Lecture Notes Combinatorics
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
LNCS 7215, Pp
ACoreCalculusforProvenance Umut A. Acar1,AmalAhmed2,JamesCheney3,andRolyPerera1 1 Max Planck Institute for Software Systems umut,rolyp @mpi-sws.org { 2 Indiana} University [email protected] 3 University of Edinburgh [email protected] Abstract. Provenance is an increasing concern due to the revolution in sharing and processing scientific data on the Web and in other computer systems. It is proposed that many computer systems will need to become provenance-aware in order to provide satisfactory accountability, reproducibility,andtrustforscien- tific or other high-value data. To date, there is not a consensus concerning ap- propriate formal models or security properties for provenance. In previous work, we introduced a formal framework for provenance security and proposed formal definitions of properties called disclosure and obfuscation. This paper develops a core calculus for provenance in programming languages. Whereas previous models of provenance have focused on special-purpose languages such as workflows and database queries, we consider a higher-order, functional language with sums, products, and recursive types and functions. We explore the ramifications of using traces based on operational derivations for the purpose of comparing other forms of provenance. We design a rich class of prove- nance views over traces. Finally, we prove relationships among provenance views and develop some solutions to the disclosure and obfuscation problems. 1Introduction Provenance, or meta-information about the origin, history, or derivation of an object, is now recognized as a central challenge in establishing trust and providing security in computer systems, particularly on the Web. Essentially, provenance management in- volves instrumenting a system with detailed monitoring or logging of auditable records that help explain how results depend on inputs or other (sometimes untrustworthy) sources. -
Computer Graphics and Visualization
European Research Consortium for Informatics and Mathematics Number 44 January 2001 www.ercim.org Special Theme: Computer Graphics and Visualization Next Issue: April 2001 Next Special Theme: Metacomputing and Grid Technologies CONTENTS KEYNOTE 36 Physical Deforming Agents for Virtual Neurosurgery by Michele Marini, Ovidio Salvetti, Sergio Di Bona 3 by Elly Plooij-van Gorsel and Ludovico Lutzemberger 37 Visualization of Complex Dynamical Systems JOINT ERCIM ACTIONS in Theoretical Physics 4 Philippe Baptiste Winner of the 2000 Cor Baayen Award by Anatoly Fomenko, Stanislav Klimenko and Igor Nikitin 38 Simulation and Visualization of Processes 5 Strategic Workshops – Shaping future EU-NSF collaborations in in Moving Granular Bed Gas Cleanup Filter Information Technologies by Pavel Slavík, František Hrdliãka and Ondfiej Kubelka THE EUROPEAN SCENE 39 Watching Chromosomes during Cell Division by Robert van Liere 5 INRIA is growing at an Unprecedented Pace and is starting a Recruiting Drive on a European Scale 41 The blue-c Project by Markus Gross and Oliver Staadt SPECIAL THEME 42 Augmenting the Common Working Environment by Virtual Objects by Wolfgang Broll 6 Graphics and Visualization: Breaking new Frontiers by Carol O’Sullivan and Roberto Scopigno 43 Levels of Detail in Physically-based Real-time Animation by John Dingliana and Carol O’Sullivan 8 3D Scanning for Computer Graphics by Holly Rushmeier 44 Static Solution for Real Time Deformable Objects With Fluid Inside by Ivan F. Costa and Remis Balaniuk 9 Subdivision Surfaces in Geometric -
11 Practice Lesso N 4 R O U N D W H Ole Nu M Bers U N It 1
CC04MM RPPSTG TEXT.indb 11 © Practice and Problem Solving Curriculum Associates, LLC Copying isnotpermitted. Practice Lesson 4 Round Whole Numbers Whole 4Round Lesson Practice Lesson 4 Round Whole Numbers Name: Solve. M 3 Round each number. Prerequisite: Round Three-Digit Numbers a. 689 rounded to the nearest ten is 690 . Study the example showing how to round a b. 68 rounded to the nearest hundred is 100 . three-digit number. Then solve problems 1–6. c. 945 rounded to the nearest ten is 950 . Example d. 945 rounded to the nearest hundred is 900 . Round 154 to the nearest ten. M 4 Rachel earned $164 babysitting last month. She earned $95 this month. Rachel rounded each 150 151 152 153 154 155 156 157 158 159 160 amount to the nearest $10 to estimate how much 154 is between 150 and 160. It is closer to 150. she earned. What is each amount rounded to the 154 rounded to the nearest ten is 150. nearest $10? Show your work. Round 154 to the nearest hundred. Student work will vary. Students might draw number lines, a hundreds chart, or explain in words. 100 110 120 130 140 150 160 170 180 190 200 Solution: ___________________________________$160 and $100 154 is between 100 and 200. It is closer to 200. C 5 Use the digits in the tiles to create a number that 154 rounded to the nearest hundred is 200. makes each statement true. Use each digit only once. B 1 Round 236 to the nearest ten. 1 2 3 4 5 6 7 8 9 Which tens is 236 between? Possible answer shown. -
