The Unexpected Hanging and Other Mathematical Diversions
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On Ramified Covers of the Projective Plane I: Segre's Theory And
ON RAMIFIED COVERS OF THE PROJECTIVE PLANE I: SEGRE’S THEORY AND CLASSIFICATION IN SMALL DEGREES (WITH AN APPENDIX BY EUGENII SHUSTIN) MICHAEL FRIEDMAN AND MAXIM LEYENSON1 Abstract. We study ramified covers of the projective plane P2. Given a smooth surface S in Pn and a generic enough projection Pn → P2, we get a cover π : S → P2, which is ramified over a plane curve B. The curve B is usually singular, but is classically known to have only cusps and nodes as singularities for a generic projection. Several questions arise: First, What is the geography of branch curves among all cuspidal-nodal curves? And second, what is the geometry of branch curves; i.e., how can one distinguish a branch curve from a non-branch curve with the same numerical invariants? For example, a plane sextic with six cusps is known to be a branch curve of a generic projection iff its six cusps lie on a conic curve, i.e., form a special 0-cycle on the plane. We start with reviewing what is known about the answers to these questions, both simple and some non-trivial results. Secondly, the classical work of Beniamino Segre gives a complete answer to the second question in the case when S is a smooth surface in P3. We give an interpretation of the work of Segre in terms of relation between Picard and Chow groups of 0-cycles on a singular plane curve B. We also review examples of small degree. In addition, the Appendix written by E. Shustin shows the existence of new Zariski pairs. -
Iterations of the Words-To-Numbers Function
1 2 Journal of Integer Sequences, Vol. 21 (2018), 3 Article 18.9.2 47 6 23 11 Iterations of the Words-to-Numbers Function Matthew E. Coppenbarger School of Mathematical Sciences 85 Lomb Memorial Drive Rochester Institute of Technology Rochester, NY 14623-5603 USA [email protected] Abstract The Words-to-Numbers function takes a nonnegative integer and maps it to the number of characters required to spell the number in English. For each k = 0, 1,..., 8, we determine the minimal nonnegative integer n such that the iterates of n converge to the fixed point (through the Words-to-Numbers function) in exactly k iterations. Our travels take us from some simple answers, to an etymology of the names of large numbers, to Conway/Guy’s established system representing the name of any integer, and finally use generating functions to arrive at a very large number. 1 Introduction and initial definitions The idea for the problem originates in a 1965 book [3, p. 243] by recreational linguistics icon Dmitri Borgmann (1927–85) and in a 1966 article [16] by another icon, recreational mathematician Martin Gardner (1914–2010). Gardner’s article appears again in a book [17, pp. 71–72, 265–267] containing reprinted Gardner articles from Scientific American along with his comments on his reader’s responses. Both Borgmann and Gardner identify the only English number that requires exactly the same number of characters to spell the number as the number represents. Borgmann goes further by identifying numbers in 16 other languages with this property. Borgmann calling such numbers truthful and Gardner calling them honest. -
Intransitivity and Future Generations: Debunking Parfit's Mere Addition Paradox
Journal of Applied Philosophy, Vol. 20, No. 2,Intransitivity 2003 and Future Generations 187 Intransitivity and Future Generations: Debunking Parfit’s Mere Addition Paradox KAI M. A. CHAN Duties to future persons contribute critically to many important contemporaneous ethical dilemmas, such as environmental protection, contraception, abortion, and population policy. Yet this area of ethics is mired in paradoxes. It appeared that any principle for dealing with future persons encountered Kavka’s paradox of future individuals, Parfit’s repugnant conclusion, or an indefensible asymmetry. In 1976, Singer proposed a utilitarian solution that seemed to avoid the above trio of obstacles, but Parfit so successfully demonstrated the unacceptability of this position that Singer abandoned it. Indeed, the apparently intransitivity of Singer’s solution contributed to Temkin’s argument that the notion of “all things considered better than” may be intransitive. In this paper, I demonstrate that a time-extended view of Singer’s solution avoids the intransitivity that allows Parfit’s mere addition paradox to lead to the repugnant conclusion. However, the heart of the mere addition paradox remains: the time-extended view still yields intransitive judgments. I discuss a general theory for dealing with intransitivity (Tran- sitivity by Transformation) that was inspired by Temkin’s sports analogy, and demonstrate that such a solution is more palatable than Temkin suspected. When a pair-wise comparison also requires consideration of a small number of relevant external alternatives, we can avoid intransitivity and the mere addition paradox without infinite comparisons or rejecting the Independence of Irrelevant Alternatives Principle. Disagreement regarding the nature and substance of our duties to future persons is widespread. -
