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500 Natural Sciences and Mathematics
500 500 Natural sciences and mathematics Natural sciences: sciences that deal with matter and energy, or with objects and processes observable in nature Class here interdisciplinary works on natural and applied sciences Class natural history in 508. Class scientific principles of a subject with the subject, plus notation 01 from Table 1, e.g., scientific principles of photography 770.1 For government policy on science, see 338.9; for applied sciences, see 600 See Manual at 231.7 vs. 213, 500, 576.8; also at 338.9 vs. 352.7, 500; also at 500 vs. 001 SUMMARY 500.2–.8 [Physical sciences, space sciences, groups of people] 501–509 Standard subdivisions and natural history 510 Mathematics 520 Astronomy and allied sciences 530 Physics 540 Chemistry and allied sciences 550 Earth sciences 560 Paleontology 570 Biology 580 Plants 590 Animals .2 Physical sciences For astronomy and allied sciences, see 520; for physics, see 530; for chemistry and allied sciences, see 540; for earth sciences, see 550 .5 Space sciences For astronomy, see 520; for earth sciences in other worlds, see 550. For space sciences aspects of a specific subject, see the subject, plus notation 091 from Table 1, e.g., chemical reactions in space 541.390919 See Manual at 520 vs. 500.5, 523.1, 530.1, 919.9 .8 Groups of people Add to base number 500.8 the numbers following —08 in notation 081–089 from Table 1, e.g., women in science 500.82 501 Philosophy and theory Class scientific method as a general research technique in 001.4; class scientific method applied in the natural sciences in 507.2 502 Miscellany 577 502 Dewey Decimal Classification 502 .8 Auxiliary techniques and procedures; apparatus, equipment, materials Including microscopy; microscopes; interdisciplinary works on microscopy Class stereology with compound microscopes, stereology with electron microscopes in 502; class interdisciplinary works on photomicrography in 778.3 For manufacture of microscopes, see 681. -
Math 788M: Computational Number Theory (Instructor’S Notes)
Math 788M: Computational Number Theory (Instructor’s Notes) The Very Beginning: • A positive integer n can be written in n steps. • The role of numerals (now O(log n) steps) • Can we do better? (Example: The largest known prime contains 258716 digits and doesn’t take long to write down. It’s 2859433 − 1.) Running Time of Algorithms: • A positive integer n in base b contains [logb n] + 1 digits. • Big-Oh & Little-Oh Notation (as well as , , ∼, ) Examples 1: log (1 + (1/n)) = O(1/n) Examples 2: [logb n] + 1 log n Examples 3: 1 + 2 + ··· + n n2 These notes are for a course taught by Michael Filaseta in the Spring of 1996 and being updated for Fall of 2007. Computational Number Theory Notes 2 Examples 4: f a polynomial of degree k =⇒ f(n) = O(nk) Examples 5: (r + 1)π ∼ rπ • We will want algorithms to run quickly (in a small number of steps) in comparison to the length of the input. For example, we may ask, “How quickly can we factor a positive integer n?” One considers the length of the input n to be of order log n (corresponding to the number of binary digits n has). An algorithm runs in polynomial time if the number of steps it takes is bounded above by a polynomial in the length of the input. An algorithm to factor n in polynomial time would require that it take O (log n)k steps (and that it factor n). Addition and Subtraction (of n and m): • We are taught how to do binary addition and subtraction in O(log n + log m) steps. -
Number Theory
“mcs-ftl” — 2010/9/8 — 0:40 — page 81 — #87 4 Number Theory Number theory is the study of the integers. Why anyone would want to study the integers is not immediately obvious. First of all, what’s to know? There’s 0, there’s 1, 2, 3, and so on, and, oh yeah, -1, -2, . Which one don’t you understand? Sec- ond, what practical value is there in it? The mathematician G. H. Hardy expressed pleasure in its impracticality when he wrote: [Number theorists] may be justified in rejoicing that there is one sci- ence, at any rate, and that their own, whose very remoteness from or- dinary human activities should keep it gentle and clean. Hardy was specially concerned that number theory not be used in warfare; he was a pacifist. You may applaud his sentiments, but he got it wrong: Number Theory underlies modern cryptography, which is what makes secure online communication possible. Secure communication is of course crucial in war—which may leave poor Hardy spinning in his grave. It’s also central to online commerce. Every time you buy a book from Amazon, check your grades on WebSIS, or use a PayPal account, you are relying on number theoretic algorithms. Number theory also provides an excellent environment for us to practice and apply the proof techniques that we developed in Chapters 2 and 3. Since we’ll be focusing on properties of the integers, we’ll adopt the default convention in this chapter that variables range over the set of integers, Z. 4.1 Divisibility The nature of number theory emerges as soon as we consider the divides relation a divides b iff ak b for some k: D The notation, a b, is an abbreviation for “a divides b.” If a b, then we also j j say that b is a multiple of a. -
