2 (n 1). f (i 1, i 2,..., i k )
|
|
- Martin Shelton
- 5 years ago
- Views:
Transcription
1 Math Time problem proposal #1 Darij Grinberg version 5 December 2010 Problem. Let x 1, x 2,..., x n be real numbers such that x 1 x 2...x n 1 and such that x i < 1 for every i {1, 2,..., n}. Prove that 1 x i 1 x j n 2 n 1. 1 n Soluon. First, for the sake of brevity 1, we introduce a notaon: If i 1, i 2,..., i k are some free variables, if f is a funcon of k variables, and if A 1, A 2,..., A l are some asserons containing some of the variables i 1, i 2,..., i k, then i 1, i 2,..., i k A 1, A 2,..., A l f i 1, i 2,..., i k will mean the sum of the values f i 1, i 2,..., i k over all good k-tuples i 1, i 2,..., i k ; hereby, a k-tuple i 1, i 2,..., i k is called good if it sasfies i u {1, 2,..., n} for every u {1, 2,..., k} and also sasfies all the asserons A 1, A 2,..., A l. Using this notaon, we can write many typical sums in a simpler form: For instance, f i i f i, and f i, j f. 1 i n 1 n Note that we won t use the notaon v ku f k for f u f u 1... f v in this soluon, since it theorecally could be misunderstood because different meaning in our above notaon. With the notaon defined above, we have 1 n 1 x i 1 x j 1 x i 1 x j. Hence, the inequality in queson, 1 x i 1 x j n 2 n 1, rewrites as 1 n 1 x i 1 x j n 2 n 1. Mulplicaon by 2 n 1 2 transforms this inequality into n x i 1 x j 1 In as far as brevity is possible for this soluon... n n 1. v f k has a ku 1
2 Since 2 1 x i 1 x j this inequality rewrites as n x i 1 x j 1 x i 1 x j j<i 1 x i 1 x j x j x i 1 x j 1 x i at this step we have interchanged i with j in the second sum 1 x i 1 x j 1 x i 1 x j, j<i 1 x i 1 x j 1 x i 1 x j n n 1. 1 Thus, in order to solve the problem, it remains to prove this inequality 1. Set 1 x i for every i {1, 2,..., n}. Then, > 0 for every i {1, 2,..., n} since x i < 1 and thus 1 x i > 0, so that 1 x i > 0. Besides, 1 x i yields x i 1. Finally, k tk k 1 xk n k xk n x 1 x 2... x n n
3 Consequently, n x i 1 x j n 1 x i n 1 x j 1 x i 1 x j n 1 1 n 1 1 k t k n 1 m t m n 1 since 2 yields n 1 k tk n 1 n 1 n 1 n 1 k t k and n 1 k t k k m t m m 1 since n 1 k and n 1 k t k m m k t k m k t k 1 Now, k t k k t k m m k tk m k m t k 1 n n 1 k t k m m tm. 3 because n n 1 is the number of all pairs with i and j being elements of the m t m 3
4 set {1, 2,..., n} and sasfying i j. Besides, k t k m m m here we have interchanged i with j and renamed k by m in the first sum m t m i 1 j m i j 1 m j m t m i m 1 j j m i m 1 j 2 t m 2 n 1 t m n i j 1 n 1 j i i j 1 n 1 Hence, 3 becomes n }{{} nn 1 n n 1 j n 1 j n x i 1 x j k t k m }{{} 0 k m t k. here we have renamed m by j in the second sum i k t k Therefore, the inequality that we have to prove, namely 1, rewrites as n n 1 This is obviously equivalent to k t k k t k m n n 1. m m 0. 4 Thus, we only have to prove 4 in order to complete the soluon of the problem because 4 is equivalent to 1, and proving 1 solves the problem. 4
5 Now, k t k, k, m m,,, k i, m j, k, m,,, k, m,,, ki mj }{{} 0, since we have ki or mj and thus t k 0 or t m 0 because every quadruple, k, m sasfies exactly one of the two asserons k i m j and k i m j, k, m,,, k i, m j, k, m,,, k i, m j, k m, k, m,,, k i, m j, k m 0, k, m,,, k i, m j, k, m,,, k i, m j, km, k,, k i t }{{ j } t k t k, since km t k t k. Thus, in order to prove 4, is enough to prove the two inequalies, k, m,,, k i, m j, k m, k,, k i 0; 5 t k t k 0. 6 Proving 5 is rather easy: First, the six condions i j, k j, m i, k i, m j, k m altogether connected by a logical and are equivalent to the condion that the four numbers, k, m are pairwisely disnct, i. e. that the set {, k, m} 5
