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Fractions of Numerical Semigroups
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2010 Fractions of Numerical Semigroups Harold Justin Smith University of Tennessee - Knoxville, [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Algebra Commons, and the Number Theory Commons Recommended Citation Smith, Harold Justin, "Fractions of Numerical Semigroups. " PhD diss., University of Tennessee, 2010. https://trace.tennessee.edu/utk_graddiss/750 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Harold Justin Smith entitled "Fractions of Numerical Semigroups." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Mathematics. David E. Dobbs, Major Professor We have read this dissertation and recommend its acceptance: David F. Anderson, Pavlos Tzermias, Michael W. Berry Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) To the Graduate Council: I am submitting herewith a dissertation written by Harold Justin Smith entitled \Fractions of Numerical Semigroups." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the require- ments for the degree of Doctor of Philosophy, with a major in Mathematics. -
Tempered Monoids of Real Numbers, the Golden Fractal Monoid, and the Well-Tempered Harmonic Semigroup
Tempered Monoids of Real Numbers, the Golden Fractal Monoid, and the Well-Tempered Harmonic Semigroup Published in Semigroup Forum, Springer, vol. 99, n. 2, pp. 496-516, November 2019. ISSN: 0037-1912. Maria Bras-Amoros´ November 5, 2019 Abstract This paper deals with the algebraic structure of the sequence of harmonics when combined with equal temperaments. Fractals and the golden ratio appear surprisingly on the way. + The sequence of physical harmonics is an increasingly enumerable submonoid of (R ; +) whose pairs of consecutive terms get arbitrarily close as they grow. These properties suggest the definition of a new mathematical object which we denote a tempered monoid. Mapping the elements of the tempered monoid of physical harmonics from R to N may be considered tantamount to defining equal temperaments. The number of equal parts of the octave in an equal temperament corresponds to the multiplicity of the related numerical semigroup. Analyzing the sequence of musical harmonics we derive two important properties that tempered monoids may have: that of being product-compatible and that of being fractal. We demonstrate that, up to normal- ization, there is only one product-compatible tempered monoid, which is the logarithmic monoid, and there is only one nonbisectional fractal monoid which is generated by the golden ratio. The example of half-closed cylindrical pipes imposes a third property to the sequence of musical har- monics, the so-called odd-filterability property. We prove that the maximum number of equal divisions of the octave such that the discretizations of the golden fractal monoid and the logarithmic monoid coincide, and such that the discretization is odd- filterable is 12. -
Program of the Sessions San Diego, California, January 9–12, 2013
Program of the Sessions San Diego, California, January 9–12, 2013 AMS Short Course on Random Matrices, Part Monday, January 7 I MAA Short Course on Conceptual Climate Models, Part I 9:00 AM –3:45PM Room 4, Upper Level, San Diego Convention Center 8:30 AM –5:30PM Room 5B, Upper Level, San Diego Convention Center Organizer: Van Vu,YaleUniversity Organizers: Esther Widiasih,University of Arizona 8:00AM Registration outside Room 5A, SDCC Mary Lou Zeeman,Bowdoin upper level. College 9:00AM Random Matrices: The Universality James Walsh, Oberlin (5) phenomenon for Wigner ensemble. College Preliminary report. 7:30AM Registration outside Room 5A, SDCC Terence Tao, University of California Los upper level. Angles 8:30AM Zero-dimensional energy balance models. 10:45AM Universality of random matrices and (1) Hans Kaper, Georgetown University (6) Dyson Brownian Motion. Preliminary 10:30AM Hands-on Session: Dynamics of energy report. (2) balance models, I. Laszlo Erdos, LMU, Munich Anna Barry*, Institute for Math and Its Applications, and Samantha 2:30PM Free probability and Random matrices. Oestreicher*, University of Minnesota (7) Preliminary report. Alice Guionnet, Massachusetts Institute 2:00PM One-dimensional energy balance models. of Technology (3) Hans Kaper, Georgetown University 4:00PM Hands-on Session: Dynamics of energy NSF-EHR Grant Proposal Writing Workshop (4) balance models, II. Anna Barry*, Institute for Math and Its Applications, and Samantha 3:00 PM –6:00PM Marina Ballroom Oestreicher*, University of Minnesota F, 3rd Floor, Marriott The time limit for each AMS contributed paper in the sessions meeting will be found in Volume 34, Issue 1 of Abstracts is ten minutes. -