Combination of Cubic and Quartic Plane Curve
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 43-53 www.iosrjournals.org Combination of Cubic and Quartic Plane Curve C.Dayanithi Research Scholar, Cmj University, Megalaya Abstract The set of complex eigenvalues of unistochastic matrices of order three forms a deltoid. A cross-section of the set of unistochastic matrices of order three forms a deltoid. The set of possible traces of unitary matrices belonging to the group SU(3) forms a deltoid. The intersection of two deltoids parametrizes a family of Complex Hadamard matrices of order six. The set of all Simson lines of given triangle, form an envelope in the shape of a deltoid. This is known as the Steiner deltoid or Steiner's hypocycloid after Jakob Steiner who described the shape and symmetry of the curve in 1856. The envelope of the area bisectors of a triangle is a deltoid (in the broader sense defined above) with vertices at the midpoints of the medians. The sides of the deltoid are arcs of hyperbolas that are asymptotic to the triangle's sides. I. Introduction Various combinations of coefficients in the above equation give rise to various important families of curves as listed below. 1. Bicorn curve 2. Klein quartic 3. Bullet-nose curve 4. Lemniscate of Bernoulli 5. Cartesian oval 6. Lemniscate of Gerono 7. Cassini oval 8. Lüroth quartic 9. Deltoid curve 10. Spiric section 11. Hippopede 12. Toric section 13. Kampyle of Eudoxus 14. Trott curve II. Bicorn curve In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation It has two cusps and is symmetric about the y-axis. -
Mathematical Circus & 'Martin Gardner
MARTIN GARDNE MATHEMATICAL ;MATH EMATICAL ASSOCIATION J OF AMERICA MATHEMATICAL CIRCUS & 'MARTIN GARDNER THE MATHEMATICAL ASSOCIATION OF AMERICA Washington, DC 1992 MATHEMATICAL More Puzzles, Games, Paradoxes, and Other Mathematical Entertainments from Scientific American with a Preface by Donald Knuth, A Postscript, from the Author, and a new Bibliography by Mr. Gardner, Thoughts from Readers, and 105 Drawings and Published in the United States of America by The Mathematical Association of America Copyright O 1968,1969,1970,1971,1979,1981,1992by Martin Gardner. All riglhts reserved under International and Pan-American Copyright Conventions. An MAA Spectrum book This book was updated and revised from the 1981 edition published by Vantage Books, New York. Most of this book originally appeared in slightly different form in Scientific American. Library of Congress Catalog Card Number 92-060996 ISBN 0-88385-506-2 Manufactured in the United States of America For Donald E. Knuth, extraordinary mathematician, computer scientist, writer, musician, humorist, recreational math buff, and much more SPECTRUM SERIES Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications ANDREW STERRETT, JR.,Chairman Spectrum Editorial Board ROGER HORN, Chairman SABRA ANDERSON BART BRADEN UNDERWOOD DUDLEY HUGH M. EDGAR JEANNE LADUKE LESTER H. LANGE MARY PARKER MPP.a (@ SPECTRUM Also by Martin Gardner from The Mathematical Association of America 1529 Eighteenth Street, N.W. Washington, D. C. 20036 (202) 387- 5200 Riddles of the Sphinx and Other Mathematical Puzzle Tales Mathematical Carnival Mathematical Magic Show Contents Preface xi .. Introduction Xlll 1. Optical Illusions 3 Answers on page 14 2. Matches 16 Answers on page 27 3. -
Type Title Author Publisher ISBN # Pages Length PFLAG Cincinnati Lending Library
PFLAG Cincinnati Lending Library # Type Title Author Publisher ISBN Pages Length Audio Accepting your Gay or Lesbian Child: Parents Share their Stories Sounds True Recordings Audio Coming Out: Kate Ramsdell Audio Reconciling Ministry Conference April 2002 MUSE, Jimmy Creech, Tanya Pike Audio Steve Schalchlin – PFLAG Cincinnati Annual Fundraiser 02/27/99 Audio Stranger in Our Midst (2 volume) Bishop Paul Egertson Audio What Matters: A Benefit CD for Matthew Shepard Foundation Randi Driscoll Audio CD Boy Meets Boy Levithan, David Full Cast Audio 345 Book (Lost)Found #1 Adams, Marc Window Books 1-889829-10-2 127 Book (Lost)Found #2 Adams, Marc Window Books 1-889829-10-2 127 Book (Lost)Found #3 Adams, Marc Window Books 1-889829-10-2 127 Book A Mother Looks At the Gay Child Davis, Jesse New Falcon Publications 1-56184-126-9 160 Book A Face in the Crowd, Expressions of Gay Life in America Peterson, John and Martin Prospect Publishing 0-9719618-0-8 132 Bedogne Book A Fire Engine for Ruthie Newman, Leslea Clarion Books 0-618-15989-4 32 Book A Journey to Moriah Murray, Rhea Banta & Pool, Publishers 0-9667237-0-8 149 Book A Member of the Family: Gay Men Write about Their Families Edit: Preston, John A Dutton Book 0-525-93549-5 309 Book A Place at the Table: The Gay Individual in American Society (2 Bawer, Bruce Poseidon Press 0-671-79533-3 269 Copies) Book A Promise To Remember The Names Project Book of Letters: Edit: Brown, Joe Avon Books 0-380-76711-2 323 Remembrances of Love from the Contributors to the Quilt Book About Face: A Gay Officer’s Account of How He Stopped Kennedy, James E A Birch Lane Press Book 1-55972-281-9 302 Prosecuting Gays in the Army and Started Fighting for Their Rights (autographed) Book Acts of Disclosure: The Coming-Out Process of Contemporary Vargo MS, Marc E. -