6 has exactly 4 elements, i. e. that {, k, m} 4. Hence,, k, m,,, k i, m j, k m , k, m {i,j,k,m} 4, k, m {i,j,k,m} 4, k, m {i,j,k,m} 4, k, m {i,j,k,m} 4 t k t m t m, k, m {i,j,k,m} 4, k, m {i,j,k,m} 4, k, m {i,j,k,m} 4, k, m {i,j,k,m} 4, k, m {i,j,k,m} 4 t k t k t k t m t k t m at this step, we have renamed some variables: in the second sum, we interchanged i with k; in the third sum, we interchanged j with m; in the fourth sum, we interchanged both i with k and j with m, k, m {i,j,k,m} 4, k, m {i,j,k,m} 4, k, m {i,j,k,m} 4, k, m {i,j,k,m} 4 tk t k t k t k t m t m t k t m t k t m tk t k t k t m t k t m t k t m t k t m t k t m 1 4, k, m {i,j,k,m} 4 t k 2 2 t k t m 0 because squares are nonnegave; thus, 5 is proven. In order to prove 6, we proceed similarly: First, the three condions i j, k j, k i altogether connected by a logical and are equivalent to the condion that the three numbers, k are pairwisely disnct, i. e. that the set {, k} has exactly 6
7 3 elements, i. e. that {, k} 3. Hence,, k,, k i t k t k, k {i,j,k} 3, k {i,j,k} 3, k {i,j,k} 3, k {i,j,k} 3, k {i,j,k} 3 t k t k t k t k t k t k, k {i,j,k} 3, k {i,j,k} 3 t k t k t k t k, k {i,j,k} 3, k {i,j,k} 3 at this step, we have renamed some variables: in the second sum, we renamed, k by j, k, i, respecvely; in the third sum, we renamed, k by k,, respecvely tk t k t k t k t k t k t k t k t k t k t k t k 0, t k t k t k t k because the Schur inequality yields t k t k t k t k t k 0 for any triple, k. Thus, the inequality 6 is proven. As both inequalies 5 and 6 are verified now, the inequality 4 follows, and thus the problem is solved. Remark. The above problem is a generalizaon of the Schur inequality which states that a a b a c b b c b a c c a c b 0 for any three nonnegave reals a, b, c to n variables in fact, if you set n 3 in the problem, you get the Schur inequality for the numbers 1 x 1, 1 x 2, 1 x 3. In the parcular case when the reals x 1, x 2,..., x n are nonnegave, the problem can be solved much more easily - a soluon for this case was given in [1] proof of Theorem 4.1. References [1] Darij Grinberg, An inequality involving 2n numbers. 7
This is an auxiliary note; its goal is to prove a form of the Chinese Remainder Theorem that will be used in [2].
Witt vectors. Part 1 Michiel Hazewinkel Sidenotes by Darij Grinberg Witt#5c: The Chinese Remainder Theorem for Modules [not completed, not proofread] This is an auxiliary note; its goal is to prove a form
More informationExample 3.3 Moving-average filter
3.3 Difference Equa/ons for Discrete-Time Systems A Discrete-/me system can be modeled with difference equa3ons involving current, past, or future samples of input and output signals Example 3.3 Moving-average
More informationCSCI1950 Z Computa4onal Methods for Biology Lecture 4. Ben Raphael February 2, hhp://cs.brown.edu/courses/csci1950 z/ Algorithm Summary
CSCI1950 Z Computa4onal Methods for Biology Lecture 4 Ben Raphael February 2, 2009 hhp://cs.brown.edu/courses/csci1950 z/ Algorithm Summary Parsimony Probabilis4c Method Input Output Sankoff s & Fitch
More informationCombinatorics Sec/on of Rosen Ques/ons
Combinatorics Sec/on 5.1 5.6 7.5 7.6 of Rosen Spring 2011 CSCE 235 Introduc5on to Discrete Structures Course web- page: cse.unl.edu/~cse235 Ques/ons: cse235@cse.unl.edu Mo5va5on Combinatorics is the study
More informationTwo common difficul:es with HW 2: Problem 1c: v = r n ˆr
Two common difficul:es with HW : Problem 1c: For what values of n does the divergence of v = r n ˆr diverge at the origin? In this context, diverge means becomes arbitrarily large ( goes to infinity ).
More informationRadical axes revisited Darij Grinberg Version: 7 September 2007
Radical axes revisited Darij Grinberg Version: 7 September 007 1. Introduction In this note we are going to shed new light on some aspects of the theory of the radical axis. For a rather complete account
More informationProofs Sec*ons 1.6, 1.7 and 1.8 of Rosen
Proofs Sec*ons 1.6, 1.7 and 1.8 of Rosen Spring 2013 CSCE 235 Introduc5on to Discrete Structures Course web- page: cse.unl.edu/~cse235 Ques5ons: Piazza Outline Mo5va5on Terminology Rules of inference:
More informationW3203 Discrete Mathema1cs. Coun1ng. Spring 2015 Instructor: Ilia Vovsha.
