Modular Arithmetic
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A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints
A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints by Mehrdad Sabetzadeh A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Computer Science University of Toronto Copyright c 2003 by Mehrdad Sabetzadeh Abstract A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints Mehrdad Sabetzadeh Master of Science Graduate Department of Computer Science University of Toronto 2003 Eliciting the requirements for a proposed system typically involves different stakeholders with different expertise, responsibilities, and perspectives. This may result in inconsis- tencies between the descriptions provided by stakeholders. Viewpoints-based approaches have been proposed as a way to manage incomplete and inconsistent models gathered from multiple sources. In this thesis, we propose a category-theoretic framework for the analysis of fuzzy viewpoints. Informally, a fuzzy viewpoint is a graph in which the elements of a lattice are used to specify the amount of knowledge available about the details of nodes and edges. By defining an appropriate notion of morphism between fuzzy viewpoints, we construct categories of fuzzy viewpoints and prove that these categories are (finitely) cocomplete. We then show how colimits can be employed to merge the viewpoints and detect the inconsistencies that arise independent of any particular choice of viewpoint semantics. Taking advantage of the same category-theoretic techniques used in defining fuzzy viewpoints, we will also introduce a more general graph-based formalism that may find applications in other contexts. ii To my mother and father with love and gratitude. Acknowledgements First of all, I wish to thank my supervisor Steve Easterbrook for his guidance, support, and patience. -
8.6 Modular Arithmetic
“mcs” — 2015/5/18 — 1:43 — page 263 — #271 8.6 Modular Arithmetic On the first page of his masterpiece on number theory, Disquisitiones Arithmeticae, Gauss introduced the notion of “congruence.” Now, Gauss is another guy who managed to cough up a half-decent idea every now and then, so let’s take a look at this one. Gauss said that a is congruent to b modulo n iff n .a b/. This is j written a b.mod n/: ⌘ For example: 29 15 .mod 7/ because 7 .29 15/: ⌘ j It’s not useful to allow a modulus n 1, and so we will assume from now on that moduli are greater than 1. There is a close connection between congruences and remainders: Lemma 8.6.1 (Remainder). a b.mod n/ iff rem.a; n/ rem.b; n/: ⌘ D Proof. By the Division Theorem 8.1.4, there exist unique pairs of integers q1;r1 and q2;r2 such that: a q1n r1 D C b q2n r2; D C “mcs” — 2015/5/18 — 1:43 — page 264 — #272 264 Chapter 8 Number Theory where r1;r2 Œ0::n/. Subtracting the second equation from the first gives: 2 a b .q1 q2/n .r1 r2/; D C where r1 r2 is in the interval . n; n/. Now a b.mod n/ if and only if n ⌘ divides the left side of this equation. This is true if and only if n divides the right side, which holds if and only if r1 r2 is a multiple of n. -
Discrete Mathematics
Slides for Part IA CST 2016/17 Discrete Mathematics <www.cl.cam.ac.uk/teaching/1617/DiscMath> Prof Marcelo Fiore [email protected] — 0 — What are we up to ? ◮ Learn to read and write, and also work with, mathematical arguments. ◮ Doing some basic discrete mathematics. ◮ Getting a taste of computer science applications. — 2 — What is Discrete Mathematics ? from Discrete Mathematics (second edition) by N. Biggs Discrete Mathematics is the branch of Mathematics in which we deal with questions involving finite or countably infinite sets. In particular this means that the numbers involved are either integers, or numbers closely related to them, such as fractions or ‘modular’ numbers. — 3 — What is it that we do ? In general: Build mathematical