Ω-Primality in Arithmetic Leamer Monoids
!-PRIMALITY IN ARITHMETIC LEAMER MONOIDS SCOTT T. CHAPMAN AND ZACK TRIPP s Abstract. Let Γ be a numerical semigroup. The Leamer monoid SΓ, for s 2 NnΓ, is the monoid consisting of arithmetic sequences of step size s contained s in Γ. In this note, we give a formula for the !-primality of elements in SΓ when Γ is an numerical semigroup generated by a arithmetic sequence of positive integers. 1. Preliminaries A numerical monoid S is an additive submonoid of the nonnegative integers N0 under regular addition such that jN0 − Sj < 1 ([11] is a good general reference on this subject). A great deal of literature has appeared over the past 15 years which studies the nonunique factorization properties of these monoids (for instance, see [4], [6], and [5] and the references therein). Among the factorization constants studied on these objects is the !-primality function, which in some sense measures how far an element x 2 S is from being a prime element. A general survey of these results can be found in [16], while the papers [2], [3], and [9] all consider issues related to algorithms for computing speciic values of the !-function. Other papers that touch on this subject in more specific terms are [7], [8], [14], and [17]. In this paper, we pick up on the study begun in [12] of the factorization properties of Leamer monoids, which are constructed using numerical monoids. Leamer monoids first appeared in [10] and were used in that paper to study the Huenke-Wiegand conjecture from commutative algebra. In our current work, we address Problem 5.4 of [12] and completely determine the behavior of the ! function on a Leamer monoid generated by an arithmetic numerical monoid (i.e, a numerical monoid generated by an arithmetic sequence of integers). -
Probabilistic Models on Fibre Bundles by Shan Shan
Probabilistic Models on Fibre Bundles by Shan Shan Department of Mathematics Duke University Date: Approved: Ingrid Daubechies, Supervisor Sayan Mukherjee, Chair Doug Boyer Colleen Robles Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 ABSTRACT Probabilistic Models on Fibre Bundles by Shan Shan Department of Mathematics Duke University Date: Approved: Ingrid Daubechies, Supervisor Sayan Mukherjee, Chair Doug Boyer Colleen Robles An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 Copyright c 2019 by Shan Shan All rights reserved Abstract In this thesis, we propose probabilistic models on fibre bundles for learning the gen- erative process of data. The main tool we use is the diffusion kernel and we use it in two ways. First, we build from the diffusion kernel on a fibre bundle a projected kernel that generates robust representations of the data, and we test that it outperforms regular diffusion maps under noise. Second, this diffusion kernel gives rise to a nat- ural covariance function when defining Gaussian processes (GP) on the fibre bundle. To demonstrate the uses of GP on a fibre bundle, we apply it to simulated data on a M¨obiusstrip for the problem of prediction and regression. Parameter tuning can also be guided by a novel semi-group test arising from the geometric properties of dif- fusion kernel. For an example of real-world application, we use probabilistic models on fibre bundles to study evolutionary process on anatomical surfaces. -
An Adiabatic Quantum Algorithm for the Frobenius Problem