W3203 Discrete Mathema1cs Coun1ng Spring 2015 Instructor: Ilia Vovsha h@p://www.cs.columbia.edu/~vovsha/w3203 Outline Bijec1on rule Sum, product, division rules Permuta1ons and combina1ons Sequences with
More informationW3203 Discrete Mathema1cs. Logic and Proofs. Spring 2015 Instructor: Ilia Vovsha. hcp://www.cs.columbia.edu/~vovsha/w3203
W3203 Discrete Mathema1cs Logic and Proofs Spring 2015 Instructor: Ilia Vovsha hcp://www.cs.columbia.edu/~vovsha/w3203 1 Outline Proposi1onal Logic Operators Truth Tables Logical Equivalences Laws of Logic
More informationNonlinear Shi, Registers: A Survey and Open Problems. Tor Helleseth University of Bergen NORWAY
Nonlinear Shi, Registers: A Survey and Open Problems Tor Helleseth University of Bergen NORWAY Outline ntroduc9on Nonlinear Shi> Registers (NLFSRs) Some basic theory De Bruijn Graph De Bruijn graph Golomb
More informationThe Absolute Value Equation
The Absolute Value Equation The absolute value of number is its distance from zero on the number line. The notation for absolute value is the presence of two vertical lines. 5 This is asking for the absolute
More informationODEs + Singulari0es + Monodromies + Boundary condi0ons. Kerr BH ScaRering: a systema0c study. Schwarzschild BH ScaRering: Quasi- normal modes
Outline Introduc0on Overview of the Technique ODEs + Singulari0es + Monodromies + Boundary condi0ons Results Kerr BH ScaRering: a systema0c study Schwarzschild BH ScaRering: Quasi- normal modes Collabora0on:
More informationCylindric Young Tableaux and their Properties
Cylindric Young Tableaux and their Properties Eric Neyman (Montgomery Blair High School) Mentor: Darij Grinberg (MIT) Fourth Annual MIT PRIMES Conference May 17, 2014 1 / 17 Introduction Young tableaux
More informationA hyperfactorial divisibility Darij Grinberg *long version*
A hyperfactorial divisibility Darij Grinberg *long version* Let us define a function H : N N by n H n k! for every n N Our goal is to prove the following theorem: Theorem 0 MacMahon We have H b + c H c
More informationMatrix products in integrable probability
Matrix products in integrable probability Atsuo Kuniba (Univ. Tokyo) Mathema?cal Society of Japan Spring Mee?ng Tokyo Metropolitan University 27 March 2017 Non-equilibrium sta?s?cal mechanics Stochas?c
More informationarxiv: v4 [math.oc] 19 Sep 2017
arxiv:777259v4 [matho] 9 Sep 27 Hierarchical Plug-and-Play oltage/urrent ontroller of D Microgrid lusters with Grid-Forming/Feeding onverters: Line-independent Primary Stabilizaon and Leader-based Distributed
More informationLeast Mean Squares Regression. Machine Learning Fall 2017
Least Mean Squares Regression Machine Learning Fall 2017 1 Lecture Overview Linear classifiers What func?ons do linear classifiers express? Least Squares Method for Regression 2 Where are we? Linear classifiers
More informationMATH 271 Summer 2016 Practice problem solutions Week 1
Part I MATH 271 Summer 2016 Practice problem solutions Week 1 For each of the following statements, determine whether the statement is true or false. Prove the true statements. For the false statement,
More informationMath 5707: Graph Theory, Spring 2017 Midterm 3
University of Minnesota Math 5707: Graph Theory, Spring 2017 Midterm 3 Nicholas Rancourt (edited by Darij Grinberg) December 25, 2017 1 Exercise 1 1.1 Problem Let G be a connected multigraph. Let x, y,
More information= W z1 + W z2 and W z1 z 2
Math 44 Fall 06 homework page Math 44 Fall 06 Darij Grinberg: homework set 8 due: Wed, 4 Dec 06 [Thanks to Hannah Brand for parts of the solutions] Exercise Recall that we defined the multiplication of
More informationWitt#5: Around the integrality criterion 9.93 [version 1.1 (21 April 2013), not completed, not proofread]
Witt vectors. Part 1 Michiel Hazewinkel Sienotes by Darij Grinberg Witt#5: Aroun the integrality criterion 9.93 [version 1.1 21 April 2013, not complete, not proofrea In [1, section 9.93, Hazewinkel states
More informationQuantum mechanics with indefinite causal order
Quantum mechanics with indefinite causal order Flaminia Giacomini, Esteban Castro- Ruiz, Časlav rukner University of Vienna Ins@tute for Quantum Op@cs and Quantum Informa@on, Vienna Vienna Theory Lunch
More informationQuiz 1, Mon CS 2050, Intro Discrete Math for Computer Science
Quiz 1, Mon 09-6-11 CS 050, Intro Discrete Math for Computer Science This quiz has 10 pages (including this cover page) and 5 Problems: Problems 1,, 3 and 4 are mandatory ( pages each.) Problem 5 is optional,
More informationNoncommutative Abel-like identities
Noncommutative Abel-like identities Darij Grinberg detailed version draft, January 15, 2018 Contents 1. Introduction 1 2. The identities 3 3. The proofs 7 3.1. Proofs of Theorems 2.2 and 2.4......................
