Mathematics I College of Arts and Sciences
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Zero-One Laws
1 THE LOGIC IN COMPUTER SCIENCE COLUMN by 2 Yuri GUREVICH Electrical Engineering and Computer Science UniversityofMichigan, Ann Arb or, MI 48109-2122, USA [email protected] Zero-One Laws Quisani: I heard you talking on nite mo del theory the other day.Itisinteresting indeed that all those famous theorems ab out rst-order logic fail in the case when only nite structures are allowed. I can understand that for those, likeyou, educated in the tradition of mathematical logic it seems very imp ortantto ndoutwhichof the classical theorems can b e rescued. But nite structures are to o imp ortant all by themselves. There's got to b e deep nite mo del theory that has nothing to do with in nite structures. Author: \Nothing to do" sounds a little extremist to me. Sometimes in nite ob jects are go o d approximations of nite ob jects. Q: I do not understand this. Usually, nite ob jects approximate in nite ones. A: It may happ en that the in nite case is cleaner and easier to deal with. For example, a long nite sum may b e replaced with a simpler integral. Returning to your question, there is indeed meaningful, inherently nite, mo del theory. One exciting issue is zero- one laws. Consider, say, undirected graphs and let be a propertyofsuch graphs. For example, may b e connectivity. What fraction of n-vertex graphs have the prop erty ? It turns out that, for many natural prop erties , this fraction converges to 0 or 1asn grows to in nity. If the fraction converges to 1, the prop erty is called almost sure. -
On the Normality of Numbers
ON THE NORMALITY OF NUMBERS Adrian Belshaw B. Sc., University of British Columbia, 1973 M. A., Princeton University, 1976 A THESIS SUBMITTED 'IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Mathematics @ Adrian Belshaw 2005 SIMON FRASER UNIVERSITY Fall 2005 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. APPROVAL Name: Adrian Belshaw Degree: Master of Science Title of Thesis: On the Normality of Numbers Examining Committee: Dr. Ladislav Stacho Chair Dr. Peter Borwein Senior Supervisor Professor of Mathematics Simon Fraser University Dr. Stephen Choi Supervisor Assistant Professor of Mathematics Simon Fraser University Dr. Jason Bell Internal Examiner Assistant Professor of Mathematics Simon Fraser University Date Approved: December 5. 2005 SIMON FRASER ' u~~~snrllbrary DECLARATION OF PARTIAL COPYRIGHT LICENCE The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. The author has further granted permission to Simon Fraser University to keep or make a digital copy for use in its circulating collection, and, without changing the content, to translate the thesislproject or extended essays, if technically possible, to any medium or format for the purpose of preservation of the digital work. -
Diophantine Approximation and Transcendental Numbers
Diophantine Approximation and Transcendental Numbers Connor Goldstick June 6, 2017 Contents 1 Introduction 1 2 Continued Fractions 2 3 Rational Approximations 3 4 Transcendental Numbers 6 5 Irrationality Measure 7 6 The continued fraction for e 8 6.1 e's Irrationality measure . 10 7 Conclusion 11 A The Lambert Continued Fraction 11 1 Introduction Suppose we have an irrational number, α, that we want to approximate with a rational number, p=q. This question of approximating an irrational number is the primary concern of Diophantine approximation. In other words, we want jα−p=qj < . However, this method of trying to approximate α is boring, as it is possible to get an arbitrary amount of precision by making q large. To remedy this problem, it makes sense to vary with q. The problem we are really trying to solve is finding p and q such that p 1 1 jα − j < which is equivalent to jqα − pj < (1) q q2 q Solutions to this problem have applications in both number theory and in more applied fields. For example, in signal processing many of the quickest approximation algorithms are given by solutions to Diophantine approximation problems. Diophantine approximations also give many important results like the continued fraction expansion of e. One of the most interesting aspects of Diophantine approximations are its relationship with transcendental 1 numbers (a number that cannot be expressed of the root of a polynomial with rational coefficients). One of the key characteristics of a transcendental number is that it is easy to approximate with rational numbers. This paper is separated into two categories. -
Network Intrusion Detection with Xgboost and Deep Learning Algorithms: an Evaluation Study