An adiabatic quantum algorithm for the Frobenius problem J. Ossorio-Castillo∗1,2 and José M. Tornero y2 1Instituto Tecnolóxico de Matemática Industrial (ITMATI), 15782 Santiago de Compostela, Spain 2IMUS & Departamento de Álgebra, Universidad de Sevilla, 41012 Sevilla, Spain Abstract The (Diophantine) Frobenius problem is a well-known NP-hard problem (also called the stamp problem or the chicken nugget problem) whose origins lie in the realm of combinatorial number theory. In this paper we present an adiabatic quantum algorithm which solves it, using the so- called Apéry set of a numerical semigroup, via a translation into a QUBO problem. The algorithm has been specifically designed to run in a D-Wave 2X machine. 1 Numerical semigroups and the Frobenius problem The study of numerical semigroups has its origins at the end of the 19th Century, when English mathematician James Joseph Sylvester (1814 – 1897) and German mathematician Ferdinand Georg Frobenius (1849 – 1917) were both interested in what is now known as the Frobenius problem, which we proceed to enunciate. Definition 1.1. Let a1; a2; : : : ; an 2 Z≥0 with gcd(a1; a2; : : : ; an) = 1, the Frobenius problem, or FP, is the problem of finding the largest positive integer that cannot be expressed as an integer conical combination of these numbers, i.e., as a sum n X λiai with λi 2 Z≥0: j=1 This problem, so easy to state, can be extremely complicated to solve in the majority of cases, as will be seen later. It can be found in a wide variety of contexts, being the most famous the problem of finding the largest amount of money which cannot be obtained with a certain set of coins: if, for example, we have an unlimited amount of coins of 2 and 5 units, we can represent any quantity except 1 and 3. -
2. Superoscillations.Pdf
When the Whole Vibrates Faster than Any of its Parts Computational Superoscillation Theory Nicholas Wheeler December 2017 Introduction. The phenomenon/theory of “superoscillations”—brought to my attention by my friend Ahmed Sebbar1—originated in the time-symmetric formulation of quantum mechanics2 that gave rise to the theory of “weak measurements.” Recognition of the role that superoscillations play in that work was brought into explicit focus by Aharonov et al in 1990, and since that date the concept has found application to a remarkable variety of physical subject areas. Yakir Aharonov took his PhD in 1960 from the University of Bristol, where he worked with David Bohm.3 In 1992 a conference was convened at Bristol to celebrate Aharonov’s 60th birthday. It was on that occasion that the superoscillation phenomenon came first to the attention of Michael Berry (of the Bristol faculty), who in an essay “Faster than Fourier” contributed to the proceedings of that conference4 wrote that “He [Aharonov] told me that it is possible for functions to oscillate faster than any of their Fourier components. This seemed unbelievable, even paradoxical; I had heard nothing like it before. ” Berry was inspired to write a series of papers relating to the theory of superoscillation and its potential applications, as also by now have a great many other authors. 1 Private communication, 17 November 2017. 2 Y. Aharonov, P. G. Bergmann & J. L. Libowitz, “Time symmetry in the quantum meassurement process,” Phys. Rev. B 134, 1410–1416 (1964). 3 It was, I suspect, at the invitation of E. P Gross—who had collaborated with Bohm when both were at Princeton—that Aharonov spent 1960–1961 at Brandeis University, from which I had taken the first PhD and departed to Utrecht/CERN in February of 1960, so we never met. -
EMS Newsletter September 2012 1 EMS Agenda EMS Executive Committee EMS Agenda
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Editorial Obituary Feature Interview 6ecm Marco Brunella Alan Turing’s Centenary Endre Szemerédi p. 4 p. 29 p. 32 p. 39 September 2012 Issue 85 ISSN 1027-488X S E European M M Mathematical E S Society Applied Mathematics Journals from Cambridge journals.cambridge.org/pem journals.cambridge.org/ejm journals.cambridge.org/psp journals.cambridge.org/flm journals.cambridge.org/anz journals.cambridge.org/pes journals.cambridge.org/prm journals.cambridge.org/anu journals.cambridge.org/mtk Receive a free trial to the latest issue of each of our mathematics journals at journals.cambridge.org/maths Cambridge Press Applied Maths Advert_AW.indd 1 30/07/2012 12:11 Contents Editorial Team Editors-in-Chief Jorge Buescu (2009–2012) European (Book Reviews) Vicente Muñoz (2005–2012) Dep. Matemática, Faculdade Facultad de Matematicas de