url
stringlengths 15
1.13k
| text
stringlengths 100
1.04M
| metadata
stringlengths 1.06k
1.1k
|
|---|---|---|
https://ahilado.wordpress.com/2017/06/24/adeles-and-ideles/ | In Valuations and Completions we introduced the $p$-adic numbers $\mathbb{Q}_{p}$, which, like the real numbers, are the completion of the rational numbers under a certain kind of valuation. There is one such valuation for each prime number $p$, and another for the “infinite prime”, which is just the usual absolute value. Each valuation may be thought of as encoding number theoretic information related to the prime $p$, or to the “infinite prime”, for the case of the absolute value (more technically, the $p$-adic valuations are referred to as nonarchimedean valuations, while the absolute value is an example of an archimedean valuation).
We can consider valuations not only for the rational numbers, but for more general algebraic number fields as well. In its abstract form, given an algebraic number field $K$, a (multiplicative) valuation of $K$ is simply any function $|\ |$ from $K$ to $\mathbb{R}$ satisfying the following properties:
(i) $|x|\geq 0$, where $x=0$ if and only if $x=0$
(ii) $|xy|=|x||y|$
(iii) $|x+y|\leq|x|+|y|$
If this seems reminiscent of the discussion in Metric, Norm, and Inner Product, it is because a valuation does, in fact, define a metric on $K$, and by extension, a topology. Two valuations are equivalent if they define the same topology; another way to phrase this statement is that two valuations $|\ |_{1}$ and $|\ |_{2}$ are equivalent if $|x|_{1}=|x|_{2}^{s}$ for some positive real number $s$, for all $x\in K$. The valuation is nonarchimedean if $|x+y|\leq\text{max}\{|x|,|y|\}$; otherwise, it is archimedean.
Just as in the case of rational numbers, we also have an exponential valuation, defined as a function $v$ from the field $K$ to $\mathbb{R}\cup \infty$ satisfying the following conditions:
(i) $v(x)=\infty$ if and only if $x=0$
(ii) $v(xy)=v(x)+v(y)$
(iii) $v(x+y)\geq\text{min}\{v(x),v(y)\}$
Two exponential valuations $v_{1}$ and $v_{2}$ are equivalent if $v_{1}(x)=sv_{2}(x)$ for some real number $s$, for all $x\in K$.
The idea of valuations allows us to make certain concepts in algebraic number theory (see Algebraic Numbers) more abstract. We define a place $v$ of an algebraic number field $K$ as an equivalence class of valuations of $K$. We write $K_{v}$ to denote the completion of $K$ under the place $v$; these are the generalizations of the $p$-adic numbers and real numbers to algebraic number fields other than $\mathbb{Q}$. The nonarchimedean places are also called the finite places, while the archimedean places are also called the infinite places. To express whether a place $v$ is a finite place or an infinite place, we write $v|\infty$ or $v\nmid\infty$ respectively.
The infinite places are of two kinds; the ones for which $K_{v}$ is isomorphic to $\mathbb{R}$ are called the real places, while the ones for which $K_{v}$ is isomorphic to $\mathbb{C}$ are called the complex places. The number of real places and complex places of $K$, denoted by $r_{1}$ and $r_{2}$ respectively, satisfy the equation $r_{1}+2r_{2}=n$, where $n$ is the degree of $K$ over $\mathbb{Q}$, i.e. $n=[K:\mathbb{Q}]$.
By the way, in some of the literature, such as in the book Algebraic Number Theory by Jurgen Neukirch, “places” are also referred to as “primes“. This is intentional – one may actually think of our definition of places as being like a more abstract replacement of the definition of primes. This is quite advantageous in driving home the concept of primes as equivalence classes of valuations; however, to avoid confusion, we will stick to using the term “places” here, along with its corresponding notation.
When $v$ is a nonarchimedean valuation, we let $\mathfrak{o}_{v}$ denote the set of all elements $x$ of $K_{v}$ for which $|x|_{v}\leq 1$. It is an example of a ring with special properties called a valuation ring. This means that, for any $x$ in $K$, either $x$ or $x^{-1}$ must be in $\mathfrak{o}_{v}$. We let $\mathfrak{o}_{v}^{*}$ denote the set of all elements of $\mathfrak{o}_{v}$ for which $|x|_{v}=1$, and we let $\mathfrak{p}_{v}$ denote the set of all elements of $\mathfrak{o}_{v}$ for which $|x|_{v}< 1$. It is the unique maximal ideal of $\mathfrak{o}_{v}$.
Now we proceed to consider the modern point of view in algebraic number theory, which is to consider all these equivalence classes of valuations together. This will lead us to the language of adeles and ideles.
An adele $\alpha$ of $K$ is a family $(\alpha_{v})$ of elements $\alpha_{v}$ of $K_{v}$ where $\alpha_{v}\in K_{v}$, and $\alpha_{v}\in\mathfrak{o}_{v}$ for all but finitely many $v$. We can define addition and multiplication componentwise on adeles, and the resulting ring of adeles is then denoted $\mathbb{A}_{K}$. The group of units of the ring of adeles is called the group of ideles, denoted $I_{K}$. For a finite set of primes $S$ that includes the infinite primes, we let
$\displaystyle \mathbb{A}_{K}^{S}=\prod_{v\in S}K_{v}\times\prod_{v\notin S}\mathfrak{o}_{v}$
and
$\displaystyle I_{K}^{S}=\prod_{v\in S}K_{v}^{*}\times\prod_{v\notin S}\mathfrak{o}_{v}^{*}$.
We denote the set of infinite primes by $S_{\infty}$. Then $\mathfrak{o}_{K}$, the ring of integers of the number field $K$, is given by $K\cap\mathbb{A}_{K}^{S_{\infty}}$, while $\mathfrak{o}_{K}^{*}$, the group of units of $\mathfrak{o}_{K}$, is given by $K^{*}\cap I_{K}^{S_{\infty}}$.
Any element of $K$ is also an element of $\mathbb{A}_{K}$, and any element of $K^{*}$ (the group of units of $K$) is also an element of $I_{K}$. The elements of $I_{K}$ which are also elements of $K^{*}$ are called the principal ideles. This should not be confused with the concept of principal ideals; however the terminology is perhaps suggestive on purpose. In fact, ideles and fractional ideals are related. Any fractional ideal $\mathfrak{a}$ can be expressed in the form
$\displaystyle \mathfrak{a}=\prod_{\mathfrak{p}}\mathfrak{p}^{\nu_{\mathfrak{p}}}$.
Therefore, we have a mapping
$\displaystyle \alpha\mapsto (\alpha)=\prod_{\mathfrak{p}}\mathfrak{p}^{v_{\mathfrak{p}}(\alpha_v)}$
from the group of ideles to the group of fractional ideals. This mapping is surjective, and its kernel is $I_{K}^{S_{\infty}}$.
The quotient group $I_{K}/K^{*}$ is called the idele class group of $K$, and is denoted by $C_{K}$. Again, this is not to be confused with the ideal class group we discussed in Algebraic Numbers, although the two are related; in the language of ideles, the ideal class group is defined as $I_{K}/I_{K}^{S_{\infty}}K^{*}$, and is denoted by $Cl_{K}$. There is a surjective homomorphism $C_{K}\mapsto Cl_{K}$ induced by the surjective homomorphism from the group of ideles to the group of fractional ideals that we have described in the preceding paragraph.
An important aspect of the concept of adeles and ideles is that they can be equipped with topologies (see Basics of Topology and Continuous Functions). For the adeles, this topology is generated by the neighborhoods of $0$ in $\mathbb{A}_{K}^{S_{\infty}}$ under the product topology. For the ideles, this topology is defined by the condition that the mapping $\alpha\mapsto (\alpha,\alpha^{-1})$ from $I_{K}$ into $\mathbb{A}_{K}\times\mathbb{A}_{K}$ be a homeomorphism onto its image. Both topologies are locally compact, which means that every element has a neighborhood which is compact, i.e. every open cover of that neighborhood has a finite subcover. For the group of ideles, its topology is compatible with its group structure, which makes it into a locally compact topological group.
In this post, we have therefore seen how the theory of valuations can allow us to consider a more abstract viewpoint for algebraic number theory, and how considering all the valuations together to form adeles and ideles allows us to rephrase the usual concepts related to algebraic number fields, such as the ring of integers, its group of units, and the ideal class group, in a new form. In addition, the topologies on the adeles and ideles can be used to obtain new results; for instance, because the group of ideles is a locally compact topological (abelian) group, we can use the methods of harmonic analysis (see Some Basics of Fourier Analysis) to study it. This is the content of the famous thesis of the mathematician John Tate. Another direction where the concept of adeles and ideles can take us is class field theory, which relates the idele class group to the other important group in algebraic number theory, the Galois group (see Galois Groups). The language of adeles and ideles can also be applied not only to algebraic number fields but also to function fields of curves over finite fields. Together these fields are also known as global fields.
References: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 116, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9590016007423401, "perplexity": 100.12146202606944}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655906214.53/warc/CC-MAIN-20200710050953-20200710080953-00396.warc.gz"} |
https://id.scribd.com/doc/217792562/Graph-Theory-Douglas-B-West-2-Ed | Anda di halaman 1dari 610
# Glossary of Notation
---+ ::::
Non-alphabetic notation
adjacency relation successor relation (digraph) isomorphism relation congruence relation implication floor of number ceiling of number
{l, ... , n)
Roman alphabet
adjacency matrix adjugate matrix bandwidth bases of matroid BM circuits of matroid CM cycle with n vertices Cn Cd power of a cycle n number of components c(G) c(G) circumference (Hamiltonian) closure C(G) cost or capacity c(e) cap(S, T) capacity of a cut degree sequence di, .. ., dn degree of vertex d(v), dc(v) d+(v), d-(v) out-degree, in-degree digraph D D(G) distance sum distance from u to v d(u, v) diamG diameter determinant detA E(G) edge set E(X) expected value e(G) size (number of edges) j+(v), j+(S) total exiting flow l-(v), l-(S) total entering flow function, flow l number of faces l graph (or digraph! G GP random graph in Model A Harary graph Hk.n independent sets of matroid IM I identity matrix matrix of all l's J complete graph Kn complete bipartite graph Kr.s L(G) line graph l(e) lower bound on flow A(G) AdjA B(G)
a= b
=}
mod n
LxJ rxl
[n]
## !xi {x: P(x))
00
ISi
0 u n
A s; B G s; H G[S] G,X
G*
Gk
5k
[S,S]
[S, T] G- v
G-e
G e
G+H
GvH
GOH GtoH, AtoB G ox Goh AxB A-B
ln
~:~"nJ
YIX
absolute value of number size of set set description infinity empty set union intersection subset subgraph subgraph of G induced by S complement of graph or set (planar) dual kth power of graph set of k-tuples from S edge cut source-sink cut deletion of vertex deletion of edge contraction of edge disjoint union of graphs join of graphs cartesian product of graphs symmetric difference vertex duplication vertex multiplication cartesian product of sets difference of sets binomial coefficient multinomial coefficient n-vector with all entries 1 conditional variable or event
## continued on inside back cover
Introduction to
Graph Theory
Second Edition
Douglas B. West
University of Illinois - Urbana
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs. Copyright 2001 by Pearson Education, Inc. This edition is published by arrangement with Pearson Education, Inc. All rights reserved. No part of this publication may be reproduced, stored in a database or retrieval system, or transmitted in any form or by auy means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. ISBN 81-7808-830-4
## First Indian Reprint, 2002
This edition is manufactured in India and is authorized for sale only in India, Bangladesh, Pakistan, Nepal, Sri Lanka and the Maldives.
Published by Pearson Education (Singapore) Pte. Ltd., Indian Braneh, 482 F.I.E. Patparganj, Delhi 110 092, India Printed in India by Rashtriya Printers.
For my dear wife Ching and for all lovers of graph theory
Contents
## Preface Chapter 1 Fundamental Concepts
1.1 What Is a Graph? The Definition, 1 Graphs as Models, 3 Matrices and Isomorphism, 6 Decomposition and Special Graphs, 11 Exercises, 14 1.2 Paths, Cycles, and Trails Connection in Graphs, 20 Bipartite Graphs, 24 Eulerian Circuits, 26 Exercises, 31 1.3 Vertex Degrees and Counting Counting and Bijections, 35 Extremal Problems, 38 Graphic Sequences, 44 Exercises, 4 7 1.4 Directed Graphs Definitions and Examples, 53 Vertex Degrees, 58 Eulerian Digraphs, 60 Orientations and Tournaments, 61 Exercises, 63
v
xi
1
1
19
34
53
vi
Contents
## Chapter 2 Trees and. Distance
2.1 Basic Properties Properties of Trees, 68 Distance in Trees and Graphs, 70 Disjoint Spanning Trees (optional), 73 Exercises, 75 2.2 Spanning Trees and Enumeration Enumeration of Trees, 81 Spanning Trees in Graphs, 83 Decomposition and Graceful Labelings, 87 Branchings and Eulerian Digraphs (optional), 89 Exercises, 92 2.3 Optimization and Trees Minimum Spanning Tree, 95 Shortest Paths, 97 Trees in Computer Science (optional), 100 Exercises, 103
67
67
81
95
## Chapter 3 Matchings and Factors
3.1 Matchings and Covers Maximum Matchings, 108 Hall's Matching Condition, 110 Min-Max Theorems, 112 Independent Sets and Covers, 113 Dominating Sets (optional), 116 Exercises, 118 3.2 Algorithms and Applications Maximum Bipartite Matching, 123 Weighted Bipartite Matching, 125 Stable Matchings (optional), 130 Faster Bipartite Matching (optional), 132 Exercises, 134 3.3 Matchings in General Graphs Tutte's 1-factor Theorem, 136 /-factors of Graphs (optional), 140 Edmonds' Blossom Algorithm (optional), 142 Exercises, 145
107
107
123
136
Contents
vii
## Chapter4 Connectivity and Paths
4.1 Cuts and Connectivity Connectivity, 149 Edge-connectivity, 152 Blocks, 155 Exercises, 158
149
149
4.2 k-connected Graphs 2-connected Graphs, 161 Connectivity of Digraphs, 164 k-connected and k-edge-connected Graphs, 166 Applications ofMenger's Theorem, 170 Exercises, 172 4.3 Network Flow Problems Maximum Network Flow, 176 Integral Flows, 181 Supplies and Demands (optional), 184 Exercises, 188
161
176
## Chapter 5 Coloring of Graphs
5.1 Vertex Colorings and Upper Bounds Definitions and Examples, 191 Upper Bounds, 194 Brooks' Theorem, 197 Exercises, 199 5.2 Structure of k-chromatic Graphs Graphs with Large Chromatic Number, 205 Extremal Problems and Turan's Theorem 207 Color-Critical Graphs, 2iO Forced Subdivisions, 212 Exercises, 214 5.3 Enumerative Aspects Counting Proper Colorings, 219 Chordal Graphs, 224 A Hint of Perfect Graphs, 226 Counting Acyclic Orientations (optional), 228 Exercises, 229
i91
191
204
219
viii
Contents
## Chapter 6 Planar Graphs
6.1 Embeddings and Euler's Formula Drawings in the Plane, 233 Dual Graphs, 236 Euler's Formula, 241255 Exercises, 243 6.2 Characterization of Planar Graphs Preparation for Kuratowski's Theorem, 24 7 Con'V'e:ll.'. Embeddings, 248 Planarity Testing (optional), 252 . Exercises, 255 6.3 Parameters of Planarity Coloring of Planar Graphs, 257 Crossing Number, 261 Surfaces of Higher Genus (optional), 266 Exercises, 269
233
233
246
257
## Chapter 7 Edges and Cycles
7.1 Line Graphs and Edge-coloring Edge-co}Qrings, 27 4 Characterization of Line Graphs (optional), 279 Exercises, 282 7.2 Hamiltonian Cycles Necessary Conditions, 287 Sufficient Conditions, 288 Cycles in Directed Graphs (optional), 293 Exercises, 294 7.3 Planarity, Coloring, and Cycles Tait's Theorem, 300 Grinberg's Theorem, 302 Snarks (optional), 304 Flows and Cycle Covers (optional), 307 Exercises, 314
273
273
286
299
Contents
ix
## Chapter 8 Additional Topics (optional)
8.1 Perfect Graphs The Perfect Graph Theorem, 320 Chordal Graphs Revisited, 323 Other Classes of Perfect Graphs, 328 Imperfect Graphs, 334 The Strong Perfect Graph Conjecture, 340 Exercises, 344 8.2 Matroids Hereditary Systems and Examples, 349 Properties of Matroids, 354 The Span Function, 358 The Dual of a Matroid, 360 Matroid Minors and Planar Graphs, 363 Matroid Intersection, 366 Matroid Union, 369 Exercises, 372 8.3 Ramsey Theory The Pigeonhole Principle Revisited, 378 Ramsey's Theorem, 380 Ramsey Numbers, 383 Graph Ramsey Theory, 386 Sperner's Lemma and Bandwidth, 388 Exercises, 392 8.4 More Extremal Problems Encodings of Graphs, 397 Branchings and Gossip, 404 List Coloring and Choosability, 408 Partitions Using Paths and Cycles, 413 Circumference, 416 Exercises, 422 8.5 Random Graphs Existence and Expectation, 426 Properties of Almost All Graphs, 430 Threshold Functions, 432 Evolution and Graph Parameters, 436 Connectivity, Cliques, and Coloring, 439 Martingales, 442 Exercises, 448
319
319
349
378
396
425
Contents
8.6 Eigenvalues of Graphs The Characteristic Polynomial, 453 Linear Algebra of Real Symmetric Matrices, 456 Eigenvalues and Graph Parameters, 458 Eigenvalues of Regular Graphs, 460 Eigenvalues and Expanders, 463 Strungly Regular Graphs, 464 Exercises, 467
452
## Appendix A Mathematical Background
Sets, 471 Quantifiers and Proofs, 475 Induction and Recurrence, 479 Functions, 483 Counting and Binomial Coefficients, 485 Relations, 489 The Pigeonhole Principle, 491
471
Appendix B
## Optimization and Complexity
493
Intractability, 493 Heuristics and Bounds, 496 NP-Completeness Proofs, 499 Exercises, 505
## Appendix C Hints for Selected Exercises
General Discussion, 507 Supplemental Specific Hints, 508
507
Appendix D
Glossary of Terms
## Appendix E Supplemental Reading Appendix F Author Index Subject Index References
Preface
Graph theory is a delightful playground for the exploration of proof techniques in discrete mathematics, and its results have applications in many areas of the computing, social, and natural sciences. The design of this book permits usage in a one-semester introduction at the undergraduate or beginning graduate level, or in a patient two-semester introduction. No previous knowledge of graph theory is assumed. Many algorithms and applica~ions are included, but the focus is on understanding the structure of graphs and the techniques used to analyze problems in graph theory. Many textbooks have been written about graph theory. Due to its emphasis on both proofs and applications, the initial model for this book was the elegant text by J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan/North-Holland [1976]). Graph theory is still young, and no consensus has emerged on how the introductory material should be presented. Selection and order of topics, choice of proofs, objectives, and underlying themes are matters of lively debate. Revising this book dozens of times has taught me the difficulty of these decisions. This book is my contribution to the debate.
## The Second Edition
The revision for the second edition emphasizes making the text easier for the students to learn from and easier for the instructor to teach from. There have not been great changes in the overall content of the book, but the presentation has been modified to make the material more accessible, especially in the early parts of the book. Some of the changes are discussed in more detail later in this preface; here I provide a brief summary. Optional material within non-optional sections is now designated by(*); such material is not used later and can be skipped. Most of it is intended to be skipped in a one-semester course. When a subsection is marked "optional", the entire subsection is optional, and hence no individual items are starred. For less-experienced students, Appendix A has been added as a reference summary of helpful material on sets, logical statements, induction, counting arguments, binomial coefficients, relations, and the pigeonhole principle.
xi
xii
Preface
Many proofs have been reworded in more patient language with additional details, and more examples have been added. More than 350 exercises have been added, mostly easier exercises in Chapters 1-7. There are now more than 1200 exercises. More than 100 illustrations have been added; there are now more than 400. In illustrations showing several types of edges, the switch to bold and solid edges instead of solid and dashed edges has increased clarity. Easier problems are now grouped at the beginning of each exercise section, usable as warm-ups. Statements of some exercises have been clarified. In addition to hints accompanying the exercise statements, there is now an appendix of supplemental hints. For easier access, terms being defined are in bold type, and the vast majority of them appear in Definition items. For easier access, the glossary of notation has been placed on the inside covers. Material involving Eulerian circuits, digraphs, and Turan's Theorem has been relocated to facilitate more efficient learning. Chapters 6 and 7 have been switched to introduce the idea of planarity earlier, and the section on complexity has become an appendix. The glossary has been improved to eliminate errors and to emphasize items more directly related to the text.
Features
Various features of this book facilitate students' efforts to understand the material. There is discussion of proof techniques, more than 1200 exercises of varying difficulty, more than 400 illustrations, and many examples. Proofs are presented in full in the text. Many undergraduates begin a course in graph theory with little exposure to proof techniques. Appendix A provides background reading that will help them get started. Students who have difficulty understanding or writing proofs in the early material should be encouraged to read this appendix in conjunction with Chapter 1. Some discussion of proof techniques still appears in the early sections of the text (especially concerning induction), but an expanded treatment of the basic background (especially concerning sets, functions, relations, and elementary counting) is now in Appendix A. Most of the exercises require proofs. Many undergraduates have had little practice at presenting explanations, and this hinders their appreciation of graph theory and other mathematics. The intellectual discipline ofjustifying an argument is valuable.independently of mathematics; I hope that students will appreciate this. In writing solutions to exercises, students should be careful in their use of language ("say what you mean"), and they should be intellectually honest ("mean what you say"). Although many terms in graph theory suggest their definitions, the quantity of terminology remains an obstacle to fluency. Mathematicians like to gather definitfons at the start, but most students succeed better if they use a
Preface
xiii
concept before receiving the next. This, plus experience and requests from reviewers, has led me to postpone many definitions until they are needed. For example, the definition of cartesian product appears in Section 5.1 with coloring problems. Line graphs are defined in Section 4.2 with Menger's Theorem and in Section 7.1 with edge-coloring. The definitions of induced subgraph and join have now been postponed to Section 1.2 and Section 3.1, respectively. I have changed the treatment of digraphs substantially by postponing their introduction to Section 1.4. Introducing digraphs at the same time as graphs tends to confuse or overwhelm students. Waiting to the end of Chapter 1 allows them to become comfortable with basic concepts in the context of a single model. The discussion of digraphs then reinforces some of those concepts while clarifying the distinctions. The two models are still discussed together in the material on connectivity. This book contains more material than most introductory texts in graph theory. Collecting the advanced material as a final optional chapter of "additional topics" permits usage at different levels. The undergraduate introduction consists of the first seven chapters (omitting most optional material), leaving Chapter 8 as topical reading for interested students. A graduate course can treat most of Chapters 1 and 2 as recommended reading, moving rapidly to Chapter 3 in class and reaching some topics in Chapter 8. Chapter 8 can also be used as the basis for a second course in graph theory, along with material that was optional in earlier chapters. Many results in graph theory have several proofs; illustrating this can increase students' flexibility in trying multiple approaches to a problem. I include some alternative proofs as remarks and others as exercises. Many exercises have hints, some given with the exercise statement and others in Appendix C. Exercises marked "( - )" or "( +)" are easier or more difficult, respectively, than unmarked problems. Those marked "( +)" should not be assigned as homework in a typical undergraduate course. Exercises marked "(!)" are especially valuable, instructive, or entertaining. Those marked"(*)" use material labeled optional in the text. Each exercise section begins with a set of"( - )"exercises, ordered according to the material in the section and ending with a line of bullets. These exercises either check understanding of concepts or are immediate applications ofresults in the section. I recommend some of these to my class as "warmup" exercises to check their understanding before working the main homework problems, most of which are marked "(!)". Most problems marked "(-)" are good exam questions. When using other exercises on exams, it may be a good idea to provide hints from Appendix C. Exercises that relate several concepts appear when the last is introduced. Many pointers to exercises appear in the text where relevant concepts are discussed. An exercise in the current section is cited by giving only its item number among the exercises of that section. Other cross-references are by Chapter.Section.Item.
xiv
Preface
## Organization and Modifications
In the first edition, I sought a development that was intellectually coherent and displayed a gradual (not monotonic) increase in difficulty of proofs and in algorithmic complexity. Carrying this further in the second edition, Eulerian circuits and Hamiltonian cycles are now even farther apart. The simple characterization of Eulerian circuits is now in Section 1.2 with material closely related to it. The remainder of the former Section 2.4 has been dispersed to relevant locations in other sections, with Fleury's Algorithm dropped. Chapter 1 has been substantially rewritten. I continue to avoid the term "multigraph"; it causes more trouble than it resolves, because many students assume that a multigraph must have multiple edges. It is less distracting to append the word "simple" where needed and keep "graph" as the general object, with occasional statements that in particular contexts it makes sense to consider only simple graphs. The treatment of definitions in Chapter 1 has been made i:ore friendly and precise, particularly those involving paths, trails, and walks. The informal groupings of basic definitions in Section 1.1 have been replaced by Definition items that help students find definitions more easily. In addition to the material on isomorphism, Section 1.1 now has a more precise treatment of the Petersen graph and an explicit introduction of the notions of decomposition and girth. This provides language that facilitates later discussion in various places, and it permits interesting explicit questions other than isomorphism. Sections 1.2-1.4 have become more coherent. The treatment of Eulerian circuits motivates and completes Section 1.2. Some material has been removed from Section 1.3 to narrow its focus to degrees and counting, and this section has acquired the material on vertex degrees that had been in Section 1.4. Section 1.4 now provides the introduction to digraphs and can be treated lightly. Trees and distance appear together in Chapter 2 due to the many relations between these topics. Many exercises combine these notions, and algorithms to compute distances produce or use trees. Most graph theorists agree that the Konig-Egervary Theorem deserves an independent proof without network flow. Also, students have trouble distinguishing "k-connected" from "connectivity k", which have the same relationship as "k-colorable" and "chromatic number k". I therefore treat matching first an.d later use matching to prove Menger's Theorem. Both matching and cor,mectivity are used in the coloring material. In response to requests from a number of users, I have added a short optional subsection on dominating sets at the end of Section 3.1. The material on weighted bipartite matching has been clarified by emphasis on vertex cover instead of augmenting path and by better use of examples. Turan's Theorem uses only elementary ideas about vertex degrees and induction and hence appeared in Chapter 1 in the first edition. This caused some difficulties, because it was the most abstract item up to that point and students
Preface
xv
felt somewhat overwhelmed by it. Thus I have kept the simple triangle-free case (Mantel's Theorem) in Section 1.3 and have moved the full theorem to Section 5.2 under the viewpoint of extremal problems related to coloring. The chapter on planarity now comes before that on "Edges and Cycles". When an instructor is short of time, planarity is more important to reach than the material on edge-coloring and Hamiltonian cycles. The questions involved in planarity appeal i:q.tuitively to students due to their visual aspects, and many students have encountered these questions before. Also, the ideas involved in discussing planar graphs seem more intellectually broadening in relation to the earlier material of the course than the ideas used to prove the basic results on edge-coloring and Hamiltonian cycles. Finally, discussing planarity first makes the material of Chapter 7 more coherent. The new arrangement permits a more thorough discussion of the relationships among planarity, edge-coloring, and Hamiltonian cycles, leading naturally beyond the Four Color Theorem to the optional new material on nowhere-zero flows as a dual concept to coloring. When students discover that the coloring and Hamiltonian cycle problems lack good algorithms, many become curious about NP-completeness. Appendix B satisfies this curiosity. Presentation of NP-completeness via formal languages can be technically abstract, so some students appreciate a presentation in the context of graph problems. NP-completeness proofs also illustrate the variety and usefulness of "graph transformation" arguments. The text explores relationships among fundamental results. Petersen's Theorem on 2-factors (Chapter 3) uses Eulerian circuits and bipartite matching. The equivalence between Menger's Theorem and the Max Flow-Min Cut Theorem is explored more fully than in the first edition, and the "Baseball Elimination" application is now treated in more depth. The k - !-connectedness of k-color-critical graphs (Chapter 5) uses bipartite matching. Section 5.3 offers a brief introduction to perfect graphs, emphasizing chordal graphs. Additional features of this text in comparison to some others include the algorithmic proof of Vizing's Theorem and the proof of Kuratowski's Theorem by Thomassen's methods. There are various other additions and improvements in the first seven chapters. There is now a brief discussion of Heawood's Formula and the Robertson-Seymour Theorem at the end of Chapter 6. In Section 7.1, a proof of Shannon's bound on the edge-chromatic number has been added. In Section 5.3, the characterization of chordal graphs is somewhat simpler than before by proving a stronger result about simplicial vertices. In Section 6.3, the proof of the reducibility of the Birkhoff diamond has been eliminated, but a brief discussion of discharging has been added. The material discussing issues in the p:t:oofofthe theorem is optional, and the aim is to give the flavor of the approach without getting into detailed arguments. From this viewpoint the reducibility proof seemed out of focus. Chapter 8 contains highlights of advanced material and is not intended for an undergraduate course. It assumes more sophistication than earlier chapters and is written more tersely. Its sections are independent; each selects appeal-
XVI
Preface
ing results from a large topic that merits a chapter of its own. Some of these sections become more difficult near the end; an instructor may prefer to sample early material in several sections rather than present one completely. There may be occasional relationships between items in Chapter 8 and items marked optional in the first seven chapters, but generally cross-references indicate the connections. The material of Chapter 8 has not changed substantially since the first edition, although many corn;ctions have been made and the presentation has been clarified in many places. I will treat advanced graph theory more thoroughly in The Art of Combinatorics. Volume I is devoted to extremal graph theory and Volume II to structure of graphs. Volume III has chapters on matroids and integer programming (including network flow). Volume IV emphasizes methods in cpmbinatorics and discusses various aspects of graphs, especially random graphs.
Design of Courses
I intend the 22 sections in Chapters 1-7 for a pace of slightly under two lectures per section when most optional material (starred items and optional subsections) is skipped. When I teach the course I spend eight lectures on Chapter 1, six lectures each on Chapters 4 and 5, and five lectures on each of Chapters 2, 3, 6, and 7. This presents the fundamental material in about 40 lectures. Some instructors may want to spend more time on Chapter 1 and omit more material from later chapters. In chapters after the first, the most fundamental material is concentrated in the first section. Emphasizing these sections (while skipping the optional items) still illustrates the scope of graph theory in a slower-paced one-semester course. From the second sections of Chapters 2, 4, 5, 6, and 7, it would be beneficial to include Cayley's Formula, Menger's Theorem, Mycielski's construction, Kuratowski's Theorem, and Dirac's Theorem (spanning cycles), respectively. Some optional material is particularly appealing to present in class. For example, I always present the optional subsections on Disjoint Spanning Trees (in Section 2.1) and Stable Matchings (in Section 3.2), and I usually present the optional subsection on /-factors (in Section 3.3). Subsections are marked optional when no later material in the first seven chapters requires them and they are not part of the central development of elementary graph theory, but these are nice applications that engage students' interest. In one sense, the "optional" marking indicates to students that the final examination is unlikely to have questions on these topics. Graduate courses skimming the first two chapters might include from them such topics as graphic sequences, kernels of digraphs, Cayley's Formula, the Matrix Tree Theorem, and Kruskal's algorithm. Courses that introduce graph theory in one term under the quarter system must aim for highlights; I suggest the following rough syllabus: 1.1: adjacency matrix, isomorphism, Petersen graph. 1.2: all. 1.3: degree-sum formula, large bipartite S,ubgraphs. 1.4: up to strong components, plus tournaments. 2.1: up
Preface
XVll
to centers of trees. 2.2: up to statement of Matrix Tree Theorem. 2.3: Kruskal's algorithm. 3.1: almost all. 3.2: none. 3.3: statement ofTutte's Theorem, proof of Petersen's results. 4.1: up to definition of blocks, omitting Harary graphs. 4.2: up to open ear decomposition, plus statement of Menger's Theorem(s). 4.3: duality between flows and cuts, statement of Max-flow = Min-cut. 5.1: up to Szekeres-Wilf theorem. 5.2: Mycielski's construction, possibly Turan's Theorem. 5.3: up to chromatic recurrence, plus perfection of chordal graphs. 6.1: non-planarity of K5 and K3. 3 , examples of dual graphs, Euler's formula with applications. 6.2: statement and examples ofKuratowski's Theorem and Tutte's Theorem. 6.3: 5-color Theorem, plus the idea of crossing number. 7.1: up to Vizing's Theorem. 7.2: up to Ore's condition, plus the Chvatal-Erdos condition. 7.3: Tait's Theorem, Grinberg's Theorem.
## Further Pedagogical Aspects
In the revision I have emphasized some themes that arise naturally from the material; underscoring these in lecture helps provide continuity. More emphasis has been given to the theme of TONCAS-"The obvious necessary condition is also sufficient." Explicit mention has been added that many of the fundamental results can be viewed in this way. This both provides a theme for the course and clarifies the distinction between the easy direction and the hard direction in an equivalence. Another theme that underlies much of Chapters 3-5 and Section 7.1 is that of dual maximization and minimization problems. In a graph theory course one does not want to delve deeply into the nature of duality in linear optimization. It suffices to say that two optimization problems form a dual pair when every feasible solution to the maximization problem has value at most the value of every feasible solution to the minimization problem. When feasible solutions with the same value are given for the two problems, this duality implies that both solutions are optimal. A discussion of the linear programming context appears in Section 8.1. Other themes can be identified among the proof techniques. One is the use of extremality to give short proofs and avoid the use of induction. Another is the paradigm for proving conditional statements by induction, as described explicitly in Remark 1.3.25. The devefopment leading to Kuratowski's Theorem is somewhat long. Nevertheless, it is preferable to present the proof in a single lecture. The preliminary lemmas reducing the problem to the 3-connected case can be treated lightly to save time. Note that the induction paradigm leads naturally to the two lemmas proved for the 3-connected case. Note also that the proof uses the notion of S-lobe defined in Section 5.2. The first lecture in Chapter 6 should not belabor technical definitions of drawings and regions. These are better left as intitive notions unless students ask for details; the precise statements appear in the text.
xviii
Preface
The motivating applications of digraphs in Section 1.4 have been marked optional because they are not needed in the rest of the text, but they help clarify that the appropriate model (graph or digraph) depends on the application. Due to its reduced emphasis on numerical computation and increased emphasize on techniques of proof and clarity of explanations, graph theory is an excellent subject in which to encourage students to improve their habits of communication, both written and oral. In addition to assigning written homework that requires carefully presented arguments, I have found it productive to organize optional "collaborative study" sessions in which students can work together on problems while I circulate, listen, and answer questions. It should not be forgotten that one of the best ways to discover whether one understands a proof is to try to explain it to someone else. The students who participate find these sessions very beneficial.
Acknowledgments
Preface
xix
Isaak, Steve Kilner, Alexandr Kostochka, Andre Kiindgen, Peter Kwok, JeanMarc Lanlignel, Francois Margot, Alan Mehlenbacher, Joel Miller, Zevi Miller, Wendy Myrvold, Charles Parry, Robert Pratt, Dan Pritikin, Radhika Ramamurthi, Craig Rasmussen, Bruce Reznick, Jian Shen, Tom Shermer, Warren Shreve, Alexander Strehl, Tibor Szabo, Vitaly Voloshin, and C.Q. Zhang. Several students found numerous typographical errors in the pre-publication version of the second edition (thereby earning extra credit!): Jaspreet Bagga, Brandon Bowersox, Mark Chabura, John Chuang, Greg Harfst, Shalene Melo, Charlie Pikscher, and Josh Reed. The cover drawing for the first edition was produced by Ed Scheinerman using BRL-CAD, a product of the U.S. Army Ballistic Research Laboratory. For the second edition, the drawing was produced by Maria Muyot using CorelDraw. Chris Hartman contributed vital assistance in preparing the bibliography for the first edition; additional references have now been added. Ted Harding helped resolve typesetting difficulties in other parts of the first edition. Students who helped gather data for the index of the first edition included Maria Axenovich, Nicole Henley, Andre Kiindgen, Peter Kwok, Kevin Leuthold, John Jozwiak, Radhika Ramamurthi, and Karl Schmidt. Raw data for the index of the second edition was gathered using scripts I wrote in perl; Maria Muyot and Radhika Ramamurthi assisted with processing of the index and the bibliography. I prepared the second edition in '!EjX, the typesetting system for which the scientific world owes Donald E. Knuth eternal gratitude. The figures were generated using gpic, a product of the Free Software Foundation.
Feedback
I welcome corrections and suggestions, including comments on topics, attributions of results, updates, suggestions for exercises, typographical errors, omissions from the glossary or index, etc. Please send these to me at [email protected] In particular, I apologize in advance for missing references; please inform me of the proper citations! Also, no changes other than corrections of errors will be made between printings of this edition. I maintain a web site containing a syllabus, errata, updates, and other material. Please visit!
http://www.math.uiuc.edu/~west/igt
I have corrected all typographical and mathematical errors known to me before the time of printing. Nevertheless, the robustness of the set of errors and the substantial rewriting and additions make me confident thRt some error remains. Please find it and tell me so I can correct it! Douglas B. West Urbana, Illinois
Chapter 1
Fundamental Concepts
1.1. What Is a Graph?
How can we lay cable at minimum cost to make every telephone reachable from every other? What is the fastest route from the national capital to each state capital? How can n jobs be filled by n people with maximum total utility? What is the maximum fiow per unit time from source to sink in a network of pipes? How many layers does a computer chip need so that wires in the same layer don't cross? How can the season of a sports league be scheduled into the minimum number of weeks? In what order should a traveling salesman visit cities to minimize travel time? Can we color the regions of every map using four colors so that neighboring regions receive different colors? These and many other practical problems involve graph theory. In this book, we develop the theory of graphs and apply it to such problems. Our starting point assumes the mathematical background in Appendix A, where basic objects and language of mathematics are discussed.
THE DEFINITION
The problem that is often said to have been the birth of graph theory will suggest our basic definition of a graph.
1.1.1. Example. The Konigsberg Bridge Problem. The city of Konigsberg was located on the Pregel river in Prussia. The city occupied two islands plus areas on both banks. These regions were linked by seven bridges as shown on the left below. The citizens wondered whether they could leave home, cross every bridge exactly once, and return home. The problem reduces to traversing the figure on the right, with heavy dots representing land masses and curves representing bridges.
1
## Chapter 1: Fundamental Concepts
z
The model on the right makes it easy to argue that the desired traversal does not exist. Each time we enter and leave a land mass, we use two bridges ending at it. We can also pair the first bridge with the last bridge on the land mass where we begin and end. Thus existence of the desired traversal requires that each land mass be involved in an even number of bridges. This necessary condition did not hold in Konigsberg. The Konigsberg Bridge Problem becomes more interesting when we show in Section 1.2 which configurations have traversals. Meanwhile, the problem suggests a general model for discussing such questions. 1.1.2. Definition. A graph ti is a triple consisting of a vertex set V(G), an edge set E(G), and a relation that associates with each edge two vertices (not necessarily distinct) called its endpoints. We draw a graph on paper by placing each vertex at a point and representing each edge by a curve joining the locations of its endpoints. 1.1.3. Example. In the graph in Example 1.1.1, the vertex set is {x, y, z, w}, the edge set is {ei, ez, e3 , e4, e5, e6, e7 }, and the assignment of endpoints to edges can be read from the picture. Note that edges e1 and e2 have the same endpoints, as do e3 and e4. Also, if we had a bridge over an inlet, then its ends would be in the same land mass, and we would draw it as a curve with both ends at the same point. We have appropriate terms for these types of edges in graphs. 1.1.4. Definition. A loop is an edge whose endpoints are equal. Multiple edges are edges having the same pair of endpoints. A simple graph is a graph having no loops or multiple edges. We specify a simple graph by its vertex set and edge set, treating the edge set as a set of unordered pairs of vertices and writing e =UV (ore= VU) for an edge e with endpoints u and v. When u and v are the endpoints of an edge, they are adjacent and are neighbors. We write u ~ v for "u is adjacent to v".
In many important applications, loops and multiple edges do not arise, and we restrict our attention to simple graphs. In this case an edge is determined by
## Section 1.1: What Is a Graph?
its endpoints, so we can name the edge by its endpoints, as stated in Definition 1.1.4. Thus in a simple graph we view an edge as an unordered pair of vertices and can ignore the formality of the relation associating endpoints to edges. This book emphasizes simple graphs.
1.1.5. Example. On the left below are two drawings of a simple graph. The vertex set is {u, v, w, x, y}, and the edge set is (uv, uw, ux, vx, vw,xw, xy}. The terms "vertex" and "edge" arise from solid geometry. A cube has vertices and edges, and these form the vertex set and edge set of a graph. It is drawn on the right below, omitting the names of vertices and edges.
"[8l
x
A graph is finite if its vertex set and edge set are finite. We adopt the convention that every graph mentioned in this book is finite, unless explicitly constructed otherwise.
1.1.6. * Remark. The null graph is the graph whose vertex set and edge set are empty. Extending general theorems to the null graph introduces unnecessary distractions, so we ignore it. All statements and exercises should be considered only for graphs with a nonempty set of vertices.
GRAPHS AS MODELS
Graphs arise in many settings. The applications suggest useful concepts and terminology about the structure of graphs.
1.1.7. Example. Acquaintance relations and subgraphs. Does every set of six people contain three mutual acquaintances or three mutual strangers? Since "acquaintance" is symmetric, we model it using a simple graph with a vertex for each person and an edge for each acquainted pair. The "nonacquaintance" relation on the same set yields another graph with the "complementary" set of edges. We introduce terms for these concepts.
## Chapter 1: Fundamental Concepts
1.1.8. Definition. The complement G of a simple graph G is the simple graph with vertex set V(G) defined by uv E E(G) if and only if uv tf. E(G). A clique in a graph is a set of pairwise adjacent vertices. An independent set (or stable set) in a graph is a set of pairwise nonadjacent vertices. In the graph G of Example 1.1.7, {u, x, y} is a clique of size 3 and {u, w} is an independent set of size 2, and these are the largest such sets. These values reverse in the complement G, since cliques become independent sets (and vice versa) under complementation. The question in Example 1.1.7 asks whether it is true that every 6-vertex graph has a clique of size 3 or an independent set of size 3 (Exercise 29). Deleting edge ux from G yields a 5-vertex graph having no clique or independent set of size 3. 1.1.9. Example. Job assignments and bipartite graphs. We have m jobs and n people, but not all people are qualified for all jobs. Can we fill the jobs with qualified people? We model this using a simple graph H with vertices for the jobs and people; job j is adjacent to person p if p can do j. Each job is to be filled by exactly one person, and each person can hold at most one of the jobs. Thus we seek m pairwise disjoint edges in H (viewing edges as pairs of vertices). Ch2pter 3 shows how to test for this; it can't be done in the graph below. The use of graphs to model relations between two disjoint sets has many important applications. These are the graphs whose vertex sets can be partitioned into two independent sets; we need a name for them. people jobs
1.1.10. Definition. A graph G is bipartite if V ( G) is the union of two disjoint (possibly empty) independent sets called partite sets of G. 1.1.11. Example. Scheduling and graph coloring. Suppose we must schedule Senate committee meetings into designated weekly time periods. We cannot assign two committees to the same time if they have a common member. How many different time periods do we need? We create a vertex for each committee, with two vertices adjacent when their committees have a common member. We must assign labels (time periods) to the vertices so the endpoints of each edge receive different labels. In the graph below, we can use one label for each of the three independent sets of vertices grouped closely together. The members of a clique must receive distinct labels, so in this example the minimum number of time periods is three. Since we are only interested in partitioning the vertices, and the labels have no numerical value, it is convenient to call them colors.
## Section l'.1: What Is a Graph?
1.1.12. Definition. The chromatic number of a graph G, written x(G), is the minimum number of colors needed to label the vertices so that adjacent verti~es receive different colors. A graph G is k-partite if V(G) can be expressed as the union of k (possibly empty) independent sets.
This generalizes the idea of bipartite graphs, which are 2-partite. Vertices given the same cofor must form an independent set, so x (G) is the minimum number of independent sets needed to partition V ( G). A graph is k-partite if and only if its chromatic number is at most k. We use the term "partite set" when referring to a set in a partition into independent sets. We study chromatic number and graph colorings in Chapter 5. The most (in)famous probiem in graph theory involves coloring of"maps".
1.i.13. Example. Maps and coloring, Roughly speaking, a map is a partition of the plane into connected regions. Can we color the regions of every map using at most four colors so that neighboring regions have different colors? To relate map coloring to graph colori~g, we introduce a vertex for each region and an edge for regions sharing a bmindary. The map question asks whether the resulting graph must have chromatic number at most 4. The graph can be drawn in the plane without crossing edges; such graphs are planar. The graph before Definition 1.1.12 is planar; that drawing has a crossing, but another drawing has no crossings. We study planar graphs in Chapter 6. 1.1.14. Example. Routes in road networks. We can model a road network using a graph with edges corresponding to road segments between intersections. We can assign edge weights to measure distance or travel time. In this context edges do represent physical links. How can we find the shortest route from x toy? We show how to compute this in Chapter 2. If the vertices of the graph represent our house and other places to visit, then we may want to follow a route that visits every vertex exactly once, so as to visit everyone without overstaying our welcome. We consider the existence of such a route in Chapter 7. We need terms to describe these two types of routes in graphs. 1.1.15. Definition. A path is a simple graph whose vertices can be ordered so that two vertices are adjacent if and only if they are consecutive in the list. A cycle is a graph with an equal number of vertices and edges whose vertices can be placed around a circle so that two vertices are adjacent if and only if they appear consecutively along the circle.
## Chapter 1: Fundamental Concepts
x
y
)<>a
b
V(>a
y b
Above we show a P!lth and a cycle, as demonstrated by listing the vertices in the order x, b, a, z, y. Dropping one edge from a cycle produces a path. In studying the rm1tes in road networks, we think of paths and cycles contained in the graph. Also, we hope that every vertex in the network can be reached from every other. The next definition makes these concepts precise.
1.1.16. Definition. A subgraph of a graph G is a graph H such that V (H) s; V(G) and E(H) s; E(G).and the assignment of endpoints to edges in His the same as m G. We then write H s; G and say that "G contains H". A graph G is connected if each pair of vertices in G belongs to a path; otherwise, G is disconnected.
The graph before Definition 1.1.12 has three subgraphs that are cycles. It is a connected graph, but the graph in Example 1.1.9 is not .
## MATRICES AND ISOMORPHISM
How do we specify a graph? We can list the vertices and edges (with endpoints), but there are other useful representations. Saying that a graph is loopless means that multiple edges are allowed but loops are not.
1.1.17. Definition. Let G be a loopless graph with vertex set V ( G) = { v1, ... , Vn} and edge set E(G) = {e 1 , ... , em}. The adjacency matrix of G, written A ( G), is the n-by-n matrix in which entry a;,j is the number of edges in G with endpoints {v;, Vj}. The incidence matrix M ( G) is the n-by-m matrix in which entry m;,j is 1 if v; is an endpoint of ej and otherwise is 0. If vertex v is an endpoint of edge e, then v and e are incident. The degree of vertex v (in a loopless graph) is the number of incident edges.
The appropriate way to define adjacency matri~, incidence matrix, or vertex degrees for graphs with loops depends on the application; Sections 1.2 and 1.3 discuss this.
1.1.18. Remark. An adjacency matrix is determined by a vertex ordering. Every adjacency matrix is symmetric (a;,j = aj,i for all i, j). An adjacency matrix of a simple graph G has entries 0 or 1, with Os on the diagonal. The degree of vis the sum of the entries in the row for v in either A(G) or M(G). 1.1.19. Example. For the loopless graph G below, we exhibit the adjacency matrix and incidence matrix that result from the vertex ordering w, x, y, z and
## Section 1.1: What Is a Graph?
the edge ordering a, b, c, d, e. The degree of y is 4, by viewing the graph or by summing the row for y in either matrix.
w x
(0
w
1 1
z
a
0 0 1 0
A(G)
H~)
~z o---~
x
; z
(H : :
0 0 0 0
M(G)
Presenting an adjacency matrix for a graph implicitly names the vertices by the order of the rows; the ith vertex corresponds to the ith row and column. Storing a graph in a computer requires naming the vertices. Nevertheless, we want to study properties (like connectedness) that do not depend on these names. Intuitively, the structural properties of G and H will be the same if we can rename the vertices of G using the vertices in H so that G will actually become H. We make the definition precise for simple graphs. The renaming is a function from V(G) to V(H) that assigns each element of V(H) to one element of V(G), thus pairing them up. Such a function is a on~-to-one correspondence or bijection (see Appendix A). Saying that the renaming turns G into H is saying that the vertex bijection preserves the adjacency relation.
1.1.20. Definition. An isomorphism from a simple graph G to a simple graph His a bijection/: V(G)-+ V(H) such that uv E E(G) ifand only if f(u)f(v) E E(H). We say"G is isomorphic to H", written G ~ H, ifthere is an isomorphism from G to H. 1.1.21. Example. The graphs G and H drawn below are 4-vertex paths. Define , the function/: V(G) -+ V(H) by f(w) =a, f(x) = d, f(y) = b, f(z) = c. To show that f is an isomorphism, we check that f preserves edges and nonedges. Note that rewriting A(G) by placing the rows in the order w, y, z, x and the columns also in that order yields A(H), as illustrated below; this verifies that f is an isomorphism. Another isomorphism maps w, x, y, z to c, b, d, a, respectively.
w y c
VI
x
)<l
a
z w y z x
b
a a b
w x y z
WC
1 0 0
1 0 1 0
0 1 0 1
WC
y z x
0 0 1
0 0 1 1
0 1 0 0
~)
(0
0 0 0 0 1 0 1 0 1 1 0
## Chapter 1: Fundamental Concepts
1.1.22. Remark. Finding isomorphisms. As suggested in Example 1.1.21, presenting the adjacency matrices with vertices ordered so that the matrices are identical is one way to prove that two graphs are isomorphic. Applying a permutation a to both the rows and the columns of A ( G) has the effect ofreordering the vertices of G. If the new matrix equals A(H), then the permutation yields an isomorphism. One can also verify preservation of the adjacency relation without writing out the matrices. In order for an explicit vertex bijection to be an isomorphism from G to H, the image in H of a vertex v in G must behave in H as v does in G. For example, they must have the same degree. 1.1.23. * Remark. Isomorphism for non-simple graphs. The definition of isomorphism extends to graphs with loops and multiple edges, but the precise statement needs the language of Definition 1.1.2. An isomorphism from G to H is a bijection f that maps V(G) to V(H) and E(G) to E(H) such each edge ofG with endpoints u and vis mapped to an edge with endpoints f(u) and f (v). This technicality will not concern us, because we will study isomorphism only in the context of simple graphs. Since H is isomorphic to G whenever G is isomorphic to H, we often say
"G and Hare isomorphic" (meaning to each other). The adjective "isomorphic" applies only to pairs of graphs; "G is isomorphic" by itself has no meaning (we
respond, "isomorphic to what?"). Similarly, we may say that a set of graphs is "pairwise isomorphic" (taken two at a time), but it doesn't make sense to say "this set of graphs is isomorphic". A relation on a set S is a collection of ordered pairs from S. An equivalence relation is a relation that is reflexive, symmetric, and transitive (see Appendix A). For example, the adjacency relation on the set of vertices of a graph is symmetric, but it is not reflexive and rarely is transitive. On the other hand, the isomorphism relation, consisting of the set ofordered pairs (G, H) such that G is isomorphic to H, does have all three properties. 1.1.24. Proposition. The isomorphism relation is an equivalence relation on the set of(simple) graphs. Proof: Reflexive property. The identity permutation on V (G) is an isomorphism from G to itself. Thus G ~ G. Symmetric property. If/: V(G) ~ V(H) is an isomorphism from G to H, then 1-1 is an isomorphism from H to G, because the statement "uv E E(G) if and only if f(u)f(v) E E(H)" yields "xy E E(H) if and only if 1- 1 (x)f- 1 (y) E E(H)". Thus G ~ H implies H ~ G. Transitive property. Suppose that/: V(F) ~ V(G) and g: V(G) ~ V(H) are isomorphisms. We are given "uv E E(F) if and only if f(u)f(v) E E(G)" and "xy E E(G) if and only if g(x)g(y) E E(H)". Since f is an isomorphism, for every xy E E(G) we can find uv E E(F) such that f(u) = x and f(v) = y. This
## Section 1,1: What Is a Graph?
yields "uv E E(F) if and only if g(f(u))g(f(v)) E E(H)". Thus the composition g o f is an isomorphism from F to H. We have proved that F ~ G and G ~ H together imply F ~ H.
An equivalence relation partitions a set into equivalence classes; two elements satisfy the relation if and only if they lie in the same class.
1.1.25. Definition. An isomorphism class of graphs is an equivalence class of graphs under the isomorphism relation. Paths with n vertices are pairwise isomorphic; the set of all n-vertex paths forms an isomorphism class. 1.1.26. Remark. "Unlabeled" graphs and isomorphism classes. When discussing a graph G, we have a fixed vertex set, but our structural comments apply also to every graph isomorphic to G. Our conclusions are independent of the names (labels) of the vertices. Thus, we use the informal expression "unlabeled graph" to mean an isomorphism class of graphs. When we draw a graph, its vertices are named by their physical locations, even if we give them no other names. Hence a drawing of a graph is a member of its isomorphism class, and we just call it a graph. When we redraw a graph to display some structural aspect, we have chosen a more convenient member of the isomorphism class, still discussing the same "unlabeled graph". When discussing structure of graphs, it is convenient to have names and notation for important isomorphism classes. We want the flexibility to refer to the isomorphism class or to any representative of it. 1.1.27. Definition. The (unlabeled) path and cycle with n vertices are denoted Pn and Cn, respectively; an n-cycle is a cycle with n vertices. A complete graph is a simple graph whose vertices are pairwise adjacent; the (unlabeled) complete graph with n vertices is denoted Kn. A complete bipartite graph or biclique is a simple bipartite graph such that two vertices are adjacent if and only if they are in different partite sets. When the sets have sizes rands, the (unlabeled) biclique is denoted Kr,s
1.1.28.* Remark. We have defined a complete graph as a graph whose vertices are pairwise adjacent, while a clique is a set of pairwise adjacent vertices in a graph. Many authors use the terms interchangeably, but the distinction allows us to discuss cliques in the same language as independent sets.
10
## Chapter 1: Fundamental Concepts
In the bipartite setting, we simply use "biclique" to abbreviate "complete bipartite graph". The alternative name "biclique" is a reminder that a complete bipartite graph is generally not a complete graph (Exercise 1).
1.1.29. Remark. A graph by any other name . . . When we name a graph without naming its vertices, we often mean its isomorphism class. Technically, "H is a subgraph of G" means that some subgraph of G is isomorphic to H (we say "G contains a copy of H"). Thus Ca is a subgraph of K 5 (every complete graph with 5 vertices has 10 subgraphs isomorphic to Ca) but not of K2,a. Similarly, asking whether G "is" Cn means asking whether G is isomorphic to a cycle with n vertices.
The structural properties of a graph are determined by its adjacency relation and hence are preserved by isomorphism. We can prove that G and ,H are not isomorphic by finding some structural property in which they differ. If they have different number of edges, or different subgraphs, or different complements, etc., then they cannot be isomorphic. On the other hand, checking t:hat a few structural properties are the same does not imply that G ~ H. To prove that G ~ H, we must ptesent a bijection f: V(G) ~ V(H) that preserves the adjacency relation.
1.1.30. Example. Isomorphic or not? Each graph below has six vertices and nine edges and is connected, but these graphs are not pairwise isomorphic. To prove that G 1 ~ G2 , we give names to the vertices, specify a bijection, and check that it preserves the adjacency relation. As labeled below, the bijection that sends u, v, w, x, y, z to 1, 3, 5, 2, 4, 6, respectively, is an isomorphism froi:p. G 1 to G 2 The map sending u, v, w, x, y, z to 6, 4, 2, 1, 3, 5, respectively, is another isomorphism. Both G 1 and G2 are bipartite; they are drawings of Ka.a (as is G4). The graph Ga contains Ka, so its vertices cannot be partitioned into two independent sets. Thus Ga is not isomorphic to the others. Sometimes we can test isomorphism quickly using the complements. Simple graphs G and H are isomorphic if and only if their complements are isomorphic (Exercise 4). Here Gi. G 2, G 4 all consist of two disjoint 3-cycles and are not connected, but Ga is a 6-cycle and is connected.
1.1.31. Example. The number of n-vertex graphs. When choosing two vertices from a set of size n, we can pick one and then the other but don't care about the order, so the number of ways is n(n -1)/2. (The notation for the number of ways
## Section 1.1: What Is a Graph?
11
to choose k elements from n elements is G), read "n choose k". These numbers are called binomial coefficients; see Appendix A for further background.) In a simple graph with a vertex set X of size n, each vertex pair may form an edge or may not. Making the choice for each pair specifies the graph, so the number of simple graphs with vertex set X is 2G). For example, there are 64 simple graphs on a fixed set of four vertices. These graphs form only 11 isomorphism classes. The classes appear below in complementary pairs; only P4 is isomorphic to its complement. Isomorphism classes have different sizes, so we cannot count the isomorphism classes of nvertex simple graphs by dividing 2G> by the size of a class.
::~~X/1
IBl I2l IZ D D: N
DECOMPOSITION AND SPECIAL GRAPHS
The example P4
~
## P 4 suggests a family of graph problems.
1.1.32. Definition. A graph is self-complementary if it is isomorphic to its complement. A decomposition of a graph is a list of subgraphs such that each edge appears in exactly one subgraph in the list.
An n-vertex graph H is self-complementary if and only if Kn has a decomposition consisting of two copies of H.
1.1.33. Example. We can decompose K5 into two 5-cycles, and thus the 5-cycle is self-complementary. Any n-vertex graph and its complement decompose Kn. Also Kl,n-1 and Kn-1.decompose Kn, even though one of these subgraphs omits a vertex. On the right below we show a decomposition of K4 using three copies of P3 Exercises 31-39 consider graph decompositions.
1.1.34.* Example. The question of which complete graphs decompose into copies of K 3 is a fundamental question in the theory of combinatorial designs.
12
## Chapter 1: Fundamental Concepts
On the left below we suggest such a decomposition for K1. Rotating the triangle through seven positions uses each edge exactly once. On the right we suggest a decomposition of K 6 into copies of P4 . Placing one vertex in the center groups the edges into three types: the outer 5-cycle, the inner (crossing) 5-cycle on those vertices, and the edges involving the central vertex. Each 4-vertex path in the decomposition uses one edge of each type; we rotate the picture to get the next path.
We referred to a copy of K3 as a triangle. Short names for graphs that arise frequently in structural discussions can be convenient.
1.1.35. Example. The Graph Menagerie. A catchy "name" for a graph usually comes from some drawing of the graph. We also use such a name for all other drawings, and hence it is best viewed as a name for the isomorphism class. Below we give names to several graphs with at most five vertices. Among these the most important are the triangle (K3 ) and the claw (Kl.3). We also sometimes discuss the paw (Ki. 3 + e)and the kite (K4 - e); the others arise less frequently. The complements of the graphs in the first row are disconnected. The complement of the house is P5 , and the bull is self-complementary. Exercise 39 asks which of these graphs can be used to decompose K6.
triangle
claw
paw
kite
bowtie
dart
In order to decompose H into copies of G, the number of edges of G must divide the number of edges of H. This is not sufficient, since K5 does not decompose into two copies of the kite.
## Section 1.1: What Is a Graph?
13
1.1.36. Definition. The Petersen graph is the simple graph whose vertices are the 2-element subsets of a 5-element set and whose edges are the pairs of disjoint 2-element subsets.
12
We have drawn the Petersen graph in three ways above. It is a useful example so often that an entire book was devoted to it (Holton-Sheehan f1993]). Its properties follow from the statement of its adjacency relation that we have used as the definition.
1.1.37. Example. Structure of the Petersen graph. Using [5] = {l, 2, 3, 4, 5) as our 5-element set, we write the pair {a, b) as ab or ba. Since 12 and 34 are disjoint, they are adjacent vertices when we form the graph, but 12 and 23 are not. For each 2-set ab, there are three ways to pick a 2-set from the remaining three elements of [5], so every ve:rtex has degree 3. The Petersen graph consists of two disjoint 5-cycles plus edges that pair up vertices on the two 5-cycles. The disjointness defir..ition tells us that 12, 34, 51, 23, 45 in order are the vertices of a 5-cycle, and similarly this holds for the remaining vertices 13, 52, 41, 35, 24. Also 13 is adjacent to 45, and 52 is adjacent to 34, and so on, as shown on the left above. We use this name even when we do not specify the vertex labeling; in essence, we use "Petersen graph" to name an isomorphism class. To show that the graphs above are pair;1ise isomorphic, it suffices to name the vertices of each using the 2-element subsets of [5] so that in each case the adjacency relation is disjointness (Exercise 24). I 1.1.38. Proposition. If two vertices are nonadjacent in the Petersen graph, then they have exactly one common neighbor. Proof: Nonadjacent vertices are 2-sets sharing one element; their union S has size 3. A vertex adjacent to both is a 2-set disjoint from both. Since the 2-sets are chosen from {1, 2, 3, 4, 5}, there is exactly one 2-set disjoint from S. I 1.1.39. Definition. The girth of a graph with a cycle is the length of its shortest cycle. A graph with no cycle has infinite girth. i.1.40. Corollary. The Petersen graph has girth 5. Proof: The graph is simple, so it has no 1-cycle or 2-cycle. A 3-cycle would require three pairwise-disjoint 2-sets, which can't occur among 5 elements.
14
## Chapter 1: Fundamental Concepts
A 4-cycle in the absence of 3-cycles would require nonadjacent vertices with two common neighbors, which Proposition 1.1.38 forbids. Finally, the vertices 12, 34, 51, 23, 45 yield a 5-cycle, so the girth is 5. The Petersen graph is highly symmetric. Every permutation of {l, 2, 3, 4, 5} generates a permutation of the 2-subsets that preserves the disjointness relation. Th<Js there are at least 5! = 120 isomorphisms from the Petersen graph to itself. Exercise 43 confirms that there are no others.
1.1.41. * Definition. An automorphism of G is an isomorphism from G to G. A graph G is vertex-transitive iffor every pair u, v E V(G) there is an automorphism that maps u to v.
The automorphisms of G are the permutations of V ( G) that can be applied to both the rows and the columns of A(G) without changing A(G).
1.1.42. * Example. Automorphisms. Let G be the path with vertex set {1, 2, 3, 4} and edge set {12, 23, 34}. This graph has two automorphisms: the identity permutation and the permutation that switches 1 with 4 and switches 2 with 3. Interchanging vertices 1 and 2 is not an automorphism of G, although G is isomorphic to the graph with vertex set {l, 2, 3, 4} and edge set {21, 13, 34}. In Kr,s. permuting the vertices of one partite set does not change the adjacency matrix; this leads to r!s! automorphisms. When r = s, we can also interchange the partite sets; K1,1 has 2(t!) 2 automorphisms. The biclique Kr,s is vertex-transitive if and only if r = s. If n > 2, then Pn is not vertex-transitive, but every cycle is vertex-transitive. The Petersen graph is vertex-transitive.
We can prove a statement for every vertex in a vertex-transitive graph by proving it for one vertex. Vertex-transitivity guarantees that the graph "looks the same" from each vertex.
EXERCISES
Solutions to probk.ns generally require clear explanations written in sentences. The designations on problems have the following meanings: "(-)" = easier or shorter than most, "(+)" = harder or longer than most, ~'(!)"=particularly useful or instructive, "(*)"=involves concepts marked optional in the text. The exercise sections begin with easier problems to check understanding, ending with a line of dots. The remaining problems roughly follow the order of material in the text.
1.1.1. (-) Determine which complete bipartite graphs are complete graphs. 1.1.2. (- ) Write down all possible adjacency matrices and incidence matrices for a 3vertex path. Also write down an adjacency matrix for a path with six vertices and for a cycle with six vertices.
## Section 1.1: What Is a Graph?
15
1.1.3. ( - ) Using rectangular blocks whose entries are all equal, write down an adjacency matrix for Km.11 1.1.4. (-) From the definition of isomorphism, prove that G ~ H if and only ifG ~Ii. 1.1.5. (-) Prove or disprove: If every vertex of a simple graph G has degree 2, then G is a cycle. 1.1.6. (-) Determine whether the graph below decomposes into copies of P4
1.1.7. (-) Prove that a graph with more than six vertices of odd degree cannot be decomposed into three paths. 1.1.8. (-) Prove that the 8-vertex graph on the left below decomposes into copies of K 1. 3 and also into copies of P4
1.1.9. (-) Prove that the graph on the right above is isomorphic to the complement of the graph on the left. 1.1.10. (-) Prove or disprove: The complement of a simple disconnected graph must be connected.
1.1.11. Determine the maximum size of a clique and the maximum size of an independent set in the graph below.
1.1.12. Determine whether the Petersen graph is bipartite, and find the size of its largest independent set. 1.1.13. Let G be the graph whose vertex set is the set of k-tuples with coordinates in {O, l}, with x adjacent toy when x and y differ in exactly one position. Determine whether G is bipartite. 1.1.14. (!) Prove that removing opposite comer squares from an 8-by-8 checkerboard leaves a subboard that cannot be partitioned into 1-by-2 and 2-by-1 rectangles. Using the same argument, make a general statement about all bipartite graphs.
16
## Chapter 1: Fundamental Concepts
1.1.15. Consider the following four families of graphs: A = {paths}, B = {cycles}, C = {complete graphs), D = {bipartite graphs). For each pair of these families, determine all isomorphism classes of graphs that belong to both families. 1.1.16. Determine whether the graphs below are isomorphic.
1.1.17. Determine the number of isomorphism classes of simple 7-vertex graphs in which every vertex has degree 4. 1.1.18. Determine which pairs of graphs below are isomorphic.
:: a~b <~W
d~c :~:
' '
## 1.1.20. Determine which pairs of graphs below are isomorphic.
1.1.21. Determine whether the graphs below are bipartite and whether they are isom:>rphic. (The graph on the left appears on the cover of Wilson-Watkins [1990].)
## Section 1.1: What Is a Graph?
17
1.1.22. (!) Determine which pairs of graphs below are isomorphic, presenting the proof by testing the smallest possible number of pairs.
1.1.23. In each class below, determine the smallest n such that there exist nonisomorphic n-vertex graphs having the same list of vertex degrees. (a) all graphs, (b) loopless graphs, (c) simple graphs.
(Hint: Since each class contains the next, the answers form a nondecreasing triple. For part (c), use the list of isomorphism classes in Example 1.1.31.) 1.1.24. (!) Prove that the graphs below are all drawings of the Petersen graph (Definition 1.1.36). (Hint: Use the disjointness definition of adjacency.)
1.1.25. (!)Prove that the Petersen graph has no cycle oflength 7. 1.1.26. (!)Let G be a graph with girth 4 in which every vertex has degree k. Prove that G has at least 2k vertices. Determine all such graphs with exactly 2k vertices. 1.1.27. (!) Let G be a graph with girth 5. Prove that if every vertex of G has degree at least k, then G has at least k2 + 1 vertices. Fork = 2 and k = 3, find one such graph with exactly k 2 + 1 vertices. 1.1.28. (+) The Odd Graph Ok. The vertices of the graph Ok are the k-element subsets of {l, 2, ... , 2k + l}. Two vertices are adjacent if and only if they are disjoint sets. Thus 0 2 is the Petersen graph. Prove that the girth of Ok is 6 if k :::: 3. 1.1.29. Prove that every set of six people contains (at least) three mutual acquaintances or three mutual strangers. 1.1.30. Let G be a simple graph with adjacency matrix A and incidence matrix M. Prove that the degree of v; is the ith diagonal entry in A 2 and in M MT. What do the entries in position (i, j) of A 2 and M MT say about G? 1.1.31. (!) Prove that a self-complementary graph with n vertices exists if and only if n or n - 1 is divisible by 4. (Hint: When n is divisible by 4, generalize the structure of P4 by splitting the vertices into four groups. For n = 1 mod 4, add one vertex to the graph constructed for n - 1.) 1.1.32. Determine which bicliques decompose into two isomorphic subgraphs. 1.1.33. For n = 5, n = 7, and n = 9, decompose Kn into copies of Cn.
18
## Chapter 1: Fundamental Concepts
1.1.34. (!) Decompose the Petersen graph into three connected subgraphs that are pairwise isomorphic. Also decompose it into copies of P4 1.1.35. (!) Prove that Kn decomposes into three pairwise-isomorphic subgraphs if and only if n + 1 is not divisible by 3. (Hint: For the case where n is divisible by 3, split the vertices into three sets of equal size.) 1.1.36. Prove that if Kn decomposes into triangles, then n - 1 or n - 3 is divisible by 6. 1.1.37. Let G be a graph in which every vertex has degree 3. Prove that G has no decomposition into paths that each have at least 5 vertices. 1.1.38. (!) Let G be a simple graph in which every vertex has degree 3. Prove that G decomposes into claws if and only if G is bipartite. 1.1.39. (+) Determine which of the graphs in Example 1.1.35 can be used to form a decomposition of K 6 into pairwise-isomorphic subgraphs. (Hint: Each graph that is not excluded by some divisibility condition works.) 1.1.40. (*)Count the automorphisms of Pn, Cn, and Kn. 1.1.41. (*) Construct a simple graph with six vertices that has only one automorphism. Construct a simple graph that has exactly three automorphisms. (Hint: Think of a rotating triangle with appendages to prevent flips.) 1.1.42. (*) Verify that the set of automorphisms of G has the following properties: a) The composition of two automorphisms is an automorphism. b) The identity permutation is an automorphism. c) The inverse of an automorphism is also an automorphism. d) Composition of automorphisms satisfies the associative property. (Comment: Thus the set of automorphisms satisfies the defining properties for a group.) 1.1.43. (*) Automorphisms of the Petersen graph. Consider the Petersen graph as defined by disjointness of 2-sets in {l, 2, 3, 4, 5}. Prove that every automorphism maps the 5-cycle with vertices 12, 34, 51, 23, 45 to a 5-cycle with vertices ab, cd, ea, be, de determined by a permutation of {l, 2, 3, 4, 5} taking elements 1,2,3,4,5 to a, b, c, d, e, respectively. (Comment: This implies that there are only 120 automorphisms.) 1.1.44. (*)The Petersen graph has even more symmetry than vertex-transitivity. Let P = (u 0 , ui. u2 , u 3 ) and Q = (v0 , v1 , v2 , v3 ) be paths with three edges in the Petersen graph. Prove that there is exactly one automorphism of the Petersen graph that maps u; into v; for i = 0, 1, 2, 3. (Hint: Use the disjointness description.) 1.1.45. (*) Construct a graph with 12 vertices in which every vertex has degree 3 and the only automorphism is the identity. 1.1.46. (*)Edge-transitivity. A graph G is edge-transitive iffor all e, f E E(G) there is an automorphism of G that maps the endpoints of e to the endpoints off (in either order). Prove that the graphs of Exercise 1.1.21 are vertex-transitive and edge-transitive. (Comment: Complete graphs, bicliques, and the Petersen graph are edge-transitive.) 1.1.47. (*)Edge-transitive versus vertex-transitive. a} Let G be obtained from Kn with n ::=:: 4 by replacing each edge of Kn with a path of two edges through a new vertex of degree 2. Prove that G is edge-transitive but not vertex-transitive. b) Suppose that G is edge-transitive but not vertex-transitive and has no vertices of degree 0. Prove that G is bipartite. c) Prove that the graph in Exercise 1.1.6 is vertex-transitive but not edge-transitive.
19
## 1.2. Paths, Cycles, and Trails
In this section we return to the Konigsberg Bridge Problem, determining when it is possible to traverse all the edges of a graph. We also we develop useful properties of connection, paths, and cycles. Before embarking on this, we review an important technique of proof Many statements in graph theory can be proved using the principle of induction. Readers unfamiliar with induction should read the material on this proof technique in Appendix A. Here we describe the form of induction that we will use most frequently, in order to familiarize the reader with a template for proof.
1.2.1. Theorem. (Strong Principle oflnduction). Let P(n) be a statement with an integer parameter n. If the following two conditions hold, then P(n) is true for each positive integer n. 1) P(l) is true. 2) For all n > 1, "P(k) is true for 1 ~ k < n" implies "P(n) is true". Proof: We ASSUME the Well Ordering Property for the positive integers: every nonempty set of positive integers has a least element. Given this, suppose that P(n) fails for some n. By the Well Ordering Property, there is a least n such that P(n) fails. Statement (1) ensures that this value cannot be 1. Statement (2) ensures that this value cannot be greater than 1. The contradiction implies that P(n) holds for every positive integer n.
In order to apply induction, we verify (1) and (2) for our sequence of statements. Verifying (1) is the basis step of the proof; verifying (2) is the induction step. The statement "P(k) is true for all k < n" is the induction hypothesis, because it is the hypothesis of the implication proved in the induction step. The variable that indexes the sequence of statements is the induction parameter. The induction parameter may be any integer function of the instances of our problem, such as the number of vertices or edges in a graph. We say that we are using "induction on n~ when the induction pa:t;'ameter is n. There are many ways to phrase inductive proofs. We can start at 0 to prove a statement for nonnegative integers. When our proof of P(n) in the induction step makes use only of P (n - 1) from the induction hypothesis, the technique is called "ordinary" induction; making use of all previous statements is "strong" induction. We seldom distinguish between strong induction and ordinary induction; they are equivalent (see Appendix A). Most students first learn ordinary induction in the following phrasing: 1) verify that P(n) is true when n = 1, and 2) prove that if P(n) is true when n is k, then P(n) is also true when n is k + 1. Proving P(k + 1) from P(k) fork ::=: 1 is equivalent to proving P(n) from P(n - 1) for n > 1.
20
## Chapter 1: Fundamental Concepts
When we focus on proving the statement for the parameter value n in the induction step, we need not decide at the outset whether we are using strong induction or ordinary induction. The language is also simpler, since we avoid introducing a new name for the parameter. In Section 1.3 we will explain why this phrasing is also less prone to error.
CONNECTION IN GRAPHS
As defined in Definition 1.1.15, paths and cycles are graphs; a path in a graph G is a subgraph of G that is a path (similarly for cycles). We introduce further definitions to model other movements in graphs. A tourist wandering in a city (or a Konigsberg pedestrian) may want to allow vertex repetitions but avoid edge repetitions. A mail carrier delivers mail to houses on both sides of the street and hence traverses each edge twice. 1.2.2. Definition. A walk is a list vo, ei, vi, ... , ek> vk of vertices and edges such that, for 1 :=: i s k, the edge ei has endpoints Vi-1 and vi. A trail is a walk with no repeated edge. Au, v-walk or u, v-trail has first vertex u and last vertex v; these are its endpoints. Au, v-path is a path whose vertices of degree 1 (its endpoints) are u and v; the others are internal vertices. The length of a walk, trail, path, or cycle is its number of edges. A walk or trail is closed if its endpoints are the same. 1.2.3. Example. In theKonigsberggraph(Example 1.1.1), thelistx, e 2 , w, e 5 , y, a closed walk oflength 5; it repeats edge e2 and hence is not a trail. Deleting the last edge and vertex yields a trail of length 4; it repeats vertices but not edges. The subgraph consisting of edges e1 , e5 , e6 and vertices x, w, y is a cycle oflength 3; deleting one of its edges yields a path. Two edges with the same endpoints (such as e 1 and e2 ) form a cycle of length 2. A loop is a cycle oflength 1.
e6, x, e 1, w, e 2 , xis
The reason for listing the edges in a walk is to distinguish among multiple edges when a graph is not simple. In a simple graph, a walk (or trail) is completely specified by its ordered list of vertices. We usually name a path, cycle, trail, or walk in a simple graph by listing only its vertices in order, even though it consists of both vertices and edges. When discussing a cycle, we can start at any vertex and do not repeat the first vertex at the end. We can use parentheses to clarify that this is a cycle and not a path. 1.2.4. Example. We illustrate the simplified notation in a simple graph. In the graph below, a, x, a, x, u, y, c, d, y, v, x, b, a specifies a closed walk oflength 12. Omitting the first two steps yields a closed trail. The graph has five cycles: (a, b, x), (c, y, d), (u, x, y), (x, y, v), (u, x, v, y). The u, v-trail u, y, c, d, y, x, v contains the edges of the u, v-path u, y, x, v, but not of the u, v-path u, y, v.
## Section 1.2: Paths, Cycles, and Trails
a
21
u
Suppose we follow a path from u to v in a graph and then follow a path from v tow. The result need not beau, w-path, because the u, v-path and v, wpath may have a common internal vertex. Nevertheless, the list of vertices and edges that we visit does form au, w-walk. In the illustration below, the u, wwalk contains au, w-path. Saying that a walk W contains a path P means that the vertices and edges of P occur as a sublist of the vertices and edges of W, in order but not necessarily consecutive.
1.2.5. Lemma. Every u, v-walk contains au, v-path. Proof: We prove the statement by induction on the length l of au, v-walk W. Basis step: l = 0. Having no edge, W consists of a single vertex (u = v). This vertex is a u, v-path oflength 0. Induction step: l :::: 1. We suppose that the claim holds for walks oflength less than l. If W has no repeated vertex, then its vertices and edges form a u, v-path. If W has a repeated vertex w, then deleting the edges and vertices between appearances of w (leaving one copy of w) yields a shorter u, v-walk W' contained in W. By the induction hypothesis, W' contains a u, v-path P, and this path P is contained in W.
uo .... o..
O......~p
p:
Exercise 13b develops a shorter proof. We apply the lemma to properties of connection.
1.2.6. Definition. A graph G is connected if it has a u, v-path whenever u, v E V(G) (otherwise, G is disconnected). If G has au, v-path, then u is connected to v in G. The connection relation on V ( G) consists of the ordered pairs (u, v) such that u is connected to v.
22
## Chapter 1: Fundamental Concepts
"Connected" is an adjective we apply only to graphs and to pairs of vertices (we never say "vis disconnected" when vis a vertex). The phrase "u is connected to v" is convenient when writing proofs, but in adopting it we must clarify the distinction between connection and adjacency: G has au, v-path
u and v are connected u is connected to v
uv
E(G)
## u and v are adjacent u is joined to v u is adjacent to v
1.2.7. Remark. By Lemma 1.2.5, we can prove that a graph is connected by showing that from each vertex there is a walk to one particular vertex. By Lemma 1.2.5, the connection relation is transitive: if G has a u, v-path and av, w-path, then G has au, w-path. It is also reflexive (paths oflength 0) and symmetric (paths are reversible), so it is an equivalence relation.
Our next definition leads us to the equivalence classes of the connection relation. A maximal connected subgraph of G is a subgraph that is connected and is not contained in any other connected subgraph of G.
1.2.8. Definition. The components of a graph G are its maximal connected subgraphs. A component (or graph) is trivial if it has no edges; otherwise it is nontrivial. An isolated vertex is a vertex of degree 0.
The equivalence classes of the connection relation on V ( G) are the vertex sets of the components of G. An isolated vertex forms a trivial component, consisting of one vertex and no edge.
1.2.9. Example. The graph below has four components, one being an isolated vertex. The vertex sets of the components are {p}, {q, r}, {s, t, u, v, w}, and {x, y, z}; these are the equivalence classes of the connection relation.
r s u
t
\W:
q p
1.2.10. Remark. Components are pairwise disjoint; no two share a vertex. Adding an edge with endpoints in distinct components combines them into one component. Thus adding an edge decreases the number of components by 0 or 1, and deleting an edge increases the number of components by 0 or 1. 1.2.11. Proposition. Every graph with n vertices and k edges has at least n -k components. Proof: Ann-vertex graph with no edges has n components.. By Remark 1.2.10, each edge added reduces this by at most 1; so when k edges have been added the number of components is still at least n - k.
## Section 1.2: Paths, Cycles, and Trails
23
Deleting a vertex or an edge can increase the number of components. Although deleting an edge can only increase the number of components by 1, deleting a vertex can increase it by many (consider the biclique Ki.in). When we obtain a subgraph by deleting a vertex, it must be a graph, so deleting the vertex also deletes all edges incident to it.
1.2.12. Definition. A cut-edge or cut-vertex of a graph is an edge or vertex whose deletion increases the number of components. We write G - e or G - M for the subgraph of G obtained by deleting an edge e or set of edges M. We write G - v or G - S for the subgraph obtained by deleting a vertex v or set of vertices S. An induced subgraph is a subgraph obtained by deleting a set of vertices. We write G[T] for G - T, where T = V(G) - T; this is the subgraph of G induced by T.
When T ~ V(G), the induced subgraph G[T] consists of T and all edges whose endpoints are contained in T. The full graph is itself an induced subgraph, as are individual vertices. A set S of vertices is an independent set if and only if the subgraph induced by it has no edges.
1.2.13. Example. The graph of Example 1.2.9 has cut-vertices v and y. Its cutedges are qr, vw, xy, and yz. (When we delete an edge, its endpoints remain.) This graph has C4 and P5 as subgraphs but not as induced subgraphs. The subgraph induced by {s, t, u, v} is a kite; the 4-vertex paths on these vertices are not induced subgraphs. The graph P4 does occur as an induced subgraph; it is the subgraph induced by {s, t, v, w} (also by {s, u, v, w}).
Next we characterize cut-edges in terms of cycles.
1.2.14. Theorem. An edge is a cut-edge if and only if it belongs to no cycle. Proof: Let e be an edge in a graph G (with endpoints x, y ), and let H be the component containing e. Since deletion of e affects no other component, it suffices to prove that H - e is connected if and only if e belongs to a cycle. First suppose that H - e is connected. This implies that H - e contains an x, y-path, and this path completes a cycle withe. Now suppose that e lies in a cycle C. Choose u, v E V(H). Since H is connected, H has au, v-path P. If P does not contain e, than P exists in H - e. If P contains e, suppose by symmetry that x is between u and y on P. Since H - e contains au, x-path along P, an x, y-path along C, and a y, v-path along P, the transitivity of the connection relation implies that H - e has au. v-path. We did this for all u, v E V(H), so H - e is connected.
c
Ue
24
## Chapter 1: Fundamental Concepts
BIPARTITE GRAPHS
Our next goal is to characterize bipartite graphs using cycles. Characterizations are equivalence statements, like Theorem 1.2.14. When two conditions are equivalent, checking one also yields the other for free. Characterizing a class G by a condition P means proving the equivalence "G E G if and only if G satisfies P". In other words, Pis both a necessary and a sufficient condition for membership in G. Necessity
G E G only if G satisfies P G E G => G satisfies P
Sufficiency
G E G if G satisfies P G satisfies P => G E G
Recall that a loop is a cycle of length 1; also two distinct edges with the same endpoints form a cycle of length 2. A walk is odd or even as its length is odd or even. As in Lemma 1.2.5, a closed walk contains a cycle C if the vertices and edges of C occur as a sublist of W, in cyclic order but not necessarily consecutive. We can think of a closed walk or a cycle as starting at any vertex; the next lemma requires this viewpoint. 1.2.15. Lemma. Every closed odd walk contains an odd cycle. Proof: We use induction on the length l of a closed odd walk W. Basis step: l = 1. A closed walk oflength 1 traverses a cycle oflength 1. Induction step: l > 1. Assume the claim for closed odd walks shorter than W. If W has no repeated vertex (other than first = last), then W itself forms a cycle of odd length. If vertex v is repeated in W, then we view W as starting at v and break W into two v, v-walks. Since W has odd length, one of these is odd and the other is even. The odd one is shorter than W. By the induction hypothesis, it contains an odd cycle, and this cycle appears in order in W.
even
1.2.16. Remark. A closed even walk need not contain a cycle; it may simply repeat. Nevertheless, if an edge e appears exactly once. in a closed walk W, then W does contain a cycle through e. Let x, y be the endpoints of e. Deleting e from W leaves an x, y-walk that avoids e. By Lemma 1.2.5, this walk contains an x, y-path, and this path completes a cycle withe. (See Exercises 15-16.) Lemma 1.2.15 will help us characterize bipartite graphs. 1.2.17. Definition. A bipartition of G is a specification of two disjoint independent sets in G whose union is V ( G). The statement "Let G be a bipartite graph with bipartition X, Y" specifies one such partition. An X, Y-bigraph is a bipartite graph with bipartition X, Y.
## Section 1.2: Paths, Cycles, and Trails
25
The sets of a bipartition are partite sets (Definition 1.1.10). A disconnected bipartite graph has more than one bipartition. A connected bipartite graph has only one bipartition, except for interchanging the two sets (Exercise 7).
1.2.18. Theorem. (Konig [1936]) A graph is bipartite if and only if it has no odd cycle. Proof: Necessity. Let G be a bipartite graph. Every walk alternates between the two sets of a bipartition, so every return to the original partite set happens after an even number of steps. Hence G has no odd cycle. Sufficiency. Let G be a graph with no odd cycle. We prove that G is bipartite by constructing a bipartition of each nontrivial component. Let u be a vertex in a nontrivial component H. For each v E V(H), let f(v) be the minimum length ofa u, v-path. Since His connected, f(v) is defined for each v E V(H). Let X = {v E V(H): f(v) is even} and Y = {v E V(H): f(v) is odd}. An edge v, v' within X or Y would create a closed odd walk using a shortest u, vpath, the edge vv', and the reverse of a shortest u, v' -path. By Lemma 1.2.15, such a walk must contain an odd cycle, which contradicts our hypothesis. Hence X and Y are independent sets. Also XU Y = V(H), so His an X, Y-bigraph.
1.2.19. Remark. Testing whether a graph is bipartite. Theorem 1.2.18 implies that whenever a graph G is not bipartite, we can prove this statement by presenting an odd cycle in G. This is much easier than examining all possible bipartitions to prove that none work. When we want to prove that G is bipartite, we define a bipartition and prove that the two sets are independent; this is easier than examining all cycles.
We consider one application.
1.2.20. Definition. The union of graphs Gi, ... , Gk. written Gi U U Gk. is k k . the graph with vertex set LJi=l V(G;) and edge set LJ;= 1E(G;). 1.2.21. Example. Below we show K4 as the union of two 4-cycles. When a graph G is expressed as the union of two or more subgraphs, an edge of G can belong to many of them. This distinguishes union from decomposition, where each edge belongs to only one subgraph in the list.
Oli':?fll
00
10
26
## Chapter 1: Fundamental Concepts
1.2.22. Example. Consider an air traffic system with k airlines. Suppose that 1) direct service between two cities means round-trip direct service, and 2) each pair of cities has direct service from at least one airline. Suppose also that no airline can schedule a cycle through an odd number of cities. In terms of k, what is the maximum number of cities in the system? By Theorem 1.2.18, we seek the largest n such that Kn can be expressed as the union of k bipartite graphs, one for each airline. The answer is 2k. 1.2.23. Theorem. The complete graph Kn can be expressed as the union of k bipartite graphs if and only if n :'.S 2k. Proof: We use induction on k. Basis step: k = 1. Since Ka has an odd cycle and K 2 does not, Kn is itself a bipartite graph if and only if n :'.S 2. Induction step: k > 1. We prove each implication using the induction hypothesis. Suppose first that Kn = G1 U U Gb where each G; is bipartite. We partition the vertex set into two sets X, Y such that Gk has no edge within X or within Y. The union of the other k - 1 bipartite subgraphs must cover the complete subgraphs induced by X and by Y. Applying the induction hypothesis to each yields IXI :'.S 2k-l and IYI :'.S 2k-l, son :'.S 2k-l + 2k-l = 2k. Conversely, suppose that n :'.S 2k. We partition the vertex set into subsets X, Y, each of size at most 2k-l. By the induction hypothesis, we can cover the complete subgraph induced by either subset with k - 1 bipartite subgraphs. The union of the ith such subgraph on X with the ith such subgraph on Y is a bipartite graph. Hence we obtain k - 1 bipartite graphs whose unior! consists of the complete subgraphs induced by X and Y. The remaining edges are those of the biclique with bipartition X, Y. Letting this be the kth bipartite subgraph completes the construction. This theorem can also be proved without induction by encoding the vertices as binary k-tuples (Exercise 31).
EULERIAN CIRCUITS
We return to our analysis of the Konigsberg Bridge Problem. What the people of Konigsberg wanted was a closed trail containing all the edges in a graph. As we have observed, a necessary condition for existence of such a trail .is that all vertex degrees be ev.en. Also it is necessary that all edges belong to the same component of the graph. The Swiss mathematician Leonhard Euler (pronounced "oiler") stated [1736) that these conditions are also sufficient. In honor of his contribution, we associate his name with such graphs. Euler's paper appeared in 1741 but gave no proof that the obvious necessary conditions are sufficient. Hierholzer [1873) gave the first complete published proof. The graph we drew in Example 1.1.1 to model the city did not appear in print until 1894 (see Wilson [1986) for a discussion of the historical record).
## Section 1.2: Paths, Cycles, and Trails
1~2.24.
27
Definition. A graph is Eulerian if it has a closed trail containing all edges. We call a closed trail a circuit when we do not specify the first vertex but keep the list in cyclic order. An Eulerian circuit or Eulerian trail in a graph is a circuit or trail containing all the edges. An even graph is a graph with vertex degrees all even. A vertex is odd [even] when its degree is odd [even].
Our discussion of Eulerian circuits applies also to graphs with loops; we extend the notion of vertex degree to graphs with loops by letting each loop contribute 2 to the degree of its vertex. This does not change the parity of the degree, and the presence of a loop does not affect whether a graph has an Eulerian circuit unless it is a loop in a component with one vertex. Our proof of the characterization of Eul~rian graphs uses a lemma. A maximal path in a graph G is a path P in G that is not contained in a longer path. When a graph is finite, no path can extend forever, so maximal (nonextendible) paths exist.
1.2.25. L~mma. If every vertex of a graph G has degree at least 2, then G contains a cycle. Proof: Let P be a maximal path in G, and let u be an endpoint of P. Since P cannot be extended, every neighbor of u must already be a vertex of P. Since u has degree at least 2, it has a neighbor v in V(P) via an edge not in P. The edge uv completes a cycle with the portion of P from v to u.
.~. u
+-=- v
Notetheimportanceoffiniteness. IfV(G) = ZandE(G) = {ij: Ii - j l = 1}, then every vertex of G has degree 2, but G has no cycle (and no non-extendible path). We avoid such examples by assuming that all graphs in this book are finite, with rare explicit exceptions.