More informationAn adventitious angle problem concerning
n adventitious angle problem concerning and 7 / Darij Grinberg The purpose of this note is to give two solutions of the following problem (Fig 1): Let be an isosceles triangle with and 1 Let be a point
More informationProbability and Structure in Natural Language Processing
Probability and Structure in Natural Language Processing Noah Smith Heidelberg University, November 2014 Introduc@on Mo@va@on Sta@s@cal methods in NLP arrived ~20 years ago and now dominate. Mercer was
More informationTHE NUMBER OF DIOPHANTINE QUINTUPLES. Yasutsugu Fujita College of Industrial Technology, Nihon University, Japan
GLASNIK MATEMATIČKI Vol. 45(65)(010), 15 9 THE NUMBER OF DIOPHANTINE QUINTUPLES Yasutsugu Fujita College of Industrial Technology, Nihon University, Japan Abstract. A set a 1,..., a m} of m distinct positive
More informationOn a class of three-variable inequalities. Vo Quoc Ba Can
On a class of three-variable inequalities Vo Quoc Ba Can 1 Theem Let a, b, c be real numbers satisfying a + b + c = 1 By the AM - GM inequality, we have ab + bc + ca 1, therefe setting ab + bc + ca = 1
More informationComputer Vision. Pa0ern Recogni4on Concepts Part I. Luis F. Teixeira MAP- i 2012/13
Computer Vision Pa0ern Recogni4on Concepts Part I Luis F. Teixeira MAP- i 2012/13 What is it? Pa0ern Recogni4on Many defini4ons in the literature The assignment of a physical object or event to one of
More informationStreaming - 2. Bloom Filters, Distinct Item counting, Computing moments. credits:www.mmds.org.
Streaming - 2 Bloom Filters, Distinct Item counting, Computing moments credits:www.mmds.org http://www.mmds.org Outline More algorithms for streams: 2 Outline More algorithms for streams: (1) Filtering
More informationMath 361: Homework 1 Solutions
January 3, 4 Math 36: Homework Solutions. We say that two norms and on a vector space V are equivalent or comparable if the topology they define on V are the same, i.e., for any sequence of vectors {x
More informationA.1 Appendix on Cartesian tensors
1 Lecture Notes on Fluid Dynamics (1.63J/2.21J) by Chiang C. Mei, February 6, 2007 A.1 Appendix on Cartesian tensors [Ref 1] : H Jeffreys, Cartesian Tensors; [Ref 2] : Y. C. Fung, Foundations of Solid
More informationPropositional Equivalence
Propositional Equivalence Tautologies and contradictions A compound proposition that is always true, regardless of the truth values of the individual propositions involved, is called a tautology. Example:
More informationDiophantine triples in a Lucas-Lehmer sequence
Annales Mathematicae et Informaticae 49 (01) pp. 5 100 doi: 10.33039/ami.01.0.001 http://ami.uni-eszterhazy.hu Diophantine triples in a Lucas-Lehmer sequence Krisztián Gueth Lorand Eötvös University Savaria
More informationLecture 4 Introduc-on to Data Flow Analysis
Lecture 4 Introduc-on to Data Flow Analysis I. Structure of data flow analysis II. Example 1: Reaching defini?on analysis III. Example 2: Liveness analysis IV. Generaliza?on 15-745: Intro to Data Flow
More informationApproximability of word maps by homomorphisms
Approximability of word maps by homomorphisms Alexander Bors arxiv:1708.00477v1 [math.gr] 1 Aug 2017 August 3, 2017 Abstract Generalizing a recent result of Mann, we show that there is an explicit function
More informationCLEP College Algebra - Problem Drill 21: Solving and Graphing Linear Inequalities
CLEP College Algebra - Problem Drill 21: Solving and Graphing Linear Inequalities No. 1 of 10 1. Which inequality represents the statement three more than seven times a real number is greater than or equal
More informationDiscrete Math, Spring Solutions to Problems V
Discrete Math, Spring 202 - Solutions to Problems V Suppose we have statements P, P 2, P 3,, one for each natural number In other words, we have the collection or set of statements {P n n N} a Suppose
More informationUnit 3: Ra.onal and Radical Expressions. 3.1 Product Rule M1 5.8, M , M , 6.5,8. Objec.ve. Vocabulary o Base. o Scien.fic Nota.
