Factoring in the Chicken Mcnugget Monoid
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Trees of Irreducible Numerical Semigroups
Rose-Hulman Undergraduate Mathematics Journal Volume 14 Issue 1 Article 5 Trees of Irreducible Numerical Semigroups Taryn M. Laird Northern Arizona University, [email protected] Jose E. Martinez Northern Arizona University Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Laird, Taryn M. and Martinez, Jose E. (2013) "Trees of Irreducible Numerical Semigroups," Rose-Hulman Undergraduate Mathematics Journal: Vol. 14 : Iss. 1 , Article 5. Available at: https://scholar.rose-hulman.edu/rhumj/vol14/iss1/5 Rose- Hulman Undergraduate Mathematics Journal Trees of Irreducible Numerical Semigroups Taryn M. Laird and Jos´eE. Martinez a Volume 14, No. 1, Spring 2013 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 Email: [email protected] a http://www.rose-hulman.edu/mathjournal Northern Arizona University Rose-Hulman Undergraduate Mathematics Journal Volume 14, No. 1, Spring 2013 Trees of Irreducible Numerical Semigroups Taryn M. Laird and Jos´eE. Martinez Abstract.A 2011 paper by Blanco and Rosales describes an algorithm for construct- ing a directed tree graph of irreducible numerical semigroups of fixed Frobenius numbers. This paper will provide an overview of irreducible numerical semigroups and the directed tree graphs. We will also present new findings and conjectures concerning the structure of these trees. Acknowledgements: We would like to give a special thanks to our mentor, Jeff Rushall, for his help and guidance, and to Ian Douglas for sharing his expertise in programming. Page 58 RHIT Undergrad. Math. J., Vol. 14, No. 1 1 Introduction There are many problems in which we unknowingly encounter numerical semigroups in mathematics. -
The Frobenius Problem in a Free Monoid
The Frobenius Problem in a Free Monoid by Zhi Xu A thesis presented to the University of Waterloo in ful¯llment of the thesis requirement for the degree of Doctor of Philosophy in Computer Science Waterloo, Ontario, Canada, 2009 °c Zhi Xu 2009 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required ¯nal revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract Given positive integers c1; c2; : : : ; ck with gcd(c1; c2; : : : ; ck) = 1, the Frobenius prob- lem (FP) is to compute the largest integer g(c1; c2; : : : ; ck) that cannot be written as a non-negative integer linear combination of c1; c2; : : : ; ck. The Frobenius prob- lem in a free monoid (FPFM) is a non-commutative generalization of the Frobe- nius problem. Given words x1; x2; : : : ; xk such that there are only ¯nitely many words that cannot be written as concatenations of words in f x1; x2; : : : ; xk g, the FPFM is to ¯nd the longest such words. Unlike the FP, where the upper bound 2 g(c1; c2; : : : ; ck) · max1·i·k ci is quadratic, the upper bound on the length of the longest words in the FPFM can be exponential in certain measures and some of the exponential upper bounds are tight. For the 2FPFM, where the given words over § are of only two distinct lengths m and n with 1 < m < n, the length of the longest omitted words is · g(m; m j§jn¡m + n ¡ m). -
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Visit : https://hemanthrajhemu.github.io Join Telegram to get Instant Updates: https://bit.ly/VTU_TELEGRAM Contact: MAIL: [email protected] INSTAGRAM: www.instagram.com/hemanthraj_hemu/ INSTAGRAM: www.instagram.com/futurevisionbie/ WHATSAPP SHARE: https://bit.ly/FVBIESHARE Introduction to The Design & ~ ~ Analysis of Algorithms IND EDITION " ~ Anany Levitin Villanova University Boston San Francisco New York London Toronto Sydney Tokyo Singapore Madrid Mexico City Munich Paris Cape Town Hong Kong Montreal https://hemanthrajhemu.github.io Contents ix 2.6 Empirical Analysis of Algorithms 84 Exercises 2.6 90 2.7 Algorithm Visualization 