1.2.26. Theorem. A graph G is Eulerian if and on,ly if it has at most one nontrivial component and its vertices all have even degree. Proof: Necessity. Suppose that G has an Eulerian circuit C. Each passage of C through a vertex uses two inddent edges, and the first edge is paired with the last at the first vertex. Hence every vertex has even degree. Also, two edges can be in the same trail only when they lie in the same component, so there is at most one nontrivial component. Sufficiency. Assuming that the condition holds, we obtain an Eulerian circuit using induction on the number of edges, m. Basis step: m = 0. A closed trail consisting of one vertex suffices. Induction step: m > 0. With even degrees, each vertex in the nontrivial component of G has degree at least 2. By Lemma 1.2.25, the nontrivial component has a cycle C. Let G' be the graph obtained from G by deleting E(C).
28
## Chapter 1: Fundamental Concepts
Since C has 0 or 2 edges at each vertex, each component of G' is also an even graph. Since each cQmponent also is connected and has fewer than m edges, we can apply the induction hypothesis to conclude that each component of G' has an Eulerian circuit. To combine these into an Eulerian circuit of G, we traverse C, but when a component of G' is entered for the first time we detour along an Eulerian circuit of that component. This circuit ends at the vertex where we began the detour. When we complete the traversal of C, we have completed an Eulerian circuit of G.
Perhaps as important as the characterization of Eulerian graphs is what the method of proof says about even graphs.
1.2.27. Proposition. Every even graph decomposes into cycles. Proof: In the proof of Theorem 1.2.26, we noted that every even nontrivial graph has a cycle, and that the deletion of a cycle leaves an even graph. Thus this proposition follows by Induction on the number of edges.
In the characterization of Eulerian circuits, the necessity of the condition is easy to see. This also holds for the characterization of bipartite graphs by absence of odd cycles and for many other characterizations. Nash-Williams and others popularized a mnemonic for such theorems: TONCAS, meaning "The Obvious Necessary Conditions are Also Sufficient''. The proof of Lemma 1.2.25 is an example of an important technique of proof in graph theory that we call extremality. When considering structures of a given type, choosing an example that is extreme in some sense may yield useful additional information. For example, since a maximal path P cannot be extended, we obtain the extra information that every neighbor of an endpoint of P belongs to V(P). In a sense, making an extremal choice goes directly to the important case. In Lemma 1.2.25, we could start with any path. Ifit is extendible, then we extend it. If not, then something important happens. We illustrate the technique with several examples, and Exercises 37-42 also use extremality. We begin by strengthening Lemma 1.2.25 for simple graphs.
1.2.28. Proposition. If G is a simple graph in which every vertex has degree at least k, then G contains a path oflength at least k. If k :::: 2, then G also contains a cycle oflength at least k + 1. Proof: Let u be an endpoint of a maximal path P in G. Since P does not extend, every neighbor of u is in V (P). Since u has at least k neighbors and G is simple,
## Section 1.2: Paths, Cycles, and Trails
29
P therefore has at least k vertices other than u and has length at least k. If k :=:: 2, then the edge from u to its farthest neighbor v along P completes a
## sufficiently long cycle with the portion of P from v to u.
.~.
:;-:---...._
v
1.2.29. Proposition. Every graph with a nonloop edge has at least two vertices that are not cut-vertices. Proof: If u is an endpoint of a maximal path P in G, then the neighbors of u lie on P. Since P - u is connected in G - u, the neighbors of u belong to a single component of G - u, and u is not a cut-vertex. 1.2.30. Remark. Note the difference between "maximal" and "maximum". As adjectives, maximum means "maximum-sized", and maximal means "no larger one contains this one". Every maximum path is a maximal path, but maximal paths need not have maximum length. Similarly, the biclique Kr.s has two maximal independent sets, but when r i= s it has only one maximum independent set. When describing numbers rather than containment, the meanings are the same; maximum vertex degree = maximal vertex degree. Besides maximal or maximum paths or independent sets, other extremal aspects include vertices of minimum or maximum degree, the first vertex where two paths diverge, maximal connected subgraphs (components), etc. In a connected graph G with disjoint sets S, Tc V(G), we can obtain a path from S to T having only its endpoints in Su T by choosing a shortest path from S to T; Exercise 40 applies this. Exercise 37 uses extremality for a short proof of the transitivity of the connection relation.
Many prpofs using induction can be phrased using extremality, and many proofs using extremality can b~ done by induction. To underscore the interplay, we reprove the characterization of Eulerian graphs using extremality directly.
1.2.31. Lemma. In an even graph, every maximal trail is closed. Proof: Let T be a maximal trail in an even graph. Every passage of T through a vertex v uses two edges at v, none repeated. T,h.us when arriving at a vertex v other than its initial vertex, T has used an odd number of edges incident to v. Since v has even degree, there remains an edge on which T can continue. Hence T can only end at its initial vertex. In a finite graph, T must indeed end. We conclude that a maximal trail must be closed. 1.2.32. Theorem 1.2.26-Second Proof. We prove TONCAS. In a graph G satisfying the conditions, let T be a trail of maximum length; T must also be a maximal trail. By Lemma 1.2.31, T is closed. Suppose that T omits some edge e of G. Since G has only one nontrivial component, G has a shortest path from e to the vertex set of I'. Hence some edge e' not in T is incident to some vertex v of T.
30
## Chapter 1: Fundamental Concepts
Since Tis closed, thereis a trail T' that starts and ends at v and uses the same edges as T. We now extend T' along e' to obtain a longer trail than T. This contradicts the choice of T, and hence T traverses all edges of G. This proof and the resulting construction procedure (Exercise 12) are similar to those of Hierholzer [1873]. Exercise 35 develops another proof. Later chapters contain several applications of the statement that every connected even graph has an Eulerian circuit. Here we give a simple one. When drawing a figure G on paper, how many times must we stop and move the pen? We are not allowed to repeat segments of the drawing, so each visit to the paper contributes a trail. Thus we seek a decomposition of G into the minimum number of trails. We may reduce the problem to connected graphs, since the number of trails needed to draw G is the sum of the number needed to draw each component. For example, the graph G below has four odd vertices and decomposes into two trails. Adding the dashed edges on the right makes it Eulerian.
Ge
[QJG'
1.2.33. Theqrem. For a connected nontrivial graph with exactly 2k odd vertices, the minimum number of trails that decompose it is max{k, l}. Proof: A trail contributes even degree to every vertex, except that a non-clo.sed trail contributes odd degree to its endpoints. Therefore, a partition of the edges into trails must have some non-closed trail ending at each odd vertex. Since each trail has only two ends, 'we must use at least k trails to satisfy 2k odd vertices. We also need at least one. trail since G has an edge, and Theorem 1.2.26 implies that one trail suffices when k = 0. It remains to prove that k trails suffice when k > 0. Given such a graph G, we pair up the odd vertices in G (in any way) and form G' by adding for each pair an edge joining its two vertices, as illustrated above. The resulting graph G' is connected and even, so by Theorem 1.2.26 it has an Eulerian circuit C. As we traverse C in G', we start a new trail in G each time we traverse an edge of G' - E(G). This yields k trails decomposing G.
We prove theorems in general contexts to avoid work. The proof of Theorem 1.2.33 illustrates this; by transforming G into a graph where Theorem 1.2.26 applies, we avoid repeating the basic argument of Theorem 1.2.26. Exercise 33 requests a proofofTheorem 1.2.33 dil'ectly by induction. Note that Theorem 1.2.33 considers only graphs having an even number of vertices of odd degree. Our first result in the next section explains why.
## Section i.2: Paths, Cycles, and Trails
31
EXERCISES
Most problems in this book require proofs. Words like "construct", "show", "obtain", "determine", etc., explicitly state that proof is required. Disproof by providing a counterexample requires confirming that it is a counterexample. 1.2.1. ( - ) Determine whether the statements below are true or false. a) Every disconnected graph has an isolated vertex. b) A graph is connected if and only if some vertex is connected to all other vertices. c) The edge set of every closed trail can be partitioned into edge sets of cycles. d) If a maximal trail in a graph is not closed, then its endpoints have odd degree. 1.2.2. (-) Determine whether K4 contains the following (give an example or a proof of non-existence). a) A walk that is not a trail. b) A trail that is not closed and is not a path. c) A closed trail that is not a cycle. 1.2.3. (-)Let G be the graph with vertex set {l, ... , 15} in which i and j are adjacent if and only if their greatest common factor exceeds 1. Count the components of G and determine the maximum length of a path in G. 1.2.4. (-)Let G be a graph. For v E V(G) and e E E(G), describe the adjacency and incidence matrices of G - v and G - e in terms of the corresponding matrices for G. 1.2.5. (- ) Let v be a vertex of a connected simple graph G. Prove that v has a neighbor in every component of G - v. Conclude that no graph has a cut-vertex of degree 1. 1.2.6. (-) In the graph below (the paw), find all the maximal paths, maximal cliques, and maximal independent sets. Also find all the maximum paths, maximum cliques, and maximum independent sets.
1.2.7. (-) Prove that a bipartite graph has a unique bipartition (except for interchanging the two partite sets) if and only if it is connected. 1.2.8. (-) Determine the values of m and n such that Km.n is Eulerian. 1.2.9. (- ) What is the minimum number of trails needed to decompose the Petersen graph? Is there a decomposition into this many trails using only paths? 1.2.10. (- ) Prove or disprove: a) Every Eulerian bipartite graph has an even number of edges. b) Every Eulerian simple graph with an even number of vertices has an even number of edges. 1.2.11. (- ) Prove or disprove: If G is an Eulerian graph with edges e, f that share a vertex, then G has an Eulerian circuit in which e, f appear consecutively. 1.2.12. (- ) Convert the proof at 1.2.32 to an procedure for finding an Eulerian circuit in a connected even graph.
32
## Chapter 1: Fundamental Concepts
1.2.13. Alternative proofs that every u, v-walk contains au, v-path (Lemma 1.2.5). a) (ordinary induction) Given that every walk of length I - 1 contains a path from its first vertex to its last, prove that every walk of length I also satisfies this. b) (extremality) Given au, v-walk W, consider a shortest u, v-walk contained in W. 1.2.14. Prove or disprove the following statements about simple graphs. (Comment: "Distinct" does not mean "disjoint".) a) The union of the edge sets of distinct u, v-walks must contain a cycle. b) The union of the edge sets of distinct u, v-paths must contain a cycle. 1.2.15. (!) Let W be a closed walk of length at least 1 that does not contain a cycle. Prove that some edge of W repeats immediately (once in each direction). 1.2.16. Let e be an edge appearing an odd number of times in a closed walk W. Prove that W contains the edges of a cycle through e. 1.2.17. (!) Let G,, be the graph whose vertices are the permutations of {1, ... , n), with two permutations ai, ... , a,, and bi. ... , b,, adjacent if they differ by interchanging a pair of adjacent entries (G 3 shown below). Prove that G,, is connected.
132~231
312 321
123
213
1.2.18. (!) Let G be the graph whose vertex set is the set of k-tuples with elements in {O, 1), with x adjacent to y if x and y differ in exactly two positions. Determine the number of components of G. 1.2..19. Let r and s be natural numbers. Let G be the simple graph with vertex set v0 , , v,,_ 1 such that vi *+ Vj if and only if jj - i I E {r, s ). Prove that S has exactly k components, where k is the greatest common divisor of {n, r, s ). 1.2.20. (!) Let v be a cut-vertex of a simple graph G. Prove that G - v is connected. 1.2.21. Let G be a self-complementary graph. Prove that G has a cut-vertex if and only if G has a vertex of degree 1. (Akiyama-Harary [1981]) 1.2.22. Prove that a graph is connected if and only if for every partition of its vertices into two nonempty sets, there is an edge with endpoints in both sets. 1.2.23. For each statement below, determine whether it is true for every connected simple graph G that is not a complete graph. a) Every vertex of G belongs to an induced subgraph isomorphic to P3 b) Every edge of G belongs to an induced subgraph isomorphic to P3 1.2.24. Let G be a simple graph having no isolated vertex and no induced subgraph with exactly two edges. Prove that G is a complete graph. 1.2.25. (!) Use ordinary induction on the number of edges to prove that absence of odd cycles is a sufficient condition for a graph to be bipartite. 1.2.26. (!) Prove that a graph G is bipartite if and only if every subgraph H of G has an independent set consisting of at least half of V (H). ,
## Section 1.2: Paths, Cycles, and Trails
33
1.2.27. Let G. be the graph whose vertices are the permutations of {l, ... , n), with two permutations a 1 , , a. and b1 , , bn adjacent if they differ by switching two entries. Prove that G. is bipartite (Gs shown below). (Hint: For each permutation a, count the pairs i, j such that i < j and a; > a1 ; these are called inversions.)
132~231
312 321
1.2.28. (!) In each graph below, find a bipartite subgraph with the maximum number of edges. Prove that this is the maximum, and determine whether this is the only bipartite subgraph with this many edges.
123
213
1.2.29. (!) Let G be a connected simple graph not having P4 or Cs as an induced subgraph. Prove that G is a biclique (complete bipartite graph). 1.2.30. Let G be a simple graph with vertices vi. ... , v. Let Ak denote the kth power of the adjacency matrix of G under matrix multiplication. Prove that entry i, j of Ak is the number of v;, vrwalks of length k in G. Prove that G is bipartite if and only if, for the odd integer r nearest ton, the diagonal entries of A' are all 0. (Reminder: A walk is an ordered list of vertices and edges.) 1.2.31. (!)Non-inductive proof of Theorem 1.2.23 (see Example 1.2.21). a) Given n :::: 2\ encode the vertices of K. as distinct binary k-tuples. Use this to construct k bipartite graphs whose union is K . b) Given that K. is a union of bipartite graphs G 1 , .. , Gb encode the vertices of K. as distinct binary k-tuples. Use this to prove that n :::: 2k. 1.2.32. The statement below is false. Add a hypothesis to correct it, and prove the corrected statement. "Every maximal trail in an even graph is an Eulerian circuit." 1.2.33. Use ordinary induction on k or on the number of edges (one by one) to prove that a connected graph with 2k odd vertices decomposes into k trails if k > 0. Does this remain true without the connectedness hypothesis? 1.2.34. Two Eulerian circu~ts are equivalent if they have the same unordered pairs of .consecutive edges, viewed cyclically (the starting point and direction are unimportant). A cycle, for example, has only one equivalence class of Eulerian circuits. How many equivalence classes of Eulerian circuits are there in the graph drawn below?
34
## Chapter 1: 'Fundamental Concepts
1.2.35. Tucker's Algorithm. Let G be a connected even graph. At each vertex, partition the incident edges into pairs (each edge appears in a pair for each of its endpoints). Starting along a given edge e, form a trail by leaving each vertex along the edge paired with the edge just used to enter it, ending with the edge paired withe. This decomposes G into closed trails. As long as there is more than one trail in the decomposition, find two trails with a common vertex and combine them into a longer trail by changing the pairing at a common vertex. Prove that this procedure works and produces an Eulerian circuit as its final trail. (Tucker [1976]) 1.2.36. ( +) Alternative characterization of Eulerian graphs. a) Prove that if G is Eulerian and G' = G - uv, then G' has an odd number of u, vtrails that visit v only at the end. Prove also that the number of the trails in this list that are not paths is even. (Toida [1973]) b) Let v be a vertex of odd degree in a graph. For each edge e incident to v, let c(e) be the number of cycles containing e. Use c(e) to prove that c(e) is even for some e incident to v. (McKee [1984]) c) Use part (a) and part (b) to conclude that a nontrivial connected graph is Eulerian if and only if every edge belongs to an odd number of cycles.
Le
1.2.37. (!) Use extremality to prove that the connection relation is transitive. (Hint: Given au, v-path Panda v, w-path Q, consider the first vertex of Pin Q.) 1.2.38. (!) Prove that every 11-vertex graph with at least n edges contains a cycle. 1.2.39. Suppose that every vertex of a loopless graph G has degree at least 3. Prove that G has a cycle of even length. (Hint: Consider a maximal path.) (P. Kwok) 1.2.40. (!) Let P and Q be paths of maximum length in a connected graph G. Prove that P and Q have a common vertex. 1.2.41. Let G be a connected graph with at least three vertices. Prove that G has two vertices x, y such that 1) G - {x, y) is connected and 2) x, y are adjacent or have a common neighbor. (Hint: Consider a longest path.) (Chung [1978a]) 1.2.42. Let G be a connected simple graph that does not have P4 or C4 as an induced subgraph. Prove that G has a vertex adjacent to all other vertices. (Hint: Consider a vertex of maximum degree.) (Wolk [1965]) 1.2.43. (+)Use induction on k to prove that every connected simple graph with an even number of edges decomposes into paths oflength 2. Does the'conclusion remain true if the hypothesis of connectedness is omitted?
## 1.3. Vertex Degrees and Counting
The degrees of the vertices are fundamental parameters of a graph. We repeat the definition in order to introduce important notation. 1.3.1. Definition. The degree of vertex v in a graph G, written da(v) or d(v), is the number of edges incident to v, except that each loop at v counts twice. The maximum degree is t.(G), the minimum degree is 8(G), and G is regular if t. (G) = 8( G). It is k-regular if the common degree is k. The neighborhood of v, written NG(v) or N(v), is the set of vertices adjacent to v.
## Section 1.3: Vertex Degrees and Counting
35
1.3.2. Definition. The order of a graph G, written 11 ( G), is the number of vertices in G. Ann-vertex graph is a graph of order n. The size of a graph G, written e(G), is the number of edges in G. For /1 EN, the notation [n] indicates the set {l, ... , n). Since our graphs are finite, n(G) and e(G) are well-defined nonnegative integers. We also often use "e" by itself to denote an edge. When e denotes a particular edge, it is not followed by the name of a graph in parentheses, so the context indicates the usage. We have used "11-cycle" to denote a cycle with n vertices; this is consistent with "n-vertex graph".
## COUNTING AND BIJECTIONS
We begin with counting problems about subgraphs in a graph. The first such problem is to count the edges; we do this using the vertex degrees. The resulting formula is an essential tool of graph theory, sometimes called the "First Theorem of Graph Theory" or the "Handshaking Lemma". 1.3.3. Proposition. (Degree-Sum Formula) If G is a graph, then
L:,,EV<Gl d(v)
= 2e(G).
Proof: Summing the degrees counts each edge twice, since each edge has two ends and contributes to the degree at each endpoint. The proof holds even when G has loops, since a loop contributes 2 to the degree of its endpoint. For a loopless graph, the two sides of the formula count the set of pairs (v, e) such that v is an endpoint of e, grouped by vertices or grouped by edges. "Counting two ways" is an elegant technique for proving integer identities (see Exercise 31 and Appendix A). The degree-sum formula has several immediate corollaries. Corollary 1.3.5 applies in Exercises 9-13 and in many arguments oflater chapters. 1.3.4. Corollary. In a graph G, the average vertex degree is :~<:Jll, and hence
8(G) <
2e(G) n(G) -
< ti.(G).
1.3.5. Corollary. Every graph has an even number of vertices of odd degree. No graph of odd order is regular with odd degree. 1.3.6. Corollary. A k-regular graph with n vertices has nk/2 edges. We next introduce an important family of graphs. 1.3.7. Definition. The k-dimensional cube or hypercube Qk is the simple graph whose vertices are the k-tuples with entries in {O, 1) and whose
36
## Chapter 1: Fundamental Concepts
edges are the pairs of k-tuples that differ in exactly one position. A } dimensional subcube of Qk is a subgraph of Qk isomorphic to Q;. 011 111
## 101 000 100
Above we show Q 3 . The hypercube is a natural computer architecture. Processors can communicate directly if they correspond to adjacent vertices in Qk. The k-tuples that name the vertices serve as addresses for the processors.
1.3.8. Example. Structure of hypercubes. The parity of a vertex in Qk is the parity of the number of ls in its name, even or odd. Each edge of Qk has an even vertex and an odd vertex as endpoints. Hence the even vertices form an independent set, as do the odd vertices, and Qk is bipartite. Each position in the k-tuples can be specified in two ways, son( Qk) = 2k. A neighbor of a vertex is obtained by changing one of the k positions in its name to the other value. Thus Qk is k-regular. By Corollary 1.3.6, e( Qk) = 1&k-l. The bold edges above show two subgraphs of Q3 isomorphic to Q2 , formed by keeping the last coordinate fixed at 0 or at 1. We can form a }-dimensional subcube by keeping any k - j coordinates fixed and letting the values in the remaining j coordinates range over all 2j possible }-tuples. The subgraph inways duced by such a set of vertices is isomorphic to Qj. Since there are to pick j coordinates to vary and 2k- j ways to specify the values in the fixed coordinates, this specifies C)2k- j such subcubes. In fact, there are no other }-dimensional subcubes (Exercise 29). The copies of Q 1 are simply the edges in Qk. Our formula reduces to k2k-l when j = 1, so we have found another counting argument to compute e(Qk). When j = k - 1, our discussion suggests a recursive description of Qk. Append 0 to the vertex names in a copy of Qk~ 1 ; append 1 in another copy. Add edges joining vertices from the two copies whose first k - 1 coordinates are equal. The result is Qk. The basis of the construction is the 1-vertex graph Q0 This description leads to inductive proofs for many properties of hypercubes, including e( Qk) = k2k-l (Exercise 23). I
C)
A hypercube is a regular bipartite graph. A simple counting argument proves a fundamental observatio~ about such graphs.
1.3.9. Proposition. If k > 0, then a k-regular bipartite graph has the same number of vertices in each partite set.
## Section 1.3: Vertex Degrees and Counting
37
Proof: Let G be an X, Y-bigraph. Counting the edges according to their endpoints in X yields e(G) = k IXI. Counting them by their endpoints in Y yields e(G) = k IYI. Thus k IXI = k IYI, which yields IXI = IYI when k > 0.
Another technique for counting a set is to establish a bijection from it to a set of known size. Our next example uses this approach. Other examples of combinatorial arguments for counting problems appear in Appendix A. Exercises 18-35 involve counting.
1.3.10. Example. The Petersen graph has ten 6-cycles. Let G be the Petersen graph. Being 3-regular, G has ten claws (copies of Kl.3). We establish a one-toone correspondence between the 6-cycles and the claws. Since G has girth 5, every 6-cycle F is an induced subgraph. Each vertex of F has one neighbor outside F. Since nonadjacent vertices have exactly one common neighbor (Proposition 1.1.38), opposite vertices on F have a common neighbor outside F. Since G is 3-regular, the resulting three vertices outside Fare distinct. Thus deleting.V(F) leaves a subgraph with three vertices of degree 1 and one vertex of degree 3; it is a claw.
We show that each claw H in G arises exactly once in this way. Let S be the set of vertices with degree 1 in H; Sis an independent set. The central vertex of H is already a common neighbor, so the six other edges from S reach distinct vertices. Thus G - V(H) is 2-regular, Since G has girth 5, G - V(H) must be a 6-cycle. This 6-cycle yields H when its vertices are deleted. We present one more counting argument related to a long-standing conjecture. Subgraphs obtained by deleting a single vertex are called vertex-deleted subgraphs. These subgraphs need not all be distinct; for example, then vertexdeleted subgraphs of Cn are all isomorphic to Pn-1
1.3.11.* Proposition. For a simple graph G with vertices v 1 , , v,, and n ::: 3,
e(G) =
:Ee(G-v;)
n- 2
and
da(v;) =
## :Ee(G - V;) - e(G n-2
Vj).
Proof: An edge e of G appears in G - v; if and only if v; is not an endpoint of e. Thus L(G - v;) counts each edge exactly n - 2 times. Once we know e(G), the degree of Vj can be computed as the number of edges lost when deleting vj to form G - vj.
38
## Chapter 1: Fundamental Concepts
Typically, we are given the vertex-deleted subgraphs as unlabeled graphs; we know only the list of isomorphism classes, not which vertex of G - v; corresponds to which vertex in G. This can make it very difficult to tell what G is. For example, K 2 and its complement have the same list of vertex-deleted subgraphs. For larger graphs we have the Reconstruction Conjecture, formulated in 1942 by Kelly and Ulam. 1.3.12.* Conjecture. (Reconstruction Conjecture) If G is a simple graph with at least three vertices, then G is uniquely determined by the list of (isomorphism classes of) its vertex-deleted subgraphs. The list of vertex-deleted subgraphs of G has n (G) items. Proposition 1.3.11 shows that e(G) and the list of vertex degrees can be reconstructed. The latter implies that regular graphs can be reconstructed (Exercise 37). We can also determine whether G is connected (Exercise 38); using this, disconnected graphs can be reconstructed (Exercise 39). Other sufficient conditions for reconstructibility are known, but the general conjecture remains open.
EXTREMAL PROBLEMS
An extremal problem asks for the maximum or minimum value of a function over a dass of objects. For example, the maximum number of edges in a simple graph with n vertices is G).
1.3.13. Proposition. The minimum number of edges in a connected graph with n vertices is n - 1. Proof: By Proposition 1.2.11, every graph with n vertices and k edges has at least n - k components. Hence every n-vertex graph with fewer than n - 1 edges has at least two components and is disconnected. The contrapositive of this is that every connected n-vertex graph has at least n - 1 edges. This lower bound is achieved by the path Pn. 1.3.14. Remark. Proving that f3 is the minimum of f(G) for graphs in a class G requires showing two things: 1) /(G) 2: f3 for all G E G. 2) /(G) = f3 for some GE G. The proof of the bound must apply to every G E G. For equality it suffices to obtain an example in G with the desired value off. Changing":::" to"~" yields the criteria for a maximum. Next we solve a maximization problem that is not initially phrased as such. 1.3.15. Proposition. If G is a simple n-vertex graph with 8(G) 2:: (n - 1)/2, then G is connected.
## Section 1.3: Vertex Degrees and Counting
39
Proof: Choose u, v E V (G). It suffices to show that u, v have a common neighbor if they are not adjacent. Since G is simple, we have IN(u)I :=:: 8(G) :=:: (n - 1)/2, and similarly for v. When u ~ v, we have IN(u) U N(v)I .:::: n - 2, since u and v are not in the union. Using Remark A.13 of Appendix A, we thus compute
IN(u)
n N(v)I =
IN(u)I
+ IN(v)I -
IN(u)
u N(v)I
::::
n2 1 + n2 1 -
(n - 2) = 1.
We say that a result is best possible or sharp when there is some aspect of it that cannot be strengthened without the statement becoming false. As shown by the next example, this holds for Proposition 1.3.15; when o(G) is smaller than (n(G) - 1)/2, we cannot still conclude that G must be connected. 1.3.16. Example. Let G be the n-vertex graph with components isomorphic to Kln/2J and Krn; 2i. where the floor LxJ of xis the largest integer at most x and the ceiling fxl of xis the smallest integer at least x. Since 8(G) = Ln/2J - 1 and G is disconnected, the inequality in Proposition 1.3.15 is sharp. We use the floor and ceiling functions here in order to describe a single family of graphs providing an example for each n.
By providing a family of examples to show that the bound is best possible, we have solved an extremal problem. Together, Proposition 1.3.15 and Example 1.3.16 prove "The minimum value of 8(G) that forces an n-vertex simple graph G to be connected is Ln /2J ," or "The maximum value of 8 ( G) among disconnected n-vertex simple graphs is Ln/2J - l." We introduce compact notation to describe the graph of Example 1.3.16.
1.3.17. Definition. The graph obtained by taking the union of graphs G and H with disjoint vertex sets is the disjoint union or sum, written G + H. In general, mG is the graph consisting of m pairwise disjoint copies of G. 1.3.18. Example. If G and H are connected, then G + H has components G and H, so the graph in Example 1.3.16 is K Ln/2J + K rn/21 This notation is convenient when we have not named the vertices. Note that Km + K" = K m.n. _ The graph mK2 consists of m pairwise disjoint edges. In graph theory, we use "extremal problem" for finding an optimum over a class of graphs. When seeking extremes in a single graph, such as the maximum size of an independent set, or maximum size of a bipartite subgraph, we have a different problem for each graph. To distinguish these from the earlier type of problem, we call them optimization problems. Since an optimization problem has an instance for each graph, we usually can't list all solutions. We may seek a solution procedure or bounds on the
40
## Chapter 1: Fundamental Concepts
answer in terms of other aspects of the input graph. In this light we consider the problem of finding a large bipartite subgraph. It allows us to introduce the technique of constructive or "algorithmic" proof. (An algorithm is a procedure for performing some task.) One way to prove that something exists is to build it. Such proofs can be viewed as algorithms. To complete an algorithmic proof, we must prove that the algorithm terminates and yields the desired result. This may involve induction,contradiction, finiteness, etc. We prove that every graph has a large bipartitesubgraph by providing an algorithm to find one. Exercises 45-49 arerelated to finding large bipartite subgraphs.
l.3.19. Theorem. Every loopless graph G has a bipartite subgraph with at least e(G)/2 edges. Proof: We start with any partition of V(G) into two sets X, Y. Using the edges having one endpoint in each set yields a bipartite subgraph H with bipartition X, Y. If H contains fewer than half the edges of G incident to ~ vertex v, then v has more edges to vertices in its own class than in the other class, as illustrated below. Moving v to the other class gains more edges of G than it loses. We move one vertex in this way as long as the current bipartite subgraph captures less than half of the edges at some vertex. Each such switch increai;;es the size of the subgraph, so the proce8s must terminate. When it terminates, we have dH(v) ::: dc(v)/2 for every v E V(G). Summing this and applying the degree-sum formula yields e(H):;::: e(G)/2.
Algorithmic proofs often correspond to proofs by induction or extremality. Such proofs are shorter and may be easier to find, so we may seek such a proof and later convert it to an algorithm. For example, here is the proof of Theorem 1.3.19 in the language of extremality and contradiction; in effect, the extremal choice of H goes directly to the end of the algorithm: Let H be the bipartite subgraph of G that has the most edges. If dH(v) ::: dc(v)/2 for all v E V(G), then the degree-sum formula yields e(H) '.'.: e(G)/2. Otherwise, dH(v) < dG(v)/2 for some v E V(G), and then switching v in the bipartition contradicts the choice of H.
1.3.20. Example. Local maximum. The algorithm in Theorem 1.3.19 need not produce a bipartite subgraph with the most edges, merely one with at least half the edges. The graph below is 5-regular with 8 vertices and hence has 20 edges. The bipartition X = {a, b, c, d) and Y = {e, f, g, h) yields a 3-regular bipartite
## Section 1.3: Vertex Degrees and Counting
41
subgraph with 12 edges. The algorithm terminates here; switching one vertex would pick up two edges but lose three. Nevertheless, the bipartition X ={a, b, g, h) and Y = {c, d, e, fl yields a 4regular bipartite subgraph with 16 edges. An algorithm seeking the maximum by local changes may get stuck in a local maximum.
h
a
1.3.21. Remark. In a graph G, the (global) maximum number cf edges in a bipartite subgraph is e(G) minus the minimum number of edges needed to obtain at least one edge from every odd cycle. Our next extremal problem doesn't start with bipartite graphs, but it winds up there. In politics and warfare, seldom do two enemies have a common enemy; usually two of the three combine against the third. Given n factions, how many pairs of enemies can there be if no two enemies have a common enemy? In the language of graphs, we are asking for the maximum number of edges in a simple n-vertex graph with no triangle. Bipartite graphs l)ave no triangles, but also many non-bipartite graphs (such as the Petersen graph) have no triangles. Using extremality (by choosing a vertex of maximum degree), we will prove that the maximum is indeed achieved by a complete bipartite graph. 1.3.22. Definition. A gTaph G is H -free if G has no induced subgraph isomorphic to H. 1.3.23. Theorem. (Mantel [1907]) The maximum number of edges in an n vertex triangle-free simple graph is l_n 2 /4 Proof: Let G be an n-vertex triangle-free simple graph. Let x be a vertex of maximum degree, with k = d(x). Since G has no triangles, there are no edges among neighbors of x. Hence summing the degrees of x and its nonneighbors counts at least one endpoint of every edge: LvN(x) d(v) :'.: e(G). We sum over n - k vertices, each having degree at most k, so e(G) :::: (n - k)k.
J.
Since (n - k)k counts the edges in Kn-k.h we have now proved that e(G) is bounded by the size of some biclique with n vertices. Moving a vertex of Kn,k,k
:)
42
## Chapter 1: Fundamental Concepts
from the set of size k to the set of size n - k gains k - 1 edges and loses n - k edges. The net gain is 2k - 1 - n, which is positive for 2k > n + 1 and negative for 2k < n + 1. Thus e(Kn-k.k) is maximized when k is fn/21 or Ln/2J. The product is then n 2 /4 for even n and (n 2 - 1)/4 for odd n. Thus e(G) :':":: Ln 2/4J. To prove that the bound is best possible, we exhibit a triangle-free graph with Ln 2/4J edges: Kln/2J.rn/2l Although (n - k)k can be maximized over k using calculus, the discrete approach is preferable in some ways. it directly restricts k to be an integer and generalizes easily to more variables. The switching idea used is that of Theorem 1.3.19; here we have used it to find the largest bipartite subgraph of Kn In Theorem 5.2.9 we generalize Theorem 1.3.23 to K,+1-free graphs. Mantel's result leads us to another reason for phrasing inductive proofs in the format that we have used. The reason is safety.
1.3.24. Example. A failed proof. Let us try to prove Theorem 1.3.23 by induction on n. Basis step: n :':":: 2. Here the complete graph Kn has the most edges and has no triangles. Induction step: n > 2. We try "Suppose that the claim is true when n = k, so KLk/ 2J.rk; 21 is the largest triimgle-free graph with k vertices. We add a new vertex x to form a trig.ngle-free graph with k + 1 vertices. Making x adjacent to vertices from both partite sets would create a triangle. Hence we add the most edges by making x adjacent to all the vertices in the larger partite set of KLk/2J.rk/21 Doing so creates KL<k+l)/2J.r<k+l)/21 This completes the proof." This argument is wrong, because we did not consider all triangle-free graphs with k + 1 vertices. We considered only those containing the extremal kvertex graph as an induced subgraph. This graph does appear in the extremal graph with k + 1 vertices, but we cannot use that fact before proving it. It remains possible that the largest example with k + 1 vertices arises by adding a new vertex of high degree to a non-maximal example with k vertices. Exercise 51 develops a correct proof by induction on n.
The error in Example 1.3.24 was that our induction step did not consider all instances of the statement for the new larger value of the parameter. We call this error the induction trap. If the induction step grows an instance with the new value of the parameter from a sm<tiler instance, then we must prove that all instances with the new value have been considered. When there is only one instance for each value of the induction parameter (as in summation formulas), this does not cause trouble. With more than one instance, it is safer and simpler to start with an arbitrary instance for the larger parameter value. This explicitly considers each instance G for the larger value, so we don't need to prove that we have generated them all. However, when we obtain from G a smaller instance, we must confirm that the induction hypothesis applies to it. For example, in the inductive proof of the characterization of Eulerian circuits (Theorem 1.2.26), we must apply the
## Section 1.3: Vertex Degrees and Counting
43
induction hypothesis to each component of the graph obtained by deleting the edges of a cycle, not to the entire graph at once.
1.3.25. Remark. A template for induction. Often the statement we want to prove by induction on n is an implication: A(n) => B(n). We must prove that every instance G satisfying A(n) also satisfies B(n). Our induction step follows a typical format. From G we obtain some (smaller) G'. If we show that G' satisfies A(n -1) (for ordinary induction), then the induction hypothesis implies that G' satisfies B(n -1). Now we use the information that G' satisfies B(n -1) to prove that G satisfies B(n).
G satisfies A(n)
_JJ.
G satisfies B(n)
G' satisfies A (n - 1)
## fr' => G' satisfies
B(n -
1)
Here the central implication is the statement of the induction hypothesis, and the others are the work we must do. Our induction proofs have followed this format.
'1.3.26. * Example. The induction trap. The induction trap can lead to a false conclusion. Let us try to prove by induction on the number of vertices that every 3-regular connected simple graph has no cut-edge. By the degree-sum formula, every regular graph with odd degree has even order, so we consider graphs with 2m vertices. The smallest 3-tegular simple graph, K4 , is connected and has no cut-edge; this proves the basis step with m = 2. Now consider the induction step. Given a simple 3-regular graph G with 2k vertices, we can obtain a simple 3-regular graph G' with 2(k + 1) vertices (the next larger possible order) by "expansion": take two edges of G, replace them by paths of length 2 through new vertices, and add an edge joining the two new vertices. As illustrated below, K 3 ,3 arises from K4 by one expansion on two disjoint edges.
If G is connected, then the expanded graph G' is also connected: a path between old vertices that traversed a replaced edge has merely lengthened, and a path to a new vertex in G' is obtained from a path in G to a neighbor. If G has no cut-edge, then every edge lies on a cycle (Theorem 1.2.14). These cycles remain in G' (those using replaced edges become longer). The edge joining the two new vertices in G' also lies on a cycle using a path in G between the edges that were replaced. Theorem 1.2:14 now implies that G' has no cut-edge. We have proved that if G is connected and has no cut-edge, then the same holds for G'. We might think we have proved by induction on m that every 3regular simple connected graph with 2m vertices has no cut-edge, but the graph
I-
expansion erasure
44
## Chapter 1: Fundamental Concepts
below is a counterexample. The proof fails because we cannot build every 3reg-ular simple connected graph from K4 by expansions. We cannot even obtain all those without cut-edges, as shown in Exercise 66.
## Appendix A presents another example of the induction trap.
GRAPHIC SEQUENCES
Next we consider all the vertex degrees together. 1.3.27. Definition. The deg1:ee sequence of a graph is the list of vertex degrees, usually written in nonincreasing order, as di ::::_ ::::_ d11 Every graph has a degree sequence, but which sequences occur? That is, given nonnegative integers di, ... , d11 , is there a graph with these as the vertex degrees? The degree-sum formula implies that 'L d; must be even. When we allow loops and multiple edges, TONCAS. 1.3.28. Proposition. The nonnegative integers di, ... , d 11 are the vertex degrees of some graph if and only if 'L d; is even. Proof: Necessity. When some graph G has these numbers as its vertex degrees, the degree-sum formula implies that 'L d; = 2e(G), which is even. Sufficiency. Suppose that 'L d; is even . .We construct a graph with vertex set vi, ... , v11 and d(v;) = d; for all{. Since 'Ld; is even, the number of odd values is even. First form an arbitrary pairing of the vertices in {v;: d; is odd}. For each resulting pair, form an edge having these two vertices as its endpoints. The remaining degree needed at each vertex is even and nonnegative; satisfy this for each i by placing Ld;/2J loops at v;. This proof is constructive; we could also use induction (Exercise 56). The construction is easy with loops available. Without them, (2, 0, 0) is not realizable and the condition is not sufficient. Exercise 63 characterizes the degree sequences ofloopless graphs. We next characterize degree sequences of simple graphs by a recursive condition that readily yields an algorithm. Many other characterizations are known; Sierksma-Hoogeveen [1991] lists seven. 1.3.29. Definition. A graphic sequence is a list of nonneg?tive numbers that is the degree sequence of some simple graph. A simple graph with degree sequence d '.'realizes" d.
## Section 1.3: Vertex Degrees and Counting
45
1.3.30. Example. A recursive condition. The lists 2, 2, 1, 1 and 1, 0, 1 are graphic. The graph K2 +Ki realizes 1, 0, 1. Adding a new vertex adjacent to vertices of degrees 1 and 0 yields a graph with degree sequence 2, 2, 1, 1, as shown below. Conversely, if a graph realizing 2, 2, 1, 1 has a vertex w with neighbors of degrees 2 and 1, then deleting w yields a graph with degrees 1, 0, 1.
Similarly, to test 33333221, we seek a realization wi~h a vertex w of degree 3 having three neighbors of degree 3. This exists if and only if 2223221 is graphic. We reorder this and test 3222221. We continue deleting and reordering until we can tell whether the remaining list is realizable. If it is, then we insert vertices with the desired neighbors to work back to a realization of the original list. The realization is not unique. The next theorem implies that this recursive test works. 33333221 2223221
w
/
/'
3222221 111221 v
/'
221111 10111
v '
=ru...
'~
## ... ;. +-v ... w
~/~:Ju
' ., ....._
+--
:Ju
........
/'
11110
+--
........
........
1.3.31. Theorem. (Havel [1955], Hakimi [1962]) For n > 1, an integer list d of size n is graphic if and only if d' is graphic, where d' is obtained from d by deleting its largest element~ and subtracting 1 from its~ next largest elements. The only 1-element graphic sequence is di = 0. Proof: For n = 1, the statement is trivial. For n > 1, we first prove that the condition is sufficient. Given d with di ::: 2: dn and a simple graph G' with degree sequence d', we add a new vertex adjacent to vertices in G' with degrees d2 -1, ... , dA+i - 1. These diare the~ largest elements of dafter (one copy of) ~itself, but d2 - 1, ... , dA+ 1 - 1 need not be the~ largest numbers ind'. To prove necessity, w~ begin with a simple graph G realizing d and produce a simple graph G' realizing d'. Let w be a vertex of degree ~ in G. Let S be a set of~ vertices in G having the "desired degrees" d2 , , dA+t If N(w) = S, then we delete w to obtain G'. Otherwise, some vertex of Sis missing from N(w). In this case, we modify G to increase IN(w) n SI without changing any vertex degree. Since IN(w) n SI can increase at most ~ times, repeating this converts G into another graph G* that realizes d and has S as the neighborhood of w. From G* we then delete w to obtain the desired graph G' realizing d'.
46
## Chapter 1: Fundamental Concepts
To find the modification when N(w) =f. S, we choose x E Sand z !!/. S so that w ~ z and w ~ x. We want to add wx and delete wz, but we must preserve vertex degrees. Since d(x) ~ d(z) and already w is a neighbor of z but not x, there must be a vertex y adjacent to x but not to z. Now we delete {wz, xy} and add {wx, yz} to increase IN(w) n SI.
s
x
z
Theorem 1.3.31 tests a list ofn numbers by testing a list ofn - 1 numbers; it yields a recursive algorithm to test whether d is graphic. The necessary condition "I: d; even" holds implicitly: I: d! = <I: d;) - 2~ implies that I: d! and I: d; have the same parity. An algorithmic proof using "local change" pushes an object toward a desired condition. This can be phrased as proofby induction, where the induction parameter is the "distance" from the desired condition. In the proof of Theorem 1.3.31, this distance is the number of vertices in S that are missing from N(w). We used edge switches to transform an arbitrary graph with degree sequence d into a graph satisfying the desired condition. Next we will show that every simple graph with degree sequenced can be transformed by such switches into every other.
1.3.32. Definition. A 2-switch is the replacement of a pair of edges xy and :::win a simple graph by the edges yz and wx, given that yz and wx did not appear in the graph originally.
l::J
w
:::
-+
"
I
.,,
The dashed lines above indicate nonadjacent pairs. If y ~ z or w ~ x, then the 2-switch cannot be performed, because the resulting graph would not be simple. A 2-switch preserves all vertex degrees. If some 2-switch turns H into H*, then a 2-switch on the same four vertices turns H* into H. Below we illustrate two successive 2-switches.
m--+ CZ--+
y
## Section 1.3: Vertex Degrees and Counting
47
1.3.33. * Theorem. (Berge [1973, pl53-154]) If G and H are two simple graphs with vertex set V, then dG(v) = dH(v) for every v E V if and only if there is a sequence of 2-switches that transforms G into H. Proof: Every 2-switch preserves vertex degrees, so the condition is sufficient. Conversely, when da( v) = dH ( v) for all v E V, we obtain an appropriate sequence of 2-switches by induction on the number of vertices, n. If n :s 3, then for each di, ... , dn there is at most one simple graph with d(vi) = d;. Hence we can use n = 3 as the basis step. Consider n ~ 4, and let w be a vertex of maximum degree, ~- Let S = (v1 , ... , v 6 } be a fixed set of vertices with the ~ highest degrees other than w. As in the proof of Theorem 1.3.31, some sequence of 2-switches transforms G to a graph G* such that NG ( w) = S, and some such sequence transforms H to a graph H* such that NH(w) = S.
~ 1s
G
~-~
~
--~
~
s
G' G*
s
H' H*
Since NG(W) NH.(w), deleting w leaves simple graphs G' = G* - w and H' = H* - w with dG'(v) = dw(v) for every vertex v. By the induction hypothesis, some sequence of2-switches transforms G' to H'. Since these do net involve w, and w has the same neighbors in G* and H*, applying this sequence transforms G* to H*. Hence we can transform G to H by transforming G to G*, then G* to H*, then (in reverse order) the transformation of H to H*. n We could also phrase this using induction on the number of edges appearing in exactly one of G and H, which is 0 if and only if they are already the same. In this approach, it suffices to find a 2-switch in G that makes it closer to H or a 2-switch in H that makes it closer to G.
EXERCISES
A statement with a parameter must be proved for all values of the parameter; it cannot be proved by giving examples. Counting a set includes providing proof.
1.3.1. (-) Prove or disprove: If u and v are the only vertices of odd degree in a graph G, then G contains a u, v-path. 1.3.2. (-) In a class with nine students, each student sends valentine cards to three others. Determine whether it is possible that each student receives cards from the same three students to whom he or she sent cards.
48
## Chapter 1: Fundamental Concepts
1.3.3. (- ) Let u and v be adjacent vertices in a simple graph G. Prove that u v belongs to at least d(u) + d(v) - n(G) triangles in G. 1.3.4. (-) Prove that the graph below is isomorphic to Q4.
1.3.5. (-)Count the copies of P3 and C4 in Qk. 1.3.6. (-) Given graphs G and H, determine the number of components and maximum degree of G + H in terms of the those parameters for G and H. 1.3. 7. (-) Determine the maximum number of edges in a bipartite subgraph of P,,, of C,,, and of K 11 1.3.8. (-) Which of the following are graphic sequences? Provide a construction or a proof of impossibility for each. a) (5,5,4,3,2,2,2,1), b) (5,5,4,4,2,2,l,1), c) (5,5,5,3,2,2,l,ll, d> <5,5,5,4,2,1,1,n
1.3.9. In a league with two divisions of 13 teams each, determine whether it is possible to schedule a season with each team playing nine games against teams within its division and four games against teams in the other division. 1.3.10. Let/, m, n be nonnegative integers with I + /11 = n. Find necessary and sufficient conditions on I. m, n such that there exists a connected simple 11-vertex graph with I vertices of even degree and /11 vertices of odd degree. 1.3.11. Let W be a closed walk in a graph G. Let H be the subgraph of G consisting of edges used an odd number of times in W. Prove that dH(v) is even for every v E V(G). 1.3.12. (!) Prove that an even graph has no cut-edge. For each k 2'.: 1, construct a 2k + 1-regular simple graph having a cut-edge. 1.3.13. (+)A mountain range is a polygonal curve from (a, 0) to (b, 0) in the upper half-plane. Hikers A and B begin at (a. 0) and (b, 0), respectively. Prove that A and B can meet by traveling on the mountain range in such a way that at all times their heights above the horizontal axis are the same. (Hint: Define a graph to model the movements, and use Corollary 1.3.5.) (Communicated by D.G. Hoffman.)
## Section 1.3: Vertex Degrees and Counting
49
1.3.14. Prove that every simple graph with at least two vertices has two vertices of equal degree. Is the conclusion true for loopless graphs? 1.3.15. For each k ::::: 3, determine the smallest n such that a) there is a simple k-regular graph .vith n vertices. b) there exist nonisomorphic simple k-regular graphs with n vertices. 1.3.16. (+)Fork::::: 2 and g::::: 2, prove that there exists an k-regular graph with girth
g. (Hint: To construct such a graph inductively, make use of an k - 1-regular graph H
with girth g and a graph with girth fg/21 that is n(H)-regular. Comment: Such a graph with minimum order is a (k, g)-cage.) (Erdos-Sachs [1963])
1.3.17. (!)Let G be a graph with at least two vertices. Prove or disprove: a) Deleting a vertex of degree ~(G) cannot increase the average degree. b) Deleting a vertex of degree 8(G) cannot reduce the average degree. 1.3.18. (!) Fork ?:: 2, prove that a k-regular bipartite graph has no cut-edge. 1.3.19. Let G be a claw-free graph. Prove that if ~(G) ?:: 5, then G has a 4-cycle. For all n E N, construct a 4-regular claw-free graph of order at least n that has no 4-cycle. 1.3.20. (!) Count the cycles oflength n in Kn and the cycles oflength 2n in Kn.n 1.3.21. Count the 6-cycles in Km.n 1.3.22. (!) Let G be a nonbipartite graph with n vertices and minimum degree k. Let I be the minimum length of an odd cycle in G. a) Let C be a cycle of length l in G. Prove that every vertex not in V ( C) has at most two neighbors in V(C). b) By counting the edges joining V(C) and G- V(C) in two ways, prove thatn :::: kl/2 (and thus l .:::: 2n/ k). (Campbell-Staton [1991]) c) When k is even, prove that the inequality of part (b) is best possible. (Hint: form a graph having k/2 pairwise disjoint /-cycles.) 1.3.23. Use the recursive description of Qk (Example 1.3.8) to prove that e(Qk) = k2k-l. 1.3.24. Prove that K2 ,3 is not contained in any hypercube Qk. 1.3.25. (!) Prove that every cycle of length 2r in a hypercube is contained in a subcube of dimension at most r. Can a cycle of length 2r be contained in a subcube of dimension less than r? 1.3.26. (!) Count the 6-cycles in Q 3. Prove that every 6-cycle in Qk lies in exactly one 3-dimensional subcube. Use this to count the 6-cycles in Qk fork?:: 3. 1.3.27. Given k EN, let G be the subgraph of Q 2k+l induced by the vertices in which the number of ones and zeros differs byl. Prove that G is regular, and compute n(G), e(G), and the girth of G. 1.3.28. Let V be the set of binary k-tuples. Define a simple graph Q~ with vertepc set V by putting u ~ v if and only if u and v agree in exactly one coordinate. Prove that Q~ is isomorphic to the hypercube Qk if and only if k is even. (D.G. Hoffman) 1.3.29. (*+)Automorphisms of the k-dimensional cube Qk. a) Prove that every copy of Qi in Qk is a subgraph induced by a set of 2i vertices having specified values on a fixed set of k - j coordinates. (Hint: Prove that a copy of Qi must have two vertices differing in j coordinates.) b) Use part (a) to count the automorphisms of Qk.
50
## Chapter 1: Fundamental Concepts
1.3.30. Prove that every edge in the Petersen graph belongs to exactly four 5-cycles, and use this to show that the Petersen graph has exactly twelve 5-cycles. (Hint: For the first part, extend the edge to a copy of P4 and apply Proposition 1.1.38.) 1.3.31. (!)Use complete graphs and counting arguments (not algebra!) to prove that a) (~) = (~) + k(n - k) + (n;k) for 0 :5 k :5 n. b) IfI:n; = n, then L (~) :5 (~). 1.3.32. (!) Prove that the number of simple even graphs with vertex set [n] is 2<"2 (Hint: Establish a bijection to the set of all simple graphs with vertex set [n - 1] .)
1 ).
1.3.33. (+) Let G be a triangle-free simple n-vertex graph such that every pair of nonadjacent vertices has exactly two common neighbors. a) Prove that n(G) = 1 + (d~)), where x E V(G). Conclude that G is regular. b) When k = 5, prove that deleting any one vertex and its neighbors from G leaves the Petersen graph. (Comment: When k = 5, the grap:!:J. Gisin fact the graph obtained from Q4 by adding edges joining complementary vertices.) 1.3.34. (+) Let G be a kite-free simple n-vertex graph such that every pair of nonadjacent vertices has exactly two common neighbors. Prove that G is regular. (Galvin) 1.3.35. (+) Let n and k be integers such that 1 < k < n - 1. Let G be a simple n-vertex graph such that every k-vertex induced subgraph of G has m edges. a) Let G' be an inducgd subgraph of G with I vertices, where l > k. Prove that
e(G')
= mG)1G=;).
b) Use part (a) to prove that G E {K,,, K,,). (Hint: Use part (a) to prove that the number of edges with endpoints u, v is independent of the choice of u and v.)
1.3.36. Let G be a 4-vertex graph whose list of subgraphs obtained by deleting one vertex appears below. Determine G.
1.3.37. Let H be a graph formed by deleting a vertex from a loopless regular graph G with n(G) '.::: 3. Describe (and justify) a method for obtaining G from H. 1.3.38. Let G be a graph with at least 3 vertices. Prove that G is connected if and only if at least two of the subgraphs obtained by deleting one vertex of G are connected. (Hint: Use Proposition 1.2.29.) 1.3.39. (*+)Prove that every disconnected graph G with at least three vertices is reconstructible. (Hint: Having used Exercise 1.3.38 to determine that G is disconnected, use G 1 , ... , G,, to find a component M ofG that occurs the most times among the components with the maximum number of vertices, use Proposition 1.2.29 to choose v so that L = M - v is connected, and reconstruct G by finding some G - v; in which a copy of M became a copy of L.) 1.3.40. (!) Let G be an 11-vertex simple graph, where n '.::: 2. Determine the maximum possible number of edges in G under each of the following conditions. a) G has an independent set of size a. b) G has exactly k components. c) G is disconnected.
## Section 1.3: Vertex Degrees and Counting
51
1.3.41. (!) Prove or disprove: If G is an n-vertex simple graph with maximum degree ln/21 and minimum degree Ln/2j - 1, then G is connected. 1.3.42. Let S be a set of vertices in a k-regular graph G such that no two vertices in Sare adjacent or have a common neighbor. Use the pigeonhole principle to prove that ISi ::=: Ln(G)/(k + l)j. Show that the bound is best possible for the cube Q3 . (Comment: The bound is not best possible for Q 4 .) 1.3.43. (+)Let G be a simple graph with no isolated vertices, and let a = 2e(G)/n(G) be the average degree in G. Let t ( v) denote the average of the degrees of the neighbors of v. Prove that t(v) :::: a for some v E V (G). Construct an infinite family of connected graphs such that t(v) > a for every vertex v. (Hint: For the first part, compute the average of t(v), using that x/y + y /x :::: 2 when x. y > 0.) (Ajtai-Koml6s-Szemeredi [1980]) 1.3.44. (!)Let G be a loopless graph with average vertex degree a = 2e(G)/n(G). a) Prove that G -x has average degree at least a if and only if d(x) ::;:: a/2. b) Use part (a) to give an algorithmic proof that if a > 0, then G has a subgraph with minimum degree greater than a /2. c) Show that there is no constant c greater than 1/2 such that G must have a subgraph with minimum degree greater than ca; this proves that the bound in part (b) is best possible. (Hint: Use K 1.11-1.) 1.3.45. Determine the maximum number of edges in a bipartite subgraph of the Petersen graph. 1.3.46. Prove or disprove: Whenever the algorithm of Theorem 1.3.19 is applied to a bipartite graph, it finds the bl.partite subgraph with the most edges (the full graph). 1.3.47. Use induction on n(G) to prove that every nontrivial loopless graph G has a bipartite subgraph H such that H has more than e(G)/2 edges. 1.3.48. Construct graphs G 1 , G 2 ... , with G 11 having 2n vertices, such that lim,Hoo j,, = 1/2, where j;, is the fraction of (G 11 ) belonging to the largest bipartite subgraph of G11 1.3.49. For each k E N and each loopless graph G, prove that G has a k-partite subgraph H (Definition 1.1.12) such that e(H) '.::': (1 - 1/ k)e(G). 1.3.50. (+) For n :::: 3, determine the minimum number of edges in a connected n-vertex graph in which every edge belongs to a triangle. (Erdos [1988]) 1.3.51. (+) Let G be a simple n-vertex graph, where n > 3. a) Use Proposition 1.3.11 to prove that if G has more than n 2 /4 edges, then G has a vertex whose deletion leaves a graph with more than (n - 1) 2 /4 edges. (Hint: In every graph, the number of edges is an integer.) b) Use part (a) to prove by induction that G contains a triangle if e(G) > n 2 /4. 1.3.52. Prove that every n-vertex triangle-free simple graph with the maximum number of edges is isomorphic to KL11 ; 2J.r11121 (Hint: Strengthen the proofofTheorem 1.3.23.) 1.3.53. (!) Each game of bridge involves two teams of two partners each. Consider a club in which four players cannot play a game if two of them have previously been partners that night. Suppose that 15 members arrive, but one decides to study graph theory. The other 14 people play until each has been a partner with four others. Next they succeed in playing six more games (12 partnerships), but after that they cannot find four players containing no pair of previous partners. Prove that if they can convince the graph theorist to play, then at least one more game can be played. (Adapt.ed from Bondy-Murty [1976, plll]).
52
## Chapter 1: Fundamental Concepts
1.3.54. (+) Let G be a simple graph with n vertices. Let t(G) be the total number of triangles in G and G together. a) Prove that t(G) = (;) - (n - 2)e(G) + LvEV(G) (d~l) triangles. (Hint: Consider the contribution made to each side by each triple of vertices. b) Prove that t(G) ?: n(n - l)(n - 5)/24. (Hint: Use a lower bound on LvEV(G) (d~l) in terms of average degree.) c) When n - 1 is divisible by 4, construct a graph achieving equality in part (b). (Goodman [1959]) 1.3.55. (+)Maximum size with no induced P4 a) Let G be the complement of a disconnected simple graph. Prove that e(G) ::: t.(G) 2 , with equality only for KMGJ,t><GJ b) Let G be a simple connected P4 -free graph with maximum degree k. Prove that e(G)::: k2 (Seinsche [1974], Chung-West [1993]) 1.3.56. Use induction (on n or on L d;) to prove that if d1 , .. , d. are nonnegative integers and Ld; is even, then there is an n-vertex graph with vertex degrees di. ... , d. (Comment: This requests an alternative proof of Proposition 1.3.28.) 1.3.57. (!) Let n be a positive integer. Let d be a list of n nonnegative integers with even sum whose largest entry is less than ~ and differs from the smallest entry by at most 1. Prove that dis graphic. (Hint: Use the Havel-Hakimi Theorem. Example: 443333 is such a list, as is 33333322.) 1.3.58. Generalization of Havel-Hakimi Theorem. Given a nonincreasing list d of nonnegative integers, let d' be obtained by deleting dk and subtracting 1 from the k largest elements remaining in the list. Prove that d is graphic if and only if d' is graphic. (Hint: Mimic the proof of Theorem 1.3.31.) (Wang-Kleitman [1973]) 1.3.59. Define d = (di. ... , d2k) by d2 ; = d2;- 1 = i for 1 ::: i ::: k. Prove that d is graphic. (Hint: Do not use the Havel-Hakimi Theorem.) 1.3.60. (+) Let d be a list of integers consisting of k copies of a and n - k copies of b, with a ?: b ~ 0. Determine necessary and sufficient conditions ford to be graphic. 1.3.61. (!)Suppose that G ~ G and that n(G) vertex of degree (n(G) -1)/2.
J
## =1mod4. Prove that G has at least one
1.3.62. Suppose that n is congruent to 0 or 1 modulo 4. Construct an n-vertex simple graph G with H;) edges such that t.(G) - 8(G) ::: 1. 1.3.63. (!) Let d 1 , , d. be integers such that d 1 ::::: ::::: d. ~ 0. Prove that there is a loopless graph (multiple edges allowed) with degree sequence d 1 , , d. if and only if d; is even and d 1 ::: d2 + + d. (Hakimi [1962])
1.3.64. (!) Let d 1 ::: ::: d. be the vertex degrees of a simple graph G. Prove that G is connected if dj ~ j when j ::: n - 1 - d. (Hint: Consider a component that omits some vertex of maximum degree.) 1.3.65. (+)Let a 1 < < ak. be dl.stinct positive integers. Prove that there is a simple graph with ak + 1 verticl'ls whose set of distinct vertex degrees is a 1 , , ak. (Hint: Use induction on k to construct such a graph.) (Kapoor-Polimeni-Wall [1977]) 1.3.66. (*)Expansion of 3-regular graphs (see Example 1.3.26). For n = 4k, where k ~ 2, construct a connected 3-regular simple graph with n vertices that has no cut-
edge but cannot be obtained from a smaller 3-regular simple graph by expansion. (Hint:
## Section 1.4: Directed Graphs
53
The desired graph must have no edge to which the inverse "erasure" operation can be applied to obtain a smaller simple graph.) 1.3.67. (*) Construction of 3-regular simple graphs a) Prove that a 2-switch can be performed by performing a sequence of expansions and erasures; these operations are defined in Example 1.3.26. (Caution: Erasure is not allowed when it would produce multiple edges.) b) Use part (a) to prove that every 3-regular simple graph can be obtained from K4 by a sequence of expansions and erasures. (Batagelj [1984]) 1.3.68. (*) Let G and H be two simple bipartite graphs, each with bipartition X, Y. Prove that dc(v) = dH(v) for all v EX UY if and only ifthere is a sequence of 2-switches that transforms G into H without ever changing the bipartition (each 2-switch replaces two edges joining X aJ}d Y by two other edges joining X and Y).
## 1.4. Directed Graphs
We have used graphs to model symmetric relations. Relation need not be symmetric; in general, a relation on S can be any set of ordered pairs in S x S (see Appendix A). For such relations, we need a more general model.
## DEFINITIONS AND EXAMPLES
Seeking a graphical representation of the information in a general relation on S leads us to a model of directed graphs.
1.4.1. Example. For natural numbers x, y, we say that x is a "maximal divisor" ofyify/xisaprimenumber. Fors~ N, thesetR = {(x,y) E S2 : xis a maximal divisor of y} is a relation on S. To represent it graphically, we name a point in the plane for each element of S and draw an arrow from x to y whenever (x, y) ER. Below we show the result when S = [12].
11
1.4.2. Definition. A directed graph or digraph G is a triple consisting of a vertex set V(G), an edge set E(G), and a function assigning each edge an ordered pair of vertices. The first vertex of the ordered pair is the tail of the edge, and the second is the head; together, they are the endpoints. We say that an edge is an edge from its tail to its head.
54
## Chapter 1: Fundamental Concepts
The terms "head" and "tail" come from the arrows used to draw digraphs. As with graphs, we assign each vertex a point in the plane and each edge a curve joining its endpoints. When drawing a digraph, we give the curve a direction from the tail to the head. When a digraph models a relation, each ordered pair is the (head, tail) pair for at most one edge. In this setting as with simple graphs, we ignore the technicality of a function assigning endpoints to edges and simply treat an edge as an ordered pair of vertices.
1.4.3. Definition. In a digraph, a loop is an edge whose endpoints are equal. Multiple edges are edges having the same ordered pair of endpoints. A digraph is simple if each ordered pair is the head and tail of at most one edge; one loop may be present at each vertex. In a simple digraph, we write uv for an edge with tail u and head v. If there is an edge from u to v, then v is a successor of u, and u is a predecessor of v. We write u --+ v for "there is an edge from u to v". 1.4.4. Application. A finite state machine (also called finite automaton or discrete system) has a number of possible "states". Such a system can be modeled using a digraph in which vertices represent the states and edges represent the possible transitions between states. Transitions inherently move in one direction, so digraphs provide the appropriate model. Labels on the edges can be used to record the events that cause the transitions. When an event causes the system to remain in the same state, we have a loop. When two types of events can cause a particular transition, we might use multiple edges. Consider a light controlled by two switches, often called a "three-way switch". The first switch can be up or down, the second switch can be up or down, and the light can be on (+) or off (-). Thus there are eight states. Transitions between stat~s result by flipping switches. In the drawing below, the horizontal edges represent transitions caused by flipping the first switch, and the vertical edges represent transitions caused by flipping the second switch. (Drawing vertices large enough to put labels inside is not uncommon when discussing finite state machines, but we will stick with filled dots.)
1.4.5. * Application. Edge labels can be used to record transition probabilities when a system operates randomly. The probabilities on the edges leaving a
## Section .1.4: Directed Graphs
55
vertex sum to 1, and the system is called a Markov chain. Methods oflinear algebra can be used to compute the long-term fraction of time spent in each state. For example, suppose that weather has two states: good and bad. Air masses move slowly enough that tomorrow's weather tends to be like today's. In most places, storm systems don't linger long, so we might have transition probabilities as shown below. If we record states hourly instead of daily, then the probably ofremaining in the same state is much higher.
.8~ .3
~
.7
1.4.6. Definition. A digraph is a path if it is a simple digraph whose vertices can be linearly ordered so that there is an edge with tail u and head v if and only if v immediately follows u in the vertex ordering. A cycle is defined similarly using an ordering of the vertices on a circle. 1.4.7. Example. Functional digraphs. We can study a function f: A ---+ A using digraphs. The functional digraph off is the simple digraph with vertex set A and edge set {(x, f(x): x E A}. For each x, the single edge with tail x points to the image of x under f. Following a path in a functional digraph corresponds to iterating the function. In a permutation, each element is the image of exactly one element, so the functional digraph has one head and one tail at each vertex. Hence the functional digraph of a permutation consists of disjoint cycles. Below we show the functional digraph for a permutation of [7].
6b_V
1
1.4.8. * Remark. We often use the same names for corresponding concepts in the graph and digraph models. Many authors replace "vertex" and "edge" with "node" and "arc" to discuss digraphs, but this obscures the analogies. Some results have the same statements and proofs; it would be wasteful to repeat them just to change terminology (especially in Chapter 4). Also, a graph G can be modeled using a digraph D in which each edge uv E E(G) is replaced with uv, vu E E(D). In this way, results about digraphs can be applied to graphs. Since the notion of "edge" in digraphs extends the notion of "edge" in graphs, using the same name makes sense. Some authors write "directed path" and "directed cycle" for our concepts of path and cycle in digraphs, but the distinction is unnecessary; for the "weak" version that does not follow the arrows, we can speak of a path or cycle in the graph obtained by ignoring the directions, which we define next.
56
## Chapter 1: Fundamental Concepts
1.4.9. Definition. The underlying graph of a digraph D is the graph G obtained by treating the edges of D as unordered pairs; the vertex set and edge set remain the same, and the endpoints of an edge are the same in G as in D, but in G they become an unordered pair.
<I>
D
Most ideas and methods of graph theory arise in the study of ordinary graphs. Digraphs can be a useful additional tool, especially in applications, as we have tried to suggest. We hope that describing the analogies and contrasts between graphs and digraphs will help clarify the concepts. When comparing a digraph with a graph, we usually use G for the graph and D for the digraph. When discussing a single digraph, we often use G.
1.4.10. Definition. The definitions of subgraph, isomorphism, decomposition, and union are the same for graphs and digraphs. In the adjacency matrix A(G) of a digraph G, the entry in position i, j is the number of edges from v; to Vj. In the incidence matrix M(G) of a loopless digraph G, we set m;,j = +1 if v; is the tail of ej and m;,j = -1 if v; is the head of ej. 1.4.11. Example. The underlying graph of the digraph below is the graph of Example 1.1.19; note the similarities and differences in their matrices. II
w x 0 1 0 0 1 0 0
A(G)
y z 1 1 0 0 1 0 o
WC
x y
0) ~ wcl
a
e
c
a
y
b +1
Y.
+1 0
0
-1
0 +1
-1
a
x
G
0
M(G)
0 -1 +1 0
11)
-1
To define connected digraphs, two options come to mind. We could require only that the underlying graph be connected. However, this does not capture the most useful sense of connection for digraphs.
1.4.12. Definition. A digraph is weakly connected if its underlying graph is connected. A digraph is strongly connected or strong if for each ordered pair u, v of vertices; there is a path from u to v. The strong components of a digraph are its maximal strong subgraphs. 1.4.13. Example. The 2-vertex digraph consisting only of the edge xy has an x, y-path but no y, x-path and is not strongly connected. AS a digraph, an nvertex path has n strong components, but a cycle has only one. In the digraph
## Section 1.4: Directed Graphs
57
below, the three circled subdigraphs are the strong components. Properties of strong components are discussed in Exercises 10-13. El
1.4.14.* Application. Games. Many games with two players can be described as finite state machines. The vertex set is the set of possible states of the game. There is an edge from state x to state y if some move can be made (by the player whose turn it is to play) to reach state y from state x. Let W be the set of vertices for winning positions; a player who brings the game to such a state wins. No edges leave W. A player who brings the game to a state with an edge to W loses, since the other player then reaches W. One way to analyze the game is to seek a set S of pairwise nonadjacent vertices containing W such that every vertex outside S has an edge to a vertex in S. A player who can bring the game to a position in S wins, but one who must move from a position in S loses. For example, consider a game with two piles of pennies. At his or her turn, each player can remove any portion of a single pile. The player who removes the last coin wins. The possible game positions are the nonnegative integer pairs (r, s). The definjtion of the game specifies (0, 0) as the only winning position. However, the set S of desirable positions is {(r, r): r ~ O}. Since only one coordinate can decrease on a move, there is no edge within S. For each vertex (r, s) S, a player can remove Ir - sl from the larger pile to reach S. The general game of Nim starts with an arbitrary number of piles with arbitrary sizes, but otherwise the rules of the game are the same as this. Exercise 18 guarantees that Nim always has a winning strategy set S, since the digraph for this game has no cycles. If the initial position is in S, then the second player wins (assuming optimal play). Otherwise, the first player wins. I 1.4.15. * Definition. A kernel in the digraph D is a set S ~ V (D) such that S induces no edges and every vertex outside S has a successor in S.
A digraph that is an odd cycle has no kernel (Exercise 17), but forbidding odd cycles as subdigraphs always yields a kernel. In proving this, all uses of paths, cycles, and walks are in th~, directed sense. We need several statements about movement in digraphs that hold by the same proofs as in graphs. For example, every u, v-walk in a digraph contains a u, v-path (Exercise 3), and every closed odd walk in a. digraph contains an odd cycle (Exercise 4). The concept of distance from x to y will be explored more fully in Section 2.1; it is the least length of an x, y-path.
58
## Chapter 1: Fundamental Concepts
1.4.16.* Theorem. (Richardson [1953]) Every digraph having no odd cycle has a kernel. Proof: Let D be such a digraph. We first consider the case that D is strongly connected; see the figure on the left below. Given an arbitrary vertex ye V(D), let S be the set of vertices with even distance to y. Every vertex with odd distance to y has a successor in S, as desired. If the vertices of Sare not pairwise nonadjacent, then there is an edge uv witl;l u, v e S. By the definition of S, there is au, y-path P of even length and a v, y-path P' of even length. Adding uv at the start of P' yields au, y-walk W of odd length. Because D is strong, D has a y, u-path Q. Combining Q with one of P or W yields a closed odd walk in D. This is impossible, since a closed odd walk contains an odd cycle. Thus Sis a kernel in D.
For the general case, we use induction on n(D). Basis step: n(D) = 1. The only example is a single vertex with no loop. This vertex is a kernel by itself. Induction step: n(D) > 1. Since we have already proved the claim for strong digraphs, we may assume that D is not strong. For some strong component D' of D, there is no edge from a vertex of D' to a vertex not in D' (Exercise 11). We have shown that D' has a kernel; let S' be a kernel of D'. Let D" be the subdigraph obtained from D by deleting D' and all the predecessors of S'. By the induction hypothesis, D" has a kernel; let S" be a kernel of D". We claim that S' U S" is a kernel of D. Since D" has no predecessor of S', there is no edge within S' US". Every vertex in D" - S" has a successor in S", and all other vertices not in S' U S" have a successor in S'.
VERTEX DEGREES
In a digraph, we use the same notation for number of vertices and number of edges as in graphs. The notation for vertex degrees incorporates the distinction between heads and tails of -edges.
1.4.17. Definition. Let v be a vertex in a digraph. The outdegree d+(v) is the number of edges with tail v. The indegree d-(v) is the number of edges with head v. The out-neighborhood or successor set N+(v) is {x e V(G): v .-+ x}. The in-neighborhood or predecessor set N-(v) is {x e V(G): x -+ v}. The minimum and maximum indegree are 8-(G) and !:l. - ( G); for outdegree we use 15+ (G) and !:l. + (G).
The digraph analogue of the degree-sum formula for graphs is easy.
## Section 1.4: Directed Graphs
59
LveV(G) d-(v).
1.4.18. Proposition. In a digraph G, LveV(G) d+(v) = e(G) = Proof: Every edge has exactly one tail and exactly one head.
(d+(v;), d-(v;)).
The digraph analogue of degree sequence is the list of "degree pairs" When is a list of pairs realizable as the degree pairs of a digraph? As with graphs, this is easy when we allow multiple edges.
1.4.19.* Proposition. A list of pairs of nonnegative integers is realizable as the degree pairs of a digraph if and only if the sum of the first coordinates equals the sum of the second coordinates. Proof: The condition is necessary because every edge has one tail and one head, contributing once to each sum. For sufficiency, consider the pairs {(dt, d;-): 1 ~ i ~ n} and vertices v1, ... , Vn. Let m = 'L dt = 'L dj-. Consider m dots. Give the dots positive labels, with dt of them having label i. Also give the dots negative labels, with dj- of them having label - j. For each dot with labels i and - j, place an edge from v; to Vj This creates a digraph with d+(v;) = dt and d-(v;) = d;-.
The analogous question for simple digraphs is har.der. The question can be rephrased in terms of bipartite graphs via a transformation that is useful in many problems about digraphs.
1.4.20.* Definition. The split of a digraph D is a bipartite graph G whose partite sets V +, v- are copies of V ( D). For each vertex x E V (D), there is one vertex x+ E v+ and one vertex x- E v-. For each edge from u to v in D, there is an edge with endpoints u+, v- in G.
w
1.4.21.* Remark. The degrees of the vertices in the split of D are the indegrees and outdegrees of the vertices in D. Furthermore, an X, Y-bigraph G with IXI = IYI = n can be transformed into an n-vertex digraph D by putting an edge V;Vj in D for each edge x;yj in G; now G is the split of D. (This is one reason to allow loops in simple digraphs.) Thus there is a simple digraph with degree pairs {(dt, d;-): 1 ~ i ~ n} if and only ifthere is a simple bipartite graph Gin which the vertex degrees are di, ... , d;i in one partite set and d!, ... , d;; in the other partite set. Exercise 32 obtains a recursive test for existence of such a bipartite graph. The statement and proof are like that of the Havel-Hakimi Theorem, so we leave further discussion to the exercise.
60
## Chapter 1: Fundamental Concepts
EULERIAN DIGRAPHS
The definitions of trail, walk, circuit, and the connection relation are the same in graphs and digraphs when we list edges as ordered pairs of vertices. In a digraph, the successive edges must "follow the arrows". In a walk v0 , e 1 , ... , ek. vki the edge e; has tail v;_ 1 and head v;.
1.4.22. Definition. An Eulerian trail in a digraph (or graph) is a trail containing all edges. An Eulerian circuit is a closed trail containing all edges. A digraph is Eulerian if it has an Eulerian circuit.
The characterization of Eulerian digraphs is analogous to the characterization of Eulerian graphs. The proof is essentially the same as for graphs, so we leave it to the exercises.
## 1, then G contains a cycle. The
Proof: Let P be a maximal path in G, and let u be the last vertex of P. Since P cannot be extended, every successor of u must already be a vertex of P. Since 8+(G) ~ 1, u has a successor v on P. The edge uv completes a cycle with the portion of P from v to u.
...
1.4.24. Theorem. A digraph is Eulerian if and only if d+ ( v) = d- ( v) for each vertex v and the underlying graph has at most one nontrivial component. Proof: See Exercise 19 or Exercise 20.
Every Eulerian digraph with no isolated vertices is strongly connected, although the characterization states that being weakly connected is sufficient.
1.4.25. Application. de Bruijn cycles. There are 2n binary strings oflength n. Is there a cyclic arrangement of 2n binary digits such that the 2n strings of n consecutive digits are all distinct? For n = 4, (0000111101100101) works. We can use such an arrangement to keep track of the position of a rotating drum (Good [1946]). Our drum has 2n rotational positions. A band around the circumference is split into 2n portions that can be coded 0 or 1. Sensors read n consecutive portions. If the coding has the property specified above, then the position of the drum is determined by the string read by the sensors. To obtain such a circular arrangement, define a digraph Dn whose vertices are the binary (n - 1)-tuples. Put an edge from a to b if the last n - 2 entries ofa agree with the first n - 2 entries of b. Label the edge with the last entry of b. Below we show D4 We next prove that Dn is Eulerian and show how an Eulerian circuit yields the desired circular arrangement.
## Section 1.4: Directed Graphs
61
001
1
011
1
1
0t
000
0 111 0
-1,
0 100 0
110
1.4.26. Theorem. The digraph D 11 of Application 1.4.25 is Eulerian, and the edge labels on the edges in any Eulerian circuit of Dn form a cyclic arrangement in which the 211 consecutive segments oflength n are distinct. Proof: We show first that D 11 is Eulerian. Every vertex has outdegree 2, because we can append a 0 or a 1 to its name to obtain the name of a successor vertex. Similarly, every vertex has indegree 2, because the same argument applies when moving in reverse and putting a 0 or a 1 on the front of the name. Also, Dn is strongly connected, because we can reach the vertex b = (b1, ... , b11 _1) from any vertex by successively following the edges lab~led b1 , ... , b11 _ 1 Thus D11 satisfies the hypotheses of Theorem 1.4.24 and is Eulerian. Let C be an Eulerian circuit of Dn. Arrival at vertex a = (a1, ... , a11 _1) must be along an edge with label a 11 _ 1 , because the label on an edge entering a vertex agrees with the last entry of the name of the vertex. Since we delete the front and shift the rest to obtain the rest of the name at the head, the successive earlier labels (looking backward) must have been a 11 _ 2 , . , a 1 in order. If C next uses an edge with label an, then the list consisting of the n most recent edge labels at that time is al, ... , an. Since the 2n-l vertex labels are distinct, and the two edges leaving each vertex have distinct labels, and we traverse each edge from each vertex exactly once along C, we have shown that the 211 strings of length n in the circular arrangement given by the edge labels along C are distinct.
The digraph D 11 is the de Bruijn graph of order n on an alphabet of size 2. It is useful for other purposes, because it has many vertices and few edges (only twice the number of vertices) and yet we can reach each vertex from any other by a short path. We can reach any desired vertex in n - 1 steps by introducing the bits in its name in order from the current vertex.
## ORIENTATIONS AND TOURNAMENTS
There are n 2 ordered pairs of elements that can be form~d from a vertex set of size n. A simple digraph allows loops but uses each ordered pair at most once as an edge. Thus there are n 2 ordered pairs that may or may not be present as 2 edges, and there are 2n simple digraphs with vertex set vi. ... , v11 Sometimes we want to forbid loops.
62
## Chapter 1: Fundamental Concepts
1.4.27. Definition. An orientation of a graph G is a digraph D obtained from G by choosing an orientation (x ---+ y or y ---+ x) for each edge xy E E(G). An oriented graph is an orientation of a simple graph. A tournament is an orientation of a complete graph.
An ori~nted graph is the same thing as a loopless simple digraph. When the edges of a graph represent comparisons to be performed among items corresponding to the vertices, we can record the results by putting x ---+ y when x does better than y in the comparison. The outcome is an orientation of G. The number of oriented graphs with vertices vi, ... , Vn is 3G); the number of tournaments is 2G).
1.4.28. Example. Orientations of complete graphs model "round-robin tournaments". Consider an n-team league where each team plays every other exactly once. For each pair u, v, we include the edge uv if u wins or vu if v wins. At the end of the season we have an orientation of Kn. The "score" of a team is its outdegree, which equals its number of wins. We therefore call the outdegree sequence of a tournament its score sequence. The outdegrees determine the indegrees, since d+ ( v) + d- ( v) = n - 1 for every vertex v. It is easier to characterize the score sequences of tournaments than the degree sequences of simple graphs (Exercise 35).
A tournament may have more than one vertex with maximum outdegree, so there may be no clear "winner"-in the example below, every vertex has outdegree 2 and indegree 2. Choosing a champion when several team!? have the maximum number of wins can be difficult. Although there need not be a clear winner, we show next that there must always be a team x such that, for every other team z, either x beats z or x beats some team that beats z.
1.4.29. Definition. In a digraph, a king is a vertex from which every vertex is r~achable by a path oflength at most 2. 1.4.30. Proposition. (Landau [1953)) Every tournament has a king. Proof: Let x be a vertex in a tournament T. If x is not a king, then some vertex y is not reachable from x by a path of length at most 2. Hence no successor of x is a predecessor of y. Since T is an orientation of a clique, every successor of x must therefore be a successor of y. Also y---+ x. Hence d+(y) > d+(x).
## Section 1.4: Directed Graphs
63
If y is not a king, then we repeat the argument to find z with yet larger outdegree. Since T is finite, we cannot forever obtain vertices of successively higher outdegree. The procedure must terminate, and it can terminate only when we have found a king.
In the language of extremality, we have proved that every vertex of maximum oq.tdegree in a tournament is a king. Exercises 36-38 ask further questions about kings (see also Maurer [1980]). Exercise 39 generalizes the result to arbitrary digraphs.
EXERCISES
1.4.1. (-) Describe a relation in the real world whose digraph has no cycles. Describe another that has cycles but is not symmetric. 1.4.2. (-) In the lightswitch system of Application 1.4.4, suppose the first switch becomes disconnected from the wiring. Draw the digraph that models the resulting system. 1.4.3. (-) Prove that every u, v-walk in a digraph contains au, v-path. 1.4.4. (-) Prove that every closed walk of odd length in a digraph -contains the edges of an odd cycle. (Hint: Follow Lemma 1.2.15.) 1.4.5. ( - ) Let G be a digraph in which indegree equals outdegree at each vertex. Prove that G decomposes into cycles. 1.4.6. (-) Draw the de Bruijn graphs D2 and D4 1.4.7. (-) Prove or disprove: If D is an orientation of.a simple graph with 10 vertices, then the vertices of D cannot have distinct outdegrees. 1.4.8. (- ) Prove that there is an n-vertex tournament with indegree equal to outdegree at every vertex if and only if n is odd .
1.4.9. For each n ~ 1, prove or disprove: Every simple digraph with n vertices has two vertices with the same outdegree or two vertices with the same indegree. 1.4.10. (!) Prove that a digraph is strongly connected if and only if for each partition of the vertex set into nonempty sets S and T, there is an edge from S to T. 1.4.11. (!) Prove that in every digraph, some strong component has no entering edges, and some strong component has no exiting edges. 1.4.12. Prove that in a digraph the connection relation is an equivalence relation, and its equivalence classes are the vertex sets of the strong components.
64
## Chapter 1: Fundamental Concepts
1.4.13. a) Prove that the strong components of a digraph are pairwise disjoint. b) Let D1 , .. , Dk be the strong components of a digraph D. Let D* be the loopless digraph with vertices v1 , . , vk such that v; ~ Vj if and only if i # j and D has an edge from D; to Dj. Prove that D* has no cycle.
D@]
z.
<}
1.4.14. (!) Let G be an n-vertex digraph with no cycles. Prove that the vertices of G can be ordered as vi. ... , v. so that ifv;Vj E E(G), then i < j. 1.4.15. Let G be the simple digraph with vertex set {(i, j) E IV: 0 .:5 i .:5 m and 0 .:5 n} and an edge from (i, j) to (i', j') if and only if (i', j') is obtained from (i, j) by adding 1 to one coordinate. Prove that the number of paths from (0, 0) to (m, n) in G is (m~"). 1.4.16. (+)Fermat's Little Theorem. Let denote the set of congruence classes of integers modulo n (see Appendix A). Let a be a natural number having no common prime factors with n; multiplication by a defines a permutation of Zn. Let l be the least natural number such that a1 =a mod n. a) Let G be the functional digraph with vertex set for the permutation defined by multiplication by a. Prove that all cycles in G (except the loop on n) have length/ - l. b) Conclude from part (a) that a- 1 1 mod n.
z.
1.4.17. (*)Prove that a (directed) odd cycle is a digraph with no kernel. Construct a digraph that has an odd cycle as an induced subgraph but does have a kernel. 1.4.18. (*)Prove that a digraph having no cycle has a unique kernel. 1.4.19. Use Lemma 1.4.23 and induction on the number of edges to prove the characterization of Eulerian digraphs (Theorem 1.4.24). (Hint: Follow Theorem 1.2.26.) 1.4.20. Prove the characterization of Eulerian digraphs (Theorem 1.4.24) using the notion of maximal trails. (Hint: Follow 1.2.32, the second proof of Theorem 1.2.26.) 1.4.21. Theorem 1.4.24 establishes necessary and sufficient conditions for a digraph to have an Eulerian circuit. Determine (with proof), the necessary and sufficient conditions for a digraph to have an Eulerian trail (Definition 1.4.22). (Good (1946]) 1.4.22. Let D be a digraph with d-(v) = d+(v) for every vertex v, except that d+(x) d-(x) = k = d-(y) - d+(y). Use the characterization of Eulerian digraphs to prove that
D contains k pairwise edge-disjoint x, y"-paths.
1.4.23. Prove that every gra~h G has an orientation D that is "balanced" at each vertex, meaning that ldZ(v) - d; (v) I .:5 1 for every v E V(G). 1.4.24. Prove or disprove: Every graph G has an orientation such that for every S V (G), the number of edges entering Sand leaving S differ by at most 1.
~
1.4.25. (!)Orientations and P3 -decomposition. a) Prove that every connected graph has an orientation in which the number of vertices with odd outdegree is at most 1. (Rotman (1991]) b) Use part (a) to conclude that a simple connected graph with an even number of edges can be decomposed into paths with two edges.
## Section 1.4: Directed Graphs
65
1.4.26. Arrange seven O's and seven l's cyclically so that the 14 strings of four consecutive bits are all the 4-digit binary strings other than 0101and1010. 1.4.27. DeBruijn sequence for any alphabet and length. Let A be an alphabet of size k. Prove that there exists a cyclic arrangement of k1 characters chosen from A such that the k1 str~gs oflength l in the sequence are all distinct. (Good [1946], Rees [1946))
m4
1.4.28. Let S be an alphabet of size m. Explain how to produce a cyclic arrangement of - m letters from S such that all four-letter strings of consecutive letters are different and contain at least two distinct letters.
1.4.29. (!) Suppose that G is a graph and D is an orientation of G that is strongly connected. Prove that if G has an odd cycle, then D has an odd cycle. (Hint: Consider each pair {v;, V;+d in an odd cycle (vi. ... , vk) of G.) 1.4.30. (+)Given a strong digraph D, let f(D) be the length of the shortest closed walk visiting every vertex. Prove that the maximum value of f(D) over all strong digraphs with n vertices isl (n + 1) 2 /4 Jifn:;:: 2. (Cull [1980]) 1.4.31. Determine the minimum n such that there is a pair of nonisomorphic n-vertex tournaments with the same list of outdegrees. 1.4.32. Let p = pi, ... , Pm and q = q1 , , q. be lists of nonnegative integers. The pair (p, q) is bigraphic ifthere is a simple bipartite graph in which Pi. ... , Pm are the degrees for one partite set and q 1 , .. , q. are the degrees for the other. When p has positive sum, prove that (p, q) is bigraphic if and only if (p', q') is bigraphic, where (p', q') is obtained from (p, q) by deleting the largest element /j,_ from p and subtracting 1 from each of the /j,_ largest elements of q. (Hint: Follow the method of Theorem 1.3.31.) 1.4.33. (*) Let A and B be two m by n matrices with entries in {O, 1). An exchange operation substitutes a submatrix of the form (~~)for a submatrix of the form(~~) or vice versa. Prove that if A and B have the same list of row sums and have the same list of column sums, then A can be transformed into B by a sequence of exchange operations. Interpret this conclusion in the context of bipartite graphs. (Ryser [1957]) 1.4.34. (!) Let G and H be two tournaments on a vertex set V. Prove that d~ (v) = d~ (v) for all v e V if and only if G can be turned into H by a sequence of direction-reversals on cycles of length 3. (Hint: Consider a vertex of maximum outdegree in the subgraph of G consisting of edges oriented oppositely in H .) (Ryser [1964])
p. be nonnegative integers with p Pn. Let p~ = p;. 1.4.35. (+) Let p Prove that there exists a tournament with outdegrees p1, .. , Pn if and only if p~ '.:::: for 1:::: k < n and p~ = (;). (Hint: Use induction on r:;= 1 [p~ - {;)J.) (Landau [1953])
,... , 1
1: : : :
r::=l
1.4.36. By Proposition 1.4.30, every tournament has a king. Let T be a tournament having no vertex with indegree 0. a) Prove that if x is a king in T, then T has another king in N-(x ). b) Use part (a) to prove that T has at least three kings. c) For each n ::::: 3, construct a tournament T with s- (T) > 0 and only 3 kings. (Comment: There exists an n-vertex tournament having exactly k kings whenever n ::::: k ::::: 1 except when k = 2 and when n = k = 4.) (Maurer [1980]) 1.4.37. Consider the following algorithm whose input is a tournament T. 1) Select a vertex x in T. 2) If x has indegree 0, call x a king of T and stop.
66
## Chapter 1: Fundamental Concepts
3) Otherwise, delete {x} U N+(x) from T to form T'. 4) Run the algorithm on T'; call the output a king in T and stop. Prove that this algorithm terminates and produces a king in T 1.4.38. (+) For n E N, prove that there is an n-vertex tournament in which every vertex is a king if and only if n : {2, 4}. 1.4.39. (+) Prove that every loopless digraph D has a set S of pairwise nonadjacent vertices such that every vertex outside S is reached from S by a path of length at most 2. (Hint: Use strong induction on n(D). Comment: This generalizes Proposition 1.4.30.) (Chvatal-Lovasz [1974]) 1.4.40. A directed graph is unipathic if for every pair of vertices x, y there is at most one (directed) x, y-path. Let T. be the tournament on n vertices with the edge between v; and Vj directed toward the vertex with larger index. What is the maximum number of edges in a unipathic subgraph of T.? How many unipathic subgraphs are there with the maximum number of edges? (Hint: Show that the underlying graph has no triangles.) (Maurer-Rabinovitch-Trotter [1980]) 1.4.41. Let G be a tournament. Let Lo be a listing of V ( G) in some order. If y immediately follows x in Lo but y -+ x in G, then yx is a reverse edge. We can interchange x and y in the order when yx is a reverse edge (this may increase the number of reverse edges). Suppose that a sequence L 0 , L 1 , is produced by successively switching one reverse edge in the current order. Prove that this always leads to a list with no reverse edges. Determine the maximum number of steps to termination. (Comment: In the special case where the vertices are numbers and each edge points to the higher number of the pair, the result says that successively switching adjacent numbers that are out of order always eventually sorts the list.) (Locke [1995]) 1.4.42. (!) Given an ordering u = v1 , , v. of the vertices of a tournament, let f (u) be the sum of the lengths of the feedback edges, meaning the sum of j - i over edges Vj v; such that j > i. Prove that every ordering minimizing f(u) places the vertices in nonincreasing order of outdegree. (Hint: Determine how f(u) changes when consecutive elements of u are exchanged.) (Kano-Sakamoto [1983], Isaak-Tesman [1991])
Chapter2
## Trees and Distance
2.1. Basic Properties
The word "tree" suggests branching out from a root and never completing a cycle. Trees as graphs have many applications, especially in data storage, searching, and communication.
