Canadam 2019 the Canadian Discrete and Algorithmic Mathematics Conference 2019.Canadam.Math.Ca
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Combinatorial Proof of an Abel-Type Identity∗
Combinatorial Proof of an Abel-type Identity∗ P. Mark Kayll Department of Mathematical Sciences, University of Montana Missoula MT 59812-0864, USA [email protected] David Perkins Department of Mathematics and Computer Science Houghton College, Houghton NY 14744, USA [email protected] Identity (1) below resulted from our investigation in [21] of chip-firing games on complete graphs Kn, for n ≥ 1; see, e.g., [2] for antecedents. The left side expresses the sum of the probabilities of a game experiencing firing sequences of each possible length ℓ = 0, 1,...,n. This note gives a combinatorial proof that these probabilities sum to unity. We first manipulate n n − 1 n ℓℓ−1(n + 1 − ℓ)n−1−ℓ + = 1 (1) n + 1 µℓ¶ n(n + 1)n−1 Xℓ=1 into a form amenable to combinatorial proof. Multiplying by n(n+1)n and n n+1 using the relation ℓ = ℓ (n + 1 − ℓ)/(n + 1), we transform (1) to the equivalent form ¡ ¢ ¡ ¢ n n + 1 ℓℓ−2(n + 1 − ℓ)n−1−ℓℓ(n + 1 − ℓ) = 2n(n + 1)n−1. (2) µ ℓ ¶ Xℓ=1 To see that (2) holds, first observe that the right side enumerates the pairs (T,~e ), where T is a spanning tree of Kn+1 for which one edge e (of 2000 MSC: Primary 05A19; Secondary 05C30, 60C05. ∗Preprint to appear in J. Combin. Math. Combin. Comput. 1 its n edges) has been distinguished and oriented (in one of two possible directions). The left side also enumerates these pairs. Given (T,~e ), notice that deleting the oriented edge ~e from T leaves behind a spanning forest of Kn+1 with two components L, R (that we may consider ordered from left to right). -
Negotiating a Quantum Computation Network: Mechanics, Machines, Mindsets
Negotiating a Quantum Computation Network: Mechanics, Machines, Mindsets by Derek Noon A thesis submitted to the Faculty of Graduate and Postdoctoral Affairs in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Communication Carleton University Ottawa, Ontario © 2016, Derek Noon Abstract This dissertation describes the origins, development, and distribution of quantum computing from a socio-technical perspective. It depicts quantum computing as a result of the negotiations of heterogeneous actors using the concepts of ANT and socio-technical analyses of computing and infrastructure more generally. It draws on two years of participant observation and interviews with the hardware and software companies that developed, sold, and distributed both machines and a mindset for a new approach to computing: adiabatic quantum computation (AQC). It illustrates how a novel form of computation and software writing was developed by challenging and recoding the usual distinctions between digital and analogue computing, and discusses how the myriad controversies and failures attending quantum computing were resolved provisionally through a series of human and non-human negotiations. These negotiations disrupted, scrambled, and reconstituted what we usually understand as hardware, software, and mindset, and permitted a ‗disruptive‘ technology to gain common acceptance in several high profile scientific, governmental, and financial institutions. It is the relationalities established across these diverse processes that constitute quantum computing, and consequences of this account of computation are considered in the context of digital media theory, industrial histories of computing, and socio-technical theories of technological innovation. Noon ii Acknowledgements Many sources of support helped me through the PhD program. I‘m grateful to Mitacs for its financial support of this research, and for providing me such good STEM peers/research subjects. -
Math 958–Topics in Discrete Mathematics Spring Semester 2018 TR 09:30–10:45 in Avery Hall (AVH) 351 1 Instructor
Math 958{Topics in Discrete Mathematics Spring Semester 2018 TR 09:30{10:45 in Avery Hall (AVH) 351 1 Instructor Dr. Tri Lai Assistant Professor Department of Mathematics University of Nebraska - Lincoln Lincoln, NE 68588-0130, USA Email: [email protected] Website: http://www.math.unl.edu/$\sim$tlai3/ Office: Avery Hall 339 Office Hours: By Appointment 2 Prerequisites Math 450. 3 Contacting me The best way to contact with me is by email, [email protected]. Please put \[MATH 958]" at the beginning of the title and make sure to include your whole name in your email. Using your official UNL email to contact me is strongly recommended. My office is Avery Hall 339. My office hours are by appoinment. 4 Course Description Bijective Combinatorics is a branch of combinatorics focusing on bijections between mathematical objects. The fact is that there many totally different objects in mathematics that are actually equinumerous, in this case, we always want to seek for a simple bijection between them. For example, the number of ways to divide that a convex (n + 2)-gon into triangles by non-intersecting diagonals is equal to the number of full binary trees with n + 1 leaves. Both objects are counted 1 2n by the well known Catalan number Cn = n+1 n . Do you know that there are more than two hundred (!!) mathematical objects are counted by the Catalan number? In this course, we will go over a number of such \Catalan objects" and investigate the beautiful bijections between them. One more example is the well-known theorem by Euler about integer partitions stating that the number of ways to write a positive integer n as the sum of distinct positive integers is equal to the number of ways to write n as the sum of odd positive integers. -