Nieuw Archief Voor Wiskunde
Nieuw Archief voor Wiskunde Boekbespreking Kevin Broughan Equivalents of the Riemann Hypothesis Volume 1: Arithmetic Equivalents Cambridge University Press, 2017 xx + 325 p., prijs £ 99.99 ISBN 9781107197046 Kevin Broughan Equivalents of the Riemann Hypothesis Volume 2: Analytic Equivalents Cambridge University Press, 2017 xix + 491 p., prijs £ 120.00 ISBN 9781107197121 Reviewed by Pieter Moree These two volumes give a survey of conjectures equivalent to the ber theorem says that r()x asymptotically behaves as xx/log . That Riemann Hypothesis (RH). The first volume deals largely with state- is a much weaker statement and is equivalent with there being no ments of an arithmetic nature, while the second part considers zeta zeros on the line v = 1. That there are no zeros with v > 1 is more analytic equivalents. a consequence of the prime product identity for g()s . The Riemann zeta function, is defined by It would go too far here to discuss all chapters and I will limit 3 myself to some chapters that are either close to my mathematical 1 (1) g()s = / s , expertise or those discussing some of the most famous RH equiv- n = 1 n alences. Most of the criteria have their own chapter devoted to with si=+v t a complex number having real part v > 1 . It is easily them, Chapter 10 has various criteria that are discussed more brief- seen to converge for such s. By analytic continuation the Riemann ly. A nice example is Redheffer’s criterion. It states that RH holds zeta function can be uniquely defined for all s ! 1. -
The Modal Logic of Potential Infinity, with an Application to Free Choice
The Modal Logic of Potential Infinity, With an Application to Free Choice Sequences Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ethan Brauer, B.A. ∼6 6 Graduate Program in Philosophy The Ohio State University 2020 Dissertation Committee: Professor Stewart Shapiro, Co-adviser Professor Neil Tennant, Co-adviser Professor Chris Miller Professor Chris Pincock c Ethan Brauer, 2020 Abstract This dissertation is a study of potential infinity in mathematics and its contrast with actual infinity. Roughly, an actual infinity is a completed infinite totality. By contrast, a collection is potentially infinite when it is possible to expand it beyond any finite limit, despite not being a completed, actual infinite totality. The concept of potential infinity thus involves a notion of possibility. On this basis, recent progress has been made in giving an account of potential infinity using the resources of modal logic. Part I of this dissertation studies what the right modal logic is for reasoning about potential infinity. I begin Part I by rehearsing an argument|which is due to Linnebo and which I partially endorse|that the right modal logic is S4.2. Under this assumption, Linnebo has shown that a natural translation of non-modal first-order logic into modal first- order logic is sound and faithful. I argue that for the philosophical purposes at stake, the modal logic in question should be free and extend Linnebo's result to this setting. I then identify a limitation to the argument for S4.2 being the right modal logic for potential infinity. -
Sabermetrics: the Past, the Present, and the Future
Sabermetrics: The Past, the Present, and the Future Jim Albert February 12, 2010 Abstract This article provides an overview of sabermetrics, the science of learn- ing about baseball through objective evidence. Statistics and baseball have always had a strong kinship, as many famous players are known by their famous statistical accomplishments such as Joe Dimaggio’s 56-game hitting streak and Ted Williams’ .406 batting average in the 1941 baseball season. We give an overview of how one measures performance in batting, pitching, and fielding. In baseball, the traditional measures are batting av- erage, slugging percentage, and on-base percentage, but modern measures such as OPS (on-base percentage plus slugging percentage) are better in predicting the number of runs a team will score in a game. Pitching is a harder aspect of performance to measure, since traditional measures such as winning percentage and earned run average are confounded by the abilities of the pitcher teammates. Modern measures of pitching such as DIPS (defense independent pitching statistics) are helpful in isolating the contributions of a pitcher that do not involve his teammates. It is also challenging to measure the quality of a player’s fielding ability, since the standard measure of fielding, the fielding percentage, is not helpful in understanding the range of a player in moving towards a batted ball. New measures of fielding have been developed that are useful in measuring a player’s fielding range. Major League Baseball is measuring the game in new ways, and sabermetrics is using this new data to find better mea- sures of player performance. -