2.1.1. Definition. A graph with no cycle is acyclic. A forest is an acyclic graph. A tree is a connected acyclic graph. A leaf (or pendant vertex) is a vertex of degree 1. A spanning subgraph of G is a subgraph with vertex set V ( G). A spanning tree is a spanning subgraph that is a tree.
2.1.2. Example. A tree is a connected forest, and every component of a forest is a tree. A graph with no cycles has no odd cycles; hence trees and forests are Bipartite. Paths are trees. A tree is a path if and only if its maximum degree is 2. A star is a tree consisting of one vertex adjacent to all the others. Then-vertex star is the biclique Ki,n-1 A graph that is a tree has exactly one spanning tree; the full graph itself. A spanning subgraph of G need not be connecteq, and a connected subgraph of G need not be a spanning subgraph. For example: Ifn(G) > 1, then the empty subgraph with vertex set V(G) and edge set 0 is spanning but not connected. If n(G) > 2, then a subgraph consisting of one edge and its endpoints is connected but not spanning.
67
68
## Chapter 2: Trees and Distance
PROPERTIES OF TREES
Trees have many equivalent characterizations, any of which could be taken as the definition. Such characterizations are useful because we need only verify that a graph satisfies any one of them to prove that it is a tree, after which we can use all the other properties. We first prove that deleting a leaf from ~ tree yields a smaller tree.
2.1.3. Lemma. Every tree with at least two vertices has at least two leaves. Deleting a leaf from an n-vertex tree produces a tree with n - 1 vertices. Proof: A connected graph with at least two vertices has an edge. In an acyclic graph, an endpoint of a maximal nontrivial path has no neighbor other than its neighbor on the path. Hence the endpoints of a such a path are leaves. Let v be a leaf of a tree G, and let G' = G - v. A vertex of degree 1 belongs to no path connecting two other vertices. Therefore, for u, w E V(G'), every u, w-path in G is also in G'. Hence G' is connected. Since deleting a vertex cannot create a cycle, G' also is acyclic. Thus G' is a tree with n - 1 vertices.
Lemma 2.1.3 implies that every tree with more than one vertex arises from a smaller tree by adding a vertex of degree 1 (all our graphs are finite). This rescues some proofs from the induction trap: growing an n + 1-vertex tree from an arbitrary 11-vertex tree by adding a new neighbor at an arbitrary old vertex generates all trees with n + 1 vertices. The word "arbitrary" means that the discussion considers all ways of making the choice. Our proof of equivalence of characterizations of trees uses induction, prior results, a counting argument, extremality, and contradiction.
2.1.4. Theorem. For an n-vertex graph G (with 11 '.:'.: 1), the following are equivalent (and characterize the trees with 11 vertices). A) G is connected and has no cycles. B) G is connected and has 11 - 1 edges. C) G has 11 - 1 edges and no cycles. D) For u, v E V(G), G has exactly one u, v-path. Proof: We first demonstrate the equivalence of A, B, and C by proving that any two of {connected, acyclic, n - 1 edges} together imply the third. A::::} {B, C}. We use induction on 11. For n = 1, an acyclic 1-vertex graph has no edge. For 11 > 1, we suppose that the implication holds for graphs with fewer than n vertices. Given an acyclic connected graph G, Lemma 2.1.3 provides a leaf v and states that G' = G - v also is acyclic and connected (see figure above). Applying the induction hypothesis to G' yields e(G') = 11 - 2. Since only one edge is incident to v, we have e(G) = 11 - 1. B ::::} {A, C}. Delete edges from cycles of G one by one until the resulting graph G' is acyclic. Since no edge ofa cycle is a cut-edge (Theorem 1.2.14), G' is
## Section 2.1: Basic Properties
69
connected. Now the preceding paragraph implies that e(G') = n - 1. Since we are given e(G) = n - 1, no edges were deleted. Thus G' = G, and G is acyclic .. C =>(A, B}. Let Gi, ... , Gk be the components of G. Since every vertex appears in one component, L; n(G;) = n. Since G has no cycles, each component satisfies property A. Thus e(G;) = n(G;) - 1. Summing over i yields e(G) = L;[n(G;) - 1] = n - k. We are given e(G) = n -1, so k = 1, and G is connected. A => D. Since G is connected, each pair of vertices is connected by a path. If some pair is connected by more than one, we choose a shortest (total length) pair P, Q of distinct paths with the same endpoints. By this extremal choice, no internal vertex of P or Q can belong to the other path (see figure below). This implies that P U Q is a cycle, which contradicts the hypothesis A. D =>A. If there is au, v-path for every u, v E V(G), then G is connected. If G has a cycle C, then G has two u, v-paths for u, v E V(C); hence G is acyclic (this also forbids loops). p Q
-ex>
0v
2.1.5. Corollary. a) Every edge of a tree is a cut-edge. b) Adding one edge to a tree forms exactly one cycle. c) Every connected graph contains a spanning tree. Proof: (a) A tree has no cycles, so Theorem 1.2.14 implies that every edge is a cut-edge. (b) A tree has a unique path linking each pair of vertices (Theorem 2.l.4D), so joining two vertices by an edge creates exactly one cycle. (c) As in the proofof B=:>A,C in Theorem 2.1.4, iterativsly deleting edges from cycles in a connected graph yields a connected acyclic subgraph.
We apply Corollary 2.1.5 to prove two results about pairs of spanning trees. We use subtraction and addition to indicate deletion and inclusion of edges. 2.1.6. Proposition, If T, T' are spanning trees of a connected graph G and e E E(T) - E(T'), then there is an edge e' E E(T') - E(T) such that T - e + e' is a spanning tree of G. Proof: By Corollary 2.l.5a, every edge of T is a cut-edge of T. Let U and U' be the two components of T - e. Since T' is connected, T' has an edge e' with endpoints in U and U'. Now T - e + e' is connected, has n(G) - 1 edges, and is a spanning tree of G. (In the figure below, T is bold, T' is solid, and they share two edges.)
70
## Chapter 2: Trees and Distance
2.1.7. Proposition. If T, T' are spanning trees of a connected graph G and e E E(T) - E(T'), then there is an edge e' E E(T') - E(T) such that T' + e - e' is a spanning tree of G. Proof: By Corollary 2.l.5b, The graph T' + e contains a unique cycle C. Since Tis acyclic, there is an edge e' E E(C) - E(T). Deleting e' breaks the only cycle in T' + e. Now T' + e - e' is connected and acyclic and is a spanning tree of G. (In the figure above, adding e to T creates a cycle C of length five; all four edges of C - e belong to E(T) - E(T') and can serVe as e'.)
The edge e' can be chosen to satisfy the conclusions of Propositions 2.1.62.1.7 simultaneously, as illustrated in the figure between them (Exercise 37). The next result illustrates proof by induction using deletion of a leaf.
2.1.8. Proposition. If T is a tree with k edges and G is a simple graph with 8(G) ::::_ k, then Tis a subgraph of G. Proof: We use induction on k. Basis step: k = 0. Every simple graph contains Ki, which is the only tree with no edges. Induction step: k > 0. We assume that the claim holds for trees with fewer thank edges. Since k > 0, Lemma 2.1.3 allows us to choose a leaf v in T; let u be its neighbor. Consider the smaller tree T' = T - v. By.the induction hypothesis, G contains T' as a subgraph, since 8(G) ::::_ k > k - 1. Let x be the vertex in this copy of T' that corresponds to u (see illustration). Because T' has only k - 1 vertices other than u and de (x) ::::_ k, x has a neighbor y in G that is not in this copy of T'. Adding the edge xy expands this copy of T' into a copy of T in G, with y playing the role of v.
The inequality of Proposition 2.1.8 is sharp; the graph Kk has minimum degree k - 1, but it contains no tree with k edges. The proposition implies that every n-vertex simple graph G with more than n(k - 1) edges has T as a subgraph (Exercise 34). Erdos and Sos conjectured the stronger statement that e(G) > n(k - 1)/2 forces T as a subgraph (Erdos [1964]). This has been proved for graphs without 4-cycles (Sacle-Wo:lniak [1997]). Ajtai, Komlos, and Szemeredi proved an asymptotic version, as reported in Soffer [2000].
## DISTANCE IN TREES AND GRAPHS
When using graphs to model communication networks, we want vertices to be close together to avoid communication delays. We measure distance using lengths of paths.
## Section 2.1: Basic Properties
71
2.1.9. Definition. If G has au, v-path, then the distance from u to v, written dG(u, v) or simply d(u, v), is the least length ofa u, v-path. If G has no such path, then d(it, v) = oo. The diameter (diam G) is max,,,veV(G) d(u, v). The eccentricity of a vertex u, written E(u), is maxveV(G) d(u, v). The radius of a graph G, written rad G, is minueV(G) E(u).
The diameter equals the maximum of the vertex eccentricities. In a disconnected graph, the diameter and radius (and every eccentricity) are infinite, because distance between vertices in different components is infinite. We use the word "diameter" due to its use in geometry, where it is the greatest distance between two elements of a set.
2.1.10. Example. The Petersen graph has diameter 2, since nonadjacent vertices have a common neighbor. The hypercube Qk has diameter k, since it takes k steps to change all k coordinates. The cycle Cn has diameter Ln/2J. In each of these, every vertex has the same eccentricity, and diam G = rad G. For n 2'.: 3, the n-vertex tree of least diameter is the star, with diameter 2 and radius 1. The one oflargest diameter is the path, with diameter n - 1 and radius f(n - 1)/21. Every path in a tree is the shortest (the only!) path between its endpoints, so the diameter of a tree is the length of its longest path. In the graph below, each vertex is labeled with its eccentricity. The radius is 2, the diameter is 4, and the length of the longest path is 7. II
~:
3 4 3
To have large diameter, many edges must be missing. Thus we expect the complement of a graph with large diameter to have small diameter. We use the simple observation that a graph has diameter at most 2 if and only if nonadjacent vertices always have common neighbors (see also Exercise 15).
## 2.1.11. Theorem. If G is a simple graph, then diam G 2'.: 3 :::} diam G
3.
Proof: When diam G > 2, there exist nonadjacent vertices u, v E V (G) with no common neighbor. Hence every x E: V(G) - {u, v} has at least one of {u, v} as a nonneighbor. This makes x adjacent in G to at least one of {u, v} in G. Since also uv E E(G), for every pair x, y there is an x, y-path oflength at most 3 in G through {u, v}. Hence diam G ~ 3.
72
## Chapter 2: Trees and Distance
2.1.12. Definition. The center of a graph G is the subgraph induced by the vertices of minimum eccentricity. The center of a graph is the full graph if and only ifthe radius and diameter are equal. We next describe the centers of trees. In the induction step, we delete all leaves instead of just one. 2.1.13. Theorem. (Jordan [1869]) The center of a tree is a vertex or an edge. Proof: We use induction on the number of vertices in a tree T. Basis step: n(T) ~ 2. With at most two vertices, the center is the entire tree. Induction step: n (T) > 2. Form T' by deleting every leaf of T. By Lemma 2.1.3, T' is a tree. Since the internal vertices on paths between leaves of T remain, T' has at least one vertex. Every vertex at maximum distance in T from a vertex u E V (T) is a leaf (otherwise, the path reaching it from u can be extended farther). Since all the leaves have been removed and no path between two other vertices uses a leaf, Er(u) = ET(u) - 1 for every u E V(T'). Also, thll eccentricity of a leaf in T is greater than the eccentricity of its neighbor in T. Hence the vertices minimizing ET(u) are the same as the vertices minimizing ET'(u). We have shown that T and T' have the same center. By the induction hypothesis, the center of T' is a vertex or an edge.
In a communication network, large diameter may be acceptable if most pairs can communicate via short paths. This leads us to study the average distance instead of the maximum. Since the average is the sum divided by (;) (the number of vertex pairs), it is equivalent to study D(G) = Lu,veV(G) dG(u, v). The sum D(G) has been called the WienerindexofG (also written W(G)). Wiener used it to study the boiling point of paraffin. Molecules can be modeled by graphs with vertices for atoms and edges for atomic bonds. Many chemical properties of molecules are related to the Wiener index of the corresponding graphs. We study the extreme values of D(G).
2.1.14. Theorem. Among trees with n vertices, the Wiener index D(T) = Lu.v d(u, v) is minimized by stars and maximized by paths, both uniquely. Proof: Since a tree has n - 1 edges, it has n - 1 pairs of vertices at distance 1, and all other pairs have distance at least 2. The star achieves this and hence minimizes D(T). To show that no other tree achieves this, consider a leaf x in T, and let v be its neighbor. If all other vertices have distance 2 from x, then they must be neighbors of v, and T is the star. The value is
D(K1,n-1) = (n - 1) + 2(n; 1) = (n - 1) 2
## Section 2.1: Basic Properties
73
For the maximization, consider first D(Pn). This equals the sum of the distances from an endpoint u to the other vertices, plus D(Pn_ 1). We have LveV(P.) d(u, v) = L:7,;;-~ i = (;). Thus D(Pn) = D(Pn-1) + (;). With Pascal's 1 1 Formula(~)+ (k~ 1 ) = ) (see Appendix A), induction yields D(Pn) = (n~ ).
et
## <::: Pn-1 ::>
We prove by induction on n that among n-vertex tree, Pn is the only tree that maximizes D(T). Basis step: n = 1. The only tree with one vertex is P1 . Induction step: n > 1. Let u be a leaf of an n-vertex tree T. Now D(T) = D(T - u) + LveV(T) d(u, v). By the induction hypothesis, D(T - u) ~ D(Pn-1), with equality if and only if T - u is a path. Thus it suffices to show that LveV(T) d(u, v) is maximized only when Tis a path and u is an endpoint of T. Consider the list of distances from u. In Pn, this list is 1, 2, ... , n - 1, all distinct. A shortest path from u to a vertex farthest from u contains vertices at all distances from u, so in any tree the set of distances from u to other vertices has no gaps. Thus any repetition makes LveV(T) d(u, v) smaller than when u is a leaf of a path. When T is not a path, such a repetition occurs. Over all connected n-vertex graphs, D( G) is minimized by Kn. The maximization problem reduces to what we have already done with trees. 2.1.15. Lemma. If His a subgraph of G, then da(u, v) ~ dH(u, v). Proof: Every u, v-path in H appears also in G, so the shortest u, v-path in G is no longer than the shortest u, v-path in H. 2.1.16. Corollary. If G is a connected n-vertex graph, then D(G) Proof: Let T be a spanning tree of G. By Lemma 2.1.15, D(G) Theorem 2.1.14, D(T) ~ D(Pn).
~ D(Pn). ~ D(T).
By
## DISJOINT SPANNING TREES (optional)
We have seen that .every connected graph has a spanning tree. Edgedisjoint spanning trees provide alternate routes when an edge in the primary tree fails. Tutte [1961a] and Nash-Williams (1961] independently characterized graphs having k pairwise edge-disjoint spanning trees (see Exercise 67). We describe one application of edge-disjoint spanning trees. David Gale devised a game marketed under the name "Bridg-it" (copyright 1960 by Hassenfeld Bros., Inc.-"Hasbro Toys"). Each of two players owns a rectangular grid of posts. The players move alternately, at each move joining two of their own posts by a unit-length bridge. The figure on the left below illustrates the board; Player l's posts are solid, and Player 2's are hollow. The object of Player 1 is to construct a path of bridges from the left column to the right column; Player 2 wants a path of bridges from the top row to the bottom row.
74
## Chapter 2: Trees and Distance
Bridges cannot cross. Therefore, every bridge that is played eliminates a potential move for the other player. Since every path from left to right crosses every path from top to bottom, the players cannot both win. Note also that the layout of the board is sytnmetric in the two players. We argue that Player 2 cannot have a winning strategy. Suppose otherwise. Because the board is symmetric, Player 1 can start with any move and then followthe strategy of Player 2, making '!n arbitrary move if the strategy of Player 2 ever calls for a bridge that has already been played. Before Player 2 can win, Player 1 wins by using the same strategy. If the game is played until no further moves are possible, then some player must have won (Exercise 70). Since Player 2 has no winning strategy, this implies that Player 1 has a winning strategy. Here we give an explicit strategythat Player 1 '.!an use to win. (The argument holds more generally in the contextof "matroi<ls"-see Theorem 8.2.46.)
0 0 0 0
fF:
~
~
---\
\
\
\
\ \ \ \
\ \
''
\ \
'
\
2.1.17. Theorem. Player 1 has a winning strategy in Bridg-' t. Proof: We form a graph of the potential connections for Player 1. Posts on the same end are equivalent, so we collect the (solid) posts from the end columns as single vertices. We add an auxiliary edge between the ends. The figure on the right above illustrates that this graph is the union of two edge-disjoint spanning trees; we omit a technical description of the two trees. Together, the two trees contain edge-disjoint paths between the goal vertices. Since the auxiliary edge doesn't really exist, we prf'tend Piayer 2 moved first and took that edge. A move by Player 2 cuts one edge e in the graph and makes it no longer available. This cuts one of the trees into two components. By Proposition 2.1.6, some edge e' from the other tree reconnects it. Player 1 chooses such an edge e'. This makes e' uncuttable, in effect putting e' in both spanning trees. After deleting e and making e' a double edge with one copy in each tree, our graph still consists of two edge-disjoint spanning trees. Since Player 2 cannot cut a double edge, Player 2 cannot cut both trees. Thus Player 1 can always defend. The figure below illustrates the strategy. The process stops when Player 1 has won or when no single edges remain to be cut. In the latter case the remaining edges are double edges and form a
## Section 2.1: Basic Properties
75
spanning tree of bridges built by Player L 'I'hus in either case Player 1 has constructed a path connecting the special vertices. a
,, ,,
'
,~........
, ..... ..,
,'
'
----
----
:
I
'\
',
,~
,, ,,
1
..... "'
I,,...,
, ,, '
'
## ,' ---- ----I"
I
I
,,.....
- - --
''
'
'\
\
'
'
\~......
''
''
''
'
Player 1 reconnects
Player 2 cuts
Player 1 reconnects
EXERCISES
2.1.1. (-) For each k, list the isomorphism ciasses of trees with maximum degree k ar.d at most six vertices. Do the same for diameter k. (Explain why there are no others.) 2.1.2. (- ) Let G be a graph. a) Prove that G is a tree if and only if G is connected and every edge is a cut-edge. b) Prove that G is a tree if and only if adding any edge with endpoints in V(G) creates exactly one cycle. 2.1.3. (-) Prove that a graph is a tree if and only if it is loopless and has exactly one spanning tree. 2.1.4. ( - ) Prove or disprove: Every graph with fewer edges than vertices has a component that is a tree. 2.1.5. ( - ) Let G be a graph. Prove that a maximal acyclic subgraph of G consists of a spar.ning tree from each component of d. 2.1.6. (-) Let T be a tree with average degree a. In terms of a, determine n(T). 2.1. 7. (-) Prove that every n-vertex graph with m edges has at least m - n + 1 cycles. 2.1.8. ( - ) Prove that each property below characterizes forests. a) Every induced subgraph has a vertex of degree at most 1. b) Every connected subgraph is an induced subgraph. c) The number of components is the number of vertices minus the number of edges. 2.1.9. (-) For 2 .:'.':: k .:'.':: n - 1, prove that then-vertex graph formed by adding one vertex adjacent to every vertex of Pn-l has a spanning tree with diameter k.
d(u, v) > 2, then d(u)
2.1.10. (-)Let u and v be vertices in a connected n-vertex simple graph. Prove that if + d(v) .:'.':: n + 1- d(u, v). Construct exampl~s to show that this can fail whenever n :'.':: 3 and d(u, v) .:'.':: 2. 2.1.11. (-)Let x and y be adjacent vertices in a graph G. For all ldG(x, z) - da(y, z)I .:'.':: 1.
zE
## V(G), prove that
2.1.12. (-)Compute the diameter and radius of the biclique Km.n 2.1.13. (-) Prove that every graph with diameter d has an independent set with at least (1 + d) /21 vertices.
76
## Chapter 2: Trees and Distance
2.1.14. (-) Suppose that the processors in a computer are named by binary k-tuples, and pairs can communicate directly if and only if their names are adjacent in the kdimensional cube Qk. A processor with name u wants to send a message to the processor with name v. How can it find the first step on a shortest path to v? 2.1.15. (-) Let G be a simple graph with diameter at least 4. Prove that G has diameter at most 2. (Hint: Use Theorem 2.1.11.) 2.1.16. (-) Given a simple graph G, define G' to be the simple graph on the same vertex set such that xy E E(G') if and only if x and y are adjacent in G or have a common neighbor in G. Prove that diam (G') = fdiam (G)/21.
2.1.17. (!) Prove C =? {A, BJ in Theorem 2.1.4 by adding edges to connect components. 2.1.18. (!) Prove that every tree with maximum degree ~ > 1 has at least ~ vertices of degree 1. Show that this is best possible by constructing an n-vertex tree with exactly ~ leaves, for each choice of n, ~with n > ~ ::: 2. 2.1.19. Prove or disprove: If n; denotes the number of vertices of degree i in a tree T, then z)n; depends only on the number of vertices in T. 2.1.20. A saturated hydrocarbon is a molecule formed from k carbon atoms and l hydrogen atoms by adding bonds between atoms such that each carbon atom is in four bonds, each hydrogen atom is in one bond, and no sequence of bonds forms a cycle of atoms. Prove that l = 2k + 2. (Bondy-Murty [1976, p27]) 2.1.21. Let G be an n-vertex simple graph having a decomposition into k spanning trees. Suppose also that ~(G) = 8(G) + 1. For 2k::: n, show that this is impossible. For 2k < n, determine the degree sequence of G in terms of n and k. 2.1.22. Let T be an n-vertex tree having one vertex of each degree i with 2 remaining n - k + 1 vertices are leaves. Determine n in terms of k.
~
k; the
2.1.23. Let T be a tree in which every vertex has degree 1 or degree k. Determine the possible values of n(T). 2.1.24. Prove that every nontrivial tree has at least two maximal independent sets, with equality only for stars. (Note: maximal # maximum.) 2.1.25. Prove that among trees with n vertices, the star has the most independent sets. 2.1.26. (!) For n ::: 3, let G be an n-vertex graph such that every graph obtained by deleting one vertex is a tree. Determine e(G), and use this to determine G itself. 2.1.27. (!) Let di. ... , dn be positive integers, with n ::: 2. Prove that there exists a tree with vertex degrees di, ... , dn if and only if L d; = 2n - 2. 2.1.28. Let di ::: ::: dn be nonnegative integers. Prove that there exists a connected graph (loops and multiple edgei;; allowed) with degree sequence di, ... , dn if and only if L d; is even, dn ::: 1, and L d; ::: 2n - 2. (Hint: Consider a realization with the fewest components.) Is the statement true for simple graphs? 2.1.2if. (!) Every tree is bipartite. Prove that every tree has a leaf in its larger partite set.(in both if they have equal size). 2.1.30. Let T be a tree in which all vertices adjacent to leaves have degree at least 3. Prove that T has some pair of leaves with a common neighbor.
## Section 2.1: Basic Properties
77
2.1.31. Prove that a simple connected graph having exactly two vertices that are not cut-vertices is a path. 2.1.32. Prove that an edge e of a connected graph G is a cut-edge if and only if e belongs to every spanning tree. Prove that e is a loop if and only if e belongs to no spanning tree. 2.1.33. (!) Let G be a connected n-vertex graph. Prove that G has exactly one cycle if and only if G has exactly n edges. 2.1.34. (!) Let T be a tree with k edges, and let G be a simple n-vertex graph with more than n(k - 1) - G) edges. Use Proposition 2.1.8 to prove that T <:; G ifn > k. 2.1.35. (!) Let T be a tree. Prove that the vertices of T all have odd degree if and only iffor all e E E(T), both components of T - e have odd order. 2.1.36. (!) Let T be a tree of even order. Prove that T has exactly one spanning subgraph in which every vertex has odd degree. 2.1.37. (!)Let T, T' be two spanning trees ofa connected graph G. Fore E E(T)- E(T'), prove that there is an edge e' E E(T') - E(T) such that T' + e - e' anci T - e + e' are both spanning trees of G. 2.1.38. Let T, T' be two trees on the same vertex set such that dr(v) = d~.(v) for each vertex v. Prove that T' can be obtained from T' using 2-switches (Definition 1.3.32) so that every graph along the way is also a tree. (Kelmans [1998)) 2.1.39. (!) Let G be a tree with 2k vertices of odd degree. Prove that G decomposes into k paths. (Hint: Prove the stronger result that the claim holds for all forests.) 2.1.40. (!)Let G be a tree with k leaves. Prove that G is the union of paths such that P; n Pj =j:. 0 for all i =j:. j. (Ando-Kaneko-Gervacio [1996])
Pi, .. , Prk/Zl
2.1.41. For n '.::: 4, let G be a simple n-vertex graph with e(G) '.::: 2n - 3. Prove that G has two cycles of equal length. (Chen--Jacobson-Lehel-Shreve [1999] strengthens this.) 2.1.42. Let G be a connected Eulerian graph with at least three vertices. A vertex v in G is extendible if every trail beginning at v can be extended to form an Eulerian circuit of G. For example, in the graphs below only the marked vertices are extendible. Prove the following statements about G (adapted from Chartrand-Lesniak [1986, p61]). a) A vertex v E V(G) is extendible if and only if G - vis a forest. (Ore [1951]) b) If vis extendible, then d(v) = L'.l(G). (Bahler [1953]) c) All vertices of G are extendible if and only if G is a cycle. d) If G is not a cycle, th~n G has at most two extendible vertices.
[X]
2.1.43. Let u be a vertex in a connected graph G. Prove that it is possible to select shortest paths from u to all other vertices of G so that the union of the paths is a tree. 2.1.44. (!) Prove or disprove: If a simple graph with diameter 2 has a cut-vertex, then its complement has an isolated vertex. 2.1.45. Let G be a graph having spanning trees with diameter 2 and diameter l. For 2 < k < l, prove that G also has a spanning tree with diameter k. (Galvin)
78
## Chapter 2: Trees and Distance
2.1.46. (!) Prove that the trees with diameter 3 are the double-stars (two central vertices plus leaves). Count the isomorphism classes of double-stars with n vertices.
## Section 2.1: Basic Properties
79
2.1.56. Let T be a tree. Prove that T has a vertex v such that for all e E E(T), the component of T - e containing v has at least fn(T)/21 vertices. Prove that either v is unique or there are just two adjacent such vertices. 2.1.57. Let ni. ... , nk be positive integers with sum n - 1. a) By counting edges in complete graphs, prove that L~=l (nd) 5 (n; 1). b) Use part (a) to prove that LvEV(T) d(u, v) 5 (;) when u is a vertex of a tree T. (Hint: Use strong induction on the number of vertices.) 2.1.58. (+)Let Sand T be trees with leaves {x 1 , . , xd and {y1 , .. , yd, respectively. Suppose that ds(x;, Xj) = dr(y;, Yi) for each pair i, j . .Prove that Sand Tare isomorphic. (Smolenskii [1962]) 2.1.59. (!) Let G be a tree with n vertices, k leaves, and maximum degree k. a) Prove that G is the union of k paths with a common endpoint. b) Determine the maximum and minimum possible values of diam G. 2.1.60. Let G be a graph with diameter d and maximum degree k. Prove that n(G) 5 1 + [(k - l)d - l]k/(k - 2). (Comment: Equality holds for the Petersen graph.) 2.1.61. (+) Let G be a graph with smallest order among k-regular graphs with girth at least g (Exercise 1.3.16 establishes the existence of such graphs). Prove that G has diameter at most g. (Hint: If da(x, y) > g, modify G to obtain a smaller k-regular graph with girth at least g.) (Erdos-Sachs [1963]) 2.1.62. (!) Let G be a connected graph with n vertic~s. Define a new graph G' having one vertex for each spanning tree of G, with vertices adjacent in G' if and only if the corresponding trees have exactly n(G) - 2 common edges. Prove that G' is connected. Determine the diameter of G'. An example appears below.
G'
2.1.63. (!) Prove that every n-vertex graph with n + 1 edges has girth at most L(2n + 2)/3j. For each n, construct an example achieving this bound. 2.1.64. (!) Let G be a connected graph that is not a tree. Prove that some cycle in G has length at most 2(diam G) + 1. For each k E N, show that this is best possible by exhibiting a graph with diameter k and girth 2k + 1. 2.1.65. (+) Let G be a connected n-vertex graph with minimum degree k, where k ::: 2 and n - 2 ::: 2(k + 1). Prove that diam G 5 3(n - 2)/(k + 1) - 1. Whenever k ::: 2 and (n - 2)/(k + 1) is an integer greater than 1, construct a graph where the bound holds with equality. (Moon [1965b]) 2.1.66. Let Fi' ... ' Fm be forests whose union is G. Prove that m ::: maXHs;G .~~~~1 (Comment: Nash-Williams [1964] and Edmonds [1965b] proved that this bound is always achieveable-Corollary 8.2.57).
80
Chapt~r
## 2: Trees and Distance
2.1.67. Prove that the following is a necessary condition for the existence of k pairwise edge-disjoint spanning trees in G: for every partition of the vertices of G into r sets, there are at least k(r -1) edges of G whose endpoints are in different sets of the partition. (Comment: Corollary 8.2.59 shows that this condition is also sufficient - Tutte [1961a], Nash-Williams [1961], Edmonds [1965c].) 2.1.68. Can the graph below be decomposed into edge-disjoint spanning trees? Into isomorphi<' tidge-disjoint spanning trees?
2.1.69. (*) Consider the graph before Theorem 2.1.17 with 12 vertical edges and 16 edges that are horizontal or slanted. Let g;, 1 be the ith edge from the top in the jth column of vertical edges. Let h;, 1 be the jth edge from the left in the ith row of horizontal/diagonal edges. Suppose that Player 1 follows the strategy of Theorem 2.1.17 and first takes hu. Player 2 deletes g2, 2 , and Player 1 takes h 2, 3 Next Player 2 deletes v3.2, and Player 1 takes h4.2. Draw the two spanning trees at this point. Given that Player 2 next deletes g2,i. list all moves available to Player 1 within the strategy. (Pritikin) 2.1.70. (*) Prove that Bridg-it cannot end in a tie no matter how the moves are made. That is, prove that when no further moves can be made, one of the players must have built a path connecting his/her goals. 2.1.71. (*)The players change the rules ofBridg-it so that a player with path between friendly ends is the loser. It is forbidden to stall by building a bridge joining end posts or joining posts already connected by a path. Show that Player 2 has a strategy that forces Player 1 to lose. (Hint: Use Proposition 2.1.7 instead of Proposition 2.1.6.) (Pritikin) 2.1.72. (+)Prove that ifGi. ... , Gk are pairwise-intersecting subtrees ofa tree G, then G has a vertex that belongs to all of Gi. ... , Gk. (Hint: Use induction on k. Comment: This result is the Belly property for trees.) 2.1. 73. (+) Prove that a simple graph G is a forest if and only if for every pairwise intersecting family of paths in G, the paths have a common vertex. (Hint: For sufficiency, use induction on the size of the family of paths.) 2.1.74. Let G be a simple n-vertex graph having n -2 edges. Prove that G has an isolated vertex or has two components that are nontrivial trees. Use this to prove indu~tively that G is a subgraph ofG. tComment: The claim is not true for all graphs with n - 1 edges.) (Bums-Schuster [1977]) 2.1.75. (+)Prove that every n-vertex tree other than Ki,n-l is contained in its complement. (Hint: Use induction on n to prove a stronger statement: if T is an n-vertex tree other than a star, then Kn contains two edge-disjoint copies of T in which the two copies of each non-leaf vertex of T appear at distinct vertices.) (Bums-Schuster [1978]) 2.1.76. (+) Let S be an n-element set, and let {A 1 , ... , An) be n distinct subsets of S. Prove that S has an element x such that the sets A1 U {x }, ... , An U {x} are distinct. (Hint: Define a graph with vertices ai. ... , an such that a; ~ a1 if and only if one of {A;, A1 } is obtained from the other by adding a single element y. Use y as a label on the edge. Prove that there is a forest consisting of one edge with each label used. Use this to obtain the desired x.) (Bondy [1972a])
81
## 2.2. Spanning Trees and Enumeration
There are 2(;) simple graphs with vertex set [n] = { 1, ... , n}, since each pair may or may not form an edge. How many of these are trees? In this section, we solve this counting problem, count spanning trees in arbitrary graphs, and discuss several applications.
ENUMERATION OF TREES
With one or two vertices, only one tree can be formed. With three vertices there is still only one isomorphism class, but the adjacency matrix is determined by which vertex is the center. Thus there are three trees with vertex set [3]. With vertex set [4], there are four stars and 12 paths, yielding 16 trees. With vertex set [5], a careful study yields 125 trees.
a
~ c d
/V
c
Now we may see a pattern. With vertex set [n], there are nn- 2 trees; this is Cayley's Formula. Priifer, Kirchhoff, P6lya, Renyi, and others found proofs. J.W. Moon [1970] wrote a book about enumerating classes of trees. We present a bijective proof, establishing a one-to-one correspondence between the set of trees with vertex set [n] and a set of known size. Given a set S of n numbers, there are exactly nn- 2 ways to form a list of length n - 2 with entries in S. The set oflists is denoted sn- 2 (see Appendix A). We use sn- 2 to encode the trees with vertex set S. The list that results from a tree is its Priifer code.
2.2.1. Algorithm. (Priifer code) Production off (T) = (ai. ... , an-2). Input: A tree T with vertex set S s; N. Iteration: At the ith step, delete the least remaining leaf, and let neighbor of this leaf.
a;
be the
2.2.2. Example. After n - 2 iterations, only one of the original n - 1 edges remains, and we have produced a list f(T) oflength n - 2 with entries in S. In the tree below, the least leaf is 2; we delete it and record 7. After deleting 3 and 5 and recording 4 each time, the least leaf in the remaining 5-vertex tree is 4. The full code is (744171), and the vertices remaining at the end are 1 and 8. After the first step, the remainder of the Priifer code is the Priifer code of the subtree T' with vertex set [8] - {2}.
82
2
## Chapter 2: Trees and Distance
5 If we know the vertex set S, then we can retrieve the tree from the code a. The idea is to retrieve all the edges. We start with the set S of isolated vertices. At each step we create one edge and mark one vertex. When we are ready to consider a;, there remain n - i + 1 unmarked vertices and n - i -:- 1 entries of a (including a;). Thus at least two of the unmarked vertices do not appear among the remaining entries of a. Let x be the least of these, add xa; to the list of edges, and mark x. After repeating this n - 2 times, two unmarked vertices remain;. we join them to form the final edge. In the example above, the least element of S not in the code is 2, so the first edge added joins 2 and 7, and we mark 2. Now the least unmarked element absent from the rest is 3, and we join it to 4, which is a 2 As we continue, we reconstruct edges in the order they were deleted to obtain a from T. Throughout the process, each component of the graph we have grown has one unmarked vertex. This is true initially, and thus adding an edge with two unmarked endpoints combines two components. After marking one vertex of the new edge, again each component has one unmarked vert~. After n - 2 steps, we have two unmarked vertices and therefore two components. Adding the last edge yields a connected graph. We have built a connected graph with n vertices and n - 1 edges. By Theorem 2.l.4B, it is a tree, but we have not yet proved that its Priifer code is a.
6
8
III
2.2.3. Theorem. (Cayley's Formula [1889]). For a set S nn- 2 trees with vertex set S.
## s; N of size n, there are
Proof: (Priifer [1918]). This holds for n = 1, so we assume n 2:: 2. We prove that Algorithm 2.2.1 defines a bijection f. from the set of trees with vertex set S to the set sn- 2 of lists of length n - 2 from S. We. must show for each a = (a1, ... , an-2) E sn- 2 that exactly one tree T with vertex set S satisfies f(T) =a. We prove this by induction on n. Basis step: n = 2. There is tree with two vertices. The Priifer code is a list oflength 0, and it is the only such list. Induction step: n > 2. Computing f (T) reduces each vertex to degree 1 and then possibly deletes it. Thus every nonleafvertex in T appears in f(T). No leaf appears, because recording a leaf as a neighbor of a leaf would require reducing the tree to one vertex. Hence the leaves of T are the elements of S not in f(T). If f(T) =a, then the first leaf deleted is the least element of Snot in a (call it x), and the neighbor of xis a 1 . We are given a E sn- 2 and seek all solutions to f (T) =a. We have shown that every such tree has x as its least leaf and has the edge xa 1. Deleting x leaves a tree with vertex set S' = S - {x}. Its Priifer code is a'= (a 2, ... , an_ 2), an n - 3-tuple formed from S'.
## Section 2.2: Spanning Trees and Enumeration
83
By the induction hypothesis, there exists exactly one tree T' having vertex set S' and Prilfer code a'. Since every tree with Prii.fer code a is formed by adding the edge xa1 to such a tree, there is at most one solution to f(T) = a. Furthermore, adding xa 1 to T' does create a tree with vertex set Sand Prii.fer code a, so there is at least one solution. Cayley approached the problem algebraically and counted the trees by their vertex degrees. Prtifer's bijection also provides this information. 2.2.4. Corollary. Given positive integers d1 , ... , d11 summing to 2n - 2, there are exactly n<;;~~i')! trees with vertex set such that vertex i has degree d;, for each i. Proof: While constructing the Prtifer code of a tree T, we record x each time we delete a neighbor of x, until we delete x itself or leave x among the last two vertices. Thus each vertex x appears dT(x) - 1 times in the Prilfer code. Therefore, we count trees with these vertex degrees by counting lists of length n - 2 that for each i have d; - 1 copies of i. If we assign subscripts to the copies of eaCh i to distinguish them, then we are permuting n - 2 distinct objects and there are (Ii - 2)! lists. Since the copies ofi are not distinguishable, we have counted each desired arrangement fl(d; -1)! times, once for each way to order the subscripts on each type of label. (Appendix A discusses further aspects of this counting problem.)
[n]
2.2.5. Example. Trees with fixed degrees. Consider trees with vertices (1, 2, 3, 4, 5, 6, 7} that have degrees (3, 1, 2, 1, 3, 1, 1), respectively. We compute n<;;~il! = 30; the trees are suggested below. Only the vertices (1, 3, 5} are non-leaves. Deleting the leaves yields a subtree on (1, 3, 5}. There are three such subtrees, determined by which of the three is in the m'.iddle.
I I
III
III
To complete each tree, we add the appropriate number ofleafneighbors for each non-leaf to give it the desired degree. There are six ways to complete the first tree (pick from the remaining four vertices the two adjacent to vertex 1) and twelve ways to complete each of the others (pick the neighbor of vertex 3 from the remaining four, and then pick the neighbor of the central vertex from the remaining three).
## SPANNING TREES IN GRAPHS
We can interpret Cayley's Formula in another way. Since the complete graph with vertex set [n] has all edges that can be used in forming trees with vertex set [n], the number of trees with a specified vertex set of size n equals the number of spanning trees in a complete graph on n vertices.
84
## Chapter 2: Trees and Distance
We now consider the more general problem of computing the number of spanning trees in any graph G. In general, G will not have as much symmetry as a complete graph, so it is unreasonable to expect as simple a formula as for Kn, but we can hope for an algorithm that provides a simple way to compute the answer when given a graph G.
2.2.6. Example. Below is the kite. To count foe spanning trees, observe that four are paths around the outside cyde in the drawing. The remaining spanning trees use the diagonal edge. Since we must include an edge to each vertex of degree 2, we obtain four more spanning trees. The total is eight.
u c
V1 /1 /1
In Example 2.2.6, we counted separately the trees that did or did not contain the diagonal edge. This suggests a recursive prqcedure to count spanning trees. It is clear that the spanning trees of G not containing e are simply the spanning trees of G - e, but how do we count the trees that contain e? The answer uses an elementary operation on graphs.
2.2.7. Definition. In a graph G, contraction of edge e with endpoints u, vis the replacement of u and v with a single vertex whose incident edges are the edges other thane that were incident to u or v. The resulting graph G e has one less edge than G.
In a drawing of G, contraction of e shrinks the edge to a single point. Con:tracting an edge can produce multiple edges or loops. To count spanning trees correctly, we must keep multiple edges (see Example 2.2.9). In other applications of contraction, the multiple edges may be irrelevant. The recurrence applies for all graphs.
2.2.8. Proposition. Let r ( G) denote the number of spanning trees of a graph G. If e E E(G) is not a loop, then r(G) = r(G - e) + r(G e). Proof: The spanning trees of G that omit e are precisely the spanning trees of G - e. To show that G has r(G e) spanning trees containing e, we show that contraction of e defines a bijection from the set of spanning trees of G containing e to the set of spanning trees of G e.
## SectiQn 2.2: Spanning Trees and Enumeration
85
When we contract e in a spanning tree that contains e, we obtain a spanning tree of G e, because the resulting subgraph of G e is spanning and connected and has the right number of edges. The other edges maintain their identity under contraction, so no two trees are mapped to the same spanning tree of G e by this operation. Also, each spanning tree of G e arises in this way, since exparlding the new vertex back into e yields a spanning tree of G. Since each spanning tree of G e arises exactly once, the function is a bijection. 2.2.9. Example. A step in the recurrence. The graphs on the right each have four spanning trees, so Proposition 2.2.8 implies that the kite has eight spanning trees. Without the multiple edges, the computation would fail.
IZJ G
D+~
G-e Ge
We can save some computation time by recognizing special graphs G where we know r(G), such as the graph on the right above. 2.2.10. Remark. If G is a connected loopless graph with no cycle of length at least 3, then r (G) is the product of the edge multiplicities. A disconnected graph has no spanning trees. We cannot apply the recurrence of Proposition 2.2.8 when e is a loop. For example, a graph consisting of one vertex and one loop has one spanning tree, but deleting and contracting the loop would count it twice. Since loops do not affect the number of spanning trees, we can delete loops as they arise. Counting trees recursively requires initial conditions for graphs in which all edges are loops. Such a graph ~as one spanning tree ifit has only one vertex, and it has no spanning trees if it has more than one vertex. If a computer completes the computation by deleting or contracting every edge in a loopless graph G, then it may compute as many as 2eCG) terms. Even with s~vings from Remark 2.2.10, the amount of computation grows exponentially with the size of the graph; this is impractical. Another technique leads to a much faster computation. The Matrix Tree Theorem, implicit in the work of Kirchhoff [18471, computes r(G) using a determinant. This is much faster, because determinants of n-by-n matrices can be computed using fewer than n 3 operations. Also, Cayley's Formula follows from the Matrix Tree Theorem with G = Kn (Exercise 17), but it does not follow easily from Proposition 2.2.8. Before stating the theorem, we illustrate the computation it specifies. 2.2.11. Example. A Matrix Tree computation. Theorem 2.2.12 instructs us to form a matrix by putting the vertex degrees on the diagonal and subtracting
86
## Chapter 2: Trees and Distance
the adjacency matrix. We then delete a row and a column and take the determinant. When G is the kite of Example 2.2.9, the vertex degrees are 3, 3, 2, 2. We form the matrix on the left below and take the determinant of the matrix in the middle. The result is the number of spanning trees! 3 -1 ( -1 -1 -1 3 -1 -1 -1 -1 2 0 -1) -1 0 2
Loops don't affect spanning trees, so we delete them befote the computation. The proof of the theorem uses properties of determinants. 2.2.12. Theorem. (Matrix Tree Theorem) Given a loopless graph G with vertex set vi. ... , Vn, let a;,j be the number of edges with endpoints v; and Vj. Let Q be the matrix in which entry (i, j) is -a;,j when i # j and is d(v;) when i = j. If Q* is a matrix obtained by deleting row s and column t of Q, then r(G) = (-1r+1 det Q*. Proof*: We prove this only whens = t; the general statement follows from a result in linear algebra (when the columns of a matrix sum to the vector 0, the cofactors are constant in each row-Exercise 8.6.18). Step 1. If Dis an orientation of G, and Mis the incidence matrix of D, then Q = M Mr. With edges ei, ... , em, the entries of Mare m;,j = 1 when v; is the tail of ej, m;,j = -1 whe~ v; is the head of ej, and m;,j = 0 otherwise. Entry i, j in MMT is the dot product of rows i and j of M. When i # j, the product counts -1 for every edge of G joining the two vertices; when i = j, it counts 1 for every incident edge and. yields the degree.
M=2
## 1 (-1 ~ ~1 ~1 ~1) bl[! d ~
3
0 0
de
-1
3 -1: ( 0
-1 2 -1
~1 ~2)
2 -1
3 -1
-2
Step 2. If Bis an (n - 1)-by-(n - 1) submatrix of M, then detB = 1 ifthe corresponding n - 1 edges form a Bpanning tree of G, and otherwise det B = 0. In the first case, we use induction on n to prove that det B = 1. For n = 1, by convention a 0 x 0 matrix has determinant 1. For n > 1, let T be the spanning tree \Yhose edges are the columns of B. Since T has at least two leaves and only one row is deleted, B has a row corresponding to a leaf x of T. This row has only one nonzero entry in B. When computing the determinant by expanding along this row, the only submatrix B' with nonzero weight in the expansion corresponds to the spanning subtree of G - x obtained by deleting x and its incident edge from T, Since B' is an (n - 2)-by-(n -2) submatrix of the incidence matrix for an orientation of G -x, the induction hypothesis yields det B' = 1. Since the nonzero entry in row xis 1, we obtain the same result for B.
## Section 2.2: Spanning Trees and Enumeration
87
If then -1 edges correspondingto columns of B do not form a spanning tree, then by Theorem 2.1.4C they contain a cycle C. We form a linear combination of the columns with coefficient 0 if the corresponding edge is not in C, +1 if it is followed forward by C, and -1 if it is followed backward by C. The result is total weight 0 at each vertex, so the columns are linearly dependent, which yields det B = 0. Step 3. Computation of det Q*. Let M* be the result of deleting row t of M, so Q* = M*(M*)T. If m < n - 1, then the determinant is 0 and there are no spanning subtrees, so we assume that m 2:: n - 1. The Binet-Cauchy Formula (Exercise 8.6.19) computes the determinant of a product of non-square matrices using the determinants of square submatrices of the factors. When m 2:: p, A is p-by-m, and Bis m-by-p, it states that detAB = I:sdetAsdetBs, where the summ runs over all p-sets Sin [m], As is the submatrix of A consisting of the columns indexed by S, and Bs is the submatrix of B consisting of the rows indexed by S. When we apply the formula to Q* = M*(M*)T, the submatrix As is an (n - 1)-by-(n - 1) submatrix of M as discussed in Step 2, and Bs = AI. Hence the summation counts 1 = (1) 2 for each set of n-1 edges corresponding to a spanning tree and 0 for all other sets of n - 1 edges.
## DECOMPOSITION AND GRACEFUL LABELINGS
We consider another problem about graph decomposition (Definition 1.1.32). We can always decompose G into single edges; can we decompose G into copies of a larger tree T? This requires that e(T) divides e(G) and .::l(G) 2:: .::l(T); is that sufficient? Even when G is e(T)-regular, this may fail (Exercise 20); for example, the Petersen graph does not deoompose into claws. Haggkvist conjectured that if G is a 2m-regular graph and T is a tree with m edges, then E(G) decomposes into n(G) copies ofT. Even the "simplest" case when G is a clique is still unsettled and notorious. 2.2.13. Conjecture. (Ringel [1964]) If T is a fixed tree with m edges, then K2m+l decomposes into 2m + 1 copies of T. Attempts to prove Ringel's conjecture have focused on the stronger Graceful Tree Conjecture. This implies Ringel's conjecture and a similar statement about decomposing complete graphs of even order (Exercise 23). 2.2.14. Definition. A graceful labeling of a graph G with m edges is a function f: V ( G) --+ {O, ... , m} such that distinct vertices receive distinct numbers and {lf(u) - f(v)I: uv E E(G)} = {l, ... , m}. A graph is graceful if it has a graceful labeling. 2.2.15. Conjecture. (Graceful Tree Conjecture-Kotzig, Ringel [1964]) Every tree has a graceful labeling.
88
## Chapter 2: Trees and Distance
2.2.16. Theorem. (Rosa [1967]) If a tree T with m edges has a graceful labeling, then Kzm+l has a decomposition into 2m + 1 copies of T. Proof: View the vertices of Kzm+l as the congruence classes modulo 2m + 1, arranged circularly. The difference between two congruence classes is 1 if they are consecutive, 2 if one class is between them, and so on up to difference m. We group the edges of Kzm+l by the difference between the endpoints. For 1 :::; j :::: m, there are 2m + 1 edges with difference j.
0
8
0
I
1
7
6.
5
From a graceful labeling of T, we define copies of T in Kzm+l; the copies are To, ... , T2m. The vertices of Tk are k, ... , k + m (mod 2pi + 1), with k + i adjacent to k + j if and only if i is adjacent to j in the graceful labeling of T. The copy To looks just like the graceful labeling and has one edge with each difference. Moving to the next copy shifts each edge to another having the same difference by adding one to the name of each endpoint. Each difference class of edges has one edge in each Tk. and thus To, ... , Tzm decompose Kzm+i Graceful labelings are known to exist for some types of trees and for some other families of graphs (see Gallian [1998]). It is easy to find graceful labelings for stars and paths. We next define a family of trees that generalizes both by permitting the addition of edges incident to a path.
2.2.17. Definition. A caterpillar is a tree in which a single path (the spine) is incident to (or contains) every edge. 2.2.18. Example. The vertices not on the spine of a caterpillar (the "feet") are leaves. Below w"' show a graceful labeling of a caterpillar; in fact, every caterpillar is graceful (Exercise 31). The tree Y below is not a caterpillar.
2 13 12 11
2.2.19. Theorem. A tree is a caterpillar if and only if it does not contain the tree Y above.
## Section 2.2: Spanning Trees and Enumeration
89
Proof: Let G' denote the tree obtained from a tree G by deleting each leaf of G. Since all vertices that survive in G' are non-leaves in G, G' has a vertex of degree at least 3 if and only if Y appears in G. Hence G has no copy of Y if and only if ti(G') ~ 2. This is equivalent to G' being a path, which is equivalent to G being a caterpillar.
## BRANCHINGS AND EULERIAN DIGRAPHS (optional)
Tutte extended the Matrix Tree Theorem to digraphs. His theorem reduces to the Matrix Tree Theorem when the digraph is symmetric (a digraph is symmetric if its adjacency matrix is symmetric, and then it models a graph). There is a surprising connection between this theorem and Eulerian circuits.
2.2.20. Definition. A branching or out-tree is an orientation of a tree having a root of indegree 0 and all other vertices of indegree 1. An in-tree is an out-tree with edges reversed.
A branching with root v is a union of paths from v (Exercise 33). Each vertex is reached by exactly one path. The analogous result holds for in-trees; an in-tree is a union of paths to the root, one from each vertex. We state without proof Tutte's theorem to count branchings.
2.2.21. Theorem. (Directed Matrix Tree Theorem-Tutte [1948]) Given a loopless digraph G, let Q- = v- - A' and Q+ = v+ - A', where v- and v+ are the diagonal matrices ofindegrees and outdegrees in G, and the i, jthentry of A' is the number of edges from Vj to v;. The number of spanning out-trees (in-trees) of G rooted at v; is the value of each cofactor in the ith row of Q- (ith column of Q+).
2.2.22. Example. The digraph above has two ~panning out-trees rooted at 1 and two spanning in-trees rooted at 3. Every cofactor in the first row of Q- is 2, arid every cofactor in the third column of Q+ is 2.
Isolated vertices don't affect Eulerian circuits. After discarding these, a digraph is Eulerian if and only if indegree equals outdegree at every vertex and the underlying graph is connected (Theorem 1.4.24). Such a digraph also is strongly connected, which allows us to find a spanning in-tree. We will describe Eulerian circuits in terms of a spanning in-tree.
90
## Chapter 2: Trees and Distance
2.2.23. Lemma. In a strong digraph, every vertex is the root of an out-tree (and an in-tree). Proof: Consider a vertex v. We iteratively add edges to grow a branching from v. Let Si be the set of vertices reached when i edges have been added; initialize So = {v}. Because the digraph is strong, there is an edge leaving Si (Exercise 1.4.10). We add one such edge to the branching and add its head to S; to obtain S;+l This repeats until we have reached all vertices. To obtain an in-tree of paths to v, reverse all edges and apply the same procedure; the reverse of a strong digraph is also strong. The lemma constructively produces a search tree of paths from a root. The next section discusses search trees in more generality. 2.2.24. Algorithm. (Eulerian circuit in directed graph) Input: An Eulerian digraph G without isolated vertices and a spanning in-tree T consisting of paths to a vertex v. Step 1: For ea~h u e V(G), specify an ordering of the edges that leave u, such that for u :j:. v the edge leaving u in T comes last. Step 2: Beginning at v, construct an Eulerian circuit by always exiting the current vertex u along the next unused edge in the ordering specified at u. 2.2.25. Example. In the digraph below, the bold edges form an in-tree T of paths to v. The edges labeled in order starting with 1 form an Eulerian circuit. It leaves a vertex along an edge of T only where there is no alternative. If the ordering at v places 1 before 10 before 13, then the algorithm traverses the edges in the order indicated. 11
2.2.26. Theorem. Algorithm 2.2.24 always produces an Eulerian circuit. Prooft Using Lemma 2.2.23, we construct an in-tree T to a vertex v. We then apply Algorithm 2.2.24 to construct a trail. It suffices to show that the trail can end only at v and does so only after traversing all edges. When we enter a vertex u :j:. v, the edge leaving u in T has not yet been used, since d+(u) = d-(u). Thus whenever we enter u there is still a way out. Therefore the trail can only end at v. We end when we cannot continue; we are at v and have used all exiting edges. Since d-(v) = d+(v), we must also have used all edges entering v. Since
## Section 2.2; Spanning Trees and Enumeration
91
we cannot use an edge of T until it is the only remaining edge leaving its tail, we cannot use all edges entering v until we have finished all the other vertices, since T contains a path from each vertex to v. 2.2.27. Example. In the digraph below, every in-tree to v contains all of uv, yz, wx, exactly one of {zu, zv}, and exactly one of {xy, xz}. There are four in-
trees to v. For each in-tree, we consider CT(d; - 1)! = (0!) 3 (1!) 3 = 1 orderings of the edges leaving the vertices. Hence we can obtain one Eulerian circuit from each in-tree, starting along the edge e = vw from v. The four in-trees and the corresponding circuits appear below.
v~y
w
x
Yi\
In-tree has
zu zu zv zv &xy &xz &xy &xz
(v, (v, (v, (v,
Circuit
w, x, z, v, x, y, z, u) W, x, y, Z, V, x, z, u) w, x, z, u, v, x, y, z) w, x, y, z, u, v, x, z)
Two Eulerian circuits are the same if the successive pairs of edges are the same. From each in-tree to V, Algorithm 2.2.24 generates nuEV(G)(d+(u) - 1)! different Eulerian circuits. The last out-edge is fixed by the tree for vertices other than v, and since we consider only the cyclic order of the edges we may also choose a particular edge e to start the ordering of edges leaving v. Any change in the exit orderings at vertices specifies at some point different choices for the next edge, so the circuits are distinct. Similarly, circuits obtained from distinct in-trees are distinct. Hence we have generated c nueV(G)(d+(u) - 1)! distinct Eulerian circuits, where c is the number of in-trees to v. In fact, these are all the Eulerian circuits. This yields a combinatorial proof that the number of in-trees to each vertex of an Eulerian digraph is the same. Tlie graph obtained by reversing all the edges has the same number of Eulerian circuits, so the number of out-trees from any vertex also has this value, c. Theorem 2.2.21 provides a computation of c. 2.2.28. Theorem. (van Aardenne-Ehrenfest and de Bruijn [1951]). In an Eulerian digraph with d; = d+(v;) = d-(v;) the number of Eulerian circuits is c CTJd; - 1)!, where c counts the in-trees to or out-trees from any vertex. Proof: We have argued that Algorithm 2.2.24 generates this many distinct Eulerian circuits using in-trees to vertex v (starting from v alond e). We need only show that this produces all Eulerian circuits. To find the tree and ordering that generates an Eulerian circuit C, follow C from e, and record the order of the edges leaving each vertex. Let T be the subdigraph consisting of the last edge on C leaving each vertex other than v. Since the last edge leaving a vertex occurs in C after all edges entering it, each edge in T extends to a path in T that reaches v. With n -1 edges, T thus forms an in-tree to v. Furthermore, C is the circuit obtained by Algorithm 2.2.24 from T and the orderings of exiting edges that we recorded.
92
## Chapter 2: Trees and Distance
EXERCISES
2.2.1. (-) Determine which trees have Priifer codes that (a) contain only one value, (b) contain exactly two values, or (c) have distinct values in all positions. 2.2.2. (-) Count the spanning trees in the graph on the left below. (Proposition 2.2.8 provides a systematic approach, and then Remark 2.2.10 and Example 2.2.6 can be used to shorten the computation.)
2.2.3. (-)Let G be the graph on the right above. Use the Matrix Tree Theorem to find a matrix whose determinant is r(G). Compute r(G). 2.2.4. (-) Let G be a simple graph with m edges. Prove that if G has a graceful labeling, then K2m+l decomposes into copies of G. (Hint: Follow the proof of Theorem 2.2.16.)
2.2.5. The graph on the left below was the logo of the 9th Quadrennial International Conference in Graph Theory, held in Kalamazoo in 2000. Count its spanning trees.
2.2.6. (!) Let G be the 3-regular graph with 4m vertices formed from m pairwise disjoint kites by adding m edges to link them in a ring, as shown on the right above for m = 6. Prove that r(G) = 2m8m 2.2.7. (!)Use Cayley's Formula to prove that the graph obtained from K. by deleting an edge has (n - 2)n- 3 spanning trees. 2.2.8. Count the following sets of trees with vertex set [n], giving two proofs for each: one using the Priifer correspondence and one by direct counting arguments. a) trees that have 2 leaves. b) trees that have n - 2 leaves. 2.2.9. Let S(m, r) denote the number of partitions of an m-element set into r nonempty subsets. In terms of these numbers, count the trees with vertex set {v 1 , . , v.} that have exactly k leaves. (Renyi [1959]) 2.2.10. Compute r(K2.,.). Also compute the number of isomorphism classes of spanning trees of Kz.,,,.
Section 2.2: Spanning Trees and Enumeration 2.2.11. (+)Compute r(K3 .111 ).
93
2.2.12. From a graph G we define two new graphs. Let G' be the graph obtained by replacing each edge of G with k copies of that edge. Let G" be the graph obtained by replacing each edge uv E E(G) with au, v-path of length k through k - 1 new vertices. Determine r(G') and r(G") in terms of r(G) and k.
G'
G"
2.2.13. Consider K11 11 with bipartition X, Y, where X ={xi. ... , x.} and Y = {yi. ... , y.}. For each spanning tree T, we form a list f (T) of ordered pairs (written vertically). Having generated part of the list, let u be the least-indexed leaf in X in the remaining subtree, and similarly let v be the least-indexed leaf in Y. Append the pair (~) to the list, where a is the index of the neighbor of u and b is the index of the neighbor of v. Delete {u, v}. Iterate until n - 1 pairs have been generated to form f(T) (one edge remains). Part (a) shows that f is well-defined. a) Prove that every spanning tree of K11 11 has a leaf in each partite set. b) Prove that f is a bijection from the set of spanning trees of K. 11 to ([n] x [n])- 1 Thus K11 11 has n211 - 2 spanning trees. (Renyi [1966], Kelmans [1992], Pritikin [1995])
x
y
~ 1 2 3 4 5
## - G). (i). (~). m
2.2.14. (+)Let f(r, s) be the number of trees with vertex [n] that have partite sets of sizes r ands (with r + s = n). Prove that f(r, s) = (';"')s'- 1,-- 1 if r # s. What is the formula when r = s? (Hint: First show that the Priifer sequence for such a tree will haver -1 of its terms from the partite set of sizes ands -1 of its terms from the partite set of sizer.) (Scoins [1962], Glicksman [1963]) 2.2.15. Let G 11 be the graph with 2n vertices and 3n - 2 edges pictured below, for n '.::: 1. Prove for n > 2 that r(G.) = 4r(Gn-1) - r(G 11 _ 2 ). (Kelmans [1967a])
l l l
l l l
2.2.16. For n '.::: 1, let a. be the number of spaming trees in the graph formed from P. by addix~g one vertex adjacent to all of V(P,.). For example, a 1 = 1, a2 = 3, and a3 = 8. Prove for n > 1 that an = an-l + 1 + r:,;~: a;. Use this to prove for n > 2 that a11 = 3an-l - an-2 (Comment: It is also possible to argue direa_tly that an = 3an-l - an-2.)
94
## Chapter 2: Trees and D~stance
2.2.18. Use the Matrix Tree Theorem to compute r(K,,s). (Lovasz [1979, p223]-see Kelmans [1965] for a generalization) 2.2.19. (+)Prove combinatorially that the number tn of trees with vertex set [n] satisfies the recurrence tn = k(~:::~)tktn-k (Comment: Since t. = nn- 2 , this proves the identity nn- 2 = I:;~:: (~:::~)kk-l (n - k)n-k- 2.) (Dziobek [1917]; see Lovasz [1979, p219])
I:::::
2.2.20. (!)Prove that ad-regular simple graph G has a decomposition into copies of Kl.d if and only if it is bipartite. 2.2.21. (+)Prove that
K2m-I,2m
## decomposes into m spanning paths.
2.2.22. Let G be an n-vertex simple graph that decomposes into k spanning trees. Given also that t.(G) = 8(G) + 1, determine the degree sequence of Gin terms of n and k. 2.2.23. (!) Prove that if the Graceful Tree Conjecture is true and T is a tree with m edges, then K 2m decomposes into 2m -1 copies of T. (Hint: Apply the cyclically invariant decomposition of K2m-l for trees with m - 1 edges from the proof of Theorem 2.2.16.) 2.2.24. Of the nn- 2 trees with vertex set {O, ... , n by their vertex names?
-
## 1), how many are gracefully labeled '
2.2.25. (!) Prove that if a graph G is graceful and Eulerian, then e(G) is congruent to 0 or 3 mod 4. (Hint: Sum the absolute edge differences (mod 2) in two different ways.) 2.2.26. (+) Prove that Cn is graceful if and only if 4 divides n or n + 1. (Frucht [1979]) 2.2.27. (+) Let G be the graph consisting of k 4-cycles with one common vertex. Prove that G is graceful. (Hint: Put 0 at the vertex of degree 2k.) 2.2.28. Let d1 , , d. be positive integers. Prove directly that there exists a caterpillar with vertex degrees d 1 , , d. if and only if L d; = 2n - 2. 2.2.29. Prove that every tree can be turned into a caterpillar with the same degree -sequence using 2-switches (Definition 1.3.32) such that each intermediate graph is a tree. 2.2.30. A bipartite graph is drawn on a channel if the vertices of one.partite set are placed on one line in the plane (in some order) and the:vertices of the other partite--se_t are placed on a line parallel to it and the edgei; .are drawn as straight-line segments--. between them. Prove that a connected graph G can be drawn on a channel without edge crossings if and only if G is a caterpillar. 2.2.31. (!)An up/down labeling is a grac~ful labeling for which there exists a critical value a such that every edge joins vertices with labels above and below a. Prove that every caterpillar has an up/down labeling. Prove that the 7-vertex tree that is not a caterpillar has no up/down-labeling. 2.2.32. (+) Prove that the number of isomorphism classes of n-vertex caterpillars is 2- 4 + 2L/ 2J- 2 ifn::::: 3. (Harary-Schwenk [1973], Kimble-Schwenk [1981]) 2.2.33. (!) Let T be an orientation of a tree such that the heads of the edges are all distinct; the one vertex that is not a head is the root. Prove. that T _is a union of paths from the root. Prove that for each vertex of T, exactly one path reaches it from the root. 2.2.34. (*)Use Theorem 2.2.26 to prove that the algorithm below generates a binary deBruijn cycle of length 2" (the cycle in Application 1.4.25 arises in this way). Start with n O's. Subsequently, append a 1 if doing so does not repeat a previous string of length n, otherwise append a 0.
## Section 2.3: Optimization and Trees
95
2.2.35. (*) Tarry's Algorithm (as presented by D.G. Hoffman). Consider a castle with finitely many rooms and corridors. Each corridor has two ends; each end has a door into a room. Each room has door(s), each of which leads to a corridor. Each room can be reached from any other by traversing corridors and rooms. Initially, no doors have marks. A robot started in some room will explore the castle using the following rules. 1) After entering a corridor, traverse it and enter the room at the other end. 2) Upon entering a room with all doors unmarked, mark I on the door of entry. 3) In a room with an unmarked door, mark 0 on such a door and use it. 4) In a room with all doors marked, exit via a door not marked 0 if one exists. 5) In a room with all doors marked 0, stop. Prove that the robot traverses each corridor exactly twice, once in each direction, and then stops. (Hint: Prove that this holds for the corridors at every reached vertex, and prove that every vertex is reached. Comment: All decisions are completely local; the robot sees nothing other than the current room or corridor. Tarry's Algorithm [1895] and others are described by Konig [1936, p35-56] and by Fleischner [1983, 1991].)
## 2.3. Optimization and Trees
"The best spanning tree" may have various meanings. A weighted graph is a graph with numerical labels on the edges. When building links to connect locations, the costs of potential links yield a weighted graph. The minimum cost of connecting the system is the minimum total weight of its spanning trees. Alternatively, the weights may represent distances. In these case we define the length of a path to be the sum of its edge weights. We may seek a spanning tree with small distances. When discussing weighted graphs, we consider only nonnegative edge weights. We also study a problem about finding good trees to eneode messages.
## MINIMUM SPANNING TREE
In a' connected weighted graph of possible communication links, al~ spanning trees have n - 1 edges; we seek one that minimizes or maximizes the sum of the edge weights. For these problems, the most naive heuristic quickly pi>oduces an optimal solution.
2.3.1. Algorithm. (Kruskal's Algorithm - for minimum spanning trees.) Input: A weighted connected graph. Idea: Maintain an acyclic spanning subgraph H, enlarging it by edges with low weight to form a spanning tree. Consider edges in nondecreasing order of weight, breaking ties arbitrarily. Initialization: Set E(H) = 0. Iteration: If the next cheapest edge joins two components of H, then include it; otherwise, discard it. Terminate when H is connected.
96
## Chapter 2: Trees and Distance
Theorem 2.3.3 verifies that Kruskal's Algorithm produces an optimal tree. Unsophisticated locally optimal heuristics are called greedy algorithms. They usually don't guarantee optimal solutions, but this one does. In a computer, the weights appear in a matrix, with huge weight on "unavailable" edges. Edges of equal weight may be examined in any order; the resulting trees.have the same cost. Kruskal's Algorithm begins with a forest of n isolated vertices. Each selected edge combines two components. 2.3.2. Example. Choices in Kruskal's Algorithm depend only on the order of the weights, not on their values. In the graph below we have used positive integers as weights to emphasize the order of examination of edges. The four cheapest edges are selected, but then we cannot take the edges of weight 5 or 6. We can take the edge of weight 7, but then not those of weight 8 or 9.
9
12
2.3.3. Theorem. (Kruskal [1956]). In a connected weighted graph G, Kruskal's Algorithm constructs a minimum-weight spanning tree. Proof: We show first that the algorithm produces a tree. It never chooses an edge that completes a cycle. If the final graph has more than one component, then we considered no edge joining two of them, because such an edge would be accepted. Since G is connected, some such edge exists and we considered it. Thus the final graph is connected and acyclic, which makes it a tree. Let T be the resulting tree, and let T* be a spanning tree of minimum weight. It T = T*, we are done. If T =f:. T*, let e be the first edge chosen for T that is not in T*. Adding e to T* creates one cycle C. Since T has no cycle, C has an edge e' fl. E(T). Consider the spanning tree T* + e - e'. Since T* contains e' and all the edges of T chosen before e, both e' and e are available when the algorithm chooses e, and hence w(e) :::; w(e'). Thus T* + e - e' is a spanning tree with weight at most T* that agrees with T for a longer initial list of edges than T* does. Repeating this argument eventually yields a minimum-weight spanning tree that agrees completely with T. Phrased extremally, we have proved that the minimum spanning tree agreeing with T the longest is T itself. 2.3.4. * Remark. To implement Kruskal's Algorithm, we first sort the m edge weights. We then maintain for each vertex the label of the component containing it, accepting the next cheapest edge ifits endpoints have different labels. We
## Section 2.3: Optimization and Trees
97
merge the two components by changing the label of each vertex in the smaller component to the label of the larger. Since the size of the component at least doubles when a label changes, each label changes at most lg n times, and the total number of changes is at most n lg n (we use lg for the base 2 logarithm). With this labeling method, the running time for large graphs depends on the time to sort m numbers. With this cost included, other algorithms may be faster than Kruskal's Algorithm. In Prim's Algorithm (Exercise 10, due also to Jarnik), a spanning tree is grown from a single vertex by iteratively adding the cheapest edge that incorporates a new vertex. Prim's and Kruskal's Algorithms have similar running times when edges are pre-sorted by weight. Both Boruvka [1926] and Jarnik [1930] posed and solved the minimum spanning tree problem. Boruvka's algorithm picks the next edge by .considering the cheapest edge leaving each component of the current forest. Modern improvements use clever data structures to merge components quickly. Fast versions appear in Tarjan [1984] for when the edges are pre-sorted and in Gabow-Galil-Spencer-Tarjan [1986] for when they are not. Thorough discussion and further references appear in Ahuja-Magnanti-Orlin [1993, Chapter 13]. More recent developments appear in Karger-Klein-Tarjan [1995].
SHORTEST PATHS
How can we find the shortest route from one location to another? How can we find the shortest routes from our home to every place in town? This requires finding shortest paths from one vertex to all other vertices in a weighted graph. Together, these paths form a spanning tree. Dijkstra's Algorithm (Dijkstra [1959] and Whiting-Hillier [1960]) solves this problem quickly, using the observation that the u, v-portion of a shortest u, z-path must be a shortest u, v-path. It finds optimal routes from u to other vertices z in increasing order of d(u, z). The distance d(u, z) in a weighted graph is the minimum sum of the weights on the edges in au, z-path (we consider only nonnegative weights).
2.3.5. Algorithm. (Dijkstra's Algorithm-distances from one vertex.) Input: A graph (or digraph) with nonnegative edge weights and a starting vertex u. The weight of edge xy is w(xy); let w(xy) = oo if xy is not an edge. Idea: Maintain the set S of vertices to which a shortest path from u is known, -enlarging S to include all vertices. To do this, maintain a tentative distance t(z) from u to each z ~ S, being the length of the shortest u, z-path yet found. Initialization: Set S = {u}; t(u) = O; t(z) = w(uz) for z-:/:- u. Iteration: Select a vertex v outside S such that t(v) = minzS t(z). Add v to S. Explore edges from v to update tentative distances: for each edge vz with z ~ S, update t(z) to min{t(z), t(v) + w(vz)}. The iteration continues until S = V(G) or until t(z) = oo for every z ~ S. At the end, set d(u, v) = t(v) for all v.
98
## Chapter 2: Trees and Distance
2.3.6. Example. In the weighted graph below, shortest paths from u are found to the other vertices in the order a, b, c, d, e, with distances 1,3,5,6,8, respectively. To reconstruct the paths, we only need the edge on which each shortest path arrives at its destination, because the earlier portion of a shortest u, z-path that reaches z on the edge vz is a shortest u, v-path. The algorithm can maintain this information by recording the identity of the "selected vertex" whenever the tentative distance to z is updated. When z is selected, the vertex that was recorded when t(z) was last updated is the predecessor of z on the u, z-path oflength d(u, z). In this example, the final edges on the paths to a, b, c, d, e generated by the algorithm are ua, ub, ac, ad, de, respectively, and these are the edges of the spanning tree generated from u.
With the phrasing given in Algorithm 2.3.5, Oijkstra's Algorithm works also for digraphs, generating an out-tree rooted at u if every vertex is reachable from u. The proof works for graphs and for digraphs. The technique of proving a stronger statement in order to make an indctive proof work is called "loading the induction hypothesis". 2.3.7. Theorem. Given a (di)graph G and a vertex u E V(G), Dijkstra's Algorithm computes d(u, z) for every z E V(G). Proof: We prove the stronger statement that at each iteration, 1) for z E S, t(z) = d(u, z), and 2) for z fl. S, t(z) is the least length ofa u, z-path reaching z directly from S. We use induction on k = ISi. Basis step: k = 1. From the initialization, S = {u}, d(u, u) = t(u) = 0, and the least length of au, z-path reaching z directly from S is t(z) = w(u, z), which is infinite when uz is not an edge. Induction step: Suppose that when ISi = k, (1) and (2) are true. Let v be a vertex among z fl. S such that t (z) is smallest. The algorithm now chooses v; let S' = SU {v}. We first argue that d(u, v) = t(v). A shortest u, v-path must exit S before reaching v. The induction hypothesis states that the length of the shortest path going directly to v from Sis t(v). The induction hypothesis and choice of v also guarantee that a path visiting any vertex outside S and later reaching v has length at least t(v). Hence d(u, v) = t(v), and (1) holds for S'. To prove (2) for S', let z be a vertex outside S other than v. By the hypothesis, the shortest u, z-path reaching z directly from S has length t(z) (oo if there is no such path). When we add v to S, we must also consider paths reaching z from v. Since we have now computed d(u, v) = t(v), the shortest such path has length t(v) + w(vz), and we compare this with the previous value of t(z) to find the shortest path reaching z directly from S'.
## Section 2.3: Optimization and Trees
99
We have verified that (1) and (2) hold for the new set S' of size k + 1; this completes the induction step.
s
t(v):::; t(z)
z
The algorithm maintains the condition that d(u, x) :::; t(z) for all x E S and z ~ S; hence it selects vertices in nondecreasing order of distance from u. It computes d(u, v) = oo when v is unreachable from u. The special case for unweighted graphs is Breadth-First Search from u. Here both the algorithm and the proof (Exercise 17) have simpler descriptions.
2.3.8. Algorithm. (Breadth-First Search-BFS) Input: An unweighted graph (or digraph) and a start vertex u. Idea: Maintain a set R of vertices that have been reached but not searched and a set S of vertices that have been searched. The set R is maintained as a First-In First-Out list (queue), so the first vertices found are the first vertices explored. Initialization: R = {u}, S = 0, d(u, u) = 0. Iteration: As long as R -:j:. 0, we search from the first vertex v of R. The neighbors of v not in S U R are added to the bacly of R and assigned distance d(u, v) + 1, and then vis removed from the front of Rand placed in S.
The largest distance from a vertex u to another vertex is the eccentricity
E(u). Hence we can compute the diameter of a graph by running Breadth-First
Search from each vertex. Like Dijkstra's Algorithm, BFS from u yields a tree T in which for each vertex v, the u, v-path is a shortest u, v-path. Thus the graph has no additional edges joining vertices of a u, v-path in T. Dijkstra's Algorithm figures prominently in the solution of another wellknown optimization problem.
2.3.9. Application. A mail carrier must traverse all edges in a road network, starting and ending at the Post Office. The edges have nonnegative weights representing distance or time. We.seek a closed walk of minimum total length that uses all the edges. This is the Chines.e Postman Problem, named in honor of the Chinese mathematician Guan Meigu [1962], who proposed it. If every vertex is even, then the graph is Eulerian and the answer is the sum of the edge weights. Otherwise, we must repeat edges. Every traversal is an Eulerian circuit of a graph obtained by duplicating edges. Finding the shortest traversal is equivalent to finding the minimum total weight of edges
100
## Chapter 2: Trees and Distance
whose duplication will make all vertex degrees even. We say "duplication" because we need not use an edge more than twice. Ifwe use an edge three or more times in making all vertices even, then deleting two of those copies will leave all vertices even. There may be many ways to choose the duplicated edges. 2.3.10. Example. In the example below, the eight outer vertices have odd degree. Ifwe match them around the outside to make the degrees even, the extra cost is 4 + 4 + 4 + 4 = 16 or 1 + 7 + 7 + 1 = 16. We can do better by using all the vertical edges, which total only 10.
7 2
"'
2
1
1
3
4
3 4 1
Adding an edge from an odd vertex to an even vertex makes the even vertex odd. We must continue adding edges until we complete a trail to an odd vertex. The duplicated edges must consist of a collection of trails that pair the odd vertices. We may restrict our attention to paths pairing up the odd vertices (Exercise 24), but the paths may need to intersect. Edmonds and Johnson [1973] described a way to solve the Chinese Postman Problem. If there are only two odd vertices, then we can use Dijkstra's Algorithm to find the shortest path between them and solve the problem. If there are 2k odd vertices, then we still can use Dijkstra's Algorithm to find the shortest paths connecting each pair of odd vertices; these are candidates to use in the solution. We use these lengths as weights on the edges of K 2k. and then our problem is to find the minimum total weight of k edges that pair up these 2k vertices. This is a '\\veighted version of the maximum matching problem discussed in Section 3.3. An exposition appears in Gibbons [1985, pl63-165].
## TREES IN COMPUTER SCIENCE (optional)
Most applications of trees in computer science use rooted trees. 2.3.11. Definition. A rooted tree is a tree with one vertex r chosen as root. For each vertex v, let P(v) be the unique v, r-path. The parent of vis its neighbor on P(v); its children are its other neighbors. Its ancestors are the vertices of P(v) - v. Its H.escendants are the vertices u such that P(u)
## Section 2.3: Optimization and Trees
101
contains v. The leaves are the vertices with no children. A rooted plane tree or planted tree is a rooted tree with a left-to-right ordering specified for thei children of each vertex.
After a BFS from u, we view the resulting tree T as rooted at u. 2.3.12. Definition. A binary tree is a rooted plane tree where each vertex has at most two children, and each child of a v~rtex is designated as its left child or right child. The subtrees rooted at the children of the root are the left subtree and the right subtree of the tree. A k-ary tree allows each vertex up to k children. In many applications of binary trees, all non-leaves have exactly two children (Exercise 26). Binary trees permit storage of data for quick access. We store each item at a leaf and access it by following the path from the root. We encode the path by recording 0 when we move to a left child and 1 when we move to a right child. The search time is the length of this code word for the leaf. Given access probabilities among n items, we want to place them at the leaves of a rooted binary tree to minimize the expected search time. Similarly, given large computer files and limited storage, we want to encode characters as binary lists to minimize total length. Dividing the frequencies by the total length of the file yields probabilities. This encoding problem then reduces to the problem above. The length of code words may vary; we need a way to recognize the end of the current word. If no code word is an initial portion of another, then the current word ends as soon as the bits since the end of the previous word form a code word. Under this prefix-free condition, the binary code words correspond to the leaves of a binary tree using the left/right encoding described above. The expected length of a message is I: p;l;, where the ith item has probability p; and its code has length l;. Constructing the optimal code is surprisingly easy. 2.3.13. Algorithm. (Huffman's Algorithm [l952]-prefix-free coding) Input: Weights (frequencies or probabilities) P1, ... , Pn. Output: Prefix-free code (equivalently, a binary tree). Idea: Infrequent itenis should have longer codes; put infrequent items deeper by combining them into parent nodes. Initial case: When n = 2, the optimal length is one, with 0 and 1 being the codes assigned to the two items (the tree has a root and two leaves; n = 1 can also be used as the initial case).
102
## Chapter 2: Trees and Distance
Recursion: When n > 2, replace the two least likely items p, p' with a single item q of weight p + p'. Treat the smaller set as a problem with n - 1 items. After solving it, give children with weights p, p' to the resulting leaf with weight q. Equivalently, replace the code computed for the combined item with its extensions by 1 and 0, assigned to the items that were replaced.
2.3.14. Example. Huffman coding. Consider eight items with frequencies 5, 1, 1, 7, 8, 2, 3, 6. Algorithm 2.3.13 combines items according to the tree on the left below, working from the bottom up. First the two items of weight 1 combine to form one of weight 2. Now this and the original item of weight 2 are the least likely and combine to form an item of weight 4. The 3 and 4 now combine, after which the least likely elements are the original items of weights 5 and 6. The remaining combinations in order are 5 + 6 = 11, 7 + 7 = 14, 8 + 11 = 19, and 14 + 19 = 33. From the drawing of this tree on the right, we obtain code words. In their original order, the items have code words 100, 00000, 00001, 01) 11, 0001, 001, and 101. The expected length is 'L p;l; = 90/33. This is less than 3, which would be the expected length of a code using the eight words of length 3. 33
2.3.15. Theorem. Given a probability distribution {p;} on n items, Huffman's Algorithm produces the prefix-free code with minimum expected length. Proof: We use induction on n. Basis step: n = 2. We mu:st send a bit to send a message, and the algorithm encodes Mch item as a single bit, so the optimum is expected length 1. Induction step: n > 2. Suppose that the algorithm computes the optimal code when given a distribution for n - 1 items. Every code assigns items to the leaves of a binary tree. Given a fixed tree with n leaves, we minimize the expected length by greedily assigning the messages with probabilities p 1 2: 2: Pn to leaves in increasing order of depth. Thus every optimal code has least likely messages assigned to leaves of greatest depth. Since every leaf at maximum depth has another leaf as its sibling and permuting the items at a given depth does not change the expected length, we may assume that two least likely messages appear as siblings at greatest depth. Let T be an optimal tree for pi, ... , Pn. with the least likely items Pn and Pn-1 located as sibling leaves at greatest depth. Let T' be the tree obtained from T by deleting these leaves, and let qi, ... , qn-1 be the probability distribution obtained by replacing {Pn-1' Pn} by qn-1 = Pn-1 + Pn The tree T' yields a code
## Section 2.3: Optimization and Trees
103
for {q; }. The expected length for Tis the expected length for T' plus q11 _i, since if k is the depth of the leaf assigned q11 _i, we lose kq11 _1 and gain (k + l)(p11 _1 + p 11 ) in moving from T' to T. This holds for each choice of T', so it is best to use the tree T' that is optimal for {q; }. By the induction hypothesis, the optimal choice for T' is obtained by applying Huffman's Algorithm to {qi}. Since the replacement of {Pn-1 p 11 } by q11 -1 is the first step of Huffman's Algorithm for {Pi}, we conclude that Huffman's Algorithm generates the optimal tree T for {p;}.
+- k--+
Pn
Pn-1
Huffman's Algorithm computes an optimal prefix-free code, and its expected length is close to the optimum over all types of binary codes. Shannon [1948] proved that for every code with binary digits, the expected length is at least the entropy of the discrete probability distribution {Pi}, defined to be - L Pi lg Pi (Exercise 31). When each Pi is a power of 1/2, the Huffman code meets this bound exactly (Exercise 30).
EXERCISES
2.3.1. (-) Assign integer weights to the edges of K.. Prove that the total weight on every cycle is even if and only if the total weight on every triangle is even. 2.3.2. (-) Prove or disprove: If T is a minimum-weight spanning tree of a weighted graph G, then the u, v-path in Tis a minimum-weight u, v-path in G. 2.3:3. ( - ) There are five cities in a network. The cost of building a road directly between i and j is the entry a;,j in the matrix below. An infinite entry indicates that there is a mountain in the way and the road cannot be built. Determine the least cost of making all thefilies reachable from each other.
u,~~f {o)
2.3.4. (-) In the graph below, assign weights (1, 1, 2, 2, 3, 3, 4, 4) to the edges in two ways: one way so that the minimum-weight spanning tree is unique, and another way so that the minimum-weight spanning tree is not unique.
i to j is the entry
2.3.5. (-) There are five cities in a network. The travel time for traveling directly from a;,j in the matrix below. The matrix is not symmetric (use directed
104
## Chapter 2: Trees and Distance
graphs), and ai.J = oo indicates that there is no direct route. Determine the least travel time and quickest route from i to j for each pair i, j.
J4 ( 30
00
17 15 12
00
~:
y~ ~~)
0 8 10 0
2.3.6. (!)Assign integer weights to the edges of Kn. Prove that on every cycle the total weight is even if and only if the subgraph consisting of the edges with odd weight is a spanning complete bipartite subgraph. (Hint: Show that every component of the subgraph consisting of the edges with even weight is a complete graph.) 2.3.7. Let G be a weighted connected graph with distinct edge weights. Without using Kruskal's Algorithm, prove that G has only one minimum-weight spanning tree. (Hint: Use Exercise 2.1.34.) 2.3.8. Let G be a weighted connected graph. Prove that no matter how ties are broken in choosing the next edge for Kruskal's Algorithm, the list of weights of a minimum spanning tree (in nondecreasing order) is unique. 2.3.9. Let F be a spanning forest of a connected weighted graph G. Among all edges of G having endpoints in different components of F, let e be one of minimum weight. Prove that among all the spanning trees of G that contain F, there is one of minimum weight that contains e. Use this to give another proof that Kruskal's Algorithm works. 2.3.10. (!) Prim's Algorithm grows a spanning tree from a given vertex of a conected weighted graph G, iteratively adding the cheapest edge from a vertex already/reached to a vertex not yet reached, finishing when all the vertices of G have been reached. (Tie.s are broken arbitrarily.) Prove that Prim's Algorithm produces a minimum-weight spanning tree of G. (Jamik [1930], Prim [1957], Dijkstra [1959], independently). 2.3.11. For a spanning tree Tin a weighted graph, let in(T) denote the maximum among the weights of the edges in T. Let x denote the minimum of m (T) over all spanning trees of a weighted graph G. Prove that if T is a spanning tree in G with minimum total weight, then m(T) = x (in other words, T also minimizes the maximum weight). Construct an example to show that the converse is false. (Comment: A tree that minimizes the maximum weight is called a bottleneck or minimax spanning tree.) 2.3.12. In a weighted complete graph, iteratively select the edge of least weight such that the edges selected so far form a disjoint union of paths. Aft.er n -1 steps, the result is a spanning path. Prove that this algorithm always gives a minimum-weight SJ>anning path, or give an infinite family of counterexamples where it fails. 2.3.13. (!) Let T be a minimum-weight spanning tree in G, and let T' be another spanning tree in G. Prove tha.t T' can be transformed into T by a list of steps that exchange one edge of T' for one edge of T, such that the edge set is always a spanning tree and the total weight never increases. 2.3.14. (!)Let C be a cycle in a connected weighted graph. Let e be an edge of maximum weight on C. Prove that there is a minimum spanning tree not containing e. Use this to prove that iteratively deleting a heaviest non-cut-edge until the remaining graph is acyclic produces a minimum-weight spanning tree.
## Section 2.3: Optimization and Trees
105
2.3.15. Let T be a minimum-weight spanning tree in a weighted connected graph G. Prove that T omits some heaviest edge from every cycle in G. 2.3.16. Four people m~st cross a canyon at night on a fragile bridge. At most two people can be on the bridge at once. Crossing requires carrying a flashlight, and there is only one flashlight (which can cross only by being carried). Alone, the four people cross in 10, 5, 2, 1 minutes, respectively. When two cross together, they move at the speed of the slower person. In 18 minutes, a flash flood coming down the canyon will wash away the bridge. Can the four people get across in time? Prove your answer without using graph theory and describe how the answer can be found using graph theory. 2.3.17. Given a starting vertex u in an unweighted graph or digraph G, prove directly (without Dijkstra's Algorithm) that Algorithm 2.3.8 computes d(u, z) for all z E V(G). 2.3.18. Explain how to use Breadth-First Search to compute the girth of a graph. 2.3.19. (+) Prove that the following algorithm correctly finds the diameter of a tree. First, run BFS from an arbitrary vertex w to find a vertex u at maximum distance from w. Next, run BFS from u to reach a vertex v at maximum distance from u. Report diam T = d(u, v). (Cormen-Leiserson-Rivest [1990, p476]) 2.3.20. Minimum diameter spanning tree. An MDST is a spanning tree where the maximum length of a path is as small as possible. Intuition suggests that running Dijkstra's Algorithm from a vertex of minimum eccentricity (a center) will produce an MDST, but this may fail. a) Construct a 5-vertex example of an unweighted graph (edge weights all equal 1) such that Dijkstra's Algorithm can be run from some vertex of minimum eccentricity and produce a spanning tree that does not have minimum diameter. b) Construct a 4-vertex example of a weighted graph such that Dijkstra's algorithm cannot produce an MDST when run from any vertex. 2.3.21. Develop a fast algorithm to test whether a graph is bipartite. The graph is given by its adjacency matrix or by lists of vertices and their neighbors. The algorithm should not need to consider an edge more than twice. 2.3.22. (-)Solve the Chinese Postman Problem in the k-dimensional cube Qk under the condition that every edge has weight 1. 2.3.23. Every morning the Lazy Postman takes the bus to the Post Office. From there, he chooses a route to reach home as quickly as possible (NOT ending at the Post Office). Below is a map of the streets along which he must deliver mail, giving the number of minutes required to walk each block whether delivering or not. P denotes the post office and H denotes home. What must the edges traveled more than once satisfy? How many times will each edge be traversed' in the optimal route?
c~~8_""'n~--8~---::.- H
7
B 2 .__--'=----...--=--'*---=----
106
## Chapter 2: Trees and Distance
2.3.24. (- ) Explain why the optimal trails pairing up odd vertices in an optimal solution to the Chinese Postman Problem may be assumed to be paths. Construct a weighted graph with four odd vertices where the optimal solution to the Chinese Postman Problem requires duplicating the edges on two paths that have a common vertex. 2.3.25. Let G be a rooted tree where every vertex has 0 or k children. Given k, for what values of n(G) is this possible? 2.3.26. Find a recurrenc~ relation to count the binary trees with n + 1 leaves (here each non-leaf vertex has exactly two children, and the left-to-right order of children matters). When n = 2, the possibilities are the two trees below.
2.3.27. Find a recurrence relation for the number of rooted plane trees with n vertices.
(As in a rooted binary tree, the subtrees obtained by deleting the root of a rooted plane
## tree are distinguished by their order from left to right.)
2.3.28. (- ) Compute a code with minimum expected length for a set of ten messages whose relative frequencies are 1, 2, 3, 4, 5, 5, 6, 7, 8, 9. What is the expected length of a message in this optimal code? 2.3.29. ( - ) The game of Scrabble has 100 tiles as listed below. This does not agree with English; "S" is less frequent here, for example, to improve the game. Pretend that these are the relative frequencies in English, and compute a prefix-free code of minimum expected length for transmitting messages. Give the answer by listing the relative frequency for each length of code word. Compute the expected length of the code (per text character). (Comment: ASCII coding uses five bits per letter; this code will beat that. Of course, ASCII suffers the handicap of including codes for punctuation.)
