6.3 Modular Exponentiation Most Technological Applications of Modular Arithmetic Involve Exponentials with Very Large Numbers
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Common Course Outline MATH 257 Linear Algebra 4 Credits
Common Course Outline MATH 257 Linear Algebra 4 Credits The Community College of Baltimore County Description MATH 257 – Linear Algebra is one of the suggested elective courses for students majoring in Mathematics, Computer Science or Engineering. Included are geometric vectors, matrices, systems of linear equations, vector spaces, linear transformations, determinants, eigenvectors and inner product spaces. 4 Credits: 5 lecture hours Prerequisite: MATH 251 with a grade of “C” or better Overall Course Objectives Upon successfully completing the course, students will be able to: 1. perform matrix operations; 2. use Gaussian Elimination, Cramer’s Rule, and the inverse of the coefficient matrix to solve systems of Linear Equations; 3. find the inverse of a matrix by Gaussian Elimination or using the adjoint matrix; 4. compute the determinant of a matrix using cofactor expansion or elementary row operations; 5. apply Gaussian Elimination to solve problems concerning Markov Chains; 6. verify that a structure is a vector space by checking the axioms; 7. verify that a subset is a subspace and that a set of vectors is a basis; 8. compute the dimensions of subspaces; 9. compute the matrix representation of a linear transformation; 10. apply notions of linear transformations to discuss rotations and reflections of two dimensional space; 11. compute eigenvalues and find their corresponding eigenvectors; 12. diagonalize a matrix using eigenvalues; 13. apply properties of vectors and dot product to prove results in geometry; 14. apply notions of vectors, dot product and matrices to construct a best fitting curve; 15. construct a solution to real world problems using problem methods individually and in groups; 16. -
The Enigmatic Number E: a History in Verse and Its Uses in the Mathematics Classroom
To appear in MAA Loci: Convergence The Enigmatic Number e: A History in Verse and Its Uses in the Mathematics Classroom Sarah Glaz Department of Mathematics University of Connecticut Storrs, CT 06269 [email protected] Introduction In this article we present a history of e in verse—an annotated poem: The Enigmatic Number e . The annotation consists of hyperlinks leading to biographies of the mathematicians appearing in the poem, and to explanations of the mathematical notions and ideas presented in the poem. The intention is to celebrate the history of this venerable number in verse, and to put the mathematical ideas connected with it in historical and artistic context. The poem may also be used by educators in any mathematics course in which the number e appears, and those are as varied as e's multifaceted history. The sections following the poem provide suggestions and resources for the use of the poem as a pedagogical tool in a variety of mathematics courses. They also place these suggestions in the context of other efforts made by educators in this direction by briefly outlining the uses of historical mathematical poems for teaching mathematics at high-school and college level. Historical Background The number e is a newcomer to the mathematical pantheon of numbers denoted by letters: it made several indirect appearances in the 17 th and 18 th centuries, and acquired its letter designation only in 1731. Our history of e starts with John Napier (1550-1617) who defined logarithms through a process called dynamical analogy [1]. Napier aimed to simplify multiplication (and in the same time also simplify division and exponentiation), by finding a model which transforms multiplication into addition. -
21. Orthonormal Bases
21. Orthonormal Bases The canonical/standard basis 011 001 001 B C B C B C B0C B1C B0C e1 = B.C ; e2 = B.C ; : : : ; en = B.C B.C B.C B.C @.A @.A @.A 0 0 1 has many useful properties. • Each of the standard basis vectors has unit length: q p T jjeijj = ei ei = ei ei = 1: • The standard basis vectors are orthogonal (in other words, at right angles or perpendicular). T ei ej = ei ej = 0 when i 6= j This is summarized by ( 1 i = j eT e = δ = ; i j ij 0 i 6= j where δij is the Kronecker delta. Notice that the Kronecker delta gives the entries of the identity matrix. Given column vectors v and w, we have seen that the dot product v w is the same as the matrix multiplication vT w. This is the inner product on n T R . We can also form the outer product vw , which gives a square matrix. 1 The outer product on the standard basis vectors is interesting. Set T Π1 = e1e1 011 B C B0C = B.C 1 0 ::: 0 B.C @.A 0 01 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 0 . T Πn = enen 001 B C B0C = B.C 0 0 ::: 1 B.C @.A 1 00 0 ::: 01 B C B0 0 ::: 0C = B. .C B. .C @. .A 0 0 ::: 1 In short, Πi is the diagonal square matrix with a 1 in the ith diagonal position and zeros everywhere else. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
Sequence Rules
SEQUENCE RULES A connected series of five of the same colored chip either up or THE JACKS down, across or diagonally on the playing surface. There are 8 Jacks in the card deck. The 4 Jacks with TWO EYES are wild. To play a two-eyed Jack, place it on your discard pile and place NOTE: There are printed chips in the four corners of the game board. one of your marker chips on any open space on the game board. The All players must use them as though their color marker chip is in 4 jacks with ONE EYE are anti-wild. To play a one-eyed Jack, place the corner. When using a corner, only four of your marker chips are it on your discard pile and remove one marker chip from the game needed to complete a Sequence. More than one player may use the board belonging to your opponent. That completes your turn. You same corner as part of a Sequence. cannot place one of your marker chips on that same space during this turn. You cannot remove a marker chip that is already part of a OBJECT OF THE GAME: completed SEQUENCE. Once a SEQUENCE is achieved by a player For 2 players or 2 teams: One player or team must score TWO SE- or a team, it cannot be broken. You may play either one of the Jacks QUENCES before their opponents. whenever they work best for your strategy, during your turn. For 3 players or 3 teams: One player or team must score ONE SE- DEAD CARD QUENCE before their opponents. -