Unit 3: Ra.onal and Radical Expressions M1 5.8, M2 10.1-4, M3 5.4-5, 6.5,8 Objec.ve 3.1 Product Rule I will be able to mul.ply powers when they have the same base, including simplifying algebraic expressions
More informationCSE 473: Ar+ficial Intelligence
CSE 473: Ar+ficial Intelligence Hidden Markov Models Luke Ze@lemoyer - University of Washington [These slides were created by Dan Klein and Pieter Abbeel for CS188 Intro to AI at UC Berkeley. All CS188
More informationContradiction MATH Contradiction. Benjamin V.C. Collins, James A. Swenson MATH 2730
MATH 2730 Contradiction Benjamin V.C. Collins James A. Swenson Contrapositive The contrapositive of the statement If A, then B is the statement If not B, then not A. A statement and its contrapositive
More informationMath 421, Homework #7 Solutions. We can then us the triangle inequality to find for k N that (x k + y k ) (L + M) = (x k L) + (y k M) x k L + y k M
Math 421, Homework #7 Solutions (1) Let {x k } and {y k } be convergent sequences in R n, and assume that lim k x k = L and that lim k y k = M. Prove directly from definition 9.1 (i.e. don t use Theorem
More informationBellman s Curse of Dimensionality
Bellman s Curse of Dimensionality n- dimensional state space Number of states grows exponen
More informationLearning Sums of Independent Integer Random Variables
Learning Sums of Independent Integer Random Variables Cos8s Daskalakis (MIT) Ilias Diakonikolas (Edinburgh) Ryan O Donnell (CMU) Rocco Servedio (Columbia) Li- Yang Tan (Columbia) FOCS 2013, Berkeley CA
More informationFormalism of the Tersoff potential
Originally written in December 000 Translated to English in June 014 Formalism of the Tersoff potential 1 The original version (PRB 38 p.990, PRB 37 p.6991) Potential energy Φ = 1 u ij i (1) u ij = f ij
More informationCS6750: Cryptography and Communica7on Security
CS6750: Cryptography and Communica7on Security Class 6: Simple Number Theory Dr. Erik- Oliver Blass Plan 1. Role of number theory in cryptography 2. Classical problems in computa7onal number theory 3.
More informationPhysics 116A Determinants
Physics 116A Determinants Peter Young (Dated: February 5, 2014) I. DEFINITION The determinant is an important property of a square n n matrix. Such a matrix, A say, has n 2 elements, a ij, and is written
More informationModule 4: Linear Equa1ons Topic D: Systems of Linear Equa1ons and Their Solu1ons. Lesson 4-24: Introduc1on to Simultaneous Equa1ons
Module 4: Linear Equa1ons Topic D: Systems of Linear Equa1ons and Their Solu1ons Lesson 4-24: Introduc1on to Simultaneous Equa1ons Lesson 4-24: Introduc1on to Simultaneous Equa1ons Purpose for Learning:
More information40h + 15c = c = h
Chapter One Linear Systems I Solving Linear Systems Systems of linear equations are common in science and mathematics. These two examples from high school science [Onan] give a sense of how they arise.
More informationAggrega?on of Epistemic Uncertainty
Aggrega?on of Epistemic Uncertainty - Certainty Factors and Possibility Theory - Koichi Yamada Nagaoka Univ. of Tech. 1 What is Epistemic Uncertainty? Epistemic Uncertainty Aleatoric Uncertainty (Sta?s?cal
More informationON A PROBLEM OF PILLAI AND ITS GENERALIZATIONS
ON A PROBLEM OF PILLAI AND ITS GENERALIZATIONS L. HAJDU 1 AND N. SARADHA Abstract. We study some generalizations of a problem of Pillai. We investigate the existence of an integer M such that for m M,
More informationMATH 225: Foundations of Higher Matheamatics. Dr. Morton. 3.4: Proof by Cases
MATH 225: Foundations of Higher Matheamatics Dr. Morton 3.4: Proof y Cases Chapter 3 handout page 12 prolem 21: Prove that for all real values of y, the following inequality holds: 7 2y + 2 2y 5 7. You
More informationJBMO 2016 Problems and solutions
JBMO 016 Problems and solutions Problem 1. A trapezoid ABCD (AB CD, AB > CD) is circumscribed. The incircle of the triangle ABC touches the lines AB and AC at the points M and N, respectively. Prove that
More informationMonadic Predicate Logic is Decidable. Boolos et al, Computability and Logic (textbook, 4 th Ed.)
Monadic Predicate Logic is Decidable Boolos et al, Computability and Logic (textbook, 4 th Ed.) These slides use A instead of E instead of & instead of - instead of Nota>on Equality statements are atomic
More informationReduced Models for Process Simula2on and Op2miza2on
Reduced Models for Process Simulaon and Opmizaon Yidong Lang, Lorenz T. Biegler and David Miller ESI annual meeng March, 0 Models are mapping Equaon set or Module simulators Input space Reduced model Surrogate
More informationThe Clifford algebra and the Chevalley map - a computational approach (detailed version 1 ) Darij Grinberg Version 0.6 (3 June 2016). Not proofread!