ABCDEFGHIJKLMNOPQRSTUVWXYZ0
9 2 2 4 12 2 3 2 9 1 1 4 2 6 8 2 1 6 4 6 4 2 2 1 2 1 2 2.3.30. Consider n messages occurring with probabilities p 1 , ... , p., such that each Pi is a power of 1/2 (each Pi :'.':: 0 and L Pi = 1). a) Prove that the two least likely messages have equal pr~bability. b) Prove that the expected message length of the Huffman code for this distribution is - L Pi lg p;. 2.3.31. (+) Suppose that n messages occur with probabilities p 1 , , p. and that the words are assigned distinct binary code words. Prove that for every code, the expected length of a code word with respect to this distribution is at least - L p; lg p;, (Hint: Use induction on n.) (Shannon [1948])
Chapter3
## Matchings and Factors
3.1. Matchings and Covers
Within a set of people, some pairs are compatible as roommates; under what conditions can we pair them all up? Many applications of graphs involve such pairings. In Example 1.1.9 we considered the problem of filling jobs with qualified applicants. Bipartite graphs have a natural vertex partition into two sets, and we. want to know whether the two sets can be paired using edges. In the roommate question, the graph need not be bipartite.
3.1.1. Definition. A matching in a graph G is a set of rton-loop edges with no shared endpoints. The vertices incident to the edges of a matching M are saturated by M; the others are unsaturated (we say M -saturated and M-unsaturated). A perfect matching in a graph is a matching that saturates every vertex. 3.1.2. Example. Perfect matchings in Kn,n Consider Kn,n with partite sets X = {x1, ... , Xn} and Y = {y1, ... , Yn}. A perfect matching defines a bijection from X to Y. Successively finding mates for xi, x2 , yields n! perfect matchings. Each matching is represented by a permutation of [n], mapping i to j when x; is matched to Yj. We can express the matchings as matrices. With X and Y indexing the rows and columns, we let position i, j be 1 for each edge x;yj in a matching M to obtain the corresponding matrix. There is one 1 in each row and each column.
x
y
Y1 Y2 Ya Y4
107
108
## Chapter 3: Matchings and Factors
3.1.3. Example. Perfect matchings in complete graphs. Since it has odd order, K2n+1 has no perfect matching. The number f 11 of perfect matchings in K2n is the number of ways to pair up 2n distinct people. There are 2n - 1 choices for the partner of V2n, and for each such choice there are fn-1 ways to complete the matching. Hence fn = (2n - l)fn-1 for n ~ 1. With fo = 1, it follows by induction that fn = (2n - 1) (2n - 3) (1). There is also a counting argument for fn. From an ordering of2n people, we form a matching by pairing the first two, the next two, and so on. Each ordering thus yields one matching. Each matching is generated by 2n n ! orderings, since change the order of the pairs or the order within a pair does not change the resulting matching. Thus there are fn = (2n)!/(2nn!) perfect matchings. The usual drawing of the Petersen graph shows a perfect matching and two 5-cycles; counting the perfect matchings takes some effort (Exercise 14). The inductive construction of the hypercube Qk readily yields many perfect matchings (Exercise 16), but counting them exactly is difficult. The graphs below have even order but no perfect matchings.
>-<
MAXIMUM MATCHINGS
A matching is a set of edges, so its size is the number of edges. We can seek a large matching by iteratively selecting edges whose endpoints are not used by the edges already selected, until no more are available. This yields a maximal matching but maybe not a maximum matching. 3.1.4. Definition. A maximal matching in a graph is a matching that cannot be enlarged by adding an edge. A maximum matching is a matching of maximum size among all matchings in the graph. A matching M is maximal if every edge not in M is incident to an edge already in M. Every maximum matching is a maximal matching, but the converse need not hold. 3.1.5. Example. Maximal =j:. maximum. The smallest graph having a maximal matching that is not a maximum matching is P4 lfwe take the middle edge, then we can add no other, but the two end edges form a larger matching. Below we show this phenomenon in P4 and in P6.
## Section 3.1: Matchings and Covers
109
In Example 3.1.5, replacing the bold edges by the solid edges yields a larger matching. This gives us a way to look for larger matchings.
3.1.6. Definition. Given a matching M, an M-alternating path is a path that alternates between edges in M and edges not in M. An M-alternating path whose enpoints are unsaturated by M is an M-augmenting path.
Given an M-augmenting path P, we can replace the edges of M in P with the other edges of P to obtain a new matching M' with one more edge. Thus when M is a maximum matching, there is no M-augmenting path. In fact, we prove next that maximum matchings are characterized by the absence of augmenting paths. We prove this by considering two matchings and examining the set of edges belonging to exactly one of them. We define this operation for any two graphs with the same vertex set. (The operation is defined in general for any two sets; see Appendix A.)
3.1.7. Definition. If G and H are graphs with vertex set V, then the symmetric difference Gt::.H is the graph with vertex set V whose edges are all those edges appearing in exactly one of G and H. We also use this notation for sets of ec;l.ges; in particular, if M and M' are matchings, then
Mt::.M' = (M- M') U (M' - M).
3.1.8. Example. In the graph below, M is the matching with five solid edges, M' is the one with six bold edges, and the dashed edges belong to neither M nor M'. The two matchings have one common edge e; it is not in their symmetric difference. The edges of M 1::.M' form a cycle oflength 6 and a path oflength 3.
-;z
I
3.1.9. Lemma. Every component of the symmetric difference of two matchings is a path or an even cycle. Proof: Let M and M' be matchings, and let F = M 1::.M'. Since M and M' are matchings, every vertex has at most one incident edge from each of them. Thus F has at most two edges at each vertex. Since tl(F) s 2, every component of F is a path or a cycle. Furthermore, every path .or cycle in F alternates between edges of M - M' and edges of M' - M. Thus each cycle has even length, with an equal number of edges from M and from M'. 3.1.10. Theorem. (Berge [1957)) A matching M in a graph G is a maximum matching in G if and only if G has no M-augmenting path.
110
## Chapter 3: Matchings and Factors
Proof: We prove the contrapositive of each direction; G has a matching larger than M if and only if G has an M-augmenting path. We have observed that an M-augmenting path can be used to produce a matching larger than M. For the converse, let M' be a matching in G larger than M; we construct an M-augmentingpath. Let F = Mt::.M'. By Lemma 3.1.9, F consists of paths and even cycles; the cycles have the same number of edges from M and M'. Since IM'I > !}Ir/I, F must have a component with more edges of M' than of M. Such a component can only be a path that starts and ends with an edge of M'; thus it is an M -augmenting path in G.
## llALUS MATCHING CONDITION
When we are filling jobs with applicants, there may be many more applicants than jobs; successfully filling the jobs will not use all applicants. To model this problem, we consider an X, Y-bigraph (bipartite graph with bipartition X, Y-Definition 1.2.17), and we seek a matching that saturates X. If a matching M saturates X, then for every S ~ X there must be at least ISi vertices that have neighbors in S, because the vertices matched to S must be chosen from that set. We use NG(S) or simply N(S) to denote the set of vertices having a neighbor in S. Thus IN(S)I 2: ISi is a necessary condition. The condition "For all S ~ X, IN(S)I 2: ISi" is Hall's Condition. Hall proved that this obvious necessary condition is also sufficient (TONCAS).
3.1.11. Theorem. (Hall's Theorem-P. Hall (1935]) An X, Y-bigraph G has a matching that saturates X if and only if IN (S) I 2: IS I for all S ~ X.
Proof: Necessity. The ISi vertices matched to S must lie in N(S). Sufficiency. To prove that Hall's Condition is sufficient, we prove the contrapositive. If Mis a maximum matching in G and M does not saturate X, then we obtain a set S ~ X such that IN(S)I < ISi. Let u E X be a vertex unsaturated by M. Among all the vertices reachable from u by M-alternating paths in G, let S consist of those in Jt, and let T consist of those in Y (see figure below with Min bold). Note that u ES.
s
u
y
T'= N(S)
We claim that M matches T with S - {u}. The M-alternating paths from u rel!lch Y along edges not in M and. return to X along edges in M. Hence every vertex of S - {u} is reached by an edge in M from a vertex in T. Since there is no M-augmenting path, every vertex of T is saturated; thus an M-alternating
## Section 3.1: Matchings and Covers
111
path reaching y E T extends via M to a vertex of S. Hence these edges of M yield a bijection from T to S - {u}, and we have ITI =IS - {u}I. The matching between T and S - {u} yields T s; N(S). In fact, T = N(S). Suppose that y E Y - T has a neighbor v E S. The edge vy cannt>t be in M, since u is unsaturated and the rest of S is matched to T by M. Thus adding vy to an M -alternating path reaching v yields an M -alternating path to y. This contradicts y ~ T, and hence vy cannot exist. With T = N(S), we have proved that IN(S)I = ITI = ISi - 1 < ISi for this choice of S. This completes the proof of the contrapositive. One can also prove sufficiency by assuming Hall's Condition, supposing that no matching saturates X, and obtaining a contradiction. As we have seen, lack of a matching saturating X yields a violation of Hall's Condition. Contradicting the hypothesis usually means that the contrapositive of the desired implication has been proved. Thus we have stated the proof in that language. 3.1.12. Remark. Theorem 3.1.11 implies that whenever an X, Y-bigraph has no matching saturating X, we can verify this by exhibiting a subset of X with too few neighbors. Note also that the statement and proof permit multiple edges. Many proofs ofHall's Theorem have been published; see Mirsky [1971, p38] and Jacobs [1969] for summaries. A proof by M. Hall [1948] leads to a lower bound on the number of matchings that saturate X, as a function of the vertex degrees. We consider algorithmic aspects in Section 3.2. When the sets of the bipartition have the same size, Hall's Theorem is the Marriage Theorem, proved originally by Frobenius [1917]. The name arises from the setting of the compatibility relation between a set of n men and a set of n women. If every man is compatible with k women and every woman is compatible with k men, then a perfect matching must exist. Again multiple edges are allowed, which enlarges the scope of applications (see Theorem 3.3.9 and Theorem 7.1.7, for example). 3.1.13. Corollary. For k > 0, every k-regular bipartite graph has a perfect matching. Proof: Let G beak-regular X, Y-bigraph. Counting the edges by endpoints' in X and by endpoints in Y shows that k IXI = k IYI, so IXI = IYI. Hence it suffices to verify Hall's Condition; a matching that saturates X will also saturate Y and be a perfect matching. Consider S s; X. Let m be the number of edges from S to N(S). Since G is k-regular, m = k ISi~ These m edges are incident to N(S), so m ~ k IN(S)I. Hence k ISi ~ k IN(S)I, which yields IN(S)I :::: ISi when k > 0. Having chosen S s; X arbitrarily, we have established Hall's condition. One can also use contradiction here. Assuming that G has no perfect matching yields a set S s; X such that IN(S)I < ISi-. The argument obtaining a contradiction amounts to a rewording of the direct proof given above.
112
## Chapter 3: Matchings and Factors
MIN-MAX THEOREMS
When a graph G does not have a perfect matching, Theorem 3.1.10 allows us to prove that M is a maximum matching by proving that G has no M-augmenting path. Exploring all M-alternating paths to eliminate the possibility of augmentation could take a long time. We faced a similar situation when proving that a graph is not bipartite. Instead of checking all possible bipartitions, we can exhibit an odd cycle. Here again, instead of exploring all M -alternating paths, we would prefer to exhibit an explicit structure in G that forbids a matching larger than M. 3.1.14. Definition. A vertex cover of a graph G is a set Q V(G) that contains at least one endpoint of every edge. The vertices in Q cover E (G). In a graph that represents a road network (with straight roads and no isolated vertices), we can interpret the problem of finding a minimum vertex cover as the problem of placing the minimum number of policemen to guard the entire road network. Thus "cover" means "watch" in this context. Since no vertex can cover two edges of a matching, the size of every vertex cover is at least the size of every matching. Therefore, obtaining a matching and a vertex cover of the same size PROVES that each is optimal. Such proofs exist for bipartite graphs, but not for all graphs. 3.1.15. Example. Matchings and vertex covers. In the graph on the left below we mark a vertex cover of size 2 and show a matching of size 2 in bold. The vertex cover of size 2 prohibits matchings with more than 2 edges, and the matching of size 2 prohibits vertex covers with fewer than 2 vertices. As illustrated on the right, the optimal values differ by 1 for an odd cycle. The difference can be arbitrarily large (Exercise 3.3.10).
0
3.1.16. Theorem. (Konig [1931], Egervary [1931]) If G is a bipartite graph, then the maximum size of a matching in G equals the minimum size of a vertex cover of G. Proof: Let G be an X, Y-bigraph. Since distinct vertices must be used to cover the edges of a matching, IQI ~ IM I whenever Q is a vertex cover and M is a matching in G. Given a smallest vertex cover Q of G, we construct a matching of"size IQI to prove that equality can always be achieved. Partition Q by letting R = Q n X and T = Q n Y. Let Hand H' be the subgraphs of G induced by R u (Y - T) and T u (X - R), respectively. We use
## Section 3.1: Matchings and Covers
113
Hall's Theorem to show that H has a matching that saturates R into Y - T and H' has a matching that saturates T. Since Hand H' are disjoint, the two matchings together form a matching of size IQ I in G. Since R U T is a vertex cover, G has no edge from Y - T to X - R. For each S ~ R, we consider NH(S), which is contained in Y - T. If INH(S)I < ISi, then we can substitute NH(S) for Sin Q to obtain a smaller vertex cover, since NH(S) covers all edges incident to S that are not covered by T. The minimality of Q thus yields Hall's Condition in H, and hence H has a matching that saturates R. Applying the same argument to H' yields the matching that saturates T.
x
H'
y
As graph theory continues to develop, new proofs of fundamental results like the Konig-Egervary Theorem appear; see Rizzo (2000].
3.1.17. Remark. A min-max relation is a theorem stating equality between the answers to a minimization problem and a maximization problem over a class of instances. The Konig-Egervary Theorem is such a relation for vertex covering and matching in bipartite graphs. For the discussions in this text, we think of a dual pair of optimization problems as a maximization problem Mand a minimization problem N, defined on the same instances (such as graphs), such that for every candidate solution M to M and every candidate solution N to N, the value of M is less than or equal to the value of N. Often the "value" is cardinality, as above where M is maximum matching and N is minimum vertex cover. When M and N are dual problems, obtaining candidate solutions M and N that have the same value PROVES that M and N are optimal solutions for that instance. We will see many pairs of dual problems in this book. A min-max relation states that, on some class of instances, these short proofs of optimality exist. These theorems are desirable because they save work! Our next objective is another such theorem for independent sets in' bipartite graphs.
## INDEPENDENT SETS AND COVERS
We now tum from matchings to independent sets. The independence number of a graph is the maximum size of an independent set of vertices.
114
## Chapter 3: Matchings and Factors
3.1.18. Example. The independence number of a bipartite graph does not always equal the size of a partite set. In the graph below, both partite sets have size 3, but we have marked an independent set of size 4.
No vertex covers two edges of a matching. Similarly, no edge contains two vertices of an independent set. This yields another dual covering problem. 3.1.19. Definition. An edge cover of G is a set L of edges such that every vertex of G is incident to some edge of L. We say that the vertices of G are covered by the edges of L. In Example 3.1.18, the four edges incident to the marked vertices form an edge cover; the remaining two vertices are covered "for free". Only graphs without isolated vertices have edge covers. A perfect matching forms an edge cover with n(G)/2 edges. In general, we can obtain an edge cover by adding edges to a maximum matching. 3.1.20. Definition. For the optimal sizes of the sets in the independence and covering problems we have defined, we use the notation below. maximum size of independent set maximum size of matching minimum size of vertex cover minimum size of edge cover
a(G) a'(G)
/3(G) /3'(G)
A graph may have many independent sets of maximum size (C5 has five of them), but the independence number a(G) is a single integer (a(C5 ) = 2). The notation treats the numbers that answer these optimizatjon problems as graph parameters, like the order, size, maximum degree, diameter, etc. Our use of a'(G) to count the edges in a maximum matching suggests a relationship with the parameter a( G) that counts the vertices in a maximum independent set. We explore this relationship in Section 7.1. We use f3(G) for minimum vertex cover due to its interaction with maximum matching. The "prime" goes on /J'(G) rather than on /J(G) because f3(G) counts a set of vertices and /3' (G) counts a set of edges. In this notation, the Konig-Egervary Theorem states that a'(G) = f3(G) for every bipartite graph G. We will prove that also a(G) = f3'(G) for bipartite graphs without isolated vertices. Since no edge can coyer two vertices of an independent set, the inequality /J'(G) ~ a(G) is immediate. (When S 5; V(G), we often use S to denote V ( G) - S, the remaining vertices).
## Section 3.1: Matchings and Covers
115
3.1.21. Lemma. In a graph G, S ~ V ( G) is an independent set if and only if S is a vertex cover, and hence a(G) + {J(G) = n(G). Proof: If S is an independent set, then every edge is incident to at least one vertex ofS. Conversely, ifS covers all the edges, then there are no edges joining vertices of S. Hence every maximum independent set is the complement of a minimum vertex cover, and a(G) + {J(G) = n(G).
'.l'he relationship between matchings and edge coverings is more subtle. Nevertheless, a similar formula holds.
3.1.22. Theorem. (Gallai [1959]) If G is a graph without isolated vertices, then a'(G) + {J'(G) = n(G). Proof: From a maximum matching M, we will construct an edge cover of size n ( G) - IM 1. Si~ce a smallest edge cover is no bigger than this cover, this will imply that {J'(G) ~ n(G) - a'(G). Also, from a minimum edge cover L, we will construct a matching of size n(G) - ILi. Since a largest matching is no smaller than this matching, this will imply that a'(G) 2: n(G) - fJ'(G). These
two inequalities complete the proof. Let M be a maximum matching in G. We construct an edge cover of G by adding to M one edge incident to each unsaturated vertex. We have used one edge for each vertex, except that each edge of M takes care of two vertices, so the total size of this edge cover is n(G) - IMI, as desired. Now let L be a minimum edge cover. If both endpoints of an edge e belong to edges in L other than e, then e L, since L - {e} is also an edge cover. Hence each component formed by edges of L has at most one vertex of degree exceeding 1 and is a star (a tree with at most one non-leaf). Let k be the number of these components. Since L has one edge for each non-central vertex in each star, we have ILi = n(G) - k. We form a matching M of size k = n(G) - ILi by choosing one edge from each star in L.
3.1.23. Example. The graph below has 13 vertices. A matching of size 4 ap-
pears in bold, and adding the solid edges yields an edge cover of size 9. The dashed edges are not needed in the cover. The edge cover consists of four stars; from each we extract one edge (bold) to form the matching.
3.1.24. Corollary. (Konig [1916]) If G is a bipartite graph with no isolated vertices, then a(G) = {J'(G). Proof: By Lemma 3.1.21 and Theorem 3.1.22, a(G)
+ {J(G)
= a'(G)
+ {J'(G).
116
## DOMINATING SETS (optional)
The edges covered by one vertex in a vertex cover are the edges incident to it; they form a star. The vertex cover problem can be described as covering the edge set with the fewest stars. Sometimes we instead want to cover the vertex set with fewest stars. This is equivalent to our next graph parameter.
3.1.25. Example. A company wants to establish transmission towers in a remote region. The towers are located at inhabited buildings, and each inhabited building must be reachable. If a transitter at x can reach y, then also one at y can reach x. Given the pairs that can reach each other, how many transmitters are needed to cover all the buildings? A similar problem comes from recreational mathematics: How many queens are needed to attack all squares on a chessboard? (Exercise 56). 3.1.26. Definition. In a graph G, a set S ~ V ( G) is a dominating set if every vertex not in S has a neighbor in S. The domination number y(G) is the minimum size of a dominating set in G. 3.1.27. Example. The graph G below has a minimal dominating set of size 4 (circles) and a tninimum dominating set of size 3 (squares): y ( G) = 3.
Berge [1962] introduced the notion of domination. Ore [1962] coined this terminology, and the notation y(G) appeared in an early survey (CockayneHedetniemi [1977]). An entire book (Haynes-Hedetniemi-Slater [1998]) is devoted to domination and its variations.
3.1.28. Example. Covering the vertex set with stars may not require as many stars as covering the edge set. When a graph G has no isolated vertices, every vertex cover is a dominating set, so y(G) ~ f3(G). The difference can be large; y(Kn) = 1, but /3(Kn) = n - 1.
When studying domination as an extremal problem, we try to obtain bounds in terms of other graph parameters, such as the order and the minimum degree. A vertex of degree k dominates itself and k other vertices; thus every dominating set in a k-regular graph G has size at least n(G)/(k + 1). For every graph with minimum degree k, a greedy algorithm produces a dominating set not too much bigger than this.
3.1.29. Definition. The closed neighborhood N[v] of a vertex v in a graph is N(v) U {v}; it is the set of vertices dominated by v.
## Section 3.1: Matchings and Covers
117
3.1.30. Theorem. (Arnautov [1974], Payan [1975]) Every n-vertex graph with minimum degree k has a dominating set of size at most n l+l~l;+ll. Proof: (Alon [1990J) Let G be a graph with minimum degree k. Given S s; V ( G), let U be the set of vertices not dominated by S. We claim that some vertex y outside S dominates at least IUI (k + 1)/n vertices of U. Each vertex in Uhas at.least k neighbors, so LvEU IN[vJI 2: IUI (k + 1). Each vertex of G is counted at most n times by these IUI sets, so some vertex y appears at least IUI (k + 1)/n times and satisfies the claim. We iteratively select a vertex that dominates the most of the remaining undominated vertices. We have proved that when r undominated vertices remain, after the next selection at most r(l - (k + 1)/n) undominated vertices remain. Hence after n lni~~l) steps the number of undominated vertices is at most
k+l The selected vertices and these remaining undominated vertices together form a dominating set of size at most n 1 +1~l;+ 1 l.
n
## n(l - k+l)nln(k+l)/(k+l) < ne-ln(k+l) =
_n_
3.1.31. Remark. This bound is also proved by probabilistic methods in Theorem 8.5.10. Caro-Yuster-West [2000J showed that for large k the total domination number satisfies a bound asymptotic to this. Alon [1990J used probabilistic methods to show that this bound is asymptotically sharp when k is large. Exact bounds remain of interest for small k. Among connected n-vertex graphs, 8(G) 2: 2 implies y(G) ~ 2n/5 (McCuaig-Shepherd [1989], with seven small exceptions), and 8(G) 2: 3 implies y(G) ~ 3n/8 (Reed [1996]). Exercise 53 requests constructions achieving these bounds.
Many variations on the concept of domination are studied. In Example 3.1.25, for example, one might want the transmitters to be able to communicate with each other, which requires that they induce a connected subgraph.
3.1.32. Definition. A dominating set S in G is a connected dominating set if G [SJ is connected, an independent dominating set if G[SJ is independent, and a total dominating set if G [SJ has no isolated vP-rtex.
Each variation adds a constraint, so dominating sets of these types are at least as large as y ( G). Exercises 54-60 explore these variations. Studying independent dominating sets amounts to studying maximal independent sets. This leads to a nice result about claw-free graphs.
3.1.33. Lemma. A set of vertices in a graph is an independent dominating set if and only ifit is a maximal independent set. Proof: Among independent sets, Sis maximal if and only if every vertex outside S has a neighbor in S, which is the condition for S to be a dominating set.
118
## Chapter 3: Matchings and Factors
3.1.34. Theorem. <Bollobas-Cockayne [1979]) Every claw-free graph has an independent dominating set of size y ( G). Proof: Let S be a minimum dominating set in a claw-free graph G. Let S' be a maximal independent subset of S. Let T = V(G) - N(S'). Let T' be a maximal independent subset of S. Since T' contains no neighbor of S', S' U T' is independent. Since S' is maximal in S, we have S ~ N(S'). Since T' is maximal in T, T' dominates T. Hence S' u T' is a dominating set. It remains to show that IS' U T'I : :; y(G). Since S' is maximal in S, T' is independent, and G is claw-free, each vertex of S - S' has at most one neighbor in T'. Since S is dominating, each vertex of T' has at least one neighbor in S - S'. Hence IT'I : :; IS - S'I, which yields IS' U T'I : :; ISi = y(G).
EXERCISES
3.1.1. (- ) Find a maximum matching in each graph below. Prove that it is a maximum matching by exhibiting an optimal solution to the dual problem (minimum vertex cover). Explain why this proves that the matching is optimal.
3.1.2. (-) Determine the minimum size of a maximal matching in the cycle Cn. 3.1.3. (-) Let S be the set of vertices saturated by a matching M in a graph G. Prove that some maximum matching also saturates all of S. Must the statement be true for every maximum matching? 3.1.4. (- ) For each of a, a', fi, {3', characterize the simple graphs for which the value of the parameter is 1. 3.1.5. (-)Prove that a(G) ~ t>~~~~ 1 for every graph G. 3.1.6. (-) Let T be a tree with n vertices, and let k be the maximum size of an independent set in T. Determine a'(T) in terms of n and k. 3.1.7. (-)Use Corollary 3.1.24 to prove that a graph G is bipartite if and only if a(H) =
{3' (H) for every subgraph H of G with no isolated vP.rtices.
3.1.8. (!)Prove or disprove: Every tree has at most one perfect matching. 3.1.9. (!)Prove that every maximal matching in a graph G has at least a'(G)/2 edges.
## Section 3.1: Matchings and Covers
119
3.1.10. Let Mand N be matchings in a graph G, with IMI > INI. Prove that there exist matchings M' and N' in G such that IM'I = IMI - 1, IN'I = INI + 1, and M', N' have the same union and intersection (as edge sets) as M, N. 3.1.11. Let C and C' be cycles in a graph G. Prove that C 1' decomposes into cycles. 3.1.12. Let C and C' be cycles oflength k in a graph with girth k. Prove that C 6C' is a single cycle if and only if C n C' is a single path. (Jiang [2001]) 3.1.13. Let Mand M' be matchings in an X, Y-bigraph G. Suppose that M saturates S ~ X and that M' saturates T ~ Y. Prove that G has a matching that saturates S U T. For example, below we show M as bold edges and M' as thin edges; we can saturate S U T by using one edge from each.
xx
T
E
3.1.14. Let G be the Petersen graph. In Example 7.1.9, analysis by cases is used to show that if Mis a perfect matching in G, then G - M = C5 + C5 . Assume this. a) Prove that every edge of G lies in four 5-cycles, and count the 5-cycles in G. b) Determine the number of perfect matchings in G. 3.1.15. a) Prove that for every perfect matching M in Qk and every coordinate i there are an even number of edges in M whose endpoints differ in coordinate i. b) Use part (a) to count the perfect matchings in Q 3 3.1.16. Fork ~ 2, prove that Qk has at least 2< 2k- l perfect matchings.
2
[kl,
3.1.17. The weight of a vertex in Qk is the number of ls in its label. Prove that for every perfect matching in Qk. the number of edges matching words of weight i to words 1 of weight i + 1 is e~ ), for 0 :5 i :5 k - 1. 3.1.18. (!) Two people play a game on a graph G, alternately choosing distinct vertices. Player 1 start by choosing any vertex. Each subsequent choice must be adjacent to the preceding choice (of the other player). Thus together they follow a path. The last player able to move wins. Prove that the second player has a winning strategy if G has a perfect matching, and otherwise the first player has a winning strategy. (Hint: For the second part, the first player should start with a vertex omitted by some maximum matching.) 3.1.19. (!) Let A = (A 1 , . , Am) be a collection of subsets of a set Y. A system of distinct representatives (SDR) for A is a set of distinct elements a 1 , .. , am in Y such that a; E A;. Prove that A has an SDR if and only if IU;EsA;I ~ ISi for every S ~ {l, ... , m}. (Hint: Transform this to a graph problem.) 3.1.20. The people in a club are planning their summer vacations. Trips t 1 , , tn are available, but trip t; has capacity n;. Each person likes some of the trips and will travel on at most one. In terms of which people like which trips, derive a necessary and sufficient condition for being able to fill all trips (to capacity) with people who like them. 3.1.21. (!)Let G be an X, Y-bigraph such that IN(S)I .> ISi whenever 0 #Sc X. Prove that every edge of G belongs to some matching that saturates X.
120
## Chapter 3: Matchings and Factors
3.1.22. Prove that a bipartite graph G has a perfect matching if and only if IN(S)I =::: ISi for all S ~ V(G), and present an infinite class of examples to prove that this characterization does not hold for all graphs. 3.1.23. (+) Alternative proof of Hall's Theorem. Consider a bipartite graph G with bipartition X, Y, satisfying IN(S)I =::: ISi for every S ~ X. Use induction on IXI to prove that G has a matching that saturates X. (Hint: First consider the case where IN(S)I > IS I for every proper subset S of X. When this does not hold, consider a minimal nonempty T ~ X such that IN(T)I = !Tl.) (M. Hall [1948), Halmos-Vaughan [1950]) 3.1.24. (!)A permutation matrix Pis a 0,1-matrix having exactly one 1 in each row and column. Prove that a square matrix of nonnegative integers can be expressed as the sum of k permutation matrices if and only if all row sums and column sums equal k. 3.1.25. (!) A doubly stochastic matrix Q is a nonnegative real matrix in which every row and every column sums to 1. Prove that a doubly stochastic matrix Q can be expressed Q = c1 P1 ++cm Pm, where c1 , .'.,Cm are nonnegative real numbers summing to 1 and Pi. ... , Pm are permutation matrices. For example, ( 1/2 1/2
## 1~2 ij~ ~j~) = !2 ( ~ ~ ~) + _31 ( ~ ~ ~) + _61 ( ~ ~ ~)
0 0 1 0 1 0 0 1 0 0
(Hint: Use induction on the number of nonzero entries in Q.) (Birkhoff [1946), von Neumann [1953]) 3.1.26. (!) A deck of mn cards with m values and n suits consists of one card of each value in each suit. The cards are dealt into an n-by-m array. a) Prove that there is a set of m cards, one in each column, having distinct values. b) Use part (a) to prove that by a sequence of exchanges of cards of the same value, the cards can be rearranged so that each column consists of n cards of distinct suits. (Enchev [1997]) 3.1.27. (!)Generalizing Tic-Tac-Toe. A positional game consists ofa set X = x 1 , , Xn of positions and a family Wi. ... , Wm of winning sets of positions (Tic-Tac-Toe has nine positions and eight winning sets). Two players alternately choose positions; a player wins by collecting a winning set. Suppose that each winning set has size at least a and each position appears in at most b winning sets (in Tic-Tac-Toe, a= 3 and b = 4). Prove that Player 2 can force a draw if a~ 2b. (Hint: Form an X, Y-bigraph G, where Y = {wi. ... , Wm} U {w~, ... , w~}, with edges x; Wj and x; wj whenever x; E Wi. How can Player 2 use a matching in G? Comment: This result implies that Player 2 can force a draw ind-dimensional Tic-TacToe when the sides are long enough.) 3.1.28. (!) Exhibit a perfect matching in the graph below or give a short proof that it has none. (Lovasz-Plummer [1986, p7])
## Section 3.1: Matchings and Covers
121
3.1.29. (!) Use the Konig-Egervary Theorem to prove that every bipartite graph G has a matching of size at least e(G)/ 6.(G). Use this to conclude that every subgraph of K. with more than (k - l)n edges has a matching of size at least k. 3.1.30. (!) Determine the maximum number of edges in a simple bipartite graph that contains no matching with k edges and no star with I edges. (Isaak) 3.1.31. Use the Konig-Egervary Theorem to prove Hall's Theorem. 3.1.32. (!)In an X, Y-bigraph G, the deficiency of a set Sis def(S) = ISi - IN(S)!; note that def(0) = 0. Prove that a'(G) = !XI - maxss;x def(S). (Hint: Form a bipartite graph G' such that G' has a matching that saturates X if and only if G has a matching of the desired size, and prove that G' satisfies Hall's Condition.) (Ore [1955]) 3.1.33. (!)Use Exercise 3.1.32 to prove the Konig-Egervary Theorem. (Hint: Obtain a matching and a vertex cover of the same size from a set with maximum deficiency.) 3.1.34. (!)Let G be an X, Y-bigraph with no isolated vertices, and define deficiency as in Exercise 3.1.32. Prove that Hall's Condition holds for a matching saturating X if and only if each subset of Y has deficiency at most IYI - IX 1. 3.1.35. Let G be an X, Y-bigraph. Prove that G is (k + l)Krfree if and only if each S ~ X has a subset of size at most k with neighborhood N(S). (Liu-Zhou [1997]) 3.1.36. Let G be an X, Y-bigraph having a matching that saturates X. Letting m = IXI, prove that G has at most (~) edges belonging to no matching of size m. Construct examples to show that this is best possible for every m. 3.1.37. (+)Let G be an X, Y-bigraph having a matching that saturates X. a) Let Sand T be subsets of X such that IN(S)I = ISi and IN(T)I = ITI. Prove that IN(SnT)I = 1snn b) Prove that X has some vertex x such that every edge incident to x belongs to some maximum matching. (Hint: Consider a minimal nonempty set S ~ X such that IN(S)I = ISi, if any exists.) 3.1.38. (+) An island of area n has n married hunter/farmer couples. The Ministry of Hunting divides the island into n equal-sized hunting regions. The Ministry of Agriculture divides it into n equal-sized farming regions. The Ministry of Marriage requires that each couple receive two overlapping regions. By Exercise 3.1.25, this is always possible, Prove a stronger result: guarantee a pairing where each couple's two regions share area at least 4/(n + 1)2 when n is odd and 4/[n(n + 2)] when n is even. Prove also that no larger commQn area can be guaranteed; the example below achieves equality for n = 3. (Marcus-Ree [1959], Floyd [1990]) 1 b 2 b
I I I I I I I I
a a
I I I I
I I I I I I I I
## c c .5 .5 ( 0 .25 .25 .5 .25) .25 .5
3.1.39. Let G be a nontrivial simple graph. Prove that a(G) :::: n(G) - e(G)/ 6.(G). Conclude that a(G}::; n(G)/2 when G also is regular. (P. Kwok) 3.1.40. Let G be a bipartite graph. Prove that a(G) = n(G)/2 if and only if G has a perfect matching.
122
## Chapter 3: Matchings and Factors
3.1.41. A connected n-vertex graph has exactly one cycle if and only if it has exactly n edges (Exercise 2.1.30). Let C be the cycle in such a graph G. Assuming the result of Exercise 3.1.40 for trees, prove that a(G) :::: Ln(G)/2J, with equality if and only if G - V ( C) has a perfect matching. 3.1.42. (!) An algorithm to greedily build a large independent set iteratively selects a vertex of minimum degree in the remaining graph and deletes it and its neighbors. Prove that this algorithm produces an independent set of size at least LvEV(GJ d(v~+I in a graph G. (Caro [1979], Wei [1981]) 3.1.43. Let M be a maximal matching and L a minimal edge cover in a graph with no isolated vertices. Prove the statements below. (Norman-Rabin [1959], Gallai [1959]) a) M is a maximum matching if and only if M is contained in a minimum edge cover. b) L is a minimum edge cover if and only if L contains a maximum matching. 3.1.44. (-) Let G be a simple graph in which the sum of the degrees of any k vertices is less than n - k. Prove that every maximal independent set in G has more than k vertices. (Meyer [1972]) 3.1.45. An edge e of a graph G is a-critical if a(G - e) > a(G). Suppose that xy and xz are a-critical edges in G. Prove that G has an induced subgraph that is an odd cycle containing xy andxz. (Hint: Let Y, Z be maximum independent sets in G-xy and G-xz, respectively. Let H = G [Y l>.Z]. Prove that every component of H has the same number of vertices from Y and from Z. Use this to prove that y and z belong to the same component of H.) (Berge [1970], with a difficult generalization in Markossian-Karapetian [1984]) 3.1.46. (*-)Characterize the graphs with domination number 1. 3.1.47. (*-) Find the smallest tree where the domination number and the v.;;i;ex cover number are not equal. 3.1.48. (*-)Determine y(C.) and y(P.). 3.1.49. (*)Let G be a graph without isolated vertices, and let S be a minimal dominating set in G. Prove that Sis a dominating set. Conclude that y(G) .:::: n(G)/2. (Ore [1962]) 3.1.50. (*)Prove that y(G) .:::: n - f3'(G) .:::: n/2 when G is an n-vertex graph without isolated vertices. For 1 .:::: k .:::: n/2, construct a connected n-vertex graph G with y(G) = k. 3.1.51. (*)Let G be an n-vertex graph. a) Prove that rn/(l + ~(G)l '.::: y(G) _:: : n - ~(G). b) Prove that (1 +diam G)/3.:::: y(G).:::: n - Ldiam G/3j. 3.1.52. (*)Prove that ifthe diameter of G is at least 3, then y(G) .:::: 2. 3.1.53. (*) For all k E N, construct a connected graph with 5k vertices and domination number 2k. Construct a single 3-regular graph G such that y(G) = 3n(G)/8. 3.1.54. (*)Determine the domination number of the Petersen graph, and determine the minimum size of a total dominating.set in the Petersen graph. 3.1.55. (*) In the hypercube Q4 , determine the minimum sizes of a dominating set, an independent dominating set, a connected dominating set, and a total dominating set. 3.1.56. (*)Find a way to place five queens on an eight-by-eight chessboard that attack all other squares. Show that the five queens cannot be placed so that also they do not attack each other. (Comment: Thus the independent domination number of the "queen's graph" exceeds its domination number; it is 7.)
## Section 3.2: Algorithms and Applications
123
3.1.57. (*)For all n EN, construct an n-vertex tree with domination number 2 in which the minimum size of an independent dominating set is Ln/2j. 3.1.58. (*) Prove that a K1,,-free graph G has an independent dominating set of size at most (r - 2)y(G) - (r - 3). (Hint: Generalize the argument of Theorem 3.1.34.) (Bollobas-Cockayne [1979]) 3.1.59. (*) In a graph G of order n, prove that the minimum size of a connected dominating set is n minus the maximum number of leaves in a spanning tree. 3.1.60. (*) Fork :'.S 5, every graph G with 8(G) :'.S k has a connected dominating set of size at most 3n(G)/(k + 1) (Kleitman-West [1991], Griggs-Wu [1992]). Prove that this is sharp using a graph formed from a cyclic arrangement of 3m pairwise-disjoint cliques by making each vertex adjacent to every vertex in the clique before it and the clique after it. Let the clique sizes be rk/21 , Lk/2J , 1, rk/21 , Lk/2J , 1, ....
## 3.2. Algorithms and Applications
MAXIMUM BIPARTITE MATCHING
To find a maximum matching, we iteratively seek augmenting paths to enlarge the current matching. In a bipartite graph, if we don't find an augmenting path, we will find a vertex cover with the same size as the current matching, thereby proving that the current matching has maximum size. This yields both an algorithm to solve the maximum matching problem and ah algorithmic proof of the Konig-Egervary Theorem. Given a matching M in an X, Y -bigraph G, we search for M -augmenting paths from each M-unsaturated vertex in X. We need only search from vertices in X, because every augmenting path has odd length and thus has ends in both X and Y. We will search from the unsaturated vertices in X simultaneously. Starting with a matching of size 0, a' (G) applications. of the Augmenting Path Algorithm produce a maximum matching.
3.2.1. Algorithm. (Augmenting Path Algorithm). Input: An X, Y-bigraph G, a matching Min G, and the set U of M-unsaturated vertices in X. Idea: Explore M -alternating paths from U, letting S ~ X and T ~ Y be the sets of vertices reached. Mark vertices of S that have been explored for path extensions. As a vertex is reached, record the vertex from which it is reached. Initialization: S = U and T = 0. Iteration: If S has no unmarked vertex, stop and report Tu (X - S) as a minimum cover and M as a maximum matching. Otherwise, select an unmarked x E S. To explore x, consider each y E N(x) such that xy M. If y is unsaturated, terminate and report an M-augmenting path from U toy. Otherwise, y is matched to some w E X by M. In this case, include yin T (reached from x)
124
## Chapter 3: Matchings and Factors
and include w in S (reached from y ). After exploring all such edges incident to x, mark x and iterate.
x
y
When exploring x in the iterative step, we may reach a vertex y E T that we have reached previously. Recording x as the previous vertex on the path may change which M-augmenting path we report, but it won't change whether such a path exists.
3.2.2. Theorem. Repeatedly applying the Augmenting Path Algorithm to a bipartite graph produces a matching and a vertex cover of equal size. Proof: We need only verify that the Augmenting Path Algorithm proquces an M-augmenting path or a vertex cover of size IMI. If the algorithm produces an M-augmenting path, we are finished. Otherwise, it terminates by marking all vertices of S and claiming that R = T U ( X - S) is a vertex cover of size IM 1. We must prove that R is a vertex cover and has size IM 1. To show that R is a vertex cover, it suffices to show that there is no edge joining S to Y - T. An M -alternating path from U enters X only on an edge of M. Hence every vertex x of S - U is matched via M to a vertex of T, and there is no edge of M from S to Y - T. Also there is no such edge outside M. When the path reaches x E S, it can continue along any edge not in M, and exploring x puts all other neighbors of x into T. Since the algorithm marks all of S before terminating, all edges from S go to T. Now we study the size of R. The algorithm puts only saturated vertices in T; each y E Tis matched via M to a vertex of S. Since U s; S, also each vertex of X - S is saturated, and the edges of M incident to X - S cannot involve T. Hence they are different from the edges saturating i, and we find that M has at least ITI +IX - SI edges. Since there is no matching larger than this vertex cover, we have IMI = ITI +IX - SI.= IRI.
In addition to studying the correctness of algorithms, we are co:rwerned about the time (number of computational steps) they use. We measure this as a function of the size of the input. For graph problems, we usually use the order n(G) and/or size e(G) to measure the input size.
3.2.3. Definition. The running time of an algorithm is the maximum number of computational steps used, expressed as a function of the size of the input. A good algorithm is one that has polynomial running time. Running time is often expressed as "O(f)", where f is a function of the
## Section 3.2: Algorithms and Applications
125
size of the input. Here 0(/) denotes the set offunctions g such that lg(x)j is bounded by a constant multiple of If (x) I when x is sufficiently large (that is, there exist c, a such that lg(x)I ~ c lf(x)I when lxl 2: a). Many problems we study in Chapters 1-4 have good algorithms; other notions of complexity (Appendix B) need not trouble us yet. Since we don't know how long a particular operation may take on a particular computer, constant factors in running time have little meaning. Hence the "Big Oh" notation 0 (f) is convenient. When f is a quadratic polynomial, we typically abuse notation by writing O(n 2 ) instead of O(f) to describe functions that grow at most quadratically in terms of n.
3.2.4. Remark. Let G be an X, Y -bigraph with n vertices and m edges. Since a'(G) ~ n/2, we find a maximum matching in G by applying Algorithm 3.2.1 at most n/2 times. Each application explores a vertex of X at most once, just
before marking it; thus it considers each edge at most once. If the time for one edge exploration is bounded by a constant, then this algorithm to find a maximum matching runs in time O(nm). Theorem 3.2.22 presents a faster algorithm, with running time O(,J"ii,m). Section 3.3 discusses a good algorithm for maximum matching in general graphs.
## WEIGHTED BIPARTITE MATCHING
Our results on maximum matching generalize to weighted X, Y-bigraphs, where we seek a matching of maximum total weight. If our graph is not all of Kn,n. then we insert the missing edges and assign them weight 0. This does not affect the numbers we can obtain as the weight of a matching. Thus we assume that our graph is Kn,n. Since we consider only nonnegative edge weights, some maximum weighted matching is a perfect matching; thus we seek a perfect matching. We solve both the maximum weighted matching problem and its dual.
3.2.5. Example. Weighted bipartite matching and its dual. A farming company owns n farms and n processing plants. Each farm can produce corn to the capacity of one plant. The profit that results from sending the output of farm i to plant j is w;,j. Placing weight w;,j on edge x; Yj gives us a weighted bipartite graph with partite sets X = {x1; ... , Xn} and Y = {y1, ... , Yn}. The company wants to select edges forming a matching to maximize total profit. The government claims that too much corn is being produced, so it will pay the company not to process corn. The government will pay u; if the company agrees not to use farm i and Vj ifit agrees not to use plant j. Ifu,+vj < w;,;, then the company makes more by using the edge x; yj than by taking the government payments for those vertices. In order to stop all production, the government must offer amounts such that u; + Vj 2: w;,j for all i, j. The government wants to find such values to minimize I: u; + I: Vj. Ill
126
## Chapter 3: Matchings and Factors
3.2.6. Definition. A transversal of an n-by-n matrix consists of n positions, one in each row and each column. Finding a transversal with maximum sum is the Assignment Problem. This is the matrix formulation of the maximum weighted matching problem, where nonnegative weight w;,j is assigned to edge x; Yj of Kn,n and we seek a perfect matching M to maximize the total weight w(M). With these weights, a (weighted) cover is a choice oflabels u;, ... , Un and Vj, .. , Vn such that u; + Vj ~ w;,j for all i, j. 'I'he cost c(u, v) of a cover (u, v) is Lu;+ L Vj. The minimum weighted cover problem is that of finding a cover of rrinimum cost.
Note that the problem of minimum weight perfect matching can be solved using maximum weight matching; simply replace each weight w;,j with M - w;,j for some large number M. The next lemma shows that the weighted matching and weighted cover problems are dual problems.
3.2.7. Lemma. For a perfect matching Mand cover (u, v) in a weighted bipartite graph G, c(u, v) ~ w(M). Also, c(u, v) = w(M) if and only if M consists of edges X;Yj such that U; + Vj = wi,j In this case, Mand (u, v) are optimal. Proof: Since M saturates each vertex, summing the constraints u; + Vj ~ w;,j that arise from its edges yields c(u, v) '.: :'.: w(M) for every cover (u, v). Furthermore, if c(u:, v) = w(M), then equality must hold in each of then inequalities summed. Finally, since c(u, v) ~ w(M) for every matching and every cover, c(u, v) = w(M) implies tha,t there is no matching with weight greater than c(u, v) and no cover with cost less than w(M).
A matching and a cover have the same value only when the edges of the matching are covered with equality. This lead& us to an algorithm.
3.2.8. Definition. The equality subgraph Gu,v for a cover (u, v) is the spanning subgraph of Kn,n having the edges x;yj such that u; + Vj = w;,j
IfGu,v has a perfect matching, then its weight is L:u;+ L Vj. and by Lemma 3.2. 7 we have the optimal solution. Otherwise, we find a matching M and avertex cover Q of the same size in Gu,v (by using the Augmenting Path Algorithm, for example). Let R = Q n X and T = Q n Y. Our matching of size IQI consists of IR I edges from R to Y - T and IT I edges from T to X - R, as shown below. To seek a larger matching in the equality subgraph, we change (u, v) to introduce an edge from X - R to Y - T while maintaining equality on all edges of M. A cover requires u; + Vj ~ w;,j for all i, j; the difference u; + Vj - w;,j is the excess for i, j. Edges joining X - Rand Y - Tare not in Gu,v and have positive e~cess. Let E be the minimum excess on the edges from X - R to Y - T. Reducing u; by E for all x; E X - R maintains the cover condition for these edges while bringing at least one into the equality subgraph. To maintain the cover condition for the edges from X - R to T, we also increase Vj by E for yj E T.
## Section 3.2: Algorithms and Applications
127
We repeat the procedure with the new equality subgraph; eventually we obtain a cover whose equality subgraph has a perfect matching. The resulting algorithm was named the Hungarian Algorithm by Kuhn in honor of the work of Konig and Egervary on which it is based.
x
y
3.2.9. Algorithm. (Hungarian Algorithm-Kuhn [1955], Munkres [1957]). Input: A matrix of weights on the edges of K 11 11 with bipartition X, Y. Idea: Iteratively adjusting the cover (u, v) until the equality subgraph Gu.v has a perfect matching. Initialization: Let (u, v) be a cover, such as u; = max1 w;.J and v_; = 0. Iteration: Find a maximum matching Min Gu.i If Mis a perfect matching, stop and report M as a maximum weight matching. Otherwise, let Q be a vertex cover of size IMI in Gu.v Let R = X n Q and T = Y n Q. Let
E
= min{u;
+ Vj
W;,{ X; Ex -
R, YJ E y - T}.
vJ
by
for Y;
## T. Form the new
We have presented the algorithm using bipartite graphs, but repeatedly drawing a changing equality subgraph is awkward. Therefore, we compute with matrices. The initial weights form a matrix A with w;.J in position i, j. We associate the vertices and the labels (u, v) with the rows and columns, which serve as X and Y, respectively. We subtract w;.J from u; + v1 to obtain the excess matrix: c;.J = u; +VJ - w;. 1 . The edges of the equality subgraph correspond to Os in the excess matrix. 3.2.10. Example. Solving the Assignment Problem. The first matrix below is the matrix of weights. The others display a cover (u, v) and the corresponding excess matrix. We underscore entries in the excess matrix to mark a maximum matching M of Gu.v' which appears as bold edges in the equality subgraphs drawn for the first two excess matrices. (Drawing the equality subgraphs is not necessary.) A matching in Gu.v corresponds to a set of Os in the excess matrix with no two in any row or column; call this a partial transversal. A set of rows and columns covering the Os in the excess mat:rix is a covering set; this corresponds to a vertex cover in Gu.i A covering set of size less than n yields progress toward a solution, since the next weighted cover costs less. We study the Os in the excess matrix and find a partial transversal and a covering set of the same size. In a small matrix, we can do this by inspection.
128
## Chapter 3: Matchings and Factors
We underscore the Os of a partial transversal, and we use Rs and Ts to label the rows and columns of the covering set. At each iteration, we compute the minimum excess on the positions not in a covered row or column (in rows X - Rand columns Y - T). These uncovered positions have positive excess (the corresponding edges are not in the equality subgraph). The value E defined in Algorithm 3.2.9 is the minimum of these excesses. We reduce the label u; by E on rows not in R and increase the label Vj by E on columns in T. In the example below, the covering set used in the first iteration reduces the cost of the coverbut does not augment the maximum matching in the equality subgraph. The second iteration produces a perfect matching. Using the last three columns as a covering set in the first iteration would augment the matching immediately. The transversal of Os after the final iteration identifies a perfect matching whose total weight equals the cost of the final cover. The corresponding edges .have weights 5, 4, 6, 8, 8 in the original data, which sum to 31. The labels 4, 5, 7, 4, 6 and 0, 0, 2, 2, 1 in the final cover satisfy each edge ~xactly and also sum to 31. The.value of the optimal solution is unique, but the solution itself is not; this example has many maximum weight matchings and many minimum cost covers, but all have total weight 31.
0 0 0 0 0
; (~ ~ ~ ~
8 6 5 4 3
2 2 0 3 6 8 3 4 3 0
T T
T T
0 0 2 2
1
5 6 8
(1
0 0 1 1 0 4 Q 4
1 6 4 Q 6 5 5 4
2 1 0 3 7 3 1 3 0 T T T
I) :~
:(g
7 4
6 2
5 1
T T T
3.2.11. Theorem. The Hungarian Algorithm finds a maximum weight matching and a minimum cost cover. Proof: The algorithm begins with a cover. It can terminate only when the equality subgraph has a perfect matching, which guarantees equal value for the current matching and cover. Suppose that (u, v) is the current cover and that the equality subgraph has no perfect matching. Let (u', v') denote the new lists of numbers assigned to the vertices. Because E is the minimum of a nonempty finite set of positive numbers, E > 0.
## Section 3.2: Algorithms and Applications
129
covered. If x; EX - Rand Yj E Y- T, then u; + vj equals u; + Vj - E, which by the choice of E is at least w;,j. The algorithm terminates only when the equality subgraph has a perfect matching, so it suffices to show that it does terminate. Suppose that the weights w;,j are rational. Multiplying the weights by their least common denominator yields an equivalent problem with integer weights. We can now assume that the labels in the current cover also are integers. Thus each excess is also an integer, and at each iteration we reduce the cost of the cover by an integer amount. Since the cost starts at some value and is bounded below by the weight ofa perfect matching, after finitely many iterations we have equality. For real-valued weights in general, see Remark 3.2.12).
We verify first that (u', v') is a cover. The change of labels on vertices of X - Rand T yields u; + vj = u; + Vj for edges x;yj from X - R to Tor from R to Y - T. If x; E R and yj E T, then u; + vj = u; + Vj + E, and the weight remains
3.2.12. * Remark. When the weights are real numbers, the algorithm still works if we obtain vertex covers in the equality subgraph more carefully. We show that the algorithm terminates within n 2 iterations. Because the edges of M remain in the new equality subgraph, the size of the current matching never decreases. Since the size of the matching can increase at most n times, it suffices to show that it must increase within n iterations. If we find the maximum matching M by iterating the Augmenting Path Algorithm, then the last iteration presents us with a vertex cover. We find it by exploring M-alternating paths from the set U of M-unsaturated vertices in X. With S and T denoting the sets of vertices reachable in X and T, we obtain the vertex cover RUT, where R = X - S. Applying a step of the Hungarian Algorithm using the vertex cover R U T maintains equality on M and all the edges in M-alternating paths from U. Edges from T to R disappear from the equality subgraph, but we don't care because they don't appear in M -alternating paths from U. Introducing an edge from S to Y - T either creates an M -augmenting path or increases T while leaving U unchanged. Since we can increase T at most n times, we obtain a larger matching in the equality subgraph within n iterations. 3.2.13.* Remark. The maximum matching and vertex cover problems in bipartite graphs are special cases of the weighted problems. Given a bipartite graph G, form a weighted graph with weight 1 on the edges of G and weight 0 on the edges of Kn,n. The maximum weight of a matching is a' (G). Given integer weights, the Hungarian algorithm always maintains integer labels in the weighted cover. Hence in this weighted cover problem we may restrict the values (labels) used to be integers. Further thought shows that these integers will always be 0 or 1. The vertices receiving label 1 must cover the weight on the edges of G, so they form a vertex cover for G. Minimizing the sum of labels under the integer restriction is equivalent to finding the minimum number of vertices in a vertex cover for G. Hence the answer to the weighted cover problem is f3 (G).
130
## STABLE MATCHINGS (optional)
Instead of optimizing total weight for a matching, we may try to optimize using preferences. Given n men and n women; we want to establish n "stable" marriages. If man x and woman a are paired with other partners, but x prefers a to his current partner and a prefers x to her current partner, then they might leave their current partners and switch to each other. In this situation we say that the unmatched pair (x, a) is an unstable pair.
## Section 3.2: Algorithms and Applications
131
3.2.15. Definition. A perfect matching is a stable matching if it yields no unstable unmatched pair. 3.2.16. Example. Given men x, y, z, w, women a, b, c, d, and preferences listed below, the matching {xa, yb, zd, we} is a stable matching.
Men {x, y, z, w} x:a>b>c>d y:a>c>b>d z:c>d>a>b w:c>b>a>d Women {a, b, c, d}
a:z>x>y>w b:y>w>x>z c:w>x>y>z d:x>y>z>w
In their paper "College admissions and the stability of marriage", Gale and Shapley proved that a stable matching always exists and can be found using a relatively simple algorithm. In the algorithm, men and women do not play symmetric roles; we will discuss this importance of this difference later. The algorithm below generates the matching of Example 3.2.16.
3.2.17. Algorithm. (Gale-Shapley Proposal Algorithm) Input: Preference rankings by each of n men and n women. Idea: Produce a stable matching using proposals by maintaining information about who has proposed to whom and who has rejected whom. Iteration: Each man proposes to the highest woman 0n his preference list who has not previously rejected him. If each woman receives exactly one proposal, stop and use the resulting matching. Otherwise, every woman receiving more than one proposal rejects all of them except the one that is highest on her preference list. Every woman receiving a proposal says "maybe" to the most attractive proposal received. 3.2.18. Theorem. (Gale-Shapley [1962]) The Proposal Algorithm produces a stable matching. Proof: The algorithm terminates (with some matching), because on each nonterminal iteration, the total length of the lists of potential mates for the men decreases. This can happen only n 2 times. Key Observation: the sequence of proposals made by each man is nonincreasing in his preference list, and the sequence of men to whorn a woman says "maybe" is nondecreasing in her preference list, culminating in the man assigned. This holds because men propose repeatedly to the same woman until rejected, and women say "maybe" to the same man until a better offer arrives. If the result is not stable, then there is an unstable unmatched pair (x, a), with x matched to b and y matched to a. By the key observation, x never proposed to a during the algorithm, since a received a mate less desirable than x. The key observation also implies that x would not have proposed to b without earlier proposing to a. This contradiction confirms the stability of the result.
The asymmetry of the proposal algorithm suggests asking which sex is happier. When the first choices of the men are distinct, they all get their first
132
## Chapter 3: Matchings and Factors
choice, and the women are stuck with whomever proposed. When the algorithm runs with women proposing, every woman is at least as happy as when men do the proposing, and every man is at least as unhappy. In Example 3.2.16, running the algorithm with women proposing immediately yields the matching {xd, yb, ca, we}, in which all women are matched to their first choices. In fact, among all stable matchings, every man is happiest in the one produced by the male-proposal algorithm, and every woman is happiest under the femaleproposal algorithm (Exercise 11). Societal conventions thus favor men. The algorithm is used in another setting. Each year, the graduates of medical schools submit preference lists of hospitals where they wish to be residents. The hospitals have their own preferences; we model a hospital with multiple openings as several hospitals with thP same preference list. Chaos in the market for residents (then called interns) forced hospitals to devise and implement the algorithm ten years before the Gale-Shapley paper defined and solved the problem! The result was the National Resident Matching Program, a nonprofit corporation established in 1952 to provide a uniform appointment date and matching procedure. Who is happier with the outcome? Since the medical organizations ran the algorithm, it is not surprising that initially they did the proposing and were happier with the outcome. The distinction is even clearer in another setting; students applying for jobs have preferences, but the employers make the proposals, called ''job offers". Unhappiness with the NRMP caused the system to be changed in 1998 to a student-proposing algorithm. In 1998 the system processed 35,823 applicants for 22,451 positions. Additional details about the system can be found at nrmp.aamc.org/nrmp/mainguid/ on the World Wide Web. There may be stable matchings other than those found by the two versions of the proposal algorithm. To seek a "fair" stable matching, we could give each person a number of points with which to rate preferences. The weight for the pair xa is then the sum of the points that x gives to a and a gives to x. The Hungarian Algorithm would yield a matching of maximum total weight, but this might not be a stable matching (Exercise 10). Other approaches appear in the books Knuth [1976] and Gusfield-Irving [1989], which discuss stable marriages and related topics.
## FASTER BIPARTITE MATCHING (optional)
We began this section with an algorithm for finding maximum matchings in bipartite graphs. The running time can be improved by seeking augmenting paths in a clever order; when short augmenting paths are available, we needn't explore many edges to find one. Using a Breadth-First Search simultaneously from all the unsaturated vertices of X, we can find many paths of the same length with one examination of the edge set. Hopcroft and Karp [1973] proved that subsequent augmentations must use longer paths, so the searches can be grouped in phases finding paths of the same lengths. They combined these
## Section 3.2: Algorithms and Applications
133
ideas to show that few phases are needed, enabling maximum matchings in n-vertex bipartite graphs to be found in O(n 25 ) time. 3.2.19. Remark. If M is a matching of size r and M* is a matching of sizes > r, then there exist at leasts - r vertex-disjoint M-augmenting paths. At least this many such paths can be found in M 1::,.M*. The next lemma impliei;; that the sequence of path lengths in successive shortest augmentations is nondecreasing. Here we treat paths as sets of edges, and cardinality indicates number of edges. 3.2.20. Lemma. If P is a shortest M-augmenting path and P' is M1::,.Paugmenting, then IP'I ~ IPI + 2 IP n P'I (treating Pas an edge set). Proof: Note that M 1::,. P is the matching obtained by using P to augment M. Let N be the matching (M 1::,.P)1::,.P' obtained by using P' to augment M 1::,.P. Since INI = IMI + 2, Remark 3.2.19 guarantees that M1::,.N contains two disjoint Maugmenting paths P1 and P2 Each of these is at least as long as P, since Pis a shortest M-augmentating path. Since N is obtained from M by switching the edges in P and then switching the edges in P', an edge belongs to exactly one of M and N if and only if it belongs to exactly one of P and P'. Therefore, M 1::,.N = P 1::,.P'. This yields IP1::,.P'I ~ IP1I + IP2I ~ 2 IPI. Thus
2 IPI ~ IP1::,.P'I = IPI + IP'I - 2 IP n P'j.
## We conclude that IP'I ~ IPI + 2 IP
n P'I
3.2.21. Lemma. If Pi. P2 , is a list of successive shortest augmentations, then the augmentations of the same length are vertex-disjoint paths. Proof: We use .the method of contradiction. Let Pb P1 with l > k be a closest pair in the list that have the same size but are not vertex-disjoint. By Lemma 3.2.20, the lengths of successive shortest augmenting .paths are nondecreasing, so Pb ... , P1 all have the same length. Since Pk> P1 is a closest intersecting pair with the same length, the paths Pk+l ... , P1 are pairwise disjoint. Let M' be the matching given by the augmentations P1, ... , Pk. Since Pk+l ... , P1 are pairwise disjoint, P1 is an M'-augmenting path. By Lemma 3.2.20, IPil ~I Pk I+ IP1 n Pkl Since IPil =I Pk I, there is no common edge. On the other hand, there must be a common edge. Each vertex of Pk is saturated in M' using an edge of Pk, and every vertex of an M' -augmenting path P1 that is saturated in M' (such as a vertex common to P1 and Pk) must contribute its saturated edge to P1 The contradiction implies that there is no such pa~r Pk. P1 3.2.22. Theorem. (Hopcroft-Karp [1973]) The breadth-first phased maximum matching algorithm runs in 0(...fom) time on bipartite graphs with n vertices and m edges.
134
## Chapter 3: Matchings and Factors
Proof: By Lemmas 3.2.20-3.2.21, searching simultaneously from all unsaturated vertices of X for shortest augmentations yields vertex-disjoint paths, after which all other augmenting paths are longer. Hence the augmentations of each length are found in one examination of the edge set, running in time O(m). It suffices to prove that there are at most 2 jn72 + 2 phases. List the augmenting paths as P1, ... , Ps in order by length, withs = a' (G) :;:: n /2. Since paths of the same length are vertex-disjoint, each Pi+l is an augmenting path for the matching Mi formed by using P1, ... , Pi. It suffices to prove the more general statement that whenever P1, ... , Ps are successive shortest augmenting paths that build a maximum matching, the number of distinct lengths among these paths is at most 2 L .JsJ + 2. Let r = Ls - .JsJ. Because IM,I =rand the maximum matching has size s, Remark 3.2.19 yields at least s - r vertex-disjoint M,-augmenting paths. The shortest of these paths uses at most Lr/ (s - r )j edges from M,. Hence IP,+1 I :S: 2 Lr/(s - r)j + 1. Since Lr /(s - r)j < Ls/ I.JslJ ::: L .JsJ, the paths up to P, provide all but the last I.Jsl augmentations using length at most 2L.JsJ+1. There are at most L .JsJ + 1 distinct odd integers up to this value, and even ifthe last I.Jsl paths have distinct lengths, they provide at most L.JsJ + 1 additional .JSJ + 2 distinct lengths. lengths, so altogether we use at most 2 L
l J
Even and Tarjan [1975] extended this to solve in time 0(,./llm) a more general problem that includes maximum bipartite matching.
EXERCISES
3.~.1. (-) Using nonnegative edge weights, construct a 4-vertex weighted graph in which the matching of maximum weight is not a matching of maximum size.
3.2.2. (- ) Show how to use the Hungarian Algorithm to test for the existence of a perfect matching in a bipartite graph. 3.2.3. (*-) Give an example of the stable matching problem with two men and two women in which there is more than one stable matching. 3.2.4. (*-)Determine the stable matchings resulting from the Proposal Algorithmrun with men proposing and with women proposing, given the preference lists below. Men {u, v, w, x, y, z)
u: v: w: x: y: z:
Women {a, b, c, d, e, fl a>b>d>c>f>e a: z>x>y>u>v>w a>b>c>f>e>d b: y>z>W>X>V>U c>b>d>a>f>e c: v>x>w>y>u>z c>a>d>b>e>f d: w>y>u>X>Z>V c>d>a>b>f>e e: U>V>X>W>y>z d > e > f > c > b > a f: U>W>X>v>z>y
## Section 3.2: Algorithms and Applications
135
3.2.5. Find a transversal of maximum total sum (weight) in each matrix below. Prove that there is no larger weight transversal by exhibiting a solution to the dual problem. Explain why this proves that there is no larger transversal.
(a)
(b)
(c)
## 12345 67872 13445 36287 41354
3.2.6. Find a minimum-weight transversal in the matrix below, and use duality to prove that the solution is optimal. (Hint: Use a transformation of the problem.)
4 7 8 ( 6 4 5 6 5 6 5 8 5 10 7 12 9 13 10 7 9 11) 4 6 7 8
3.2.7. The Bus Driver Problem. Let there be n bus drivers, n morning routes with durations x1 , . , Xn, and n afternoon routes with dur(;ltions yi, ... , Yn A driver is paid overtime when the morning route and afternoon route exceed total time t. The objective is to assign one morning run and one afternoon run to each driver to minimize the total amount of overtime. Express this as a weighted matching problem. Prove that giving the ith longest morning route and ith shortest afternoon route to the same driver, for each i, yields an optimal solution. (Hint: Do not use the Hungarian Algorithm; consider the special structure of the matrix.) (R.B. Potts) 3.2.8. Let the entries in matrix A have the form w;, 1 = a;b1 , where a:i,.. . ~.,an are numbers associated with the rows and b1 , , b. are numbers associated with the columns. Determine the maximum weight of a transversal of A. What happens when w;, 1 = a; + b1? (Hint: In each case, guess the general pattern by examining the solution when n = 2.) 3.2.9. (*)A mathematics department offers k seminars in different topics to its n students. Each student will take one seminar; the ith seminar will have k; students, where Lk; = n. Each student submits a preference list ranking the k seminars. An assignment of the students to seminars is stable if no two students can both obtain more preferable seminars by switching their assignments. Show how to find a stable ~ssign ment using w~ighted bipartite matching. (Isaak) 3.2.10. (*)Consider n men and n women, each assigning n - i points to the ith person in his or her preference list. Let the weight of c. pair be the sum of the points assigned by those two people. Construct an example where no maximum weight matching is a stable matching. 3.2.11. (*!)Prove that ifman xis paired with woman a in some stable matching, then a does not reject x in the Gale-Shapley Proposal Algorithm with men proposing. Conclude that among all stable matchings, every man is happiest in the matching produced by this algorithm. (Hint: Consider the first occurrence of such a rejection.) 3.2.12. (*) In the Stable Roommates Problem, each of 2n people has a preference ordering on the other 2n - 1 people. A stable matching is a perfect matching such that no
136
## Chapter 3: Matchings and Factors
unmatched pair prefers each other to their current roommates. Prove that there is no stable matching when the preferences are those below. (Gale-Shapley [1962])
a:b>c>d b:c>a>d c:a>b>d d:a>b>c
3.2.13. (*) In the stable roommates problem, suppose that each individual declares a top portion of the preference list as "acceptable". Define the acceptability graph to be the graph whose vertices are the people and whose edges are the pairs of people who rank each other as acceptable. Prove that all sets of rankings with acceptability graph G lead to a stable matching if and only if G is bipartite. (Abeledo-Isaak [1991] ).
## 3.3. Matchings in General Graphs
When discussing perfect matchings in graphs, it is natural to consider more general spanning subgraphs. 3.3.1. Definition. A factor of a graph G is a spanning subgraph of G. A kfactor is a spanning k-regular subgraph. An odd component of a graph is a component of odd order; the number of odd components of H is o(H). 3.3.2. Remark. A 1-factor and a perfect matching are almost the same thing. The precise distinction is that "1-factor" is a spanning 1-regular subgraph of G, while "perfect matching" is the set of edges in such a subgraph. A 3-regular graph that has a perfect matching decomposes into a 1-factor and a 2-factor.
## TUTTE'S 1-FACTOR THEOREM
Tutte found a necessary and sufficient condition for which graphs have 1factors. If G has a 1-factor and we consider a set S V ( G), then every odd component of G - S has a vertex matched to something outside it, which can only belong to S. Since these vertices of S must be distinct, o(G - S) ::: ISi.
even
even
The condition "For all S V(G), o(G - S) ::=: ISi" is Tutte's Condition. Tutte proved that this obvious necessary condition is also sufficient (TONCAS).
## Section 3.3: Matchings in General Graphs
137
Many proofs are known, such as Exercise 13 and Exercise 27. We present the proof by Lovasz using the ideas of symmetric difference and extremality.
3.3.3. Theorem. (Tutte [1947]) A graph G has a 1-factor if and only if o(G S):::: ISi for every S V(G). Proof: (Lovasz [1975]). Necessity. The odd components of G - S must have vertices matched to distinct vertices of S. Sufficiency. When we add an edge joining two components of G - S, the number of odd components does not increase (odd and even together become one odd component, two components of the same parity become one even component). Hence Tutte's Condition is preserved by addition of edges: if G' = G + e and S V(G), then o(G' - S) :::: o(G - S) :::: ISi. Also, if G' = G + e has no 1-factor, then G has no 1-factor. Therefore, the theorem holds unless there exists a simple graph G such that G satisfies Tutte's Condition, G has no 1-factor, and adding any missing edge to G yields a graph with a 1-factor. Let G be such a graph. We obtain a contradiction by showing that G actually does contain a 1-factor. Let U be the set of vertices in G that have degree n ( G) - 1. Case 1: G - U consists of disjoint complete graphs. In this case, the vertices in each component of G - U can be paired in any way, with one extra in the odd components. Since o(G - U) :::: IUI and each vertex of U is adjacent to all of G - U, we can match the leftover vertices to vertices of U. The remaining vertices are in U, which is a clique. To complete the 1factor, we need only show that an even number of vertices remain in U. We have matched an even number, so it suffices to show that n(G) is even. This follows by invoking Tutte's Condition for S = 0, since a graph of odd order would have a component of odd order.
G-U
u
Case 2: G - U is not a disjoint union of cliques. In this case, G - U has two vertices at distance 2; these are nonadjacent vertices x, z with a common neighbor y fl. U (Exercise l.2.23a). Furthermore, G - U has another vertex w not adjacent toy, since y fl. U. By the choice of G, adding an edge to G creates a 1-factor; let Mi and M2 be 1-factors in G + xz and G + yw, respectively. It suffices to show that Mi1::.M2 contains a 1-factor avoiding xz and yw, because this will be a 1-factor in G.
138
## Chapter 3: Matchings and Factors
Let F = Mito.M2. Since xz E Mi - M2 and yw E M2 - Mi, both xz and yw are in F. Since every vertex of G has degree 1 in each of Mi and M 2 , every vertex of G has degree 0 or 2 in F. Hence the components of F are even cycles and isolated vertices (see Lemma 3.1.9). Let C be the cycle of F containing xz. If C does not also contain yw, then the desired 1-factor consists of the edges of M 2 from C and all of Mi not in C. If C contains both yw and xz, as shown below, then to avoid them we use y x or y z. In the portion of C starting from y along y w, we use edges of Mi to avoid using yw. When we reach {x, z}, we use zy if we arrive at z (as shown); otherwise, we use xy. In the remainder of C we use the edges of M 2 We have produced a 1-factor of C that does not use xz or yw. Combined with Mi or M 2 outside C, we have a 1-factor of G.
M2
Mi
M2
~:::~ --;;\\M,
\
M2
----
Mi
.........
M2
3.3.4. Remark. Like other characterization theorems (such as Theorem 1.2.18 and Theorem 3.1.11)), Theorem 3.3.3 yields short verifications both when the property holds and when it doesn't. We prove that G has a 1-factor exists by exhibiting one. When it doesn't exist, Theorem 3.3.3 guarantees that we can exhibit a set whose deletion leaves too many odd components. 3.3.5. Remark. For a graph G and any S s; V ( G), counting tr :wertices modulo 2 shows that ISi +o(G-S) has the same parity as n(G). Thus also the difference o(G - S) - ISi has the same parity as n(G). We conclude that if n(G) is even and G has no 1-factor, then o(G - S) exceeds ISi by at least 2 for some S.
For non-bipartite graphs (such as odd cycles), there may be a gap between
a'(G) and f3(G) (see also Exercise 10). Nevertheless, another minimization problem yields a min-max relation for a'(T) in general graphs. This mm-max
relation generalizes Remark 3.3.5. The proof uses a graph transformation that involves a general graph operation.
3.3.6. Definition. The join of simple graphs G and H, written G v H, is the graph obtained from the disjoint union G + H by adding t:P.e edges {xy : x e
V(G), y E V(H)}.
## Section 3.3: Matchings in General Graphs
139
3.3.7. Corollary. (Berge-Tutte Formula-Berge [1958]) The largest number of vertices saturated by a matching in G is mins;V(G){n(G) - d(S)}, where d(S) = o(G - S) - ISi. Proof: Given S s; V(G), at most ISi edges can match vertices of S to vertices in odd components of G - S, so every matching has at least o(G - S) - ISi unsaturated vertices. We want to achieve this bound. Ll:lt d = max{o(G - S) - ISi : S s; V(G)}. The case S = 0 yields d 2: 0. Let G' = G v Kd. Since d(S) has the same parity as n(G) for each S, we know that n(G') is even. If G' satisfies Tutte's Condition, then we obtain a matching of the desired size in G from a perfect matching in G', because deleting the d added vertices eliminates edges that saturate at most d vertices of G. The condition o(G' - S') ::: IS'I holds for S' = 0 because n(G') is even. If S' is nonempty but does not contain all of Kd, then G' - S' has only one component, and 1 ::: IS'I Finally, when Kd s; S', we let S = S' - V(Kd). We have G' - S' = G - s., so o(G' - S') = o(G - S)::: ISi + d = IS'I We have verified that G' satisfies Tutte's Condition.
Corollary 3.3.7 guarantees that there is a short PROOF that a maximum matching indeed has maximum size by exhibiting a vertex set S whose deletion leaves the appropriate number of odd components. Most applications ofTutte's Theorem involve showing that some other condition implies Tutte's Condition and hence guarantees a 1-factor. Some were proved by other means long before Tutte's Theorem was available.
3.3.8. Corollary. (Petersen [1891]) Every 3-regular graph with no cut-edge has a 1-factor. Proof: Let G be a 3-regular graph with no cut-edge. We prove that G satisfies Tutte's Condition. Given S s; V(G), we count the edges between Sand the odd components of G - S. Since G is 3-regular, each vertex of S is incident to at most three such edges. If each odd component H of G - S is incident to at least three such edges, then 3o(G - S) ::: 3 tSI and hence o(G - S) ::: ISi, as desired. Let m be the number of edges from S to H. The sum of the vertex degrees in His 3n(H) - m. Since His a graph, the sum of its vertex degrees must be even. Since n(H) is odd, we conclude that m must also be odd. Since G has no cut-edge, m cannot equal 1. We conclude-that there are at least three edges from S to H, as desired.
Proofby contradiction would also be natural here. Assuming o(G- S) > ISi also leads to o(G - S) ::: ISi, so we rewrite the proof directly. Corollary 3.3.8 is best possible; the Petersen graph satisfi~ the hypothesis but does not have two edge-disjoint 1-factors (Petersen [1898)).
140
## Chapter 3: Matchings and Factors
Petersen also proved a sufficient condition for 2-factors. A connected graph with even vertex degrees is Eulerian (Theorem 1.2.26) and decomposes into edge-disjoint cycles (Proposition 1.2.27). For regular graphs of even degree, the cycles in some decomposition can be grouped to form 2-factors.
3.3.9. Theorem. (Petersen [1891]) Every regular graph of even degree has a 2-factor. Proof: Let G be a 2k-regular graph with vertices v1 , ... , v11 Every component of G is Eulerian, with some Eulerian drcuit C. For each component, define a bipartite graph H with vertices u1 .... , u 11 and wi, ... , w11 by putting u; *+ Wj if Vj immediately follows v; so:::newhere on C. Because C enters and exits each vertex k times, His k-regular. (Actualiy, His the split of the digraph obtained by orienting Gin accordance with C-see Definition 1.4.20.) Being a regular bipartite graph, H has a 1-factor M (Corollary 3.1.13). The edge incident to w; in H corresponds to an edge entering v; in C. The edge incident to u; in H corresponds to an edge exiting v;. Thus te 1-factor in H transforms into a 2-regular spanning subgraph of this component of G. Doing this for each component of G yie!ds a 2-factor of G. 3.3.10. Example. Construction of a 2-factor. Consider the Eulerian circuit in G = K5 that successively vi:oits 1231425435. The corresponding bipartite graph His on the right. For the 1-factor whose u, w-pairs are 12, 43, 25, 31, 54, the resulting 2-factor is the cycle (1, 2, 5, 4, 3). The remaining edges form another 1-factor, which corresponds to the 2-factor (1, 4, 2, 3, 5) that remains in G.
1
w1
2 3
4
## !-FACTORS OF GRAPHS (optional)
A factor is a spanning subgraph of G; we ask about existence of factors of special types. A k-factor is a k-regular factor; we have studied 1-factors and 2-factors. We can try to specify the degree at each vertex.
3.3.11. Definition. Given a function f: V ( G) -+ NU {0}, an /-factor ofa graph G is a subgraph H such that dH(v) = f(v) for all v E V(G).
Tutte [1952] proved a necessary and sufficient condition for a graph G to have an /-factor (see Exercise 29). He later reduced the problem to checkipg for
## Section 3.3: Matchings in General Graphs
141
a 1-factor in a related simple graph. We describe this reduction; it is a beautiful example of transforming a graph problem into a previously solved problem.
3.3.12. Example. A graph transformation (Tutte [1954a]). We assume that f(w) ~ d(w) for all w; otherwise G has too few edges at w to have an !-factor. We then construct a graph H that has a 1-factor if and only if G has an !-factor. Let e(w) = d(w) - f (w); this is the excess degree at wand is nonnegative. '.l'o construct H, replace each vertex v with a biclique Kd(v),e(v) having partite sets A(v) of size d(v) and B(v) of size e(v). For each vw E E(G), add an edge joining one vertex of A(v) to one vertex of A(w). Each vertex of A(v) participates in one such edge. The figure below shows a graph G, vertex labels given by f, and the resulting simple graph H. The bold edges in H form a 1-factor that corresponds to ail !-factor of G. In this example, the /-factor is not unique.
3.3.13. Theorem. A graph G has an /-factor if and only if the graph H constructed from G and fas in Example 3.3.12 has a 1-factor. Proof: Necessity. If G has an /-factor, then the corresponding edges in H leave e(v) vertices of A(v) unmatched; match them arbitrarily to the vertices of B(v) to obtain a 1-factor of H. Sufficiency. From a 1-factor of H, deleting B(v) and the vertices of A(v) matched into B(v) leaves f (v) edges at v. Doing this for each v and mergingthe remaining f(v) vertices of each A(v) yields a subgraph of G with degree f(v) at v. It is an /-factor of G.
Tutte's Condition for a 1-factor in the derived graph Hof Example 3.3.12 transforms into a necessary and sufficient condition for an /-factor in G. Among the applications is a proof of the Erdos-Gallai [1960] characterization of degree sequences of simple graphs (Exercise 29). Given an algorithm to find a 1-factor, the correspondence in Theorem 3.3.13 provides an algorithmic test for an f-factor. Instead of just seeking a 1-factor (that is, a perfect matching), we next consider the more general problem of finding a maximum matching in a graph.
142
## EDMONDS' BLOSSOM ALGORITHM (optional)
Berge's Theorem (Theorem 3.1.10) states that a matching Min G has maximum size if and only if G has no M-augmenting path. We can thus find a maximum matching using successive augmenting paths. Since we augment at most n/2 times, we obtain a good algorithm if the search for an augmenting path does not take too long. Edmonds [1965a] presented the first such algorithm in his famous paper "Paths, trees, and flowers". In bipartite graphs, we can search quickly for augmenting paths (Algorithm 3.2.1) because we explore from each vertex at most once. An M-alternating path from u can reach a vertex x in the same partite set as u only along a saturated edge. Hence only once can we reach and explore x. This property fails in graphs with odd cycles, because M -alternating paths from an unsaturated vertex may reach x both along saturated and along unsaturated edges.
3.3.14. Example. In the graph below, with M indicated in bold, a search for shortest M-augmenting paths from u reaches x via the unsaturated edge ax. If we do not also consider a longer path reaching x via a saturated edge, then we miss the augmenting path u, v, a, b, c, d, x, y.
y x
d
We describe Edmonds' solution to this difficulty. If an exploration of Malternating paths from u reaches a vertex x by an unsaturated edge in one path and by a saturated edge in another path, then x belongs to an odd cycle. Alternating paths from u can diverge only when the next edge is unsaturated (leaving vertex a in Example 3.3.14); when the next edge is saturated there is only one choice for it. From the vertex where the paths diverge, the path reaching x on an unsaturated edge has odd length, and the path reaching it on a saturated edge has even length. Together, they form an odd cycle.
3.3.15. Definition. Let M be a matching in a graph G, and let u be an Munsaturated vertex. A flower is the union of two M -alternating paths from u that reach a vertex x on steps of opposite parity (having not done so earlier). The stem of the flower is the maximal common initial path (of nonnegative even length). The blossom of the flower is the odd cycle obtained by deleting the stem.
In Example 3.3.14, the flower is the full graph except y, the stem is the path u, v, a, and the blossom is the 5-cycle. The horticultural terminology echoes the use of tree for the structures given by most search procedures.
## Section 3.3: Matchings in General Graphs
143
Blossoms do not impede our search. For each vertex z in a blossom, some M-alternating u, z-path reaches z on a saturated edge, found by traveling the proper direction around the blossom to reach z from the stem. We therefore can continue our search along any unsaturated edge from the blossom to a vertex not yet reached. Example 3.3.14 shows such an extension that immediately reaches an unsaturated vertex and completes an M -augmenting path. Since each vertex of a blossom is saturated by an edge on these paths, no saturated edge emerges from a blossom (except the stem). The effect of these two observations is that we can view the entire blossom as a single "supervertex" reached along the saturated edge at the end of the stem. We search from all vertices of the supervertex blossom simultaneously along unsaturated edges. We implement this consolidation by contracting the edges of a blossom B when we find it. The result is a new saturated vertex b incident to the last (saturated) edge of the stem. Its other incident edges are the unsaturated edges joining vertices of B to vertices outside B. We explore from b in the usual way to extend our search. We may later find another blossom containing b; we then contract again. If we find an M -alternating path in the final graph from u to an unsaturated vertex x, then we can undo the contractions to obtain an M-augmenting path to x in the original graph. Except for the treatment of blossoms, the approach is that of Algorithm 3.2.1 for exploring M-alternating paths. In the corresponding phrasing, T is the set of vertices of the current graph reached along unsaturated edges, and S is the set of vertices reached along saturated edges. The vertices that arise by contracting blossoms belong to S. 3.3.16. Example. Let M be the bold matching in the graph on the left below. We search from the unsaturated vertex u for an M -augmenting path. We first explore the unsaturated edges incident to u, reaching a and b. Since a and b are saturated, we immediately extend the paths along the edges ac and bd. Now S = {u, c, d}. Ifwe next explore from c, then we find its neighbors e and f along unsaturated edges. Since ef EM, we discover the blossom with vertex set {c, e, f}. We contract the blossom to obtain the new vertex C, changing S to (u, C, d}. This yields the graph on the right.
g
h
Suppose we now explore from the vertex C E S. Unsaturated edges take us tog and to d. Since g is saturated by the edge gh, we place h in S. Since d is already in S, we have found another bloss0m. The paths reaching d are u, b, d and u, a, C, d. We contract the blossom, obtaining the new vertex U and
144
## Chapter 3: Matchings and Factors
the graph on the right below, with S = {U, h}. We next explore from h, finding nothing new (if we exhaust S without reaching an unsaturated vertex, then there is no M-augmenting path from u). Finally, we explore from U, reaching the unsaturated vertex x.
f
u
e
x~
Having recorded the edge on which we reached each vertex, we can extract an M-augmenting u, x-path. We reached x from U, so we expand the blossom back into {u, a, C, d, b} and find that xis reached from U along bx. The path in the blossom U that reaches bona saturated edge ends with C, d, b. Since C is a blossom in the original graph, we expand C back into {c, f, e}. Note that dis reached from C by the unsaturated edge ed. The path from the ''base" of C that reaches e along a saturated edge is c, f, e. Finally, c was reached from a and a from u, so we obtain the full M-augmenting path u, a, c, f, e, d, b, x. We summarize the steps of the algorithm, glossing over the details of implementation, especially the treatment of contractions.
3.3.17. Algorithm. (Edmonds' Blossom Algorithm [1965a]-sketch). Input: A graph G, a matching Min G, an M-unsaturated vertex u. Idea: Explore M -alternating paths from u, recording for each vertex the vertex from which it was reached, and contracting blossoms when found. Maintain sets Sand T analogous to those in Algorithm 3.2.1, with S consisting of u and the vertices reached along saturated edges. Reaching an unsaturated vertex yields an augmentation. Initialization: S = {u} and T = 0. Iteration: If S has no unmarked vertex, stop; there is no M-augmenting path from u. Otherwise, select an unmarked v E S. To explore from v, successively consider each y E N (-..;) such that y fl. T. If y is unsaturated by M, then trace back from y (expanding blossoms as needed) to report an M-augmenting u, y-path. If y E S, then a blossom has been found. Suspend the exploration of v and contract the blossom, replacing its vertices in S and T by a single new vertex in S. Continue the search from this vertex in the smaller graph. Otherwise, y is matched to some w by M. Include y in T (reached from v ), and include w in S (reached from y ). After exploring all such neighbors of v, mark v and iterate.
We cannot explore all unsaturated vertices simultaneously as in Algorithm 3.2.1, because the membership of vertices in blossoms depends on the choice of
## Section 3.3: Matchings in General Graphs
145
the initial unsaturated vertex. Nevertheless, if we find no M-augmenting path from u, then we can delete u from the graph and ignore it in the subsequent search for a maximum matching (Exercise 26). 3.3.18. Remark. Edmonds' original algorithm runs in time O(n 4 ). Theimplementation in Ahuja-Magnanti-Orlin (1993, p483-494] runs in time O(n 3 ). This requires (1) appropriate data structures to represent the blossoms and to process contractions, and (2) careful analysis of the number of contractions that can be performed, the time spent exploring edges, and the time spent contracting and expanding blossoms. The first algorithm solving the maximum matching problem in less than cubic time was the O(n 512 ) algorithm in Even-Kariv (1975]. The best algorithm now known runs in time O(n 112 m) for a graph with n vertices and m edges (this is faster than O(n 512 ) for sparse graphs). The algorithm is rather complicated and appears in Micali-Vazirani (1980], with a complete proof in Vazirani (1994]. We have not discussed the weighted matching problem for general graphs. Edmonds [1965d] found an algorithm for this, which was implemented in time O(n 3 ) by Gabow (1975] and by Lawler [1976]. Faster algorithms appear in Gabow (1990] and in Gabow-Tarjan (1989].
EXERCISES
3.3.1. (-) Determine whether the graph below has a 1-factor.
3.3.2. (-) Exhibit a maximum matching in the graph below, and use a result in this section to give a short proof that it has no larger matching.
146
## Chapter 3: Matchings and Factors
3.3.3. (-) In the graph drawn below, exhibit a k-factor for each kin {O, 1, 2, 3, 4}.
3.3.4. (-) Let G be a k-regular bipartite graph. Prove that G can be decomposed into r-factors if and only if r divides k. 3.3.5. (-) Given graphs G and H, determine the number of components and maximum degree of G v H in terms of parameters of G and H.
3.3.6. (!) Prove that a tree T has a perfect matching if and only if o(T - v) = 1 for every v E V(T). (Chungphaisan) 3.3.7. (!)For each k > 1, construct a k-regular simple graph having no 1-factor. 3.3.8. Prove that if a graph G decomposes into 1-factors, then G has no cut-vertex. Draw a connected 3-regular simple graph that has a 1-factor and has a cut-vertex. 3.3.9. Prove that every graph G has a matching of size at least n(G)/(1 + L\(G)). (Hint: Apply induction on e(G).) (Weinstein [1963]) 3.3.10. (!)For every graph G, prove that f3(G) :S 2a'(G). For each k E N, construct a simple graph G with a'(G) = k and f3(G) = 2k. 3.3.11. Let T be a set of vertices fo a graph G. Prove that G has a matching that saturates T if and only if for all S V(G), the number of odd components of G - S contained in G[T] is at most ISi. 3.3.12. (!) Extension of Konig-Egeruary Theorem to general graphs. Given a graph G, let Si. ... , Sk and T be subsets of V(G) such that each S; has odd size. These sets form a generalized cover of G if every edge of G has one endpoint in T or both endpoints in some S;. The weight of a generalized cover is ITI + L LISd /2J. Let f3*(G) be the minimum weight of a generalized cover. Prove that a'(G) = f3*(G). (Hint: Apply Corollary 3.3.7. Comment: every vertex cover is a generalized cover, and thus f3*(G) :S f3(G).) 3.3.13. (+) Tutte's Theorem from Hall's Theorem. Let G be a graph such thato(G- S)::; ISi for all S V(G). Let T be a maximal vertex subset such that o(G - T) = ITI. a) Prove that every component of G - T is odd, and conclude that T =I= 0. b) Let C be a component of G - T. Prove that Tutte's Condition holds for every subgraph of C obtained by deleting one vertex. (Hint: Since C - x has even order, a violation requires o(C - x - S) ~ ISi + 2:) c) Let H be a bipartite graph with partite sets T and C,. where C is the set of components of G - T. Fort ET and CE C, put tC E E(H) ifand only if NG(t) contains a vertex of C. Prove that H satisfies Hall's Condition for a matching that saturates C. d) Use parts (a), (b), (c), and Hall's Theorem to prove Tutte's 1-factor Theorem by induction on n(G). (Anderson [1971], Mader [1973])
## Section 3.3: Matchings in General Graphs
147
3.3.14. Fork E N, let G be a simple graph such that 8(G) :::: k and n(G) :::: 2k. Prove that a'(G):::: k. (Hint: Apply Corollary 3.3.7.) (Brandt [1994]) 3.3.15. Let G be a 3-regular graph with at most two cut-edges. Prove that G has a 1-factor. (Petersen [1891]) 3.3.16. (!) Let G be a k-regular graph of even order that remains connected when any k - 2 edges are deleted. Prove that G has a 1-factor. 3.3.17. With G as in Exercise 3.3.16, use Remark 3.3.5 to prove that every edge of G belongs to some 1-factor. (Comment: This strengthens Exercise 3.3.16.) (Schonberger [1934] fork = 3, Berge [1973, p162]) 3.3.18. (+)For each odd k greater than 1, construct a graph G with no 1-factor that is k-regular and remains connected when any k - 3 edges are deleted. (Comment: Thus Exercise. 3.3.16 is best possible.) 3.3.19. (!) Let G be a 3-regular simple graph with no cut-edge. Prove that G decomposes into copies of P4 (Hint: Use Theorem 3.3.9.) 3.3.20. (!)Prove that a 3-regular simple graph has a 1-factor if and only ifit decomposes into copies of P4 3.3.21. (+) Let G be a 2m-regular graph, and let T be a tree with m edges Prove that if the diameter of T is less than the girth of G, then G decomposes into copies of T. (Hint: Use Theorem 3.3.9 to give an inductive proof of the stronger result that G has such a decomposition in which each vertex is used once as an image of each vertex of T .) (Haggkvist) 3.3.22. (!)Let G be an X, Y-bigraph. Let H be the graph obtained from G by adding one vertex to Y if n (G) is odd and then adding edges to make Y a clique. a) Prove that G has a matching of size IXI if and only if H has-a 1-factor. b) Prove that if G satisfies Hall's Condition (IN(S)I :::: ISi for all S ~ X), then H satisfies Tutte's Condition (o(H - T) ~ ITI for all T ~ V(H)). c) Use parts (a) and (b) to obtain Hall's Theorem from Tutte's Theorem. 3.3.23. Let G be a claw-free connected graph of even order. a) Let T be a spanning tree of G generated by Breadth-First Search (Algorithm 2.3.8). Let x and y be vertices that have a common paren.t in T other than the root. Prove that x and y must be adjacent. b) Use part (a) to prove that G has a 1-factor. (Comment: Without part (a), one can simply prove the stronger result that the last edge in a longest path belongs to a 1-factor.) (Sumner [1974a], Las Vergnas [1975]) 3.3.24. (!) Let G be a simple graph of even order n having a set S of si.ze k such that 1 o(G - S) > k. Prove that G has at most (~) + k(n - k) + (- 2 ; - ) edges, and that this
bound is best possible. Use this to determine the maximum size of a simple n-vertex graph_with no 1-factor. (Erdos-Gallai [1961])
3.3.25. A graph G is factor-critical if each subgraph G - v obtained by deleting one vertex has a 1-factor. Prove that G is factor-critical if and only if n(G) is odd and o(G S) ~ ISi for all nonempty S ~ V(G). (Gallai [1963a]) 3.3.26. (!)Let M be a matching in a graph G, and let u be an M-unsaturated vertex. Prove that if G has no M-augmenting path that starts at u, then u is unsaturated in some maximum matching in G.