Grade 7/8 Math Circles the Scale of Numbers Introduction
Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 7/8 Math Circles November 21/22/23, 2017 The Scale of Numbers Introduction Last week we quickly took a look at scientific notation, which is one way we can write down really big numbers. We can also use scientific notation to write very small numbers. 1 × 103 = 1; 000 1 × 102 = 100 1 × 101 = 10 1 × 100 = 1 1 × 10−1 = 0:1 1 × 10−2 = 0:01 1 × 10−3 = 0:001 As you can see above, every time the value of the exponent decreases, the number gets smaller by a factor of 10. This pattern continues even into negative exponent values! Another way of picturing negative exponents is as a division by a positive exponent. 1 10−6 = = 0:000001 106 In this lesson we will be looking at some famous, interesting, or important small numbers, and begin slowly working our way up to the biggest numbers ever used in mathematics! Obviously we can come up with any arbitrary number that is either extremely small or extremely large, but the purpose of this lesson is to only look at numbers with some kind of mathematical or scientific significance. 1 Extremely Small Numbers 1. Zero • Zero or `0' is the number that represents nothingness. It is the number with the smallest magnitude. • Zero only began being used as a number around the year 500. Before this, ancient mathematicians struggled with the concept of `nothing' being `something'. 2. Planck's Constant This is the smallest number that we will be looking at today other than zero. -
Determinants Math 122 Calculus III D Joyce, Fall 2012
Determinants Math 122 Calculus III D Joyce, Fall 2012 What they are. A determinant is a value associated to a square array of numbers, that square array being called a square matrix. For example, here are determinants of a general 2 × 2 matrix and a general 3 × 3 matrix. a b = ad − bc: c d a b c d e f = aei + bfg + cdh − ceg − afh − bdi: g h i The determinant of a matrix A is usually denoted jAj or det (A). You can think of the rows of the determinant as being vectors. For the 3×3 matrix above, the vectors are u = (a; b; c), v = (d; e; f), and w = (g; h; i). Then the determinant is a value associated to n vectors in Rn. There's a general definition for n×n determinants. It's a particular signed sum of products of n entries in the matrix where each product is of one entry in each row and column. The two ways you can choose one entry in each row and column of the 2 × 2 matrix give you the two products ad and bc. There are six ways of chosing one entry in each row and column in a 3 × 3 matrix, and generally, there are n! ways in an n × n matrix. Thus, the determinant of a 4 × 4 matrix is the signed sum of 24, which is 4!, terms. In this general definition, half the terms are taken positively and half negatively. In class, we briefly saw how the signs are determined by permutations. -
Efficient Regular Modular Exponentiation Using
J Cryptogr Eng (2017) 7:245–253 DOI 10.1007/s13389-016-0134-5 SHORT COMMUNICATION Efficient regular modular exponentiation using multiplicative half-size splitting Christophe Negre1,2 · Thomas Plantard3,4 Received: 14 August 2015 / Accepted: 23 June 2016 / Published online: 13 July 2016 © Springer-Verlag Berlin Heidelberg 2016 Abstract In this paper, we consider efficient RSA modular x K mod N where N = pq with p and q prime. The private exponentiations x K mod N which are regular and con- data are the two prime factors of N and the private exponent stant time. We first review the multiplicative splitting of an K used to decrypt or sign a message. In order to insure a integer x modulo N into two half-size integers. We then sufficient security level, N and K are chosen large enough take advantage of this splitting to modify the square-and- to render the factorization of N infeasible: they are typically multiply exponentiation as a regular sequence of squarings 2048-bit integers. The basic approach to efficiently perform always followed by a multiplication by a half-size inte- the modular exponentiation is the square-and-multiply algo- ger. The proposed method requires around 16% less word rithm which scans the bits ki of the exponent K and perform operations compared to Montgomery-ladder, square-always a sequence of squarings followed by a multiplication when and square-and-multiply-always exponentiations. These the- ki is equal to one. oretical results are validated by our implementation results When the cryptographic computations are performed on which show an improvement by more than 12% compared an embedded device, an adversary can monitor power con- approaches which are both regular and constant time. -
0.999… = 1 an Infinitesimal Explanation Bryan Dawson