The Cliord algebra and the Chevalley map - a computational approach detailed version 1 Darij Grinberg Version 0.6 3 June 2016. Not prooread! 1. Introduction: the Cliord algebra The theory o the Cliord
More informationFrom differen+al equa+ons to trigonometric func+ons. Introducing sine and cosine
From differen+al equa+ons to trigonometric func+ons Introducing sine and cosine Calendar OSH due this Friday by 12:30 in MX 1111 Last quiz next Wedn Office hrs: Today: 11:30-12:30 OSH due by 12:30 Math
More informationSpectral Algorithms for Graph Mining and Analysis. Yiannis Kou:s. University of Puerto Rico - Rio Piedras NSF CAREER CCF
Spectral Algorithms for Graph Mining and Analysis Yiannis Kou:s University of Puerto Rico - Rio Piedras NSF CAREER CCF-1149048 The Spectral Lens A graph drawing using Laplacian eigenvectors Outline Basic
More informationOn state-space reduction in multi-strain pathogen models, with an application to antigenic drift in influenza A
On state-space reduction in multi-strain pathogen models, with an application to antigenic drift in influenza A Sergey Kryazhimskiy, Ulf Dieckmann, Simon A. Levin, Jonathan Dushoff Supporting information
More informationbesides your solutions of these problems. 1 1 We note, however, that there will be many factors in the admission decision
The PRIMES 2015 Math Problem Set Dear PRIMES applicant! This is the PRIMES 2015 Math Problem Set. Please send us your solutions as part of your PRIMES application by December 1, 2015. For complete rules,
More informationPolynomials and Gröbner Bases
Alice Feldmann 16th December 2014 ETH Zürich Student Seminar in Combinatorics: Mathema:cal So
More informationPar$al Fac$on Decomposi$on. Academic Resource Center
Par$al Fac$on Decomposi$on Academic Resource Center Table of Contents. What is Par$al Frac$on Decomposi$on 2. Finding the Par$al Fac$on Decomposi$on 3. Examples 4. Exercises 5. Integra$on with Par$al Fac$ons
More informationCSE446: Linear Regression Regulariza5on Bias / Variance Tradeoff Winter 2015
CSE446: Linear Regression Regulariza5on Bias / Variance Tradeoff Winter 2015 Luke ZeElemoyer Slides adapted from Carlos Guestrin Predic5on of con5nuous variables Billionaire says: Wait, that s not what
More informationMA/CS 109 Lecture 4. Coun3ng, Coun3ng Cleverly, And Func3ons
MA/CS 109 Lecture 4 Coun3ng, Coun3ng Cleverly, And Func3ons The first thing you learned in math The thing in mathema3cs that almost everybody is good at is coun3ng we learn very early how to count. And
More informationPredicate abstrac,on and interpola,on. Many pictures and examples are borrowed from The So'ware Model Checker BLAST presenta,on.
Predicate abstrac,on and interpola,on Many pictures and examples are borrowed from The So'ware Model Checker BLAST presenta,on. Outline. Predicate abstrac,on the idea in pictures 2. Counter- example guided
More informationMATH 2030: MATRICES. Example 0.2. Q:Define A 1 =, A. 3 4 A: We wish to find c 1, c 2, and c 3 such that. c 1 + c c
MATH 2030: MATRICES Matrix Algebra As with vectors, we may use the algebra of matrices to simplify calculations. However, matrices have operations that vectors do not possess, and so it will be of interest
More informationFour-coloring P 6 -free graphs. I. Extending an excellent precoloring
Four-coloring P 6 -free graphs. I. Extending an excellent precoloring Maria Chudnovsky Princeton University, Princeton, NJ 08544 Sophie Spirkl Princeton University, Princeton, NJ 08544 Mingxian Zhong Columbia
More informationBulletin of the Iranian Mathematical Society
ISSN: 117-6X (Print) ISSN: 1735-8515 (Online) Bulletin of the Iranian Mathematical Society Vol. 4 (14), No. 6, pp. 1491 154. Title: The locating chromatic number of the join of graphs Author(s): A. Behtoei
More informationSummary of basic probability theory Math 218, Mathematical Statistics D Joyce, Spring 2016
8. For any two events E and F, P (E) = P (E F ) + P (E F c ). Summary of basic probability theory Math 218, Mathematical Statistics D Joyce, Spring 2016 Sample space. A sample space consists of a underlying
More informationRigidity of Differentiable Structure for New Class of Line Arrangements 1