A Taste of Set Theory for Philosophers
Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. A taste of set theory for philosophers Jouko Va¨an¨ anen¨ ∗ Department of Mathematics and Statistics University of Helsinki and Institute for Logic, Language and Computation University of Amsterdam November 17, 2010 Contents 1 Introduction 1 2 Elementary set theory 2 3 Cardinal and ordinal numbers 3 3.1 Equipollence . 4 3.2 Countable sets . 6 3.3 Ordinals . 7 3.4 Cardinals . 8 4 Axiomatic set theory 9 5 Axiom of Choice 12 6 Independence results 13 7 Some recent work 14 7.1 Descriptive Set Theory . 14 7.2 Non well-founded set theory . 14 7.3 Constructive set theory . 15 8 Historical Remarks and Further Reading 15 ∗Research partially supported by grant 40734 of the Academy of Finland and by the EUROCORES LogICCC LINT programme. I Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. 1 Introduction Originally set theory was a theory of infinity, an attempt to understand infinity in ex- act terms. Later it became a universal language for mathematics and an attempt to give a foundation for all of mathematics, and thereby to all sciences that are based on mathematics. -
Analysis and Verification of Arithmetic Circuits Using Computer Algebra Approach
University of Massachusetts Amherst ScholarWorks@UMass Amherst Doctoral Dissertations Dissertations and Theses March 2020 ANALYSIS AND VERIFICATION OF ARITHMETIC CIRCUITS USING COMPUTER ALGEBRA APPROACH TIANKAI SU University of Massachusetts Amherst Follow this and additional works at: https://scholarworks.umass.edu/dissertations_2 Part of the VLSI and Circuits, Embedded and Hardware Systems Commons Recommended Citation SU, TIANKAI, "ANALYSIS AND VERIFICATION OF ARITHMETIC CIRCUITS USING COMPUTER ALGEBRA APPROACH" (2020). Doctoral Dissertations. 1868. https://doi.org/10.7275/15875490 https://scholarworks.umass.edu/dissertations_2/1868 This Open Access Dissertation is brought to you for free and open access by the Dissertations and Theses at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. ANALYSIS AND VERIFICATION OF ARITHMETIC CIRCUITS USING COMPUTER ALGEBRA APPROACH A Dissertation Presented by TIANKAI SU Submitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY February 2020 Electrical and Computer Engineering c Copyright by Tiankai Su 2020 All Rights Reserved ANALYSIS AND VERIFICATION OF ARITHMETIC CIRCUITS USING COMPUTER ALGEBRA APPROACH A Dissertation Presented by TIANKAI SU Approved as to style and content by: Maciej Ciesielski, Chair George S. Avrunin, Member Daniel Holcomb, Member Weibo Gong, Member Christopher V. Hollot, Department Head Electrical and Computer Engineering ABSTRACT ANALYSIS AND VERIFICATION OF ARITHMETIC CIRCUITS USING COMPUTER ALGEBRA APPROACH FEBRUARY 2020 TIANKAI SU B.Sc., NORTHEAST DIANLI UNIVERSITY Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Maciej Ciesielski Despite a considerable progress in verification of random and control logic, ad- vances in formal verification of arithmetic designs have been lagging. -
Complexity Theory
Complexity Theory Course Notes Sebastiaan A. Terwijn Radboud University Nijmegen Department of Mathematics P.O. Box 9010 6500 GL Nijmegen the Netherlands [email protected] Copyright c 2010 by Sebastiaan A. Terwijn Version: December 2017 ii Contents 1 Introduction 1 1.1 Complexity theory . .1 1.2 Preliminaries . .1 1.3 Turing machines . .2 1.4 Big O and small o .........................3 1.5 Logic . .3 1.6 Number theory . .4 1.7 Exercises . .5 2 Basics 6 2.1 Time and space bounds . .6 2.2 Inclusions between classes . .7 2.3 Hierarchy theorems . .8 2.4 Central complexity classes . 10 2.5 Problems from logic, algebra, and graph theory . 11 2.6 The Immerman-Szelepcs´enyi Theorem . 12 2.7 Exercises . 14 3 Reductions and completeness 16 3.1 Many-one reductions . 16 3.2 NP-complete problems . 18 3.3 More decision problems from logic . 19 3.4 Completeness of Hamilton path and TSP . 22 3.5 Exercises . 24 4 Relativized computation and the polynomial hierarchy 27 4.1 Relativized computation . 27 4.2 The Polynomial Hierarchy . 28 4.3 Relativization . 31 4.4 Exercises . 32 iii 5 Diagonalization 34 5.1 The Halting Problem . 34 5.2 Intermediate sets . 34 5.3 Oracle separations . 36 5.4 Many-one versus Turing reductions . 38 5.5 Sparse sets . 38 5.6 The Gap Theorem . 40 5.7 The Speed-Up Theorem . 41 5.8 Exercises . 43 6 Randomized computation 45 6.1 Probabilistic classes . 45 6.2 More about BPP . 48 6.3 The classes RP and ZPP . -
Representation Theory* Its Rise and Its Role in Number Theory