A Set of Solutions to Parfit's Problems
NOÛS 35:2 ~2001! 214–238 A Set of Solutions to Parfit’s Problems Stuart Rachels University of Alabama In Part Four of Reasons and Persons Derek Parfit searches for “Theory X,” a satisfactory account of well-being.1 Theories of well-being cover the utilitarian part of ethics but don’t claim to cover everything. They say nothing, for exam- ple, about rights or justice. Most of the theories Parfit considers remain neutral on what well-being consists in; his ingenious problems concern form rather than content. In the end, Parfit cannot find a theory that solves each of his problems. In this essay I propose a theory of well-being that may provide viable solu- tions. This theory is hedonic—couched in terms of pleasure—but solutions of the same form are available even if hedonic welfare is but one aspect of well-being. In Section 1, I lay out the “Quasi-Maximizing Theory” of hedonic well-being. I motivate the least intuitive part of the theory in Section 2. Then I consider Jes- per Ryberg’s objection to a similar theory. In Sections 4–6 I show how the theory purports to solve Parfit’s problems. If my arguments succeed, then the Quasi- Maximizing Theory is one of the few viable candidates for Theory X. 1. The Quasi-Maximizing Theory The Quasi-Maximizing Theory incorporates four principles. 1. The Conflation Principle: One state of affairs is hedonically better than an- other if and only if one person’s having all the experiences in the first would be hedonically better than one person’s having all the experiences in the second.2 The Conflation Principle sanctions translating multiperson comparisons into single person comparisons. -
Unexpected Hanging Paradox - Wikip… Unexpected Hanging Paradox from Wikipedia, the Free Encyclopedia
2-12-2010 Unexpected hanging paradox - Wikip… Unexpected hanging paradox From Wikipedia, the free encyclopedia The unexpected hanging paradox, hangman paradox, unexpected exam paradox, surprise test paradox or prediction paradox is a paradox about a person's expectations about the timing of a future event (e.g. a prisoner's hanging, or a school test) which he is told will occur at an unexpected time. Despite significant academic interest, no consensus on its correct resolution has yet been established.[1] One approach, offered by the logical school of thought, suggests that the problem arises in a self-contradictory self- referencing statement at the heart of the judge's sentence. Another approach, offered by the epistemological school of thought, suggests the unexpected hanging paradox is an example of an epistemic paradox because it turns on our concept of knowledge.[2] Even though it is apparently simple, the paradox's underlying complexities have even led to it being called a "significant problem" for philosophy.[3] Contents 1 Description of the paradox 2 The logical school 2.1 Objections 2.2 Leaky inductive argument 2.3 Additivity of surprise 3 The epistemological school 4 See also 5 References 6 External links Description of the paradox The paradox has been described as follows:[4] A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day. -
Russell's Paradox in Appendix B of the Principles of Mathematics
HISTORY AND PHILOSOPHY OF LOGIC, 22 (2001), 13± 28 Russell’s Paradox in Appendix B of the Principles of Mathematics: Was Frege’s response adequate? Ke v i n C. Kl e m e n t Department of Philosophy, University of Massachusetts, Amherst, MA 01003, USA Received March 2000 In their correspondence in 1902 and 1903, after discussing the Russell paradox, Russell and Frege discussed the paradox of propositions considered informally in Appendix B of Russell’s Principles of Mathematics. It seems that the proposition, p, stating the logical product of the class w, namely, the class of all propositions stating the logical product of a class they are not in, is in w if and only if it is not. Frege believed that this paradox was avoided within his philosophy due to his distinction between sense (Sinn) and reference (Bedeutung). However, I show that while the paradox as Russell formulates it is ill-formed with Frege’s extant logical system, if Frege’s system is expanded to contain the commitments of his philosophy of language, an analogue of this paradox is formulable. This and other concerns in Fregean intensional logic are discussed, and it is discovered that Frege’s logical system, even without its naive class theory embodied in its infamous Basic Law V, leads to inconsistencies when the theory of sense and reference is axiomatized therein. 1. Introduction Russell’s letter to Frege from June of 1902 in which he related his discovery of the Russell paradox was a monumental event in the history of logic. However, in the ensuing correspondence between Russell and Frege, the Russell paradox was not the only antinomy discussed. -
