of Letters, Despatches, and State Papers relating to negotiations between England and Spain,Vol. XI: Edward VI and Mary. Edited by Royall Tyler. London: HMSO, 1916 and Nendeln, Lichtenstein, 1969.Calendar of Letters, Despatches, and State Papers relating to negotiations between England and Spain,Vol. XII: Mary I: January–July 1554. Edited by Royall Tyler. London: HMSO, 1916 and Nendeln, Lichtenstein, 1969.The Chronicle of the Grey Friars of London. Edited by John Gough Nichols. London: The Camden Society, 1852.The Chronicle of Queen Jane, and of Two Years of Queen Mary, and especially of the Rebellion of SirThomas Wyat. Edited by John Gough Nichols. London: Camden Society, 1850.Commendone, Giovanni Francesco. ‘Il Successi d’Inghilterra.’ The Accession, Coronation, and Marriage of Mary Tudor, as related in four manuscripts of the Escorial. Translated by Cesare V. Malfatti. Barcelona, 1956.A conference between the Lady Jane Grey and F. Fecknam a Romish priest, concerning the blessed sacrament; whilest she was prisoner in the Tower of London, and was beheaded on the Green there, Feb. 12. 1554. Together with her behaviour and last speech and prayers at her suffering. London: 1688.Edward VI. The ‘Chronicle’ and Political Papers of King Edward VI. Edited by W. K. Jordan. Ithaca: New York: Cornell University Press, 1966.Edward VI. England’s Boy King: The Diary of Edward VI, 1537–1553. Edited by Jonathan North. Welwyn Garden City, UK: Ravenhall Books, 2005.English Historical Documents 1485–1558. Edited by C. H. Williams. London: Eyre and Spottiswoode, 1967.Fabyan, Robert. The New Chronicles of England and France, in two parts. London: Printed for F. C. and J. Rivington, 1811.Florio, Michelangelo. Historia de la vita e de la morte de l’Illustrissima Signora Giovanna Graia, giaregina eletta a publicata d'Inghilterra. Middelburgh, Zeeland: Johan Radermacher, 1607.Florio, Michelangelo. Regole de la Lingua Thoscana, edited by Giuliano Pellegrini. Firenze: Sansoni, 1954.Foxe, John. Actes and Monuments of these latter and perillous dayes touching matters of the Church ....London: John Day, 1563.Grey, Jane. Brief Confessional Writings: Grey, Stubbes, Livingstone and Clarksone. Edited by Mary Ellen Lamb. Vol. 2 of The Early Modern Englishwoman: A Facsimile Library of Essential Works, Series 1, Printed Writings, 1500–1640, Part 2, edited by Betty S. Travitsky and Patrick Cullen. Aldershot, UK: Ashgate, 2001.Grey, Jane. An epistle of the Ladye Iane a righte vertuous woman, to a learned man of late falne from the truth of Gods most holy word, for fear of the worlde. London: John Daye, 1554. Accessible via Early English Books Online.Grey, Jane. ‘A French Letter Concerning Lady Jane Grey and Queen Mary’. In La révolte du conte de Warwick contre le roi Edward IV, edited by J. A. Giles. London: Caxton Society, 1849 and NewYork: B. Franklin, 1967.Grey, Jane. Here in this booke ye haue a godly epistle made by a faithful Christian A comunication betwene Feckna and the Lady Iane Dudley. A letter that she wrote to her syster Lady Katherin. The ende of the Ladye Iane vpon the scaffolde. Ye shal haue also herein a godly prayer made by maister Iohn Knokes. London: Successor of A. Scoloker, 1554. Accessible via Early English Books Online.Grey, Jane. The life, death and actions of the most chast, learned, and religious lady, the Lady Iane Gray,daughter to the Duke of Suffolke Containing foure principall discourses written with her owne hands. The first an admonition to such as are weake in faith: the second a catechisme: the third an exhortation to her sister: and the last her words at her death. London: G. Eld, for John Wright, 1615 and London: I. Haviland for John Wright, 1636. Accessible via Early English Books Online.Grey, Lady Jane. Memoirs and Literary Remains of Lady Jane Grey. Edited by Harris Nicolas. London: Henry Colburn, 1832.Haynes, Samuel. A Collection of State Papers, relating to Affairs in the reigns of King Henry the Eighth,King Edward VI, Queen Mary, and Queen Elizabeth, from the year 1542–1570. London: 1740.Hayward, John. The Life and Raigne of King Edward the Sixth. Edited by Barrett L. Beer. Kent, OH and London: Kent State University Press, 1993.Hoby, Thomas. ‘A Booke of the Travaile and Lief of Me, Thomas Hoby, w[i]th Diverse Things Woorth the Notinge.’ Edited by Edgar Powell. The Camden Miscellany, Vol. 10. London: The Royal Historical Society, 1902.The Lady Jane Grey's Prayer Book: British Library Harley Manuscript 2342, fully illustrated and transcribed. Introduction by J. Stephan Edwards. Palm Springs: Old John Publishing, 2016.Letter and Papers, Foreign and Domestic, of the Reign of HVIII, Vol. XII, Part II. Edited by James Gairdner. London: HMSO, 1891 and Vaduz: Kraus Reprint, 1965.Lettere di Principi, le qvali si scrivono o da principi, o a principi, o ragionano di principi. Libro Terzo. Venice: Appresso Giordano Ziletti. 1577.Machyn, Henry. The Diary of Henry Machyn, Citizen and Merchant-Taylor of London, from A.D. 1550 to A.D. 1563. Edited by John Gough Nichols. London: The Camden Society, 1848.Noailles, Antoine de, Mssrs. Ambassades de Messieurs de Noailles en Angleterre. 3 vols. Leyde: Dessaint and Saillant, 1763.Original Letters Relative to the English Reformation. 2 vols. Edited by Hastings Robinson. Cambridge: Cambridge University Press, 1846 and 1847.The Parliamentary History of England, from the earliest period to the year 1803. 30 vols. London: Longman, Hurst, Rees, Orme, & Browne, 1806–1820.Parliaments of England, 1213-1702, Part I: Members of Parliament. London, 1878.Records of the Borough of Leicester, Being a series of Extracts from the Archives of the Corporation of Leicester, 1509–1603. Vol. 3. Edited by Mary Bateson. Cambridge: Cambridge University Press,1905.Speed, John. The history of Great Britaine under the conquests of ye Romans, Saxons, Danes and Normans Their originals, manners, warres, coines & seales: with yesuccessions, lives, acts & issues of the English monarchs from Iulius Caesar, to our most gracious soueraigne King Iames. London: William Hall and John Beale, 1611. Accessible via Early English Books Online.The Statutes of the Realm. Printed by command of His Majesty King George the Third. From Original Records and Authentic Manuscripts. Vol. 3. London: 1817.Strype, John, ed. Ecclesiastical Memorials, relating chiefly to Religion and the Reformation of it, and the emergencies of the Church of England under King Henry VIII, King Edward VI, and Queen Mary. 3 vols. Translated by John Strype. London: John Wyat, 1721 and Oxford: Clarendon Press, 1822.Wingfield, Robert. Vita Mariæ Reginæ. Edited and translated by Diarmaid MacCulloch. Camden Miscellany XXVIII, 4th series, vol. 29. London: Royal Historical Society, 1984Wriothesley, Charles. A Chronicle of England during the Reign of the Tudors, from A.D. 1485 to 1559.Vol. 2. Edited by William Douglas Hamilton. London and New York: The Camden Society, 1872.
What are some good number theory books for beginners? Or other books that would help me prepare for the national IMO competition?