148
## Chapter 3: Matchings and Factors
3.3.27. (*)Assuming that Algorithm 3.3.17 is correct, we develop an algorithmic proof ofTutte's Theorem (Theorem 3.3.3). a) Let G be a graph with no perfect matching, and let M be a maximum matching in G. Let Sand T be the sets generated when running Algorithm 3.3.17 from u. Prove that ITI < ISi ::: o(G - T). b) Use part (a) to prove Theorem 3.3.3. 3.3.28. (*)Let No =NU {0}. Given f: V(G)-+ No, the graph G is !-soluble ifthere exists w: .(G) ~ No such that LuveE(GJ w(uv) = f(v) for every v E V(G). a) Prove that G has an !-factor if and only if the graph H obtained from G by subdividing each edge twice and defining f to be 1 on the new vertices is !-soluble. (This reduces testing for an !-factor to testing !-solubility.) b) Given G and an f: V(G)-+ N 0 , construct a graph H (with proof) such that G is !-soluble ifand only if H has a 1-factor. (Tutte [1954a]) 3.3.29. (*+) Tutte's !-factor condition and graphic seqilences. Given f: V(G) -+ No, define f(S) = Lves f(v) for S ~ V(G). For are disjoint subsets S, T of V(G), let q(S, T) denote the number of components Q of G - S - T such that e(Q, T) + f(V(Q)) is odd, where e(Q, T) is the number of edges from Q to T. Tutte [1952, 1954a] proved that G has an !-factor if and only if
q(S, T)
+ f(T)
## - LveT dG-s(V) :Sf (S)
for all choices of disjoint subsets S, T c V. a) The Parity Lemma. Let 8(S, T) = f(S)- f(T) + LveT dG-s(v)-q(S, T). Prove that 8(S, T) has the same parity as f(V) for disjoint sets S, T ~ V(G). (Hint: Use induction on ITI.) b) Suppose that G = Kn and f(v;) = d;, where Ld; is even and d1 ~ ~ dn. Use the !~factor condition and part (a) to prove that G has an !-factor if and only if d; ::; (n - 1 - s )k + L:;=n+l-s d; for all k, s with k + s ::: n. c) Conclude that d1 , , dn ~ 0 are the vertex degrees of a simple graph if and only if L d; is even and d; ::: k(k - 1) + min{k, d;} for 1 ::: k ::: n. (Erdos-Gallai [1960])
L:=l
I::=l
I:;=k+l
Chapter4
## Connectivity and Paths
4.1. Cuts and Connectivity
A good communication network is hard to disrupt. We want the graph (or digraph) of possible transmissions to remain connected even when some vertices or edges fail. When communication links are expensive, we want to achieve these goals with few edges. Loops are irrelevant for connection, so in this chapter we assume that our graphs and digraphs have no loops, especially when considering degree conditions.
CONNECTIVITY
How many vertices must be deleted to disconnect a graph? 4.1.1. Definition. A separating set or vertex cut of a graph G is a set S s; V(G) such that G - S has more than one component. The connectivity of G, written K(G), is the minimum size of a vertex set S such that G - S is disconnected or has only one vertex. A graph G is k-connected if its connectivity is at least k. A graph other than a complete graph is k-connected if and only if every separating set has size at least k. We can view "k-connected" as a structural eond~tion, while "connectivity k" is the solution of an optimization problem. 4.1.2. Example. Connectivity of Kn and Km n Because a clique has no separating set, we need to adopt a convention for its connectivity. This explains the phrase "or has only one vertex" in Definition 4.1.1. We obtain K(Kn) = n - 1, while K(G) ::=: n(G) - 2 when G is not a complete graph. With this convention, most general results about connectivity remain valid on complete graphs.
149
150
## Chapter 4: Connectivity and Paths
Consider a bipartition X, Y of Km.n Every induced subgraph that has at least one vertex from X and from Y is connected. Hence every separating set of Km,n contains X or Y. Since X and Y themselves are separating sets (or leave only one vertex), we have K(Km,n) = min{m, n). The connectivity of Ks. 3 is 3; the graph is 1-connected, 2-connected, and 3-connected, but not 4-connected. A graph with more than two vertices has connectivity 1 if and only if it is connected and has a cutvertex. A graph with more than one vertex has connectivity 0 if and only if it is disconnected. The 1-vertex graph K 1 is annoyingly inconsistent; it is connected, but for consistency in discussing connectivity we set K(K1) = 0.
4.1.3. Example. The hypercube Qk. Fork : :-: 2, the neighbors of one vertex in Qk form a separating set, so K(Qk) :::: k. To prove that K(Qk) = k, we show that every vertex cut has size at least k. We use induction on k. Basis step: k E {O, 1). Fork :::: 1, Qk is a clique with k + 1 vertices and has connectivity k. Induction step: k : :-: 2). By the induction hypothesis, K(Qk_ 1) = k - 1. Consider the description of Qk as two copies Q and Q' of Qk-l plus a matching that joins corresponding vertices in Q and Q' (Example 1.3.8). Let S be a vertex cut in Qk. If Q - Sis connected and Q' - Sis connected, then Qk - Sis also connected unless S contains at least one endpoint of every matched pair. This requires ISI : :-: 2k-l, but 2k-l : :-: k fork :::-:: 2. Hence we may assume that Q - S is disconnected, which means that S has at least k - 1 vertices in Q, by the induction hypothesis. If S contains no vertices of Q', then Q' - S is connected and all vertices of Q - S have neighbors in Q' - S, so Qk - S is connected. Hence S must also contain a vertex of Q'. This yields ISi : :-: k, as desired.
Deleting the neighbors of a vertex disconnects a graph (or leaves only one vertex), so K(G) :::: 8(G). Equality need not hold; 2Km has minimum degree m - 1 but connectivity 0. Since connectivity k requires 8(G) :::-:: k, it also requires at least fkn/21 edges. The k-dimensional cube achieves this bound, but only for the case n = 2k. The bound is best possible whenever k < n, as shown by the next example.
4.1.4. Example. Harary graphs. Given k < n, place n vertices around a circle, equally spaced. If k is even, form Hk.n by making each vertex adjacent to the nearest k/2 vertices in each direction around the circle. If k is odd and n is even, form Hk,n by making each vertex adjacent to the nearest (k-1)/2 vertices
## Section 4.1: Cuts and Connectivity
151
in each direction and to the diametrically opposite vertex. In each case, Hk,n k-regular. When k and n are both odd, index the vertices by the integers modulo Construct Hk,n from Hk-1,n by adding the edges i *+ i + (n - 1)/2 for 0 ~ i (n - 1)/2. The graphs H4,s, H5,s, and Hr. 9 appear below.
4
is
n.
<
4.1.5. Theorem. (Harary [1962a]) K(Hk.n) = k, and hence the minimum number of edges in a k-connected graph on n vertices is fkn /21. Proof: We prove only the even case k = 2r, leaving the odd case as Exercise 12. Let G = Hk,n Since 8(G) = k, it suffices to prove K(G) :=:: k. For S s; V(G) with ISi < k, we prove that G - Sis connected. Consider u, v E V(G) - S. The original circular arrangement has a clockwise u, v-path and a counterclockwise u, v-path along the circle; let A and B be the sets of internal vertices on these two paths. Since ISi < k, the pigeonhole principle implies that in one of {A, B}, S has fewer than k/2 vertices. Since in Geach vertex has edges to the next k/2 vertices in a particular direction, deleting fewer than k/2 consecutive vertices cannot block travel in that direction. Thus we can find a u, v-path in G - S via the set A or Bin which S has fewer than k/2 vertices. Harary's construction determines the degree conditions that allow a graph to be k-connected. Exercise 22 determines the degree conditions that force a simple graph to be k-connected. Since it depends on vertex deletions, connectivity is not affected by deleting extra copies of multiple edges. Hence we state degree conditions for k-coimectedness only in the context of simple graphs. 4.1.6. Remark. A direct proof of K(G) :=:: k considers a vertex cut Sand proves that ISi :=:: k, or it considers a set S with fewer thank vertices and proves that G - S is connected. The indirect approach assumes a cut of size less than k and obtains a contradiction. The indirect proof may be easier to find, but the direct proof may be clearer to state. Note also that if k < n ( G) and G has a vertex cut of size less than k, then G has a vertex cut of size k - 1 (first delete the cut, then continue deleting vertices until k - 1 are gone, retaining a vertex in each of two components). Finally, proving K(G) = k also requires presenting a vertex cut of size k; this is usually the easy part.
152
## Chapter 4: Connectivity and Paths
EDGE-CONNECTIVITY
Perhaps our transmitters are secure and never fail, but our communication links are subject to noise or other disruptions. In this situation, we want to make it hard to disconnect our graph by deleting edges.
4.1.7. Definition. A disconnecting set of edges is a set F s; E(G) such that G - F has more than one component. A graph is k-edge-connected if every disconnecting set has at least k edges. The edge-connectivity of G, written K' ( G), is the minimum size of a disconnecting set (equivalently, the maximum k such that G is k-edge-connected). Given S, T s; V(G), we write [S, Tl for the set of edges having one endpoint in S and the other in T. An edge cut is an edge set of the form [S, S], where Sis a nonempty proper subset of V(G) and S denotes V(G) - S.
r-------,
r--1
I
,__
disconnecting set
L ______ J
__..,,
edge cut
4.1.8. Remark. Disconnecting set vs. edge cut. Every edge cut is a disconnecting set, since G ..- [S, S] has no path from S to S. The converse is false, since a disconnecting set can have extra edges. Above we show a disconnecting ~et and .an edge cut in bold; see also Exercise 13. Nevertheless, every minimal disconnecting set ofedges is an edge cut (when n(G) > 1). If G - F has more than one component for some F ~ E(G), then for some component H of G - F we. have deleted all edges with exactly one endpoint in H. Hence F contains the edge cut [V(H), V(H)], and Fis not a minimal disconnecting set unless F = [V(H), V(H)].
The notation for edge-connectivity continues our convention of using a "prime" for an edge parameter analogous to a vertex parameter. Using the same base letter emphasizes the analogy and avoids the confusion of using many different letters - and running out of them. Deleting one endpoint of each edge in an edge cut F deletes every edge of F. This suggests that K(G) :=:: K'(G). However, we must be careful not to delete the only vertex of a component of G - F and thereby leave a connected subgraph.
## 4.1.9. Theorem. (Whitney [1932a]) If G is a simple graph, then
K(G):::: K'(G):::: o(G).
Proof: The edges incident to a vertex v of minimum degree form an edge cut; hence K1 (G) :=:: o(G). It remains to show that K(G) :=:: K'(G).
## Section 4.1: Cuts and Connectivity
153
We have observed that K(G) :s n(G) - 1 (see Example 4.1.2). Consider a smallest edge cut [S; SJ. If every vertex of S is adjacent to every vertex of S, then I[S, SJ I = IS 1 IS_I ~ n ( G) - 1 ~ K ( G), and the desired inequality holds. Otherwise, we choose x E S and y E S with x ~ y. Let T consist of all neighbors of x in S and all vertices of S - {x} with neighbors in S. Every x, ypath passes through T, so T is a. separating set. Also, picking the edges from x to T n S and one edge from each vertex of T n S to S (shown bold below) yields ITI distinct edges of[S, SJ. Thus K'(G) = l[S, SJI ~ ITI ~ K(G).
We have seen that K(G) = 8(G) when G is a complete graph, a biclique, a hypercube, or a Harary graph. By Theorem 4.1.9, also K'(G) = 8(G) for these graphs. Nevertheless, in many graphs the set of edges incident to a v~rtex of minimum degree is not a minimum edge cut. The situation K'(G) < 8(G) is precisely the situation where no minimum edge cut isolates a vertex. 4.1.10. Example. Possibility of K < K 1 < 8. For graph G below, K(G) = 1, K'(G) = 2, and 8(G) = 3. Note that no minimum edge cut isolates a vertex. Each inequality can be arbitarily weak. When G = Km+ Km, we have K(G) = K'(G) = 0 but o(G) = m - 1. When G consists of two m-cliques sharing a single vertex, we have K'(G) = 8(G) = m - 1 but K(G) = 1.
## K'(G) = 8(G) when G has diameter 2 (Exercise 25).
Various conditions force equalities among the parameters; for example, For 3-regular graphs, t:onnectivity and edge-connectivity are always equal.
4.1.11. Theorem. If G is a 3-regular graph, then K(G) = K'(G). Proof: Let S be a minimum vertex cut (ISi = K(G)). Since K(G) :s K'(G) always, we need only provide an edge cut of size ISi. Let Hi, H 2 be two components of G - S. Since Sis a minimum vertex cut, each v ES has a neighbor in H1 and a neighbor in H2. Since G is 3-regular, v cannot have two neighbors in. H1 and
154
## Chapter 4: Connectivity and Paths
two in H 2. For each v E S, delete the edge from v to a member of {Hi. H 2} where v has only one neighbor. These K(G) edges break all paths from H1 to H2 except in the case below, where a path can enter S via v1 and leave via v2 In this case we delete the edge to H1 for both v1 and v2 to break all paths from H1 to H2 through {vi, v2}.
When K'(G) < 8(G), a minimum edge cut cannot isolate a vertex. In fact, whenever l[S, Sll < 8(G), the set S (and also S) must be much larger than a single vertex. This follows from a simple relationship between the size of the edge cut [S, S] and the size of the subgraph induced by S.
## 4.1.12. Proposition. If Sis a set of vertices in a graph G, then
l[S, Sll = [LvEsd(v)] - 2e(G[S]).
Proof: An edge in G[S] contributes twice to LvES d(v), while an edge in [S, S] contributes only once to the sum. Since this counts all contributions, we obtain LvESd(v) = l[S, Sll + 2e(G[S]). 4.1.13. Corollary. If G is a simple graph and I[S, S] I < 8 ( G) for some nonempty proper subset S of V(G), then ISi > 8(G). Proof: By Proposition 4.1.12, we have 8(G) > LvES d(v) - 2e(G[S]). Using d(v) ~ 8(G) and 2e(G[S]) :S ISi (ISi - 1) yields
8(G) > ISi 8(G) - ISi (ISi - 1).
This inequality requires ISi > 1, so we can combine the terms involving 8(G) and cancel ISi - 1 to obtain ISi > 8(G). As a set of edges, an edge cut may contain another edge cut. For example, K 1.2 has three edge cuts, but one of them contains the other two. The minimal non-empty edge cuts of a graph have useful structural properties.
## 4.1.14. Definition. A bond is a minimal nonempty edge cut.
Here "minimal" means that no proper nonempty subset 'is also an edge cut. We characterize bonds in connected graphs.
4.1.15. Proposition. If G is a connected graph, then an edge cut Fis a bond if and only if G - F has exactly two components.
## Section 4.1: Cuts and Connectivity
155
Proof: Let F = [S, S] be an edge cut. Suppose first that G - F has exactly two components, and let F' be a proper subset of F. The graph G - F' contains the two components of G - F plus at least one edge between them, making it connected. Hence F is a minimal disconnecting set and is a bond. For the converse, suppose that G - F has more than two components. ~ince G - Fis the disjoint union of G[S] and G[J, one of these has at least two components. Assume by symmetry that it is G[S]. We can thus write S =AU B, where no edges join A and B. Now the edge cuts [A, A] and [B, B] are proper subsets of F, so F is not a bond.
BLOCKS
A connected graph with no cut-vertex need not be 2-connected, since it can be K1 or K2 . Connected subgraphs without cut-vertices provide a useful decomposition of a graph.
4.1.16. Definition. A block of a graph G is a maximal connected subgraph of G that has no cut-vertex. If G itself is connected and has no cut-vertex, then G is a block. 4.1.17. Example. Blocks. If His a block of G, then H as a graph has no cutvertex, but H may contain vertices that are cut-vertices of G. For example, the graph drawn below has five blocks; three copies of K2 , one of K3, and one subgraph that is neither a cycle nor a clique.
4.1.18. Remark. Properties of blocks. An edge of a cycle cannot itself be a block, since it is in a larger subgraph with no cut-vertex. Hence an edge is a block if and only if it is a cut-edge; the blocks of a tree are its edges. If a block has more than two vertices, then it is 2-connected. The blocks of a loopless graph are its isolated vertices, its cut-edges, and its maximal 2-connected subgraphs.
156
## Chapter 4: Connectivity and Paths
4.1.19. Proposition. Two blocks in a graph share at most one vertex. Proof: We use contradiction. Suppose that blocks B1 , B2 have at least two common vertices. We show that B1 U B2 is a connected subgraph with no cutvertex, which contradicts the maximality of B1 and B2 When we delete one vertex from B;, what remains is connected. Hence we retain a path in B; from every vert~x that remains to every vertex of V(B 1 ) n V(B 2 ) that remains. Since the blocks have at least two common vertices, deleting a single vertex leaves a vertex in the intersection. We retain paths from all vertices to that vertex, so B1 U B 2 cannot be disconnected by deleting one vertex.
Every edge by itself is a subgraph with no cut-:vertex and hence is in a block. We conclude that the blocks of a graph decompose the graph. Blocks in a graph behave somewhat like strong components of a digraph (Definition 1.4.12), but strong components share no vertices (Exercise 1.4.13a). Thus although blocks in a graph decompose the edge set, strong components in a digraph merely partition the vertex set and usually omit edges. When two blocks of G share a vertex, it must be a cut-vertex of G. The interaction between blocks and cut-vertices is described by a special graph.
4.1.20. * Definition. The block-cutpoint graph of a graph G is a bipartite graph H in which one partite set consists of the cut-vertices of G, and the other has a vertex b; for each block B; of G. We include vb; a:; :m edge of H if and only if v EB;.
b
h
g
G c
When G is connected, its block-cutpoint graph is a tree (Exercise 34) whose leaves are blocks of G. Thus a graph G that is not a single block has at least two blocks (leaf blocks) that each contain exactly one cut-vertex of G. Blocks can be found using a technique for searching graphs. In DepthFirst Search (DFS), we explore always from the most recently discovered vertex that has unexplored edges (also called bacJdracking). In contrast, Breadth-First Search (Algorithm 2.3.8) explores from the oldest vertex, so the difference between DFS and BFS is that in DFS we maintain the list of vertices to be searched as a Last-In First-Out "stack" rather than a queue.
4.1.21.* Example. Depth-First Search. In the graph below, one depth-first search from u finds the vertices in the order u, a, b, c, d, e, f, g. For both BFS and DFS, the order of discovery depends on the order of exploring edges from a searched vertex.
## Section 4.1: Cuts and Connectivity
g
157
l l VI
u a
A breath-first or depth-first search from u generates a tree rooted at u; each time exploring a vertex x yields a new vertex v, we include the edge xv. This grows a tree that becomes a spanning tree of the component containing u. Applications of depth-first search rely on a fundamental property of the resulting spanning tree.
4.1.22. * Lemma. If T is fl spanning tree of a connected graph G grown by DFS from u, then every edge of G not in T consists of two vertices v, w such that v lies on the u, w-path in T. Proof: Let vw be an edge of G, with v encountered before w in the depth-first search. Because vw is an edge, we cannot finish v before w is added to T. Hence w appears somewhere in the subtree formed before finishing v, and the path from w to u contains v. 4.1.23. * Algorithm. (Computing the blocks of a graph) Input: A connected graph G. (The blocks of a graph are the blocks of its components, which can be found by depth-first search, so we may assume that G is connected.) Idea: Build a depth-first search tree T of G, discarding portions of T as blocks are identified. Maintain one vertex called ACTIVE. Initialization: Pick a root x E V(H); make x ACTIVE; set T = {x}. Iteration: Let v denote the current active vertex. 1) If v has an unexplored incident edge vw, then lA) If w rt V (T), then add vw to T, mark vw explored, make w ACTIVE. lB) If w E V (T), then w is an ancestor of v; mark vw explored. 2) If v has no more unexplored incident edges, then 2A) If v # x, and w is the parent of v, make w ACTIVE. If no vertex in the current subtree T' rooted at 17 haa an explored edge to an ancestor above w, then V ( T') U { w} is the vertex set of a bioch.; record this information and delete V(T') from T. 2B) If v = x, terminate. 4.1.24.* Example. Finding blocks. For the graph below, crie <lepth-first traversal from x visits the other vertices in the order a, b, c, d, e, f, g, h, i, j. We find blocks in the order {a, b, c, d}, {e, f, g, h}, {a, i}, {x, a, e}, {x, }}. After finding each block, we delete the vertices other than the highest. Exercise 36 requests a proof of correctness.
158
## Chapter 4: Connectivity and Paths
h
g
f
g
EXERCISES
4.1.1. (-) Give a proof or a counterexample for each statement below. a) Every graph with connectivity 4 is 2-connected. b) Every 3-connected graph has connectivity 3. c) Every k-connected graph is k-edge-connected. d) Every k-edge-connected graph is k-connected. 4.1.2. (- ) Give a counterexample to the following statement, add a hypothesis to correct it, and prove the corrected statement: If e is a cut-edge of G, then at least one vertex of e is a cut-vertex of G. 4.1.3. (-) Let G be an n-vertex simple graph other than K.. Prove that if G is not k-connected, then G has a separating set of size k - 1. 4.1.4. (-) Prove that a graph G is k-connected if and only if G v K, (Definition 3.3.6) is k + r-connected. 4.1.5. (- ) Let G be a connected graph with at least three vertices. Form G' from G by adding an edge with endpoints x, y whenever de (x, y) = 2. Prove that G' is 2-connected. 4.1.6. (-) For a graph G with blocks Bi, ... , Bk. prove that n(G)
= (L:=i n(B;)) -
+ 1.
4.1.7. (-) Obtain a formula for the number of spanning trees of a connected graph in terms of the nunibers of spanning trees of its blocks.
4.1.8. Determine K(G), K'(G), and 8(G) for each graph G drawn below.
4.1.9. For each choice of integers k, l, m with 0 < k :'.S l :'.Sm, construct a simple graph G with K(G) = k, K'(G) = l, and 8(G) = m. (Chartrand-Harary [1968]) 4.1.10. (!) Find (with proof) the smallest 3-regular simple graph having connectivity 1. 4.1.11. Prove that K'(G)
:'.S 3.
## Section 4.1: Cuts and Connectivity
159
4.1.12. Let n, k be positive integers with n even, k odd, and n > k > 1. Let G be the kregular simple graph formed by placing n vertices on a circle and making each vertex adjacent to the opposite vertex and to the (k - 1)/2 nearest vertices in each direction. Prove that K(G) = k. (Harary [1962a)) 4.1.13. In Km.n, let S consist of a vertices from one partite set and b from the other. a) Compute I[S, SJ I in terms of a, b, m, n. b) Use part (a) to prove numerically that K 1 (Km. 11 ) = min{m, 11}. c) Prove that every set of seven edges in K3 . 3 is a disconnecting set, but no set of seven edges is an edge cut. 4.1.14. (!)Let G be a connected graph in which for every edge e, there are cycles C1 and C2 containing e whose only common edge is e. Prove that G is 3-edge-connected. Use this to show that the Petersen graph is 3-edge-connected. 4.1.15. (!)Use Proposition 4.1.12 and Theorem 4.1.11 to prove that the Petersen graph is 3-connected. 4.1.16. Use Proposition 4.1.12 to prove that the Petersen graph has an edge cut of size m ifand only if3:::: m :::: 12. (Hint: Consider I[S, SJ I for 1:::: ISi:::: 5.) 4.1.17. Prove that deleting an edge cut of size 3 in the Petersen graph isolates a vertex. 4.1.18. Let G be a triangle,-free graph with minimum degree. at least 3. Prove that if n(G) :::: 11, then G is 3-edge-connected. Show that this inequality is sharp by finding a 3-regular bipartite graph with 12 vertices that is not 3-edge-connected. (Galvin) 4.1.19. Prove that K(G) = 8(G) if G is simple and 8(G) ::::: n(G) - 2. Prove that this is best possible for each n ::::: 4 by constructing a simple n-vertex graph with minimum degree n - 3 and connectivity less than n - 3. 4.1.20. (!) Let G be a simple n-vertex graph with n/2 - 1 :::: 8(G) :::: n - 2. Prove that G is k-connected for all k with k :::: 28(G) + 2 - n. Prove that this is best possible for all 8 ?: n/2 - 1 by constructing a simple n-vertex graph with minimum degree 8 that is not k-connected fork= 28 + 3 - n. (Comment: Proposition 1.3.15 is the special case of this when 8(G) = (n - 1)/2.) 4.1.21. (+)Let G be a simple n-vertex graph with n ?: k +land 8(G) ::::: n+;~; 2 l. Prove that if G - S has more than l components, then ISi ::::: k. Prove that the hypothesis on 8(G) is best possible for n ::::: k + l by constructing an appropriate n-vertex graph with minimum degree L n+l(:;~)-l (Comment: This generalizes Exercise 4.1.20.)
J.
4.1.22. (!)Sufficient condition fork+ 1-connected graphs. (Bondy [1969]) a) Let G be a simple n-vertex graph with vertex degrees d 1 :::: :::: dn. Prove that if dj ::::: j + k whenever j :::: n - 1- dn-k> then G is k + 1-connected. (Comment: Exercise 1.3.64 is the special case of this when k = 0.) b) Suppose that 0 :::: j + k ~ n. Construct an n-vertex graph G such that K(G) :::: k and G has j vertices of degree j + k - 1, has n - j - k vertices of degree n - j - 1, and has k vertices of degree n - 1. In what sense does this show that part (a) is best possible? 4.1.23. (!) Let G be an r-connected graph of even order having no K I.r+l as an induced subgraph. Prove that G has a 1-factor. (Sumner [1974b]) 4.1.24. (!) Degree conditions for K 1 = 8. Let G be a simple n-vertex graph. Use Corollary 4.1.13 to prove the following statements. a) If 8(G)::::: Ln/2J, then K'(G) = 8(G). Prove this best possible by constructing for each n ?: 3 a simple n-vertex graph with 8(G) = Ln/2J- 1 and K'(G) < 8(G).
160
## Chapter 4: Connectivity and Paths
b) If d(x) + d(y) ::::: n - 1 whenever x fr y, then K'(G) = 8(G). Prove that this is best possible by constructing for each n ::::: 4 and 8(G) = m ;::: n/2 - 1 an nvertex graph G with K'(G) < 8(G) =min which d(x) + d(y) ::=:: n - 2 whenever x fry.
4.1.25. (!) K'(G) = 8(G) for diameter 2. Let G be a simple graph with diameter 2, and let [S, SJ be a minimum edge cut with ISi;::: ISi. a) Prove that every vertex of S has a neighbor in S. b) Use part (a) and Corollary 4.1.13 to prove that K'(G) = 8(G). (Plesnik [1975]) 4.1.26. (!) Let F be a set of edges in G. Prove that F is an edge cut if and only if F contains an even number of edges from every cycle in G. For example, when G = Cn, every even subset of the edges is an edge cut, but no odd subset is an edge cut. (Hint: For sufficiency, the task is to show that the components of G - F can be grouped into two nonempty collections so that every edge of F has an endpoint in each collection.) 4.1.27. (!)Let [S, SJ be an edge cut. Prove that there is a set ofpairwis.e edge-disjoint bonds whose union (as edge sets) is [S, SJ. (Note:'This is trivial if [S, SJ is itself a bond.) 4.1.28. (!)Prove that the symmetric difference of two different edge cuts is an edge cut. (Hint: Draw a picture illustrating the two edge cuts and use it to guide the proof.) 4.1.29. (!) Let H be a spanning subgraph of a connected graph G. Prove that H is a spanning tree if and only ifthe subgraph H* = G - E(H) is a maximal subgraph that contains no bond. (Comment: See Section 8.2 for a more general context.) 4.1.30. (-)Let G be the simple graph with vertex set {l, ... , 11} defined by i only if i, j have a common factor bigger than 1. Determine the blocks of G.
~
j if and
4.1.31. A cactus is a connected graph in which every block is an edge or a cycle. Prove that the maLmum number of edges in a simple n-vertex cactus is L3(n - 1)/2J. (Hint: LxJ + LyJ :::: Lx + yJ.)
4.1.32. Prove that every vertex of a graph has even degree if and only if every block is Eulerian. 4.1.33. Prove that a connected graph is k-edge-connected if and only if each of its blocks is k-edge-connected. 4.1.34. (!)The block-cutpointgraph (see Definition 4.1.20). Let H be the block-cutpoint graph of a graph G that has a cut-vertex. (Harary-Prins [1966]) a) Prove that H is a forest. b) Prove that G has at least two blocks each of which contains exactly one cutvertex of G. ' c) Prove that a graph G with k components has exactly k + LveV(G) (b( v) - 1) blocks, where b(v) is the number of blocks containing v. d) Prove that every graph has fewer cut-vertices than blocks. 4.1.35. Let Hand H' be two maximal k-co:rinected subgraphs ofa graph G. Prove that they have at most k - 1 common vertices. (Harary-Kodama [1964]) 4.1.36. Prove that Algorithm 4.1.23 correctly computes blocks of graphs. 4.1.37. Develop an algorithm to compute the strong components of a digraph. Prove that it works. (Hint: Model the algorithm on Algorithm 4.1.23).
161
## 4.2. k-Connected Graphs
A communication network is fault-tolerant if it has alternative paths between vertices: the more disjoint paths, the better. In this section, we prove that this alternative measure of connection is essentially the same as k-connectedness. When k = 1, the definition already states that a graph G is 1connected if and only if each pair of vertices is connected by a path. For larger k the equivalence is more subtle.
2-CONNECTED GRAPHS
We begin by characterizing 2-connected graphs.
4.2.1. Definition. Two paths from u to v are internally disjoint if they have no common internal vertex. 4.2.2. Theorem. (Whitney [1932a]) A graph G having at least three vertices is 2-connected if and only if for each pair u, v E V ( G) there exist internally disjoint u, v-paths in G. Proof: Sufficiency. When G has internally disjoint u, v-paths, deletion of one vertex cannot separate u from v. Since this condition is given for every pair u, v, deletion of one vertex cannot make any vertex unreachable from any other. We conclude that G is 2-connected. Necessity. Suppose that G is 2-connected. We prove by induction on d (u, v) that G has internally disjoint u, v-paths. Basis step (d(u, v) = 1). When d(u, v) = 1, the graph G - uv is connected, since K'(G) ~ K(G) ~ 2. Au, v-path in G - uv is internally disjoint in G from the u, v-path formed by the edge uv itself. Induction step (d(u, v) > 1). Let k = d(u. v). Let w be the vertex before v on a shortest u, v-path; we have d(u, w) = k - 1. By the induction hypothesis, G has internally disjoint u, w-paths P and Q. Ifv E V(P) U V(Q), then we find the desired paths in the cycle P U Q. Suppose not. Since G is 2-connected, G - w is connected and contains au, ti-path R. If R avoids P or Q, we are done, but R may share internal vertices with both P and Q. Let z be the last vertex of R (before v) belonging to P U Q. By symmetry, we may assume that z E P. We combine the u, z-subpathof P with the z. v-subpath of R to obtain au, v-path internally disjoint from Q U wv.
162
## Chapter 4: Connectivity and Paths
4.2.3. Lemma. (Expansion Lemma) If G is a k-connected graph, and G' is obtained from G by adding a new vertex y with at least k neighbors in G, then G' is k-connected. Proof: We prove that a separating set S of G' must have size at least k. If y E S, then S - {y} separates G, so ISi :::: k + 1. If y fj. Sand N(y) s; S, then ISi :::: k. Otherwise, y and N(y) - S lie in a single component of G' - S. Thus again S must separate G and ISi :::: k.
~y
4.2.4. Theorem. For a graph G with at least three vertices, the following conditions are equivalent (and characterize 2-connected graphs). A) G is connected and has no cut-vertex. B) For all x, y E V(G), there are internally disjoint x, y-paths. C) For all x, y E V ( G), there is a cycle through x and y. D) 8( G) :::: 1, and every pair of ~dges in G lies on a common cycle. Proof: Theorem 4.2.2 proves A<=}B. For B<=}C, note that cycles containing x and y correspond to pairs of internally disjoint x, y-paths. For D=:>C, the condition 8(G) :::: 1 implies that vertices x and y are not isolated; we then apply the last part of D to edges incident to x and y. If there is only one such edge, then we use it and any edge incident to a third vertex. To complete the proof, we assume that G satisfies the equivalent properties A and C and then derive D. Since G is connected, 8(G) :::: 1. Now consider two edges uv and xy. Add to G the vertices w with neighborhood (u, v) and z with neighborhood {x, y). Since G is 2-connected, the Expansion Lemma (Lemma 4.2.3) implies that the resulting graph G' is 2-connected. Hence condition C holds in G', so w and z lie on a cycle C in G'. Since w, z each have degree 2, C must contain the paths u, w, v and x, z, y but not the edges uv or xy. Replacing the paths u, w, v and x, z, yin C with the edges uv and xy yields the desired cycle through uv and xy in G.
4.2.5. Definition. In a graph G, subdivision of an edge uv is the operation of replacing uv with a path u, w, v through a new vertex w.
~
~
## Section 4.2: k-Connected Graphs
163
4.2.6. Corollary. If G is 2-connected, then the graph G' obtained by subdividing an edge of G is 2-connected. Proof: Let G' be formed from G by adding vertex w to subdivide uv. To show that G' is 2-connected, it suffices to find a cycle through arbitrary edges e, f of G' (by Theorem 4.2.4D). Since G is 2-connected, any two edges of G lie on a common cycle (Theorem 4.2.4D). When our given edges e, f of G' lie in G, a cycle through them in G is also in G', unless it uses uv, in which case we modify the cycle. Here "modify the cycle" means "replace the edge u v with the u, v-path oflength 2 through w ". When e E E ( G) and f E {u w, w v}, we modify a cycle passing through e and 11v in G. When {e, fl= {uw, wv}, we modify a cycle through uv.
The class of 2-connected graphs has a characterization that expresses the construction of each such graph from a cycle and paths.
4.2.7. Definition. An ear of a graph G is a maximal path whose internal vertices have degree 2 in G. An ear decomposition of G is a decomposition Po . .... P, such that Po is a cycle and P; for i :=:: 1 is an ear of Po U U P;.
Pa
4.2.8. Theorem. (Whitney [1932a]) A graph is 2-connected if and only if it has an ear decomposition. Furthermore, every cycle in a 2-connected graph is the initial cycle in some ear decomposition. Proof: Sufficiency. Since cycles are 2-connected, it suffices to show that adding an ear preserves 2-connectedness. Let u, v be the endpoints of an ear P to be added to a 2-connected graph G. Adding an edge cannot reduce connectivity, so G + uv is 2-connected. A succession of edge subdivisions converts G + uv into the graph Gu P in which P is an ear; by Corollary 4.2.6, each subdivision preserves 2-connectedness. Necessity. Given a 2-connected graph G, we build an ear decomposition of G from a cycle C in G. Let Go= C. Let G; be a subgraph obtained by successively adding i ears. If G; 'f. G, then we can choose an edge uv of G - E(G;) and an edge xy E E(G;). Because G is 2-connected, uv and xy lie on a common cycle C'. Let P be the path in C' that contains u v and exactly two vertices of G;, one at each end of P. Now P can be added to G; to obtain a larger subgraph G;+1 in which P is an ear. The process ends only by absorbing all of G.
164
## Chapter 4: Connectivity and Paths
Every 2-connected graph is 2-edge-connected, but the converse does not hold. Recall that the bowtie is the graph consisting of two triangles sharing one common vertex; it is 2-edge-connected but not 2-connected. Since more graphs are 2-edge-connected, decomposition of 2-edge-connected graphs needs a more general operation. The proof is like that of Theorem 4.2.8.
I><J
4.2.9. Definition. A closed ea.., in a graph G is a cycle C such that all vertices of C except one have degree 2 in G. A closed-ear decomposition of G is a decomposition Po, ... , Pk such that Po is a cycle and P; for i ~ 1 is either an (open) ear or a closed ear in G. 4.2.10. Theorem. A graph is 2-edge-connected if and only if it has a closed-ear decomposition, and every cycle in a 2-edge-connected graph is the initial cycle in some such decomposition. Proof: Sufficiency. Cut-edges are the edges not on cycles (Theorem 1.2.14), so a connected graph is 2-edge-co:inected if and only if every edge lies on a cycle. The initial cycle is 2-edge-connected. When we add a clo~ed ear, its edges form a cycle. When we add an open ear P to a connected graph G, a path in G connecting the endpoints of P completes a cycle containing all edges of P. In each case, the new graph also is connected. Thus adding an open or closed ear preserves 2-edge-connectedness. Necessity. Given a 2-edge-connected graph G, let Po be a cycle in G. Consider a closed-ear decomposition P0 , . , P; of a subgraph G; ofG. When G; =f. G, we find an ear to add. Since G is connected, there u v E E ( G) - E ( G;) .., is an edge . with u E V(G;). Since G is 2-edge-connected, uv lies on a cycle C. Follow Cuntil it returns to V(G;), forming up to this point a path or cycle P. Adding P to G; yields a larger subgraph G;+ 1 in which P is an open or closed ear. The process ends only by absorbing all of G. Ill
CONNECTIVITY OF DIGRAPHS
Our results about k-connected and k-Hige-connected graphs will apply as well for digraphs, where we use analogous terminology. 4.2.1 l. Definition. A separating set or vertex cut of a digraph D is a set S ~ V (D) such that D - S is not strongly connected. A digraph is k-connected if every vertex cut has at least k vertices. The minimum size of a vertex cut is the connectivity K(D). For vertex sets S, T in a digraph D, let [S, T] denote the set of edges with tail in S and head in T. An. edge cut is the set [S, S] for some 0 =f.
## Section 4.2: k-Connected Graphs
165
S c V (D). A digraph is k-edge-connected if every edge cut has at least k edges. The minimum size of an edge cut is the edge-connectivity K'(D).
4.2.12. Remark. Because j[s, SJI is the number of edges leaving S, we can restate the definition of edge-connectivity as follows: A graph or digraph G is kecige-connected if and only if for every nonempty proper vertex subset S, there are at least k edges in G leaving S. Note that [S, T] is the set of edge from S to T. The meaning of this depends on whether we are discussing a graph or a digra1;>h. In a graph, we take all edges that have endpoints in ooth sets. In a digraph, we take only the edges with tail in S and head in T.
Strong digraphs are similar to 2-edge-connected graphs.
4.2.lS. Proposition. Adding a (directed) ear to a strong digraph produces a larger strong digraph. Proof: By Remark 4.2.12; a digraph is strong if and only if for every nonempty vertex subset there is a departing edge. If we add an open ear or closed ear P to a strong digraph D, then for every set S with 0 c S c V (D) we already have an edge from S to V (D) - S. We need only consider sets that don't intersect V (D) and sets that contain all of V (D) but not all of V (P). For every such set, there is an edge leaving it along P.
\Vhe11 can the streets in a road network all be made one-way without making any location unreachable from some other location? In other words, when does a graph have a strong orientation? The graph below does not. The obvious necessary conditions are sufficient.
4.2.14. Theorem. (Robbins [1939]) A graph has a strong orientation if and only if it is 2-edge-connected. Proof: Necessity. If a graph G is disconnected, then some vertices cannot reach others in any orientation. If G has a cut-edge xy oriented from x to y in an orientation D, then y cannot reach x in D. Hence G must be connected and have no cut-edge. Sufficiency. When G is 2-edge-connected, it has a closed-ear decomposition. We orient the initial cycle consistently to.obtain a strong digraph. As we add each new ear and direct it consistently, Proposition 4.2.13 guarantees that we still have a strong digraph.
166
## Chapter 4: Connectivity and Paths
Robbins' Theorem generalizes for all k. When G has a k-edge-connected orientation, Remark 4.2.12 implies that G must be 2k-edge-connected. NashWilliams [1960] proved that this obvious necessary condition is also sufficient: a graph has a k-edge-connected orientation if and only if it is 2k-edge-connected. This is easy when G is Eulerian (Exercise 21), but the general case is difficult (see Exercises 36-38). A thorough discussion of this and other orientation theorems appears in Frank [1993].
## k-CONNECTED AND k-EDGE-CONNECTED GRAPHS
We have introduced two measures of good connection: invulnerability to deletions and multiplicity of alternative paths. Extending Whitney's Theorem, we show that these two notions are the same, for both vertex deletions and edge deletions, and for both graphs and digraphs. We first discuss the "local" problem of x, y-paths for a fixed pair x, y E V ( G). These definitions hold both for graphs and for digraphs.
4.2.15. Definition. Given x, y E V(G), a set S ~ V(G) - {x, y} is an x, yseparator or x, y-cut if G - S has no x, y-path. Let K (x, y) be the minimum size ofan x, y-cut. LetA.(x, y) be the maximum size of a set of pairwise internally disjoint x, y-paths. For X, Y ~ V(G), an X, Y-path is a path having first vertex in X, last vertex in Y, and no other vertex in X U Y.
An x, y-cut must contain an internal vertex of every x, y-path, and no vertex can cut two internally disjoint x, y-paths. Therefore, always K(x, y) :::: A.(x, y). Thus the problems of finding the smallest cut and the largest set of paths are dual problems, like the duality between matching and covering in Chapter 3.
4.2.16. Example. In the graph G below, the set S = {b, c, z, d} is an x, y-cut of size 4; thus K (x, y) :::; 4. As shown on the left, G has four pairwise internally disjoint x, y-paths; thus A.(x, y) :::: 4. Since K(x, y) :::: A.(x, y) always, we have K(x, y) = A.(x, y) =4.
x
d
''
/x
'
d'
'
/ /
Consider also the pair w, z. As shown on the right, K(w, z) = A.(w, z) = 3, with {b, c, x} being a minimum w, z-cut. The graph G is 3-connected; for every pair u, v E V(G), we can find three pairwise internally disjoint u, v-paths. From the equality for internally disjoint paths, we will obtain an analogous equality for edge-disjoint paths. Although K ( w, z) = 3 above, it takes four edges to break all w, z-paths, and there are four pairwise edge-disjoint w, z-paths.
## Section 4.2: k-Connected Graphs
167
What we call Menger's Theorem states that the local equality K(x, y) = A.(x, y) always holds. The global statement for connectivity and analogous results for edge-connectivity and digraphs were observed by others. All are considered forms of Menger's Theorem. More than 15 proofs of Menger's Theorem have been published, some yielding stronger results, some incorrect. (A gap in Menger's original argument was later repaired by Konig.)
4.2.17. The.orem. (Menger [1927]) If x, y are vertices of a graph G and xy fl. E(G), then the minimum size of an x, y-cut equals the maximum number of pairwise internally disjoint x, y-paths. Proof: An x, y-cut must contain an internal vertex from each path in a set of pairwise internally disjoint x, y-paths. These vertices must be distinct, so
K(x, y) '.'.: A.(x, y).
To prove equality, we use induction on n(G). Basis step: n(G) = 2. Here xy fj. E(G) yields K(x, y) = A.(x, y) = 0. Induction step: n(G) > 2. Let k = Kc(x, y). We construct k pairwise internally disjoint x, y-paths. Note that since N(x) and N(y) are x, y-cuts, no minimum cut properly contains N(x) or N(y). Case 1: G has a minimum x, y-cut S other than N(x) or N(y). To obtain the k desired paths, we combine x, S-paths and S, y-paths obtained from the induction hypothesis (as formed by solid edges shown below). Let Vi be the set of vertices on x, S-paths, and let V2 be the set of vertices on S, y-paths. We claim that S = Vi n V2 Since S is a minimal x, y-cut, every vertex of S lies on an x, y-path, and hence S s; V1 n V2 If v E (V1 n V2 ), then following the x, vportion of some x, S-path and then the v, y-portion of some S, y-path yields an x, y-path that avoids the x, y-cut S. This is impossible, so S = V(G 1 ) n V(G2). By the same argument, V1 omits N(y) -Sand V2 omits N(x) - S. Form H1 by adding to G[V1] a vertex y' with edges fr~m S. Form H2 by adding to G[V2 ] a vertex x' with edges to S. Every x, y-path in G starts with an x, S-path(containedin H 1 ), so every x, y'-cutin H 1 is anx, y-cutin G. Therefore, KH 1 (x, y') = k, and similarly KH2 (x', y) = k. Since Vi omits N(y)- Sand V2 omits N(x)- S, both H1 and H2 are smaller than G. Hence the induction hypothesis yields A.H1 (x, y') = k = A.H2 (x', y). Since V1 n V2 = S, deleting y' from the k paths in H 1 and x' from the k paths in H 2 yields the desired x, S-paths and S, y-paths in G that combine to form k pairwise internally disjoint x, y-paths in G.
x
G
H1
G'
Case 2 Case 2. Every minimum x, y-cut is N (x) or N (y ). Again we construct the k desired paths. In this case, every vertex outside {x U N(x) U N(y) Uy} is in no minimum x, y-cut. If G has such a vertex v, then KG-v(x, y) = k, and
Case 1
168
## Chapter 4: Connectivity and Paths
applying the induction hypothesis to G - v yields the desired x, y-paths in G. Also, if there exists u E N(x) n N(y), then u appears in every x, y-cut, and Kc-u(x, y) = k - 1. Now applying the induction hypothesis to G - v yields k - 1 paths to combine with the path x, v, y. We may thus assume that N(x) and N(y) partition V(G) - {x, y}. Let G' be the bipartite graph with bipartition N(x), N(y) and edge set [N(x), N(y)]. Every x, y-path in G uses some edge from N(x) to N(y), so the x, y-cuts in G are precisely the vertex covers of G'. Hence f3(G') = k. By the Konig-Egervary Theorem, G' has a matching of size k. These k edges yield k pairwise internally disjoint x, y-paths oflength 3. Case 2 is needed in the proof because when S = N (x), the induction hypothesis cannot be used to obtain the S, y-paths. The statement ofTheorem 4.2.17 makes sense also for digraphs. The proof of the digraph version is exactly the same; we only need to replace N(x) and N(y) with N+(x) and N-(y) throughout. We next develop the analogue of Theorem 4.2.17 for edge-disjoint paths, which we prove by applying Theorem 4.2.17 to a transformed graph. The main part of the transformation is an operation that we will use again in Chapter 7.
4.2.18. Definition. The line graph of a graph G, written L(G), is the graph whose vertices are the edges of G, with ef E E(L(G)) when e = uv and f = vw in G. Substituting "digraph" for "graph" in this sentence yields the definition of line digraph. For graphs, e and f share a vertex; for digraphs, the head of e must be the tail off.
J;
g
j
l
L(G)
L(H)
When disconnecting y from x by deleting edges, we use notation analogous to that of Definition 4.2.15: A.'(x, y) is the maximum size of a set of pairwise edge-disjoint x, y-paths, and K'(x, y) is the minimum number of edges whose deletion makes y unreachable from x. Elias-Feinstein-Shannon [1956] and Ford-Fulkerson [1956] proved that always A.'(x, y) = K'(x, y) (using the methods of Section 4.3). We allow mltiple edges and allow xy E E(G).
4.2.19. Theorem. If x and y are distinct vertices of a graph or digraph G, then the minimum size of an x, y-disconnecting set of edges equals the maximum number of pairwise edge-disjointx, y-paths. Proof: Modify G to obtain G' by adding two new vertices s, t and two new edges sx and yt. This does not change K'(x, y) or A.'(x, y), and we can think of each
## Section 4.2: k-Connected Graphs
169
path as starting from the edge sx and ending with the edge yt. A set of edges disconnects y from x in G if and only if the corresponding vertices of L ( G') form an sx, yt-cut. Similarly, edge-disjoint x, y-paths in G become internally disjoint sx, yt-paths in L(G'), and vice versa. Since x =j:. y, we have no edge from sx to ty in L(G'). Applying Theorem 4.2.17 to L(G') yields
K~(x, y)
## = KL(G'j(SX, yt) = AL(G')(SX, yt) = A.~(x, y).
a
f
yt
f
g
SX
The global version for k-connected graphs, observed first by Whitney [1932a], is also commonly called Menger's Theorem. The global versions for edges and digraphs appeared in Ford-Fulkerson [1956].
4.2.20. Lemma. Deletion of an edge reduces connectivity by at most 1. Proof: We discuss only graphs; the argument for digraphs is similar (Exercise 7). Since every separating set of G is a separating set of G - xy, we have K(G - xy) ~ K(G). Equality holds unless G - xy has a separating set S that is smaller than K ( G) and hence is not a separating set of G. Since G - S is connected, G - xy - S has two components G[X] and G[Y], with x E X and y E Y. In G - S, the only edge joining X and Y is xy. If IXI ::: 2, then SU {x} is a separating set ofG, and K(G)~ l((G-xy) + 1. If IYI ::: 2, then again the inequality holds. In the remaining case, ISi = n(G) - 2. Since we have assumed that ISi < K(G), ISi = n(G) - 2 implies that K(G) ::: n(G) - 1, which holds only for a complete graph. Thus K(G - xy) = n(G) - 2 = K ( G) - 1, as desired. 4.2.21. Theorem. The connectivity of G equals the maximum k such that A.(x, y) :::_ k for all x, y E V ( G). The edge-connectivity of G equals the maximum k such that A.' (x, y) :::_ k for all x, y E V ( G). Both statements hold for graphs and for digraphs. Proof: Since K'(G) = minx.y<;V(G) K'(x, y), Theorem 4.2.19 immediately yields the claim for edge-connectivity. For connectivity, we have K(x, y) = A.(x, y) for xy fj. E(G), and K(G) is the minimum of these values. We need only show that A.(x, y) cannot be smaller than K(G) when xy e E(G). Certainly deletion of xy reduces A.(x, y) by 1, since xy itself is anx, y-path and cannot contribute to any other x, y7path. With this, Theorem 4.2.17, and Lemma 4.2.20, we have
A.o(x, y) =
170
## APPLICATIONS OF MENGER'S THEOREM
Dirac extended Menger's Theorem to other families of paths.
4.2.22. Definition. Given a vertex x and a set U of vertices, an x, U -fan is a set of paths from x to U such that any two of them share only the vertex x. 4.2.23. Theorem. (Fan Lemma, Dirac [1960]). A graph is k-connected if and only if it has at least k + 1 vertices and, for every choice of x, U with IU I ::: k, it has an x, U-fan of size k. Proof: Necessity. .Given k-connected graph G, we construct G' from G by adding a new vertex y adjacent to all of lf. Tht: Expansion Lemma (Lemma 4.2.3) implies that G' also is k-connected, and then Menger's Theorem yields k pairwise internally disjoint x, y-paths in G'. Deleting y from these paths produces an x, U -fan of size k in G. Sufficiency. Suppose that G satisfies the fan condition. For v E V ( G) and U = V(G) - {v}, there is av, U-fan of size k; thus 8(G)::: k. Given w, z E V(G), let U = N(z). Since IUI ::: k, we have an w, U-fan of size k; extend each path by adding an edge to z. We obtain k pairwise internally disjoint w, z-paths, so A.(w, z)::: k. This holds for all w, z E V(G), so G is k-connected.
The Fan Lemma generalizes considerably. Whenever X and Y are disjoint sets of vertices in a k-connected graph G and we specify integers at X and Y summing to kin each set, there are k pairwise internally disjoint X, Y-paths with the specified number ending at each point (Exercise 28). The Fan Lemma also yields the next result.
4.2.24. * Theorem. (Dirac [1960]) If G is a k-connected graph {with k ::: 2), and S is a set of k vertices in G, then G has a cyclG including S in its vertex set. Proof: We use induction on k. Basis step (k = 2): Theorem 4.2.2 (or Theorem 4.2.21) implies that any two vertices are connected by two internally disjoint paths, which form a cycle containing them. Induction step (k > 2): With G and Sas specified, choose x ES. Since G is also k - 1-connected, the induction hypothesis implies that all of S - {x} lies on a cycle C. Suppose first that n(C) = k - 1. Since G is k - 1-connected, we have an x, V ( C)-fan of size k - 1, and the paths of the fan to two consecutive vertices of C enlarge the cycle to include x. Hence we may assume that n(C) ::: k. Since G is k-connected, G has an
## Section 4.2: k-Connected Graphs
171
x, V(C)-fan of size k. We claim that again the fan has two paths forming a
detour from C that includes x while keeping S - {x }. Let vi, ... , Vk-l be the vertices of S - {x} in order on C, and let V; be the portion of V ( C) from v; up to but not including V;+1 (here vk = v1). The sets V1, ... , Vk-1 partition V(C) into k - 1 disjoint sets. Since the x, V(C)-fan has k paths, two of them enter V(C) in one of these sets, by the pigeonhole principle. Let u, u' be the vertices where these paths reach C. Replacing the u, u' -portion of C by the x, u-path and x, u' -path in the fan builds a new cycle that contains x and all of S - {x }. u'
Many applications of Menger's Theorem involve modeling a problem so that the desired objects correspond to paths in a graph or digraph, often by graph transformation arguments. For example, given sets A= A1, ... , Am with union X, a system of distinct representatives (SDR) is a set of distinct elements x1, ... , Xm such that x; E A;. A necessary and sufficient condition for the existence of an SDR is that IUie/ A; I 2: Ill for all/ ~ [m]. It is easy to prove this from Hall's Theorem by modeling A with an appropriate bipartite graph (Exercise 3.1.19). Indeed, Hall's Theorem was originally proved in the language of SDRs and is equivalent to Menger's Theorem (Exercise 23). Ford and Fulkerson considered a more difficult problem. Let A = Ai, ... , Am and B = B1, ... , Bm be two families of sets. We may ask when there is a common system of distinct representatives (CSDR), meaning a set of m elements that is both an SDR for A and an SDR for B. They found a necessary and sufficient condition.
4.2.25.* Theorem. (Ford-Fulkerson [1958]) Families A = {Ai, ... , Am} and B ={Bi. ... , Bm} have a common system ofdistinctrepresentatiyes(CSDR) if and only if
'
jeJ
## for each pair/, J
[m].
Proof: We create a digraph G with vertices a1, ... , am and b1, ... , bm, plus a vertex for each element in the sets and special vertices s, t. The edges are
{sa;: A; EA}
{bjt: Bj EB}
{a;x:
## x EA;} {xbj: x E Bj}
Each s, t-path selects a member of the intersection of some A; and some Bj. There is a CSDR if and only if there is a set of m pairwise internally disjoint s, t-paths. By Menger's Theorem, it suffices to show that the stated condition
172
## Chapter 4: Connectivity and Paths
is equivalent to having no s, t-cut of size less than m. Given a set R s:; V ( G) {s, t}, let I= {a;} - Rand J = {bj} - R. The set R is ans, t-cut if and only if (LJ;e/ A;) n (LJjeJ Bj) s:; R. For ans, t-cut R, we thus have
IRI ~
## l(U A;) n (u Bj)' + (m iE/ jEJ
Ill)+ (m
Ill).
This lower bound is at least m for every s, t-cut ifand only if the stated condition holds.
1 4.2.26. * Example. Digraph for CSDR. In the example above, the elements are {1, 2, 3, 4}, A = {12, 23, 31}, and B = {14, 24, 1234}. Suppose that Rn {a;} = {ai, a2} and Rn {bj} = {b1, b2}. In the arguinent, we set I= {a3} and J = (b3}, and we observe that R is ans, t-cut if and only if it also contains {l, 3}, which equals (LJiE/ A;) n (LJjeJ Bj).
EXERCISES
4.2.1. (-)Determine K(u, v) and K'(u, v) in the graph drawn below. (Hint: Use the dual problemsto give short proofs of optimality.)
4.2.2. (-) Prove that if G is 2-edge-connected and G' is obtained from G by subdividing an edge of G, then G' is 2-edge-conriected. Use this to prove that every graph having a t'.:losed-ear decomposition is 2-edge-connected. (Comment: This is an alternative proof of sufficiency for Theorem 4.2.10.)
## Section 4.2: k-Connected Graphs
173
~
4.2.3. (-) Let G be the digraph with vertex set [12] in which i divides j. Determine K(l, 12) and K'(l, 12).
j if and only if i
4.2.4. (-) Prove or disprove: If P is a u, v-path in a 2-connected graph G, then there is au, v-path Q that is internally disjoint from P. 4.2.5. (-)Let G be a simple graph, and let H(G) be the graph with vertex set V(G) such that uv E E(H) ifand only ifu, v appear on a common cycle in G. Characterize the graphs G such that H is a clique. 4.2.6. (-)Use results of this section to prove that a simple graph G is 2-connected if and only if G can be obtained from Cs by a sequence of edge additions and edge subdivisions .
4.2.7. Let xy be an edge in a digraph G. Prove that K(G - xy)::::: K(G) - 1. 4.2.8. Prove that a simple graph G is 2-connected if and only if for every triple (x, y, z) of distinct vertices, G has an x, z-path through y. (Chein [1968]) 4.2.9. Prove that a graph G with at least four vertices is 2-connected if and only if for every pair X, Y of disjoint vertex subsets with IXI, IYI ::::: 2, there exist two completely disjoint paths P1 , P2 in G such that each has an endpoint in X and an endpoint in Y and no internal vertex in X or Y. 4.2.10. A greedy ear decomposition of a 2-connected graph is an ear decomposition that begins with a longest cycle and iteratively adds a longest ear from the remaining graph. Use a greedy ear decomposition to prove that every 2-connected claw-free graph G has Ln(G)/3J pairwise-disjoint copies of P3 . (Kaneko-Kelmans-Nishimura [2000]) 4.2.11. (!) For a connected graph G with at least three vertices, prove that the following statements are equivalent (use of Menger's Theorem is permitted). A) G is 2-edge-connected. B) Every edge of G appears in a cycle. C) G has a closed trail containing any specified pair of edges. D) G has a closed trail containing any specified pair of vertices. 4.2.12. (!)Use Menger's Theorem to prove that K(G) = K'(G) when G is 3-regular (Theorem 4.1.11). 4.2.13. (!)Let G be a 2-edge-connected graph. Define a relation Ron E(G) by (e, f) if e = f or if G - e - f is disconnected. (Lovasz [1979, p277]) a) Prove that (e, f) E R if and only if e, f belong to the same cycles. b) Prove that R is an equivalence relation on E(G). c) For each equivalence class F, prove that F is contained in a cycle. d) For each equivalence class F, prove that G - F has no cut-edge.
E
4.2.14. (!) A u, v-necklace is a list of cycles C1, ... , Ck such that u E Ci, v E Ck. consecutive cycles share one vertex, and nonconsecutive cycles are disjoint. Use induction on d(u, v) to prove that a graph G is 2-edge-connected ifand only iffor all u, v E V(G) there is au, v-necklace in G.
u
4.2.15. (+)Let v be a vertex of a 2-connected graph G. Prove that v has a neighbor u such that G - u - vis connected. (Chartrand-Lesniak [1986, p51])
174
## Chapter 4: Connectivity and Paths
4.2.16. (+)Let G be a 2-connected graph. Prove that if Ti, T2 are two spanning trees of G, then Ti can be transformed into T2 by a sequence of operations in which a leaf is removed and reattached using another edge of G. 4.2.17. Determine the smallest graph with connectivity 3 having a pair of nonadjacent vertices linked by four pairwise internally disjoint paths. 4.2.18. Let G be a graph without isolated vertices. Prove that if G has no even cycles, then every l.Jlock of G is an edge or an odd cycle. 4.2.19. (!) Member.ship in common cycles. a) Prove that two distinct edges lie in the same block of a graph if and only if they belong to a common cycle. b) Given e, f, g E E(G), suppose that G has a cycle through e and f and a cycle through f and g. Prove that G also has a cycle through e and g. (Comment: This problem implies that for graphs without cut-edges, ''belong to a common cycle" is an equivalence relation whose equivalence classes are the edge sets of blocks.) 4.2.20. Prove that the hypercube Qk is k-connected by constructing k pairwise internally disjoint x, y-paths for each vertex pair x, y E V ( Qk). 4.2.21. (!) Let G be a 2k-edge-connected graph with at most two vertices of odd degree. Prove that G has a k-edge-connected orientation. (Nash-Williams [1960]) 4.2.22. (!)Suppose that K(G) = k and diam G = d. Prove that n(G) ::::: k(d - 1) + 2 and a(G) ::::: rc1 + d)/21. For each k ::::: 1 and d ::::: 2, construct a graph for which equality holds in both bounds. 4.2.23. (!) Use Menger's Theorem (K(x, y) = A.(x, y) when xy E(G)) to prove the Konig-Egervary Theorem (a'(G) = f3(G) when G is bipartite). 4.2.24. (!) Let G beak-connected graph, and let S, T be disjoint subsets of V(G) with size at least k. Prove that G has k pairwise disjoint S, T-paths. 4.2.25. (*) Show that Theorem 4.2.24 is best possible by constructing for each k a kconnected graph having k + 1 vertices that do not lie on a cycle. 4.2.26. For k '.:: 2, prove that a graph with at least k + 1 vertices is k-connected if and only if for every T C:::: S C:::: V(G) with ISi = k and ITI = 2, there is a cycle in G that contains T and avoids S - T. (Lick [1973]) 4.2.27. A vertex k-split of a graph G is a graph H obtained from G by replacing one vertex x E V(G) by two adjacPnt vertices xi, x 2 such that dH(x;) :=: k and that NH(xi) U NH(x2) = Nc(x) U {xi. x2}. a) Prove that every vertex k-split of a k-connected graph is k-connected. b) Conclude that any graph obtained from a "wheel" Wn = Ki v Cn-i (Definition 3.3.6) by a sequence of edge additions and vertex 3-splits on vertices of degree at least 4 is 3-connected. (Comment: Tutte [1961b] proved also that every 3-connected graph arises in this way. The characterization does not extend easily for k > 3.)
## Section 4.2: k-Connected Graphs
175
4.2.28. (!) Let X and Y be disjoint sets of vertices in a k-connected graph G. Let u(x) for x E X and w(y) for y E Y be nonnegative integers such that LxEX u(x) = LvEY w(y) = k. Prove that G has k pairwise internally disjoint X, Y-paths so that u(x) of them start at x and w(y) of them end at y, for x EX and y E Y. 4.2.29. Given a graph G, let D be the digraph obtained by replacing each edge with two oppositely-directed edges having the same endpoints (thus Dis the symmetric digraph with underlying graph G). Assume that for all x, y E V(D) both K~(x, y) = A.~(x, y) and Kv(x, y) = A.v(x, y) hold, the latter applying only when x fr y. Use this hypothesis to prove that also K~(x, y) = A.~(x, y) and Kc(x, y) = A.c(x, y), the latter for x ~ y. 4.2.30. (!) Prove that applying the expansion operation of Example 1.3.26 to a 3connected graph yields a 3-connected graph. Qbtain the Petersen graph from K4 by expansions. (Comment: Tutte [1966a] proved that a 3-regular graph is 3-connected if and only ifit arises from K4 by a sequence of these operations.) 4.2.31. Let G beak-connected simple graph. a) Let C and D be two cycles in G of maximum length. For k = 2 and k = 3, prove that C and D share at least k vertices. (Hint: If they don't, construct a longer cycle.) b) For each k :::-: 2, construct a k-connected graph that has distinct longest cycles with only k common vertices. (Hint: K2.4 works for k = 2.) 4.2.32. Graph splices. Let G 1 and G 2 be disjoint k-connected graphs with k :::-: 2. Choose v 1 E V(G 1 ) and v2 E V(G 2 ). Let B be a bipartite graph with partite sets Na 1 (vi) and Na 2 (v2 ) that has no isolated vertex and has a matching of size at least k. Prove that (G 1 - v1 ) U (G 2 - v2 ) U Bis k-connected. 4.2.33. (*)Prove Hall's Theorem from Theorem 4.2.25. 4.2.34. A k-connected graph G is minimally k-connected if for every e E E(G), the graph G - e is not k-connected. Halin [1969] proved that 8(G) = k when G is minimally k-connected. Use ear decomposition to prove this fork= 2. Conclude that a minimally 2-connected graph G with at least 4 vertices has at most 2n(G) - 4 edges, with equality only for K2.n-2 (Dirac [1967)) 4.2.35. Prove that if G is 2-connected, then G - xy is 2-connected if and only if x and y lie on a cycle in G - xy. Conclude that a 2-connected graph is minimally 2-connected if and only if every cycle is an induced subgraph. (Dirac [1967], Plummer [1968]) 4.2.36. (!)For S ~ V(G), let d(S) = l[S, s11. -Let X and Y be nonempty proper vertex subsets of G. Prove that d(X n Y) + d(X UY)::: d(X) + d(Y). (Hint: Draw a picture and consider contributions from various types of edges.) 4.2.37. (+) A k-edge-connected graph G is minimally k-edge-connected if for every e E E(G) the graph G - e is not k-edge-connected. Prove that 8(G) = k when G is minimally k-edge-connected. (Hint: Consider a minimal set S such that I[S, SI I = k. If ISi # 1, use G - e for some e E E(G[\$]) to obtain another set T with l[T, Tll = k such that S, T contradict Exercise 4.2.36.) (Mader [1971); see also Lovasz [1979, p285)) 4.2.38. Mader [1978] proved the following: "If z is a vertex of a graph G such that dc(z) . {O, 1, 3) and z is incident to no cut-edge, then z has neighbors x and y such that KG-xz-yz+xy(u, v) = Ka(u, v) for all u, v E V(G) - {z)." Use Mader's Theorem and Exercise 4.2.37 to prove Nash-Williams' Orientation Theorem: every 2k-edge-connected graph has a k-edge-connected orientation. (Comment: A weaker version ofM.ader's Theorem, given in Lovasz [1979, p286-288], also yields Nash-Williams' Theorem in the same way.)
176
## 4.3. Network Flow Problems
Consider a network of pipes where valves allow flow in only one direction. Each piptJ has a capacity per unit time. We model this with a vertex fur each junction and a (directed) edge for each pipe, weighted by the capacity. We also assume that flow cannot accumulate at a junction. Given two locations s, t in the network, we may ask "what is the maximum flow (par unit time) from s tot?" This question arises in many contexts. The network may represent roads with traffic capacities, or links in a computer network with data transmission capacities, or currents in an electrical network. Th~re are applications in industrial settings and to combinatorial min-max theorems. The seminal book on the subject is Ford-Fulkerson [1962]. More recently, Ahuja-Magnanti-Orlin [1993] presents a thorough treatment of network flow problems.
4.3.1. Definition. A network is a digraph wiia a nonnegative capacity c(e) on each edge e and a distinguished source vertex s and sink vertex t. Vertices are also called nodes. A fl.ow I assigns a value l(e) to each dge e. We write l+(v) for the total flow on edges leaving v and 1-(v) for the total flow on edges entering v. A flow is feasible if it satisfies the capacity constraints 0.::: l(e) .::: c(e) for each edge and the conservation constraints l+(v) = 1-(v) for each node v fl. {s, t}.
## MAXIMUM NETWORK FLOW
We consider first the problem of maximizing the net flow into the sink.
4.3.2. Definition. The value val(/) of a flow I is the net flow 1-(t) - l+(t) into the sink. A maximum fl.ow is a feasible flow of maximum value. 4.3.3. Example. The zero fl.ow assigns flow 0 to each edge; this is feasible. In the network below we illustrate a nonzero feasible flow. Each capacities are shown in bold, flow values in parentheses. Our flow I assigns l(sx) = l(vt) = 0, and I (e) = 1 for every other edge e. This is a feasible flow of value 1.
s x
(1)1
s
y
A path from the source to the sink with excess capacity would allow us to increase flow. In this example, no path remains with excess capacity, but the
## Section 4.3: Network Flow Problems
i77
flow /' with f'(vx) = 0 and f'(e) = 1 fore =f:. vx has value 2. The flow f is "maximal" in that no other feasible flow can be found by increasing the flow on some edges, but f is not a maximum flow. We need a more general way to increase flow. In addition to traveling forward along edges with excess capacity, we allow traveling backward (against the arrow) along edges where the flow is nonzero. In this example, we can travel from.\$ to x to v tot. Increasing the flow by 1 on sx and vt and decreasing it by one on vx changes f into f'. 4.3.4. Definition. When f is a feasible flow in a network N, an f -augmenting path is a source-to-sink path P in the underlying graph G such that for each e E E(P), a) if P follows e in the forward direction, then f(e) < c(e). b) if P follows e in the backward direction, then f(e) > 0. Let E(e) = c(e)- f(e) when e is forward on P, and let E(e) = f(e) when e is backward on P. The tolerance of Pis mineeE(P) E(e).
As in Example 4.3.3, an /-augmenting path leads to a flow with larger value. The definition of /-augmenting path ensures that the tolerance is positive; this amount is the increase in the flow value.
4.3.5. Lemma. If Pis an /-augmenting path with tolerance z, then changing flow by + z on edges followed forward by P and by - z on edges followed backward by P produces a feasible flow/' with val(/')= val(/)+ z. Proof: The definition of tolerance ensures that 0 :=:: f'(e) :=:: c(e) for every edge e, so the capacity constraints hold. For the conservation constraints we need only check vertices of P, since flow elsewhere has not changed. The edges of P incident to an internal vertex v of P occur in one of the four ways shown below. In each case, the change to the flow out of v is the same as the change to the flow into v, so the net flow out of v remains 0 in/'. Finally, the net flow into the sink t increases by z.
+ ..
.....+
v
..
v
..
v
The flow on backward edges did not disappear; it was redirected. In effect, the augmentation in Example 4.3.3 cuts the flow path and extends each portion to become a new flow path. We will soon describe an algorithm to find augmenting paths. Meanwhile, we would like a quick way to know when our present flow is a maximum flow. In Example 4.3.3, the central edges seem to form a "bottleneck"; we only have capacity 2 from the left half of the network to the right half. This observation will give us a PROOF that the flow value can be no larger
178
## Chapter 4: Connectivity and Paths
4.3.6. Definition. In a network, a source/sink cut [S, T] consists of the edges from a source set S to a sink set T, where S and T partition the set of nodes, with s E S and t E T. The capacity of the cut [S, T], written cap(S, T), is the total of the capacities on the edges of [S, Tl.
Keep in mind that in a digraph [S, T] denotes the set of edges with tail in
S and head in T. Thus the capacity of a cut [S, T] is completely unaffected by edges from T to S.
Given a cut [S, Tl, every s, t-path uses at least one edge of [S, Tl, so intuition suggests that the value of a feasible fl.ow should be bounded by cap(S, T). To make this precise, we extend the notion of net fl.ow to sets of nodes. Let l+(U) denote the total fl.ow on edges leaving U, and let 1-(U) be the total flow on edges entering U. The net fl.ow out of U is then l+(U) - 1-(U).
4.3.7. Lemma. If U is a set of nodes in a network, then the net fl.ow out of U is the sum of the net flows out of the nodes in U. In particular, if l is a feasible fl.ow and [S, T] is a source/sink cut, then the net fl.ow out of S and net fl.ow into T equal val(f). Proof: The stated claim is that
l+(U) -
1-cu) =
Lveu[f+(v) - 1-(v)l.
We consider the contribution of the fl.ow l(xy) on an edge xy to each side of the formula. If x, y E U, then l (xy) is not counted on the left, but it contributes positively (via l+(x)) and negatively (via 1-(y)) on the right. If x, y fl. U, then l(xy) contributes to neither sum. Ifxy E [U, U], then it contributes positively to each sum. If xy E [U, U], then it contributes negatively to each sum. Summing over all edges yields the equality. When [S, Tl is a source/sink cut and I is a feasible fl.ow, net fl.ow from nodes of S sums to l+(s) - 1-(s), and net fl.ow from nodes ofT sums to l+(t) - 1-(t), which equals -val(f). Hence the ne~ fl.ow across any source/sink cut equals both the net fl.ow out of s and the net fl.ow into t.
4.3.8. Corollary. (Weak duality) If l is a feasible fl.ow and [S, Tl is a source/sink cut, then val(f) ~ cap(S, T). Proof: By the lemma, the value of I equals the net fl.ow out of S. Thus
val(f) = f+(S) - 1-(s)
~ l+(S),
since the fl.ow into S is no less than 0. Since the capacity constraints require l+(S) ~ cap(S, T), we obtain val(f) ~ cap(S, T).
## Section 4.3: Network Flow Problems
179
Among source/sink cuts, one with minimum capacity yields the best bound on the value of a flow. This defines the minimum cut problem. The max flow and min cut problems on a network are dual optimization problems. t Given a flow with value a and a cut with value a, the duality inequality in Corollary 4.3.8 PROVES that the cut is a minimum cut and the flow is a maximum flow. If every instance has solutions with the same value to both the max problem and the min problem ("strong duality"); then a short proof of optimality always exists. This does not hold for all dual pairs of problems (recall matching and covering in general graphs), but it holds for max flow and min cut. The Ford-Fulkerson algorithm seeks an augmenting path to increase the flow value. If it does not find such a path, then it finds a cut with the same value (capacity) as this flow; by Corollary 4.3.8, both are optimal. If no infinite sequence of augmentations is possible, then the iteration leads to equality between the maximum flow value and the minimum cut capacity.
4.3.9. Algorithm. (Ford-Fulkerson labeling algorithm) Input: A feasible flow f in a network. Output: An /-augmenting path or a cut with capacity val(/). Idea: Find the nodes reachable from s by paths with positive tolerance. Reaching t completes an /-augmenting path. During the search, R is the set of nodes labeled Reached, and Sis the subset of R labeled Searched. Initialization: R = {s}, S = 0. Iteration: Choose v E R - S. For each exiting edge vw with f (vw) < c(vw) and w . R, add w to R. For each entering edge uv with f (uv > 0) and u . R, add u to R. Label each vertex added to Ras "reached", and record v as the vertex reaching it. After exploring all edges at v, add. v to S. If the sink t has been reached (put in R), then trace the path reaching t to report an /-augmenting path and terminate. If R = S, then return the cut [S, S] and terminate. Otherwise, iterate. 4.3.10. Example. On the left below is the network of Example 4.3.3 with the flow f. We run the labeling algorithm. First we search from s and find excess capacity to u and x, labeling them reached. Now we have u, v E R - S. There is no excess capacity on uv or xy, so searching from u reaches nothing, and also
tThe precise notion of "dual problem" comes from linear programming. For our purposes, dual problems are a maximization problem and a minimization problem such that a ::::: b whenever a and bare the values of feasible solutions to the max problem and min problem, respectively. See Section 8.1 for further. discussion.
180
## Chapter 4: Connectivity and Paths
searching from x does not reach y. However, there is nonzero flow on vx. Thus we label v from x. Now vis the only element of R - S, and searching from v reaches t. We labeled t from v, v from x, and x from s, so we have found the augmenting path s, x, v, t. The tolerance on this path is 1, so the augmentation increases the flow value by 1. In the new flow /' shown on the right, every edge has unit flow except j'(vx) = 0. When we run the labeling algorithm again, we have excess capacity on su and sx and can label {u, x }, but from these nodes we can label no others. We terminate with R = S = {s, u, x}. The capacity of the resulting cut [S, S] is 2, which equals val(/') and proves that f' is a maximum flow. Repeated use of the labeling algorithm allows us to solve the maximum flow problem and prove the strong duality relationship.
4.3.11. Theorem. (Max-flow Min-cut Theorem-Ford and Fulkerson [1956]) In every network, the maximum value of a feasible flow equals the minimum capacity of a source/sink cut. Proof: In the max-flow problem, the zero flow (f(e) = 0 for all e) is always a feasible flow and gives us a place to start. Given a feasible flow, we apply the labeling algorithm. It iteratively adds vertices to S (each vertex at most once) and terminates with t ER ("breakthrough") or with S = R. In the breakthrough case, we have an /-augmenting path and increase the flow value. We then repeat the labeling algorithm. When the ~:.:lpacities are rational, each augmentation increases the flow by a multiple of 1/a, where a is the least common multiple of the denominators, so after finitely many augmentations the capacity of some cut is reached. The labeling algorithm then terminates with S = R. When terminating this way, we claim that [S, Tl is a source/sink cut with capacity val(/), where T =Sand f is the present flow. It is a cut because s E S and t fl. R = S. Since applying the labeling algorithm to the flow f introduces no node of T into R, no edge from S to T has excess capacity, and no edge from T to S has nonzero flow in/. Hence j+(S) = cap(S, 1/) and f-(S) = 0. Since the net flow out of any set containing the/source but not the sink is val(/), we have proved
val(/)= f+(S) - f-(S) = j+(S) = cap(S, T).
This proofofTheorem 4.3.11 requires rational capacities; otherwise, Algorithm 4.3.9 may yield augmenting paths forever! Ford and Fulkerson provided an example of this with only ten vertices (see Papadimitriou-Steiglitz [1982, p126-128]). Edmonds and Karp [1972] modi:ijed the labeling algorithm to use at most (n 3 - n)/4 augmentations in an n-vertex network and work for all real capacities. As in the bipartite matching problem (Theorem 3.2.22), this is done by searching always for shortest augmenting paths. Faster algorithms are now known; again we cite Ahuja-Magnanti-Orlin [1993] for a thorough discussion.
## Section 4.3: Network Flow Problems
181
INTEGRAL FLOWS
In combinatorial applications, we typically have integer capacities and want a solution in which the flow on each edge is an integer.
4.3.12. Corollary. (lntegrality Theorem) If all capacities in a network are integers, then there is a maximum flow assigning integral flow to each edge. Furthermore, some maximum flow can be partitioned into flows of unit value along paths from source to sink. Proof: In the labeling algorithm of Ford and Fulkerson, the change in flow value when an augmenting path is found is always a flow value or the difference between a flow value and a capacity. When these are integers, the difference is also an integer. Starting with the zero flow, this implies that there is no first time when a noninteger flow appears. The algorithm thus produces a maximum flow with integer flow on each edge. At each internal node, we now match units of entering flow to units of exiting flow. This forms s, t-paths and perhaps cycles. If a cycle arises, then we decrease flow on its edges by 1 to eliminate it without changing the flow value. This leaves val(/) paths from s to t, each corresponding to a unit of flow.
4k;
3 2
The integrality theorem yields paths of unit flow. In applications, we build networks where these units of flow have meaning. The next two remarks show that the Max-flow Min-cut Theorem for networks with integer capacities is almost the same statement as Menger's Theorem for edge-disjoint paths in digraphs.
4.3.13. Remark. Menger from Max-flow Min-cut. When x, y are vertices in a digraph D, we can view Das a network with source x and sink y and capacity 1 on every edge. Capacity 1 ensures that units of flow from x to y correspond to pairwise edge-disjoint x, y-paths in D. Thus a flow of value k yields a set of k such paths. Similarly, every source/sink partition S, T defines a set of edges whose deletion makes y unreachable from x: the set [S, T]. Since every capacity is 1, the size of this set is cap(S, T). The paths anq the edge cut we have obtained might not be optimal, but by the Max-flow Min-cut Theorem we have
A.~(x, y) ~
maxval(f) = mincap(S, T)
~ A.'(x, y),
~ Kb(x, y).
182
## Chapter 4: Connectivity and Paths
4.3.14. Remark. Max-flow Min-cut from Menger. To show that Menger's Theorem implies the Max-flow Min-cut Theorem for rational capacities, we take an arbitrary network and transform it into a digraph where we apply Menger's Theorem. By multiplying all capacities by the least common denominator, we may assume that the capacities are integers. Given a network N with integer capacities, we form a digraph D by splitting each edge of capacity j into j edges with the same endpoints. For N, duality yields maxval{f) .'.S mincap(S, T). This time we want to use Menger's Theorem on D to obtain the reverse inquality, so in contrast to Remark 4.3.13 our desired computation is
maxval{f) 2:.
A.~(s, t)
## = K~(s, t) 2:. mincap(S, T).
A set of A.'(s, t) pairwise edge-disjoints, t-paths in D collapses into a flow of value A. 1 (s, t) in N, since the number of copies of each edge in D equals the capacity of the edge in N. Thus maxval{f) 2:. A.'(s, t). Now, let F be a set of K'(s, t) edges disconnecting t from sin D. If e E F, then the minimality of F implies that D - (F - e) has ans, t-path P through e. If some other copy e' of the edge e = u v is not in F, then P can be rerouted along e' to obtain ans, t-path in D - F. Therefore, F contains all copies or no copies of each multiple edge in D. Hence K (s, t) is the sum of the capacities on a set of edges that disconnects t from s in N. Letting S be the set of vertices reachable from sin D - F, we have cap(S, T) = K (s, t). The minimum cut has at most this capacity, so mincap(S, T) .'.S K'(s, t), and we have proved all the needed inequalities.
1 1
For combinatorial applications, Menger's Theorem may yield simpler proofs than the Max-flow Min-cut Theorem (compare Theorem 4.2.25 with?? ). Nevertheless, our proof ofMenger's Theorem in Section 4.2 is awkward to implement algorithmically. For large-scale computations, network flow and the FordFulkerson labeling algorithm are more appropriate. Indeed, most algorithms that compute connectivity in graph::; and digraphs use network flow methods (Stoer-Wagner [1994] presents a different approach). We present other network models for combinatorial problems. For example, the other local versions of Menger's Theorem can also be obtained directly.
4.3.15. Remark. Other transformations. For each version of Menger's Theorem, we encode the path problem using network flows with integer capacities. To obtain a network model for the problem of internally disjoint paths in a digraph D, we must prevent two units of flow from passing through a vertex. This can be done by replacing each vertex v with two vertices v-, v+ that inherit the entering and exiting edges at v. By adding an edge of unit capacity from v- to v+, we obtain the effect oflimiting flow through v to one unit. By putting very large capacity (essentially infinite) on the edges that were in D, we ensure that a minimum cut will count only edges of the form v-v+ To obtain a network model for the problem of ~dge-disjoint paths in a graph G, we must permit flow to pass either way in an edge. This can be done by
## Section 4.3: Network Flow Problems
183
replacing each edge uv with two directed edges uv and vu. When the network sends unit fl.ow in both directions, in effect the edge is not being used at all. In each case, a fl.ow in the network provides a set of paths, and a minimum cut leads to a separating set of vertices or edges. As in Remark 4.3.13, duality then gives us the desired equality in Menger's Theorem. To model the problem of internally disjoint paths in a graph, we need both of these transformations. Exercises 5-7 request the details of these proofs.
~00~00 00~00
4.3.16. Application. Baseball Elimination Problem (Schwartz [1966]). At some time during the season, we may wonder whether team X can still win the championship. In other wqrds, can winners be assigned for the remaining games so that no team ends with more victories than X? If so, then such an assignment exists with X winning all its remaining games, reaching W wins. We want to know whether winners can be chosen for other games so that no team obtains more than W wins. To test this, we create a network where units of fl.ow correspond to the remaining games. Let Xi, ... , Xn be the other teams. Include nodes x1, ... , Xn for then teams, nodes yi,j for the (~) pairs of teams, and a sources and sink t. Put an edge from s to each team node and an edge from each pair node to t. Each pair node Yi.j is entered by edges from x; and Xj The capacities model the constraints. The capacity on edge Y;,jt is a;,j, the number of remaining games between X; and Xj. Given that X; has won w; games already, the capacity on edge sx; is W - w; to keep X in contention. The capacity on edges x; Yi.j and Xj Yi.j is oo (the number of games x; can win from Xj is constrained by the capacity on Yi,jt).
X;
00
By the integrality theorem, a maximum fl.ow breaks into fl.ow units. Each unit corresponds to one game; the first edge specifies the winner, and the last edge specifies the pair. The network has a flow of value Li,j a;,j if and only if all remaining games can be played with no team exceeding W wins; this is the condition for X remaining in contention. By the Max-fl.ow Min-cut Theorem, there is a fl.ow of value 'Lau if and only if every cut has capacity at least L a;,j. Let S, T be a cut with finite capacity,
184
## Chapter 4: Connectivity and Paths
and let Z = {i: x; E T}. Since c(x;y;,j) = oo, we cannot have x; E Sand Yi,j E T; thus Yi,j E S whenever i or j is not in Z. To minimize capacity, we put Yi,j E T whenever {i, j} ~ Z. Now cap(S, T) = L;EZ(W - w;) + 'Lu.niz ai,j The condition that every cut have capacity at least 'L a;,j becomes
l)w - w;):::: L
iEZ
{i,j};Z
a;,j
## for all Z ~ [n].
Note that this condition is obviously necessary; it states that we need enough leeway in the total wins among teams indexed by Z in order to accommodate winners for all the games among these teams. We have proved TONCAS. Combinatorial applications of network flow usually involve showing that the desired configuration exists if and only if a related network has a large enough flow. As in Application 4.3.16, the Max-flow Min-cut Theorem then yields a necessary and sufficient condition for its existence. Other examples include most of Exercises 5- and also Exercise 13 and Theorems 4.3.17-4.3.18.
## SUPPLIES AND DEMANDS (optional)
Next we consider a more general network model. We allow multiple sources and sinks, and also we associate with each source x; a supply a(x;) and with each sink Yj a demand a(yj). To the capacity constraints for edges and conservation constraints for internal nodes, we add transportation constraints for the sources and sinks. l+(x;) - 1-(x;) ::; a(x;) for each source x; 1-(yj) - l+(yj):::: a(yj) for each sink Yj The resulting configuration is a transportation network. With positive values for the demands, the zero flow is not feasible. We seek a feasible flow satisfying these additional constraints. The "supply/demand" terminology suggests the constraints; we must satisfy the demands at the sinks without exceeding the available supply at any source. This model is appropriate when a company has multiple distribution centers (sources) and retail outlets (sinks). Let X and Y denote the sets of sources and sinks, respectively. Let a(A) = LveA a(v) and a(B) = LveB a(v) denote the total supply or demand at a set A ~ X or B ~ Y. For a set F of edges, let c(F) = LeeF c(e). Given a set T of vertices, the net demand a(YnT)-a(XnT) must be satisfied by flow from the remaining vertices. Hence it is necessary that c([T, T]) be at least this large. Satisfying this for every set Tis also sufficient for a feasible flow (TONCAS).
4.3.17. Theorem. (Gale [1957)) In a transportation network N with sources X and sinks Y, a feasible flow exists if and only if
c([S, T]) :::: a(Y n T) - a(X n T) for every partition of the vertices of N into sets S and T.
## Section 4.3: Network Flow Problems
185
Proof: We have already observed the necessity of the condition. For sufficiency, construct a new network N' by adding a supersource s and a supersink t, with an edge of capacity a(xi) from s to each xi E X and an edge of capacity a(yj) from each Yj E Y to t. The transportation network N has a feasible flow if and only if N' has a flow saturating each edge tot (a flow of value a(Y)). By the Ford-Fulkerson Theorem, we know that N' has a flow of value a(Y) if and only if cap(S Us, TU t) :=:_ a(Y) for each partition S, T of V(N). The cut [SU s, TU t] in N' consists of [S, Tl from N, plus edges from s to T and edges from S tot in N'. Hence cap(S Us, TU t) = c(S, T) c(S, T)
+ a(X n T)
## which is the condition assumed.
For specific instances, the construction of N' is the key point, because we produce a feasible flow in N (when it exists) by running the Ford-Fulkerson algorithm on the network N'. When costs (per unit flow) are attached to the edges, we have the Min-cost Flow Problem, which generalizes the Transportation Problem of Application 3.2.14. Solution algorithms for the Min-cost Flow Problem appear in Ford-Fulkerson [1962] and in Ahuja-Magnanti-Orlin [1993]. We discuss several applications of Gale's condition. A pair of integer lists p =(pi. ... , Pm) and q =(qi, ... , qn) is bigraphic (Exercise 1.4.31) if there is a simple X, Y-bigraph such that the vertices of X have degrees pi, ... , Pm and the vertices of Y have degrees qi, ... , qn. Clearly I: Pi = I: qj is necessary, but this condition is not sufficient. To test whether (p, q) is bigraphic, we create a network in which units of flows will correspond to edges in the desired graph. The result is a bipartite analogue of the Erdos-Gallai condition for graphic sequences (Exercise 3.3.28). 4.3.18. Theorem. (Gale [1957], Ryser [1957]) If p, q are lists of nonnegative integers with Pl :::::; :::_ Pm and qi 2: :::::; qn, then (p, q) is bigraphic if and only ifI:~=l min{pi, k} :::_ I:~=l qj for 1:;:: k:;:: n. Proof: Necessity. Let G be a simple X, Y-bigraph realizing (p, q). Consider the edges incident to a set of k vertices in Y. Because G is simple, each Xi E X is incident to at most k of these edges, and also xi is incident to at most Pi of these edges. Hence I:~=l min{pi, k} is an upper bound on the number of edges incident to any k vertices of Y, such as those with degrees qi, ... , qk. Sufficiency. Given (p, q), create a network N with an edge of capacity 1 from xi to yj for each i, j, and let a(xi) =Pi and a(yj) = qj. Unit capacity prevents multiple edges, and (p, q) is realizable if and only if N has a feasible flow. It suffices to show that the stated condition on p and q implies the condition ofTheorem 4.3.17. For S ~ V(N), let /(S) = {i: xi ES} and J(S) = {j: Yj ES}. For a partition S, T of V(N), we now have a(X n T) = LieI(T) Pi and a(Y n T) = LjeJ(T) qj, and we have c([S, T]) = l/(S)I ll(T)I.