2020 International Conference on Computational Science and Computational Intelligence (CSCI) Network Intrusion Detection with XGBoost and Deep Learning Algorithms: An Evaluation Study Amr Attia Miad Faezipour Abdelshakour Abuzneid Computer Science & Engineering Computer Science & Engineering Computer Science & Engineering University of Bridgeport, CT 06604, USA University of Bridgeport, CT 06604, USA University of Bridgeport, CT 06604, USA [email protected] [email protected] [email protected] Abstract— This paper introduces an effective Network Intrusion In the KitNET model introduced in [2], an unsupervised Detection Systems (NIDS) framework that deploys incremental technique is introduced for anomaly-based intrusion statistical damping features of the packets along with state-of- detection. Incremental statistical feature extraction of the the-art machine/deep learning algorithms to detect malicious packets is passed through ensembles of autoencoders with a patterns. A comprehensive evaluation study is conducted predefined threshold. The model calculates the Root Mean between eXtreme Gradient Boosting (XGBoost) and Artificial Neural Networks (ANN) where feature selection and/or feature Square (RMS) error to detect anomaly behavior. The higher dimensionality reduction techniques such as Principal the calculated RMS at the output, the higher probability of Component Analysis (PCA) and Linear Discriminant Analysis suspicious activity. (LDA) are also integrated into the models to decrease the system Supervised learning has achieved very decent results with complexity for achieving fast responses. Several experimental algorithms such as Random Forest, ZeroR, J48, AdaBoost, runs confirm how powerful machine/deep learning algorithms Logit Boost, and Multilayer Perceptron [3]. Machine/deep are for intrusion detection on known attacks when combined learning-based algorithms for NIDS have been extensively with the appropriate features extracted. -
The Prime Number Theorem a PRIMES Exposition
The Prime Number Theorem A PRIMES Exposition Ishita Goluguri, Toyesh Jayaswal, Andrew Lee Mentor: Chengyang Shao TABLE OF CONTENTS 1 Introduction 2 Tools from Complex Analysis 3 Entire Functions 4 Hadamard Factorization Theorem 5 Riemann Zeta Function 6 Chebyshev Functions 7 Perron Formula 8 Prime Number Theorem © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 2 Introduction • Euclid (300 BC): There are infinitely many primes • Legendre (1808): for primes less than 1,000,000: x π(x) ' log x © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 3 Progress on the Distribution of Prime Numbers • Euler: The product formula 1 X 1 Y 1 ζ(s) := = ns 1 − p−s n=1 p so (heuristically) Y 1 = log 1 1 − p−1 p • Chebyshev (1848-1850): if the ratio of π(x) and x= log x has a limit, it must be 1 • Riemann (1859): On the Number of Primes Less Than a Given Magnitude, related π(x) to the zeros of ζ(s) using complex analysis • Hadamard, de la Vallée Poussin (1896): Proved independently the prime number theorem by showing ζ(s) has no zeros of the form 1 + it, hence the celebrated prime number theorem © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 4 Tools from Complex Analysis Theorem (Maximum Principle) Let Ω be a domain, and let f be holomorphic on Ω. (A) jf(z)j cannot attain its maximum inside Ω unless f is constant. (B) The real part of f cannot attain its maximum inside Ω unless f is a constant. Theorem (Jensen’s Inequality) Suppose f is holomorphic on the whole complex plane and f(0) = 1. -
Network Topology Generators: Degree-Based Vs
Network Topology Generators: Degree-Based vs. Structural Hongsuda Tangmunarunkit Ramesh Govindan Sugih Jamin USC-ISI ICSI Univ. of Michigan [email protected] [email protected] [email protected] Scott Shenker Walter Willinger ICSI AT&T Research [email protected] [email protected] ABSTRACT in the Internet more closely; we find that degree-based generators Following the long-held belief that the Internet is hierarchical, the produce a form of hierarchy that closely resembles the loosely hi- network topology generators most widely used by the Internet re- erarchical nature of the Internet. search community, Transit-Stub and Tiers, create networks with a deliberately hierarchical structure. However, in 1999 a seminal Categories and Subject Descriptors paper by Faloutsos et al. revealed that the Internet’s degree distri- bution is a power-law. Because the degree distributions produced C.2.1 [Computer-Communication Networks]: Network Archi- by the Transit-Stub and Tiers generators are not power-laws, the tecture and Design—Network topology; I.6.4 [Simulation and Mod- research community has largely dismissed them as inadequate and eling]: Model Validation and Analysis proposed new network generators that attempt to generate graphs with power-law degree distributions. General Terms Contrary to much of the current literature on network topology generators, this paper starts with the assumption that it is more im- Performance, Measurement portant for network generators to accurately model the large-scale structure of the Internet (such as its hierarchical structure) than to Keywords faithfully imitate its local properties (such as the degree distribu- tion). The purpose of this paper is to determine, using various Network topology, hierarchy, topology characterization, topology topology metrics, which network generators better represent this generators, structural generators, degree-based generators, topol- large-scale structure. -