Usually Number Theory is governed by some syllabus, some goal(s) and several levels ...IMO preparation has no syllabus that I know of ... so you have to look over solved problems ... while a lot of solutions are elegant and even spectacular, they are also not easy and they will not give you too much "durable/professional education" ...A brief list of Number Theory books (mostly elementary):"Ivan Morton Niven, Herbert S. Zuckerman, Hugh L. Montgomery","An Introduction to the Theory of Numbers 5e""Pierre Samuel","Algebraic Theory of Numbers""Godfrey Harold Hardy, Edward M. Wright","An Introduction to the Theory of Numbers 4e""Godfrey Harold Hardy, Edward Maitland Wright","An Introduction to the Theory of Numbers 6e 2009""Kenneth H. Rosen","Elementary Number Theory and Its Applications 5e""Bart Goddard, Kenneth H. Rosen","Elementary Number Theory and Its Applications 5e Ism Sup""David M. Burton","Elementary Number Theory 5e""David M. Burton","Elementary Number Theory""Thomas Koshy","Elementary Number Theory With Applications""Melvyn B. Nathanson","Elementary Methods in Number Theory""William Stein","Elementary Number Theory: Primes, Congruences, and Secrets""Waclaw Sierpinski","250 Problems in Elementary Number Theory""James J. Tattersall","Elementary Number Theory in Nine Chapters""Andrzej Schnizel","Andrzej Schinzel, Selecta""Matteo Baldoni, Ciro Ciliberto","Elementary Number Theory, Cryptography and Codes""Harold G. Diamond, H. Halberstam, William F. Galway","A Higher-Dimensional Sieve Method""George Bachman","Introduction to P-Adic Numbers and Valuation Theory""Paulo Ribenboim","The Little Book of Big Primes""Marcus Du Sautoy","The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics""K. Chandrasekharan","Arithmetical Functions""Ina Kersten","Symmetries in algebra and number theory (SANT)""Andre Weil","Basic Number Theory""Hans Riesel","Prime Numbers and Computer Methods for Factorization""Giulia Maria Piacentini Cattaneo","Algebra: Un Approccio Algoritmico""Hermann Weyl","Algebraic Theory of Numbers""George E. Andrews","Number Theory""Takshi Aoki, Shigeru Kanemitsu, Jianya Liu","Number Theory: Dreaming in Dreams: Proceedings of the 5th China-Japan Seminar, Higashi-Osaka, Japan, 27-31 August 2008""Dennis A. Hejhal","Emerging Applications of Number Theory""József Sándor, Dragoslav S. Mitrinović, Borislav Crstici","Handbook of Number Theory I""Liz Strachan","Numbers Are Forever""Tom Johnson, Franck Jedrzejewski","Looking at Numbers""David F. Wells","Dictionnaire Penguin Des Nombres Curieux""Harold Davenport, James A. Davenport","The Higher Arithmetic""L. K. Hua","Additive Theory of Prime Numbers""A. E. Ingham, R. C. Vaughan","The Distribution of Prime Numbers""Gerald Tenenbaum, Michel Mendès France","The Prime Numbers and Their Distribution""Mikhail J. Atallah, Marina Blanton","Algorithms and Theory of Computation Handbook, Second Edition, Volume 1: General Concepts and Techniques""M. N. Huxley","The Distribution of Prime Numbers: Large Sieves and Zero-Density Theorems""Wolfgang P. Schleich, Helmut Maier","Prime Numbers 101""David G. Wells","Prime Numbers: The Most Mysterious Figures in Math""Hans Riesel","Prime Numbers and Computer Methods for Factorization""Alexander S. Karpenko","Lukasiewicz's Logics and Prime Numbers""Richard Crandall, Carl B. Pomerance","Prime Numbers""Mikhail J. Atallah, Marina Blanton","Algorithms and Theory of Computation Handbook, Second Edition, Volume 2""Eric Bach, Jeffrey Shallit","Algorithmic Number Theory vol. I Efficient Algorithms"A brief list of IMO bibliography:"Kenneth Hardy, Kenneth S. Williams","The Red Book of Mathematical Problems""Kenneth Hardy, Kenneth S. Williams","The Green Book of Mathematical Problems""Eugene P. Northrop","Riddles in Mathematics; A Book of Paradoxes""Martin Gardner","The 2nd Scientific American Book of Mathematical Puzzles & Diversions""Titu Andreescu, Oleg Mushkarov, Luchezar N. Stoyanov","Geometric Problems on Maxima and Minima""Alan Sultan, Alice F. Artzt","The Mathematics That Every Secondary Math Teacher Needs to Know""Timothy Gowers, June Barrow-Green, Imre Leader","Princeton Companion to Mathematics""J. C. Burkill, H. M. Cundy","Mathematical Scholarship Problems""Alfred S. Posamentier, Charles T. Salkind","Challenging Problems in Algebra""Alfred S. Posamentier, Charles T. Salkind","Challenging Problems in Geometry""Charles T. Salkind","Contest Problem