models and apply methods to analyse problems that arise in computer science. In particular: Make and study mathematical constructions by means of definitions and theorems. We aim at understanding their properties and limitations. — 4 — Lecture plan I. Proofs. II. Numbers. III. Sets. IV. Regular languages and finite automata. — 6 — Proofs Objectives ◮ To develop techniques for analysing and understanding mathematical statements. ◮ To be able to present logical arguments that establish mathematical statements in the form of clear proofs. ◮ To prove Fermat’s Little Theorem, a basic result in the theory of numbers that has many applications in computer science. — 16 — Proofs in practice We are interested in examining the following statement: The product of two odd integers is odd. This seems innocuous enough, but it is in fact full of baggage. — 18 — Proofs in practice We are interested in examining the following statement: The product of two odd integers is odd. -
EE 595 (PMP) Advanced Topics in Communication Theory Handout #1
EE 595 (PMP) Advanced Topics in Communication Theory Handout #1 Introduction to Cryptography. Symmetric Encryption.1 Wednesday, January 13, 2016 Tamara Bonaci Department of Electrical Engineering University of Washington, Seattle Outline: 1. Review - Security goals 2. Terminology 3. Secure communication { Symmetric vs. asymmetric setting 4. Background: Modular arithmetic 5. Classical cryptosystems { The shift cipher { The substitution cipher 6. Background: The Euclidean Algorithm 7. More classical cryptosystems { The affine cipher { The Vigen´erecipher { The Hill cipher { The permutation cipher 8. Cryptanalysis { The Kerchoff Principle { Types of cryptographic attacks { Cryptanalysis of the shift cipher { Cryptanalysis of the affine cipher { Cryptanalysis of the Vigen´erecipher { Cryptanalysis of the Hill cipher Review - Security goals Last lecture, we introduced the following security goals: 1. Confidentiality - ability to keep information secret from all but authorized users. 2. Data integrity - property ensuring that messages to and from a user have not been corrupted by communication errors or unauthorized entities on their way to a destination. 3. Identity authentication - ability to confirm the unique identity of a user. 4. Message authentication - ability to undeniably confirm message origin. 5. Authorization - ability to check whether a user has permission to conduct some action. 6. Non-repudiation - ability to prevent the denial of previous commitments or actions (think of a con- tract). 7. Certification - endorsement of information by a trusted entity. In the rest of today's lecture, we will focus on three of these goals, namely confidentiality, integrity and authentication (CIA). In doing so, we begin by introducing the necessary terminology. 1 We thank Professors Radha Poovendran and Andrew Clark for the help in preparing this material. -
Quaternion Algebra and Calculus
Quaternion Algebra and Calculus David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. Created: March 2, 1999 Last Modified: August 18, 2010 Contents 1 Quaternion Algebra 2 2 Relationship of Quaternions to Rotations3 3 Quaternion Calculus 5 4 Spherical Linear Interpolation6 5 Spherical Cubic Interpolation7 6 Spline Interpolation of Quaternions8 1 This document provides a mathematical summary of quaternion algebra and calculus and how they relate to rotations and interpolation of rotations. The ideas are based on the article [1]. 1 Quaternion Algebra A quaternion is given by q = w + xi + yj + zk where w, x, y, and z are real numbers. Define qn = wn + xni + ynj + znk (n = 0; 1). Addition and subtraction of quaternions is defined by q0 ± q1 = (w0 + x0i + y0j + z0k) ± (w1 + x1i + y1j + z1k) (1) = (w0 ± w1) + (x0 ± x1)i + (y0 ± y1)j + (z0 ± z1)k: Multiplication for the primitive elements i, j, and k is defined by i2 = j2 = k2 = −1, ij = −ji = k, jk = −kj = i, and ki = −ik = j. Multiplication of quaternions is defined by q0q1 = (w0 + x0i + y0j + z0k)(w1 + x1i + y1j + z1k) = (w0w1 − x0x1 − y0y1 − z0z1)+ (w0x1 + x0w1 + y0z1 − z0y1)i+ (2) (w0y1 − x0z1 + y0w1 + z0x1)j+ (w0z1 + x0y1 − y0x1 + z0w1)k: Multiplication is not commutative in that the products q0q1 and q1q0 are not necessarily equal. -