Some of Erdös' Unconventional Problems in Number Theory, Thirty
Some of Erdös’ unconventional problems in number theory, thirty-four years later Gérald Tenenbaum To cite this version: Gérald Tenenbaum. Some of Erdös’ unconventional problems in number theory, thirty-four years later. Erdös centennial, János Bolyai Math. Soc., pp.651-681, 2013. hal-01281530 HAL Id: hal-01281530 https://hal.archives-ouvertes.fr/hal-01281530 Submitted on 2 Mar 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. (26/5/2014, 16h36) Some of Erd˝os’ unconventional problems in number theory, thirty-four years later G´erald Tenenbaum There are many ways to recall Paul Erd˝os’ memory and his special way of doing mathematics. Ernst Straus described him as “the prince of problem solvers and the absolute monarch of problem posers”. Indeed, those mathematicians who are old enough to have attended some of his lectures will remember that, after his talks, chairmen used to slightly depart from standard conduct, not asking if there were any questions but if there were any answers. In the address that he forwarded to Mikl´os Simonovits for Erd˝os’ funeral, Claude Berge mentions a conversation he had with Paul in the gardens of the Luminy Campus, near Marseilles, in September 1995. -
An Elementary Proof That Almost All Real Numbers Are Normal
Acta Univ. Sapientiae, Mathematica, 2, 1 (2010) 99–110 An elementary proof that almost all real numbers are normal Ferdin´and Filip Jan Sustekˇ Department of Mathematics, Department of Mathematics and Faculty of Education, Institute for Research and J. Selye University, Applications of Fuzzy Modeling, Roln´ıckej ˇskoly 1519, Faculty of Science, 945 01, Kom´arno, Slovakia University of Ostrava, email: [email protected] 30. dubna 22, 701 03 Ostrava 1, Czech Republic email: [email protected] Abstract. A real number is called normal if every block of digits in its expansion occurs with the same frequency. A famous result of Borel is that almost every number is normal. Our paper presents an elementary proof of that fact using properties of a special class of functions. 1 Introduction The concept of normal number was introduced by Borel. A number is called normal if in its base b expansion every block of digits occurs with the same frequency. More exact definition is Definition 1 A real number x ∈ (0,1) is called simply normal to base b ≥ 2 if its base b expansion is 0.c1c2c3 ... and #{n ≤ N | c = a} 1 lim n = for every a ∈ {0,...,b − 1} . N N b 2010 Mathematics→∞ Subject Classification: 11K16, 26A30 Key words and phrases: normal number, measure 99 100 F. Filip, J. Sustekˇ A number is called normal to base b if for every block of digits a1 ...aL, L ≥ 1 #{n ≤ N − L | c = a ,...,c = a } 1 lim n+1 1 n+L L = . N N bL A number is called→∞ absolutely normal if it is normal to every base b ≥ 2. -
GH HARDY and JE LITTLEWOOD, Cambridge, England. 1
Some More Integral Inequalities, by G. H. HARDYand J. E. LITTLEWOOD,Cambridge, England. 1. This is in a sense a companion to an earlier note with a similar title(1) ; and the contents of both notes were suggested by a more systematic investigation concerned primarily with series. Both in the note just referred to and here we are concerned with special cases of a general theorem which we have stated without proof elsewhere(2), and also (and more particularly) with their integral analogues. These special cases are peculiar, and deserve special treatment, because in them we can go further than in the general case and determine the best value of the constant K of the inequa- lity. The actual inequality proved here, which corresponds to the case p = q = r of " Theorem 25 ", is much easier than that proved in Note X(3), but has some distinguishing features. There is generally an attained maximum, which is unusual in such problems ; and the proof that the maximal solution is effectively unique involves points of some independent interest. Integral theorems. 2. In what follows it is assumed throughout that f and g are positive (in the wide sense) and measurable. Theorem 1. If and (2.1) then (2.2) where (1) G. H. Hardy and J. E. Littlewood : "Notes on the theory of series (X) ; Some more inequalities", Journal of London Math. Soc. 3 (1928), 294-299. (2) G. H. Hardy and J. E. Littlewood : " Elementary theorems concern- ing power series with positive coefficients and moment constants of positive func- tions ", Journal fur Math., 157 (1927), 141-158. -
Transcendental Number Theory, by Alan Baker, Cambridge Univ. Press, New York, 1975, X + 147 Pp., $13.95