Vancouver Cross-Border Investment Guide
Claire to try illustration idea as one final cover option Vancouver Cross-Border Investment Guide Essential legal, tax and market information for cross-border investment into Vancouver, Canada Digital Download 1 Vancouver Cross-Border Contents Investment Guide Published October 2020 Why Invest in Vancouver ............................................................................1 Sectors to Watch ........................................................................................... 3 About the Vancouver Economic Commission Technology ..................................................................................................3 The Vancouver Economic Commission (VEC) serves one of the world’s fastest-growing, low- Cleantech .................................................................................................... 4 carbon economies. As the economic development agency for the city’s businesses, investors and citizens, VEC works to strengthen Vancouver’s economic future by supporting local companies, attracting high-impact investment, conducting and publishing leading-edge industry research, Media and Entertainment ............................................................................5 and promoting international trade. VEC works collaboratively to position Vancouver as a global destination for innovative, creative, diverse and sustainable development. Life Sciences ............................................................................................... 6 VEC respectfully acknowledges that it is located -
Basic Combinatorics
Basic Combinatorics Carl G. Wagner Department of Mathematics The University of Tennessee Knoxville, TN 37996-1300 Contents List of Figures iv List of Tables v 1 The Fibonacci Numbers From a Combinatorial Perspective 1 1.1 A Simple Counting Problem . 1 1.2 A Closed Form Expression for f(n) . 2 1.3 The Method of Generating Functions . 3 1.4 Approximation of f(n) . 4 2 Functions, Sequences, Words, and Distributions 5 2.1 Multisets and sets . 5 2.2 Functions . 6 2.3 Sequences and words . 7 2.4 Distributions . 7 2.5 The cardinality of a set . 8 2.6 The addition and multiplication rules . 9 2.7 Useful counting strategies . 11 2.8 The pigeonhole principle . 13 2.9 Functions with empty domain and/or codomain . 14 3 Subsets with Prescribed Cardinality 17 3.1 The power set of a set . 17 3.2 Binomial coefficients . 17 4 Sequences of Two Sorts of Things with Prescribed Frequency 23 4.1 A special sequence counting problem . 23 4.2 The binomial theorem . 24 4.3 Counting lattice paths in the plane . 26 5 Sequences of Integers with Prescribed Sum 28 5.1 Urn problems with indistinguishable balls . 28 5.2 The family of all compositions of n . 30 5.3 Upper bounds on the terms of sequences with prescribed sum . 31 i CONTENTS 6 Sequences of k Sorts of Things with Prescribed Frequency 33 6.1 Trinomial Coefficients . 33 6.2 The trinomial theorem . 35 6.3 Multinomial coefficients and the multinomial theorem . 37 7 Combinatorics and Probability 39 7.1 The Multinomial Distribution . -
Vancouver Cross-Border Investment Guide
Claire to try illustration idea as one final cover option Vancouver Cross-Border Investment Guide Essential legal, tax and market information for cross-border investment into Vancouver, Canada Digital Download 1 Vancouver Cross-Border Contents Investment Guide Published October 2020 Version 1.2, released January 2021 Why Invest in Vancouver ............................................................................1 Sectors to Watch ........................................................................................... 3 About the Vancouver Economic Commission Technology ..................................................................................................3 The Vancouver Economic Commission (VEC) serves one of the world’s fastest-growing, low- Cleantech .................................................................................................... 4 carbon economies. As the economic development agency for the city’s businesses, investors and citizens, VEC works to strengthen Vancouver’s economic future by supporting local companies, attracting high-impact investment, conducting and publishing leading-edge industry research, Media and Entertainment ............................................................................5 and promoting international trade. VEC works collaboratively to position Vancouver as a global destination for innovative, creative, diverse and sustainable development. Life Sciences ............................................................................................... 6 VEC respectfully -