Ciências, Edifício C6, Universidad Complutense Piso 2 Campo Grande Mathematical de Madrid 1749-006 Lisboa, Portugal e-mail: [email protected] Plaza de Ciencias 3, 28040 Madrid, Spain Eva-Maria Feichtner e-mail: [email protected] (2012–2015) Society Department of Mathematics Lucia Di Vizio (2012–2016) Université de Versailles- University of Bremen St Quentin 28359 Bremen, Germany e-mail: [email protected] Laboratoire de Mathématiques Newsletter No. 85, September 2012 45 avenue des États-Unis Eva Miranda (2010–2013) 78035 Versailles cedex, France Departament de Matemàtica e-mail: [email protected] Aplicada I EMS Agenda .......................................................................................................................................................... 2 EPSEB, Edifici P Editorial – S. Jackowski ........................................................................................................................... 3 Associate Editors Universitat Politècnica de Catalunya Opening Ceremony of the 6ECM – M. -
April 2017 Table of Contents
ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society April 2017 Volume 64, Number 4 AMS Prize Announcements page 311 Spring Sectional Sampler page 333 AWM Research Symposium 2017 Lecture Sampler page 341 Mathematics and Statistics Awareness Month page 362 About the Cover: How Minimal Surfaces Converge to a Foliation (see page 307) MATHEMATICAL CONGRESS OF THE AMERICAS MCA 2017 JULY 2428, 2017 | MONTREAL CANADA MCA2017 will take place in the beautiful city of Montreal on July 24–28, 2017. The many exciting activities planned include 25 invited lectures by very distinguished mathematicians from across the Americas, 72 special sessions covering a broad spectrum of mathematics, public lectures by Étienne Ghys and Erik Demaine, a concert by the Cecilia String Quartet, presentation of the MCA Prizes and much more. SPONSORS AND PARTNERS INCLUDE Canadian Mathematical Society American Mathematical Society Pacifi c Institute for the Mathematical Sciences Society for Industrial and Applied Mathematics The Fields Institute for Research in Mathematical Sciences National Science Foundation Centre de Recherches Mathématiques Conacyt, Mexico Atlantic Association for Research in Mathematical Sciences Instituto de Matemática Pura e Aplicada Tourisme Montréal Sociedade Brasileira de Matemática FRQNT Quebec Unión Matemática Argentina Centro de Modelamiento Matemático For detailed information please see the web site at www.mca2017.org. AMERICAN MATHEMATICAL SOCIETY PUSHING LIMITS From West Point to Berkeley & Beyond PUSHING LIMITS FROM WEST POINT TO BERKELEY & BEYOND Ted Hill, Georgia Tech, Atlanta, GA, and Cal Poly, San Luis Obispo, CA Recounting the unique odyssey of a noted mathematician who overcame military hurdles at West Point, Army Ranger School, and the Vietnam War, this is the tale of an academic career as noteworthy for its o beat adventures as for its teaching and research accomplishments. -
1992 CUPM Report 1992
Tomorrow’s Geometry Joseph Malkevitch YORKCOLLEGE (CUNY) Introduction In A Century of Mathematics in America (Part II), Robert Osserman contributed an article entitled “The Geometry Renaissance in America: 1938-1988.” The renaissance in geometry that he recounts has not been restricted only to America and did not end in 1988. Nor, as Osserman notes in a “postscript” to the article, has this renaissance been confined only to the area of differential geometry, which is what his article deals with. There are many reasons for the renaissance in geometry, ranging from new developments in physics and biology, to the development of the digital computer, to the flowering of old and new fields within mathematics that stimulate geometric insights. Yet in terms of the way geometry is represented in the undergraduate curriculum, there has been no renaissance. On the contrary, there has been relatively little changein the nature of the geometric mathematics taught in colleges. This unchanging curriculum for geometry in college has in turn prevented any new geometry or applications stemming from geometry from being presented in high school (or earlier grades). Students planning to become high school or middle-school teachers usually see only that geometry covered in a survey course on Euclidean and non-Euclidean geometry. The content of such courses rarely discusses geometry that reaches into the period of which Osserman speaks. However, because of its unique intuitive and quick-starting nature, geometry that has been discovered in the last 30 years can be taught to undergraduates. In light of these contrasting phenomena-explosive growth in geometric met hods and applications of geometry but an unchanging undergraduate geometry curriculum-t he MAA Curriculum Action Project convened an E-mail Focus Group to debate geometry education. -