1370 BOOK REVIEWS REFERENCES 1. J. Horvâth, Topological vector spaces and distributions, Addison-Wcsley, Reading, Mass., 1966. 2. A. P. Robertson and W. J. Robertson, Topological vector spaces, Cambridge Tracts in Mathematics and Physics, no. 53 Cambridge Univ. Press, New York, 1964. 3. H. H. Schaefer, Topological vector spaces, MacMillan, New York, 1966. 4. F. Treves, Topological vector spaces, distributions and kernels, Academic Press, New York, 1967. L. WAELBROECK BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 84, Number 6, November 1978 © American Mathematical Society 1978 Transcendental number theory, by Alan Baker, Cambridge Univ. Press, New York, 1975, x + 147 pp., $13.95. Lectures on transcendental numbers, by Kurt Mahler, Edited and completed by B. Divis and W. J. LeVeque, Lecture Notes in Math., no. 546, Springer- Verlag, Berlin, Heidelberg, New York, 1976, xxi + 254 pp., $10.20. Nombres transcendants, by Michel Waldschmidt, Lecture Notes in Math., no. 402, Springer-Verlag, Berlin and New York, 1974, viii + 277 pp., $10.30. The last dozen years have been a golden age for transcendental number theory. It has scored successes on its own ground, while its methods have triumphed over problems in classical number theory involving exponential sums, class numbers, and Diophantine equations. Few topics in mathematics have such general appeal within the discipline as transcendency. Many of us learned of the circle squaring problem before college, and became acquainted with Cantor's existence proof, Liouville's construction, and even Hermite's proof of the transcendence of e well before the close of our undergraduate life. How can we learn more? Sophisticated readers may profitably consult the excellent survey articles of N. -
Essential Skills in Mathematics a Comparative Analysis of American and Japanese Assessments of Eighth-Graders
NATIONAL CENTER FOR EDUCATION STATISTICS Essential Skills in Mathematics A Comparative Analysis of American and Japanese Assessments of Eighth-Graders John A. Dossey Department of Mathematics Illinois State University Lois Peak and Dawn Nelson, Project Officers National Center for Education Statistics U.S. Department of Education Office of Educational Research and Improvement NCES 97-885 For sale by the U.S. Ckwemment Riming Office Superintendent of Dncuments, Mail Stop SSOP, Wadungton, DC 20402-932S U.S. Department of Education Richard W. Riley Secretaty Office of Educational Research and Improvement Marshall S. Smith Acting Assistant Secreta~ National Center for Education Statistics Pascal D. Forgione, Jr. Commissioner The National Center for Education Statistics (NCES) is the primary federal entity for collecting, analyzing, and reporting data related to education in the United States and other nations. It fulfills a congressional mandate to collect, collate, analyze, and report full and complete statistics on the condition of education in the United States; conduct and publish reports and specialized analyses of the meaning and significance of such statistics; assist state and local education agencies in improving their statistical systems; and review and report on education activities in foreign countries. NCES activities are designed to address high prioriy education data needs; provide consistent, reliable, complete, and accurate indicators of education status and trends; and report timely, useful, and high quality data to the U.S. Department of Education, the Congress, the states, other education policymakers, practitioners, data users, and the general public. We strive to make our products available in a variety of formats and in language that is appropriate to a variety of audiences. -
On Transcendence Theory with Little History, New Results and Open Problems
On Transcendence Theory with little history, new results and open problems Hyun Seok, Lee December 27, 2015 Abstract This talk will be dealt to a survey of transcendence theory, including little history, the state of the art and some of the main conjectures, the limits of the current methods and the obstacles which are preventing from going further. 1 Introudction Definition 1.1. An algebraic number is a complex number which is a root of a polynomial with rational coefficients. p p p • For example 2, i = −1, the Golden Ration (1 + 5)=2, the roots of unity e2πia=b, the roots of the polynomial x5 − 6x + 3 = 0 are algebraic numbers. Definition 1.2. A transcendental number is a complex number which is not algebraic. Definition 1.3. Complex numbers α1; ··· ; αn are algebraically dependent if there exists a non-zero polynomial P (x1; ··· ; xn) in n variables with rational co- efficients such that P (α1; ··· ; αn) = 0. Otherwise, they are called algebraically independent. The existence of transcendental numbers was proved in 1844 by J. Liouville who gave explicit ad-hoc examples. The transcendence of constants from analysis is harder; the first result was achieved in 1873 by Ch. Hermite who proved the transcendence of e. In 1882, the proof by F. Lindemann of the transcendence of π gave the final (and negative) answer to the Greek problem of squaring the p circle. The transcendence of 2 2 and eπ, which was included in Hilbert's seventh problem in 1900, was proved by Gel'fond and Schneider in 1934.