Representation Theory* Its Rise and Its Role in Number Theory Introduction By representation theory we understand the representation of a group by linear transformations of a vector space. Initially, the group is finite, as in the researches of Dedekind and Frobenius, two of the founders of the subject, or a compact Lie group, as in the theory of invariants and the researches of Hurwitz and Schur, and the vector space finite-dimensional, so that the group is being represented as a group of matrices. Under the combined influences of relativity theory and quantum mechanics, in the context of relativistic field theories in the nineteen-thirties, the Lie group ceased to be compact and the space to be finite-dimensional, and the study of infinite-dimensional representations of Lie groups began. It never became, so far as I can judge, of any importance to physics, but it was continued by mathematicians, with strictly mathematical, even somewhat narrow, goals, and led, largely in the hands of Harish-Chandra, to a profound and unexpectedly elegant theory that, we now appreciate, suggests solutions to classical problems in domains, especially the theory of numbers, far removed from the concerns of Dirac and Wigner, in whose papers the notion first appears. Physicists continue to offer mathematicians new notions, even within representation theory, the Virasoro algebra and its representations being one of the most recent, that may be as fecund. Predictions are out of place, but it is well to remind ourselves that the representation theory of noncompact Lie groups revealed its force and its true lines only after an enormous effort, over two decades and by one of the very best mathematical minds of our time, to establish rigorously and in general the elements of what appeared to be a somewhat peripheral subject. -
Elementary Number Theory
Elementary Number Theory Peter Hackman HHH Productions November 5, 2007 ii c P Hackman, 2007. Contents Preface ix A Divisibility, Unique Factorization 1 A.I The gcd and B´ezout . 1 A.II Two Divisibility Theorems . 6 A.III Unique Factorization . 8 A.IV Residue Classes, Congruences . 11 A.V Order, Little Fermat, Euler . 20 A.VI A Brief Account of RSA . 32 B Congruences. The CRT. 35 B.I The Chinese Remainder Theorem . 35 B.II Euler’s Phi Function Revisited . 42 * B.III General CRT . 46 B.IV Application to Algebraic Congruences . 51 B.V Linear Congruences . 52 B.VI Congruences Modulo a Prime . 54 B.VII Modulo a Prime Power . 58 C Primitive Roots 67 iii iv CONTENTS C.I False Cases Excluded . 67 C.II Primitive Roots Modulo a Prime . 70 C.III Binomial Congruences . 73 C.IV Prime Powers . 78 C.V The Carmichael Exponent . 85 * C.VI Pseudorandom Sequences . 89 C.VII Discrete Logarithms . 91 * C.VIII Computing Discrete Logarithms . 92 D Quadratic Reciprocity 103 D.I The Legendre Symbol . 103 D.II The Jacobi Symbol . 114 D.III A Cryptographic Application . 119 D.IV Gauß’ Lemma . 119 D.V The “Rectangle Proof” . 123 D.VI Gerstenhaber’s Proof . 125 * D.VII Zolotareff’s Proof . 127 E Some Diophantine Problems 139 E.I Primes as Sums of Squares . 139 E.II Composite Numbers . 146 E.III Another Diophantine Problem . 152 E.IV Modular Square Roots . 156 E.V Applications . 161 F Multiplicative Functions 163 F.I Definitions and Examples . 163 CONTENTS v F.II The Dirichlet Product . -
The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems
The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them? Antonio J. Durán, Mario Pérez, and Juan L. Varona Introduction by a typical research mathematician, but let us Nowadays, mathematicians often use a computer explain it briefly. It is not necessary to completely algebra system as an aid in their mathematical understand the mathematics, just to realize that it research; they do the thinking and leave the tedious is typical mathematical research using computer calculations to the computer. Everybody “knows” algebra as a tool. that computers perform this work better than Our starting point is a discrete positive measure people. But, of course, we must trust in the results on the real line, µ n 0 Mnδan (where δa denotes = ≥ derived via these powerful computer algebra the Dirac delta at a, and an < an 1) having P + systems. First of all, let us clarify that this paper is a sequence of orthogonal polynomials Pn n 0 f g ≥ not, in any way, a comparison between different (where Pn has degree n and positive leading computer algebra systems, but a sample of the coefficient). Karlin and Szeg˝oconsidered in 1961 (see [4]) the l l Casorati determinants current state of the art of what mathematicians × can expect when they use this kind of software. Pn(ak)Pn(ak 1):::Pn(ak l 1) Although our example deals with a concrete system, + + − Pn 1(ak)Pn 1(ak 1):::Pn 1(ak l 1) 0 + + + + + − 1 we are sure that similar situations may occur with (1) det . -
Addition and Subtraction