Book I ANNUAL HIGH SCHOOL MATHEMATICS EXAMINATIONS 1950-1960""Charles T. Salkind","Contest Problem Book II: Annual High School Contests of the Mathematical Association of America, 1961-1965""James M. Earl, Charles T. Salkind","Contest Problem Book III""Ralph A. Artino, Anthony M. Gaglione, Niel Shell","Contest Problem Book IV: Annual High School Examinations, 1973-1982""Titu Andreescu, Dorin Andrica","Complex Numbers From a To--Z""N. N. Chentzov, D. O. Shklarsky, Isaak Moiseevich Yaglom","The USSR Olympiad Problem Book""Heinrich Dörrie","100 Great Problems of Elementary Mathematics""A. M. Yaglom, Isaak Moiseevich Yaglom","Challenging Mathematical Problems With Elementary Solutions, Vol. 1 Combinatorial Analysis and Probability Theory""A. M. Yaglom, Isaak Moiseevich Yaglom","Challenging Mathematical Problems With Elementary Solutions, vol II: Problems From Various Branches of Mathematics""Samuel L. Greitzer","International Mathematical Olympiads 1959-1977""Murray S. Klamkin","International Mathematical Olympiads 1978-1985 and Forty Supplementary Problems""Murray S. Klamkin","U.S.A. Mathematical Olympiads, 1972-1986""Paul Richard Halmos","Problems for Mathematicians, Young and Old""Thomas Hull","Project Origami""Edward B. Burger, Michael Starbird","The Heart of Mathematics""Elwyn R. Berlekamp, Tom Rodgers","The Mathemagician and Pied Puzzler""N. N. Chentsov, D. O. Shklyarsky, Isaak Moiseevich Yaglom","Selected Problems and Theorems in Elementary Mathematics: Arithmetic and Algebra""A. Domoryad, H. Moss","Mathematical Games and Pastimes""Charles W. Trigg","Mathematical Quickies""G. Ţiţeica","Probleme de Geometrie""Radu Gologan, Dan Schwarz","ROMANIAN MATHEMATICAL COMPETITIONS 2008""Jiagu Xu","Lecture Notes on Mathematical Olympiad Courses For Junior Section vol. 1 vol. 2""Daniel J. Velleman","How to Prove It 2e 2006""Titu Andreescu, Zuming Feng","101 Problems in Algebra: From the Training of the USA IMO Team""Xiong Bin, Lee Peng Yee","Mathematical Olympiad in China (2007-2008): Problems and Solutions 2008""Christopher J. Bradley","Challenges in Geometry: For Mathematical Olympians Past and Present""Titu Andreescu, Răzvan Gelca","Putnam and Beyond""Andrew M. Gleason","William Lowell Putnam Mathematical Competition Problems and Solutions: 1938-1964""Gerald L. Alexanderson, Leonard F. Klosinski, Loren C. Larson","The William Lowell Putnam Mathematical Competition Problems and Solutions 1965-1984""Kiran S. Kedlaya, Bjorn Poonen, Ravi Vakil","The William Lowell Putnam Mathematical Competition 1985-2000""Gustave Choquet","What Is Modern Mathematics""Constance Reid","From Zero to Infinity 3e 1965""Martin Gardner","The Colossal Book of Mathematics""Dan Brânzei, loan Şerdean, Vasile Şerdean","Junior Balkan mathematical olympiads""Hans Lausch, Peter J. Taylor","Australian Mathematical Olympiads 1979-1995""Marcin E. Kuczma","International Mathematical Olympiads, 1986-1999""Arkadii M. Slinko","USSR Mathematical Olympiads 1989-1992""Titu Andreescu, Zuming Feng, Gerald L. Alexanderson, Roger B. Nelsen","Mathematical Olympiads 1998-1999 Problems and Solutions from around the World""Marcin E. Kuczma, Erich Windischbacher","Polish and Austrian Mathematics Olympiads: 1981-1995 Selected Problems with Multiple Solutions""Titu Andreescu, Zuming Feng, Gerald L. Alexanderson, Roger B. Nelsen","Mathematical Olympiads 1999-2000 Problems and Solutions from around the World""Titu Andreescu, Zuming Feng, George L. Lee","Mathematical Olympiads 2000-2001 Problems and Solutions from around the World""le Hai Chau, le Hai Khoi","Selected Problems of the Vietnamese Mathematical Olympiad""Arthur Engel","Problem-Solving Strategies""Dusan Djukic, Vladimir Z. Jankovic, Ivan Matic, Nikola Petrovic","The IMO Compendium - A Collection of Problems Suggested for the International Mathematical Olympiads: 