Math 253: Mathematical Methods for Data Visualization – Course Introduction and Overview (Spring 2020)
Math 253: Mathematical Methods for Data Visualization – Course introduction and overview (Spring 2020) Dr. Guangliang Chen Department of Math & Statistics San José State University Math 253 course introduction and overview What is this course about? Context: Modern data sets often have hundreds, thousands, or even millions of features (or attributes). ←− large dimension Dr. Guangliang Chen | Mathematics & Statistics, San José State University2/30 Math 253 course introduction and overview This course focuses on the statistical/machine learning task of dimension reduction, also called dimensionality reduction, which is the process of reducing the number of input variables of a data set under consideration, for the following benefits: • It reduces the running time and storage space. • Removal of multi-collinearity improves the interpretation of the parameters of the machine learning model. • It can also clean up the data by reducing the noise. • It becomes easier to visualize the data when reduced to very low dimensions such as 2D or 3D. Dr. Guangliang Chen | Mathematics & Statistics, San José State University3/30 Math 253 course introduction and overview There are two different kinds of dimension reduction approaches: • Feature selection approaches try to find a subset of the original features variables. Examples: subset selection, stepwise selection, Ridge and Lasso regression. ←− Already covered in Math 261A • Feature extraction transforms the data in the high-dimensional space to a space of fewer dimensions. ←− Focus of this course Examples: principal component analysis (PCA), ISOmap, and linear discriminant analysis (LDA). Dr. Guangliang Chen | Mathematics & Statistics, San José State University4/30 Math 253 course introduction and overview Dimension reduction methods to be covered in this course: • Linear projection methods: – PCA (for unlabled data), – LDA (for labled data) • Nonlinear embedding methods: – Multidimensional scaling (MDS), ISOmap – Locally linear embedding (LLE) – Laplacian eigenmaps Dr. -
Useful Relations in Permutations and Combination 1. Useful Relations
Useful Relations in Permutations and Combination 1. Useful Relations - Factorial n! = n.(n-1)! 2. n퐶푟= n푃푟/r! n n-1 3. Pr = n( Pr-1) 4. Useful Relations - Combinations n n 1. Cr = C(n - r) Example 8 8 C6 = C2 = 8×72×1 = 28 n 2. Cn = 1 n 3. C0 = 1 n n n n n 4. C0 + C1 + C2 + ... + Cn = 2 Example 4 4 4 4 4 4 C0 + C1 + C2 + C3+ C4 = (1 + 4 + 6 + 4 + 1) = 16 = 2 n n (n+1) Cr-1 + Cr = Cr (Pascal's Law) n퐶푟 =n/퐶푟−1=n-r+1/r n n If Cx = Cy then either x = y or (n-x) = y. 5. Selection from identical objects: Some Basic Facts The number of selections of r objects out of n identical objects is 1. Total number of selections of zero or more objects from n identical objects is n+1. 6. Permutations of Objects when All Objects are Not Distinct The number of ways in which n things can be arranged taking them all at a time, when st nd 푃1 of the things are exactly alike of 1 type, 푃2 of them are exactly alike of a 2 type, and th 푃푟of them are exactly alike of r type and the rest of all are distinct is n!/ 푃1! 푃2! ... 푃푟! 1 Example: how many ways can you arrange the letters in the word THESE? 5!/2!=120/2=60 Example: how many ways can you arrange the letters in the word REFERENCE? 9!/2!.4!=362880/2*24=7560 7.Circular Permutations: Case 1: when clockwise and anticlockwise arrangements are different Number of circular permutations (arrangements) of n different things is (n-1)! 1. -
Molecular Symmetry
Molecular Symmetry Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy) – used with group theory to predict vibrational spectra for the identification of molecular shape, and as a tool for understanding electronic structure and bonding. Symmetrical : implies the species possesses a number of indistinguishable configurations. 1 Group Theory : mathematical treatment of symmetry. symmetry operation – an operation performed on an object which leaves it in a configuration that is indistinguishable from, and superimposable on, the original configuration. symmetry elements – the points, lines, or planes to which a symmetry operation is carried out. Element Operation Symbol Identity Identity E Symmetry plane Reflection in the plane σ Inversion center Inversion of a point x,y,z to -x,-y,-z i Proper axis Rotation by (360/n)° Cn 1. Rotation by (360/n)° Improper axis S 2. Reflection in plane perpendicular to rotation axis n Proper axes of rotation (C n) Rotation with respect to a line (axis of rotation). •Cn is a rotation of (360/n)°. •C2 = 180° rotation, C 3 = 120° rotation, C 4 = 90° rotation, C 5 = 72° rotation, C 6 = 60° rotation… •Each rotation brings you to an indistinguishable state from the original. However, rotation by 90° about the same axis does not give back the identical molecule. XeF 4 is square planar. Therefore H 2O does NOT possess It has four different C 2 axes. a C 4 symmetry axis. A C 4 axis out of the page is called the principle axis because it has the largest n . By convention, the principle axis is in the z-direction 2 3 Reflection through a planes of symmetry (mirror plane) If reflection of all parts of a molecule through a plane produced an indistinguishable configuration, the symmetry element is called a mirror plane or plane of symmetry . -