A Addition and Subtraction SUMMARY (475– 221 BCE), when arithmetic operations were per- formed by manipulating rods on a flat surface that was Addition and subtraction can be thought of as a pro- partitioned by vertical and horizontal lines. The num- cess of accumulation. For example, if a flock of 3 sheep bers were represented by a positional base- 10 system. is joined with a flock of 4 sheep, the combined flock Some scholars believe that this system— after moving will have 7 sheep. Thus, 7 is the sum that results from westward through India and the Islamic Empire— the addition of the numbers 3 and 4. This can be writ- became the modern system of representing numbers. ten as 3 + 4 = 7 where the sign “+” is read “plus” and The Greeks in the fifth century BCE, in addition the sign “=” is read “equals.” Both 3 and 4 are called to using a complex ciphered system for representing addends. Addition is commutative; that is, the order numbers, used a system that is very similar to Roman of the addends is irrelevant to how the sum is formed. numerals. It is possible that the Greeks performed Subtraction finds the remainder after a quantity is arithmetic operations by manipulating small stones diminished by a certain amount. If from a flock con- on a large, flat surface partitioned by lines. A simi- taining 5 sheep, 3 sheep are removed, then 2 sheep lar stone tablet was found on the island of Salamis remain. In this example, 5 is the minuend, 3 is the in the 1800s and is believed to date from the fourth subtrahend, and 2 is the remainder or difference. -
Arxiv:Math/0412262V2 [Math.NT] 8 Aug 2012 Etrgae Tgte Ihm)O Atnscnetr and Conjecture fie ‘Artin’S Number on of Cojoc Me) Domains’
ARTIN’S PRIMITIVE ROOT CONJECTURE - a survey - PIETER MOREE (with contributions by A.C. Cojocaru, W. Gajda and H. Graves) To the memory of John L. Selfridge (1927-2010) Abstract. One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the lit- erature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contribu- tions in the survey on ‘elliptic Artin’ are due to Alina Cojocaru. Woj- ciec Gajda wrote a section on ‘Artin for K-theory of number fields’, and Hester Graves (together with me) on ‘Artin’s conjecture and Euclidean domains’. Contents 1. Introduction 2 2. Naive heuristic approach 5 3. Algebraic number theory 5 3.1. Analytic algebraic number theory 6 4. Artin’s heuristic approach 8 5. Modified heuristic approach (`ala Artin) 9 6. Hooley’s work 10 6.1. Unconditional results 12 7. Probabilistic model 13 8. The indicator function 17 arXiv:math/0412262v2 [math.NT] 8 Aug 2012 8.1. The indicator function and probabilistic models 17 8.2. The indicator function in the function field setting 18 9. Some variations of Artin’s problem 20 9.1. Elliptic Artin (by A.C. Cojocaru) 20 9.2. Even order 22 9.3. Order in a prescribed arithmetic progression 24 9.4. Divisors of second order recurrences 25 9.5. Lenstra’s work 29 9.6. -
Interactions of Computational Complexity Theory and Mathematics
Interactions of Computational Complexity Theory and Mathematics Avi Wigderson October 22, 2017 Abstract [This paper is a (self contained) chapter in a new book on computational complexity theory, called Mathematics and Computation, whose draft is available at https://www.math.ias.edu/avi/book]. We survey some concrete interaction areas between computational complexity theory and different fields of mathematics. We hope to demonstrate here that hardly any area of modern mathematics is untouched by the computational connection (which in some cases is completely natural and in others may seem quite surprising). In my view, the breadth, depth, beauty and novelty of these connections is inspiring, and speaks to a great potential of future interactions (which indeed, are quickly expanding). We aim for variety. We give short, simple descriptions (without proofs or much technical detail) of ideas, motivations, results and connections; this will hopefully entice the reader to dig deeper. Each vignette focuses only on a single topic within a large mathematical filed. We cover the following: • Number Theory: Primality testing • Combinatorial Geometry: Point-line incidences • Operator Theory: The Kadison-Singer problem • Metric Geometry: Distortion of embeddings • Group Theory: Generation and random generation • Statistical Physics: Monte-Carlo Markov chains • Analysis and Probability: Noise stability • Lattice Theory: Short vectors • Invariant Theory: Actions on matrix tuples 1 1 introduction The Theory of Computation (ToC) lays out the mathematical foundations of computer science. I am often asked if ToC is a branch of Mathematics, or of Computer Science. The answer is easy: it is clearly both (and in fact, much more). Ever since Turing's 1936 definition of the Turing machine, we have had a formal mathematical model of computation that enables the rigorous mathematical study of computational tasks, algorithms to solve them, and the resources these require.