Articles, Etc
ARTICLES, ETC. Articles "A summation formula and an identity for a class of Dirichlet series", Acta Arithmetica, Vol. XI, No. 4; 1966, 443-449. "Point groups and color symmetry", Zeitschrift fur Kristallographie, 142, 1975, 1- 23. "The mechanism of formation of certain growth twins of the penetration type", Neues Jarbuch fur Mineralogie, 11, 1976, 518-524. "A prophet without honor, Dorothy Wrinch 1894-1976", Smith Alumnae Quarterly, April 1977, 18-23. "Number, the ruler of all things", Smith Alumnae Quarterly, February, 1978, 12- 18. "Themes and variations", Criss-Cross Art Communications, 7,8,9; 1979, 40-45. "Color groups", Discrete Applied Mathematics, 1, 1979, 51-73. "Recent advances and unsolved problems in symmetry theory", in Uzori Symmetrii. In Russian. (with R.V. Galiulin) "The genesis of growth twins in crystals", Kristallografia, 25 (5), 1980, 908-915, in Russian. English translation in Soviet Physics - Crystallography, 25 (5), September-October 1980, 520-524. "A simple characterization of the subgroups of space groups", Acta Crystallographica, A36, 1980, 845-850. "Which tetrahedra fill space?", Mathematics Magazine, 54 (5), November 1981, 227-243. "Coloring symmetrical objects symmetrically", Mathematics Magazine, 56 (1), January 1983, 3-16. "Color symmetry and colored polyhedra", Acta Crystallographica, A39, 1983, 505-511. "On colored lattices and lattice preservation", Acta Crystallographica, A39, 1983. (with M.Rolley-LeCoz and Y. Billiet) "An introduction to the theory of figures: the geometry of E.S. Fedorov", Structural Topology, 10, 1984, 5-22. (with R.V. Galiulin) "Space groups and their isotropy subgroups, Journal of Mathematical Physics, 25 (1984) 3148-3154. (with M.V. Jaric) "Morphisms of crystallographic groups: kernels and images", Journal of Mathematical Physics, 26 (1985) 219-228. -
Ingrid Daubechies Receives NAS Award in Mathematics
Ingrid Daubechies Receives NAS Award in Mathematics INGRID DAUBECHIES has received the 2000 National well as the 1997 Ruth Lyttle Academy of Sciences (NAS) Award in Mathematics. Satter Prize. From 1992 to The $5,000 award, established by the AMS in com- 1997 she was a fellow of the memoration of its centennial in 1988, is presented John D. and Catherine T. every four years for excellence in published math- MacArthur Foundation. ematical research. Daubechies was chosen “for fun- The previous recipients damental discoveries on wavelets and wavelet ex- of the NAS Award in Math- pansions and for her role in making wavelet methods ematics are Robert P. Lang- a practical basic tool of applied mathematics.” lands (1988), Robert D. Ingrid Daubechies received both her bachelor’s MacPherson (1992), and An- and Ph.D. degrees (in 1975 and 1980) from the drew J. Wiles (1996). Free University in Brussels, Belgium. She held a re- —Allyn Jackson search position at the Free University until 1987. From 1987 to 1994 she was a member of the tech- nical staff at AT&T Bell Laboratories, during which time she took leaves to spend six months (in 1990) at the University of Michigan and two years (1991–93) at Rutgers University. She is now a pro- fessor in the Mathematics Department and in the Program in Applied and Computational Mathematics at Princeton University. Her research Ingrid Daubechies interests focus on the mathematical aspects of time-frequency analysis, in particular wavelets, as well as applications. In 1993 Daubechies was elected as a member of the American Academy of Arts and Sciences, and in 1998 she was elected as a member of the NAS and as a fellow of the Institute of Electrical and Electronics Engineers.