1959-2004""Alexander Soifer","The Colorado Mathematical Olympiad and Further Explorations""Edward Lozansky, Cecil Rousseau","Winning Solutions""Titu Andreescu, BOGDAN ENESCU","Mathematical Olympiad Treasures 2e 2011""Ross Honsberger","From Erdos to Kiev - Problems of Olympiad Caliber""György Hajós, Jjózsef Kürschák, G. NEUKOMM, János Surányi, Elvira Rapaport","Hungarian Problem Book II Based on the EOTVOS Competitions, 1906-1928""György Hajós, G. NEUKOMM, János Surányi, Chiang-Fung Andrew Liu","Hungarian Problem Book III Based on the EOTVOS Competitions, 1929-1943""Titu Andreescu, Zuming Feng","USA and International Mathematical Olympiads 2003""Titu Andreescu, Zuming Feng, Po-Shen Loh","USA and International Mathematical Olympiads 2004""MIRCEA BECHEANU","International Mathematical Olympiads 1959-2000 - Problems - Solutions - Results""Michael Th. Rassias","Problem-Solving and Selected Topics in Number Theory: In the Spirit of the Mathematical Olympiads""Dusan Djukic, Vladimir Z. Jankovic, Ivan Matic, Nikola Petrovic","The Imo Compendium - A Collection of Problems Suggested for the International Mathematical Olympiads: 1959-2009 2e 2011""Andy Liu","Chinese Mathematics Competitions & Olympiads. [1]. 1981 - 1993""Andy Liu","Chinese Mathematics Competitions and Olympiads, 1993-2001""István Reiman","International Mathematical Olympiad 1959 - 1999""Radmila Bulajich Manfrino, Rogelio Valdez Delgado","Inequalities - A Mathematical Olympiad Approach""Hans-Dietrich Gronau, Hanns-Heinrich Langmann, Dierk Schleicher","50th IMO - 50 Years of International Mathematical Olympiads""C. Pranesachar, B. Venkatachala, C. Yogananda","Problem primer for the olympiad""Andrei Neguț","Problems for the Mathematical Olympiads: From the First Team Selection Test to the IMO""Yao Zhang","Combinatorial Problems in Mathematical Competitions""Organising Committee ***","44th International Mathematical Olympiad Short-listed problems and solutions 2003""Titu Andreescu","Mathematical Olympiads 1997-1998 Problems and Solutions from around the World""Titu Andreescu, Răzvan Gelca, Mark Saul","Mathematical Olympiad Challenges""Titu Andreescu","Mathematical Olympiads 1996-1997 Problems and Solutions from around the World""Cao Minh Quang","The best problems from around the world""Kiran S. Kedlaya","Notes on Euclidean Geometry""Samin Riasat","Basics of olympiad inequalities""Stefan Straszewicz","Mathematical Problems and Puzzles - from the Polish Mathematical Olympiads""Thomas J. Mildorf","Olympiad inequalities""Titu Andreescu, Kiran Kedlaya, Paul Zeitz","Mathematical Contests 1995 - 1996: Olympiad Problems and Solutions from Around the World""Miklós Schweitzer, Gábor J. Székely","Contests in Higher Mathematics""Ralph A. Artino, Anthony M. Gaglione, Niel Shell","The Contest Problem Book IV""Steven R. Conrad, Daniel Flegler","Math Contests 1982-1991 vol. 2""Steven R. 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Solutions""Anany Levitin, Maria Levitin","Algorithmic Puzzles""William L Briggs","Ants, Bikes, & Clocks""Wolfgang Schwarz","40 Puzzles and Problems in Probability and Mathematical Statistics""Paul Zeitz","The Art and Craft of Problem Solving 2e 2007""Paulo Ney de Souza, Jorge-Nuno Silva","Berkeley Problems in Mathematics""Wayne A. Wickelgren","How to Solve Mathematical Problems""Eugene Don, Benay Don","How to Solve Word Problems in Calculus""Stephen Ainley","Mathematical Puzzles""Peter Winkler","Mathematical Puzzles""Alexander Soifer","Mathematics as Problem Solving 2e 2009""Ji-Xiu Chen, Jiang Guo-Ying, Pan Yang-Lian, Qin Tie-Hu, Tong Yu-Sun, Wu Quan-Shui, Xu Shen-Zhi, Ta-Chien Li","Problems and Solutions in Mathematics""J. Marshall Unger","A Collection of Sangaku Problems - A Set of Beautiful Japanese Geometry Theorems""Christopher G. Small","Functional Equations and How to Solve