communications in analysis and geometry Volume 13, Number 5, 1057-1075, 2005 Rigidity of Differentiable Structure for New Class of Line Arrangements 1 Shaobo Wang, Stephen S.-T. Yau 2 An arrangement of
More informationCh 3.2: Direct proofs
Math 299 Lectures 8 and 9: Chapter 3 0. Ch3.1 A trivial proof and a vacuous proof (Reading assignment) 1. Ch3.2 Direct proofs 2. Ch3.3 Proof by contrapositive 3. Ch3.4 Proof by cases 4. Ch3.5 Proof evaluations
More information#A69 INTEGERS 13 (2013) OPTIMAL PRIMITIVE SETS WITH RESTRICTED PRIMES
#A69 INTEGERS 3 (203) OPTIMAL PRIMITIVE SETS WITH RESTRICTED PRIMES William D. Banks Department of Mathematics, University of Missouri, Columbia, Missouri bankswd@missouri.edu Greg Martin Department of
More informationAnalysis III. Exam 1
Analysis III Math 414 Spring 27 Professor Ben Richert Exam 1 Solutions Problem 1 Let X be the set of all continuous real valued functions on [, 1], and let ρ : X X R be the function ρ(f, g) = sup f g (1)
More informationON THE EXTENSIBILITY OF DIOPHANTINE TRIPLES {k 1, k + 1, 4k} FOR GAUSSIAN INTEGERS. Zrinka Franušić University of Zagreb, Croatia
GLASNIK MATEMATIČKI Vol. 43(63)(2008), 265 291 ON THE EXTENSIBILITY OF DIOPHANTINE TRIPLES {k 1, k + 1, 4k} FOR GAUSSIAN INTEGERS Zrinka Franušić University of Zagreb, Croatia Abstract. In this paper,
More informationFrom average to instantaneous rates of change. (and a diversion on con4nuity and limits)
From average to instantaneous rates of change (and a diversion on con4nuity and limits) Extra prac4ce problems? Problems in the Book Problems at the end of my slides Math Exam Resource (MER): hcp://wiki.ubc.ca/science:math_exam_resources
More informationSolving and Graphing Inequalities Joined by And or Or
Solving and Graphing Inequalities Joined by And or Or Classwork 1. Zara solved the inequality 18 < 3x 9 as shown below. Was she correct? 18 < 3x 9 27 < 3x 9 < x or x> 9 2. Consider the compound inequality
More informationQuantum Energy Inequali1es
Quantum Energy Inequali1es Elisa Ferreira PHYS 731 Ian Morrison Ref: Lectures on Quantum Energy Inequali1es, C. Fewster 1208.5399 Introduc1on and Mo1va1on Classical energy condi1ons (CEC) of GR- WEC, NEC,
More informationDiskrete Mathematik und Optimierung
Diskrete Mathematik und Optimierung Steffen Hitzemann and Winfried Hochstättler: On the Combinatorics of Galois Numbers Technical Report feu-dmo012.08 Contact: steffen.hitzemann@arcor.de, winfried.hochstaettler@fernuni-hagen.de
More informationAppendix C Vector and matrix algebra
Appendix C Vector and matrix algebra Concepts Scalars Vectors, rows and columns, matrices Adding and subtracting vectors and matrices Multiplying them by scalars Products of vectors and matrices, scalar
More informationLast Lecture Recap UVA CS / Introduc8on to Machine Learning and Data Mining. Lecture 3: Linear Regression
UVA CS 4501-001 / 6501 007 Introduc8on to Machine Learning and Data Mining Lecture 3: Linear Regression Yanjun Qi / Jane University of Virginia Department of Computer Science 1 Last Lecture Recap q Data
More informationPerfect Zero-Divisor Graphs of Z_n
Rose-Hulman Undergraduate Mathematics Journal Volume 17 Issue 2 Article 6 Perfect Zero-Divisor Graphs of Z_n Bennett Smith Illinois College Follow this and additional works at: http://scholar.rose-hulman.edu/rhumj
More informationGroup B Heat Flow Across the San Andreas Fault. Chris Trautner Bill Savran
Group B Heat Flow Across the San Andreas Fault Chris Trautner Bill Savran Outline Background informa?on on the problem: Hea?ng due to fric?onal sliding Shear stress at depth Average stress drop along fault
More informationMath 4242 Fall 2016 (Darij Grinberg): homework set 8 due: Wed, 14 Dec b a. Here is the algorithm for diagonalizing a matrix we did in class:
Math 4242 Fall 206 homework page Math 4242 Fall 206 Darij Grinberg: homework set 8 due: Wed, 4 Dec 206 Exercise Recall that we defined the multiplication of complex numbers by the rule a, b a 2, b 2 =
More informationSkills Practice Skills Practice for Lesson 10.1
Skills Practice Skills Practice for Lesson.1 Name Date Higher Order Polynomials and Factoring Roots of Polynomial Equations Problem Set Solve each polynomial equation using factoring. Then check your solution(s).