WORKSHEET # 7 QUATERNIONS the Set of Quaternions Are the Set Of
WORKSHEET # 7 QUATERNIONS The set of quaternions are the set of all formal R-linear combinations of \symbols" i; j; k a + bi + cj + dk for a; b; c; d 2 R. We give them the following addition rule: (a + bi + cj + dk) + (a0 + b0i + c0j + d0k) = (a + a0) + (b + b0)i + (c + c0)j + (d + d0)k Multiplication is induced by the following rules (λ 2 R) i2 = j2 = k2 = −1 ij = k; jk = i; ki = j ji = −k kj = −i ik = −j λi = iλ λj = jλ λk = kλ combined with the distributive rule. 1. Use the above rules to carefully write down the product (a + bi + cj + dk)(a0 + b0i + c0j + d0k) as an element of the quaternions. Solution: (a + bi + cj + dk)(a0 + b0i + c0j + d0k) = aa0 + ab0i + ac0j + ad0k + ba0i − bb0 + bc0k − bd0j ca0j − cb0k − cc0 + cd0i + da0k + db0j − dc0i − dd0 = (aa0 − bb0 − cc0 − dd0) + (ab0 + a0b + cd0 − dc0)i (ac0 − bd0 + ca0 + db0)j + (ad0 + bc0 − cb0 + da0)k 2. Prove that the quaternions form a ring. Solution: It is obvious that the quaternions form a group under addition. We just showed that they were closed under multiplication. We have defined them to satisfy the distributive property. The only thing left is to show the associative property. I will only prove this for the elements i; j; k (this is sufficient by the distributive property). i(ij) = i(k) = ik = −j = (ii)j i(ik) = i(−j) = −ij = −k = (ii)k i(ji) = i(−k) = −ik = j = ki = (ij)i i(jj) = −i = kj = (ij)j i(jk) = i(i) = (−1) = (k)k = (ij)k i(ki) = ij = k = −ji = (ik)i i(kk) = −i = −jk = (ik)k j(ii) = −j = −ki = (ji)i j(ij) = jk = i = −kj = (ji)j j(ik) = j(−j) = 1 = −kk = (ji)k j(ji) = j(−k) = −jk = −i = (jj)i j(jk) = ji = −k = (jj)k j(ki) = jj = −1 = ii = (jk)i j(kj) = j(−i) = −ji = k = ij = (jk)j j(kk) = −j = ik = (jk)k k(ii) = −k = ji = (ki)i k(ij) = kk = −1 = jj = (ki)j k(ik) = k(−j) = −kj = i = jk = (ki)k k(ji) = k(−k) = 1 = (−i)i = (kj)i k(jj) = k(−1) = −k = −ij = (kj)j k(jk) = ki = j = −ik = (kj)k k(ki) = kj = −i = (kk)i k(kj) = k(−i) = −ki = −j = (kk)j 1 WORKSHEET #7 2 3. -
Math 412. §3.2, 3.3: Examples of Rings and Homomorphisms Professors Jack Jeffries and Karen E. Smith
Math 412. x3.2, 3.3: Examples of Rings and Homomorphisms Professors Jack Jeffries and Karen E. Smith DEFINITION:A subring of a ring R (with identity) is a subset S which is itself a ring (with identity) under the operations + and × for R. DEFINITION: An integral domain (or just domain) is a commutative ring R (with identity) satisfying the additional axiom: if xy = 0, then x or y = 0 for all x; y 2 R. φ DEFINITION:A ring homomorphism is a mapping R −! S between two rings (with identity) which satisfies: (1) φ(x + y) = φ(x) + φ(y) for all x; y 2 R. (2) φ(x · y) = φ(x) · φ(y) for all x; y 2 R. (3) φ(1) = 1. DEFINITION:A ring isomorphism is a bijective ring homomorphism. We say that two rings R and S are isomorphic if there is an isomorphism R ! S between them. You should think of a ring isomorphism as a renaming of the elements of a ring, so that two isomorphic rings are “the same ring” if you just change the names of the elements. A. WARM-UP: Which of the following rings is an integral domain? Which is a field? Which is commutative? In each case, be sure you understand what the implied ring structure is: why is each below a ring? What is the identity in each? (1) Z; Q. (2) Zn for n 2 Z. (3) R[x], the ring of polynomials with R-coefficients. (4) M2(Z), the ring of 2 × 2 matrices with Z coefficients. -
Modulo of a Negative Number1 1 Modular Arithmetic