0 1 2 0.9999999999999999 0.999… = 1 An Infinitesimal Explanation Bryan Dawson know the proofs, but I still don’t What exactly does that mean? Just as real num- believe it.” Those words were uttered bers have decimal expansions, with one digit for each to me by a very good undergraduate integer power of 10, so do hyperreal numbers. But the mathematics major regarding hyperreals contain “infinite integers,” so there are digits This fact is possibly the most-argued- representing not just (the 237th digit past “Iabout result of arithmetic, one that can evoke great the decimal point) and (the 12,598th digit), passion. But why? but also (the Yth digit past the decimal point), According to Robert Ely [2] (see also Tall and where is a negative infinite hyperreal integer. Vinner [4]), the answer for some students lies in their We have four 0s followed by a 1 in intuition about the infinitely small: While they may the fifth decimal place, and also where understand that the difference between and 1 is represents zeros, followed by a 1 in the Yth less than any positive real number, they still perceive a decimal place. (Since we’ll see later that not all infinite nonzero but infinitely small difference—an infinitesimal hyperreal integers are equal, a more precise, but also difference—between the two. And it’s not just uglier, notation would be students; most professional mathematicians have not or formally studied infinitesimals and their larger setting, the hyperreal numbers, and as a result sometimes Confused? Perhaps a little background information wonder . -
Miller-Rabin Primality Test (Java)
Miller-Rabin primality test (Java) Other implementations: C | C, GMP | Clojure | Groovy | Java | Python | Ruby | Scala The Miller-Rabin primality test is a simple probabilistic algorithm for determining whether a number is prime or composite that is easy to implement. It proves compositeness of a number using the following formulas: Suppose 0 < a < n is coprime to n (this is easy to test using the GCD). Write the number n−1 as , where d is odd. Then, provided that all of the following formulas hold, n is composite: for all If a is chosen uniformly at random and n is prime, these formulas hold with probability 1/4. Thus, repeating the test for k random choices of a gives a probability of 1 − 1 / 4k that the number is prime. Moreover, Gerhard Jaeschke showed that any 32-bit number can be deterministically tested for primality by trying only a=2, 7, and 61. [edit] 32-bit integers We begin with a simple implementation for 32-bit integers, which is easier to implement for reasons that will become apparent. First, we'll need a way to perform efficient modular exponentiation on an arbitrary 32-bit integer. We accomplish this using exponentiation by squaring: Source URL: http://www.en.literateprograms.org/Miller-Rabin_primality_test_%28Java%29 Saylor URL: http://www.saylor.org/courses/cs409 ©Spoon! (http://www.en.literateprograms.org/Miller-Rabin_primality_test_%28Java%29) Saylor.org Used by Permission Page 1 of 5 <<32-bit modular exponentiation function>>= private static int modular_exponent_32(int base, int power, int modulus) { long result = 1; for (int i = 31; i >= 0; i--) { result = (result*result) % modulus; if ((power & (1 << i)) != 0) { result = (result*base) % modulus; } } return (int)result; // Will not truncate since modulus is an int } int is a 32-bit integer type and long is a 64-bit integer type. -
Three Methods of Finding Remainder on the Operation of Modular
Zin Mar Win, Khin Mar Cho; International Journal of Advance Research and Development (Volume 5, Issue 5) Available online at: www.ijarnd.com Three methods of finding remainder on the operation of modular exponentiation by using calculator 1Zin Mar Win, 2Khin Mar Cho 1University of Computer Studies, Myitkyina, Myanmar 2University of Computer Studies, Pyay, Myanmar ABSTRACT The primary purpose of this paper is that the contents of the paper were to become a learning aid for the learners. Learning aids enhance one's learning abilities and help to increase one's learning potential. They may include books, diagram, computer, recordings, notes, strategies, or any other appropriate items. In this paper we would review the types of modulo operations and represent the knowledge which we have been known with the easiest ways. The modulo operation is finding of the remainder when dividing. The modulus operator abbreviated “mod” or “%” in many programming languages is useful in a variety of circumstances. It is commonly used to take a randomly generated number and reduce that number to a random number on a smaller range, and it can also quickly tell us if one number is a factor of another. If we wanted to know if a number was odd or even, we could use modulus to quickly tell us by asking for the remainder of the number when divided by 2. Modular exponentiation is a type of exponentiation performed over a modulus. The operation of modular exponentiation calculates the remainder c when an integer b rose to the eth power, bᵉ, is divided by a positive integer m such as c = be mod m [1]. -
The Exponential Function
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 5-2006 The Exponential Function Shawn A. Mousel University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Mousel, Shawn A., "The Exponential Function" (2006). MAT Exam Expository Papers. 26. https://digitalcommons.unl.edu/mathmidexppap/26 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. The Exponential Function Expository Paper Shawn A. Mousel In partial fulfillment of the requirements for the Masters of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor May 2006 Mousel – MAT Expository Paper - 1 One of the basic principles studied in mathematics is the observation of relationships between two connected quantities. A function is this connecting relationship, typically expressed in a formula that describes how one element from the domain is related to exactly one element located in the range (Lial & Miller, 1975). An exponential function is a function with the basic form f (x) = ax , where a (a fixed base that is a real, positive number) is greater than zero and not equal to 1. The exponential function is not to be confused with the polynomial functions, such as x 2. One way to recognize the difference between the two functions is by the name of the function.