More informationRandom Sets versus Random Sequences
Random Sets versus Random Sequences Peter Hertling Lehrgebiet Berechenbarkeit und Logik, Informatikzentrum, Fernuniversität Hagen, 58084 Hagen, Germany, peterhertling@fernuni-hagende September 2001 Abstract
More informationG. Appa, R. Euler, A. Kouvela, D. Magos and I. Mourtos On the completability of incomplete orthogonal Latin rectangles
G. Appa R. Euler A. Kouvela D. Magos and I. Mourtos On the completability of incomplete orthogonal Latin rectangles Article (Accepted version) (Refereed) Original citation: Appa Gautam Euler R. Kouvela
More informationMathematical Induction
Mathematical Induction Let s motivate our discussion by considering an example first. What happens when we add the first n positive odd integers? The table below shows what results for the first few values
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics POLYNOMIALS AND CONVEX SEQUENCE INEQUALITIES A.McD. MERCER Department of Mathematics and Statistics University of Guelph Ontario, N1G 2W1, Canada.
More informationIntroduc)on to Ar)ficial Intelligence
Introduc)on to Ar)ficial Intelligence Lecture 9 Logical reasoning CS/CNS/EE 154 Andreas Krause First order logic (FOL)! Proposi)onal logic is about simple facts! There is a breeze at loca)on [1,2]! First
More informationOn the Maximum Number of Hamiltonian Paths in Tournaments
On the Maximum Number of Hamiltonian Paths in Tournaments Ilan Adler, 1 Noga Alon,, * Sheldon M. Ross 3 1 Department of Industrial Engineering and Operations Research, University of California, Berkeley,
More informationRG Flows, Entanglement & Holography Workshop. Michigan Center for Theore0cal Physics September 17 21, 2012
RG Flows, Entanglement & Holography Workshop Michigan Center for Theore0cal Physics September 17 21, 2012 Intersec0ons in Theore0cal Physics: Par$cle Physics Sta$s$cal Mechanics Renormalization Group Flows
More informationW3203 Discrete Mathema1cs. Number Theory. Spring 2015 Instructor: Ilia Vovsha. hcp://www.cs.columbia.edu/~vovsha/w3203
W3203 Discrete Mathema1cs Number Theory Spring 2015 Instructor: Ilia Vovsha hcp://www.cs.columbia.edu/~vovsha/w3203 1 Outline Communica1on, encryp1on Number system Divisibility Prime numbers Greatest Common
More informationUnit 6 Sequences. Mrs. Valen+ne CCM3
Unit 6 Sequences Mrs. Valen+ne CCM3 6.1 Sequences and Series Genera&ng a Sequence Using an Explicit Formula Sequence: ordered list of numbers (each one is a term) Terms are symbolized with a variable and
More informationAntimagic Properties of Graphs with Large Maximum Degree
Antimagic Properties of Graphs with Large Maximum Degree Zelealem B. Yilma Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 1513 E-mail: zyilma@andrew.cmu.edu July 6, 011 Abstract
More informationQuasi DG categories and mixed motivic sheaves
Quasi DG categories and mixed motivic sheaves Masaki Hanamura Department of Mathematics, Tohoku University Aramaki Aoba-ku, 980-8587, Sendai, Japan Abstract We introduce the notion of a quasi DG category,
More informationd ORTHOGONAL POLYNOMIALS AND THE WEYL ALGEBRA
d ORTHOGONAL POLYNOMIALS AND THE WEYL ALGEBRA Luc VINET Université de Montréal Alexei ZHEDANOV Donetsk Ins@tute for Physics and Technology PLAN 1. Charlier polynomials 2. Weyl algebra 3. Charlier and Weyl
More information2016 EF Exam Texas A&M High School Students Contest Solutions October 22, 2016
6 EF Exam Texas A&M High School Students Contest Solutions October, 6. Assume that p and q are real numbers such that the polynomial x + is divisible by x + px + q. Find q. p Answer Solution (without knowledge
More informationMATH 3511 Lecture 1. Solving Linear Systems 1
MATH 3511 Lecture 1 Solving Linear Systems 1 Dmitriy Leykekhman Spring 2012 Goals Review of basic linear algebra Solution of simple linear systems Gaussian elimination D Leykekhman - MATH 3511 Introduction
More informationIS A CI-GROUP. RESULTS TOWARDS SHOWING Z n p
RESULTS TOWARDS SHOWING Z n p IS A CI-GROUP JOY MORRIS Abstract. It has been shown that Z i p is a CI-group for 1 i 4, and is not a CI-group for i 2p 1 + ( 2p 1 p ; all other values (except when p = 2
More informationUnit 5. Matrix diagonaliza1on
Unit 5. Matrix diagonaliza1on Linear Algebra and Op1miza1on Msc Bioinforma1cs for Health Sciences Eduardo Eyras Pompeu Fabra University 218-219 hlp://comprna.upf.edu/courses/master_mat/ We have seen before
More information