91 Summary 95 3 Brute Force 97 3.1 Selection Sort and Bubble Sort 98 Selection Sort 98 Bubble Sort 100 Exercises 3.1 102 3.2 Sequential Search and Brute-Force String Matching 103 Sequential Search 103 Brute-Force String Matching 104 Exercises 3.2 105 3.3 Closest-Pair and Convex-Hull Problems by Brute Force 107 Closest-Pair Problem 107 Convex-Hull Problem 108 Exercises 3.3 112 3.4 Exhaustive Search 114 Traveling Salesman Problem 114 Knapsack Problem 115 Assignment Problem 116 Exercises 3.4 119 Summary 120 4 Divide-and-Conquer 123 4.1 Mergesort 125 Exercises 4.1 128 4.2 Quicksort 129 Exercises 4.2 134 4.3 Binary Search 135 Exercises 4.3 138 https://hemanthrajhemu.github.io r 111 )( Contents I ' i::! 4.4 Binary Tree Traversals and Related Properties 139 Exercises 4.4 142 I:],, 4.5 Multiplication of Large Integers and Strassen's Matrix Multiplication 144 Multiplication of Large Integers 144 Strassen's Matrix -
On the Change-Making Problem
On the Change-Making Problem Timothy M. Chan∗ Qizheng He∗ Abstract (There are a number of other interesting problems Given a set of n non-negative integers representing a coin related to coin systems, such as the Frobenius problem, system, the change-making problem seeks the fewest number but these are not relevant to the present paper.) of coins that sum up to a given value t, where each Partially spurred by a renewed interest in fine- type of coin can be used an unlimited number of times. grained complexity, there have been a flurry of re- This problem is a popular homework exercise in dynamic cent works that improve upon the standard O(nt) so- programming, where the textbook solution runs in O(nt) lution for the subset sum problem,p starting with Koil- time. iaris and Xu's deterministic O( nt polylog t)-time al- It is not hard to solve this problem in O(t polylog t) time gorithm [8, 9], followed by Bringmann's randomized by using convolution. In this paper, we present a simple O(t polylog t)-time algorithm [3] and Jin and Wu's deterministic O(t log t log log t) time algorithm, and later randomized O(t log t)-time algorithm from SOSA last improve the running time to O(t log t) by randomization. year [7]. All these improved algorithms use convolu- tions (i.e., FFT) or other algebraic techniques. There 1 Introduction are also conditional lower bounds [1] to suggest that get- ting better than t1−" running time might not be possible In the change-making (or coin changing) problem, a for subset sum (ignoring nO(1) factors). -
The Frobenius Problem in a Free Monoid
The Frobenius Problem in a Free Monoid by Zhi Xu A thesis presented to the University of Waterloo in ful¯llment of the thesis requirement for the degree of Doctor of Philosophy in Computer Science Waterloo, Ontario, Canada, 2009 °c Zhi Xu 2009 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required ¯nal revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract Given positive integers c1; c2; : : : ; ck with gcd(c1; c2; : : : ; ck) = 1, the Frobenius prob- lem (FP) is to compute the largest integer g(c1; c2; : : : ; ck) that cannot be written as a non-negative integer linear combination of c1; c2; : : : ; ck. The Frobenius prob- lem in a free monoid (FPFM) is a non-commutative generalization of the Frobe- nius problem. Given words x1; x2; : : : ; xk such that there are only ¯nitely many words that cannot be written as concatenations of words in f x1; x2; : : : ; xk g, the FPFM is to ¯nd the longest such words. Unlike the FP, where the upper bound 2 g(c1; c2; : : : ; ck) · max1·i·k ci is quadratic, the upper bound on the length of the longest words in the FPFM can be exponential in certain measures and some of the exponential upper bounds are tight. For the 2FPFM, where the given words over § are of only two distinct lengths m and n with 1 < m < n, the length of the longest omitted words is · g(m; m j§jn¡m + n ¡ m).