Them""Daniel Shanks","Solved and Unsolved Problems in Number Theory""B. A. Kordemskii","The Moscow Puzzles: 359 Mathematical Recreations""G. Polya, J. Kilpatrick","The Stanford Mathematics Problem Book - With Hints and Solutions""Loren C. Larson","Problem Solving Through Problems""Paul Vaderlind, Loren C. Larson, Richard K. Guy","The Inquisitive Problem Solver""George Thomas Gilbert, Mark Krusemeyer, Loren C. Larson","Wohascum County Problem Book""Xiong Bin, Lee Peng Yee","Mathematical Olympiad in China: Problems and Solutions""Shay Gueron","Hungary-Israel Mathematics Competition""György Hajós, József Kürschák, G. NEUKOMM, J. SURANYI, Elvira Rapaport","Hungarian Problem Book I BASED ON THE EOTVOS COMPETITIONS, 1894-1905""Peter J. Taylor, Andrei M. Storozhev","Tournament of the Towns 1993-1997""Peter J. Taylor","International Mathematics Tournament of Towns, Book 2: 1984-1989""A Storozhev","International Mathematics Tournament of Towns. 5. 1997 - 2002""Peter J. Taylor","International Mathematics Tournament of the Towns, Book 3: 1989-1993""Michael Doob","The Canadian Mathematical Olympiad 1969-1993""MIRCEA BECHEANU, MIHAl BĂLUNĂ, BOGDAN ENESCU","ROMANIAN MATHEMATICAL COMPETITIONS 1996""Richard Rusczyk","Introduction to Algebra""G. M. Kendall P.m.h.; Thomas","Mathematical Puzzles for the Connoisseur""Marcin E Kuczma","Problems: 144 Problems of the Austrian-Polish Mathematics Competition, 1978-1993""Paul Zeitz","The Art and Craft of Problem Solving""Cao Minh Quang","Best Problems From Around the World - Mathematical Olympiads""Ross Honsberger","Mathematical Diamonds""Terence Tao","Solving Mathematical Problems: A Personal Perspective""Titu Andreescu, Dorin Andrica, Zuming Feng","One Hundred Four Number Theory Problems""Michael Holt","Math Puzzles and Games""Michael Holt","More Math Puzzles and Games""Jack Frohlichstein","Mathematical Fun, Games and Puzzles""National Council Of Teachers Of Mathematics ***","Enrichment Mathematics for the Grades 27e""Terry H. Stickels","Math Puzzles and Brainteasers, Grades 6-8""Bradley W. Jackson, Dmitri Thoro","Applied Combinatorics With Problem Solving""Peter Boyvalenkov, Oleg Mushkarov, Emil Kolev, Nikolai Nikolov","Bulgarian Mathematical Competitions [2003-2006] - et al., (GIL, 2007) WW""Dudeney Henry Ernest","536 Puzzles and Curious Problems.""George Berzsenyi, Stephen B. Maurer","The Contest Problem Book V - AHSME and AIME 198S-1988""Steven George Krantz","Techniques of Problem Solving""Vasile Cirtoaje","Algebraic inequalities""Alfred S. Posamentier, Wolfgang Schulz","The Art of Problem Solving""Alfred S. Posamentier, Stephen Krulik","Problem-Solving Strategies for Efficient and Elegant Solutions""Alfred S. Posamentier, Ingmar Lehmann","Mathematical Amazements and Surprises""W. P. Ball, H. S. M. Coxeter","Mathematical Recreations and Essays""Alfred S. Posamentier","Math Wonders to Inspire Teachers and Students""Alfred Steven Posamentier, Stephen Krulik","Problem Solving in Mathematics, Grades 3-6""Robert P. Allen, Mensa","Ultimate Mental Challenge""Charles Townsned","World's Hardest Puzzles""Michael Holt","More Math Puzzles and Games""Peter Gordon, Mike Shenk, Mayme Allen, Janine Kelsch, Mark Danna, James Richard Sukach","Great Book of Puzzles""Muriel Mandell, Elise Chanowitz","Fantastic Book of Logic Puzzles""Lazlo C. Bardos, Samuel Carbaugh","Amazing Math Projects You Can Build Yourself""André Sainte-Laguë","Avec des nombres et des lignes: Récréations mathématiques""Allison Bartl, Klaus Puth","101 Quick Thinking Games and Riddles""James G. Nourse, Dusan Krajan","The Simple Solution to Rubik's Cube""Jerry Slocum, Dic Sonneveld","The 15 Puzzle Book""Persi Diaconis, Ron Graham","Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks""Arcturus Pub Group","Huge Book of Puzzles"
How can I heal my spine without surgery?