An Exploration of the Relationship Between Mathematics and Music
An Exploration of the Relationship between Mathematics and Music Shah, Saloni 2010 MIMS EPrint: 2010.103 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 An Exploration of ! Relation"ip Between Ma#ematics and Music MATH30000, 3rd Year Project Saloni Shah, ID 7177223 University of Manchester May 2010 Project Supervisor: Professor Roger Plymen ! 1 TABLE OF CONTENTS Preface! 3 1.0 Music and Mathematics: An Introduction to their Relationship! 6 2.0 Historical Connections Between Mathematics and Music! 9 2.1 Music Theorists and Mathematicians: Are they one in the same?! 9 2.2 Why are mathematicians so fascinated by music theory?! 15 3.0 The Mathematics of Music! 19 3.1 Pythagoras and the Theory of Music Intervals! 19 3.2 The Move Away From Pythagorean Scales! 29 3.3 Rameau Adds to the Discovery of Pythagoras! 32 3.4 Music and Fibonacci! 36 3.5 Circle of Fifths! 42 4.0 Messiaen: The Mathematics of his Musical Language! 45 4.1 Modes of Limited Transposition! 51 4.2 Non-retrogradable Rhythms! 58 5.0 Religious Symbolism and Mathematics in Music! 64 5.1 Numbers are God"s Tools! 65 5.2 Religious Symbolism and Numbers in Bach"s Music! 67 5.3 Messiaen"s Use of Mathematical Ideas to Convey Religious Ones! 73 6.0 Musical Mathematics: The Artistic Aspect of Mathematics! 76 6.1 Mathematics as Art! 78 6.2 Mathematical Periods! 81 6.3 Mathematics Periods vs. -
18.703 Modern Algebra, Permutation Groups
5. Permutation groups Definition 5.1. Let S be a set. A permutation of S is simply a bijection f : S −! S. Lemma 5.2. Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S. (2) Let f be a permutation of S. Then the inverse of f is a permu tation of S. Proof. Well-known. D Lemma 5.3. Let S be a set. The set of all permutations, under the operation of composition of permutations, forms a group A(S). Proof. (5.2) implies that the set of permutations is closed under com position of functions. We check the three axioms for a group. We already proved that composition of functions is associative. Let i: S −! S be the identity function from S to S. Let f be a permutation of S. Clearly f ◦ i = i ◦ f = f. Thus i acts as an identity. Let f be a permutation of S. Then the inverse g of f is a permutation of S by (5.2) and f ◦ g = g ◦ f = i, by definition. Thus inverses exist and G is a group. D Lemma 5.4. Let S be a finite set with n elements. Then A(S) has n! elements. Proof. Well-known. D Definition 5.5. The group Sn is the set of permutations of the first n natural numbers. We want a convenient way to represent an element of Sn. The first way, is to write an element σ of Sn as a matrix. -
Combinatorics
Combinatorics Problem: How to count without counting. I How do you figure out how many things there are with a certain property without actually enumerating all of them. Sometimes this requires a lot of cleverness and deep mathematical insights. But there are some standard techniques. I That's what we'll be studying. We sometimes use the bijection rule without even realizing it: I count how many people voted are in favor of something by counting the number of hands raised: I I'm hoping that there's a bijection between the people in favor and the hands raised! Bijection Rule The Bijection Rule: If f : A ! B is a bijection, then jAj = jBj. I We used this rule in defining cardinality for infinite sets. I Now we'll focus on finite sets. Bijection Rule The Bijection Rule: If f : A ! B is a bijection, then jAj = jBj. I We used this rule in defining cardinality for infinite sets. I Now we'll focus on finite sets. We sometimes use the bijection rule without even realizing it: I count how many people voted are in favor of something by counting the number of hands raised: I I'm hoping that there's a bijection between the people in favor and the hands raised! Answer: 26 choices for the first letter, 26 for the second, 10 choices for the first number, the second number, and the third number: 262 × 103 = 676; 000 Example 2: A traveling salesman wants to do a tour of all 50 state capitals. How many ways can he do this? Answer: 50 choices for the first place to visit, 49 for the second, .