Algologic Technical Report # 01/2017 Modulo of a negative number1 S. Parthasarathy [email protected] Ref.: negamodulo.tex Ver. code: 20170109e Abstract This short article discusses an enigmatic question in elementary num- ber theory { that of finding the modulo of a negative number. Is this feasible to compute ? If so, how do we compute this value ? 1 Modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value - the modulus (plural moduli). The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. Modular arithmetic is a fascinating aspect of mathematics. The \modulo" operation is often called as the fifth arithmetic operation, and comes after + − ∗ and = . It has use in many practical situations, particu- larly in number theory and cryptology [1] [2]. If x and n are integers, n < x and n =6 0 , the operation \ x mod n " is the remainder we get when n divides x. In other words, (x mod n) = r if there exists some integer q such that x = q ∗ n + r 1This is a LATEX document. You can get the LATEX source of this document from [email protected]. Please mention the Reference Code, and Version code, given at the top of this document 1 When r = 0 , n is a factor of x. When n is a factor of x, we say n divides x evenly, and denote it by n j x . The modulo operation can be combined with other arithmetic operators. -
The Development and Understanding of the Concept of Quotient Group
HISTORIA MATHEMATICA 20 (1993), 68-88 The Development and Understanding of the Concept of Quotient Group JULIA NICHOLSON Merton College, Oxford University, Oxford OXI 4JD, United Kingdom This paper describes the way in which the concept of quotient group was discovered and developed during the 19th century, and examines possible reasons for this development. The contributions of seven mathematicians in particular are discussed: Galois, Betti, Jordan, Dedekind, Dyck, Frobenius, and H61der. The important link between the development of this concept and the abstraction of group theory is considered. © 1993 AcademicPress. Inc. Cet article decrit la d6couverte et le d6veloppement du concept du groupe quotient au cours du 19i~me si~cle, et examine les raisons possibles de ce d6veloppement. Les contributions de sept math6maticiens seront discut6es en particulier: Galois, Betti, Jordan, Dedekind, Dyck, Frobenius, et H61der. Le lien important entre le d6veloppement de ce concept et l'abstraction de la th6orie des groupes sera consider6e. © 1993 AcademicPress, Inc. Diese Arbeit stellt die Entdeckung beziehungsweise die Entwicklung des Begriffs der Faktorgruppe wfihrend des 19ten Jahrhunderts dar und untersucht die m/)glichen Griinde for diese Entwicklung. Die Beitrfi.ge von sieben Mathematikern werden vor allem diskutiert: Galois, Betti, Jordan, Dedekind, Dyck, Frobenius und H61der. Die wichtige Verbindung zwischen der Entwicklung dieses Begriffs und der Abstraktion der Gruppentheorie wird betrachtet. © 1993 AcademicPress. Inc. AMS subject classifications: 01A55, 20-03 KEY WORDS: quotient group, group theory, Galois theory I. INTRODUCTION Nowadays one can hardly conceive any more of a group theory without factor groups... B. L. van der Waerden, from his obituary of Otto H61der [1939] Although the concept of quotient group is now considered to be fundamental to the study of groups, it is a concept which was unknown to early group theorists. -
Grade 7/8 Math Circles Modular Arithmetic the Modulus Operator
Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 7/8 Math Circles April 3, 2014 Modular Arithmetic The Modulus Operator The modulo operator has symbol \mod", is written as A mod N, and is read \A modulo N" or "A mod N". • A is our - what is being divided (just like in division) • N is our (the plural is ) When we divide two numbers A and N, we get a quotient and a remainder. • When we ask for A ÷ N, we look for the quotient of the division. • When we ask for A mod N, we are looking for the remainder of the division. • The remainder A mod N is always an integer between 0 and N − 1 (inclusive) Examples 1. 0 mod 4 = 0 since 0 ÷ 4 = 0 remainder 0 2. 1 mod 4 = 1 since 1 ÷ 4 = 0 remainder 1 3. 2 mod 4 = 2 since 2 ÷ 4 = 0 remainder 2 4. 3 mod 4 = 3 since 3 ÷ 4 = 0 remainder 3 5. 4 mod 4 = since 4 ÷ 4 = 1 remainder 6. 5 mod 4 = since 5 ÷ 4 = 1 remainder 7. 6 mod 4 = since 6 ÷ 4 = 1 remainder 8. 15 mod 4 = since 15 ÷ 4 = 3 remainder 9.* −15 mod 4 = since −15 ÷ 4 = −4 remainder Using a Calculator For really large numbers and/or moduli, sometimes we can use calculators to quickly find A mod N. This method only works if A ≥ 0. Example: Find 373 mod 6 1. Divide 373 by 6 ! 2. Round the number you got above down to the nearest integer ! 3. -
Vector Spaces
Chapter 1 Vector Spaces Linear algebra is the study of linear maps on finite-dimensional vec- tor spaces. Eventually we will learn what all these terms mean. In this chapter we will define vector spaces and discuss their elementary prop- erties. In some areas of mathematics, including linear algebra, better the- orems and more insight emerge if complex numbers are investigated along with real numbers. Thus we begin by introducing the complex numbers and their basic✽ properties. 1 2 Chapter 1. Vector Spaces Complex Numbers You should already be familiar with the basic properties of the set R of real numbers. Complex numbers were invented so that we can take square roots of negative numbers. The key idea is to assume we have i − i The symbol was√ first a square root of 1, denoted , and manipulate it using the usual rules used to denote −1 by of arithmetic. Formally, a complex number is an ordered pair (a, b), the Swiss where a, b ∈ R, but we will write this as a + bi. The set of all complex mathematician numbers is denoted by C: Leonhard Euler in 1777. C ={a + bi : a, b ∈ R}. If a ∈ R, we identify a + 0i with the real number a. Thus we can think of R as a subset of C. Addition and multiplication on C are defined by (a + bi) + (c + di) = (a + c) + (b + d)i, (a + bi)(c + di) = (ac − bd) + (ad + bc)i; here a, b, c, d ∈ R. Using multiplication as defined above, you should verify that i2 =−1. -
RING THEORY 1. Ring Theory a Ring Is a Set a with Two Binary Operations
CHAPTER IV RING THEORY 1. Ring Theory A ring is a set A with two binary operations satisfying the rules given below. Usually one binary operation is denoted `+' and called \addition," and the other is denoted by juxtaposition and is called \multiplication." The rules required of these operations are: 1) A is an abelian group under the operation + (identity denoted 0 and inverse of x denoted x); 2) A is a monoid under the operation of multiplication (i.e., multiplication is associative and there− is a two-sided identity usually denoted 1); 3) the distributive laws (x + y)z = xy + xz x(y + z)=xy + xz hold for all x, y,andz A. Sometimes one does∈ not require that a ring have a multiplicative identity. The word ring may also be used for a system satisfying just conditions (1) and (3) (i.e., where the associative law for multiplication may fail and for which there is no multiplicative identity.) Lie rings are examples of non-associative rings without identities. Almost all interesting associative rings do have identities. If 1 = 0, then the ring consists of one element 0; otherwise 1 = 0. In many theorems, it is necessary to specify that rings under consideration are not trivial, i.e. that 1 6= 0, but often that hypothesis will not be stated explicitly. 6 If the multiplicative operation is commutative, we call the ring commutative. Commutative Algebra is the study of commutative rings and related structures. It is closely related to algebraic number theory and algebraic geometry. If A is a ring, an element x A is called a unit if it has a two-sided inverse y, i.e.