2019-2020 Award #1440415 August 10, 2020
Institute for Pure and Applied Mathematics, UCLA Annual Progress Report for 2019-2020 Award #1440415 August 10, 2020 TABLE OF CONTENTS EXECUTIVE SUMMARY .................................................................................................................... 2 A. PARTICIPANT LIST ....................................................................................................................... 4 B. FINANCE SUPPORT LIST ............................................................................................................. 4 C. INCOME AND EXPENDITURE REPORT .................................................................................... 4 D. POSTDOCTORAL PLACEMENT LIST ........................................................................................ 5 E. MATH INSTITUTE DIRECTORS’ MEETING REPORT ............................................................. 6 F. PARTICIPANT SUMMARY .......................................................................................................... 20 G. POSTDOCTORAL PROGRAM SUMMARY............................................................................... 23 H. GRADUATE STUDENT PROGRAM SUMMARY ...................................................................... 24 I. UNDERGRADUATE STUDENT PROGRAM SUMMARY .......................................................... 25 J. PROGRAM DESCRIPTION .......................................................................................................... 25 K. PROGRAM CONSULTANT LIST ............................................................................................... -
Annual Report Astc Science World Society 2019/20 Stats
2019/20 ANNUAL REPORT ASTC SCIENCE WORLD SOCIETY 2019/20 STATS 1,007,272 Visitors on-site and in outreach programs 145,475 Participants in outreach programs outside the dome 472 Volunteers 34,757 Volunteer hours 52,609 Members OUR MISSION Through science and nature, we ignite wonder and empower dreams. OUR VISION Within a generation, Canada will be a country of thriving, sustainable communities rooted in science, innovation and a deep connection to nature. 1 CHAIR & CEO REPORT It is an honour to present to you Science World’s 2019/2020 Annual Report! In my third year as Chair, we celebrated and achieved many remarkable feats. In 2019, we celebrated 30 Years of Wonder under our dome and around the province, having served over 18 million people over the past three decades. Several foundational members of Science World, who were instrumental in creating this organization and leading us to the dome in the first place, re-engaged with us by sharing their stories and bringing awareness to the inspiring history of this unique charitable organization. These reminders of how important Science World is to the story of Vancouver and BC peaked in September when a CBC poll, with over 25,000 votes, declared our geodesic dome to be Vancouver’s Most Iconic Building. People shared memories of entering the dome and being ignited with a sense of wonder. We are proud of what Science World means to the people of this province, and we thank you for helping shape who we are and how we serve learners of all ages. -
Undergraduate Degree Programs in Physics and Astronomy 12:40
Beyond First Year: Undergraduate Degree Programs in Physics and Astronomy 12:40 – 13:45, Thursday March 14, 2018 Hennings 202 Sandwiches and drinks will be served 12:40 Colin Gay, Department Head and Salena Li, Undergraduate Coordinator - Introduction to the Department 12:45 Chris Waltham, Undergraduate Chair 0 12:55 Janis McKenna, 2nd-4th Year Advisor 1 13:00 Vesna Sossi, Biophysics Program Chair 2 13:05 Ingrid Stairs, Astronomy Program Chair 3 13:10 Javed Iqbal, Science Coop Program Director 4 Club activities 13:20 Physsoc – Ella Meyer 5 13:25 Astronomy – Katie Rink 6 13:30 Biophysics – Chantal Percival 7 QUESTION Period Physics is Hard So why do it? • Because you love it • Because you want to know how things work • Because you want to know how the Universe works • Because it opens up a wide variety of career opportunities What can you do with a physics degree? What can you do with a physics degree? Online Physics and Astronomy (PHAS) Graduate Survey • 235 participants • Most participants graduated in the last decade, with a few much earlier. • Careers 30% PHAS, 40% PHAS-related, 30% unrelated to PHAS. • Highest PHAS degree is a B.Sc. – 43% Which aspects of your studies have been most relevant to your career? More than 80% of respondents declared all of these as “Very Important” or “Important”: • Assessing the value of information critically • Learning on your own • Speaking clearly/effectively • Writing clearly/effectively • Working with others • Using computers • Solving numerical problems • Research experience • Managing projects UBC PHAS B.Sc grads in non- Law • Partner, McMillan LLP academic/teaching work • Barrister & Solicitor Manager Administrators • Technical project manager, D-Wave • Program Administrator, Canadian Mental Health • Business Development Lead at Abcellera, a UBC start-up Association Military Airline • Pilot, RCAF • Pilot • Ottawa/Defence Scientist Consultants Owner/President • Dubin Environmental • Owner of Murray Johnson Engineering Ltd. -