R2AI realise you have tried some form of chiropractic and some exercise based physiotherapy and that is the logical choice before any major surgery.I have written on similar questions on Quora beforeJoel V Simon's answer to My mom has a terrible back ache. It's due to minimal slipped disc in L4 and L5. What can cure her?Joel V Simon's answer to What is the best remedy for a slipped disc?Please read through them and see if you have any doubts. I have seen really ugly looking bulges way worse than yours which present with NO pain and some with scans which are way more innocuous than yours with life altering pain, numbness, tingling into the feet and weakness. All is not as it seems!If you are looking for empirical evidence to what I'm saying, there is a ton of it.Myth – Moving will make my back pain worse.Although it is true that some movements can be uncomfortable when you have back pain, it is well established that returning to movement and work as soon as you are able, is better for recovery and preventing recurrence than bed rest [1][7].This is not a new concept by any means [6], but it is an unfortunate misconception which is continues to endure, due in part, to the complex nature of pain [2][3][4][5].#MotionIsLotionBalagu, F. et al., 2012. Non-specific low back pain. The Lancet, 379(9814), pp.482–491.Darlow, B. et al., 2015. Easy to Harm, Hard to Heal. Spine, (August 2016), p.1.Picavet, H.S.J., Vlaeyen, J.W.S. & Schouten, J.S.A.G., 2002. Pain catastrophizing and kinesiophobia: Predictors of chronic low back pain. American Journal of Epidemiology, 156(11), pp.1028–1034.Pincus, T. et al., 2002. A systematic review of psychological factors as predictors of chronicity/disability in prospective cohorts of low back pain. Spine (Phila Pa 1976), 27(5), pp.E109–20.Swinkels-Meewisse, I.E.J. et al., 2006. Acute low back pain: Pain-related fear and pain catastrophizing influence physical performance and perceived disability. Pain, 120(1-2), pp.36–43.Waddell, G., 1993.Simple low back pain: rest or active exercise? Annals of the rheumatic diseases, 52(5), p.317.Wynne-Jones, G. et al., 2014. Absence from work and return to work in people with back pain: a systematic review and meta-analysis. Occupational and environmental medicine, 71(6), pp.448–56.Myth – I should avoid exercise, especially weight trainingExercise is generally accepted amongst all respected authorities to be the best modality for treating low back pain in both the acute and chronic phases [1].Studies have shown great benefits and long-term safety of various types of exercises [2][3][4][5] including high load resistance training [5][6].Interestingly, no one type of exercise proves to be better or worse, [7] so simply do what you enjoy and can tolerate! Gradually build up as your confidence and ability improves.#StrongerForLongerO’Sullivan and Lin (2014) Acute low back pain Beyond drug therapies; Pain Management Today, Volume 1, Number 1.Steele et al (2015) A Review of the Clinical Value of Isolated Lumbar Extension Resistance Training for Chronic Low Back Pain; American Academy of Physical Medicine and Rehabilitation Volume 7, Issue 2, Pages 169–187.Searle et al (2015) Exercise interventions for the treatment of chronic low back pain: a systematic review and meta-analysis of randomised controlled trials; Clinical Rehabilitation 2015, Vol. 29(12) 1155 –1167.Bjorn et al (2015) Individualized Low-Load Motor Control Exercises and Education Versus a High-Load Lifting Exercise and Education to Improve Activity, Pain Intensity, and Physical Performance in Patients With Low Back Pain: A Randomized Controlled Trial; Journal of Orthopaedic & Sports Physical Therapy, Volume:45 Issue:2 Pages:77-85.Pieber et al (2014) Long-term effects of an outpatient rehabilitation program in patients with chronic recurrent low back pain; Eur Spine J 23:779–785.Vincent et al (2014) Resistance Exercise, Disability, and Pain Catastrophizing in Obese Adults with Back Pain; Med Sci Sports Exerc. 