186
## Chapter 4: Connectivity and Paths
Letting k = IJ(T)I, this last quantity becomes c([S, T]) = ll(S)lk = LiE/(S)k '.'.: LiE/(S)min{p;,k}. Also LiE/(T) p; '.'.: LiE/(T) min{p;, k}, and LjeJ(T) qj ~ L,~=l qj. Combining these inequalities, the condition L,~=l min{p;, k} ::: L,~=l qj implies c([S, T]) ::: a(Y n T)-a(XnT). Since this holds for each partition S, T, the network has a feasible flow, which yields the desired bipartite graph. We can extend the maximum flow problem by imposing a nonnegative lower bound on the permitted flow in each edge. The capacity constraint remains as an upper bound, so we require l(e) ~ f(e) ~ u(e) for the flow f(e). We still impose conservation constraints on the internal nodes. Ifwe have a feasible flow, then an easy modification of the Ford-Fulkerson labeling algorithm allows us to find a maximum (or minimum) feasible flow (Exercise 4). The difficulty is finding an initial feasible flow. First we present an application.
4.3.19. Application. Matrix rounding (Bacharach [1966]). We may want to round the entries of a data matrix up or down to integers. We also want to present integers for the row sums and column sums. 'fhe sum of each rounded row or column should be a rounding of the original sum. The resulting integer matrix, ifit exists, is a consistent rounding. We can represent the consistent rounding problem as a feasible flow problem. Establish vertices x1, ... , Xn for the rows and vertices Y1, ... ; Yn for the columns of the matrix. Add a sources and a sink t. Add edges sx;, x;yj. yjt for all values ofi and j. Ifthe matrix has entries a;,j with row-sums ri, ... , rn and column-sums s1, ... , Sn, set
l(sx;) = Lr;J u(sx;) l(x;yj) = u(x;yj)
La;,jj
zcyjt) = u(yjt) =
LcjJ
fr; l
Ia;.1 l
ICj l
We test for a feasible flow by transforming again to an ordinary maximum flow problem. With these two transformations, we can use network flow to test for the existence of a consistent rounding.
X3
Ya
## Section 4.3: Network Flow Problems
187
4.3.20. Solution. Circulations and flows with lower bounds. In a maximum flow problem with upper and lower bounds on edge capacities, the zero flow is not feasible, so the Ford-Fulkerson labeling algorithm has no place to start. We must first obtain a feasible flow, after which an easy modification of the labeling algorithm applies (Exercise 4). The first step is to add an edge of infinite capacity from the sink to the source. The resulting network has a feasible flow with conservation at every node (called a circulation) if and only if the original network has a feasible flow. In a circulation problem, there is no source or sink. . Next, we convert a feasible circulation problem C into a maximum flow problem N by introducing supplies or demands at the nodes and adding a source and sink to satisfy the supplies and demands. Given the flow constraints l(e) :::; f(e) :::; u(e), let c(e) = u(e) - l(e) for each edge e. For each vertex v, let
z-(v) z+(v) b(v)
= = =
## LeE[V(C)-v.v] l(e), LeE[v.V(C)-v] l(e),
z-(v) _ z+(v).
Since each l(uv) contributes to z+(u) and z-(v), we have 'L b(v) = 0. A feasible circulation f must satisfy the flow constraints at each edge and satisfy j+(v) f-(v) = 0 at each node. Letting f'(e) = f(e) - l(e), we find that f is a feasible circulation in C if and only if f' satisfies 0 :::; f'(e) :::; c(e) on each edge and j'+(v) - f'-(v) = b(v) at each vertex. This transforms the feasible circulation problem into a flow problem with supplies and demands. If b(v) 2:: 0, then v supplies flow lb(v)I to the network; otherwise v demands lb(v)I. To restore conservation constraints, we add a sources with an edge of capacity b(v) to each v with b(v) 2:: 0, and we add a sink t with an edge of capacity -b(v) from each v with b(v) < 0. This completes the construction of N. Let a be the total capacity on the edges leavings; since 'Lb(v) = 0, the edges entering t also have total capacity a. Now Chas a feasible circulation f if and only if N has a flow of value a (saturating all edges out of s or into t).
4.3.21. Corollary. A network D with conservation constraints at every node has a feasible circulation if and only if LeE[S.'SJ l(e) :::; LeE[S.sJ u(e) for every S V(D). . Proof: We ~an stop before the fast step in the discussion of Solution 4.3.20 and interpret our problem with supplies and demands in the model of Theorem 4.3.17. Since 'Lb( v) = 0, the only way to satisfy all the demands is to use up all the supply. Hence there is a circulation if and only if the supply/demand problem with supplies a(v) = b(v) for {v E V(D) : b(v) 2:: O} and demands 8(v) = -b(v) for {v E V(D): b(v) < O} has a solution. Theorem 4.3.17 characterizes when this problem has a solution. Translated back into the lower and upper bounds on flow in the original problem (Exercise 22), the criterion of Theorem 4.3.17 becomes LeE[S,SJ l(e) :'S LeE[S.sJ u(e) for every S s; V(D).
188
## Chapter 4: Connectivity and Paths
EXERCISES
4.3.1. (-) In the network below, list all integer-valued feasible flows and select a flow of maximum value (this illustrates the advantage of duality over exhaustive search). Prove that this flow is a maximum flow by exhibiting a cut with the same value. Determine the number of source/sink cuts. (Comment: There is a nonzero fl.ow with value 0.)
1
4,3.2. (-) In the network below, find a maximum flow from s to t. Prove that your answer is optimal by using the dual problem, and ex:plain why this proves optimality.
14
4
12
4.3.3. (-) A kitchen sink draws water from two tanks according to the network of pipes with capacities per unit time shown below. Find the maximum fl.ow. Prove that your answer is optimal by using the dual problem, and explain why this proves optimality.
## Tank2 Sink Tankl
4.3.4:. Let N be a network with edge capacity and node conservation constraints plus lower bound constraints l(e) on the :flow in edges, meaning that f (e) ::::; l(e) is required. If an initial feasible flow is given, how can the Ford-Fulkerson labeling algorithm be modified to search for a maximum feasible flow in this network?
## Section 4.3: Network Flow Problems
l89
4.3.5. (!)Use network flows to prove Menger's Theorem for internally-disjoint paths in digraphs: K(x, y) = A.(x, y) when xy is not an edge. (Hint: Use the first transformation suggested in Remark 4.3.15.) 4.3.6. (!) Use network flows to prove Menger's Theorem for edge-disjoint paths in graphs: K'(~. y) = A.'(x, y). (Hint: Use the second transformation suggested in Remark 4.3.15.) 4.3.7. (!) Use network flows to prove Menger's Theorem for nonadjacent vertices in graphs: K(x, y)::::;: A.(x, y). (Hint: Use both transformations suggested in Remark 4.3.15.) 4.3.8. Let .G be a directed graph with x, y E V(G). Suppose that capacities are specified not on the edges of G, but rather on the vertices (other than x, y ); for each vertex there is a fixed limit on the total flow through it. There is no restriction on flows in edges. Show how to use ordinary network flow theory to determine the maximum value of a feasible flow from x toy in the vertex-capacitated graph G. 4.3.9. Use network flows to prove that a graph G is connected if and only if for every partition of V(G) into two nonempty sets S, T, there is an edge with one endpoint in Sand one endpoint in T. (Comment: Chapter 1 contains an easy direct proof of the conclusion, so this is an example of"using a sledgehammer to squash a bug''.) 4.3.10. (!)Use network flows to prove the Konig-Egervary Theorem (a'(G) = f3(G) ifG is bipartite!. 4.3.11. Show that the Augmenting Path Algorithm for bipartite graphs (Algorithm 3.2.1) is a special case of the Ford-Fulkerson Labeling Algorithm. 4.3.12. Let [S, SJ and [T, Tl be source/sink cuts in a network N. a) Prove that cap(S UT, Su T) + cap(S n T, Sn T) ::: cap([S, S]) + cap(T, T). (Hint: Draw a picture and consider contributions from various types of edges.) b) Suppose that [S, SJ and [T, Tl are minimum cuts. Conclude from part (a) that [SU T, SU Tl and [Sn T, Sn Tl are also minimum cuts. Conclude also that no edge between S - T and T - S has positive capacity. 4.3.13. (!).Several companies send representatives to a conference; the ith company sends m; representatives. The organizers of the conference conduct simultaneous networking groups; the jth group can accommodate up to nj participants. The organizers want to schedule all the participants into groups, but the participants from the same company must be in different groups. The groups need not all be filled. a) Show how to use network flows to test whether the constraints can be satisfied. b) Let p be the number of companies, and let q be the number of groups, indexed so that m 1 ?:: ?::mp and n 1 ::: ::: nq. Prove that there exists an assignm~nt of participants to groups that satisfies all the constraints if and only if, for all 0 ::: k ::: p and 0::: l ::: q, it holds that k(q - l) + L~=l nj ?:: L~=l m;. 4.3.14. In a large university with k academic departments, we must appoint an important committee. One professor will be chosen from each department. Some professors have joint appointments in two or more departments, but each must be the designated representative of at most one department. We must use equally many assistant professors, associate professors, and full professors among the chosen representatives (assume that k is divisible by 3). How can the committee be found? (Hint: Build a network in which units of flow correspond to professors chosen for the committee and capacities enforce the various constraints. Explain how to use the network to test whether such a committee exists and find it if it does.) (Hall [1956])
190
## Chapter 4: Connectivity and Paths
4.3.15. Let G be a weighted graph. Let the value of a spanning tree be the minimum weight of its edges. Let the cap from a edge cut [S, S] be the maximum weight of its edges. Prove that the maximum value of a spanning tree of G equals the minimum cap of an edge cut in G. (Ahuja-Magnanti-Orlin [1993, p538]) 4.3.16. (+)Let x be a vertex of maximum outdegree in a tournament T. Prove that T has a spanning directed tree rooted at x such that every vertex has distance at most 2 from x and every vertex other than x has outdegree at most 2. (Hint: Create a network to model the desired paths to the non-successors of x, and show that every cut has enough capacity. Comment: This strengthens Proposition 1.4.30 about kings in tournaments; no vertex need be an intermediate vertex for more than two others.) (Lu [1996])
4.3.17. (*-) Use the Gale-Ryser Theorem (Theorem 4.3.18) to determine whether there is a simple bipartite graph in which the vertices in one partite set have degrees (5, 4A, 2, 1) and the vertices in the other partite set also have degrees (5, 4, 4, 2, 1). 4.3.18. (*-)Given list r = (r 1 , . , r 11 ) ands= (si. ... , s.), obtain necessary and sufficient conditions for the existence of a digraph D with vertices v1 , . , v. such that each ordered pair occurs at most once as an edge and d+(v;) = r; and d-(v;) = s; for all i. 4.3.19. (*-) Find a consistent rounding of the data in the matrix below. Is it unique? (Every entry must be 0 or 1.)
( .55 .55 .6 .65 .65
.6 .6)
.7 .7
4.3.20. (*) Prove that every two-by-two matrix can be consistently rounded. 4.3.21. (*) Suppose that every entry in an n-by-n matrix is strictly between l/n and l/(n - 1). Describe all consistent roundings. 4.3.22. (*) Complete the details of proving Corollary 4.3.21, proving the necessary and sufficient condition for a circulation in a network with lower and upper bounds. 4.3.23. (*!) A (k +/)-regular graph G is (k, /)-orientable if it can be oriented so that each indegree is k or l. a) Prove that G is (k, /)-orientable if and only if there is a partition X, Y of V (G) such that for every S ~ V(G),
(k -/)(IX
## n SI- IY n SI):::: l[S, s11.
(Hint: Use Theorem 4.3.17.) b) Conclude that if G is (k, /)-orientable and k > l, then G 'is also (k - 1, l + 1)orientable. (Bondy-Murty [1976, p210-211])
Chapter5
Coloring of Graphs
5.1. Vertex Coloring and Upper Bounds
The committee-scheduling example (Example 1.1.11) used graph coloring to model avoidance of conflicts. Similarly, in a university we want to assign time slots for final examinations so that two courses with a common student have different slots. The number of slots needed is the chromatic number of the graph in which two courses are adjacent if they have a common student. Coloring the regions of a map with different colors on regions with common boundaries is another example; we return to it in Chapter 6. The map on the left below has five regions, and four colors suffice. The graph on the right models the "common boundary" relation and the corresponding coloring. Labeling of vertices is our context for coloring problems.
## DEFINITIONS AND EXAMPLES
Graph coloring takes its name from the map-coloring application. We assign labels to vertices. When the numerical value of the labels is unimportant, we call them "colors" to indicate that they may be elements of any set.
5.1.1. Definition. A k-coloring of a graph G is a labeling f: V ( G) ~ S, where ISi = k (often we use S = [k]). The labels are colors; the vertices of one color form a color class. A k-coloring is proper if adjacent vertices have different labels. A graph is k-colorable if it has a proper k-coloring. The chromatic number x (G) is the least k such that G is k-colorable.
191
192
## Chapter 5: Coloring of Graphs
5.1.2. Remark. In a proper coloring, each color class is an independent set, so G is k-colorable if and only if V(G) is the union of k independent sets. Thus "k-colorable" and "k-partite" have the same meaning. (The usage of the two terms is slightly different. Often "k-partite" is a structural hypothesis, while "k-colorable" is the result of an optimization problem.) Graphs with loops are uncolorable; we cannot make the color of a vertex different from itself. Therefore, in this chapter all graphs are loopless. Also, multiple edges are irrelevant; extra copies don't affect colorings. Thus we usually think in terms of simple graphs when dis~ussing colorings, and we will name edges by their endpoints. Most of the statements made without restriction to simple graphs remain valid when multiple edges are allowed. 5.1.3. Example. Since a graph is 2-colorable if and only if it is bipartite, C5 and the Petersen graph have chromatic number at least 3. Since they are 3colorable, as shown below, they have chromatic number exactly 3.
a
c b b
5.1.4. Definition. A graph G is k-chromatic if x(G) = k. A proper k-coloring of a k-chromatic graph is an optimal coloring. If x (H) < x (G) = k for every proper subgraph H of G, then G is color-critical or k-critical. 5.1.5. Example. k-critical graphs for small k. Properly coloring a graph needs at least two colors if and only if the graph has an edge. Thus K 2 is the only 2-critical graph (similarly, K1 is the only 1-critical graph). Since 2-colorable is the same as bipartite, the characterization of bipartite graphs implies that the 3-critical graphs are the odd cycles. We can test 2-colorability of a graph G by computing distances from a vertex x (in each component). Let X = {u e V(G): d(u, x) is even}, and let Y = {u e V(G): d(u, x) is odd}. The graph G is bipartite if and only if X, Y is a bipartition, meaning that G[X] and G[Y] are independent sets. No good characterization of 4-critical graphs or test for 3-colorability is known. Appendix B discusses the computational ramifications. I 5.1.6. Definition. The clique number of a graph G, written w(G), is the maximum size of a set of pairwise adjacent vertices (clique) in G.
We have used a(G) for the independence number of G; the usage of w(G) is analogous. The letters a and w are the first and last in the Greek alphabet.
## Section 5.1: Vertex Coloring and Upper Bounds
193
This is consistent with viewing independent sets and cliques as the beginning and end of the "evolution" of a graph (see Section 8.5).
## x (G) :=:: w ( G) and x (G)
:=:: ~.
Proof: The first bound holds because vertices of a clique require distinct colors. The second bound holds because each color class is an independent set and thus has at most a(G) vertices.
Both bounds in Proposition 5.1.7 are tight when G is a complete graph.
5.1.8. Example. x(G) may exceed w(G). For r :=:: 2, let G = C2r+1 v Ks (the join of C2r+1 and Ks-see Definition 3.3.6). Since C2,+1 has no triangle, w(G) = s+2. Properly coloring the induced cycle requires at least three colors. The sclique needs s colors. Since every vertex of the induced cycle is adjacent to every vertex of the clique, these s colors must differ from the first three, and x(G) :=:: s + 3. We conclude that x(G) > w(G).
Exercises 23-30 discuss the chromatic number for special families of graphs. We can also ask how it behaves under graph operations. For the disjoint union, x(G + H) = max{x(G), x(H)}. For the join, x(G v H) = x(G) + x(H). Next we introduce another combining operation.
5.1.9. Definition. The cartesian product of G and H, written G o H, is the graph with vertex set V(G) x V(H) specified by putting (u, v) adjacent to (u', v') ifand only if (1) u = u' and vv' e E(H), or (2) v = v' and uu' e E(G). 5.1.10. Example. The cartesian product operation is symmetric; GoH :::::= HoG. Below we show Cs o C4. The hypercube is another familiar example: Qk = Qk-1 D K2 when k :=:: 1. The m-by-n grid is the cartesian proP,uct Pm D Pn. In general, G o H decomposes into copies of H for each vertex of G and copies of G for each vertex of H (Exercise 10). We use o instead of x to avoid confusion with other product operations, reserving x for the cartesian product of vertex sets. The symbol o, due to ROdl, evokes the identity K2 o K2 = C4
a
(x, d)
GoH
( z. d)
194
## Chapter 5: Coloring of Graphs
5.1.11. Proposition. (Vizing [1963], Aberth [1964]) x(GoH) =max:{x(G),x(H)}. Proof: The cartesian product G o H contains copies of G and H as subgraphs, so x(GoH):::: max{x(G), x(H)}. Let k = max{x(G), x(H)}. To prove the upper bound, we produce a proper k-coloring of Go H using optimal colorings of G and H. Let g be a proper x (G)coloring of G, and let h be a proper x (H)-coloring of H. Define a coloring f of Go H by letting f (u, v) be the congruence class of g(u) + h(v) modulo k. Thus f assigns colors to V (Go H) from a set of size k. We claim that f properly colors Go H. If (u, v) and (u', v') are adjacent in Go H, then g(u) + h(v) and g(u') + h(v') agree in one summand and differ by between 1 and k in the other. Since the difference of the two sums is between 1 and k, they lie in different congruence classes modulo k.
3
3
1 1
2 3
3 1
2
G
1
2 GoH
3
The cartesian product allows us to compute chromatic numbers by computing independence numbers, because a graph G is m-colorable if and only if the cartesian product Go Km has an independent set of size n(G) (Exercise 31).
UPPER BOUNDS
Most upper bounds on the chromatic number come from algorithms that produce colorings. For example, assigning distinct colors to the vertices yields x(G) s n(G). This bound is best possible, since x(Kn) = n, but it holds with equality only for complete graphs. We can improve a "best-possible" bound by obtaining another bound that is always at least as good. For example, x (G) _::: n(G) uses nothing about the structure of G; we can do better by coloring the vertices in some order and always using the "least available" color.
5.1.12. Algorithm. (Greedy coloring) The greedy coloring relative to a vertex ordering v1 , . , Vn of V ( G) is obtained by coloring vertices in the order vi. ... , Vn, assigning to vi the smallest-indexed color not already used on its lower-indexed neighbors. 5.1.13. Proposition. x(G) S ~(G) + 1. Proof: In a vertex ordering, each vertex has at most t.(G) earlier neighbors, so the greedy coloring cannot be forced to use more than t. (G) + 1 colors. This proves constructively that x (G) s t. (G) + 1.
## Section 5.1: Vertex Coloring and Upper Bounds
195
The bound ~(G) + 1 is the worst upper bound that greedy coloring could produce (although optimal for cliques and odd cycles). Choosing the vertex ordering carefully yields improvements. We can avoid the trouble caused by vertices of high degree by putting them at the beginning, where they won't have 'JUany earlier neighbors (see Exercise 36 for a better ordering).
5.1.14. Proposition. (Welsh-Powell [1967]) If a graph G has degree sequence di 2::. 2::. dn, then x(G) :'.:: 1 +max; min{d;, i - l}. Proof: We apply greedy coloring to the vertices in nonincreasing order of degree. When we color the ith vertex v;, it has at most min{d;, i - 1} earlier neighbors, so at most this many colors appear on its earlier neighbors. Hence the color we assign to v; is at most 1 + min {d;, i - 1}. This holds for each vertex, so we maximize over i to obtain the upper bound on the maximum color used.
The bound in Proposition 5.1.14 is always at most 1 + ~(G), so this is always at least as good as Proposition 5.1.13. It gives the optimal upper bound in Example 5.1.8, while 1 + ~(G) does not. In Proposition 5.1.14, we use greedy coloring with a well-chosen ordering. In fact, every graph G has some vertex ordering for which the greedy algorithm uses only x(G) colors (Exercise 33). Usually it-is hard to find such an ordering. Our next example introduces a class of graphs where such an ordering is easy to find. The ordering produces a coloring that achieves equality in the bound x(G) 2::. w(G).
5.1.15. Example. Register allocation and interval graphs. A computer program stores the values of its variables in memory. For arithmetic computations, the values must be entered in easily accessed locations called registers. Registers are expensive, so we want to use them efficiently. If two variables are never used simultaneously, then we can allocate them to the same register. For each variable, we compute the first and last time when it is used. A variable is active during the interval between these times. We define a graph whose vertices are the variables. Two vertices are adjacent if they are active at a common time. The number of registers needed is the chromatic number of this graph. The time when a variable is active is an interval, so we we obtain a special type of representation for the graph. An interval represeptation of a graph is a family of intervals assigned to the vertices so that vertices are adjacent if and only if the corresponding intervals intersect. A graph having such a representation is an interval graph. For the vertex ordering a, b, c, d, e, f, g, h of the interval graph below, greedy coloring assigns 1, 2, 1, 3, 2, 1, 2, 3, respectively, which is optimal. Greedy colorings relative to orderings starting a, d, ... use four colors.
c b---- e--a d
g_ __
f----_ __ h_
196
## Chapter 5: Coloring of Graphs
5.1.16. Proposition. If G is an interval graph, then x(G) = w(G). Proof: Order the vertices according to the left endpoints of the intervals in an interval representation. Apply greedy coloring, and suppose that x receives k, the maximum color assigned. Since x does not receive a smaller color, the left endpoint a of its interval belongs also to intervals that already have colors 1 through k - 1. These intervals all share the point a, so we have a k-clique consisting of x and neighbors of x with colors 1 through k - 1. Hence w(G) ::: k ::: x (G). Since x (G) ::: w( G) always, this coloring is optimal. 5.1.17. * Remark. The greedy coloring algorithm runs rapidly. It is "on-line" in the sense that it produces a proper coloring even if it sees only one new vertex at each step and must color it with no option to change earlier colors. For a random vertex ordering in a random graph (see Section 8.5), greedy coloring almost always uses only about twice as many colors as the minimum, although with a bad ordering it may use many colors on a tree (Exercise 34).
We began with greedy coloring to underscore the constructive aspect of upper bounds on chromatic number. Other bounds follow from the properties of k-critical graphs but don't produce proper colorings: every k-chromatic graph has a k-critical subgraph, but we have no good algorithm for finding one. We derive the next bound ucing- critical subgraphs; it can also be proved using greedy coloring (Exercise 36).
5.1.18. Lemma. If His a k-critical graph, then o(H)::: k - 1. Proof: Let x be a vertex of H. Because H is k-critical, Ii - x is k - 1-colorable. If dH(x) < k - 1, then the k - 1 colors used on H -x do not all appear on N(x). We can assign x a color not used on N (x) to obtain a proper k - 1-coloring of H. This contradicts our hypothesis that x(H) = k. We conclude that dH(x) ::: k - 1 (for each x E V(H)). 5.1.19. Theorem. (Szekeres-Wilf [1968]) If G is a graph, then x (G) :::; 1 + maxHc;G o(H). Proof: Let k = x(G), and let H' beak-critical subgraph of G. Lemma 5.1.18 yields x(G) - 1 = x(H') - 1:::; o(H'):::; maxHc;G o(H).
5.1.20. Example. If G is bipartite, then the orientation of G that directs every edge from one partite set to the other has no path (in the directed sense) of length more than 1. The next theorem thus implies that x (G) :::; 2. Every orientation of an odd cycle must somewhere have two consecutive edges in the same direction. Thus each orientation has a path oflength at least two, and the theorem confirms that an odd cycle is 3-chromatic. 5.1.21. Theorem. Gallai-Roy-Vitaver Theorem (Gallai [1968], Roy [1967], Vitaver [1962]) If Dis an orientation of G with longest path length l (D), then x(G) :=: 1 + l(D). Furthermore, equality holds for some orientation of G.
## Section 5.1: Vertex Coloring and Upper Bounds
197
Proof: Let D be an orientation of G. Let D' be a maximal subdigraph of D that contains no cycle (in the example below, uv is the only edge of D not in D'). Note that D' includes all vertices of G. Color V(G) by letting f(v) be 1 plus the length of the longest path in D' that ends at v. Let P be a path in D', and let u be the first vertex of P. Every path in D' ending at u has no other vertex on P, since D' is acyclic. Therefore, each path ending at u (including the longest such path) can be lengthened along P. This implies that f strictly increases along each path in D'. The coloring fuses colors 1 through l+l(D') on V(D') (which is also V(G)). We claim that f is a proper coloring of G. For each uv E E(D), there is a path in D' between its endpoints (since uv is an edge of D' or its addition to D' creates a cycle). This implies that f (u) =f:. f (v), since f increases along paths of D'. To prove the second statement, we construct an orientation D* such that l(D*) ::S x(G) - 1. Let f be an optimal coloring of G. For each edge uv in G, orient it from u to v in D* ifand only if f(u) < f (v). Since f is a proper coloring, this defines an orientation. Since the labels used by f increase along each path in D*, and there are only x(G) labels in/, we have l(D*) ::s x(G) -1.
2
1 1
4
5
D' and D
D*
BROOKS' THEOREM
The bound x(G) ::s 1 + .!l(G) holds with equality for complete graphs and odd cycles. By choosing the vertex ordering more carefully, we can show that these are essentially the only such graphs. This implies, for example, that the Petersen graph is 3-colorable, without finding an explicit coloring. To avoid unimportant complications, we phrase the statement only for connected graphs. It extends tO all graphs because the chroi:uatic number of a graph is the maximum chromatic number ofits components. Many proofs are known; we present a modification of the proof by Lovasz [1975]. 5.1.22. Theorem. (Brooks [1941J) If G is a connected graph other than a complete graph or an odd cycle, then x(G) ::s .!l(G). Proof: Let G be a connected graph, and let k = .!l(G). We may assume that k ::::. 3, since G is a complete graph wh,en k ::s 1, and G is an odd cycle or is bipartite when k = 2, in which case the bound holds.
198
## Chapter 5: Coloring of Graphs
Our aim is to order the vertices so that each has at most k-1 lowerindexed neighbors; greedy coloring for such an ordering yields the bound. When G is not k-regular, we can choose a vertex of degree less thank as Vn. Since G is connected, we can grow a spanning tree of G from Vn, assigning indices in decreasing order as we reach vertices. Each vertex other than Vn in the resulting ordering v1 , ... , Vn has a higher-indexed neighbor along the path to Vn. in the tree. Hence each vertex has at most k - 1 lower-indexed neighbors, and the greedy coloring uses at most k colors.
V;
## In the remaining case, G is k-regular. Suppose first that G has a cut-vertex
x, and let G' be a subgraph consisting of a component of G - x together with its edges to x. The degree of x in G' is less than k, so the method above provides
a proper k-coloring of G'. By permuting the names of colors in the subgraphs resulting in this way from components of G - x, we can make the colorings agree on x to complete a proper k-coloring of G. We may thus assume that G is 2-connected. In every vertex ordering, the last vertex has k earlier neighbors. The greedy coloring idea may still work if we arrange that two neighbors of Vn get the same color. In particular, suppose that some vertex Vn has neighbors vi. v2 such that v1 ~ v2 and G - {v 1, v2} is connected. In this case, we index the vertices of a spanning tree of G - {v1, v2} using 3, ... , n such that labels increase along paths to the root Vn. As before, each vertex before Vn has at most k - 1 lower indexed neighbors. The greedy coloring also uses at most k - 1 colors on neighbors of Vn, since v1 and v2 receive the same color. Hence it suffices to show that every 2-connected k-regular graph with k ~ 3 has such a triple vi. v2, Vn Choose a vertex x. If K(G - x) ~ 2, let v1 be x and let v2 be a vertex with distance 2 from x. Such a vertex v2 exists because G is regular and is not a complete graph; let Vn be a common neighbor of v 1 and v2 If K(G - x) = 1, let Vn = x. Since G has no cut-vertex, x has a neighbor in every leaf block of G - x . Neighbors vi. v2 of x in two such blocks are nonadjacent. Also, G - {x, vi. v2 } is connected, since blocks have no cut-vertices. Since k ~ 3, vertex x has another neighbor, and G - {v1 , v2 } is connected.
Vn
=x
5.1.23.* Remark. The bound x(G) :=::: D.(G) can be improved when G has no large clique (Exercise 50). Brooks' Theorem implies that the complete graphs and odd cycles are the only k - 1-regular k-critical graphs (Exercise 47). Gallai
## Section 5.1: Vertex Coloring and Upper Bounds
199
[1963b] strengthened this by proving that in the subgraph of a k-critical graph induced by the vertices of degree k - 1, every block is a clique or an odd cycle. Brooks' Theorem states that x(G) ~ ~(G) whenever 3 ~ w(G) ~ ~(G). Borodin and Kostochka [1977] conjectured that w(G) < ~(G) implies x(G) < ~(G) if ~(G) 2: 9 (examples show that the condition ~(G) 2: 9 is needed). Reed [1999] proved that this is true when ~(G)::: 1014 Reed [1998] also conjectured that the chromatic number is bounded by the average of the trivial upper and lower bounds; that is, x(G) ~ j 8<GH~+w(G)l Because the idea of partitioning to satisfy constraints is so fundamental, there are many, many variations and generalizations of graph coloring. In Chapter 7 we consider coloring the edges of a graph. Sticking to vertices, we could allow color classes to induce subgraphs other than independent sets ("generalized coloring"-Exercises 49-53). We could restrict the colors allowed to be used on each vertex ("list coloring''-Section 8:4). We could ask questions involving numerical values of the colors (Exercise 54). We have only touched the tip of the iceberg on coloring problems.
EXERCISES
5.1.1. (-) Compute the clique number, the independence number, and the chromatic number of the graph below. Does either bound in Proposition 5.1. 7 prove optimality for some proper coloring? Is the graph color-critical?
5.1.2. (-) Prove that the chromatic number of a graph equals the maximum of the chromatic numbers of its components. 5.1.3. (-) Let G 1 , . , Gk be the blocks of a graph G. Prove that 5.1.4. (-)Exhibit a graph G with a vertex v so that x (G-v) < 5.1.5. (-) Given graphs G and H, prove that x(G x(G v H) = x(G) + x(H).
x (G) =
max;
x (G; ).
+ H)
## = max{x(G), x(H)} and that
5.1.6. (-)Suppose that x(G) = w(G) + 1, as in Example 5.1.8. Let H1 = G and Hk = Hk-l v G fork > 1. Prove that x (H)k == w(H)k + k. 5.1.7. (-) Construct a graph G that is neither a clique nor an odd cycle but has a vertex ordering relative to which greedy coloring uses ~(G) + 1 colors. 5.1.8. (-)Prove that maxH<;;a8(H) _::: ~(G) to explain why Theorem 5.1.19 is better than Proposition 5.1.13. Determine all graphs G such that maxH<;;G 8(H) = ~(G). 5.1.9. (-)Draw the graph Kl.3 o P3 and exhibit an optimal coloring of it. Draw C5 oC5 and find a proper 3-coloring of it with color classes of sizes 9, 8, 8.
200
## Chapter 5: Coloring of Graphs
5.1.10. (-)Prove that GD H decomposes into n(G) copies of Hand n(H) copies of G. 5.1.11. (-) Prove that each graph below is isomorphic to Cs o Cs.
5.1.12. (-) Prove or disprove: Every k-chromatic graph G has a proper k-coloring in which some color class has a(G) vertices. 5.1.13. (-) Prove or disprove: If G 5.1.14.
## 5.1.15. a(G) is the average of the vertex degrees in G.
= FU H, then x (G) :S x (F) + x (H). (-)Prove or disprove: For every graph G, x(G) :S n(G) - a(G) + 1. (-) Prove or disprove: If G is a connected graph, then x (G) :s 1 + a(G), where
5.1.16. (-)Use Theorem 5.1.21 to prove that every tournament has a spanning path. (Redei [1934]) 5.1.17. (-)Use Lemma 5.1.18 to prove that x(G) :s 4 for the graph G below.
5.1.18. (-)Determine the number of colors needed to label V(K.) such that each color class induces a subgraph with maximum degree at most k. 5.1.19. (-) Find the error in the false argument below for Brooks' Theorem (Theorem 5.1.22). "We use induction on n(G); the statement holds when n(G) = 1. For the induction step, suppose that G is not a complete graph or an odd cycle. Since K(G) :S 8(G), the graph G has a separating set S of size at most ~(G). Let Gi. ... , Gm be the components of G - S, and let H; = G[V(G;) U SJ. By the induction hypothesis, each H; is MG)colorable. Permute the names of the colors used on these subgraphs to agree on S. This yields a proper ~(G)-coloring of G ."
5.1.20. (!) Let G be a graph whose odd cycles are pairwise intersecting, meaning that every two odd cycles in ahave a common vertex. Prove that x(G) :s 5. 5.1.21. Suppose that every edge of a graph G appears in at most one cycle. Prove that every block of G is an edge, a cycle, or an isolated vertex. Use this to prove that x (G) :S 3. 5.1.22. (!) Given a set of lines in the plane with no three meeting at a point, form a graph G whose vertices are the intersections of the lines, with two vertices adjacent if they appear consecutively on one of the lines. Prove that x(G) :S 3. (Hint: This
## Section 5.1: Vertex Coloring and Upper Bounds
201
can be solved by using the Szekeres-WilfTheorem or by using greedy colofr ; with an appropriate vertex ordering. Comment: The conclusion may fail when three lines are allowed to share a point.) (H. Sachs)
5.1.23. (!)Place n points on a circle, where n :=::: k(k+ 1). Let Gn,k be the 2k-regular graph obtained by joining each point to the k nearest points in each direction on the circle. For example, Gn,1 = Cn, and G7,2 appears below. Prove that x (Gn,k) = k + 1ifk+1 divides n and x(Gn,d = k + 2ifk+1 does not divide n. Prove that the lower bound on n cannot be weakened, by proving that x (Gk(k+lJ-u) .,. k + 2 if k :'.:'.: 2.
5.1.24. (+) Let G be any 20-regular graph with 360 vertices formed in the following way. The vertices are evenly-spaced around a circle. Vertices separated by 1 or 2 degrees are nonadjacent. Vertices separated by 3, 4, 5 or 6 degrees are adjacent. No information is given about other adjacencies (except that G is 20-regular). Prove that x(G) ~ 19. (Hint: Color successive vertices in order around the circle.) (Pritikin) 5.1.25. (+)Let G be the unit-distance graph in the plane; V(G) = IR2 , and two points are adjacent if their Euclidean distance is 1 (this is an infinite graph). Prove that 4 ~ x(G) ~ 7. (Hint: For the upper bound, present an explicit coloring by regions, paying attention to the boundaries.) (Hadwiger [1945, 1961], Moser-Moser [1961]) 5.1.26. Given finite sets Si, ... , Sm, let U = S1 x x Sm. Define a graph G with vertex set U by putting u *+ v if and only if u and v differ in every coordinate. Determine x (G). 5.1.27. Let H be the complement of the graph in Exercise 5.1.26. Determine x(H). 5.1.28. Consider a traffic signal controlled by two switches, each of which can be set in
n positions. For each setting of the switches, the traffic signal shows one of its n possible
colors. Whenever the setting of both switches changes, the color changes. Prove that the color ;hown is determined by the position of one of the switches. Interpret this in terms of the chromatiC numb~r of some graph. (Greenwell-Lovasz [197 4]) 5.1.29. For the graph G below, compute x(G) and find a )((G)-critical subgraph.
202
## Chapter 5: Coloring of Graphs
5.1.30. (+)Let S = (l;1) denote the collection of2-sets of then-element set [n]. Define the graph G. by V(G.) =Sand E(G.) = {(ij, jk): 1::: i < j < k::: n} (disjoint pairs, for example, are nonadjacent). Prove that x(G.) = flgnl. (Hint: Prove that G. is rcolorable if and only if [r] has at least n distinct subsets. Comment: Gn is called the shift graph of K. ) (attributed to A. Hajnal) 5.1.31. (!) Prove that a graph G is m-colorable if and only if a(G D Km) [1973, p379-80])
~
n(G). (Berge
5.1.32. (!)Prove that a graph G is 2k-colorable if and only if G is the union of k bipartite graphs. (Hint: This generalizes Theorem 1.2.23.) 5.1.33. (!) Prove that every graph G has a vertex ordering relative to which greedy coloring uses x (G) colors. 5.1.34. (!) For all k E N, construct a tree Tk with maximum degree k and an ordering a of V (Tk) such that greedy coloring relative to the ordering a uses k + 1 colors. (Hint: Use induction and construct the tree and ordering simultaneously. Comment: This result shows that the performance ratio of greedy coloring to optimal coloring can be as bad as (ti.(G) + 1)/2.) (Bean [1976]) 5.1.35. Let G be a graph having no induced subgraph isomorphic to P4 Prove that for every vertex ordering, greedy coloring produces an optimal coloring of G. (Hint: Suppose that the algorithm uses k colors for the ordering v1 , , v., and let i be the smallest integer such that G has a clique consisting of vertices assigned colors i through kin this coloring. Prove that i = 1. Comment: P4 -free graphs are also called cographs.) 5.1.36. Given a vertex ordering a = vi. ... , Vn of a graph G, let G; = G [{vi. ... , v;}] and f(a) = 1 +max; dG,(V;). Greedy coloring relative to a yields x(G) ::: f(a). Define a* by letting v. be a minimum degree vertex of G and letting V; for i < n be a minimum degree vertex of G-{V;+i, ... , v.}. Show that f (a*) = l+maxH<;G 8(H), and thus that a* minimizes f(a). (Halin [1967], Matula [1968], Finck-Sachs [1969], Lick-White [1970]) 5.1.37. Prove that V(G) can be partitioned into 1+maxHc;;G8(H)/r classes such that every subgraph whose vertices lie in a single class has a vertex 01 degree less than r. (Hint: Consider ordering a* of Exercise 5.1.36. Comment: This generalizes Theorem 5.1.19. See also Chartrand-Kronk [1969] when r = 2.) 5.1.38. (!) Prove that x(G) = w(G) when G is bipartite. (Hint: Phrase the claim in terms ofG and apply results on bipartite graphs.) 5.1.39. (!) Prove that every k-chromatic graph has at least (;) edges. Use this to prove that if G is the union of m complete graphs of order m, then x (G) ::: 1 + mJm - 1. (Comment: This bound is near tight, but the Erdos-Faber-Lovasz Conjecture \see Erdos [1981]) asserts that x(G) = m when the complete graphs are pairwise edge-disjoint.) 5.1.40. Prove that x (G) x (G) ~ n(G), use this to prove that x (G) + x (G) ~ 2Jn(G), and provide a construction achieving these bounds whenever Jn(G) is an integer. (Nordhaus-Gaddum [1956], Finck [1968]) 5.1.41. (!) Provethatx(G)+x(G)::: n(G)+l. (Hint: Useinductiononn(G).) (NordhausGaddum [1956]) 5.1.42. (!) Looseness of x (G) ~ n(G)/a(G). Let G be an n-vertex graph, and let c = (n + 1)/a(G). Use Exercise 5.1.41 to prove that x (G) x (G) ::: (n -+ 1) 2 /4, and use this to prove that x(G) ::: c(n + 1)/4. For each odd n, construct a graph such that x(G) = c(n + 1)/4. (Nordhaus-Gaddum [1956], Finck [1968])
## Section 5.1: Vertex Coloring and Upper Bounds
203
5.1.43. (!) Paths and chromatic number in digraphs. a) Let G =FU H. Prove that x(G)::;:: x(F)x(H). b) Consider an orientation D of G and a function f: V(G) --+ JR. Use part (a) and Theorem 5.1.21 to prove that if x(G) > rs, then D has a path Uo --+ ... --+ u, with f(uo) ::5: ::5: f(u,) or a path Vo--+--+ Vs with f(vo) > > f(vs) c) Use part (b) to prove that every sequence of rs+ 1 distinct real numbers has an increasing subsequence of size r + 1 or a decreasing subsequence of size s + 1. (ErdosSzekeres [1935]) 5.1.44. (!) Minty's Theorem (Minty [1962] ). An acyclic orientation of a loopless graph is an orientation having no cycle. For each acyclic orientation D of G, let r(D) = maxc ra/bl, where C is a cycle in G and a, b count the edges of C that are forward in Dor backward in D, respectively. Fix a vertex x E V(G), and let W be a walk in G beginning at x. Let g(W) =a - b r(D), where a is the number of steps along W that are forward edges in D and bis the number that are backward in D. For each y E V(G), let g(y) be the maximum of g(W) such that Wis anx, y-walk(assume that G is connected). a) Prove that g(y) is finite and thus well-defined, and use g(y) to obtain a proper 1 + r(D)-coloring of G. Thus G is 1 + r(D)-colorable. b) Prove that x (G) = minven, where Dis the set of acyclic orientations of G. 5.1.45. (+) Use Minty's Theorem (Exercise 5.1.44) to prove Theorem 5.1.21. Prove that l (D) is maximized by some acyclic orientation of G)
(Hin~:
5.1.46. (+) Prove that the 4-regular triangle-free graphs below are 4-chromatic. (Hint: Consider the maximum independent sets. Comment: Chvatal [1970] showed that the graph on the left is the smallest triangle-free 4-regular 4-chromatic graph.)
5.1.47. (!)Prove that Brooks' Theorem is equivalent to the following statement: every
k - 1-regular k-critical graph is a complete graph or an odd cycle.
5.1.48. Let G be a simple graph with n vertices and m edges and maximum degree at most 3. Suppose that no component of G is a complete graph on 4 vertices. Prove that G contains a bipartite subgraph with at least m - n /3 edges. (Hint: Apply Brooks' Theorem, and -then show how to delete a few edges to change a proper 3-coloring of G into a proper 2-coloring of a large subgraph of G.) 5.1.49. (-) Prove that the Petersen graph can be 2-colored so that the subgraph induced by each color class consists of isolated edges and vertices. 5.1.50. (!) Improvement of Brooks' Theorem. a) Given a graph G, let ki. ... , k, be nonnegative integers with I: k; ::=:: ~(G) - t + 1. Prove that V(G) can be partitioned into sets Vi, ... , V, so that for each i, the subgraph G; induced by V; has maximum degree at most k; .. (Hint: Prove that the partition minimizing I: e(G;)/ k; has the desired property.) (Lov~sz [1966])
204
## Chapter 5: Coloring of Graphs
b) For 4 :::: r :::: ~(G) + 1, use part (a) to prove that x (G) :::: '~ 1 (~(G) + 1) when G has no r-clique. (Borodin-Kostochka [1977], Catlin [1978], Lawrence [1978]) 5.1.51. (!) Let G be an k-colorable graph, and let P be a set of vertices in G such that d(x, y) '.::: 4 whenever x, y E P. Prove that every coloring of P with colors from [k + 1] extends to a proper k
+ 1 coloring of G.
(Albertson-Moore [1999])
5.1.52. Prove that every graph G can be f(~(G) + l)/jl-colored so that each color class induces a subgraph having no j-edge-connected i;:ubgraph. For j > 1, prove that no smaller number of classes suffices when G is a j-regular j-edge-connected graph or is a complete graph with order congruent to 1 modulo j. (Comment: For j = 1, the restriction reduces to ordinary proper coloring.) (Matula [1973]) 5.1.53. (+) Let G 11 .k be the 2k-regular graph of Exercise 5.1.23. Fork :::: 4, determine the values of n such that Gn,k can be 2-colored so that each color class induces a subgraph with maximum degree at most k. (Weaver-West [1994]) 5.1.54. Let f be a proper coloring of a graph G in which the colors are natural numbers. The color sum is LveV(Gl f(v). Minimizing the color sum may require using more than x(G) colors. In the tree below, for example, the best proper 2-coloring has color sum 12, while there is a proper 3-coloring with color sum 11. Construct a sequence of trees in which the kth tree Tk use k colors in a proper coloring that minimizes the color sum. (Kubicka-Schwenk [1989])
5.1.55. (+) Chromatic number is bounded by one plus longest odd cycle length. a) Let G be a 2-connected nonbipartite graph containing an even cycle C. Prove that there exist vertices x, y on C and an x, y-path P internally disjoint from C such that dc(x, y) # dp(x, y) mod 2. b) Let G be a simple graph with no odd cycle having length at least 2k + 1. Prove that if 8(G) :::: 2k, then G has a cycle oflength at least 4k. (Hint: Consider the neighbors of an endpoint of a maximal path.) c) Let G be a 2-connected nonbipartite graph with no odd cycle longer than 2k - 1. Prove that x(G):::: 2k. (Erdos-Hajnal [1966])
## 5.2. Structure of k-chromatic Graphs
We have observed that x (H) 2: w(H) for all H. When equality holds in this bound for G and all its induced subgraphs (as for interval graphs), we say that G is perfect; we discuss such graphs in Sections 5.3 and 8.1. Our concern with the bound x(G) 2: w(G) in this section is how bad it can be. Almost always x(G) is much larger than w(G), in a sense discussed precisely in Section 8.5. (The average values of w(G), a(G), and x(G) over all graphs with vertex set [n] are very close to 2 lg n, 2 lg n, and n / (2 lg n), respectively. Hence w ( G) is generally a bad lower bound on x(G), and n/a(G) is generally a good lower bound.)
205
## GRAPHS WITH LARGE CHROMATIC NUMBER
The bound x(G) 2: w(G) can be tight, but it can also be very loose. There have been many constructions of graphs without triangles that have arbitrarily large chromatic number. We present one such construction here; others appear in Exercises 12-13.
5.2.1. Definition. From a simple graph G, Mycielski's construction produces a simple graph G' containing G. Beginning with G having vertex set {vi. ... , Vn}, add vertices U = {u1, ... , Un} and one more vertex w. Add edges to make u; adjacent to all of NG(v;), and finally let N(w) = U.
5.2.2. Example. From the 2-chromatic graph Kz, one iteration of Mycielski's construction yields the 3-chromatic graph C5 , as shown above. Below we apply the construction to C5, producing the 4-chromatic Grotzsch graph.
V1
5.2.3. Theorem. (Mycielski [1955)) From a k-chromatic triangle-free graph G, Mycielski's construction produces a k + 1-chromatic triangle-fr~e graph G'. Proof: Let V(G) = {v 1 , ... , vn), and let G' be the graph produced from it by Mycielski's constructfon. Let u1, ... , Un be the copies of V1, ... , Vn, with w the additional vertex. Let U = {u1, .. .,Un}. By construction, U is an independent set in G'. Hence the other vertices of any triangle containing u; belong to V ( G) and are neighbors of v;. This would complete a triangle in G, which can't exist. We conclude that G' is triangle-free. A proper k-coloring f of G extends to a proper k + 1-coloring of G' by setting f(u;) = f(v;) and f(w) = k + 1; hence x(G'):::: x(G) -t 1. We prove equality by showing that x(G) < x(G'). To prove this we consider any proper coloring of G' and obtain from it a proper coloring of G using fewer colors. Let g be a proper k-coloring of G' By changing the names of colors, we may assume that g(w) = k. This restricts g to {1, ... , k - 1) on U. On V(G), it may
206
## Chapter 5: Coloring of Graphs
use all k colors. Let A be the set of vertices in G on which g uses color k; we change the colors used on A to obtain a proper k - 1-coloring of G. For each v; E A, we change the color of v; to g(u;). Because all vertices of A have color k under g~ no two edges of A are adjacent. Thus we need only check edges of the form v;v' with v; EA and v' E V(G) - A. Ifv' *+ v;, then by construction also v' *+ u;, which yields g(v') =f:. g(u;). Since we change the color on v; to g(u;), our change does not violate the edge v;v'. We have shown that the modified coloring of V ( G) is a proper k - 1-coloring of G.
If G is color-critical, then the graph G' resulting from Mycielski's construction is also color-critical (Exercise 9).
5.2.4. * Remark. Starting with G2 = K 2 , iterating Mycielski's construction produces a sequence G2, G3 , G4 , of graphs. The first three are K 2 , C5 , and the Grotzsch graph. These are the smallest triangle-free 2-chromatic, 3-chromatic, and 4-chromatic graphs. The graphs then grow rapidly: n(Gk) = 2n(Gk_ 1 ) + 1. With n(G2) = 2, this yields n(Gk) = 3 2k- 2 - 1 (exponential growth). Let f (k) b~ the minimum number of vertices in a triangle-free k-chromatic graph. Using probabilistic (non-constructive) methods, Erdos [1959] proved that f (k) :::; ck 2+, where Eis any positive constant and c depends on E but not on k. Using Ramsey numbers (Section 8.3), it is now known (non-constructively) that there are constants ci, c2 such that c1k2 log k .:::; f (k) :::; c 2 k 2 log k. Exercise 15 develops a quadratic lower bound. Blanche Descartest [194 7, 1954] constructed color-critical graphs with girth 6 (Exercise 13). Using probabilistic methods, Erdos [1959] proved that graphs exist with chromatic number at least k and girth at least g (Theorem 8.5.11). Later, explicit constru~tions were found (Lovasz [1968a], Nesetnl-Rodl [1979-], Lubotzsky-Phillips-Sarnak [1988], Kriz [1989]). By all these constructions, forbidding Kr from G does not place a bound on x(G). Gyarfas [1975) and Sumner [1981]) conjectured that forbidding a fixed clique and a fixed forest as an induced subgraph does bound the chromatic number. Exercise 11 proves this when the forest is 2K 2 (See also KiersteadPenrice [1990, 1994], Kierstead [1992, 1997], Kierstead-ROdl [1996])
tThis pseudonym was used by W.T. Tutte and also by three others.
207
## EXTREMAL PROBLEMS AND TuRANS THEOREM
Perhaps extremal questions can shed some light on the structure of kchromatic graphs. For example, which are the smallest and largest k-chromatic graphs with n vertices?
5.2.5. Proposition. Every k-chromatic graph with n vertices has at least edges. Equality holds for a complete graph plus isolated vertices. Proof: An optimal coloring of a graph has an edge with endpoints of colors i and j for each pair i, j of colors. Otherwise, colors i and j could be combined into a single color class and use fewer colors. Since there are (;) distinct pairs of colors, there must be (;) distinct edges.
Exercise 6 asks for the minimum size among connected k-chromatic graphs with n vertices. The maximization problem is more interesting (of course, it makes sense only when restricted to simple graphs). Given a proper k-coloring, we can continue to add edges without increasing the chromatic number as long as two vertices in different color classes are nonadjacent. Thus we may restrict our attention to graphs without such pairs.
5.2.6. Definition. A complete multipartite graph is a simple graph G whose vertices can be partitioned into sets so that u ~ v if and only if u and v belong to different sets of the partition. Equivalently, every component of G is a complete graph. When k ~ 2, we write Kn 1 , ,n* for the complete k-partite graph with partite sets of sizes ni, ... , nk and complement Kn 1 ++Kn*
We use this notation only fork > 1, since Kn denotes a complete graph. A complete k-partite graph is k-chromatic; the partite sets are the color classes in the only proper k-coloring. Also, since a vertex in a partite set of size t has degree n ( G) - t, the edges can be counted using the degree-sum formula (Exercise 18). Which distribution of vertices to partite sets maximizes e(G)?
5.2.7. Example. The Turan graph. The Turan graph Tn,r is the completerpartite graph with n vertices whose partite sets differ in size by at most 1. By the pigeonhole principle (see Appendix A), some partite set has size at least rn/rl and some has size at most Ln/r J. Therefore, differing by at most 1 means that they all have size Ln/r J or rn/rl. Let a = Ln/r J. After putting a vertices in each partite set, b = n - ra remain, so Tn,r has b partite sets of size a + 1 and r - b partite sets of size a. Thus the defining condition on Tn,r specifies a single isomorphism class. 5.2.8. Lemma. Among simple r-partite (that is, r-colorable) graphs with n vertices, the Turan graph is the unique graph with the most edges. Proof: As noted before Definition 5.2.6, we need only consider complete rpartite graphs. Given a complete r-partite graph with partite sets differing by
208
## Chapter 5: Coloring of Graphs
more than 1 in size, we move a vertex v from the largest class (size i) to the smallest class (size j). The edges not involving v are the same as before, but v gains i - 1 neighbors in its old class and loses j neighbors in its new class. Since i - 1 > j, the number of edges increases. Hence we maximize the number of edges only by equalizing the sizes as in Tn,r We used the idea of this local alteration previously in Theorem 1.3.19 and in Theorem 1.3.23; we are finding the largest r-partite subgraph of K11 What happens if we have more edges and thus force chromatic number at least r + 1? We have seen that there are graphs with chromatic number r + 1 that have no triangles. Nevertheless, if we go beyond the maximum number of edges in an r-colorable graph with n vertices, then we are forced not only to use r + 1 colors but also to have K,+1 as a subgraph. This famous result of Turan generalizes Theorem 1.3.23 and is viewed as the origin of extremal graph theory.
5.2.9. Theorem. (Turan [1941]) Among the n-vertex simple graphs with no r + 1-clique, Tn,r has the maximum number of edges. Proof: The Turan graph Tn,n like every r-colorable graph, has nor+ 1-clique, since each partite set contributes at most one vertex to each clique. If we can prove that the maximum is achieved by an r-partite graph, then Lemma 5.2.8 implies that the maximum is achieved by Tn,r Thus it suffices to prove that if G has nor+ 1-clique, then there is an r-partite graph H with the same vertex set as G and at least as many edges. We prove this by induction on r. When r = 1, G and H have no edges. For the induction step, consider r > 1. Let G be an n-vertex graph with nor+ 1clique, and let x E V ( G) be a vertex of degree k = ~ ( G). Let G' be the subgraph of G induced by the neighbors of x. Since x is adjacent to every vertex in G' and G has nor + 1-clique, the graph G' has no r-clique. We can thus apply the induction hypothesis to G'; this yields an r - 1-partite graph H' with vertex set N(x) such that e(H') :::.: e(G'). Let H be the graph formed from H' by joining all of N(x) to all of S = V(G) - N(x). Since Sis an independent set, H is r-partite. We claim that e(H) :::.: e(G). By construction, e(H) = e(H') + k(n - k). We also have e(G) :;:: e(G') + Lves dc(v), since the sum counts each edge of Gonce for each endpoint it has outside V(G'). Since ~(G) = k, we have dc(v) :::; k for each v E S, and ISi = n - k, so Lves dc(v) :S: k(n - k). As desired, we have
e(G) :;:: e(G')
+ (n -
+ k(n -
k) = e(H)
## Section 5:2: Structure of k-chromatic Graphs
209
In fact, the Turan graph is the unique extremal graph (Exercise 21). Exercises 16-24 pertain to Turan's Theorem, including alternative proofs, the value of e(Tn,r), and applications. The argument used in Theorem 1.3.23 was simply one instance of the induction step in Theorem 5.2.9. Turan's theorem applies to extremal problems when some condition forbids cliques of a given order; we describe a geometric application from Bondy-Murty [1976, p113-115].
5.2.10. * Example. Distant pairs of points. In a circular city of diameter 1, we might want to locate n police cars to maximize the number of pairs that are far apart, say separated by distance more than d = 1/ .J2. If six cars occupy equally spaced points on the circle, then the only pairs not more than d apart are the consecutive pairs around the outside: there are nine good pairs. Instead, putting two cars each near the vertices of an equilateral triangle with side-length ,J3;2 yields three bad pairs arid twelve good pairs. (This may not be the socially best criterion!) In general, with fn/31 or Ln/3j cars near each vertex of this triangle, the good pairs correspond to edges of the tripartite Turan graph. We show next that this construction is best.
5.2.11.* Application. In a set ofn points in the plane with no pair more than distance 1 apart, the maximum number of pairs separated by distance more than 1/.J2 is L n 2 /3 J. Proof: Draw a graph G on these points by making points adjacent when the distance between them exceeds l/.J2. By Turan's Theorem and the construction in Example 5.2.10, it suffices to show that G has no K 4 Among any four points, some three form an angle of at least 90: if the four form a convex quadrilateral, then the interior angles sum to 360, and if one point is inside the triangle formed by the others, then with them it forms three angles summing to 360. Suppose that G has a 4-vertex clique with points w, x, y, z, where Lxyz::::: 90. Since the lengths of xy and yz exceed 1/.J2, xz is longer than the hypotenuse of a right triangle with legs of length 1/ .J2. Hence the distance between x and z exceeds 1, which contradicts the hypothesis.
210
## Chapter 5: Coloring of Graphs
Even without the full structural statement of Turan's Theorem, one can prove directly a rough bound on the number of edges in an n-vertex graph with no K,+1 (Exercise 16). Turning this around yields a sharp lower bound on the chromatic number of a graph in terms of the number of vertices and number of edges (Exercise 17).
COLOR-CRITICAL GRAPHS
The Turan graph solves a problem that is somehow opposite to understanding what forces chromatic number k. It considers the maximal graphs that avoid needing k colors instead of the minimal graphs that do need k colors. Every k-chromatic graph has a k-critical subgraph, since we can continue discarding edges and isolated vertices without reducing the chromatic number until we reach a point where every such deletion reduces the chromatic number. Thus knowing the k-critical graphs could help us test fork - 1-colorability. We begin with elementary properties of k-critical graphs.
if x(G - e) < x(G) for every e
5.2.12. Remark. A graph G with no isolated vertices is color-critical if and only E E(G). Hence when we prove that a connected graph is color-critical, we need only compare it with subgraphs obtained by deleting a single edge.
5.2.13. Proposition. Let G beak-critical graph. a) For v E V(G), there is a proper k-coloring of Gin which the color on v appears nowhere else, and the other k - 1 colors appear on N(v). b) Fore E E(G), every proper k - 1-coloring of G - e gives the same color to the two endpoints of e. Proof: (a) Given a proper k - 1-coloring f of G - v, adding color k on v alone completes a proper k-coloring of G. The ot:l).er colors must all appear on N(v), since otherwise assigning a missing color to v would complete a proper k - 1coloring of G. (b) If some proper k-1-coloringof G-e gave distinct colors to the endpoints of e, then adding e would yield a proper k - 1-coloring of G.
For any graph G, Proposition 5.2.13a holds for every v E V(G) such that x(G - v) < x(G) = k, and Proposition 5.2.13b holds for every e e E(G) such that x(G - e) < x(G) = k.
5.2.14. Example. The graph C5 v Ks of Example 5.1.8 is color-critical. In general, the join of two color-critical graphs is always color-critical. This is easy to prove using Remark 5.2.12 and Proposition 5.2.13 by considering cases for the deletion of an edge; the deleted edge e may belong to G or H or have an endpoint in each (Exercise 3).
We proved in Lemma 5.1.18 that o(G):::.: k-1 when G is a k-critical graph. We can strengthen this to K' ( G) :::.: k - 1 by using the Konig-Egervary Theorem.
## Section 5.2: Structure of k-chromatic Graphs
211
5.2.15. Lemma. (Kainen) Let G be a graph with x(G) > k, and let X, Y be a partition of V(G). IfG[X] and G[Y] are k-colorable, then the edge cut [X, Y] has at least k edges. Proof: Let X 1 ... , Xk and Y1 , ... , Yk be the partitions of X and Y formed by the color classes in proper k-colorings of G[X] and G[Y]. Ifthere is no edge between X; and Yj, then X; U Yj is an independent set in G. We show that if I[X, Y] I < k, then we can combine color classes from G[X] and G[Y] in pairs to form a proper k-coloring of G. Form a bipartite graph H with vertices X1 , ... , Xk and Yi. ... , Yb putting X; Yj E E(H) if in G there is no edge between the set X; and the set Yj. If l[X, Y] I < k, then H has more than k(k - 1) edges. Since m vertices can cover at most km edges in a subgraph of Ku, E(H) cannot be covered by k - 1 vertices. By the Konig-Egervary Theorem, H therefore has a perfect matching M. In G, we give color i to all of X; and all of the set Yj to which it is matched by M. Since there are no edges joining X; and Yj, doing this for all i produces a proper k-coloring of G, which contradicts the hypothesis that x (G) > k. Hence we conclude that l[X, Y]I ~ k.
5.2.16. Theorem. (Dirac [1953]) Every k-critical graph is k -1-edge-connected. Proof: Let G beak-critical graph, and let [X, Y] be a minimum edge cut. Since G is k-critical, G[X] and G[Y] are k - 1-colorable. Applied with k - 1 as the parameter, Lemma 5.2.15 then states that l[X, Yll ~ k - 1.
Although a k-critical graph must be k - 1-edge-connected, it need not be k-1-connected; Exercise 32 shows how to construct k-critical graphs that have connectivity 2. Nevertheless, we can restrict the behavior of small vertex cutsets in k-critical graphs.
5.2.17. Definition. Let S be a set of vertices in a graph G. An S-lobe of G is an induced subgraph of G whose vertex set consists of S and the vertices of a component of G - S.
H3
212
## Chapter 5: Coloring of Graphs
For every S 5; V(G), the grnph G is the union of its S-lobes. We use this to prove a statement about vertex cutsets in k-critical graphs that will be useful in the next theorem. Exercise 33 strengthens the result when ISi = 2.
5.2.18. Proposition. If G is k-critical, then G has no cutset consisting of pairwise adjacent vertices. In particular, if G has a cutset S = {x, y}, then x fry and G has an S-lobe H such that x(H + xy) = k. Proof: Let S be a cutset in a k-critical graph G. Let H1 , ... , H, be the S-lobes of G. Since each H; is a proper subgraph of a k-critical graph, each H; is k - 1colorable. If each H; has a proper k - 1-coloring giving distinct colors to the vertices of S, then the names of the colors in these k - 1-colorings can be permuted to agree on S. The colorings then combine to form a k - 1-coloring of G, which is impossible. Hence some S-lobe H has no proper k - 1-coloring with distinct colors on S. This implies that Sis not a clique. If S = {x, y}, then every k - 1-coloring of H assigns the same color to x and y, and hence H + xy is not k - l-colorable.-9
FORCED SUBDIVISIONS
We need not have a k-clique to have chromatic number k, but perhaps we must have some weakened form of a k-clique.
5.2.19. Definition. An H-subdivision (or subdivision of H) is a graph obtained from a graph H by successive edge subdivisions (Definition 5.2.19). Equivalently, it is a graph obtained from H by replacing edges with pairwise internally disjoint paths.
a subdivision of K4
5.2.20. Theorem. (Dirac [1952a]) Every graph with chromatic number at least 4 contains a K4 -subdivision. Proof: We use induction on n(G). Basis step: n(G) = 4. The graph G can only be K 4 itself. Induction step: n(G) > 4. Since x(G) 2: 4, we may let H be a a 4-critical subgraph of G. By Proposition 5.2.18, H has no cut-vertex. If K(H) = 2 and S = {x, y} is a cutset of size 2, then by Proposition 5.2.18 x fr y and H has an S-lobe H' such that x (H' + xy) 2: 4. Since n(H') < n(G), we can apply the induction hypothesis to obtain a K4 -subdivision in H'.
## Section 5.2: Structure of k-chromatic Graphs
213
This K4-subdivision F appears also in G unless it contains xy (see figure below). In this case, we modify F to obtain a K4-subdivision in G by replacing the edge xy with an x, y-path through another S-lobe of H. Such a path exists because the minimality of the cutset S implies that each vertex of S has a neighbor in each component of H - S. Hence we may assume that H is 3-connected. Select a vertex x E V ( G). Since H - x is 2-connected, it has a cycle C of length at least 3. (Let x be the central vertex and C the outside cycle in the figure above.) Since H is 3connected, the Fan Lemma (Theorem 4.2.23) yields an x, V(C)-fan of size 3 in H. These three paths, together with C, form a K4-subdivision in H.
H'
5.2.21.* Remark. Haj6s (1961] conjectured that every k-chromatic graph contains a !;\Ubdivision of Kk. Fork= 2, the statement says that every 2-chromatic graph has a nontrivial path. For k = 3, it says that every 3-chromatic graph has a cycle. Theorem 5.2.20 proves it fork= 4, and it is open fork E {5, 6}. Haj6s' Conjecture is false fork :=::: 7 (Catlin (1979]-see Exercise 40). Hadwiger [1943] proposed a weak.er conjecture: every k-chromatic graph has a subgraph that becomes Kk via edge contractions. This is weaker because a Kksubdivision is a special subgraph of this type. Fork = 4, Hadwiger's Conjecture is equivalent to Theorem 5.2.20. For k = 5, it is equivalent to the Four Color Theorem (Chapter 6). Fork= 6, it was proved using the Four Color Theorem by Robertson, Seymour, and Thomas [1993]. For k :=::: 7, it remains open.
Some results about k-critical graphs extend to the larger class of graphs with 8(G) :=::: k - 1. For example, every graph with minimum degree at least 3 has a K4-subdivision (Exercise 38); this strengthens Theorem 5.2.20. Dirac (1965] and Jung (1965] proved that sufficiently large chromatic number forces a Kk-subdivision in G~ Mader improved this by weakening the hypothesis and generalizing the conclusion: for a simple graph F, every simple graph G with 8(G) :=::: 2e<Fl contains a subdivision of F. The threshold 2e<Fl is larger than necessary but permits a short proof.
5.2.22. * Lemma. (Mader [1967], see Thomassen (1988]) If G is a simple graph with minimum degree at least 2k, then G contains disjoint subgraphs G' and H such that 1) His connected, 2) 8(G') :=::: k, and 3) each vertex of G' has a neighbor in H. Proof: We may assume that G is connected. Let G H' denote the graph obtained fro'm G by contracting the edges of a connected subgraph H' and delete extra copies of multiple edges. In G H', the set V(H') becomes a single vertex. Consider all connected subgraphs H' of G such that G H' ha\$ at least
214
k(n(G) - n(H')
## Chapter 5: Coloring of Graphs
+ 1) edges. Since o(G) ::: 2k, every 1-vertex subgraph of G is such a subgraph. Since such subgraphs exist, we may choose H to be a maximal subgraph with this property. Let S be the set of vertices outside H with neighbors in H, and let G' = G[S]. We need only show that o(G') ::: k. Each x E V(G') has a neighbor y e V (H). In G (HU xy), the edges incident toxin G' collapse onto edges from V(G') t0 H that appear in G H, and the edge xy contracts. Hence e(G H) e(G .. (H Uxy)) = dG'(x) + 1. By the choice of H, this difference is more thank, and hence o(G') ::: k.
5.2.23.* Theorem. (Mader [1967], see Thomassen [1988]) If F and Gare simple graphs with e(F) = m and o(F) '.'.: 1, then o(G) '.'.: 2m implies that G contains a subdivision of F. Proof: We use induction on m. The claim is trivial form ::=: 1. Consider m ::: 2. By Lemma 5.2.22, we may choose disjoint subgraphs H and G' in G such that His connected, o(G') ::: 2m- 1, and every vertex of G' has a neighbor in H. If F has an edge e = xy such that o(F - e) ::: 1, then the induction hypothesis yields a subdivision J of F - e in G'. A path through H can be added between the vertices of J representing x and y to complete a subdivision of F. If o(F - e) = 0 for all e E E(F), then every edge of Fis incident to a leaf. Now Fis a forest of stars, and o(G)::: 2m::: 2m allows us to find F itself in G; we leave this claim to Exercise 42. 5.2.24.* Remark. The case when Fis a complete graph remains of particular interest. Let f(k) be the minimum d such that every graph with minimum degree at least d contains a Kk-subdivision. Theorem 5.2.23 yields f (k) :::: 2m ., Koml6s-Szemeredi [1996] and Bollobas-Thomason [1998] proved that f(k) < ck 2 for some constant c (the latter shows c < 256). Since Km m-1 has no K2ksubdivision when m = k(k + 1)/2 (Exercise 4l), we have f (k) ; k 2 /8. Exercise 38 yields f (4) = 3. Furthermore, f (5) = 6. The icosahedron (Exercise 7.3.8) yields f (5) ::: 6, since this graph is 5-regular and has no Kssubdivision. On the other hand, Mader [1998] proved Dirac'S conjecture [1964] that every n-vertex graph with at least 3n - 5 edges contains a Ks-subdivision. By the degree-sum formula, o(G) ::: 6 yields at least 3n edges; hence f(5Y::: 6. Finally, we note that Scott [1997] proved a subdivision version of the Gyitrfas-Sumner Conjecture (Remark 5.2.4) for each tree T. and integer k: If G has with no k-clique but x (G) is sufficiently large, then G contains a subdivision of T as an induced subgraph.
EXERCISES
5.2.1. (-)Let G be a graph such that x(G -x - y) = x(G)-2 for all pairs x, y of distinct vertices. Prove that G is a complete graph. (Comment: Lovasz conjectured that the conclusion also holds when the condition is imposed only on pairs of adjacent vertices.)
## Section 5.2: Structure of k-chromatic Graphs
215
5.2.2. ( - ) Prove that a simple graph is a complete multipartite graph if and only if it has no 3-vertex induced subgraph with one edge. 5.2.3. ( - ) The results below imply that there is no k-critical graph with k + 1 vertices. a) Let x and y be vertices in a k-critical graph G. Prove that N(x) ~ N(y) is impossible. Conclude that no k-critical graph has k + 1 vertices. b) Prove that x(G v H) = x(G) + x(H), and that G v His color-critical if and only if both G and Hare color-critical. Conclude that C5 v Kk-s, with k + 2 vertices, is k-criti 11. 5.2.4. For n. E N, let G be the graph with vertex set {vo, ... , Van} defined by v; and only if Ii - jl :=:: 2 and i + j is not divisible by 6. a) Determine the blocks of G. b) Prove that adding the edge v0 v3n to G creates a 4-critical graph. 5.2.5. (-) Find a subdivision of K 4 in the Gri:itzsch graph (Example 5.2.2) .
*+ '
if
5.2.6. Determine the minimum number of edges in a connected n-vertex graph with chromatic number k. (Hint: Consider a k-critical subgraph.) (Ersov-Kofohin [1962]see Bhasker-Samad-West [1994] for higher connectivity.) 5.2.7. (!) Given an optimal coloring of a k-chromatic graph, prove that for each color i there is a vertex with color i that is adjacent to vertices of the other k - 1 colors. 5.2.8. Use properties of color-critical graphs to prove Proposition 5.1.14 again: x(G):::: 1 +max; min{d;, i -1}, where di:=::::=::: dn are the vertex degrees in G. 5.2.9. (!) Prove that if G is a color-critical graph, then the graph G' generated from it by applying Mycielski's construction is also color-critical. 5.2.10. Given a graph G with vertex set vi, ... , Vn, let G' be the graph generated from G by Mycielski's construction. Let H be a subgraph of G. Let G" be the graph obtained from G' by adding the edges {u;uj: V;Vj E E(H)}. Prove that x(G") = x(G) + 1 and that w(G") = max{w(G), w(H) + l}. (Pritikin) 5.2.11. (!) Prove that if G has no induced 2K2, then x (G) :=:: (w(Gd" i). (Hint: Use a maximum clique to define a collection of (w~Gl) + w(G) independent sets that cover the vertices. Comment: This is a special case of the Gyarfas-Sumner Conjecture-Remark 5.2.4) (Wagon [1980]) 5.2.12. (!) Let Gi = Ki. For k > 1, construct Gk as follows. To the disjoint union Gi + + Gk-i, and add an independent set T of size n(G;). For each choice of (vi, ... , Vk-i) in V(Gi)x x V(Gk-i), letonevertexofT have neighborhood {vi, ... , Vk-i}. (In the sketch of G 4 below, neighbors are shown for only two elements of T.) a) Prove that w(Gk) = 2 arid x(Gk) = k. (Zykov [1949]) b) Prove that Gk is k-critical. (Schauble [1969])
fl:::
216
## Chapter 5: Coloring of Graphs
5.2.13. (+) Let G beak-chromatic graph with girth 6 and order n. Construct G' as follows. Let T be an independent set of kn new vertices. Take pairwise disjoint copies of G, one for each way to choose an n-set S c T. Add a matching between each copy of G and its corresponding n-set S. Prove that the resulting graph has chromatic number k + 1 and girth 6. (Comment: Since C6 is 2-chromatic with girth 6, the process can start and these graphs exist.) (Blanche Descartes [1947, 1954])
e;)
6
T
C:l oopi"'
5.2.14. Chromatic number and cycle lengths. a) Let v be a vertex in a graph G. Among all spanning trees of G, let T be one that maximizes LueV(G) dr(u, v). Prove that every edge of G joins vertices belonging to a path in T starting at v. b) Prove that if x(G) > k, then G has a cycle whose length is one more than a multiple of k. (Hint: Use the tree T of part (a) to define a k-coloring of G.) (Tuza) 5.2.15. (!) Prove that a triangle-free graph with n vertices is colorable with 2..fo colors. (Comment: Thus every k-chromatic triangle-free graph has at least k2 /4 vertices.) 5.2.16. (!) Prove that every n-vertex simple graph with no r + 1-clique has at most (1 - 1/r)n2 /2 edges. (Hint: This can be proved using Turan's Theorem or by induction on r without Turan's Theorem.) 5.2.17. (!)Let G be a simple n-vertex graph with m edges. a) Prove that w(G) ~ n 2 /(n 2 - 2m) and that this bound is sharp. (Hint: Use Exercise 5.2.16. Comment: This also yields x(G) ~ n 2 /(n 2 - 2m)l) (Myers-Liu [1972]) b) Prove that a(G) ~ rn/(d + 1)1, where dis the average vertex degree of G. (Hint: Use part (a).) (Erdos-Gallai [1961])
5.2.18. The Turan graph Tn,r (Example 5.2. 7) is the complete r-partite graph with b partite sets of size a + 1 and r - b partite sets of size a, where a = Ln / r J and b = n - ra. a) Prove that e(Tn,r) = (1- 1/r)n2 /2 - b(r - b)/(2r). b) Since e(G) must be an integer, part (a) implies e(Tn,r) ~ (1- 1/r)n2 /2 Determine the smallest r such that strict inequality occurs for some n. For this value of r, determine all n such that e(1~.r) < (1- 1/r)n2 /2
J.
J.
5.2.19. (+) Let a = Ln / r J. Compare the Turan graph prove directly that e(Tn,r) = (n; 0 ) + (r - 1)(~ 1 ).
Tn,r
## with the graph
Ka
Kn-a
to
5.2.20. Given positive integers n and k, let q = Ln/kJ, r = n - qk, s = Ln/(k + l)J, and t = n - s(k + 1). Prove that (~)k + r_q ~ G} (k + 1) +ts. (Hint: Consider the complement of the Turan graph.) (Richter [1993]) 5.2.21. Prove that among then-vertex simple graphs with nor+ 1-clique, the Turan graph Tn,r is the unique graph having the maximum number of edges. (Hint: Examine the proof of Theorem 5.2.9 more carefully.) 5.2.22. A circular city with diameter four miles will get 18 cellular-phone power stations. Each station has a transmission range of six miles. Prove that no matter where
## Section 5.2: Structure of k-chromatic Graphs
217
in the city the stations are placed, at least two will each be able to transmit to at least five others. (Adapted from Bondy-Murty [1976, pl15]) 5.2.23. (!) Turan's proof of Turan's Theorem, including uniqueness (Turan [1941]). a) Prove that a maximal simple graph with no r + 1-clique has an r-clique. b) Prove that e(T. ,) = {;) + (n - r)(r - 1) + e(Tn-r.r). c) Use parts (a) and (b) to prove Turan's Theorem by induction on n, including the characterization of graphs acheiving the bound. 5.2.24. (+)Let t,(n) = e(T. ,). Let G be a graph with n vertices that has t,(n) - k edges and at least oner+ 1-clique, where k ~ 0. Prove that G has at least f,(n) + 1-k cliques of order r + 1, where f,(n) = n - rn/rl - r. (Hint: Prove that a graph with exactly one r + 1-clique has at most t,(n) - f,(n) edges.) (Erdos [1964], Moon [1965c]) 5.2.25. Partial analogue ofTuran's Theorem for K2 ,m. a) Prove that if G is simple and LvEV(G) {d~)) > (m - l){;), then G contains Kz.m (Hint: View K 2 .m as two vertices with m common neighbors.) b) Prove that LvEV(G) (d~l) ~ e(2e/n - 1), where G has e edges. c) Use parts (a) and (b) to prove that a graph with more than ~(m -1) 112 n 312 + n/4 edges contains Kz.m d) Application: Given n points in the plane, prove that the distance is exactly 1 for at most 1n 312 + n/4 pairs. (Bondy-Murty [1976, plll-112]) 5.2.26. For n ~ 4, prove that every n-vertex graph with more than ~nJn=l edges has girth at most 4. (Hint: Use the methods of Exercise 5.2.25 5.2.27. (+) For n ~ 6, prove that the maximum number of edges in a simple m-vertex graph not having two edge-disjoint cycles is n + 3. (Posa) 5.2.28. (+) For n :::: 6, prove that the maximum number of edges in a simple n-vertex graph not having two disjoint cycles is 3n - 6. (Posa) 5.2.29. (!) Let G be a claw-free graph (no in<;luced Kl.3). a) Prove that the subgraph induced by the union of any two color classes in !l proper coloring of G consists of paths and even cycles. b) Prove that if G has a proper coloring using exactly k colors, then G has a proper k-coloring where the color classes differ in size by at most one. (Niessen-Kind [2000]) 5.2.30. (+) Prove that if G has a proper coloring g in which every color class has at least two vertices, then G has an optimal coloring f in which every color class has at least two vertices. (Hint: If j has a color class with only one vertex, use g to make an alteration inf. The proof can be given algorithmically or by induction on x(G).) (Gallai [1963c]) 5.2.31. Let G be a connected k-chromatic graph that ~snot a complete graph or a cy~e of length congruent to 3 modulo 6. Prove that every proper k-coloring of G has two vertices of the same color with a common neighbor. (Tomescu) 5.2.32. (!) The Haj6s construction (Haj6s (1961]). a) Let G and H be k-critical graphs sharing only vertex v, with vu E E(G) and vw E E(H). Prove that (G - vu) U (H - vw) U uw is k-critical. b) For all k :::: 3, use part (a) to obtain a k-critical graph other than Kk. c) For all n :::: 4 except n = 5, construct a 4-critical graph with n vertices.
218
## Chapter 5: Coloring of Graphs
5.2.33. Let G beak-critical graph having a separating set S = {x, y}. By Proposition 5.2.18, x f+ y. Prove that G has exactly two S-lobes and that they can be named G 1 , G 2 such that G 1 + xy is k-critical and G 2 xy is k-critical (here G 2 xy denotes the graph obtained from G 2 by adding xy and then contracting it). 5.2.34. (!) Let G be a 4-critical graph having a separating set S of size 4. Prove that G [SJ has at most four edges. (Pritikin) 5.2.35. (+) Alternative proof that k-critical. graphs are k - 1-edge-connected. a) Let G beak-critical graph, with k ~ 3. Prove that for every e, f E E(G) there is a k - 1-critical subgraph of G containing e but not f. (Toft [1974]) b) Use part (a) and induction on k to prove Dirac's Theorem that every k-critical graph is k - 1-edge-c.onnected. (Toft [1974]) 5.2.36. (+) Prove that if G is k-critical and every k - 1-critical subgraph of G is isomorphic to Kk_ 1, then G = Kk (if k ~ 4) (Hint: Use Toft's critical graph lemma-Exercise 5.2.35a.) (Stiebitz [1985]) 5.2.37. A graph G is vertex-color-critical if x (G - v) < x (G) for all v E V(G). a) Prove that every color-critical graph is vertex-color-critical. b) Prove that every 3-chromatic vertex-color-critical graph is color-critical. c) Prove that the graph below is vertex-color-critical but not color-critical. (Comment: This is not the Grotzsch graph.)
5.2.38. (!)Prove that every simple graph with minimum degree at least 3 contains a K4 subdivision. (Hint: Prove a stronger result-every nontrivial simple graph with at most one vertex of degree less than 3 contains a K4 -subdivision. The proof of Theorem 5.2.20 already shows that every 3-connected graph contains a K4 -subdivision.) (Dirac [1952a]) 5.2.39. (!)Given that S(G) ~ 3 forces a K4 -subdivision in G, prove that the maximum number of edges in a simple n-vertex graph with no K4 -subdivision is 2n - 3. 5.2.40. Thick edges below indicate that every vertex in one circle is adjacent to every vertex in the other. Prove that x(G) = 7 but G has no K7 -subdivision. Prove that x(H) = 8 but H has no K8 -subdivision. (Catlin [1979])
219
## + 1)/2. Prove that Km,m-l has no K2k-subdivision.
5.2.42. (+) Let F be a forest with m edges. Let G be a simple graph such that 8( G) ::::: m and n(G)::::: n(F). Prove that G contains Fas a subgraph. (Hint: Delete one leaf from each nontrivial component of F to obtain F'. Let R be the set of neighbors of the deleted vertices. Map R onto an m-set X ~ V(G) that minimizes e(G[X]). Extend X to a copy of F'. Use Hall's Theorem to show that X can be matched into the remaining vertices to complete a copy of F.) (Brandt [1994]) 5.2.43. (+) Let G beak-chromatic graph. It follows from Lemma 5.1.18 and Proposition 2.1.8 that G contains every k-vertex tree as a subgraph. Strengthen this to a labeled analogue: if f is a proper k-coloring of G and T is a tree with vertex set {w 1 , ... , wk}, then there is an adjacency-preserving map: V(T) ~ V(G) such that f (cp(w;)) = i for all i. (Gyarfas-Szemeredi-Tuza [1980], Sumner [1981]) 5.2.44. (+) Let G beak-chromatic graph of girth at least 5. Prove that G contains every k-vertex tree as an induced subgraph. (Gyarfas-Szemeredi-Tuza [1980])
## 5.3. Enumerative Aspects
Sometimes we can shed light on a hard problem by considering a more general problem. No good algorithm to test existence of a proper k-coloring is known (see Appendix B), but still we can study the number of proper k-colorings (here we fix a particular set of k colors). The chromatic number x(G) is the minimum k such that the count is positive; knowing the count for all k would tell us the chromatic number. Birkhoff [1912] introduced this counting problem as a possible way to attack the Four Color Problem (Section: 6.3). In this section, we will discuss properties of the counting function, classes where it is easy to compute, and further related topics.
## COUNTING PROPER COLORINGS
We start by defining the counting problem as a function of k.
5.3.1. Definition. Given k EN and a graph G, the value x(G; k) is the number of proper colorings /: V(G) ~ [k]. The set of available colors is [kl = {1, ... , k}; the k colors need not all be used in a coloring f. Changing the names of the colors that are used produces a different coloring. 5.3.2. Example. x(Kn; k)) =kn and x(Kn; k)) = k(k - 1) (k - n + 1). When coloring the vertices of Kn, we can use any of the k colors at each vertex no matter what colors we have used at other vertices. Each of the kn functions from the vertex set to [kl is a proper coloring, and hence xCKn; k) =kn. When we color the vertices of Kn, the colors chosen earlier cannot be used on the ith vertex. There remain k-i +1 choices for the color of the i th vertex no matter how the earlier colors were chosen. Hence x (Kn; k) = k(k - 1) (k - n + 1).
220
## Chapter 5: Coloring of Graphs
We can also count this as (~)n! by first choosing n distinct colors and then multiplying by n! to count the ways to assign the chosen colors to the vertices. For example, x(Ks; 3) = 6 and x(Ks; 4) = 24. The value of the product is 0 when k < n. This makes sense, since. Kn has no proper k-colorings when k < n.
k
k-2
k-1
5.3.3. Proposition. If T is a tree with n vertices, then x (T; k) = k(k - 1r- 1 Proof: Choose some vertex v of T as a root. We can color v in k ways. If we ext~nd a proper coloring to new vertices as we grow the tree from v, at each step only the color of the parent is forbidden, and we have k - 1 choices for the color of the new vertex. Furthermore, deleting a leaf shows inductively that every proper k-coloring arises in this way. Hence x (T; k) = k(k - 1r- 1
k-1
k-1
Another way to count the colorings is to observe that the color classes of each proper coloring of G partition V(G) into independent sets. Grouping the colorings according to this partition leads to a formula for x(G; k) that is a polynomial ink of degree n(G). Note that this holds for the answers in Example 5.3.2 and Proposition 5.3.3. Since every graph has this property, x(G; k) as a function of k is called the chromatic polynomial of G.
5.3.4. Proposition. Let X(r) = x (x - 1) (x - r + 1). If Pr ( G) denotes the number of partitions of V(G) into r .nonempty independent sets, then x(G; k) = ~;~~) Pr(G)k(r)> which is a polynomial ink of degree n(G). Proof: When r colors are actually used in a proper coloring, the color classes partitipn V(G) into exactly r independent sets, which can happen in Pr(G) ways. When k colors are available, there are exactly k(r) ways to choose colors and assign them to the classes. All the proper colorings arise in this way, so the formula for x(G; k) is correct. Since k(r) is a polynomial ink and Pr(G) is a constant for each r, this formula implies that x(G; k) is a polynomial function of k. When G has n vertices, there is exactly one partition of G into n independent sets and no partition using more sets, so the leading term is kn.
## Section 5.3: Enumerative Aspects
221
5.3.5. Example. Always p 11 ( G) = 1, using independent sets of size 1. Also Pt ( G) = 0 unless G has no edges, since only for K is the entire vertex set an independent set. Consider G = C4 There is exactly one partition into two independent sets: opposite vertices must be in the same independent set. When r = 3, we put two opposite vertices together and leave the other two in sets by themselves; we can do this in two ways. Tpus p2 = 1, p3 = 2, p4 = 1.
11
x(C4; k)
## = 1 k(k - ~ + 2 k(k - l)(k - 2) + 1 k(k = k(k - l)(k 2 - 3k + 3).
- l)(k - 2)(k - 3)
Computing the chromatic polynomial in this way is not generally feasible, since there are too many partitions to consider. There is a recursive computation much like that used in Proposition 2.2.8 to count spanning trees. Again G e denotes the graph obtained by contracting the edge e in G (Definition 2.2. 7). Since the number of proper k-colorings is unaffected by multiple edges, we discard multiple copies of edges that arise from the contraction, keeping only one copy of each to form a simple graph.
5.3.6. Theorem. (Chromatic recurrence) If G is a simple graph and e E E(G), then x(G; k) = x(G - e; k) - x(G e; k). Proof: Every proper k-coloring of G is a proper k-coloring of G - e. A proper kcoloring of G - e is a proper k-coloring of G if and only if it gives distinct colors to the endpoints u, v of e. Hence we can count the proper k-colorings of G by subtracting from x (G - e; k) the number of proper k-colorings of G - e that give u and v the same color. Colorings of G - e in which u and v have the same color correspond directly to proper k-colorings of G e, in which the color of the contracted :vertex is the common color of u and v. Such a coloring properly colors all the edges of G e if and only if it properly colors all the edges of G other thane.
0
G
--- D~
G-e Ge
5.3.7. Example. Proper k-colorings of C4 . Deleting an edge of C4 produces P4, while contracting an edge produces K3. Since P4 is a tree and K3 is a complete graph, we have x (P4 ; k) = k(k - 1) 3 and x (K 3 ; k) = k(k - l)(k - 2). Using the chromatic recurrence, we obtain
x(C4; k) = x(P4; k) - x(K3; k) = k(k -
l)(k 2 - 3k + 3).
Because both G - e and G e have fewer edges than G, we can use the chromatic recurrence inductively to compute x(G; k). We need initial conditions for graphs with no edges, which we have already computed: x(K 11 ; k) = k".
222
## Chapter 5: Coloring of Graphs
5.3.8. Theorem. (Whitney [1933c]) The chromatic polynomial x(G; k) has degree n(G), with integer coefficients alternating in sign and beginning
1, -e(G), .
Proof: We use induction on e(G). The claims hold trivially when e(G) = 0, where x(Kn; k) =kn. For the induction step, let G be an n-vertex graph with e(G) 2: 1. EachofG-e and G e has fewer edges than G, and G e has n -1 vertices. By the induction hypothesis, there are nonnegative integers {a;} and {b;} such that x(G - e; k) = I:7= 0 (-l);a;kn-i and x(G e; k) = I;7,:-~(-l);b;kn-l-i. By the chromatic recurrence,
x(G - e; k): x(G e; k): x(G; k):
k" -(
[e(G) - l]k-l
k"-1 -
a2k"- 2 b1kn-2
## a;kn-i b;-1kn-i ... )
Hence x(G; k) is a polynomial with leading coefficient a0 = 1 and next coefficient -(a1 + bo) = -e(G), and its coefficients alternate in sign.
5.3.9. Example. When adding an edge yields a graph whose' chromatic polynomial is easy to.compute, we can use the chromatic recurrence in a different way. Instead of x(G; k) = x(G - e; k) - x(G e; k), we can writ~ x(G - e; k) = x(G; k) + x(G e; k). Thus we may be able to compute x(G - e; k) using x(G; k). To compute x(Kn - e; k), for example, we let G be Kn in this alternative formula and obtain
n-3
x(Kn - e; k)
= X.(Kn; k) + x(Kn-1; k) = (k -
+ 2)
n
i=O
(k - i).
IZ1
G-e
--- ~+~
G Ge
We close our general discussion of x(G; k) with an explicit formula. It has exponentially many terms, so its uses are primarily theoretical. The formula summarizes what happens if we iterate the chromatic recurrence until we dispose of all the edges.
5.3.10. Theorem. (Whitney [1932b]) Let c{G) denote the number of components of a graph G. Given a set S s; E(G) of edges in G, let G(S) denote the spanning subgraph of G with edg~ set S. Then the number x(G; k) of proper k-colorings of G is given by
x(G; k)
= Lsc;E(G)(-l)ISIF<G<s))
Proof: In applying the chromatic recurrence, contraction may produce multiple edges. We have observed that <;lropping these does not aff~ct x(G; k). We claim that deleting extra copies of edges also does not change the claimed formula.