A Combinatorial Proof of the Log- Concavity of the Numbers Of
Journal of Combinatorial Theory, Series A 90, 293303 (2000) doi:10.1006Âjcta.1999.3040, available online at http:ÂÂwww.idealibrary.com on A Combinatorial Proof of the Log-Concavity of the Numbers of Permutations with k Runs MiklosBona1 University of Florida, Gainesville, Florida 32611 E-mail: bonaÄmath.ufl.edu and Richard Ehrenborg2 School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540 E-mail: jrgeÄmath.ias.edu Communicated by the Managing Editors Received February 26, 1999 We combinatorially prove that the number R(n, k) of permutations of length n having k runs is a log-concave sequence in k, for all n. We also give a new com- binatorial proof for the log-concavity of the Eulerian numbers. 2000 Academic Press 1. INTRODUCTION Let p= p1 p2 }}}pn be a permutation of the set [1, 2, ..., n] written in the one-line notation. We say that p get changes direction at position i, if either pi&1<pi>pi+ j ,orpi&1>pi>pi+1, in other words, when pi is either a peak or a valley. We say that p has k runs if there are k&1 indices i so that p changes direction at these positions. So, for example, p=3561247 has 3 runs as p changes direction when i=3 and when i=4. A geometric way to represent a permutation and its runs by a diagram is shown in Fig. 1. The runs are the line segments (or edges) between two consecutive entries where p changes direction. So a permutation has k runs if it can be represented by k line segments so that the segments go ``up'' and ``down'' 1 The paper was written while the author's stay at IAS was supported by Trustee Ladislaus von Hoffmann, the Arcana Foundation. -
MAT344 Week 1
MAT344 Week 1 2019/Sep/9 1 Announcements 1. Make sure you have access to quercus and piazza. 2. Enroll in a tutorial. 3. Read the Syllabus and Study tips (also uploaded to https://www.math.toronto.edu/balazse/2019_Fall_ MAT344/). 4. Read Chapter 1 of the textbook [KT17]. It is an easy and fun read, and explores many of the questions that we will study later in the class. 5. Note: for some reason the textbook .pdf and print versions numbers differ slightly from the (more interactive) html version. When using references, we will refer to the .pdf version. 6. Problem set 1 will be sent out on Thursday. 2 This week This week, we are talking about 1. Basic counting techniques, 2. Permutations, 3. Combinations 3 Strings (Chapter 2.1 in [KT17]) Exercise 3.1 ([Bog04], Chapter 1.2 Problem 6). An ice cream parlor offers 12 different flavors of ice cream, and triple decker cones are made in homemade waffle cones. Having chocolate ice cream as the bottom scoop is different from having chocolate ice cream as the top scoop. How many different triple decker cones of ice cream are available? Solution: We have 12 choices for all of the bottom, middle, and top scoops, and if two sequence of choices differ at any point, they result in different triple decker ice cream cones. So we have 12 · 12 · 12 = 1728 different kinds of triple decker cones. The above problem is a typical one that we will encounter in this class. We are asked \How many...", and with some logic we come up with a number. -
Combinatorics Combinatorics I Combinatorics II Product Rule Sum Rule I Sum Rule II
Combinatorics I Combinatorics Introduction Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Combinatorics is the study of collections of objects. Specifically, counting objects, arrangement, derangement, etc. of objects along with their mathematical properties. Counting objects is important in order to analyze algorithms and Spring 2006 compute discrete probabilities. Originally, combinatorics was motivated by gambling: counting configurations is essential to elementary probability. Computer Science & Engineering 235 Introduction to Discrete Mathematics Sections 4.1-4.6 & 6.5-6.6 of Rosen [email protected] Combinatorics II Product Rule Introduction If two events are not mutually exclusive (that is, we do them separately), then we apply the product rule. A simple example: How many arrangements are there of a deck of 52 cards? Theorem (Product Rule) In addition, combinatorics can be used as a proof technique. Suppose a procedure can be accomplished with two disjoint A combinatorial proof is a proof method that uses counting subtasks. If there are n1 ways of doing the first task and n2 ways arguments to prove a statement. of doing the second, then there are n1 · n2 ways of doing the overall procedure. Sum Rule I Sum Rule II There is a natural generalization to any sequence of m tasks; If two events are mutually exclusive, that is, they cannot be done namely the number of ways m mutually exclusive events can occur at the same time, then we must apply the sum rule. is n1 + n2 + ··· nm−1 + nm Theorem (Sum Rule) We can give another formulation in terms of sets. Let If an event e1 can be done in n1 ways and an event e2 can be done A1,A2,...,Am be pairwise disjoint sets.