Lecture 5. Zeno's Four Paradoxes of Motion
Lecture 5. Zeno's Four Paradoxes of Motion Science of infinity In Lecture 4, we mentioned that a conflict arose from the discovery of irrationals. The Greeks' rejection of irrational numbers was essentially part of a general rejection of the infinite process. In fact, until the late 19th century most mathematicians were reluctant to accept infinity. In his last observations on mathematics1, Hermann Weyl wrote: \Mathematics has been called the science of the infinite. Indeed, the mathematician invents finite constructions by which questions are decided that by their very nature refer to the infinite. That is his glory." Besides the conflict of irrational numbers, there were other conflicts about the infinite process, namely, Zeno's famous paradoxes of motion from around 450 B.C. Figure 5.1 Zeno of Citium Zeno's Paradoxes of motion Zeno (490 B.C. - 430 B.C.) was a Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides2. We know little 1Axiomatic versus constructive procedures in mathematics, The Mathematical Intelligencer 7 (4) (1985), 10-17. 2Parmenides of Elea, early 5th century B.C., was an ancient Greek philosopher born in Elea, a Greek city on the southern coast of Italy. He was the founder of the Eleatic school of philosophy. For his picture, see Figure 2.3. 32 about Zeno other than Plato's assertion that he went to Athens to meet with Socrates when he was nearly 40 years old. Zeno was a Pythegorean. By a legend, he was killed by a tyrant of Elea whom he had plotted to depose. -
Signed Modern Firsts I: Achebe–Kunzru
Pryor-Johnson Rare Books Signed Modern Firsts I: Achebe-Kunzru [1123 Broadway, Suite 517, New York, NY 10075] [email protected] What an odd time to be an antiquarian bookseller. As the whole world strives to heal while convulsing with rage against injustice, issue points and binding variants can seem pitifully insignificant. We slip into the shop masked and gloved to wrap and dispatch books, but 1123 Broadway has not now for months been a haven of bibliophilia, with baroque music playing and a pot of coffee freshly brewed. Yet we hear from our clients and feel ourselves that cultivating bibliomania — even reading the books — has been a balm. Reading has been an escape as well as a means of rediscovering some of the humanity that’s been eroded by seclusion. Signed books have always had to do with imputed distance: the author’s signature in it is a marker of its having passed through her hands, sometime somewhere. As our own geographical remit has narrowed, a signed book now feels even more like a piece of moon fallen into our hands. We feel it is vital to amplify black, brown and native voices. Thus we highlight the following authors in this list: Achebe, Borges, Danticat, Erdrich, Fuentes, Garcia-Aguilera, Iweala and Kunzru. Their work travers- es style, subject and period. There is no one “literature of color,” nor do we wish to tokenize or fetishize these authors’ work. Bringing the work of people of color to the fore nevertheless feels like a small but powerful role the world of antiquarian books can play — a safeguard against our becoming truly irrelevant or frivolous. -
Catalogue 143 ~ Holiday 2008 Contents
Between the Covers - Rare Books, Inc. 112 Nicholson Rd (856) 456-8008 will be billed to meet their requirements. We accept Visa, MasterCard, American Express, Discover, and Gloucester City NJ 08030 Fax (856) 456-7675 PayPal. www.betweenthecovers.com [email protected] Domestic orders please include $5.00 postage for the first item, $2.00 for each item thereafter. Images are not to scale. All books are returnable within ten days if returned in Overseas orders will be sent airmail at cost (unless other arrange- the same condition as sent. Books may be reserved by telephone, fax, or email. ments are requested). All items insured. NJ residents please add 7% sales tax. All items subject to prior sale. Payment should accompany order if you are Members ABAA, ILAB. unknown to us. Customers known to us will be invoiced with payment due in 30 days. Payment schedule may be adjusted for larger purchases. Institutions Cover verse and design by Tom Bloom © 2008 Between the Covers Rare Books, Inc. Catalogue 143 ~ Holiday 2008 Contents: ................................................................Page Literature (General Fiction & Non-Fiction) ...........................1 Baseball ................................................................................72 African-Americana ...............................................................55 Photography & Illustration ..................................................75 Children’s Books ..................................................................59 Music ...................................................................................80