## Section 5.3: Enumerative Aspects
223
Let e and e' be edges in G with the same endpoints. When e' E S and = c(G(S)), since both endpoints of e are in the same component of G(S). However, IS U {e}I = ISi + 1. Thus the terms for S and SU {e} in the sum cancel. Therefore, omitting all terms for sets of edges containing e' does not change the sum. This implies that we can keep or drop e' from the graph without changing the formula. When computing the chromatic recurrence, we therefore obtain the same result if we do not discard multiple edges or loops and instead retain all edges for contraction or deletion. Iterating the recurrence now yields 2e<G) terms as we dispose of all the edges; each in turn is deleted or contracted. When all edges have been deleted or contracted, the graph that remains consists of isolated vertices. Let S be the set of edges that were contracted. The remaining vertices correspond to the components of G(S); each such component becomes one vertex when the edges of Sare contracted and the other edges are deleted. The c(G(S)) isolated vertices at the end yield a term with kc(G(S)) colorings. Furthermore, the sign of the contribution changes for each contracted edge, so the contribution is positive if and only if ISi is even. Thus the contribution when Sis the set of contracted edges is (- l)ISlkc(G(S)), and this accounts for all terms in the sum.
e
## rf. S, we have c(G(S U {e}))
5.3.11. Example. A chromatic polynomial. When G is a simple graph with n vertices, every spanning subgraph with 0, 1, or 2 edges has n, n - 1, or n - 2 components, respectively. When ISi = 3, the number of components is n - 2 if and only if the three edges form a triangle; otherwise it is n - 3. For example, when G is a kite (four vertices, five edges) there are ten sets of three edges. For two of these, G (S) consists of a triangle plus one isolated vertex. The other eight sets of three edges yield spanning subgraphs with one component. Both types of triples are counting negatively, since ISi = 3. All spanning subgraphs with four or five edges have only one component. Hence Theorem 5.3.10 yields
x(G; k) = k4
-
5k 3
+ 10k2
(2k 2
+ 8k 1 ) + 5k
- k = k4
5k 3
+ 8k2
4k.
This agrees with x(G; k) = k(k-l)(k-2)(k-2), computed by counting colorings directly or by using x(G; k) = x(C4; k) - x(P3; k).
IZJ
Whitney proved Theorem 5.3.10 using the inclusion-exclusion principle of elementary counting. Among the universe of k-colorings, the proper colorings are those not assigning the same color to the endpoints of a11y edge. Letting A; be the set of k-colorings assigning the same color to the endpoints of edge e;, we want to count the colorings that lie in none of Ai, ... , Am (see Exercise 17).
224
## Chapter 5: Coloring of Graphs
CHORDAL GRAPHS
Counting colorings is easy for cliques and trees (and the kite) because each such graph arises from K1 by successively adding a vertex joined to a clique. The chromatic polynomial of such a graph is a product oflinear factors.
5.3.12. Definition. A vertex of G is simplicial ifits neighborhood in G induces a clique. A simplicial elimination ordering is an ordering Vn, . , v1 for deletion of vertices so that each vertex v; is a simplidal vertex of the remaining graph induced by {v 1 , ... , v;}. (These orderings are also called perfect elimination orderings.) 5.3.13. Example. Chromatic polynomials from simplicial elimination orderings. In a tree, a simplicial elimination ordering is a successive deletion of leaves. We have observed that x(G; k) = k(k-1r- 1 when G is an n-vertex tree. When Vn, , v 1 is a simplicial elimination ordering for G, the product rule of elementary combinatorics (Appendix,A) allows us to count proper k-colorings of G. Ifwe have colored vi, ... , v;, then when we add v; there .are k- d (i) ways to color it, where d(i) = IN(v;) n {vi, ... , v;_i}j. The factor k - d(i) is independent of how previous color choices were made, because the neighbors ofv; that have been colored form a clique of size d(i) and have distinct colors. Deleting a simplicial vertex that starts a simplicial elimination ordering yields inductively that every proper k-coloring of G arises in this way. Thus we have expressed the chromatic polynomial as a product of linear factors. In the graph below, vs, ... , v1 is a simplicial elimination ordering. When we form the graph in the order v1 , , vs, the values d (1), ... , d (6) are 0, 1, 1, 2, 3, 2, and the chromatic polynomial is k(k - l)(k - l)(k - 2)(k - 3)(k - 2).
5.3.14. Remark. It is important to note that some graphs without simplicial elimination orderings also have chromatic polynomials that can be expressed as a product of linear factors of the form k - r; with r; a nonnegative integer. Exercise 19 presents an example. Thus the existence of a simplicial elimination ordering is a sufficient but not necessary condition for the chromatic polynomial to have this nice factorization property.
Trees, cliques, near-complete graphs (Kn -e), and interval graphs (Exercise 28) all have simplicial elimination orderings. When n ~ 3, the cycle Cn has no simplicial elimination ordering, because a cycle has no simplicial vertex to start the elimination. The existence of simplicial elimination orderings is equivalent to the absence of such cycles as induced subgraphs.
## Section 5.3: Enumerative Aspects
225
5.3.15. Definition. A chord of a cycle C is an edge not in C whose endpoints lie in C. A chordless cycle in G is a cycle of length at least 4 in G that has no chord (that is, the cycle is an induced subgraph). A graph G is chordal if it is simple and has no chordless cycle.
The motivation for the term "chord" is geometric. If a cycle is drawn with its vertices in order on a circle and its chords are drawn as line segments, then the chords of the cycle are chords of the circle. It is fairly easy to show that a graph with a simplicial elimination ordering cannot have a chordless cycle. Thus our characterization of these graphs is another TONCAS theorem. We separate the substantive part of the proof of sufficiency as a lemma that is useful on its own (see also Laskar-Shier [1983]).
5.3.16. Lemma. (Voloshin (1982], Farber-Jamison (1986]) For every vertex x in a chordal graph G, there is a simplicial vertex of G among the vertices farthest from x in G Proof: We use induction on n(G). Basis step (n(G) = 1): The one vertex in K1 is simp\icial. Induction step (n(G) 2: 2): If x is adjacent to all other vertices, then we apply the induction hypothesis to the chordal graph G - x. Each simplicial vertex y of G - xis also simplicial in G, since xis adjacent to all of N(y) U {y}. Otherwise, let T be the set of vertices in G with maximum distance from x, and let H be a component of G[T]. Let S be the set of vertices in G - T having neighbors in V(H), and let Q be the component of G - S containing x.
We claim that S is a clique. Each vertex of S has a neighbor in V (H) and a neighbor in Q. For distinct vertices u, v e S, the union of shortest u, v-paths through H and through Q is a cycle of length at least 4. Since there are no edges from V(H) to V(Q), this cycle has no chord other than uv. Since G has no chordless cycle, u ~ v. Since u, v e S were chosen arbitrarily, Sis a clique. Now let G' = G[S U V(H)]; this omits x and thus is smaller than G. We apply the induction hypothesis to G' and a vertex u e S. Since Sis a clique, S - {u} s.;;; N(u). Whether G' is a clique or not, it thus has a simplicial vertex z within V(H). Since N 0 (z) s.;;; V(G'), the vertex z is also simplicial in G, and z is a vertex with maximum distance from x, as desired.
226
## Chapter 5: Coloring of Graphs
5.3.17. Theorem. (Dirac [1961]) A simple graph has a simplicial elimination ordering if and only if it is a chordal graph.
Proof: Necessity. Let G be a graph with a simplicial elimination ordering. Let C be a cycle in G oflength at least 4. At the point when the elimination ordering first deletes a vertex of C, say v, the remaining neighbors of v form a clique. The clique includes the neighbors of v on C; the resulting edge joining them is a chord of C. Hence G has no chordless cycle.
Sufficiency. By Lemma 5.3.16, every chordal graph has a simplicial vertex. This yields a simplicial elimination ordering by induction on n(G), since every induced subgraph of a chordal graph is a chordal graph.
Other properties of chordal graphs appear in Exercises 20-27.
## A HINT OF PERFECT GRAPHS
In Proposition 5.1.16, we proved that x(G) = w(G) when G is an interval graph. Furthermore, every induced subgraph of an interval graph is also an interval graph, since we can delete the interval representing v in an interval representation of G to obtain an interval representation of G - v. Thus x (H) = w(H) holds for every induced subgraph H of an interval graph. 5.3.18. Definition. A graph G is perfect if x (H) = w(H) for every induced subgraph H s; G. Equivalently, x(G[A]) = w(G[A]) for all As; V(G). The clique covernumber8(G) of a graph G is the minimum number of cliques in G needed to cover V ( G); note that e(G) = x (G). Since cliques and independent sets exchange roles under complementation, the statement of perfection for G is "a(H) = 8(H) for every induced subgraph H of G". Lovasz [1972a, 1972b] proved the Perfect Graph Theorem (PGT): G is perfect if and only if its complement G is perfect. We prove this in Theorem 8.1.6; here we merely illustrate perfect graphs. 5.3.19. Definition. A family of graphs G is hereditary if every induced subgraph of a graph in G is also a graph in G. -
5.3.20. Remark. In order to prove that every graph in a hereditary class G is perfect, it suffices to verify that x(G) = w(G) for every G E G. Doing so includes the proof of equality for the induced subgraphs of G.
## Section 5.3: Enumerative Aspects
227
5.3.21. Example. Bipartite graphs and their line graphs. Bipartite graphs form a hereditary class, and x(G) = w(G) for every bipartite graph; hence bipartite graphs are perfect. When H is bipartite, the statement of perfection for His Exercise 5.1.38 and follows from a(H) = fJ'(H) (Corollary 3.1.24). For bipartite graphs, the nontrivial a(G) = O(G) = fJ'(G) follows at once from the trivial x(G) = w(G) by the P~T. We briefly introduced line graphs in Definition 4.2.18 to prove the edge versions of Menger's Theorem; recall that the line graph L(G) has a vertex for each edge of G, withe, f E V(L(G)) adjacent in L(G) if they have a common endpoint in G. Line graphs of bipartite graphs form a hereditary family, since deleting a vertex in the line graph represents deleting the corresponding edge in the original graph. Tlierefore, proving that a(L(G)) = O(L(G)) when G is bipartite will show that complements of line graphs are perfect. A clique in L(G) (when G is bipartite) consists of edges in G with a common endpoint. Thus covering the vertices of L(G) with cliques corresponds to selecting vertices in G to form a vertex cover. Independent sets in L(G) are matchings in G. Thus perfection for complements ofline graphs ofbipartite graphs amounts to the Konig-Egervary Theorem (a'(G) = fJ(G)) for matchings and vertex covers in bipartite graphs. From this the PGT yie~ds also x(L(G)) = w(L(G)). A proper coloring of L(G) is a partition of E(G) into matchings, and w(L(G)) = ~(G) (for bipartite G). Hence x(L(G)) w(L(G)) means that the edges ofa bipartite graph G can be partitioned into ~(G) matchings. In Theorem 7.1.7, we prove directly this additional result of Konig [1916].
Since every interval graph is a chordal graph (Exercise 28), proving that all chordal graphs are perfect strengthens Proposition 5.1.16. We explore other characterizations of interval graphs and chordal graphs in Section 8.1.
5.3.22. Theorem. (Berge [1960]) Chordal graphs are perfect. Proof: Deleting vertices cannot create chordless cycles, so the family is hereditary. By Remark 5.3.20, we need only prove x (G) = w ( G) when G is chordal. In Theorem 5.3.17, we proved that G has a simplicial elimination ordering. Let v 1 , . , Vn be the reverse of such an ordering. For each i, the neighbor~ of v; among {vi. ... , v;_i} fotm a clique. We apply greedy coloring with this ordering. If v; receives color k, then colors 1, ... , k - 1 appear on earlier neighbors of v;. Since they form a clique, with v; we have a clique of size k. Thus we obtain a clique whose size equals the number of colors used.
The argument of Theorem 5.3.22 shows that greedy coloring relative to the reverse of a simplicial elimination ordering produces an optimal coloring. This generalizes Proposition 5.1.16 about interval graphs. We present one more fundamental class of perfect graphs; it includes all bipartite graphs.
228
## Chapter 5: Coloring of Graphs
5.3.23. * Definition. A transitive orientation of a graph G is an orientation D such that whenever xy and yz are edges in D, also there is an edge xz in G that is oriented from x to z in D. A simple graph G is a comparability graph if it has a transitive orientation. 5.3.24. * Example. If G is an X, Y-bigraph, then dire~ting every edge from X to Y yields a transitive orientation. Thus every bipartite graph is a comparability graph. Transitive orientations arise from order relations; x ---+ y could mean "x contains y", which is a transitive relation. 5.3.25. * Proposition. (Berge [1960]) Comparability graphs are perfect. Proof: Every induced subdigraph of a transitive digraph is transitive, so the class of comparability graphs is hereditary. Thus we need only show that each comparability graph G is w(G)-colorable. Let F be a transitive orientation of G; note that F has no cycle. As shown in proving Theorem 5.1.21, the coloring of G that assigns to each vertex v the number of vertices in the longest path of F ending at v is a proper coloring. By transitivity, the vertices of a path in F form a clique in G. Thus we have x(G) ~ w(G).
## COUNTING ACYCLIC ORIENTATIONS (optional)
Surprisingly, x (G; k) has meaning when k is a negative integer. An acyclic orientation of a graph is an orientation having no cycle. Setting k = -1 in x (G; k) enables us to count the acyclic orientations of G.
5.3.26. Example. Since C4 has 4 edges, it has 16 orientations. Of these, 14 are acyclic. In Example 5.3.7, we proved that x(C4 ; k) = k(k - l)(k 2 - 3k + 3). Evaluated at k = -1, this equals (-1)(-2)(7) = 14. 5.3.27. Theorem. (Stanley [1973]) The value of x(G; k) at k = -1 is (-l)n<G> times the number of acyclic orientations of G. Proof: We use induction one( G). Let a ( G) be the number of acyclic orientations of G. When G has no edges, a(G) = 1 and x(G; -1) = (-l)n(G), !'lo tha daim holds. We will prove that a(G) = a(G - e) + a(G e) fore E E(G). If so, then we apply the recurrence for a, the induction hypothesis a(G) in terms of x(G; k), and the chromatic recurrence to compute
for
## a(G) = (-lt<G)X(G - e; -1)
+ (-lr<GJ-lx(G e; -1) =
(-ly<G>x(G; -1).
Now we prove the recurrence for a. Every acyclic orientation of G contains an acyclic orientation of G - e. An acyclic orientation D of G - e may extend to 0, 1, or 2 acyclic orientations of"G by orienting the edge e = uv. When D has no u, v-path, we can choose v ---+ u. When D has no v, u-path, we can choose u ---+ v. Since Dis acyclic, D cannot have both au, v-path and a v, u-path, so the two choices for e cannot both be forbidden.
## Section 5.3: Enumerative Aspects
229
Hence every D extends in at least one way, and a ( G) equals a ( G -e) plus the number of orientations that extend in both ways. Those extending in both ways are the acyclic orientations of G - e with rt!) u, v-path and no v, u-path. There are exactly a(G e) of these, since au, v-path or av, u-path in an orientation of G - e becomes a cycle in G e. The interpretation of x (G; k) for general negative k (Exercise 32) is an instance of the phenomenon of"combinatorial reciprocity" (Stanley [1974]).
EXERCISES
Keep in mind that the notation x(G; k) may be viewed as a polynomial or as the number of proper k-colorings of G.
## 5.3.1. ( - ) Compute the chromatic polynomials of the graphs below.
5.3.2. (-)Use the chromatic recurrence to obtain the chromatic polynomial of every tree with n vertices. 5.3.3. (-) Prove that k4
-
4k3
## + 3k2 is not a chromatic polynomial.
5.3.4. a) Prove that x(C.; k) = (k -1)" + (-l)"(k - 1). b) For H = G v K 1 , prove that x(F; k) = kx(G; k - 1). From this and part (a), find the chromatic polynomial of the wheel c. v K1 . 5.3.5. For n ::: 1, let G. = P. o K2 ; this is the graph with 2n vertices and 3n - 2 edges shown below. Prove that x(G.; k) = (k 2 - 3k + 3)"- 1k(k - 1).
[ [ [
[ [ [
5.3.6. (!)Let G be a graph with n vertices. Use Proposition 5.3.4 to give a non-inductive proof that the coefficient of k"- 1 ih x(G; k) is -e(G). 5.3.7. Prove that the chromatic polynomial of an n-vertex graph has no real root larger than n - 1. (Hint: Use Proposition 5.3.4.) 5.3.8. (!) Prove that the number of proper k-colorings of a connected graph G is less than k(k - l)"- 1 if k '.?::: 3 and G is not a tree. What happens when k = 2? 5.3.9. (!) Prove that x(G; x + y) = Lu,;V(G) x(G[U]; x)x(G[UJ; y). (Hint: Since both sides are polynomials, ~t suffices to prove equality when x and y are positive integers; do this by counting proper x + y-colorings in a different way.)
230
## Chapter 5: Coloring of Graphs
5.3.10. Let G be a connected graph with x(G; k) = I:;,:;(-l)ia;kn-i. For 1::: i ::: n, prove that ai ::::: (n7 1). (Hint: Use the chromatic recurrence.) 5.3.11. (!)Prove that the sum of the coefficients of x(G; k) is 0 unless G has no edges. (Hint: When a function is a polynomial, how can one obtain the sum of the coefficients?) 5.3.12. -(+)Coefficients of x (G; k). a) Prove that the last nonzero term in the chromatic polynomial of G is the term whose exponent is the number of components of G. b) Use part (a) to prove that if p(k) =kn -akn-l +ck' and a > (-;+ 1), then pis not a chromatic polynomial. (For example, this immediately implies that the polynomial in Exercise 5.3.3 is not a chromatic polynomial.) 5.3.13. Let G and H be graphs, possibly overlapping. a) Prove that x(G UH' k) = x(G;k)x(H;k) when G n His a complete graph. x(GnH;k) b) Consider two paths whose union is a cycle to show that the formula may fail when G n H is not a complete graph. c) Apply part (a) to conclude that the chromatic number of a graph is the maximum of the chromatic numbers of its blocks. 5.3.14. (!) Let P be the Petersen graph. By Brooks' Theorem, the Petersen graph is 3colorable, and hence by the pigeonhole principle it has an independent set S of size 4. a) Prove th,at P - S = 3K2. b) Using part (a) and symmetry, determine the number of vertex partitions of P into three independent sets. c) In general, how can the number of partitions into the minimum number of independent sets be obtained from the chromatic polynomial of G? 5.3.15. Prove that a graph with chromatic number k has at most k"-k vertex partitions into k independent sets, with equality achieved only by Kk + (n - k)K 1 (a k-clique plus n - k isolated vertices). (Hint: Use induction on n and consider the deletion of a single vertex.) (Tomescu [1971]) 5.3.16. Let G be a simple graph with n vertices and m edges. Prove that G has at most H~) triangles. _Conclude that the coefficient of kn- 2 in x(G; k) is positive, unless G has at most one edge. (Hint: Use Theorem 5.3.10.) 5.3.17. (*)Use the inclusion-exclusion principle to prove Theorem 5.3.10 directly. 5.3.18. (!) Consider the chromatic polynomials o(the graphs below. a) Without computing them, give a short proof that they are equal. b) Express this chromatic polynomial as the sum of the chromatic polynomials of two chordal graphs, and use this to give a one-line computation of it.
5.3.19. (- ) Let G be the graph obtained from K 6 by subdividing one edge. Use the chromatic recurrence to Compute x (G; k) as a product of linear factors (factors of the form k - c;). Show that G is not a chordal graph. (Read [1975], Dmitriev [1980])
## Section 5.3: Enumerative Aspects
231
5.3.20. Let G be a chordal graph. Use a simplicial elimination ordering of G to prove the following statements. a) G has at most n maximal cliques, with equality if and only if G has no edges. (Fulkerson-Gross [1965]) b) Every maximal clique of G containing no simplicial vertex of G is a separating set. 5.3.21. The Szekeres-Wilf number of a graph G is 1 + maxH~G S(H). Prove that a graph G is chordal if and only if in every induced subgraph the Szekeres-Wilf number equals the clique number. (Voloshin [1982]) 5.3.22. Let k,. (G) be the number of r-cliques in a connected chordal graph G. Prove that L,> 1 (-1)'- 1k,(G) = 1. (Hint: Use induction on n(G). Note that the binomial formula (Appendix A) implies that 0 (-l)i (7) = 0 when m E N.)
Lj2'.
5.3.23. Let S be the vertex set of a cycle in a chordal graph G. Prove that G has a cycle whose vertex set consists of all but one element of S. (Comment: When G has a spanning cycle and Sc V(G), Hendry conjectured that G also has a cycle whose vertex set consists of S plus one vertex.) (Hendry [1990]) 5.3.24. Let e be a edge of a cycle C in a chordal graph. Prove that e forms a triangle with a third vertex of C. 5.3.25. Let Q be a maximal clique in a chordal graph G. Prove that if G - Q is connected, then Q contains a simplicial vertex. (Voloshin-Gorgos [1982]) 5.3.26. Exercise 5.3.13 establishes the formula x(G UH; k) = x~~~~x~~~k) when G n His a complete graph. a) Prove that the formula holds when GU His a chordal graph regardless of whether G n H is a complete graph. b) Prove that if x is a vertex in a chordal graph G, then
x( '
G k) =
x
2
## (G - x k)k x(G[N(x)]; k - 1). ' x(G[N(x)]; k)
(Comment: Part (b) allows the chromatic polynomial of a chordal graph to be computed via an arbitrary elimination ordering. For example, eliminating the central vertex of P5 yields x (P5 ; k) = [k(k - 1)] 2k <k~21 > = k(k - 1) 4 .) (Voloshin [1982]) 5.3.27. (+)A minimal vertex separator in a graph G is a set S ; V(G) that for some pair x, y is a minimal set whose deletion separates x and y. Every minimal separating set is a minimal vertex separator, but u, v below show that the converse need not hold. a) Prove that if every minimal vertex separator in G is a clique, then the same property holds in every induced subgraph of G. b) Prove that a graph G is chordal if and only if every minimal vertex separator is a clique. (Dirac _[1961])
5.3.28. (!) Let G be an interval graph. Prove that G is a chordal graph and that G iis a comparability graph.
232
## Chapter 5: Coloring of Graphs
5.3.29. Determine the smallest imperfect graph G such that x (G) = w(G). 5.3.30. An edge in an acyclic orientation of G is dependent ifreversing it yields a cycle. a) Prove that every acyclic orientation of a connected n-vertex graph has at least n - 1 independent edges. b) Prove that if x(G) is less than the girth ofG, then G has an orientation with no dependent edges. (Hint: Use the technique in the proof of Theorem 5.1.21.)
a(G - e)
5.3.31. (*)The number a(G) of acyclic orientations of G satisfies the recurrence a(G) = + a(G e) (Theorem 5.3.27). The number of spanning trees of G appears to satisfy the same recurrence; does the number of acyclic orientations of G always equal the number of spanning trees? Why or why not?
5.3.32. (*)Let D be an acyclic orientation of G, and let f be a coloring of V(G) from the set [k]. We say that (D, f) is a compatible pair ifu :-+ v in D implies f(u) .:'S f(v). Let ri(G; k) be the number of compatible pairs. Prove that ri(G; k) = (-1r<G>x(G; k). (Stanley [1973])
Chapter6
Planar Graphs
6.1. Embeddings and Euler's Formula
Topological graph theory, broadly conceived, is the study of graph layouts. Initial motivation involved the famous Four Color Problem: can the regions of every map on a globe be colored with four colors so that regions sharing a nontrivial boundary have different colors? Later motivation involves circuit layouts on silicon chips. Wire crossings cause problems in layouts, so we ask which circuits have layouts without crossings.
## DRAWINGS IN THE PLANE
The following brain teaser appeared as early as Dudeney [1917].
6.1.1. Example. Gas-water-electricity. Three sworn enemies A, B, Clive in houses in the woods. We must cut paths so that each has a path to each of three utilities, which by tradition are gas, water, and electricity. In order to avoid confrontations, we don't want any of the paths to cross. Can this be done? This asks whether K 3 ,3 can be drawn in the plane without edge crossings; we will give two proofs that it cannot.
Arguments about drawings of graphs in the plane are based on the fact that every closed curve in the plane separates the plane into two regions (the
233
234
## Chapter 6: Planar Graphs
inside and the outside). In elementary graph theory, we take this as an intuitive notion, but the full details in topology are quite difficult. Before discussing a way to make the arguments precise for graph theory, we show informally how this result is used to prove impossibility for planar drawings. 6.1.2. Proposition. Ks and K 3 .3 cannot be drawn without crossings. Proof: Consider a drawing of Ks or K 3 ,3 in the plane. Let C be a spanning cycle. If the drawing does not have crossing edges, then C is drawn as a closed curve. Chords of C must be drawn inside or outside this curve. Two chords conflict if their endpoints on C occur in alternating order. When two chords conflict, we can draw only one inside C and one outside C. A 6-cycle in K 3 .3 has three pairwise conflicting chords. We can put at most one inside and one outside, so it is not possible to complete the embedding. When C is a 5-cycle in Ks, at most two chords can go inside or outside. Since .there are five chords, again it is not possible to complete the embeddings. Hence neither of these graphs is planar.
We need a precise notion of"drawing". We have used curves for edges. Using only curves formed from line segments avoids topological difficulties. These can approximate any curve well enough that the eye cannot tell the difference. 6.1.3. Definition. A curve is the image of a continuous map from [0, 1] to JR2 A polygonal curve is a curve composed of finitely many line segments. It is a polygonal u, v-curve when it starts at u and ends at v. A drawing of a graph G is a function f defined on V ( G) U E ( G) that assigns each vertex v a point f (v) in the plane and assigns each edge with endpoints u, v a polygonal j(u), f(v)-curve. The images of vertices are distinct. A point in f (e) n f (e') that is not a common endpoint is a crossing.
It is common to use the same name for a g!"aph G and a particular drawing of G, referring to the points and curves in the drawing as the vertices and edges of G. Since the endpoint relation between the points and curves is the same as the incidence relation between the vertices and edges, the drawing can be viewed as a member of the isomorphism class containing G. By moving edges slightly, we can ensure that no three edges have a common internal point, that an edge contains no vertex except its endpoints, and that no two edges are tangent. If two edges cross more than once, then modifying them as shown below reduces the number of crossings; thus we also require that edges cross at most once. We consider only drawings with these properties.
235
/ / / /
"' '
''
"
/ /
''
'
vv
/ /
"
/ /
## " ' '
6.1.4. Definition. A graph is planar if it has a drawing without crossings. Such a drawing is a planar embedding of G. A plane graph is a particular planar embedding of a planar graph. A curve is closed if its first and last points are the same. It is simple if it has no repeated points except possibly first=last. A planar embedding of a graph cuts the plane into pieces. These pieces are fundamental objects of study. 6.1.5. Definition. An open set in the plane is a set U ~ JR2 such that for every p E U, all points within some small distance from p belong to U. A r.egion is an open set U that contains a polygonal u, v-curve for every pair u, v e U. The faces of a plane graph are the maximal regions of the plane that contain no point used in the embedding. A finite plane graph G has one unbounded face (also called the outer face). The faces are pairwise disjoint. Points p, q E JR2 lying in no edge of Gare in the same face if and only ifthere is a polygonal p, q-curve that crosses no edge. In a plane graph, every cycle is embedded as a simple closed curve. Some faces lie inside it, some outside. This again relies on the fact that a simple closed curve cuts the plane into two regions. As we have suggested, this is not too difficult for polygonal curves. We present some detail of this case in order to explain how to compute whether a point is in the inside or the outside. This proof appears in Tverberg [1980]. 6.1.6. * Theorem. (Restricted Jordan Curve Theorem) A simple closed polygonal curve C consisting of finitely many segments partitions the plane into exactly two faces, each having C as boundary. Proof: Because the list of segments is finite, nonintersecting segments cannot be arbitrarily close. Hence we can leave a face only by crossing C. As we follow C, the nearby points on our right are in a single face, and similarly for the points qn the left. (There is a precise algebraic definition for "left" and "right" here.) If x fl. C and y E C, the segment xy first intersects C somewhere, approaching it from the right or the left. Hence every point not along C lies in the same face with at least one of the two sets we have described. To prove that the points on the left and right lie in different faces, we consider rays in the plane. A ray emanating from a point pis "bad" ifit contains an endpoint of a segment of C. Since C has finitely many segments, there are finitely many bad rays from p.
236
## Chapter 6: Planar Graphs
Since the list of segments is finite, each good ray from p crosses C finitely often. As the direction changes, the number of crossings changes only at a bad direction. Before and after such a direction, the parity of the number of crossings is the same. We say that p is an even point when every good ray from p crosses Can even number of times; otherwise pis an odd point.
Given points x and y in the same face of C, let P be a polygonal x, y-curve that avoids C. Since C has finitely many segments, the endpoints of segments on P can be adjusted slightly so that the rays along segments on P are good for their endpoints. A segment of P belongs to a ray from one end that contains the other; both points have good rays in the same direction. Since the segment does not intersect C, the two points have the same parity. Hence every two points in the same face have the same parity Because the endpoints of a short segment intersecting C exactly once have opposite parity, there are two distinct faces. The even points and odd points form the outside face and the inside face, respectively.
DUAL GRAPHS
A map on the plane or the sphere can be viewed as a plane graph in which the faces are the territories, the vertices are places where boundaries meet, and the edges are the portions of the boundaries that join two vertices. We allow the full generality of loops and multiple edges. From any plane graph G, we can form a related plane graph called its "dual".
6.1.7. Definition. The dual graph G* of a plane gr!lph 9 is a plane graph whose vertices correspond to the faces of G. The edges of G* correspond to the edges of G as follows: if e is an edge of G with face X on one siqe and face Yon the other side, then the endpoints of the dual edge e* E E(G*) are the vertices x, y of G* that represent the faces X, Y of G. The order in the plane of the edges incident to x E V(G*) js the order of.the edges bounding the face X of G in a walk around its boundary. 6.1.8. Example. Every planar embedding of K 4 has four faces, and these pairwise share boundary edges. Hence the dual is another copy of K4. Every planar embedding of the cube Q3 has eight vertices, 12 edges, and six faces. Opposite faces have no common boundary; the dual is a planar embedding of K2.2.2, which has six vertices, 12 edges, and eight faces. Taking the dual can introduce loops and multiple edges. For example, let G be the paw, drawn below in bold edges as a plane graph. Its dual graph G* is
## Section 6: 1: Embeddings and Euler's Formula
237
drawn in solid edges. Since G has four vertices, four edges, and two faces, G* has four faces, four edges, and two vertices.
6.1.9. Remark. 1) Example 6.1.8 shows that a simple plane graph may have loops and multiple edges in its dual. A cut-edge of G becomes a loop in G*, because the faces on both sides of it are the same. Multiple edges arise in the dual when distinct faces of G have more than one common boundary edge. 2) Some arguments require more careful geometric description of the dual. For each face X of G, we place the dual vertex x in the interior of X, so each face of G contains one vertex of G*. For each edge e in the boundary of X, we draw a curve from x to a point on e; these do not cross. Each such curve meets another from the other side of e at the same point on e to form the edge of G* that is dual toe. No other edges enter X. Hence G* is a plane graph, and each edge of G* in this layout crosses exactly one edge of G. Such arguments lead to a proof that (G*)* is isomorphic to G if and only if G is connected (Exercise 18). Mathematicians often use the word "dual" in a setting when performing an operation twice returns the original object. 6.1.10. Example. Two embeddings ofa planar graph may have nonisomorphic duals. Each embedding shown below has three faces, so in each case the dual has three vertices. In the embedding on the right, the dual vertex corresponding to the outside face has degree 4. In the embedding on the left, no dual vertex has degree 4, so the duals are not isomorphic. This does not happen with 3-connected graphs. Every 3-connected planar graph has essentially one embedding (see Exercise 8.2.45).
When a plap.e graph is connected, the boundary of each face is a closed walk. When the gr~ph is not connected, there are faces whose boundary consists of more tha:p. one closed walk.
6.1.11. Definition. The length of a face in a plane graph G is the total length of the closed walk(s) in G bounding the face.
238
## Chapter 6: Planar Graphs
6.1.12. Example. A cut-edge belongs to the boundary of only one face, and it contributes twice to its length. Each graph in Example 6.1.10 has three faces. In the embedding on the left the lengths are 3, 6, 7; on the right they are 3, 4, 9. The sum of the lengths is 16 in each case, which is twice the number of edges. 6.1.13. Proposition. If l(F;) denotes the length of face F; in a plane graph G, then ~e(G) = L,l(F;). Proof: The face lengths are the degrees of the dual vertices. Since e(G) = e(G*), the statement 2e(G) = L,l(F;) is thus the same as the degree-sum formula 2e(G*) = L,dc.(x) for G*. (Both sums count each edge twice.) Proposition 6.1.13 illustrates that statements about a connected plane graph becomes statements about the dual graph when we interchange the roles of vertices and faces. Edges incident to a vertex become edges bounding a face, and vice versa, so the roles of face lengths and vertex degrees interchange. We can also interpret coloring of G* in terms of G. The edges of G* represent shared boundaries between faces of G. Hence the chromatic number of G* equals the number of colors needed to properly color the faces of G. Since the dual of the dual of a connected plane graph is the original graph, this means that four colors suffice to properly color the regions in every planar map if and only if every planar graph has chromatic number at most four. The Jordan Curve Theorem states that a simple closed curve cuts its interior from its exterior. Jn plane graphs, this duality between curve and cut becomes a duality between cycles and bonds. 6.1.14. Theorem. Edges in a plane graph G form a cycle in G if and only ifthe corresponding dual edges form a bond in G*. Proof: Consider D ~ E(G). Suppose first that Dis the edge set of a cycle in G. The corresponding edge set D* ~ E(G*) contains all dual edges joining faces inside D to faces outside D (the Jordan Curve Theorem implies that there is at least one of each). Thus D* contains an edge cut. If D contains a cycle and more, then D* contains an edge cut and more. If D contains no cycle in G, then it encloses no region (see Exercise 24a). It remains possible to reach the unbounded face of G from every other without crossing D. Hence G* - D* is connected, and D* contains no edge cut. Thus D* is a minimal edge cut if and only if D is a cycle.
The next remark yields an inductive proof of Theorem 6.1.14 (Exercise 19).
## Section 6.1: Embeddings and Euler's Formula
239
6.1.15. Remark. Deleting a non-cut edge of G has the effect of contracting an edge in G*, as two faces of G merge into one. Contracting a non-loop edge of G has the effect of deleting an edge in G*. Letting G be the central solid graph below, we have G - e on the left and G e on the right. Note that to maintain this duality, we keep multiple edges and loops that arise from edge contraction in plane graphs.
~
/
''
/
/
/
''
Face boundaries allow us to characterize bipartite planar graphs. The characterization can also be proved by inductfon (Exercise 20).
6.1.16. Theorem. The following are equivalent for a plane graph G. A) G is bipartite. B) Every face of G has even length. C) The dual graph G* is Eulerian. Proof: A :::} B. A face boundary consists of closed walks. Every odd closed walk contains an odd cycle. Therefore, in a bipartite plane graph the contributionsto the length of faces are all even. B :::} A. Let C be a cycle in G. Since G has no crossings, C is laid out as a simple closed curve; let F be the region enclosed by C. Every region of G is wholly within F or wholly outside F. If we sum the face lengths for the regions inside F, we obtain an even number, since each face length is even. This sum counts each edge of C once. It also counts each edge inside F twi(;e, since each such edge belongs twice to faces in F. Hence the parity of the length of C is the same as the parity of the full sum, which is even. B > C. The dual graph G* is connected, and its vertex degrees are the face lengths of G.
Many questions we consider for general planar graphs can be answered rather easily for a special class of planar graphs.
6.1.17. Definition. A graph is outerplanar if it has an embedding with every vertex on the boundary of the unbounded face. An outerplane graph is such an embedding of an outerplanar graph.
240
## Chapter 6: Planar Graphs
The graph in Example 6.1.10 is outerplanar, but another embedding is needed to demonstrate this.
6.1.18. Proposition. The boundary of the outer face a 2-connected outerplane graph is a spanning cycle. Proof: This boundary contains all the vertices. Ifit is not a cycle, then it passes through some vertex more than once. Such a vertex would be a cut-vertex. 6.1.19. Proposition. K 4 and K 2 ,3 are planar but not outerplanar. Proof: The figure below shows that K4 and K 2 .3 are planar. To show that they are not outerplanar, .observe that they are 2-connected. Thus an outerplane embedding requires a spanning cycle. There is no spanning cycle in K 2 .3 , since it would be a cycle oflength 5 in: a bipartite graph. There is a spanning cycle in K4 , but the endpoints of the remaining two edges alternate along it. Hence these chords conflict and cannot both be drawn inside. Drawing a chord outside separates a vertex from the outer face.
6.1.20. Proposition. Every simple outerplanar graph has a vertex of degree at most 2. Proof: It suffices to prove the statement for connected graphs. We use induction on n(G); when n(G) ~ 3, every vertex.has degree at most 2. For n(G) :::: 4, we prove the stronger statement that G has two nonadjacent vertices of degree at most 2. Basis step (n(G) = 4): Since K 4 is not outerplanar, G has nonadjacent vertices, and two nonadjacent vertices have degree at most 2. Induction step (n ( G) :::: 4): If G has a cut-vertex x, then each {x }-lobe of G has a vertex of degree at most 2 other than x, and these are nonadjacent in G. If G is 2-connected, then the outer face boundary is a cycle C. If C has no chords, then G is 2-regular. If xy is a chord of C, then the vertex sets of the two x, y-paths on C both induce outerplanar subgraphs. By the induction hypothesis, these subgraphs H, H' contain vertices z, z'of degree at most 2 that are not in {x, y} (this includes the case where Hor H' is K 3 ). Since no chord of C can be drawn outside C or cross xy, we have z ~ z'. Thus z, z' is the desired pair of vertices.
x
H'
y
z'
## Section 6.1: Embeddings and Euler's Formula
241
EULER'S FORMULA
Euler's Formula (n - e + f = 2) is the basic counting tool relating vertices, edges, and faces in planar graphs. 6.1.21. Theorem. (Euler [17 58] ): If a connected plane graph G has exactly n vertices, e edges, and f faces, then n - e + f = 2. Proof: We use induction on n. Basis step (n = 1): G is a "bouquet" of loops, each a closed curve in the embedding. If e = 0, then f = 1, and the formula holds. Each added loop passes through a face and cuts it into two faces (by the Jordan Curve Theorem). This augments the edge count and the face count each by 1. Tpus the formula holds when n = 1 for any number of edges. Induction step (n > 1): Since G is connected, we can find an edge that is not a loop. When we contract such an edge, we obtain a plane graph G' with n' vertices, e' edges, and f' faces. The contraction does not change the number of faces (we merely shortened boundaries), but it reduces the number of edges and vertices by 1, so n' = n - 1, e' = e - 1, and f' = f. Applying the induction hypothesis yields
n - e + f = n'
+1-
(e'
+ 1) + f' =
n' - e'
+ f'
= 2.
oCD
n=l
6.1.22. Remark. 1) By Euler's Formula, all planar embeddings of a connected graph G have the same number of faces. Although the dual may depend on the embedding chosen for G, the number of vertices in the dual does not. 2) Euler's Formula as stated fails for disconnected graphs. If a plane graph G has k components, then adding k-1 edges to G yields a connected plane graph without changing the number of faces. Hence Euler's Formula generalizes for plane graphs with k components as n - e + f = k + 1 (for example, consider a graph with n vertices and no edges).
Euler's Formula has many applications, particularly for simple plane graphs, where all faces have length at least 3.
6.1.23. Theorem. If G is a simple planar graph with at least three vertices, then e(G) :=:: 3n(G) - 6. If also G is triangle-free, then e(G) :=:: 2n(G) - 4. Proof: It suffices to consider connected graphs; otherwise we could add edges. Euler's Formula will relate n(G) and e(G) if we can dispose off. Proposition 6.1.13 provides an inequality between e and f. E\.'ery face boundary in a simple graph contains at least three edges (if n(G):::::: 3). Letting {/;}be the list of face lengths, this yields 2e = L, f; :::::: 3/. Substituting into n - e + f = 2 yields e :=:: 3n - 6.
242
## Chapter 6: Planar Graphs
When G is triangle-free, the faces have length at least 4. In this case 2e = L f; ~ 4/, and we obtain e ~ 2n - 4.
6.1.24. Example. Nonplanarity of K 5 and K3, 3 follows immediately from Theorem 6.1.23. For K5, we have e = 10 > 9 = 3n - 6. Since K3,3 is triangle-free, we have e = 9 > 8 = 2n - 4. These graphs have too many edges to be planar. 6.1.25. Definition. A maximal planar graph is a simple planar graph that is not a spanning subgraph of another planar graph. A triangulation is a simple plane graph where every face boundary is a 3-cycle. 6.1.26. Proposition. For a simple n-vertex plane graph G, the following are equivalent. A) G has 3n - 6 edges. B) G is a triangulation. C) G is a maximal plane graph. Proof: A <=> B. For a simple n-vertex plane graph, the proof of Theorem 6.1.23 shows that having 3n - 6 edges is equivalent to 2e = 3 f, which occurs if and only if every face is a 3-cycle. B <=> C. There is a face that is longer than a 3-cycle if and only if there is a way to add an edge to the drawing and obtain a larger simple plane graph. 6.1.27. Remark. A graph embeds in the plane if and only if it embeds on a sphere. Given an embedding on a sphere, we can puncture the sphere inside a face and project the embedding onto a plane tangent to the opposite point. This yields a planar embedding in which the punctured face on the sphere becomes the unbounded face in the plane. The process is reversible. 6.1.28. Application. Regular polyhedra. Informally, we think of a regular polyhedron as a solid whose boundary consists of regular polygons of the same length, with the same number of faces meeting at each vertex. When we expand the polyhedron out to a sphere and then lay out the drawing in the plane as in Remark 6.1.27, we obtain a regular plane graph with faces of the same length. Hence the dual also is a regular graph. Let G be a plane graph with n vertices, e edges, and f faces. Supposethat G is regular of degree k and that all faces have length I. The degree-sum formula for G and for G* yields kn = 2e = If. By substituting for n and f in Euler's Formula, we obtain e(f - 1 + = 2. Since e and 2 are positive, the other factor must also be positive, which yields (2/ k) + (2/ /) > 1, and hence 2/ + 2k > kl. This inequality is equivalent to (k - 2)(/ - 2) < 4. Because the dual of a 2-regular graph is not simple, we require that k, l ~ 3. Now (k - 2)(/ - 2) < 4 also requires k, I s 5. The only integer pairs satisfying these requirements for (k, I) are (3, 3), (3, 4), (3, 5), (4, 3), and (5, 3). Once we specify k and l, there is only one way to lay out the plane graph when we start with any face. Hence there are only the five Platonic solids listed below, one for each pair (k, l) that satisfying the requirements.
f)
## Section 6.1: Embeddings and Euler's Formula
243
k 3 3 4 3
5
l 3 4 3
5
(k - 2)(/ - 2) 1 2 2 3 3
e 6 12 12 30 30
n f 4 4 8 6 8 6 20 12 12 20
## name tetrahedron cube octahedron dodecahedron icosahedron
EXERCISES
6.1.1., (-) Prove or disprove: a) Every subgraph of a planar graph is planar. b) Every subgraph of a nonplanar graph is nonplanar. 6.1.2. (-) Show that the graphs formed by deleting one edge from K 5 and K 3 , 3 are planar. 6.1.3. (-) Determine all r, s such that
Kr,s
is planar.
6.1.4. (-) Determine the number of isomorphism classes of planar graphs that can be ,obtained as planar duals of the graph below
6.1.5. (-) Prove that a plane graph has a cut-vertex if and only if its dual has a cutvertex. 6.1.6. (-) Prove that a plane graph is 2-connected if and only if for every face, the bounding walk is a cycle. 6.1. 7. (-) A maximal outerplanar graph is a simple outerplanar graph that is not a spanning subgraph of a larger simple outerplanar graph. Let G be a maximal outerplanar graph with at least three vertices. Prove that G is 2-connected. 6.1.8. (-) Prove that every simple planar graph has a vertex of degree at most 5. 6.1.9. (-) Use Theorem 6.1.23 to prove that every simple planar graph w~th fewer than 12 vertices has a vertex of degree at most 4. 6.1.10. (-) Prove or disprove: There is no simple bipartite planar graph with minimum degree at least 4. 6.1.11. (-) Let G be a maximal planar graph. Prove that G* is 2-edge-connected and 3-regular. 6.1.12. (-)Draw the five regular polyhedra as planar graphs. Show that the octahedron is the dual of the cube and the icosahedron is the dual of the dodecahedron .
244
6.1.13. Find a planar embedding of the graph below.
## Chapter 6: Planar Graphs
6.1.14. Prove or disprove: For each n EN, there is a simple connected 4-regular planar graph with more than n vertices. 6.1.15. Construct a 3-regular planar graph of diameter 3 with 12 vertices. (Comment: T. Barcume proved that no such graph has more than 12 vertices.) 6.1.16. Let F be a figure drawn continuously in the plane without retracing any segment, ending a~ the start (this can be viewed as an Eulerian graph). Prove that F can be drawn without allowing the pencil point to cross what has already been drawn. :For example, the figure below has two traversals; one crosses itself and the other does not.
6.1.17. Prove or disprove: If G is a 2-connected simple plane graph with minimum degree 3, then the dual grapha is simple. 6.1.18. Given a.plane graph G, draw the dual graph G* so that each dual edge intersects its corresponding edge in G and no other edge. Prove the followi~g. a) G* is connected. b) If G is connected, then each face of G* contains exactly one vertex of G. c) (G*)* = G if and only if G is connected. 6.1.19. Let G be a plane graph. Use induction on e(G) to prove Theorem 6.1.14: a set D ~ E(G) is a cycle in G if and only if the corresponding set D* ~ E(G*) is a bond in G*. (Hint: Contract an edge of D and apply Remark 6.1.15.) 6.1.20. Prove by induction on the number of faces that a plane graph G is bipartite if and only if every face has even length. 6.1.21. (!) Prove that a set of edges in a connected plane graph G forms a spanning tree of G if and only if the duals of the remaining edges form a spanning tree of G*. 6.1.22. The weak dual of a plane graph G is the graph obtained from the dual G* by deleting the vertex for the unbounded face of G. Prove that the weak dual of an outerplane graph is a forest. 6.1.23. (!) Directed plane graphs. Let G be a plane graph, and let D be an orientation of G. The dual D* is an orientation of G* such that when an edge of Dis traversed from
## Section 6.1: Embeddings and Euler's Formula
245
tail to head, the dual edge in D* crosses it from right to left. For example, if the solid edges below are in D, then the dashed edges are in D*.
_,,~,
~/',/
Prove that if Dis strongly connected, then D* has no cycle, and 8-(D*) = 8+(D*) = 0. Conclude that if D is strongly connected, then D has a face on which the edges form a clockwise cycle and another face on which the edges form a counterclockwise cycle.
6.1.24. (!) Alternative proof of Euler's Formula. a) Use polygonal curves (not Euler's Formula) to prove by induction on n(G) that every planar embedding of a tree G has one face. b) Prove Euler's Formula by induction on the number of cycles. 6.1.25. (!) Prove that every n-vertex plane graph isomorphic to its dual has 2n - 2 edges. For all n ::::: 4, construct a simple n-vertex plane graph isomorphic to its dual. 6.1.26. Determine the maximum number of edges in a simple outerplane graph with n vertices, giving three proofs. a) By induction on n. b) By using Euler's Formula. c) By adding a vertex in the unbounded face and using Theorem 6.1.23. 6.1.27. Let G be a connected 3-regular plane graph in which every vertex lies on one face oflength 4, one face oflength 6, and one face oflength 8. a) In terms of n(G), determip.e the number of faces of each length. b) Use Euler's Formula and part (a) to determine the number of faces of G. 6.1.28. Let C be a closed curve bounding a convex region in the plane. Suppose that m chords of C are drawn so that no three share a point and no two share an endpoint. Let p be the number of pairs of chords that cross. In terms of m and p, compute the number of segments and the number ofregions formed inside C. (Alexanderson-Wetzel [1977]) 6.1.29. Prove that the complement of a simple planar graph with at least 11 vertices is nonplanar. Construct a self-complementary simple planar graph with 8 vertices. 6.1.30. (!) Let G be an n-vertex simple planar graph with girth k. Prove that G has at most (n - 2) k~ 2 edges. Use this to prove that the Petersen graph is nonplanar. 6.1.31. Let G be the simple graph with vertex set v1 ... , v. whose edges are {v;vj: Ii - ii::: 3}. Prove that G is a maximal planar graph. 6.1.32. Let G be a maximal planar graph. Prove that if S is a separating 3-set of G*, then G* - S has two components. (Chappell) 6.1.33. (!) Let G be a triangulation, and let n; be the number of vertices of degree i in G. Prove that ~)6 - i)n; = 12. 6.1.34. Construct an infinite family of simple planar graphs with minimum degree 5 such that each has exactly 12 vertices of degree 5. (Hint: Modify the dodecahedron.) 6.1.35. (!)Prove that every simple planar graph with at least four vertices has at least four vertices with degree less than 6. For each even value of n with n ::::: 8, construct an n-vertex simple planar graph G that has exactly four vertices with degtee less than 6. (Griinbaum-Motzkin [1963])
246
## Chapter 6: Planar Graphs
6.1.36. Let S be a set of n points in the plane such that for all x, y E S, the distance in the plane between x and y is at least 1. Prove that there are at most 3n - 6 pairs u, v in S such that the distance in the plane between u and v is exactly 1. 6.1.37. Given integers k ::::: 2, I ::::: 1, and kl even, construct a planar graph with exactly k faces in which every face has length I.
## 6.2. Characterization of Planar Graphs
Which graphs embed in the plane? We have prov~d that K 5 and Ka.a do not. In fact, these are the crucial graphs and lead to a characterization of planar graphs known as Kuratowski's Theorem. Kasimir Kuratowski once asked Frank Harary about the origin of the notation for K 5 and Ka.a. Harary replied, "'The Kin K 5 stands for Kasimir, and the Kin Ka.a stands for Kuratowski!" Recall that a subdivision of a graph is a graph obtained from it by replacing edges with pairwise internally-disjoint paths (Definition 5.2.19).
a subdivision of Ka.a
6.2.1. Proposition. If a graph G has a subgraph that is a subdivision of K 5 or Ka.a, then G is nonplanar. Proof: Every subgraph of a planar graph is planar, so it suffices to show that subdivisions of K5 and Ka.a are nonplanar. Subdividing edges does ot affect planarity; the curves in an embedding of a subdivision of G can be used to obtain an embedding of G, and vice versa.
By Proposition 6.2.1, avoiding subdivisions of K 5 and Ka.a is a necessary condition for being a planar graph. Kuratowski proved TONCAS:
6.2.2. Theorem. (Kuratowski [1930]) A graph is planar if and only if it does not contain a subdivision of K 5 or Ka.a.
Kuratowski's Theorem is our goal in the first half of this section, after which we will comment on other characterizations of planar graphs. When G is planar, we can seek a planar embedding with additional properties. Wagner [1936], Fary [1948], and Stein [1951] showed that every finite
## Section 6.2: Characterization of Planar Graphs
247
simple planar graph has an embedding in which all edges are straight line segments; this is known as Fary's Theorem (Exercise 6). For 3-connected planar graphs, we will prove the stronger property that there exists an embedding in which every face is a convex polygon.
## PREPARATION FOR KURATOWSKl'S THEOREM
We introduce short names for subgraphs that demonstrate nonplanarity. 6.2.3. Definition. A Kuratowski subgraph of G is a subgraph of G that is a subdivision of K5 or K 3 .a. A minimal nonplanar graph is a nonplanar graph such that every proper subgraph is planar. We will prove that a minimal nonplanar graph with no Kuratowski subgraph must be 3-connected. Showing that every 3-connected graph with no Kuratowski subgraph is planar then completes the proof ofKuratowski's Theorem. 6.2.4. Lemma. If F is the edge set of a face in a planar embedding of G, then G has an embedding with F being the edge set of the unbounded face. Proof: Project the embedding onto the sphere, where the edge sets of regions remain the same and all regions are bounded, and then return to the plane by projecting from inside the face bounded by F. 6.2.5. Lemma. Every minimal nonplanar graph is 2-connected. Proof: Let G be a minimal nonplanar graph. If G is disconnected, then we embed one component of G inside one face of an embedding of the rest. If G has a cut-vertex v, let Gi. ... , Gk be the {v}-lobes of G. By the minimality of G, each G; is planar. By Lemma 6.2.4, we can embed each G; with v on the outside face. We squeeze each embedding to fit in an angle smaller than 360/ k degrees at v, after which we combine the embeddings at v to obtain an embedding of G. 6.2.6. Lemma. Let S = {x, y} be a separating 2-set of G. If G is nonplanar, then adding the edge xy to some S-lobe of G yields a nonplanar graph. Proof: Let Gi. ... , Gk be the S-lobes of G, and let Hi= G; Uxy. If H; is planar, then by Lemma 6.2.4 it has an embedding with xy on the outside face. For each i > 1, this allows H; to be attached to an embedding of LJ~-=:,11 Hj by embedding H; in a face that has xy on its boundary. Afterwards, deleting the edge xy if it is not in G yields a planar embedding of G. The next lemma allows us to restrict our attention to 3-connected graphs in order to prove Kuratowski's Theorem. The hypothesized graph doesn't exist, but if it did, it would be 3-connected.
248
## Chapter 6: Planar Graphs
6.2.7. Lemma. If G is a graph with fewest edges among all nonplanar graphs without Kuratowski subgraphs, then G is 3-connected. Proof: Deleting an edge of G cannot create a Kuratowski subgraph in G. The hypothesis thus guarantees that deleting one edge produces a planar subgraph, and hence G is a minimal nonplanar graph. By Lemma 6.2.5, G is 2-connected. Suppose that G has a separating 2-set S = (x, y}. Since G is nonplanar, the union of xy with some S-lobe is nonplanar (Lemma 6.2.6); let H be such a graph. Since H has fewer edges than G, the minimality of G forces H to have a Kuratowski subgraph F. All of F appears in G except possibly the edge xy. Since Sis a minimal vertex cut, both x and y have neighbors in every Slobe. Thus we can replace xy in F with an x, y-path through another S-lobe to obtain a Kuratowski subgraph of G. This contradicts the hypothesis that G has no Kuratowski subgraph, so G has no separating 2-set.
CONVEX EMBEDDINGS
To complete the proof of Kuratowski's Theorem, it suffices to prove that 3-connected graphs without Kuratowski subgraphs are planar. We will use induction. In order to facilitate the proof of the induction step, it is helpful to prove a stronger statement.
6.2.8. Definition. A convex embedding of a graph is a planar embedding in which each face boundary is a convex polygon.
Tutte [1960, 1963] proved that every 3-connected planar graph has a convex embedding. This is best possible in terms of connectivity, since for n 2: 4 the 2connected planar graph K 2 ,n has no convex embedding. We follow Thomassen's approach to proving Kuratowski's Theorem by proving Tutte's stronger conclusion for 3-connected graphs without Kuratowski subgraphs. (Another proof of Tutte's result is based on ear decompositions-Kelmans [2000] .) We prove this theorem of Tutte by induction on n(G). The paradigm for proving conditional statements by induction (Remark 1.3.25) tells us what lemmas we need. Our hypotheses are "3-connected" and "no Kuratowski subgraph"; our conclusion is "convex embedding". For a graph G satisfying the hypotheses, we need to find a smaller graph G' that satisfies both hypotheses in order to apply the induction hypothesis.
## Section 6.2: Characterization of Planar Graphs
249
The first lemma allows us to obtain a smaller 3-connected graph G' by contracting some edge in G. The second shows that G' will also satisfy the hypothesis of having no Kuratow ski subgraph. The proof will then be completed by obtaining a convex embedding of G from a convex embedding of G'. 6.2.9. Lemma. (Thomassen [1980]) Every 3-connected graph G with at least five vertices has an edge e such that G e is 3-connected. Proof: We use contradiction and extremality. Consider an edge e with endpoints x, y. If G e is not 3-connected, then it has a separating 2-set S. Since G is 3-connected, S must include the vertex obtained by shrinking e. Let z denote th,e other vertex of Sand call it the mate of the adjacent pair x, y. Note that {x, y, z} is a separating 3-set in G. Suppose that G has no edge whose contraction yields a 3-connected graph, so every adjacent pair has a mate. Among all the edges of G, choose e = xy and their mate z so that the resulting disconnected graph G - [x, y, z} has a component H with the largest order. Let H' be another component of G {x, y, z} (see the figure below). Since {x, y, z} is a minimal separating set, each of x, y, z has a neighbor in each of H, H'. Let u be a neighbor of z in H', and let v be the :iate of u, z. By the definition of "mate", G - {z, u, v} is disconnected. However, the subgraph of G induced by V (H) u {x, y} is connected. Deleting v from this subgraph, if it occurs there, cannot disconnect it, since then G - {z, v} would be disconnected. Therefore, Gv(H)u{x,y} - vis contained in a component of G {z, u, v} that has more vertices than H, which contradicts the choice of x, y, z.
Next we need to show that edge contraction preserves the absence of Kuratowski subgraphs. We introduce a convenient term: the branch vertices in a subdivision H' of H are the vertices of degree at least 3 in H'. 6.2.10. Lemma. If G has no Kuratowski subgraph, then also G e has no Kuratowski subgraph. Proof: We prove the contrapositive: If G e contains a Kuratowski subgraph, then so does G. Let z be the vertex of G e obtained by contracting e = xy. If z is not in H, then H itself is a Kuratowski subgraph of G. If z E V (H) but z is not a branch vertex of H, then we obtain a Kuratowski subgraph of G from H by replacing z with x or y or with the edge xy. Similarly, if z is a branch vertex in H and at most one edge incident to z in
250
## Chapter 6: Planar Graphs
H is incident to x in G, then expanding z into xy lengthens that path, and y is the corresponding branch vertex for a Kuratow ski subgraph in G. In the remaining case (shown below), H is a subdivision of K 5 and z is a branch vertex, and the four edges incident to z in H consist of two incident to x and two incident toy in G. In this case, let ui. u 2 be the branch vertices of H that are at the other ends of the paths leaving z on edges incident toxin G, and let vi. v2 be the branch vertices of H that are at the other ends of the paths leaving z on edges incident toy in G. By deleting the ui. u2-path and v1, v2-path (rom H, we obtain a subdivision of K 3 ,3 in G, in which y, ui. u2 are the branch vertices for one partite set and x, vi. v2 are the branch vertices of the other.
K5 subdivision in G'
K3,3
subdivision in G
## Now we can prove Tutte's Theorem.
6.2.11. Theorem; (Tutte [1960, 1963]) If G is a 3-connected graph v:ith no subdivision of K 5 or K 3,3, then G has a convex embedding in the plane with no three vertices on a line. Proof: (Thomassen [1980, 1981]) We use induction on n(G). Basis step: n ( G) ::=:: 4. The only 3-connected graph with at most four vertices is K4, which has such an embedding. Induction step: n(G) :::: 5. Let e be an edge such that G e is 3-connected, as guaranteed by Lemma 6.2.9. Let z be the vertex obtained by contracting e. By Lemma 6.2.10, G e has no Kuratowski subgraph. By the induction hypothesis, we obtain a convex embedding of H = G e with no three vertices. on a line. In this embedding, the subgraph obtained by deleting the edges incident to z has a face containing z (perhaps unbounded). Since H - z is 2-connected, the boundary of this face is a cycle C. All neighbors of z lie on C; they may be neighbors in G of x or y or both, where x and y are the original endpoints of e. The convex embedding of H includes straight segments from z to all its neighbors. Let xi. ... , xk be the neighbors of x in cyclic order on C. If all neighbors of y lie in the portion of C from xi to Xi+l then we obtain a convex embedding of G by putting x at z in H and putting y at a point close to z in the wedge formed by xx; and xxi+i. as shown in the diagrams for Case 0 below. If this does not occur, then either 1) y shares three neighbors u, v, w with x, or 2) y has neighbors u, v that alternate on C with neighbors xi, Xi+l of x. In Case 1, C together with xy and the edges from {x, y} to {u, v, x} form a subdivision of K5. In Case 2, C together with the paths uyv, x;xx;+ 1, and xy form a
## Section 6.2: Characterization of Planar Graphs
251
subdivision of K 3 .3 . Since we are considering only graphs without Kuratowski subgraphs, in fact Case 0 must occur.
u v
Xi+l
X;
Case 0
Case 1
Case 2
Together, Lemma 6.2.7 and Theorem 6.2.11 imply Kuratowski's Theorem (Theorem 6.2.2). Fary's Theorem can be obtained separately: if a graph has a planar embedding, then it has a straight-line planar embedding (Exercise 6). For applications in computer science, we want more-a straight-line planar embedding in which the vertices are located at the integer points in a relatively small grid. Schnyder [1992] proved that every n-vertex planar graph has a straight-line embedding in which the vertices are located at integer points in the grid [n - 1] x [n - 1]. Many other characterizations of planar graphs have been proved; some are mentioned in the exercises. We describe two additional characterizations.
6.2.12. * Definitfon. A graph H is a minor of a graph G if a copy of H can be obtained from G by deleting and/or contracting edges of G.
For example, K5 is a minor of the Petersen graph, although the Petersen graph does not contain a subdivision of K 5 .
6.2.13.* Remark. Deletions and contractions can be performed in any order, as long as we keep track of which edge is which. Thus the minors of G can be described as "contractions of subgraphs of G". If G contains a subdivision of H, say H', then H also is a minor of G, obtained by deleting the edges of G not in H' and then contracting edges incident to vertices of degree 2. If H has maximum degree at most 3, then H is a minor of G if and only if G contains a subdivision of H (Exercise 11). Wagner [1937] proved that a graph G is planar if and only ifneither K 5 nor K 3 ,3 is a minor of G. Exercise 12 obtains this from Kuratowski's Theorem. 6.2.14. * Remark. Some characterizations are more closely related to actual embeddings. For example, when a 3-connected graph is drawn in the plane, deleting the vertex set of a facial cycle leaves a connected subgraph. We say that a cycle in a graph is nonseparating if its vertex set is not a separating set. Kelmans [1980, 1981b] proved that a subdivision of a 3-connected graph is planar if an<;l only if every edge e lies in exactly two nonseparating cycles. Kelmans [1993] surveys related material.
252
## PLANARITY TESTING (optional)
Dirac and Schuster [1954] gave the first short proof of Kuratowski's Theorem. Appearing in Harary [1969, 109-112], Bondy-Murty [1976, p153-156], and Chartrand-Lesniak [1986, p96-98], it uses special subgraphs of a graph.
6.2.15. Definition. When H is a subgraph of G, an H -fragment of G is either 1) an edge not in H whose endpoints are in H, or 2) a component of G - V (H) together with the edges (and vertices of attachment) that connect it to H.
Together with the subgraph H itself, the H -fragments form a decomposition of G. The H -fragments are the "pieces" that must be added to an embedding of H to obtain an embedding of G. Historically, the term "H -bridge" was used; we use "H-fragment" to avoid confusion with other uses of"bridge". An H-fragment differs from a V (H)-lobe because the H -fragment omits the edges of H. Also, an H-fragment may be a single_ edge not in H but joining vertices of H, since H need not be an induced subgraph. For the 3-connected case of Kuratowski's Theorem, Dirac and Schuster considered a minimal nonplanar 3-connected graph G with no Kuratowski subgraph. Deleting an edge e yields a planar 2-connected graph. After choosing a cycle C through the endpoints of e, we can add e to the embedding unless there is a C-fragment embedded inside C and another embedded outside C that "conflict" withe. As in the proof of Theorem 6.2.11, this produ~es a Kuratowski subgraph of G. Tutte used the idea of conflicting C-fragments to obtain another characterization of planar graphs.
o.~.16.
Definition. Let C be a cycle in a graph G. Two C-fragments A, B conflict if they have three common vertices of attachment to C or if there are four vertices vi, v2 , v3 , v4 in cyclic order on C such that vi. v3 are vertices of attachment of A and v2 , v4 are vertices of attachment of B. The conflict graph of C is a graph whose vertices are the C-fragments of G, with conflicting C-fragments adjacent.
Tutte [1958] proved that G is planar if and only ifthe conflict gr~ph of each cycle in G is bipartite (Exercise 13). We used this idea in our first proof that K5 and K3,3 are nonplanar (Proposition 6.1.2); the conflict graph of a spanning cycle in K3.3 is C3 , and the conflict graph of a spanning cycle in K 5 is C5 . Nonplanar 3-connected graphs have Kuratowski subgraphs of a special type. Kelmans [1984a] conjectured this extension of Kuratowski's Theorem, and it was proved independently by Kelmans [1983, 1984b] and Thomassen [1984]: Every 3-connected non planar graph with at least six vertices contains a cycle with three pairwise crossing chords. Characterizations of planarity lead us to ask whether we can test quickly whether a graph is planar. There are linear-time algorithms due to Hopcroft and Tarjan [1974] and to Booth and Luecker [1976], but these are very complicated (Gould [1988, pl 77-185] discusses the ideas used in the Hopcroft-Tarjan
## Section 6.2: Characterization of Planar Graphs
253
algorithm). A simpler earlier algorithm is not linear but runs in polynomial time. Due to Demoucron, Malgrange, and Pertuiset [1964], it uses H -fragments. The idea is that if a planar embedding of H can be extended to a planar embedding of G, then in that extension every H-fragment of G appears inside a single face of H. We build increasingly larger plane subgraphs H of G that can be extended to an embedding of G if G is planar. We try to enlarge H by making small decisions that won't lead to trouble. To enlarge H, we choose a face F that can accept an H-fragment B; the boundary of F must contain all vertices of attachment of B. Although we do not know the best way to embed B in F, a single path in B between vertices of attachment by itself has only one way to be added across F, so we add such a path. The details of choosing F and B appear below. Like the other algorithms menti~ned, this algorithm produces an embedding if G is planar.
6.2.17. Algorithm. (Planarity Testing) Input: A 2-connected graph. (Since G is planar if and only if each block of G is planar, and Algorithm 4.1.23 computes blocks, we may assume that G is a block with at least three vertices.) Idea: Successively add paths from current fragments. Maintain the vertex sets forming face boundaries of the subgraph already embedded. Initialization: Go is an arbitary cycle in G embedded in the plane, with two face boundaries consisting of its vertices. Iteration: Having determined G;, find G;+1 as follows. 1. Determine all G;-fragments of the input block G. 2. For each G;-fragment B, determine all faces of G; that contain all vertices of attachment of B; call this set F(B). 3. If F(B) is empty for some B, return NONPLANAR. If IF(B)I = 1 for some B, select such a B. If IF(B)I > 1 for every B, select any B. 4. Choose a path P between two vertices of attachment of the selected B. Embed P across a face in F(B). Call the resulting graph G;+1 and update the list of face boundaries. 5. If G;+i = G, return PLANAR. Otherwise, augment i and return to Step 1. 6.2.18. Example. Consider the two graphs below (from Bondy-Murty [1976, p165-166]). Algorithm 6.2.17 produces a planar embedding of the graph on the left, but it terminates in Step 3 for the graph on the right. The cycle 12348765 has three pairwise crossing chords: 14, 27, 36. -
8
1
254
## Chapter 6: Planar Graphs
6.2.19. Theorem. (Demoucron-Malgrange-Pertuiset [1964]) Algorithm 6.2.17 produces a planar embedding if G is planar. Proof: We may assume that G is 2-connected. A cycle appears as a simple closed curve in every planar embedding. Since we can reflect the plane, every embedding of a cycle in a planar graph G extends to an embedding of G. Hence Go extends to a planar embedding of G if G is planar. It suffices to show that if the plane graph Gi is extendable to a planar embedding of G and the algorithm produces a plane graph Gi+l from G;, then G;+1 also is extendable to a planar embedding of G. Note that every Gi-fragment has at least two vertices of attachment, since G is 2-connected, If some G;-fragment B has [F(B)I = 1, then there is only one face of G; that can contain P in an extension of G; to a planar embedding of G. The algorithm puts Pin that face to obtain G;+ 1, so in this case G;+ 1 is extendable. Problems can arise only if IF(B)I > 1 for all Band we select the wrong face in which to embed a path P from the selected fragment. Suppose that (1) we embed Pin face f E F(B), and (2) G; can be extended to a planar embedding G of Gin which Pis inside face f' E F(B). We modify G to show that G; can be extended to another embedding G' of G in which P is inside f. This shows that our chc.ice causes no problem, and the constructed G;+i is extendable. Let C be the set of vertices in the boundaries of both f and f'; this includes the vertices of attachment of B. We draw G' by switching between f and f' a11 G;-fragments that Gplaces inf or f' and whose vertices of attachment lie in C. We show this on the left below, where edges of G not present in G; are dashed.
f / --- D
I
'
. ---- ---.-
I I
I I I I
: ' '
/ /
' I,
'
'
\
\
I \
I B I!'
:,..\
---
I
!'
---- ----
I
B'
I!'
{; G' The change switches B and produces the desired embedding G' unless some unswitched G;-fragment B conflicts with a switched fragment. Since the switch is symmetric in f and f' and changes only their interiors, we may assume that B appears in f in G. "Conflict" means that G has some B' in f', which we are trying to move to f, such that Band B' are adjacent in the conflict graph off. Let A, A' denote the vertex sets where B, B' attach to the boundary off. -Since B and B' conflict, A, A' have three common vertices or four alternating vertices on the boundary off. Since A' C but A ~ C, the first possibility implies the second. Let x, u, y, v be the alternation, with x, y E A' C and u, v E A. We may assume that u ~ C, as shown on the right above; if there is no such alternation, then B, B' do not conflict or B can switch to f'. Since u ~ Candy is between u and v on f, no other face contains both u and v. Thus B fails to have its vertices of attachment contained in at least two faces, contradicting the hypothesis that I F(B)I > 1.
## Section 6.2: Characterization of Planar Graphs
255
We can begin by checking that G has at most 3n - 6 edges, maintain appropriate lists for the face boundaries, and perform the other operations via searches of linear size. Thus this algorithm runs in quadratic time. The proof of Ktiratowski's Theorem by Klotz [1989] also gives a quadratic algorithm to test planarity, and it finds a Kuratowski subgraph when G is not planar.
EXERCISES
6.2.1. (-) Prove that the complement of the 3-dimensional cube Q 3 is non planar. 6.2.2.' (-) Give three proofs that the Petersen graph is non planar. a) Using Kuratowski's Theorem. b) Using Euler's Formula and the fact that the Petersen graph has girth 5. c) Using the planarity-testing algorithm ofDemoucron-Malgrange-Pcrtuiset. 6.2.3. (-) Find a convex embedding in the plane for the graph below.
6.2.4. (-) For each graph below, prove nonplanarity or provide a convex embedding.
6.2.5. Determine the minimum number of edges that must be deleted from the Pete(sen graph to obtain a planar subgraph. 6.2.6. (!) Fary's Theorem. Let R be a region in the plane bounded by a simple polygon with at most five sides (simple polygon means the edges are line segments that do not cross). Prove there is a point x inside R that "sees" all of R, meaning that the segment from x to any point of R does not cross the boundary of R. Use this to prove inductively that every simple planar graph has a straight-line embedding.
256
## Chapter 6: Planar Graphs
6.2. 7. (!) Use Kuratowski's Theorem to prove that G is outerplanar if and only if it has no subgraph that is a subdivision of K4 or K2.a. (Hint: To apply Kuratowski's Theorem, find an appropriate modification of G. This is much easier than trying to mimic a proof of Kuratowski's Theorem.) 6.2.8. (!) Prove that every 3-connected graph with at least six vertices that contains a subdivision of Ks also contains a subdivision of Ka.a. (Wagner [1937]) 6.2.9. (+) For n 2: 5, prove that the maximum number of edges in a simple planar nvertex graph not having two disjoint cycles is 2n - 1. (Comment: Compare with Exercise 5.2.28.) (Markus [1999]) 6.2.10. (!) Let f (n) be the maximum number of edges in a simple n-vertex graph containing no Ka.a-subdivision. a) Given that n - 2 is divisible by 3, construct a graph to show that f (n) ::: 3n - 5. b) Prove that f(n) = 3n - 5 when n - 2 is divisible by 3 and that otherwise f(n) = 3n - 6. (Hint: Use induction on n, invoking Exercise 6.2.8 for the 3-connected case.) (Thomassen [1984]) (Comment: Mader [1998] proved the more difficult result that 3n - 6 is the maximum number of edges in an n-vertex simple graph with no Ks-subdivision.) 6.2.11. (!) Let H be a graph with maximum degree at most 3. Prove that a graph G contains a subdivision of H if and only if G contains a subgraph contractible to H. 6.2.12. (!) Wagner [1937] proved that the following condition is necessary and sufficient for a graph G to be planar: neither Ks nor Ka.a can be obtained from G by performing deletions and contractions of edges. a) Show that deletion and contraction of edges preserve planarity. Conclude from this that Wagner's condition is necessary. b) Use Kuratowski's Theorem to prove that Wagner's condition is sufficient. 6.2.13. Prove that a graph G is planar if and only if for every cycle C in G, the conflict graph for C is bipartite. (Tutte [1958]) 6.2.14. Let x and y be vertices of a planar graph G. Prove that G has a planar embedding with x and y on the same face unless G - x - y has a cycle C with x and y in conflicting C-fragments in G. (Hint: Use Kuratowski's Theorem. Comment: Tutte proved this without Kuratowski's Theorem and used it to prove Kuratowski's Theorem.) 6.2.15. Let G be a 3-connected simple plane graph containing a cycle C. Prove that C is the boundary of a face in G if and only if G has exactly one C-fragment. (Comment: Tutte [1963] proved this to obtain Whitney's [1933b] result that 3-connected planar graphs have essentially only one planar embedding. See also Kelmans [1981a]) 6.2.16. (+) Let G be an outerplanar graph with n vertices, and let P be a set of n points in the plane, no three of which lie on a line. The extreme points of P induce a convex polygon that contains- the other points in its interior. a) Let p 1 , p 2 be consecutive extreme points of P. Prove that there is a point p E P - {p1, pz) such that 1) no poip.t of Pis inside p 1 p 2 p, and 2) some line l through p separates P1 from pz, meets P only at p, and has exactly i - 2 points of P on the side of l containing pz. b) Prove that G has a straight-line embedding with its vertices mapped onto P. (Hint: Use part (a) to prove the stronger statement that if vi. v2 are two consecutive vertices of the unbounded face of a maximal outerplanar graph G, and p 1 , p 2 are consecutive vertices of the convex hull of P, then G can be straight-line embedded on P such that f (v1) = P1 and f (v2) = pz.) (Gritzmann-Mohar-Pach-Pollack [1989])
257
## 6.3. Parameters of Planarity
Every property and parameter we have studied for general graphs can be studied for planar graphs. The problem of greatest historical interest is the maximum chromatic number of planar graphs. We will also study parameters that measure how far a graph is from being a planar graph.
## COLORING OF PLANAR GRAPHS
Because every simple n-vertex planar graph has at most 3n - 6 edges, such a graph has a vertex of degree at most 5. This yields an inductive proof that planar graphs are 6-colorable (see Exercise 2). Heawood improved the bound.
6.3.1. Theorem. (Five Color Theorem-Heawood [1890]) Every planar graph is 5-colorable. Proof: We use induction on n(G). Basis step: n(G) ~ 5. All such graphs are 5-colorable. Induction step: n(G) > 5. The edge bound (Theorem 6.1.23) implies that G has a vertex v of degree at most 5. By the induction hypothesis, G - v is 5-colorable. Let f: V(G - v)---+ [5] be a proper 5-coloring of G - v. If G is not 5colorable, then f assigns each color to some neighbor of v, and hence d(v) = 5. Let v 1 , v2 , v3 , v4 , v5 be the neighbors of v in clockwise order around v. Name the colors so that f(v;) = i. Let G;,j denote the subgraph of G - v induced by the vertices of colors i and j. Switching the two colors on any component of G;,j yields another proper 5-coloring of G - v. If the component of G i,j containing v; does not contain vj, then we can switch the colors on it to remove color i from N(v). Now giving color i to v produces a proper 5-coloring of G. Thus G is 5-colorable unless, for each choice of i and j, the component of Gi,j containing v; also contains Vj. Let P;.j be a path in G;,j from v; to Vj, illustrated below for (i, j) = (1, 3).
3
5
Consider the cycle C completed with Pi.a by v; this separates v2 from V4.
258
## Chapter 6: Planar Graphs
By the Jordan Curve Theorem, the path P2 A must cross C. Since G is planar, paths can cross only at shared vertices. The vertices of P1.3 all have color 1 or 3, and the vertices of P2 ,4 all have color 2 or 4, so they have no common vertex. By this contradiction, G is 5-colorable. Every planar graph is 5-colorable, but are five colors ever needed? The history of this infamous question is discussed in Aigner [1984, 1987], Ore [1967a], Saaty-Kainen [1977, 1986], Appel-Haken [1989], and Fritsch-Fritsch [1998]. The earliest known posing of the Four Color Problem is in a letter of October 23, 1852, from Augustus de Morgan to Sir William Hamilton. The question was asked by de Morgan's student Frederick Guthrie, who later attributed it to his brother Francis Guthrie. It was phrasPd in terms of map coloring. The problem's ease of statement and geometric subtleties invite fallacious proofs; some were published and remained unexposed for years. It does not suffice to forbid five pairwise-adjacent regions, since there are 5-chromatic graphs not containing K 5 (recall Mycielski's construction, for example). Cayley announced the problem to the London Mathematical Society in 1878, and Kempe [1879] published a "solution". In 1890, Heawood published a refutation. Nevertheless, Kempe's idea of alternating paths, used by Heawood to prove the Five Color Theorem, led eventually to a proof by Appel and Haken [1976, 1977, 1986] (working ~:ith Koch). A path on which the colors alternate between two specified colors is a Kempe chain. In proving the Five Color Theorem inductively, we argued that a minimal counterexample contains a vertex of degree at most 5 and that a planar graph with such a vertex cannot be a minimal counterexample. This suggests an approach to the Four Color Problem; we seek an unavoidable set of graphs that can't be present! We need only consider triangulations, since every simple planar graph is contained in a triangulation.
6.3.2. Definition. A configuration in a planar triangulation is a separating cycle C (the ring) togeth~r withthe portion of the graph inside C ~r t!ie Four Color Problem, a set of configurations is unavoidable if a 1miriln:ial counterexample must contain a member of it. A configuration is reducible if a planar graph containing it cannot be a minimal counterexample. 6.3.3. Example. An unavoidable set. We have remarked that o(G) :::: 5 for every simple planar graph. In a triangulation, every vertex has degree at least 3. Thus the set of three configurations below is unavoidable.
The edges from the ring to the interior are drawn with dashes because a configuration (in a triangulation) is completely determined if we state the degrees of the vertices adjacent to the ring and delete the. ring (Exercise 7). Thus these configurations are written as" 3", " 4", and" 5", respectively.
## Section 6.3: Parameters of Planarity
259
When we say that a configuration cannot be in a minimal counterexample, we mean that ifit appears in a triangulation G, then it can be replaced to obtain a triangulation G' with fewer vertices such that every 4-coloring of G' can be manipulated to obtain a 4-coloring of G.
6.3.4. Remark. Kempe's proof. Let us try to prove the Four Color Theorem by induction using the unavoidable set (3, e4, 5}. The approach is similar to Theorem 6.3.1. We can extend a 4-coloring of G - v to complete a 4-coloring of G unless all four colors appear on N ( v). Thus " 3" is reducible. If d ( v) = 4, then the Kempe-chain argument works as in Theorem 6.3.1, and" 4" is reducible. Now consider" 5". When d(v) = 5, the restriction to triangulations implies that the repeated color on N(v) in the proper 4-coloring of G - v appears on nonconsecutive neighbors ofv. Let vi, v2, va, V4, v5 again be the neighbors of v in clockwise order. In the 4-coloring f of G - v, we may assume by symmetry that f(v5) = 2 and that f (v;) = i for 1 ~ i ~ 4. Define G;.j and P;,j as in Theorem 6.3.1. We can eliminate color 1 from N(v) unless the chains Pi.a and P1,4 exist from v1 to va and V4, respectively, as shown on the left below. The component H of G2,4 containing v2 is separated from V4 and V5 by the cycle completed by v with Pi.a. Also, the component H' of G2.a containing v5 is separated from v2 and va by the cycle completed by v with Pl.4. We can eliminate color 2 from N(v) by switching colors 2 and 4 in Hand colors 2 and 3 in H'. Right? This was the final case in Kempe's proof
4 3
The problem is that Pi.a and P 1,4 can intertwine, intersecting at a vertex with color 1 as shown on the right above. We can make the switch in H or in H', but making them both creates a pair of adjacent vertices with color 2. Because of this difficulty, we have not shown that " 5" is reducible, and we must consider larger configurations. Heesch [1969] contributed the idea of seeking configurations with small ring size instead of few vertices inside. It is not hard to show that every configuration having ring size 3 or 4 is reducible (Exercise 9). This is equivalent to showing that no minimal 5-chromatic triangulation has a separating cycle of length at most 4.
6.3.5. * Example. Birkhoff [1913] pushed the idea farther. He proved that every configuration with ring size 5 that has more than one vertex inside is reducible. He also proved that the configuration with ring size 6 below, called the Birkhoff diamond, is reducible.
260
## Chapter 6: Planar Graphs
Proving that the Birkhoff diamond is reducible takes a full page of detailed analysis. One approach is to try to show that all proper 4-colorings of the ring extend to the interior. Although some cases can be combined, and some do extend, in some cases it is necessary to use Kempe chains to show that the coloring can be changed into one that extends. The intricate analysis of this first nontrivial example suggests that we have barely begun. The detail remaining is enormous. From 1913 to 1950, additional 'reducible configurations were found, enough to prove that all planar graphs with at most 36 vertices are 4-colorable. This was slow progress. In the 1960s, Heesch focused attention on the size of the ring, gave heuristics for finding reducible configurations, and developed methods for generating unavoidable sets. The first proof used configurations with ring size up to 14. A ring of size 13 has 66430 distinguishable 4-colorings. Reducibility requires showing that each leads to a 4-coloring of the full graph. ~empe-chain arguments and partial collapsing of the configuration may be needed, so reducibility proofs are, not easy. Appel and Haken, working with Koch, improved upon the heuristics of Heesch and others to restrict computer searches to "promising" configurations. Using 1000 hours of computer time on three computers in 1976, they found an unavoidable set of 1936 reducible configurations, all with ring size at most.14.
6.3.6. Theorem. (Four Color Theorem-Appel-Haken-Koch [1977]) Every planar graph is 4-colorable.
By 1983, refinements led to an unavoidable set of 1258 reducible configurations. The proof was revisited by Robertson, Sanders, Seymour, and Thomas [1996], using the same approac4. They reduced the rules used for producing unavoidable sets to a set of 32 rules. Their simplifications yielded an unavoidable set of 633 reducible configurations. They made their computer code available on the Internet; in 1997, it would prove the Four Color Theorem on a desktop workstation in about three hours.
6.3.7.* Remark. Discharging. To generate unavoidable sets, we replace the problem case (vertex of degree 5) by larger configurations involving a vertex of degree 5; this can be viewed as a more detailed case analysis for the hard case. Systematic rules are needed to maintain a reasonably small exhaustive set. In a triangulation, 'Ld(v) = 2e(G) = 6n - 12. We rewrite this as 12 = L(6-d(v)) and thinkof6-d(v) as a charge on vertex v. Because 12 is positive, some vertices must have positive charge (degree 5). The rules for replacing bad
## Section 6.3: Parameters of Planarity
261
cases involve moving the charge around; they are called discharging rules. Since positive charge must remain somewhere, we obtain new unavoidable sets. The next proposition describes the effect of the simplest discharging rule.
6.3.8. * Proposition. Every planar triangulation with minimum degree 5 contains a configuration in the set below.
5--5 5--6
Proof: Start with charge defined by 6 - d(v). The first discharging rule takes the charge from each vertex of positive charge (degree 5) and distributes that charge equally among its neighbors. A vertex of degree 5 or 6 now having positive charge must have a neighbor of degree 5. A vertex of degree 7 now having positive charge must have at least six neighbors of degree 5. Since G is a triangulation, this requires adjacent vertices of degree 5. No vertex of degree 8 or more can acquire positive charge from this discharging rule. The total charge in the graph remains 12, so some vertex v has positive charge. For each case of d(v), one of the specified configurations occurs.
Discharging methods are now being applied to attack other problems using computer-assisted analysis by cases. The proof of the Four Color Theorem met with considerable uproar. Some objected in principle to the use of a computer. Others complained that the proof was too long to be verified. Others worried about computer error. A few errors were found in the original algorithms, but these were fixed (AppelHaken [1986]). Those who have checked calculations by hand recognize that the probability of human error in a mathematical proof is much higher than the probability of computer error when the algorithm has been proved correct.
CROSSING NUMBER
In the remainder of this section, we consider parameters that measure a graph's deviation from planarity. One natural parameter is the number of planar graphs needed to form the graph; Exercises 16-20 consider this.
6.3.9. Definition. The thickness of a graph G is the minimum number of planar graphs in a decomposition of G into planar graphs. 6.3.10. Proposition. A simple graph G with n vertices and m edges has thickness at least m/(3n - 6). If G has no triangles, then it has thickness at least m/(2n - 4). Proof: By Theorem 6.1.23, the denominator is the maximum size of each planar subgraph. The pigeonhole principle then yields the inequality.
262
## Chapter 6: Planar Graphs
Sometimes we simply must draw a graph in the plane, even if it is not a planar graph. For example, a circuit laid out on a chip corresponds to a drawing of a graph. Since wire crossings lessen performance and cause potential problems, we try to minimize the number of crossings. We discuss the resulting parameter in the remainder of this subsection.
6.3.11. Definition. The crossing number v (G) of a graph G is the minimum .number of crossings in a drawing of G in the plane. 6.3.12. Example. v(Ka) = 3 and v(K 3,2,2) = 2. We can determine the crossing number of some small graphs by considering maximal planar subgraphs. Consider a drawing of G in the plane. If H is a maximal plane subgraph of'this drawing, then every edge of G not in H crosses some edge of H, so the drawing has at least e(G) - e(H) crossings. If G has n vertices, then e(H) _:::: 3n - 6. If also G has no triangles, then e(H) _:: : 2n - 4. Since Ka has 15 edges, and planar 6-vertex graphs have at most 12 edges, we have v(Ka) ~ 3. The drawing on the left below proves equality. Since K 3 ,2.2 has 16 edges, and planar graphs with seven vertices have at most 15 edges, v(K3.2.2) ~ 1. The best drawing we find has two crossings, as shown on the right below. To improve the lower bound, observe that K 3 ,2 ,2 contains K3,4 Because K3,4 is triangle-free, its planar subgraphs have at most 2 7 - 4 = 10 edges, and hence v(K3,4) ~ 2. Every drawing of K 3 ,2 2 contains a drawing of K3,4, so v(K3,2,2) ~ v(K3,4) ~ 2.
6.3.13. Proposition. Let G be an n-vertex graph with m edges. If k is the maximum number of edges in a planar subgraph of G, then v(G) ~ m - k. Furthermore, v(G) ~ ;; - 'i Proof: Given a drawing of G in the plane, let H be a maximal subgraph of G whose edges do not cross in this drawing. Every edge not in H crosses at least one edge in H; otherwise, it could be added to H. Since H has at most k edges, we have at least m - k crossings between edges of H and edges of G - E(H). After discarding E(H), we have at least m - k edges remaining. The same argument yields at least (m - k) - k crossings in the drawing of the remaining graph. Iterating the argument yields at least L::=l (m - ik) crossings, where t = Lm/ kJ. The value of the sum is mt - kt(t + 1)/2. We now write m = tk + r, where 0 _:: : r _:::: k -1. We substitute t = (m - r)/ k in the value of the sum and simplify to obtain v( G) ~ ;; - ~ + r(~~r).
## Section 6.3: Parameters of Planarity
263
The first bound m - kin Proposition 6.3.13 is useful when G has few edges: the crossing number of a simple graph G is at least e(G) - 3n + 6, and when G is bipartite it is at least e(G) - 2n + 4. Iterating the argument improves the bound when e(G) is larger, but for dense graphs this lower bound is weak. Consider Kn, for example. Lacking an exact answer, we hope at least to determine the leading term in a polynomial expression for v(Kn). To indicate a polynomial of degree kin n with leading term ank, we often write ank + O(nk-l ). This is consistent with the definition of"Big Oh" notation in Definition 3.2.3. Proposition 6.3.13 yields v(Kn) ~ 4 n 3 + O(n 2 ), but actually v(Kn) grows like a polynomial of degree 4. The crossing number cannot exceed(~), since we can place the vertices on the circumference of a circle and draw chords. For Kn, each set of four vertices contributes exactly one crossing. Actually, this is the worst possible straight-line drawing of K,., since in every straight-line drawing, each set of four vertices contributes at most one crossing, depending on whether one vertex is inside the triangle formed by the other three. How many crossings can be saved by a better drawing?
6.3.14. Theorem. (R. Guy [1972]) 8~n 4 + O(n 3 ) _::: v(Kn) _::: 6~n 4 + O(n 3 ). Proof: A counting argument yields a recursive lower bound. A drawing of Kn with fewest crossings contains n drawings of K,,_i, each obtained by deleting one vertex. Each subdrawing has at least v(Kn-l crossings. The total count is at least nv(Kn_ 1), but each crossing in the full drawing has been counted (n -4) times. We conclude that (n - 4)v(Kn) ~ nv(K,,_1). From this inequality, we prove by induction on n that v(K,,) ~ (~) when n ~ 5. Basis step: n = 5. The crossing number of K 5 is 1. Induction step: n > 5. Using the induction hypothesis, we compute
l)(n -
## 2)(n - 3)(n - 4) _ ~(n)
24
- 5 4
The denominator of the quartic term in the lower bound can be improved from 120 to 80 by considering copies of K 6 ,n_6 , which has crossing number 6 l"26 L n;-7 (Exercise 26b). A better drawing lowers the upper bound from (~) to 6~ n 4 + 0 (n 3 ). Consider n = 2k. Drawing Kn in the plane is equivalent to drawing it on a sphere or on the surface of a can. Place k vertices on the top rim of the can and k vertices on the bottom rim, drawing chords on the top and bottom for those k-cliques. The edges from top to bottom fall into k natural classes. The "class number" is the circular separation between the top and bottom endpoints, ranging 1 from r-k2 +i to k2 We draw these edges to wind around the can as little as possible in passing from top to bottom, so edges in the same class don't cross. We now twist the can to make the class displacements run from 1 to k. This makes them easier to count but doesn't change the p~irs of edges that cross. Crossings 01:1 the side of the can involve two vertices on the top and two on the bottom. For top vertices x, y and bottom vertices z, w, where x z has smaller positive displacement than x w, we have a crossing for x, y, z, w if and only if the displacements to y, z, w ate distinct positive values in increasing order. (For
1 I l
264
## Chapter 6: Planar Graphs
example, this holds for x, y, z, w in the illustration, but not for x. y, z, a; the edge ya winds around the can.) Hence there are k(;) crossings on the side of the twisted can, and v(K11 ) :=: 2(~) + k(;) = hn 4 + O(n 3 ).
x
a z
6.3.15. Example. v(Km,n). The most naive drawing puts the vertices of one partite set on one side of a channel and the vertices of the other partite set on the other side, with all edges drawn straight across. This has (;) (';) crossings, but it is easy to reduce this by a factor of 4. Place the vertices of Km,n along two perpendicular axes. Put rn/21 vertices along the positive y-axis and Ln/2J along the negative y-axis; similarly split the m vertices along the positive and negative x-axis. Adding up the four types of crossings generated when we join every vertex on the x-axis to every vertex on the y-axis yields v(Km,n) :'.': ~ m2l l~J n2l (Zarankiewicz [1954]). This bound is conjectured to be optimal (Guy [1969] tells the history). Kleitman [1970] proved it for min{n, m) :=: 6. Aided by a computer search, Woodall [1993} extended this so that the smallest unknown cases are K7,11 and Kg,9, From Kleitman's result, Guy [1970] proved that v(Km,n) 2: m(~-l) l~J 21 which is not far from the upper bound (Exercise 26).