46(9): 1693–170Smith et al (2014) An update of stabilisation exercises for low back pain: a systematic review with meta-analysis. BMC Musculoskeletal Disorders 15:416 DOI: 10.1186/1471-2474-15-416Myth – A scan will tell me exactly what’s wrongThere is a large and growing body of research that shows that not only do results of scans correlate poorly with symptoms in people with Low Back Pain [2, 4, 5], but also that most people without Low Back Pain have changes on scans and x-rays that do not cause any symptoms at all [1, 3, 5].For these reasons and more, imaging alone isn’t capable of telling us exactly why someone is experiencing pain.Of course this does not mean that all MRI scans are irrelevant in all cases, but it does mean that they are not always necessary or helpful. In fact, there is evidence to suggest that in some cases, having a scan can make situations worse [6].#ThePlanNotTheScanTeraguchi et al, (2013) Prevalence and distribution of intervertebral disc degeneration over the entire spine in a population-based cohort: the Wakayama Spine Study.Videman et al, (2003) Associations Between Back Pain History and Lumbar MRI FindingsCheung et al, (2009) Prevalence and Pattern of Lumbar Magnetic Resonance Imaging Changes in a Population Study of One Thousand Forty-Three Individuals.Endcan et al, (2011) Potential of MRI findings to refine case definition for mechanical low back pain in epidemiological studies: a systematic review.Brinjikji et al, (2015) MRI Findings of Disc Degeneration are More Prevalent in Adults with Low Back Pain than in Asymptomatic Controls: A Systematic Review and Meta-AnalysisWebster et al, (2010) Relationship of early magnetic resonance imaging for work-related acute low back pain with disability and medical utilization outcomes.Myth - Pain equals damageThe level of pain experienced is very rarely proportional to the amount injury sustained to the back. Pain is far more complex than this, as pain levels are a reflection of how threatened each human perceives itself to be.For example, past experiences, general health factors, beliefs, sleep and exercise levels as well as psychological wellbeing, all play important parts in how much pain each individual might experiences. [1][2][3][4][5][6]#SoreButSafeShiri et al (2013) The role of obesity and physical activity in non-specific and radiating low back pain: The Young Finns study; Seminars In Arthritis and Rheumatism June 2013, vol.42(6):640-650Taylor et al (2014) Incidence and risk factors for first-time incident low back pain: a systematic review and meta-analysis: The Spine Journal October 2014, Vol.14(10):2299-2319Manchikanti et al (2014) Epidemiology of Low Back Pain In Adults. Neuromodulation: Technology at the Neural Interface, Vol.17: 3–10.George et al. (2012) Predictors of Occurrence and Severity of First Time Low Back Pain Episodes: Findings from a Military Inception Cohort. PLoS ONE 7(2): e30597Wilkens et al (2013) Prognostic Factors of Prolonged Disability in Patients with Low Back Pain and Lumbar Degeneration in Primary Care: A Cohort Study. Spine. Jan 1;38(1):65-74Campbell et al (2013) Prognostic Indicators of Low Back Pain in Primary Care: Five-Year Prospective Study. The Journal of Pain. August 2013. Vol.14(8):873-883You need to weigh the risks and benefits very carefully when it comes to surgery- there is no going back.Sticking with gradual exercise tolerance is evidence based and at worst you are only postponing your surgery for later! There's a lot of info in the previous Quora answers I linked above regarding gentle exercises for discogenic back pain.All the best.
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