LJ l J
l J
l J,
11
Another general lower bound for crossing number, conjectured in ErdosGuy [1973], has an appealing geometric application. Our proof is inductive, generalizing the lower bound argument in Theorem 6.3.14. There is an elegant probabilistic proof in Exercise 8.5.lland stronger results in Pach-T6th [1997]. 6.3.16. * Theorem. (Ajtai-Chvatal-Newborn-Szemeredi [1982],Leighton [1983]) Let G be a simple graph. Ife(G) 2: 4n(G), then v(G) ::=:: 6~e(G) 3 /n(G) 2 Proof: Let m = e(G) and n = n(G), We use induction on n. Basis step: m :=: 5n (this includes all simple graphs with at most 11 vertices). Note that (a - 3) 2: 4 a 3 when 4 :=: a :=: 5. Letting m = an for 4 :=: a :=: 5, we obtain v(G) 2: m - 3n 2: 4 m 3 /n 2 , as desired. Induction step: n > 11. Given an optimal drawing of G, each crossing appears in n - 4 of the drawings obtained by deleting a single vertex. By the induc. v (G - v ) 2: 1 (ni-d(v))3 Th us (n - 4) v (G) 2: " 1 (m-d(v))3 . h ypo th es1s, t ion L...veV(G) 64 (n-l)2 64 (n-l)2 . By convexity, the lower bound is always at least what results when the vertex degrees are all replaced by the average degree. In other words, L:<m d(v)) 3 2: n(m - 2m/n) 3 Also (n - 1) 2 (n - 4) :=: (n - 2) 3 Thus
l l
(n - 2) 3 m 3 v(Q) 2: 64 n n 3 (n - 1) 2 (n -
1 m3
4) 2: 64 -;;2
## Section 6.3: Parameters of Planarity
265
6.3.17.* Example. Achieving the bound. The order of magnitude in Theorem 6.3.16 is best possible. Consider G = ;~ K2m/n where 2m is a multiple of n. The total number of vertices is n, and the total number of edges is asymptotic to 2m) 2 = m. Since v(K) < 1-r 4 we have v(G) < IC1-( 2m) 4 = lm 3 This is IC1( 2m 2 n r - 64 ' - 2m 64 n 8 n2 within a constant factor of the lower bound from Theorem 6.3.16.
We apply Theorem 6.3.16 to a problem in combinatorial geometry. Erdos [1946] asked how many unit distances can occur among a set of n points in the plane. If the points occur in a unit grid, then the graph of unit distances is the cartesian product of two paths, and this produces about n - 0(,Jfi) edges. By taking all the points of a refined grid that lie within an appropriate distance from the origin, Erdos obtained about n 1+c/loglogn unit distances. This growth rate is superlinear, but it is slower than n l+ for each positive E. Erdos also proved an upper bound of O(n 312 ). Since two circles of radius 1 intersect in at most two points, the graph G of unit distances cannot contain K2, 3 Thus each pair of points has at most two common neighbors. Since each vertex v is a common neighbor for its (d~)) pairs of neighbors, L (d~)) ~ 2(;). Since 2e( G) / n is the average vertex degree, convexity yields L (d~)) ::: n (2e<~)fn). Together, these inequalities yield the desired bound (Exercise 5.2.25 considers the edge-maximization problem in general when a biclique is forbidden). Using number-theoretic arguments about incidences between lines and points in a point set, Spencer-Szemeredi-Trotter [1984] improved the upper bound to O(n 413 ). Szekely applied Theorem 6.3.16 to give an elegant and short graph-theoretic proof of this bound.
6.3.18.* Theorem. (Spencer-Szemeredi-Trotter [1984]) There are at most 4n 413 pairs of points at distance 1 among a set of n points in the plane. Proof: (Szekely [1997]) By moving points or pairs of points without reducing the number of pairs at distance 1, we can ensure that each point is involved in such a pair and that no two points have distance 1 only from each other. If any point now is involved in only one unit distance pair, we can rotate it around its mate until it is distance 1 from another point. This reduces the problem to the case that every point is involved in at least two such pairs. Let P be an optimal n-point configuration, with q unit distance pairs. We obtain a graph from P, not by using the unit distance pairs as edges, but rather by drawing a unit circle around each point. If a point in P is at distance 1 from k other points in P, then these points partition the circle into k arcs. Altogether we obtain 2q arcs. These are the edges of a loopless graph G. Since two points can appear on two (but not three) unit circles, G may have edges of multiplicity 2 but no larger multiplicity. We delete one copy of each duplicated edge to obtain a simple graph G' with at least q edges. We may assume that q ::: 4n; otherwise the bound already holds. Because these arcs lie on n circles, they cannot produce many crossings; each pair of circles crosses at most twice. Thus our layout of G' has at most 2G) crossings. By Theorem 6.3.16, G' has at least 4 q 8 /n 2 crossings. Together, these inequalities yield q ~ 4n 413 .
266
## SURFACES OF HIGHER GENUS (optional)
Instead of minimizing crossings in the plane, we could change the surface to avoid crossings. This is the effect of building overpasses and cloverleafs instead of installing traffic lights. The surface of the earth is a sphere, and for this discussion it is convenient to consider drawings on the sphere instead of in the plane. As observed in Remark 6.1.27, these settings are equivalent. To avoid creating boundaries in the surface, we add an overpass by cutting two holes in the sphere and joining the edges of the holes by a tube. By stretching the tube and squeezing the rest of the sphere, we obtain a doughnut.
6.3.19. Definition. A handle is u tube joining two holes cut in a surface. The torus is the surface obtained by adding one handle tO' a sphere.
The torus is topolcgicully the same as the sphere with one handle, in the sense that one surface can be continuously transformed into the other.t A large graph may have many crossings and need more handles. For any graph, adding enough handles to a drawing on the sphere will eliminate all crossings and produce an embedding. When we add some number of handles, it doesn't matter how we do it, because a fundamental result of topology says that two surfaces obtained by adding the same number of handles to a sphere can be continuously deformed into each other.
6.3.20. Definition. The genus of a surface obtained by _adding handles to a sphere is the number of handles added; we use Sy for the surface of genus y. The genus of a graph G is the minimum y such that G embeds on Sy. The graphs embeddable on the surfaces of genus 0, 1, 2 are the planar, toroidal, and double-toroidal graphs, respectively (the surface with two handles is the double-torus).
The theory of planar graphs extends in some ways to graphs embeddable on higher surfaces; we discuss tl~is only briefly, for cultural interest. Drawings of large graphs on surfaces of large genus are hard to follow, even on the pretzel (S3 ). Locally, the surface looks like a plane sheet of paper. To draw the graph we want to lay the entire surface flat; to do this we must cut the surface. Ifwe keep track of how the edges should be pasted back together to get the surface, we can describe the surface on a flat piece of paper. Consider first the torus.
-tThis is the source of the joke that a topologist is a person who can't tell the difference between a doughnut and a coffee cup.
267
## 6.3.21. Example. Combinatorial description of the torus.
()-~()
--
Cutting the closed tube once turns it into a cylinder, and then slitting the length of the cylinder allows us to lay it flat as a rectangle. Labeling the edges of the rectangle indicates how to paste it back together. The two sides of a cut labelf)d with the same letter are "identified". . Keeping track of the identifications is important because edges of an embedding on a surface may cross such a cut. When the edge reaches one border of the rectangle, it is reaching one side of the imagined cut. When it crosses the cut, it _emerges from the identical point on the other copy of this border. The four "corners" of the rectangle correspond to the single point on the surface through which both cuts pass. These ideas lead to nice toroidal embeddings of K 5 , K 3 .3 , and K7
r----------------,
r-------
I I I
I
'
I I I I I
r
r
L----------------~
r _______ ..J
For surfaces of higher genus, there is some flexibility in making the cuts, but each way takes two cuts per handle before we can lay the surface flat. The usual representation comes from expressing the handles as "lobes" of the surface, with the cuts having a common point on the hub.
6.3.22. Example. Laying the double torus flat. Below is a polygonal representation for the double torus. Making the cuts is equivalent to adding loops at a single vertex until we have a one-face embedding of a bouquet of loops. In general, we make 2y cuts through a single point to lay Sy flat. Keeping track of the borders from each cut leads to representing Sy by a 4y-gon in which a clockwise traversal of the boundary can be described by reading out the cuts as we traverse them. We record a cut using the notation of inverses when we traverse it in the opposite order. Since we are following the boundary of a single face, with our left hand always on the wall, each edge will be followed once forward and once backward. For the example here, the traversal is a1,B1a1 1.B'1 1a2,B2a2, 1,Bi, 1. Each surface Sy has a layout of the form a 1.Bia1 1.B1 1 ay,Bya; 1 1 Other layouts result from other ways of making the cuts -different ways of embedding
.B;
268
## Chapter 6: Planar Graphs
a bouquet of 2y loops. For example, the double torus can also be represented by an octagon with boundary af3y8a-- 1 13- 1 y-- 18-- 1
/31
a2
6.3.23. Remark. Euler's Formula for Sy. A 2-cell is a region such that every closed curve in the interior can be continuously contracted to a point. A 2-cell embedding is an embedding where every region is a 2-cell. Euler's Formula generalizes for 2-cell embeddings of connected graphs on Sy (Exercise 35) as
n- e+
= 2 - 2y.
For example, our embe<lding of K1 on the torus (y = 1) has 7 vertices, 21 edges, 14 faces, and 7 - 21+14 = 0. The proof ()f Euler's Formula for Sy is like the proof in the plane, except that the basis case of 1-vertex graphs needs more care. It requires showing that it takes 2y cuts to lay the surface flat (that is, to obtain a 2-cell embedding of a graph with one vertex and one face).
6.3.24. Lemma. Every simple n-vertex graph embedded on Sy has at most 3(n - 2 + 2y) edges. Proof: Exercise 35.
Note that K 7 satisfies Lemma 6.3;24 with equality on the torus (y = 1), as every face in the toroidal embedding of K 7 is a 3-gon. Hence K 7 is a maximal toroidal graph. Rewriting e :::=: 3(n - 2 + 2y) yields a lower bound on the number of handles we must add to obtain a surface on which G is embeddable; thus y(G) 2: 1 + (e - 3n)/6. Lemma 6.3.24 leads to an analogue of the Four Color Theorem for Sy.
6.3.25. Theorem. (Heawood's Formula-Heawood [1890]) If G is embeddable on Sy with y > 0, then x(G) :::=: (7 + + 48y)/2
Jl
J.
Proof: Let c = (7 + + 48y)/2. It suffices to prove that every simple graph embeddable on Sy has a vertex of degree at most c -1; the bound on x(G) then follows by induction on n(G). Since x(G) :::=: c for all graphs with at most c vertices, so need only consider n ( G) > c. We use Lemma 6.3.24 to show that the average (and hence minimum) degree is at most c -1. The second inequality below follows from y > 0 and n > c.
Jl
## Section 6.3: Parameters of Planarity
269
Since c satisfies c 2 - 7c + (12 - 12y) = 0, we have c - 1 = 6 - (12 - 12y)/c, so the average degree satisfies the desired bound. 6(n - 2 + 2y) 12 - 12y 2e -< <6=c-1. n n c The key inequality here fails when y = 0. Thus the argument is invalid for planar graphs, even though the formula reduces to x (G) :::=: 4 when y = 0. Proving that the Heawood bound is sharp involves embedding K 11 on Sy with y = l(n - 3)(n - 4)/121. The proof breaks into cases by the congruence class of n modulo 12 (K 7 is the first example in the easy class). Completed in RingelYoungs [1968], it comprises the book Map Color Theorem (Ringel [1974]). ~aving considered the coloring problem on Sy, one naturally wonders which graphs embed on Sy. Planar graphs have many characterizations, beginning with Kuratowski's Theorem (Theorem 6.2.2) and Wagner's Theorem (Exercise 6.2.12). On any surface, embeddability is preserved by deleting or contracting an edge. Thus every surface has a list of "minor-minimal" obstructions to embeddability. Wagner's Theorem states that the list for the plane is {K3. 3 , K5 }; every nonplanar graph has one of these as a minor. More than 800 minimal forbidden minors are known for the torus. For each surface, the list is finite; this follows from the much more general statement below (the subdivision relation in Kuratowski's Theorem leads to infinite lists). 6.3.26. Theorem. (The Graph Minor Theorem-Robertson-Seymour [1985]) In any infinite list of graphs, some graph is a minor of another. This is perhaps the most difficult theorem known in graph theory. The complete proof takes well over 500 pages (without computer assistance) in a series of 20 papers stretching beyond the year 2000. It has many ramifications about structure of graphs and complexity of computation. The techniques involved in the proof have spawned new areas of graph theory. Some aspects of these techniqmis and their relation to the proof of the Graph Minor Theorem are presented in the final chapter of the text by Diestel [1997]. ,
EXERCISES
6.3.1. (-) State a polynomial-time algorithm that takes an arbitrary planar graph as input and produces a proper 5-coloring of the graph. 6.3.2. ( - ) A graph G is k-degenerate if every sJ.bgraph of G has a vertex of degree at most k. Prove that every k-degenerate graph is k + 1-colorable. 6.3.3. (-) Use the Four Color Theorem to prove that every outerplanar graph is 3colorable. 6.3.4. (-) Determine the crossing numbers of K2 , 2. 2.2 , K4,4 , and the. Petersen graph .
270
## Chapter 6: Planar Graphs
6.3.5. Prove that every planar graph decomposes into two bipartite graphs. (Hedetniemi [1969], Mabry [1995]) 6.3.6. Without using the Four Color Theorem, prove that every planar graph with at most 12 vertices is 4-colorable. Use this to prove that every planar graph with at most 32 edges is 4-colorable. 6.3.7. (!) Let H be a configuration in a planar triangulation (Definition 6.3.2). Let H' be obtained by labeling the neighbors of the ring vertices with their degrees and then deleting the ring vertices. Prove that H can be retrieved from H'. 6.3.8. Create a configuration with ring size 5 in a planar triangulation such that every internal vertex has degree at least five. 6.3.9. (+) Prove that every planar configuration having ring size at most four is reducible. (Hint: The ring is a separating cycle C. Prove that if smaller triangulations are 4-colorable, then the C-lobes of G have 4-colorings that agree on C .) (Birkhoff [1913]) 6.3.10. Grotzsch's Theorem [1959] (see Steinberg [1993], Thomassen [1994a]) states that a triangle-free planar graph G is 3-colorable. Hence a(G) ~ n(G)/3. ToveySteinberg [1993] proved that a(G) > n(G)/3 always. Prove that this is best possible by considering the family of graphs Gk defined as follows: G 1 is the 5-cycle, with vertices a, x0 , xi. yi, z1 in order. Fork > 1, Gk is obtained from Gk-l by adding the three vertices xk. Yk> Zk and the five edges Xk-1Xk> XkYk> YkZk> ZkYk-1 ZkXk-2 The graph Ga is shown on the left below. (Fraughnaugh [1985])
Z1
Y1
z2
Y2
Zs
Ys
6.3.11. Define a sequence of plane graphs as fol~ows. Let G 1 be C4 For n > 1, obtain Gn from Gn-1 by adding a new 4-cycle surrounding Gn-h making each vertex of the new cycle also adjacent to two consecutive vertices of the previous outside face. The graph Gs is shown on the right above. Prove that if n is even, then every proper 4-coloring of Gn uses each color on exactly n vertices. (Albertson) 6.3.12. (!) Without using the Four Color Theorem, prove that every outerplanar graph is 3-colorable. Apply this to prove the Art Gallery Theorem: If an art gallery is laid out as a simple polygon with n sides, then it is possible to place Ln/3J guards such that every point of the interior is visible to some guard. Construct a polygon that requires Ln/3J guards. (Chvatal [1975], Fisk [1978])
## Section 6.3: Parameters of Planarity
271
6.3.13. An art gallery with walls is a polygon plus some nonintersecting chords called "walls" that join vertices. Each interior wall has a tiny opening called a "doorway". A guard in a doorway can see everything in the two neighboring rooms, but a guard not in a doorway cannot see.past a wall. Determine the minimum number t such that for every walled art gallery with n vertices, it is possible to place t guards so that every interior point is visible to some guard. (Hutchinson [1995], Kiindgen [1999]) 6.3.14. (+) Prove that a maximal planar graph is 3-colorable if and only ifit is Eulerian. (Hint: For sufficiency, use induction on n(G). Choose an appropriate pair or triple of adjacent vertices to replace with appropriate edges.) (Heawood [1898]) 6.3.15. (!) Prove that the vertices of an outerplanar graph can be partitioned into two sets so that the subgraph induced by each set is a disjoint union of paths. (Hint: Define the partition using the parity of the distance from a fixed vertex.) (Akiyama-EraGerv~cio [1989], Goddard [1991]) 6.3.16. (-) Prove that the 4-dimensional cube Q 4 is nonplanar. Decompose it into two isomorphic planar graphs; hence Q4 has thickness 2. 6.3.17. Prove that Kn has thickness at least n~ 7 j. (Hint: = x+~- 1 j.) Show that equality holds for K8 by finding a. self-complementary planar graph with 8 vertices. (Comment: The thickness equals n~ 7 except that Kg and K10 have thickness 3; Beineke-Harary [1965] for n . 4 mod 6, and Alekseev-Goneakov [1976] for n 4 mod 6.)
l;l
l J
6.3.18. Decompose Kg into three pairwise-isomorphic planar graphs. 6.3.19. Prove that ifG has thickness 2, then x(G) _::: 12. Use the two graphs below to show that x (G) may be as large as 9 when G has thickness 2. (Sulanke)
6.3.20. (!) When r is even and s is greater than (r - 2) 2 /2, prove that the thickness of K,,, is r /2. (Beineke-Harary-Moon [1964]) 6.3.21. Determine v(K1.2.2.2) and use it to compute v(K2.2.2.2). 6.3.22. Prove that K3 , 2 ,2 has no planar subgraph with 15 edges. Use this to give another proof that v(K 3 .2.2) ::::: 2. 6.3.23. Let Mn be the graph obtained from the cycle Cn by adding chords between vertices that are opposite (if n is even) or nearly opposite (if n is odd). The graph Mn is 3-regular ff n is even, 4-regular if n is odd. Determine v(Mn). (Guy-Harary [1967]) 6.3.24. The graph P: has vertex set [n] and edge set {ij: Ii - jl _::: k}. Prove that P! is a maximal planar graph. Use a planar embedding of P! to prove that v(Pn4 ) = n - 4. (Harary-Kainen [1993])
272
## Chapter 6: Planar Graphs
6.3.25. For every positive integer k, construct a graph that embeds on the torus but requires at least k crossings when drawn in the plane. (Hint: A single easily described toroidal family suffices; use Proposition 6.3.13.)
"26 6.3.26. (!) Use Kleitman's computation that v(Ks,n) = 6 L arguments for the following lower bounds. a) v(Km,n) ?: m m~l L~J "2 1 (Guy [1970])
b) v(Kp) ?: tfop
4
## JL"27 J to give counting
+ O(p
L J.
).
6.3.27. (!) It is conjectured that v(Km,n) = ~ m; 1 L~J "2 1 Suppose that this conjecture holds for Km,n and that m is odd. Prove that the conjecture then holds also for Km+l.n (Kleitman [1970]) 6.3.28. (!) Suppose that m and n are odd. Prove that in all drawings of Km,n, the parity of the number of pairs of edges that cross is the same. (We consider only drawings where edges cross at most once and edges sharing an endpoint do not cross.) Conclude that v(Km,m) is odd when m - 3 and n - 3 are divisible by 3 and even otherwise. 6.3.29. Suppose that n is odd. Prove that in all drawings of K., the parity of the number of pairs of edges that cross is the same. Conclude that v(K.) is even when n is congruent to 1 or 3 modulo 8 and is odd when n is congruent to 5 or 7 modulo 8. 6.3.30. (!)It is known that v(C,. oC.) = (m -2)n ifm ~ min{5, n}. Also v(K4 oC.) = 3n. a) Find drawings in the plane to establish the upper bounds. b) Prove that v(C3 D C3 ) ?: 2. (Hint: Find three subdivisions of K3 , 3 that together use each edge exactly twice.) 6.3.31. Let f(n) = v(K.. ). a) Show that 3v(K. ) ~ f (n) ~ 3(;)2_ b) Show that v(Ks.2.2) = 2 and v(K3,a.i) = 3. Show that 5 ~ v(K3 . 3 , 2) ~ 7 and 9 ~ v(Ks.s.a) ~ 15. c) Exercise 6.3.26a shows that the lower bound in part (a) is at least (3/20)n 4 + O(n 3 ). Improve it by using a recurrence to show that f(n) ?: n 3 (n - 1)/6. d) The upper bound in part (a) is ~n 4 + O(n 3 ). Improve it to f (n) ~ -fsn 4 + O(n 3 ). (Hint: One construction embeds the graph on a tetrahedron and generalizes to a construction for K 1,,..n; another uses Kn and generalizes to a construction for Kn, ... ,. ) 6.3.32. (*) Construct an embedding of a 3-regular nonbipartite simple graph on the torus so that every face has even length. 6.3.33. (*)Suppose that n is at least 9 and is not a prime or twice a prime. Construct a 6-regular toroidal graph with n vertices. 6.3.34. (*) An embedding of a graph on a surface is regular if its faces all have the same length. Construct regular embeddings of K4.4, K3 ,6 , and K3 .3 on the torus. 6.3.35. (*) Prove Euler's Formula 'for genus y: For every 2-cell embedding of a graph on the "surface Sy, the numbers of vertices, edges, and faces satisfy n - e + f = 2 - 2y. Conclude that an n-vertex graph embeddable on Sy has at most 3(n - 2 + 2y) edges. 6.3.36. (*) Use Euler's Formula for Sy to prove that y(K3 . 3 . ) ?: n - 2, and determine the value exactly for n ~ 3. 6.3.37. (*)For every positive integer k, use Euler's Formula for higher surfaces to prove that there exists a planar graph G such that y(G o K2) ?: k.
LJL J
L J.
Chapter 7
## Edges and Cycles
7.1. Line Graphs and Edge-coloring
Many questions about vertices have natural analogues for edges. Independent sets have no adjacent vertices; matchings have no "adjacent" edges. Vertex colorings partition vertices into independent sets; we can instead partition edges into matchings. These pairs of problems are related via line graphs (Definition 4.2.18). Here we repeat the definition, emphasizing our return to the context in which a graph may have multiple edges. We use "line graph" and L(G) instead of"edge graph" because E(G) already denotes the edge set.
7.1.1. Definition. The line graph of G, written L(G), is the simple graph whose vertices are the edges of G, with ef E E(L(G)) when e and f have a common endpoint in G.
~h
G
Some questions about edges in a graph G can be phrased as questions about vertices in L ( G). When extended to all simple graphs, the vertex question may be more difficult. If we can solve it, then we can answer the original question about edges in G by applying the vertex result to L(G). In Chapter 1, we studied Eulerian circuits. An Eulerian circ.uit in G yields a spanning cycle in the line graph L(G). (Exercise 7.2.10 shows that the converse need not hold!) In Section 7 .2, we study spanning cycles for graphs in general. As discussed in Appendix B, this problem is computationally difficult. In Chapter 3, we studied matchings. A matching in G becomes an independent set in L(G). Thus a'(G) = a(L(G)), and the study of a' for graphs is
273
274
## Chapter 7: Edges and Cycles
the study of a for line graphs. Computing a is harder for general graphs than for line graphs. Section 3.1 considers this for bipartite graphs, and we describe the general case briefly in Appendix B. In Chapter 4, we studied connectivity. Menger's Theorem gave a min-max relation for connectivity and internally disjoint paths in all graphs. By applying this theorem to an appropriate line graph, we proved the analogous min-max relation for edge-connectivity and edge-disjoint paths in all graphs. In Chapter 5, we studied vertex coloring. Coloring edges so that each color class is a matching amounts to proper vertex coloring of the li:r~e graph. Thus edge-coloring is a special case of vertex coloring and therefore potentially easier. We discuss edge-coloring in this section. Our main result, when stated in terms of vertex coloring ofline graphs, is an algorithm to compute x (H) within 1 when H is the line graph of a simple graph. Thus line graphs suggest the problems of edge-coloring and spanning cycles that are discussed in this chapter. We first study these separately. In Section 7 .3, we study their connections to each other and to planar graphs. In applying algorithms for line graphs, we may need to know whether G is a line graph. There are good algorithms to check this; they use characterizations ofline graphs, which we postpone to the end of this section.
EDGE-COLORINGS
In Example 1.1.11 that introduced vertex coloring, we needed to schedule Senate committees. Edge-coloring problems arise when the objects being scheduled are pairs of underlying elements.
7.1.2. Example. Edge-coloring of K2n In a league with 2n teams, we want to schedule games so that each pair of teams plays a game, but each team plays at most once a week. Since each tearp. must play 2n - 1 others, the season lasts at least 2n - 1 weeks. The games of each week must form a matching. We can schedule the season in 2n -1 weeks ifand only if we can partition E(K 2n) into 2n -1 matchings. Since K2n is 2n -1-regular, these must be perfect matchings. The figure below describes the solution. Put orie vertex in the center. Arrange the other 2n - 1 vertices cyclically, viewed as congruence classes modulo 2n - 1. As in Theorem 2.2.16, the difference between two congruence classes is 1 if they are consecutive, 2 if there is one class between them, and so on up to difference n - 1. There are 2n - 1 edges with each difference i, for 1 :=:: i :=:: n - 1.
## Section 7 .1: Line Graphs and Edge-coloring
275
Each matching consists of one edge from each difference class plus one edge involving the center vertex. We show one such matching in bold. Rotating the picture (to obtain the solid matching) yields n new edges; again they are one of each length plus one to the center. The 2n - 1 rotations of the figure yield the desired matchings, since these matchings take distinct edges from each difference class and distinct edges involving the center vertex. 7.1.3. Definition. A k-edge-coloring of G is a labeling f: E(G) --+ S, where ISi = k (often we use S = [k]). The labels are colors; the edges of one color form a color class. A k-edge-coloring is proper if incident edges have different labels; that is, if each color class is a matching. A graph is k-edgecolorable if it has a proper k-edge-coloring. The edge-chromatic number x'(G) of a loopless graph G is the least k such that G is k-edge-colorable. Chromatic index is another name for x'(G). Since edges sharing a vertex need different colors, x'(G) 2.: ~(G). Vizing [1964] and Gupta [1966] independently proved that ~(G) + 1 colors suffice when G is simple; this is our main objective. A clique in L(G) is a set of pairwise-intersecting edges of G. When G is simple, such edges form a star or a triangle in G (Exercise 9). For the hereditary class of line graphs of simple graphs, Vizing's Theorem thus states that x(H) .:=: w(H) + 1; thus line graphs are "almost" perfect. In contrast to x(G) in Chapter 5, multiple edges greatly affect x'(G). A graph with a loop has no p~oper edge-coloring; the adjective "loopless" excludes loops but allows !11Ultiple edges. 7.1.4. Definition. In a graph G with multiple edges, we say that a vertex pair x, y is an edge of multiplicity m ifthere are m edges with endpoints x, y. We write (xy) for the multiplicity of the pair, and we write (G) for the maximum of the edge multiplicities in G. 7.1.5. Example. The "Fat Triangle". For loopless graphs with multiple edges, x'(G) may exceed ~(G) + 1. Shannon [1949] proved that the maximum of x'(G) in terms of ~(G) alone is 3~(G)/2 (see Theorem 7.1.13). Vizing and Gupta proved that x'(G) .:=: ~(G) + (G), where (G) is the maximum edge multiplicity. The graph below achieves both bounds. The edges are pairwise intersecting and hence require distinct colors. Thus x'(G) = 3~(G)/2 = ~(G) + (G).
7.1.6. Remark. We have observed that always x'(G) 2.: ~(G). The upper bound x'(G) .:=: 2~(G) - 1 also follows easily. Color the edges in some order,
276
## Chapter 7: Edges and Cycles
always assigning the current edge the least-indexed color different from those already appearing on edges incident to it. Since no edge is incident to more than 2(.!l(G) - 1) other edges, this never uses more than 2.!l(G) - 1 colors. The procedure is precisely greedy coloring for vertices of L ( G). x'(G) = x(L(G)):::; Ll(L(G))
+ 1:::; 2.!l(G) -
l.
For bipartite graphs, the results of Chapter 3 improve the upper bound of Remark 7.1.6, achieving the trivial lower bound even when multiple edges are allowed. Furthermore, there is a good algorithm to produce a proper il(G)edge-coloring in a bipartite graph G.
7.1.7. Theorem. (Konig [1916]) If G is bipartite, then x'(G) = il(G). Proof: Corollary 3.1.13 states that every regular bipartite graph H has a 1factor. By induction on il(H), this yields a proper il(H)-edge-coloring. It therefore suffices to show that for every bipartite graph G with maximum degree k, there is a k-regular bipartite graph H containing G. To construct such a graph, first add vertices to the smaller partite set of G, if necessary, to equalize the sizes. If the resulting G' is not regular, then each partite set has a vertex with degree less thank. Add an edge with these two vertices as endpoints. Continue adding such edges until the graph becomes k-regular; the resulting graph is H.
For a regular graph G, proper edge-coloring with .cl ( G) colors is :)qnivalent to decomposition into 1-factors.
7.1.8. Definition. A decomposition of a regular graph G into 1-factors is a I-factorization of G. A graph with a !-factorization is 1-factorable.
An odd cycle is not 1-factorable; x'(C2m+1) = 3 > Ll(C2m+1). The Petersen graph also requires an extra color, but only. one extra color.
7.1.9. Example. The Petersen graph is 4-edge-chromatic (Petersen [1898]). The Petersen graph is 3-regular; 3-edge-colorability requires a !-factorization. Deleting a perfect matching leaves a 2-factor; all components are cycles. The !-factorization can be completed only if these are all even cycles. Thus it suffices to show that every 2-factor is isomorphic to 2C5 . Consider the drawing consisting of two 5-cycles and a mlitching (the cross edges) between them. We consider cases by the number of cross edges used.
<l>A~-,
\ '',,,,
;,.,
---
--
## Section 7.1: Line Graphs and Edge-coloring
277
Every cycle uses an even number of cross edges, so a 2-factor H has an even number m of cross edges. If m = 0 (left figure), then H = 2C5 If m = 2 (central figure), then the two cross edges have nonadjacent endpoints on the inner cycle or the outer cycle. On the cycle where their endpoints are nonadjacent, the remaining three vertices force all five edges of that cycle into H, which violates the 2-factor requirement. If m = 4 (right figure), then the cycle edges forced into H by the unused cross edges form a 2P5 whose only completion to a 2-factor in His 2C5 Note that since C5 is 3-edge-colorable, the graph is 4-edge-colorable. Now we consider all simple graphs. We make b.(G) + 1 colors available and build a proper edge-coloring, incorporating edges one by one until we have a proper D.(G) + 1-edge-coloring of G. The algorithm runs surprisingly quickly.
7.1.10. Theorem. (Vizing [1964, 1965], Gupta [1966]) If G is a simple graph, then x'(G) ~ D.(G) + 1. Proof: Let f be a proper D.(G) + 1-edge-coloring of a subgraph G' of G. If G' # G, then some edge uv is uncolored by f. After possibly recoloring some edges, we extend the coloring to include uv; call this an augmentation. After e(G) augmentations, we obtain a proper D.(G) + 1-edge-coloring of G. Since the number of colors exceeds D.(G), every vertex has some color not appearing on its incident edges. Let a0 be a color missing at u. We generate a list of neighbors of u and a corresponding list of colors. Begin with v0 = v. Let a 1 be a color missing at vo. We may assume that a 1 appears at u on some edge uv 1; otherwise, we would use a 1 on uv 0. Let a2 be a color missing at v1. We may assume that a 2 appears at u on some edge uv 2 ; otherwise, we would replace color a 1 with a 2 on uv 1 and then use a 1 on uv0 to augment the coloring. Having selected uv;-1 with color a;-1, let a; be a color missing at v;_ 1. If a; is missing at u, then we use a; on uv;-1 and shift color aj from uvj to uvj-l for 1 ~ j ~ i - 1 to complete the augmentation. We call this downshifting from i. lfa; appears at u (on some edge uv;), then the process continues.
V1
Vo= V
Sin.ce we have only D. ( G) + 1 colors to choose from, the list of selected colors eventually repeats (or we complete the augmentation by downshifting). Let l be the smallest index such that a color missing at v1 is in the list a1, ... , a1; let this color be ak. Instead of extending the list, we use this repetition to perform the augmentation in one of several ways. The color ak missing at Vt is also missing at vk-J and appears on uvk. If ao does not appear at Vt, then we downshift from v1 and use color ao on u v1 to complete the augmentation. Hence we may assume that a 0 appears at Vt.
278
## Chapter 7: Edges and Cycles
Let P be the maximal alternating path of edges colored ao and ak that begins at Vt along color ao. There is only orie such path, because each vertex has at most one incident edge in each color (we ignore edges not yet colored). To complete the augmentation, we will interchange colors a 0 and ak on P and downshift from an appropriate neighbor of u, depending on where P goes. If P reaches vk. then it arrives at vk along an edge with color a 0 , follows vku in color ck> and stops at u, which lacks color a 0 In this case, we downshift from vk and switch colors on P (left picture below). If Preaches vk-1, then it reaches Vk-1 on color ao and stops there, because ak does not appear at Vk-1 In this case, we downshift from Vk-1, give color ao to uvk-1, and switch colors on P (middle picture). If P does not reach vk or vk_ 1 , then it ends at some vertex outside {u, Vt, vb vk-d In this case, we downshift from Vt, give color a 0 to uv1, and switch colors on P (rightmost picture). In each case, the changes described yield a proper il(G) + 1-edge-coloring of G' + uv, so we have completed the desired augmentation.
ak ak ak
ao Vk Vk-1 Vk
ao
ao
Vk-1 Vk
ao
V2
V1
V2
V1
V2
ak
'
V1
a1
VI
v1
V1
Vk
Preaches
P reaches
Vk-1
otherwise
For simple graphs, we now have only two possibilities for x'.
7.1.11. Definition. A simple graph G is Class 1 if x'(G) = il(G). It is Class 2 if x'(G) = il(G) + 1.
Determining whether a graph is Class 1 or Class 2 is generally hard (Holyer [1981]; see Appendix B). Thus we seek conditions that forbid or guarantee Ll ( G)edge-colorability. Examples of such conditions include Exercises 24-27.
7.1.12.* Remark. There is an obvious necessary condition for a graph to be Class 1 that is conjectured to be sufficient when il(G) > i30 n(G). Part (a) of Exercise 27 observes that a subgraph of G with odd order is an obstruction to il (G)-edge-colorability if it has too many edges. A subgraph H of a simple graph G is an overfull subgraph ifn(H) is odd and 2e(H)/(n(H) - 1) > il(G). The Overfull Conjecture (Chetwynd-Hilton [1986]-see also Hilton [1989]) states that if il(G) > n(G)/3, then a simple graph G is Class 1 if and
## Section 7.1: Line Graphs and Edge-coloring
279
only if G has no overfull subgraph. The Petersen graph with a vertex deleted shows that the condition is not sufficient when b.(G) = n(G)/3 (Exercise 28). The Overfull Conjecture implies the I-factorization Conjecture: If r ~ m (or r ~ m - I if mis even), then every r-regular simple graph of order 2m is Class 1. This also is sharp (Exercise 29). The conclusions of the two conjectures hold when b.(G) is large enough (Chetwynd-Hilton [I989], Niessen-Volkmann [I990], Perkovic-Reed [I997], Plantholt [200I]. When G has multiple edges, x'(G) _::: L3b.(G)/2J (Shannon [1949]) and x'(G) _::: b.(G) + ,(G) (Vizing [I964, 1965], Gupta [I966]) These bounds follow (Exerc;ise 35) from that of Andersen [I977] and Goldberg [1977, I984]:
x' (G) _::: max{b.(G), maxp U<d(x) + ,(xy) + ,(yz) + d(z)) J} where P = {x, y, z E V ( G): y E N (x) n N (z)}. Proving this bound uses the methods of Theorem 7.1.10 plus counting arguments. To illustrate the use of counting arguments, we prove Shannon's Theorem from that ofVizing and Gupta.
7.I.I3.* Theorem. (Shannon [1949]) If G is a graph, then x'(G) _::: ~b.(G). Proof: Let k = x'(G), and assume k ~ (3/2)b.(G). Let G' be a minimal subgraph of G with x'(G'.) = k. Since k _::: b.(G') + ,(G') (Vizing-Gupta), we obtain ,(G') ~ b.(G)/2. Let e with endpoints x, y be an edge with multiplicity ,(G'). Let f be a proper k - I-edge-coloring of G' - e. In G' - e, both x and y have degree at most b.(G) - I, so under fat least (k - I) - (b.(G) - I) colors are missing at x, and similarly at y. No color is missing at both, since G' is not k - I-edge-colorable. Accounting for the ,(G') - I colors used on edges with endpoints x, y yields
2(k - b.(G))
+ (b.(G)/2) -
+ ,(G')
- 1 _::: k - 1,
If x'(G)
## CHARACTERIZATION OF LINE GRAPHS (optional)
Characterizations ofline graphs can lead to good algorithms to test whether a graph G is a line graph and, if so, to obtain H such that L(H) = G.
7.I.I5. Example. To illustrate the ideas, we prove that the rightmost graph below is not the line graph of a simple graph. The kite G (two triangles with a common edge) is the line graph of the paw H (a claw plus an edge). By case
280
## Chapter 7: Edges and Cycles
analysis, we find that H is the only simple graph whose line graph is G, and the edges becoming the vertices of degree 2 in G must be the dashed edges. The rightmost graph adds a vertex to G having only the vertices of degree 2 as neighbors. The result is not a line graph, because there is no way to add an edge to H that shares an endpoint with each dashed edge without sharing an endpoint with a solid edge.
G = L(H)
## not a line graph
Our first characterization encodes the process of taking the line graph. If
G = L(H) and His simple, then each v E V (H) with d(v) ::: 2 generates a clique Q(v) in G corresponding to edges incident to v. These cliques partition E(G). Furthermore, each vertex e E V ( G) belongs only to the cliques generated by the two endpoints of e E E(H). For example, when G is the kite, we can partition E(G) into three cliques
(a triangle plus two edges), each vertex covered at most twice. These three cliques correspond to the vertices of degree at least 2 in the paw. The rightmost graph above does not have such a partition.
7.1.16. Theorem. (Krausz [1943]) For a simple graph G, there is a solution to L(H) = G if and only if G decomposes into complete subgraphs, with each vertex of G appearing in at most two in the list. Proof: We argued above that the condition is necessary. Note that when G = L(H), the vertices of G that belong to only one of the cliques we have defined are those corresponding to edges of H that are incident to leaves. For sufficiency, let Si. ... , Sk be the vertex sets of the specified complete subgraphs. We construct H such that G = L(H). Isolated vertices of G become isolated edges of H, so we may assume that 8 ( G) ::: 1. Let vi. ... , v1 be the vertices of G (if any) that appear in exactly one of S;, ... , S1n. Give H one vertex for each set in the list A= Si. ... , Sb {t}i}, ... , {vi}}, and let vertices of H be adjacent if the corresponding sets intersect. Each vertex of G appears in exactly two sets in A, and no two vertices appear in the same two sets. Hence H is a simple graph with one edge for each vertex of G. If vertices are adjacent in G, then they appear together in some S;, and the corresponding edges of H share the vertex for S;. Hence G = L(H).
Krausz's characterization does not directly yield an efficient test for line graphs, because there are too many possible decompositions to test. The next characterization tests substructures of fixed size and therefore yields a good algorithm. We say that each triangle T in G is odd or even as defined below.
## Section 7.1: Line Graphs and Edge-coloring
281
Tis odd if IN(v) n V(T)I is odd for some v E V(G). Tis even if IN(v) n V(T)I is even for every v E V(G). An induced kite is a double triangle; it consists of two triangles sharing an edge, and the two vertices not in that edge are nonadjacent. 7.1.17. Theorem. (van Rooij and Wilf [1965]) For a simple graph G, there is a solution to L(H) = G if and only if G is claw-free and no double triangle of G has two odd triangles. Proof: Necessity. Suppose that G = L(H). A vertex e in G with neighbors x, y, z corresponds to an edge e in H incident to edges x, y, z. Since e has only two endpoints in H, two of x, y, z are incident at one of them and hence are adjacent in G. This forbids the claw as an induced subgraph of G. For the other condition, we saw in Example 7.1.15 that the vertices of a double triangle in G must correspond to the edges of a paw in H. In particular, the vertices of one of these triangles in G correspond to the edges of a triangle in H. This triangle must be even, because every edge in H incident to exactly one vertex of a triangle shares an endpoint with exactly two of its edges. Hence for each double triangle in G, at least one of its triangles is even. Sufficiency. Suppose that G satisfies the specified conditions. We may assume that G is connected; otherwise, we apply the construction to each component. The case where G is claw-free and has a double triangle with both triangles even is very special; there are only three such graphs (Exercise 38). Here we consider only the general case, in which every double triangle of G has exactly one odd triangle. By Theorem 7 .1.16, it suffices to decompose G into complete subgraphs, using each vertex in at most two of them. Let Si. ... , Sk be the maximal complete subgraphs of G that are not even triangles, and let T1 , . , 1i be the edges that belong to one even triangle and no odd triangle. We claim that together these form the desired decomposition B.
Every edge appears in a maximal complete subgraph, but every triangle in a complete subgraph with more than three vertices is odd. Hence each edge 1j in the list is not in any S;. Also S; and S;' share no edge, because G has no double triangles with both triangles odd. Hence the subgraphs in B are pairwise edge-disjoint. If e E E(G), then e is in some S; unless the only maximal clique containing e is an even triangle. In this case e is a 1j, since we have forbidden double triangles with both triangles even. Hence B is a decomposition. It remains to show that each v E G appears in at most two of these subgraphs. Suppose that v belongs to A, B, C e B. Edge-disjointness implies that v has neighbors x, y, z with each belonging to only one of {A, B, C}. Since G has
282
## Chapter 7: Edges and Cycles
no induced claw, we may assume that x # y. By edge-disjointness, the triangle vxy cannot belong to a member of B. Hence it must be an even triangle. Therefore, z must have exactly one other edge to vxy, say z # x and z j4 y. But now the same argument shows zvx is an even triangle, and we have a double triangle with both triangles even. Theorem 7.1.17 is close to a forbidden subgraph characterization.
7.1.18. Theorem. (Beineke [1968]) A simple graph G is the line graph of some simple graph if and only if G does not have any of the nine graphs below as an induced subgraph.
Proof: By Theorem 7.1.17, it suffices to show that the eight graphs listed other than K 1.a are the vertex-minimal claw-free graphs containing a doubl~ triangle with both triangles odd. Each such graph has a double triangle and one or two additional vertices that make the triangles odd by having one or three neighbors in the triangles. The details of showing that this is the full list are requested in Exercise 40.
The characterizations in Theorems 7.1.17-7.1.18 yield algorithms to test whether G is a line graph that run in time polynomial in n(G). In fact, there is such an algorithm that runs in linear time (Lehot [1974]) and produces a graph H such that G = L(H) when G is a line graph. This graph His unique if G has no component that is a triangle (Exercise 39).
EXERCISES
7.1.1. (-) For each graph G below, compute x'(G) and draw L(G).
## 7.1.2. (-)Give an explicit edge-coloring to prove that x'(Qk) =
~(Qk)
Section 7.1: Line Graphs and Edge-coloring 7.1.3. (-) Determine the edge-chromatic number of c. o K 2 7.1.4. (-)Obtain an inequality for x'(G) in terms of e(G) and a'(G). 7.1.5. (-)Prove that the Petersen graph is the complement of L(K5 ).
283
7.1.6. (-)Determine the number of triangles in the line graph of the Petersen graph. 7.1.7. (-)Determine whether P5 is a line graph. If so, find H such that L(H) = 7.1.8. (-)Prove that L(Km.n)
~
P5
Km 0 K.
7.1.,9. Let G be a simple graph. Prove that vertices form a clique in L(G) if and only if the corresponding edges in G have one common endpoint or form a triangle. (Comment: Thus w(L(G)) = ~(G) unless ~(G) = 2 and some component of G is a triangle.) 7.1.10. Let G be a simple graph without isolated vertices. Prove that if L(G) is connected and regular, then either G is regular or G is a bipartite graph in which vertices of the same partite set have the same degree. (Ray-Chaudhuri [1967]) 7.1.11. (!) Let G be a simple graph. a) Prove that the number of edges in L( G) is LvEV(GJ {d~l). b) Prove that G is isomorphic to L(G) if and only if G is 2-regular. 7.1.12. Let G be a connected simple graph. Use part (a) of Exercise 7.1.11 to determine when e(L(G)) < e(G). 7.1.13. (+) Prove that the graph below is the only simple graph whose line graph is isomorphic to its complement. (Albertson) '7.1.14. (!) Let G beak-edge-connected simple graph. Prove that L(G) is k-connected and is 2k - 2-edge-connected. (Hint: For a minimum edge cut [S, SJ in L(G), describe what the cut corresponds to in G and count its edges in terms of the vertices of G.) 7.1.15. (!)Use Tutte's 1-factor Theorem to prove that every connected line graph of even order has a perfect matching. Conclude from this that the edges of a simple connected graph of even size can be partitioned into paths of length 2. (Comment: Exercise 3.3.22 shows that every connected claw-free graph has a perfect matching , but that stronger result is more difficult than this.) (Chartrand-Polimeni-Stewart [1973]) 7.1.16. (*)Let G be a simple graph. Prove that y(L(G)) the genus of G (Definition 6.3.20). (D. Greenwell)
:=:::
## 7.1.17. Compute the number of proper 6-edge-colorings of the graph below.
7.1.18. (!)Give an explicit edge-coloring to prove that x'(K,,,) = fl(K,,,). 7.1.19. (!)Prove that for every simple bipartite graph G, there is a bipartite graph H that contains G.
~(G)-regular
simple
284
## Chapter 7: Edges and Cycles
7.1.20. (!)Let D be a digraph (loops allowed) such that d+(v) ::::: d and d-(v) ::::: d for all v E V(D). Prove that E(D) can be colored using at most d colors so that the edges entering each vertex have distinct colors and the edges exiting each vertex have distinct colors. (Hint: Transform the digraph into another object where a known result applies.) 7.1.21. Algorithmic proof of Theorem 7.1. 7. Let G be a bipartite graph with maximum degree k. Let f be a proper k-edge-coloring of a subgraph H of G. Let uv be an edge not in H, By using a path alternating in two colors, show that f can be altered and then extended to a proper k-edge-coloring of H +UV. Conclude that x'(G) = A(G). 7.1.22. Use Brooks' Theorem to an appropriate graph to prove that if G is a simple graph with A(G) = 3, then G is 4-edge-colorable. (Comment: The result is a special case ofVizing's Theorem; do not use Vizing's Theorem to prove this.) 7.1.23. (+)Let K(p, q) be the complete p-partite graph with q vertices in each partite set. Let G [H] denote the composition operation, in which each vertex of G expands into a copy of H. Note that K(p, q) = K(p, d)[Kq;dl when d divides q. a) Show that if G has a decomposition into copies of F, then G [K ml has a decomposition into copies of F[Kml Show also that the relation "G decomposes into spanning copies of F" is transitive. b) Cliques of even order decompose into 1-factors. Cliques of odd order decompose into spanning cycles. Use these statements and part (a) to. prove that K(p, q) decomposes into 1-factors when pq is even. (Hartman [1997]) 7.1.24. (!)Let G and H be nontrivial simple graphs. Use Vizing's Theorem to prove that x'(H) = A(H) implies x'(G 0 H) = A(G 0 H). 7.1.25. Kotzig's Theorem for cartesian products of simple graphs. a) Use Vizing's Theorem to prove that x'(G o Kz) = A(G o Kz). b) Let G 1 , G 2 be edge-disjoint graphs with vertex set V, and let Hi, H 2 be edgedisjointgraphs with vertex set W. Prove that (G 1 UG 2 )D(H1 UH2 ) = (G 1 oH2 )U (G 2 oH1 ). c) Use parts (a) and (b) to prove that x'(G o H) = A(G o H) if both G and H have 1factors. (Comment: As a result,. the product of the Petersen graph with itself is Class 1, which does not follow from Exercise 7.1.24. Here neither factor need be Class 1; there G need not have a 1-factor.) (Kotzig [1979], J. George [199iD 7.1.26. (!) Let G be a regular graph with a cut-vertex. Prove that x'(G) > A(G). 7.1.27. Density conditions for x'(G) > A(G). a) Prove that ifn(G) = 2m + 1 and e(G) > m A(G), then x'(G) > A(G). b) Prove that if G is obtained from a k-regular graph with 2m + 1 vertices by deleting fewer than k/2 edges, then x'(G) > A(G). c) Prove that if G is obtained by subdividing an edge of a regular graph with 2m vertices and degree at least 2, then x'(G) > A(G). 7.1.28. (*-) Prove that the Petersen graph has no overfull subgraph. 7.1.29. Let G be them - 1-regular connected graph formed from 2Km by deleting an edge from each compom,mt and adding two edges between the components to restore regularity. Prove that G is not 1-factorable if mis odd and greater than 3. (Comment: This shows that the 1-factorization Conjecture (Remark 7.1.12) is sharp.)
~
Km -e Km -e
## Section 7.1: Line Graphs and Edge-coloring
285
7.1.30. (*!)Overfull Conjecture=} I-factorization Conjecture (Remark 7.1.12). a) Prove that in a regular graph of even order, an induced subgraph is overfull if and only ifthe subgraph induced by the other vertices is overfull. b) Let G be an k-regular graph of order 2m having an overfull subgraph. Prove that k < m if m is odd and that k < m - 1 if m is even. 7;1.31. Given an edge-coloring of a graph G, let c(v) denote the number of distinct colors appearing on edges incident to v. Among all k-edge-colorings of G, a coloring is optimal if it.maximizes LvEV(G) c(v). a) Prove that if no component is an odd cycle, then G has a 2-edge-coloring where both colors appear at each vertex of degree at least 2. (Hint: Use Eulerian circuits.) b) Let f be an optimal k-edge-coloring of G in which color a appears at least twice at u E V(G) and color b does not appear at u. Let H be the subgraph of G consisting of edges colored a or b. Prove that the component of H containing u is an odd cycle. c) Let G be a bipartite graph. Conclude from part (b) that G is t.(G)-edge-colorable. (Comment: These ideas also lead to a proof ofVizing's Theorem.) (Fournier [1973]) 7.1.32. Let G be a bipartite graph with minimum degree k. Prove that G has a k-edgecoloring in which at each vertex v, each color appears rdCv)/ kl or Ld(v)/ kJ times. (Hint: Use a graph transformation.) (Gupta [1966]) 7.1.33. Use Vizing's Theorem to prove that every simple graph with maximum degree ~has aI)J"equitable" ti.+ 1-edge-coloring: a proper edge-coloring with each color used re(G)/(t. + 1)1 or le(G)/(t. + l)j times. (de Werra [1971], McDiarmid [1972]), 7.1.34. Use Petersen's Theorem (every 2k-regular graph has a 2-factor-Theorem 3.3.9) to prove that x'(G) ~ 3 rt.(G)/21 when G is a loopless graph. 7.1.35. Bounds on x'(G). Let P = {x, y, z E V(G): y E N(x) n N(z). Prove that the last bound below (Andersen [1977], Goldberg [1977, 1984]) implies the earlier bounds. x'(G) ~ L3t.(G)/2j. (Shannon [1949]) x'(G) ~ ~(G) + (G). (Vizing [1964, 1965], Gupta [1966]) x'(G) ~ max{t.(G), maxp UCd(x) + d(y) + d(z)) J }. (Ore [1967a]) x'(G) ~ max{~(G), maxp UCd(x) + (xy) + (yz) + d(z))
J}.
7.1.36. (:+)For n -:f:. 8, prove that L(Kn) is the only 2n - 4-regular simple graph oforder (;) in which nonadjacent vertices have four common neighbors and adjacent vertices haven - 2 common neighbors. (Comment: When n = 8, three exceptional graphs satisfy the conditions.) (Chang [1959], Hoffman [1960]) 7.1.37. (+) Forn, m not both equalling4, prove that L(Km,n) is the only (n+m-2)-regular simple graph of order mn in which nonadjacent vertices have two common neighbors, n(;) pairs of adjacent vertices. have m - 2 common neighbors, and m(~) pairs of adjacent vertices have n - 2 common neighbors. (Comment: When n = m = 4, there one exceptional graph-Shrikande [1959].) (Moon [1963], Hoffman [1964]) 7.1.38. (*) Let G be a connected, simple, claw-free graph having a double triangle H with each triangle even. Prove that G is one of the three graphs below, and conclude that G is a line graph. (Comment: This completes the proof of Theorem 7.1.17.)
286
## Chapter 7: Edges and Cycles
7.1.39. (*)A Krausz decomposition of a simple graph His a partition of E(H) into cliques such that each vertex of H appears in at most two of the cliques. a) Prove that for a connected simple graph H, two Krausz decompositions of H that have a common clique are identical. b) Find distinct Krausz decompositions for the graphs in Exercise 7.1.38. c) Prove that no other connected simple graph except K3 has two distinct Krausz decompositions (use Exercise 7.1.38 and the proof of Theorem 7.1.17). d) Conclude that K 1, 3 , K3 is the only pair of nonisomorphic connectt:d simple graphs with "isomorphic line graphs. (Whitney [1932a]) 7.1.40. (*) Complete the proof of Theorem 7.1.18 by proving that a simple graph with no induced claw has a double triangle with both triangles odd if and only if it containj'l an induced subgraph among the other eight graphs listed in the theorem statement.
## 7.2. Hamiltonian Cycles
Studied first by Kirkman [1856], Hamiltonian cycles are named for Sir William Hamilton, who described a game on the graph of the dodecahedron in which one player specifies a 5-vertex path and the other must extend it to a spanning cycle. The game was marketed as the "Traveller's Dodecahedron", a wooden version in which the vertices were named for 20 important cities.
7.2.1. Definition. A Hamiltonian graph is a graph with a spanning cycle, also called a Hamiltonian cycle.
Until the 1970s, interest in Hamiltonian cycles centered on their relationship to the Four Color Problem (Section 7.3). Later study was stimulated' by practical applications and by the issue of complexity (Appendix B). No easily testable characterization is known for Hamiltonian graphs; we will study necessary conditions and sufficient conditions. Loops and multiple edges are irrelevant; a graph is Hallliltonian if and only i:f the simple graph obtained by keeping one copy of each non-loop edge is Hamiltonian. Therefore, in this section we restrict our attention to simple graphs; this is relevant when discussing conditions involving vertex degrees. For further material on Hamiltonian cycles, see Chvatal [1985a].
## Section 7 .2: Hamiltonian Cycles
287
NECESSARY CONDITIONS
Every Hamiltonian graph is 2-connected, because deleting a vertex leaves a subgraph with a spanning path. Bipartite graphs suggest a way to strengthen this necessary condition.
7.2.2. Example. Bipartite graphs. A spanning cycle in a bipartite graph visits the two partite sets alternately, so there can be no such cycle unless the partite sets have the same size. Hence Km.n is Hamiltonian only if m = n. Alternatively, we can argue that the cycle returns to different vertices of one partite set after each visit to the other partite set. 7.2.3. Proposition. If G has a Hamiltonian cycle, then for each nonempty set S V, the graph G - S has at most ISi components. Proof: When leaving a component of G - S, a Hamiltonian cycle can go only to S, and the arrivals in S must use distinct vertices of S. Hence S must have at least as many vertices as G - S has components.
## 7.2.4. Definition. Let c(H) denote the number of components of a graph H.
Thus the necessary condition is that c(G - S) ~ ISi for all 0 =j:. S V. This condition guarantees that G is 2-connected (deleting one vertex leaves at most one component), but it does not guarantee a Hamiltonian cycle.
7.2.5. Example. The graph on th,e foft below is bipartite with partite sets of equal size. However, it fails the ne<\e111sary condition of Proposition 7 .2.3. Hence it is not Hamiltonian.
The graph on the right shows that the necessary condition is not sufficient. This graph satisfies the condition but .has no spanning cycle. All edges incident to vertices of degree 2 must be used, but in this graph that requires three edges ip.cident to the central vertex.
288
## Chapter 7: Edges and Cycles
The Petersen graph is another non-Hamiltonian graph satisfying the condition. We proved in Example 7.1.9 that 2C5 is the only 2-factor of the Petersen graph, so it has no spanning cycle.
7.2.6.* Remark. Strengthening a necessary condition may yield a sufficient condition. Perhaps requiring ISi ::: 2c(G-S) for everycutset S would guarantee a spanning cycle. A graph Gist-tough if ISI ::: tc( G - S) for every cutset S c V. The toughness of G is the maximum t such that G is t-tough. For example, the toughness of the Petersen graph is 4/3 (Exercise 23). By Proposition 7.2.3, spanning cycles require toughness at least 1. Chvatal (1974] conjectured that a sufficiently large toughness is sufficient. No value of toughness larger than 1 is necessary, since Cn itself is only 1-tough. For some years it was thought that toughness 2 would be sufficient. Enomoto-JacksonKaterinis-Saito [1985] constructed non-Hamiltonian graphs with toughness 2 - E for each E > 0. Finally, Bauer-Broersma-Veldman [2000] constructed non-Hamiltonian graphs with toughness approaching 9/4. Chvatal's conjecture that some value of toughness suffices remains open.
SUFFICIENT CONDITIONS
The number of edges needed to force an n-vertex graph to be Hamiltonian is quite large (Exercises 26-27). Under conditions that "spread out" the edges, we can reduce the number of edges while still guaranteeing Hamiltonian cycles. The simplest such condition is a lower bound on the minimum d'egree; o(G) ::: n(G)/2 suffices. We first note that no smaller minimum degree is sufficient.
7.2.7. Example. The graph consisting of cliques of orders L(n + 1)/2J and r<n + 1)/21 sharing a vertex has minimum degree L(n - l)/2J but is not Hamiltonian (not even 2-connected). . For odd order, another non-Hamiltonian graph with this minimum degree is the biclique with partite sets of sizes (n - 1)/2 and (n + 1)/2. Proving that o(G) ::: n(G)/2 forces a spanning cycle thus shows that L (n - 1) /2J is the largest value of the minimum degree among non-Hamiltonian graphs with n vertices. I
7.2.8. Theorem. (Dirac [1952b]). If G is a simple graph with at least three vertices and 8 ( G) ::: n (G) /2, then G is Hamiltonian. Proof: The condition n(G) ::: 3 is annoying but must be included, since K 2 is not Hamiltonian but satisfies 8(K2 ) = n(K2 )/2.
## Section 7.2: Hamiltonian Cycles
289
The proof uses contradiction and extremality. If there is a non-Hamiltonian graph satisfying the hypotheses, then adding edges cannot reduce the minimum degree. Thus we may restrict our attention to maximal non-Hamiltonian graphs with minimum degree at least n/2, where "maximal" means that adding any edge joining nonadjacent vertices creates a spanning cycle. When u ~ v in G, the maximality of G implies that G has a spanning path vi, ... , Vn from u = v1 to v = Vn, because every spanning cycle in G + u v contains the new edge uv. To prove the theorem, it suffices to make a small change in this cycle to avoid using the edge uv; this will build a spanning cycle in G. If a neighbor of u directly follows a neighbor of v on the path, such as u # V;+1 and v t t v;, then (u, V;+i, V;+2 ... , v, v;, V;-1, ... , v2) is a spanning cycle.
U
~
V; Vi+l
To prove that such a cycle exists, we show that there is a common index in the sets Sand T defined by S = {i: u # V;+d and T = {i: v # v;}. Summing the sizes of these sets yields IS u Tl+ IS n Tl= ISi + ITI = d(u) + d(v) ::::_ n. Neither S nor T contains the index n. Thus IS U Tl < n, and hence IS n Tl::::_ 1. We have established a contradiction by finding a spanning cycle in G; hence there is no (maximal) non-Hamiltonian graph satisfying the hypotheses. Ore observed that this argument uses 8 ( G) 2: n ( G) /2 only to show that d(u) +d(v) 2: n. Therefore, we can weaken the requirement of minimum degree n/2 to require only that d(u) + d(v) 2: n whenever u ~ v. We also did not need that G was a maximal non-Hamiltonian graph, only that G + uv was Hamiltonian and thereby provided a spanning u, v-path.
7.2.9. Lemma. (Ore [1960]) Let G be a simple graph. If u, v are distinct nonadjacent vertices of G with d(u) + d(v) 2: n(G), then G is Hamiltonian if and only if G + uv is Hamiltonian. Proof: One direction is trivial, and the proof of the other direction is the same as for Theorem 7.2.8.
Bondy and Chvatal [1976] phrased the essence of Ore's argument in a much more general form that yields sufficient conditions for cycles of length I and other subgraphs. Here we discuss only the application to spanning cycleg,_ Using Lemma 7.2.9 to add edges, we can test whether G is Hamiltonian by testing whether the larger graph is Hamiltonian.
7.2.10. Definition. The (Hamiltonian) closure ofa graph G, denoted C(G), is the graph with vertex set V(G) obtained from G by iteratively adding edges joining pairs of nonadjacent vertices whose degree sum is at least n, until no such pair remains.
290
## Chapter 7: Edges and Cycles
The graph above begins with vertices of degree 2, but its closure is K6. Ore's Lemma yields the following theorem. 7.2.11. Theorem. (Bondy-Chvatal [1976]) A simple n-vertex graph is Hamiltonian if and only if its closure is Hamiltonian. Fortunately, the closure does not depend on the order in which we choose to add edges when more than one is available. 7.2.12. Lemma. The closure of G is well-defined. Proof: Let ei, ... , e, and fi, ... , Is be sequences of edges added in forming C(G), the first yielding G 1 and the second G 2. If in either sequence nonadjacent vertices JM and v acquire degree summing to at least n(G), then the edge uvmust be added before the sequence ends. Thus Ji, being initially addable to G, must belong to G1. Similarly, if fi, ... , f;-i E E(G 1), then /; becomes addable to G 1 and therefore belongs to G 1 . Hence neither sequence contains a first edge omitted by the other sequence, and we have G1 ~ G2 and G2 ~ G1. We now have a necessary and sufficient condition to test for Hamiltonian cycles in simple graphs. lt doesn't help much, because it requires us to test whether another graph is Hamiltonian! Nevertheless, it does furnish a method for proving sufficient conditions. A condition that forces C ( G) to be Hamiltonian also forces a Hamiltonian cycle in G. For example, the condition may imply C(G) = Kn. Chvatal used this method to prove the best possible degree sequence condition for Hamiltonian cycles. Some vertex degrees can be small if others are large enough.
7.2.13. Theorem. (Chvatal [1972]) Let G be a simple graph with vertex degrees di :S :S dn, where n : :'._ 3. If i < n/2 implies that d; > i or dn-i 2: n -i
(Chvatal's condition), then G is Hamiltonian. Proof: Adding edges to form the closure reduces no entry in the degree sequence. Also, G is Hamiltonian if and only if C(G) is Hamiltonian. Thus it suffices to consider the case where C ( G) = G, which we describe by saying that G is closed. In this case, we prove that Chvatal's condition implies that G = Kn. We prove the contrapositive; if G is a closed n-vertex graph that is not a complete graph, then we construct a value of i less than n /2 for which Chvatal's eondition is violated. Violation means that at least i vertices have degree at most i and at least n - i vertices have degree less than n - i.
## Section 7.2: Hamiltonian Cycles
291
u, v with maximum degree sum. Because G is closed, u fr v implies that d(u) + d(v) < n. We choose the labels on u, v so that d(u) :::; d(v). Since d(u) +d(v) < n, we thus have d(u) < n/2. Let i = d(u).
## With. G #- Kn, we choose among the pairs of nonadjacent vertices a pair
We need to find i vertices with degree at most i. Because we chose a nonadjacent pair with maximum degree sum, every vertex of V - {v} that is not adjacent to v has degree at most d(u), which equals i. There are n - 1 - d(v) such vertices, and d(u) + d(v) :::; n - 1 yields n - 1 - d(v) ::::. i. We also need n - i vertices with degree less than n - i. Every vertex of V - {u} that is not adjacent to u has degree at most d(v), and we have d(v) < n - d (u) = n - i. There are n - 1 - d (u) such vertices. Since d (u) :::; d ( v), we can also add u itself to the set of vertices with degree at most d(v). We thus obtain n - i vertices with degree less than n - i. We have proved that d; :::; i and dn-i < n - i for this specially chosen i, which contradicts the hypothesis.
eu
N(u)
7.2.14. Example. Non-Hamiltonian graphs with "large" vertex degrees. Theorem 7.2.13 characterizes the degree sequences of simple graphs that force Hamiltonian cycles. If the degree sequence fails Chvatal's condition at i, then the largest we can make the terms in di, ... , d11 is
dj = i dj = n - i - 1 dj = n - 1
## for j :::; i, for i + 1 :::; j :::; n - i, for j > n - i.
Let G be a simple graph realizing this degree sequence (if it exists). The i vertices of degree n - 1 are adjacent to all others (the central clique in the figure). This already gives i neighbors to the i vertices of degree i, so they form an independent set and have no additional neighbors. With degree n - i - 1, each of the remaining n - 2i vertices must be adjacent to all vertices except itself and the independent set. Thus these vertices form a clique. The only possible realization is (K; + Kn_ 2;) v K;, shown below. This graph is not Hamiltonian, because deleting the i vertices of degree n - 1 leaves a subgraph with i + 1 components. If a simple graph H is nonHamiltonian and has vertex degrees d~ :::; :::; d~, then Chvatal's result implies that for some i the graph (K; + Kn-2;) v K; with vertex degrees di :::; :::; d11 satisfies dj ::::. dj for all i .
292
## 7.2.15. Definition. A Hamiltonian path is a spanning path.
Every graph with a spanning cycle has a spanning path, but P,, shows that the converse is not true. We could make arguments like those above to prove sufficient conditions for Hamiltonian paths, but it is easier to use our previous work and prove the new theorem by invoking a theorem about cycles. To do this, we use a standard transformation.
7.2.16. Remark. A graph G has a spanning path if and only if the graph Gv K1 has a spanning cycle.
Remark 7.2.16 applies in several of the exercises. Here we use it to derive the analogue for paths of Chvatal's condition for spanning cycles.
7.2.17. Theorem. Let G be a simple graph with '!'ertex degrees di :=: :=: dn. If i < (n + 1)/2 implies (d; :::: i or dn+l-i :::: n - i), then G has a spanning path. Proof: Let G' = G v K 1 , let n' = n + 1, and let d~, ... , d~, be the degree sequence of G'. Since a spanning cycle in G v K 1 becomes a spanning path in G when the extra vertex is deleted, it suffices to show that G' satisfies Chvatal's sufficient condition for Hamiltonian cycles. Since the new vertex is adjacent to all of V(G), we have d~, = n and dj = dj + 1 for j < n'. For i < n' /2 = (n + 1)/2, the hypothesis on G yields
d;
= d; + 1:::: i + 1 >
or
d~'-i
= dn+l-i + 1;::: n - i + 1 =
/
n' - i.
This is precisely Chvatal's sufficient condition, so G' has a spanning cycle, and deleting the extra vertex leaves a spanning path in G.
7.2.18. * Remark. The degree requirements can be weakened under conditions such as regularity or high toughness. Every regular simple graph G with vertex degrees at least n(G)/3 is Hamiltonian (Jackson [1980]). Only the Petersen graph prevents lowering the threshold to (n(G) - 1)/3 (Zhu-Liu-Yu [1985], partly simplified in Bondy-Kouider [1988]; see also Exercise 13). It may be possible to lower the degree condition further when connectivity is high. For example, Tutte [1971] conjectured that every 3-connected 3-regular bipartite graph is Hamiltonian. Horton [1982] found a counterexample with 96 vertices, and the smallest known counterexumple has 50 vertices (Georges [1989]), but stronger conditions of this sort may suffice. Our last sufficient condition for Hamiltonian cycles involves connectivity and independence, not degrees. The proof yields a good algorithm that constructs a Hamiltonian cycle or shows that the hypothesis is false. 7.2.19. Theorem. (Chvatal-Erdos [1972]) If K(G) :::: a(G), then G has a Hamiltonian cycle (unless G = K~). Proof: With G =j:. K 2, the conditions require K(G) > 1. Suppose that K(G) :::: a(G). Let k = K(G), and let C be a longest cycle in G. Since 8(G) :::: K(G), and
## Section 7.2: Hamiltonian Cycles
293
every graph with 8(G) :::: 2 has a cycle oflength at least 8(G) + 1 (Proposition 1.2.28), C has at least k + 1 vertices. Let H be a component of G - V ( C). The cycle C has at least k vertices with edges to H; otherwise, deleting the vertices of C with edges to H contradicts K(G) = k. Let u 1 , ... , uk be k vertices of C with edges to H, in clockwise order. For i = 1, ... , k, let a; be the vertex immediately following u; on C. If any two of these vertices are adjacent, say a; # aj, then we construct a longer cycle by using a;aj, the portions of C from a; to uj and aj to u;, and a u;, urpath through H (see illustration). If a; has a neighbor in H, then we can detour to H between u; and a; on C. Thus we also conclude that no a; has a neighbor in H. Hence {a1, ... , ak} plus a vertex of H forms an independent set of size k + 1. This contradiction implies that C is a Hamiltonian cycle.
7.2.20.* Remark. Most sufficient conditions for Hamiltonian cycles generalize to conditions for long cycles. The circumference of a graph is the length of its longest cycle. A weaker form of a sufficient condition for spanning cycles may force a long cycle. Dirac [1952b] proved the first such result: a 2-connected graph with minimum degree k has circumference at least min{n, 2k}. Proposition 1.2.28 only guarantees a cycle of length at least k + 1. Most long-cycle results are more more difficult than the corresponding sufficient conditions for Hamiltonian cycles (see Lemma 8.4.36-Theorem 8.4.37).
## CYCLES IN DIRECTED GRAPHS (optional)
The theory of cycles in digraphs is similar to that of cycles in graphs. For a digraph G, let 8-(G) = mind-(v) and 8+(G) = mind+(v). The arguments of Chapter 1 using maximal paths guarantee paths of length k and cycles of length k + 1, where k = max{8-(G), 8+(G)}. Every complete graph is Hamiltonian, but orientations of complete graphs are more complicated. The-necessary condition of 2-connectedness becomes a necessary condition of strong connectedness for spanning cycles in digraphs. For tournaments, this necessary condition is also sufficient (Exercise 45).
294
## Chapter 7: Edges and Cycles
For arbitrary digraphs, we prove an analogue of Dirac's theorem (Theorem 7 .2.8). Indeed, it yields Dirdc's theorem as a special case (Exercise 49). Meyniel [1973] subtantially strengthened the theorem by weakening the hypothesis (Theorem 8.4.42).
7.2.21. Definition. A digraph is strict if it has no loops and has at most one copy of each ordered pair as an edge. 7.2.22. Theorem. (Ghouila-Houri [1960]) If D is a strict digraph, and min{8+(D), 8-(D)} ::: n(D)/2, then Dis Hamiltonian. Proof: Again we use contradiction and extremality. In an n-vertex counterexample D, let C be a longest cycle, with length l. As we have observed, l > max{8+, 8-) ::: n/2. Let P be a longest path in D - V(C), beginning at u, ending at w, and having length m ::: 0. Now l > n/2 and n :=: l + m + 1 imply m < n/2. Let S be the set of predecessors of u on C, and let T be the set of successors of w on C. By the maximality of P, every predecessor of u and successor of w lies in V(C) u V(P). Thus Sand Teach have size at least min{8+, 8-) - m, which is at least::: n/2 - m and hence is positive. Thus Sand Tare nonempty. The maximality of C guarantees that the distance along C from a vertex u' E S to a vertex w' E T must exceed m + 1. Oth~rwise, traveling along P instead of C from u' to w' yields a longer cycle. Hence we may assume that every vertex of S is followed on C by more than m vertices not in T.
u
w
If the distance between successive vertices of S along C is always at most vertex of T. Since both S and T are nonempty, we may thus asshme there is a vertex of S followed on C by at least m + 1 vertices not in S. These are forbidden from T, as is the immediate successor on C of all the other vertices of S. Thus at least ISi - 1+m+1 ::: n/2 vertices of Care not in T. Together withthe vertices that are in T, this yields IV(C)I ::: n - m, which contradicts l s n - m - 1. The contradiction implies that C must be a spanning cycle.
m
## + 1, then there is no legal :N'ace to put a
EXERCISES
7.2.1. (-) For which values of r is K,,, Hamiltonian? 7.2.2. (-)Is the Grtitzsch graph (Example 5.2.2) Hamiltonian? 7.2.3. (-) For n > 1, prove that
Kn,n
## Section 7.2: Hamiltonian Cycles
295
7.2.4. (-)Prove that G has a Hamiltonian path only if for every S s; V(G), the number of components of G - S is at most IS I + 1.
7.2.5. Prove that every 5-vertex path in the dodecahedron lies in a Hamiltonian cycle. 7.2.6. (!)Let G be a Hamiltonian bipartite graph, and choose x, y E V(G). Prove that G - x - y has a perfect matching if and only if x and y are on opposite sides of the bipartition of G. Apply this to prove that deleting two unit squares from an 8 by 8 chessbqard leaves a board that can be partitioned into 1 by 2 rectangles if and only if the two missing squares have opposite colors. 7.2.7. A mouse eats its way through a 3 x 3 x 3 cube of cheese by eating all the 1x1x1 subcubes. If it starts at a comer subcube and always moves on to an adjacent subcube (sharing a face of area 1), can it do this and eat the center subcube last? Give a method or prove impossible. (Ignore gravity.) 7.2.8. (!)On a chessboard, a knight can move from one square to another that differs by 1 in one coordinate and by 2 in the other coordinate, as shown below. Prove that no 4 x n chessboard has a knight's tour: a traversal by knight's moves that visits each square once and returns to the start. (Hint: Find an appropriate set of vertices in the corresponding graph to violate the necessary condition.)
7.2.9. Construct an infinite family of non-Hamiltonian graphs satisfying the necessary condition of Proposition 7.2.3. 7.2.10. (!) Hamiltonian us. Eulerian. a) Find a 2-connected non-Eulerian graph whose line graph is Hamiltonian. b) Prove that L(G) is Hamiltonian if and only if G has a closed trail that contains at least one endpoint of each edge. (Harary and Nash-Williams [1965]) 7.2.11. Construct a 3-regular 3-connected graph whose line graph is not Hamiltonian. (Hint: Replace each vertex in the Petersen graph with an appropriate graph and apply Exercise 7.2.10.) 7.2.12. Determine whether the graph below is Hamiltonian.
296
## Chapter 7: Edges and Cycles
7.2.13. Let G be the 3-regular graph obtained from the Petersen graph by -replacing one vertex with a triangle, matching the vertices of the triangle to the former neighbors of the deleted vertex. Prove that G is not Hamiltonian. (Comment: Except for this graph and the Petersen graph, every 2-connected, k-regular graph with at most 3k + 3 vertices is Hamiltonian.) (Hilbig [1986]) 7.2.14. A graph G is uniquely k-edge-colorable if all proper k-edge-colorings of G induce the same partition of the edges. Prove that every uniquely 3-edge-colorable 3-regular graph is Hamiltonian. (Greenwell-Kronk [1973]) 7.2.15. Place n points around a circle. Let G n be the 4-regular graph obtained by joining each point to the nearest two points in each direction. If n '.'.'.: 5, prove that Gn is the union of two Hamiltonian cycles. 7.2.16. For k ::: 3, let Gk be the graph obtained from two disjoint copies of Kk.k_ 2 by adding a matching between the two "partite sets" of size k. Determine all values of k such that Gk is Hamiltonian.
7.2.17. (!) Prove that the cartesian product of two Hamiltonian graphs is Hamiltonian. Conclude that the k-dimensional cube Qk is Hamiltonian fork::: 2. 7.2.18. Prove that the cartesian product of two graphs with Hamiltonian paths fails to have a Hamiltonian cycle if and only if both graphs are bipartite and have odd order, in w~ich case the product has a Hamiltonian path. 7.2.19. (+)For each odd natural number k, construct a k -1-connected k-regular simple bipartite graph that is not Hamiltonian. 7.2.20. (!)The kth power of a simple graph G is the simple graph Gk with vertex set V(G) and edge set {uv: da(u, v).:::: k). a) Suppose that G - x has at least three nontrivial components in each of which x has exactly one neighbor. Prove that G 2 is not Hamiltonian. (Hint: Consider the second graph in Example 7.2.5.) b) Prove that the cube of each connected graph (with at least three vertices) is Hamiltonian. (Hint: Reduce this to the special case of trees, and prove it for trees by proving the stronger result that ifxy is an edge of the tree T, then T 3 has a Hamiltonian cycle using the edge xy. Comment: Fleischner [1974] proved that the square of each 2-connected graph is Hamiltonian.) 7.2.21. Let n = k(21 + 1). Construct a non-Hamiltonian complete k-partite graph with n vertices and minimum degree ~ kkI 21~ 1 (Snevily) 7.2.22. Let G(k, t) be the class of connected k-partite graphs in which each partite set has size t and each subgraph induced by two partite sets is a matching of size t. Fork ::: 4 and t ::: 4, construct a graph in G(k, t) that is not Hamiltonian. (Hint: There is a graph in G(4, 4) with a 3-set whose deletion leaves four components; generalize this example. Comment: G(3, t) = {C3, }, and also every graph in G(k, 3) is Hamiltonian.) (Ayel [1982])
Section 7.2: Hamiltonian Cycles 7.2.23. (*)Prove that the Petersen graph has toughness 4/3.
297
7.2.24. (*) Let t (G) denote tl;te toughness of G. a) Prove that t(G_):::: K(Q)/2. (Chvatal [1973]) b) Prove that equality holds in part (a) for claw-free graphs. (Hint: Consider a set S such that ISi = t(G) c(G - S).) (Matthews-Sumner [1984]) 7.2.25. (!) Let G be a simple graph that is not a forest and has girth at least 5. Prove that G is Hamiltonian. (Hint: Use Ore's condition.) (N. Graham) 7.2.26. (!) Prove that if G fails Chvatal's condition, then G has at least n - 2 edges. Conclude from this that the maximum number of edges in a simple non-Hamiltonian n-vertex graph is {";i) + 1. (Ore [1961], Bondy [1972b]) 7.2.27. Prove directly by induction on n that the maximum number of edges in a simple non-Hamiltonian n-vertex graph is (";i) + 1. 7.2.28. Generalization of the edge bound. a) Let f (i) = 2i 2 - i + (n - i)(n - i -1), and suppose that n ::: 6k. Prove that on the interval k:::: i :::: n/2, the maximum value of f(i) is f(k). . b) Let G be a simple graph with minimum degree k. Use part (a) and Chvatal's condition to prove that if G has at least 6k vertices and has more than ("'Gtk) + k2edges, then G is Hamiltonian. (Erdos [1962]) 7.2.29. (~j Let G be a simple graph with vertex degrees di :::: :::: d., and let d~ .:::: :::: d~ be the vertex degrees in G. Prove that if d; ::: dI for all i .:::: n/2, then G has a Hamiltonian path. Conclude that every simple graph isomorphic to its complement has a Hamiltonian path. (Clapham [1974]) 7.2.30. Obtain Lemma 7.2.9 (suf:(iciency of Ore's condition) from Theorem 7.2.13 (sufficiency of Chvatal's condition). (Bondy [1978]) 7.2.31. (!)Prove or disprove: If G is a simple graph with at least three vertices, and G has at least a(G) vertices of degree n(G) - 1, then G is Hamiltonian. 7.2.32. (+)Suppose that n is even and G is a simple bipartite graph with partite sets X, Y of size n/2. Let the vertex degrees of G be di, ... , d . Let G' be the supergraph of G obtained by adding edges so that G[Y] = K. 12 . a) Prove that G is Hamiltonian if and only if G' is Hamiltonian, and describe the relationship between the degree sequences of G and G'. b) Suppose that dk > k or d. 12 > n/2 - k whenever k .:=:: n/4. Prove that G is Hamiltonian. (Hint: Assume that the degree sequence of G' fails Chvatal's condition for some i < n/2, and obtain a contradiction.) (Chvatal [1972]) 7.2.33. (!) A graph is Hamiltonian-connected if for every pair of vertices u, v there is a Hamiltonian path from u to v. Prove that a simple graph G is Hamiltonian if 1 e(G) ;:: ((Gti) + 2 and Hamiltonian-connected if e(G) ;:: ( 111 Gt ) + 3. (Proving the two together permits a simpler proof.) (Ore [1963]) 7.2.34. Necessary condition for Hamiltonian-connected. (Moon [1965a]) a) Prove that every Hamiltonian-connected graph G with at least four vertices has at least r3n(G)/21 edges. b) Prove that the bound in part (a) is best possible by showing that Cm o K 2 is Hamiltonian-connected if m is odd. 7.2.35. (!)Sufficient condition for Hamiltonian-connected. (Ore [1963]) a) Prove that a simple graph G is Hamiltonian-connected if x f+ y implies d(x) +
298
## Chapter 7: Edges and Cycles
d(y) > n(G). (Hint: Prove that appropriate graphs related to G are Hamiltonian by
considering their closures.) b) Prove that part (a) is sharp by constructing, for each even n greater than 2, a simple n-vertex graph with minimum degree n/2 that is not Hamiltonian-connected.
7.2.36. Las Vergnas' condition for a simple n-vertex graph is the existence of a vertex ordering vi, ... , v. such that there is no nonadjacent pair v;, Vj satisfying i < j, d (v;) ~ i, d(vj) < j, d(v;) + d(vj) < n, and i + j ~ n. Las Vergnas [1971] proved that this condition is sufficient for the existence of a spanning cycle. a) Prove that Chvatal's condition (Theorem 7.2.13) implies Las Vergnas' condition, which means that Las Vergnas' theorem strengthens Chvatal's theorem. b) Prove that each of the graphs below fails Chvatal's condition but has a complete graph as its Hamiltonian closure. Prove that the smaller graph satisfies Las Vergnas' condition but the larger one does not.
7.2.37. For0 ::f s c V(G), lett(S) = 1snN(S)l/ls1. LetO(G) = mint(S). Lu [1994] proved that ifO(G)n(G) ~ a(G), then G is Hamiltonian. Prove that K(G) ~ a(G) implies O(G)n(G) ~ a(G). (Comment: This shows that Lu's theorem implies the Chvatal-Erdos Theorem and is a stronger result.) 7.2.38. (!)Long paths and cycles. Let G be a connected simple graph with 8(G) = k ~ 2 and n(G) > 2k. a) Let P be a maximal path in G (not a subgraph of any longer path). Ifn(P) ~ 2k, prove that the induced subgraph G[V(P)] has a spanning cycle (this cycle need not have its vertices in the same order as P). b) Use part (a) to prove that G has a path with at least 2k + 1 vertices. Give an example for each odd value of n to show that G need not have a cycle with more than k + 1 vertices. 7.2.39. Prove that ifa simple graph G has degree sequence di ~ ~ d. and di +d2 < n, then G has a path oflength at least di + d 2 + 1 unless G is the join of n - (di + 1) isolated vertices with a graph on di+ 1 vertices or G = pKd 1 v Ki for some p ~ 3. (Ore [1967b]) 7.2.40. (!) Dirac [1952b] proved that every 2-connected simple graph G has a cycle of length at least min{n(G), 28(G)}. Use this to prove that every 2k-regular graph with 4k + 1 vertices is Hamiltonian. (Nash-Williams) 7.2.41. Scott Smith conjectured that any two longest cycles in a k-connected graph have at least k common vertices. The approach below works for small k. a) Suppose that G is a 4-regular graph with n vertices that is the union of two cycles (multiple edges may arise). Let G' be the 4-regular graph on n + 2 vertices obtained from G by subdividing two edges and adding a double edge between the two new vertices. Show that G' is also the union of two spanning cycles if n ~ 5. b) Use part (a) to conclude that any pair of longest cycles in a k-connected graph intersect in at least k points if k ~ 6. (Smith, Burr) 7.2.42. (+) Let G be an Eulerian graph. Let V' be the set of Eulerian circuits of G, considering a circuit and its reversal to be the same. Let G' be the graph with vertex
## Section 7.3: Planarity, Coloring, and Cycles
299
set V' such that two circuits are adjacent if and only if one arises from the other by reversing the edge order on a proper closed subcircuit. Prove that G' is Hamiltohian if ~(G) :::: 4. (Hint: Use induction on the number of vertices of degree 4, proving that there is a Hamiltonian cycle through every edge of G'. Comment: The conclusion also holds without restriction on MG).) (Xia [1982), Zhang-Guo [1986])
7.2.43. Prove that the Eulerian circuit graph G' of Exercise 7.2.42 is regular, and derive a formula for its vertex degree. Compare o(G') and n(G') when n(G) = 2 to show that the preceding problem cannot be solved by applying general results on Hamiltonicity of regular graphs with specified degree. 7.2.44. Prove that every tournament has a Hamiltonian path (a spanning directed path). (Hint: Use extremality). (Redei [1934]) 7.2.45. Let T be a strong tournament. For each u E V (T) and each k such that 3 :::: k :::: n, prove that u belongs to a cycle oflength k in T. (Hint: Use induction on k .) (Moon [1966)) 7.2.46. Let. G be a 7-vertex tournament in which every vertex has outdegree 3. Use Exercise 7.2.45 to prove that G has two vertex-disjoint cycles. 7.2.47. (+)Prove that every tournament has a Hamiltonian path that is not contained in a Hamiltonian cycle, except the cyclic tournament on three vertices and the tournament Ts on five vertices drawn below. (Hint: Induction works, but some care is needed to prove the claim for six vertices. In all cases, find the desired configuration or G = Ts.) (Griinbaum, in Harary [1969, p211])
7.2.48. (*) Prove that Theorem 7.2.22 is best possible by showing that the strictness condition on the digraph cannot be weakened to allow loops. In particular, construct for each even n an n-vertex digraph D that is not Hamiltonian even though at most one copy of each ordered pair is an edge and min{8-(D), o+(D)} ::=:: n/2. 7.2.49. (*) Obtain Theorem 7.2.8 (sufficiency of Dirac's condition in graphs) from Theorem 7.2.22 (sufficiency of Ghouila-Houri's condition on digraphs). (Hint: Transform a simple graph G into a strict digraph by replacing each edge with a pair of directed edges in opposite directions.)
## 7.3. Planarity, Colorings, and Cycles
We return to the Four Color Problem to explore its historical relationship with the problems of edge-coloring and Hamiltonian cycles. We then consider ways in which the problem generalizes.
300
## Chapter 7: Edges and Cycles
TAIT'S THEOREM
In 1878, Tait proved a theorem relating face-coloring and edge-coloring of plane graphs, and he used this in an approach to the Four Color Theorem. This stimulated interest in edge-coloring. We first define face-coloring precisely.
7.3.1. Definition. A proper face-coloring of a 2-edge-connected plane graph is an assignment of colors to its faces so that faces having a common edge in their boundaries have distinct colors.
We often think of a face-coloring as a coloring of the dual graph. For this reason, we restrict our attention to face-colorings of 2-edge-connected graphs. When a plane graph has a cut-edge, its dual has a loop. We say that graphs with loops do not have proper colorings. In a plane graph with a cut-edge, a face shares a boundary with itself and is thus uncolorable. Since adding edges does not make ordinary coloring easier, to prove the Four Color Theorem it suffices to prove that all triangulations are 4-colorable. Equivalently, we could show that all duals of triangulations are 4-face-colorable. The dual G* of a plane triangulation G is a 3-regular, 2-edge-connected plane graph (Exercise 6.1.11). Tait showed that for such graphs, proper 4-facecolorings are equivalent to p:roper 3-edge-colorings.
7.3.2. Theorem. (Tait [1878]) A simple 2-edge-connected 3-regular plane graph is 3-edge-colorable if and only if it is 4-face-colc:irable. Proof: Let G be such a graph. Suppose first that G is 4-face-colorable; we obtain a 3-edge-coloring. Let the four colors be denoted by binary ordered pairs: c0 = 00, c1 = 01, c2 = 10, c3 = 11. Color E(G) by assigning to the edge between faces with colors c; and cj the color obtained by adding c; and cj coordinatewise using addition modulo 2. (Thus c2 + c3 = ci, for example.) We show that this is a proper 3-edge-coloring. Because G is 2-edge-connected, each edge bounds two distinct faces. Hence the color 00 never occurs as a sum. We check that the edges at a vertex receive distinct colors. At vertex v the faces bordering the three incident edges must have distinct colors {c;, Cj, ck), as illustrated below. If color 00 is not in this set, then the sum of any two of these is the third, and hence {c;. cj, ck} is also th~ set of colors on the edges. If ck = 00, then c; and c; i:i.ppear on two of the edges, and the third receives color c; + cj, which is the color not in {c;, Cj, ck).
11 01 10 11 10 01 01 01
10 11 11 00
M~:-~
c \
\
'
a
'
## Section 7.3: Planarity, Coloring, and Cycles
301
For the converse, suppose that G has a proper 3-edge-coloring using colors
a, b, c (shown bold, solid, and dashed). Let E 0 , Eb, Ee be the edge sets having
the three colors, respectively. We construct a 4-face-coloring using the four colors defined above. Since G is 3-regular, each color appears at every vertex, and the union of any two of Ea, Eb, Ee is 2-regular, which makes it a union of disjoint cycles. Each face of this subgraph is a union of faces of the original graph. Let H1 =Ea U Eb and H2 =Eb U Ee. To each face of G, assign the color whose ith coordinate (i E {l, 2}} is the parity of the number of cycles in H; that contain it (0 for even, 1 for odd). We claim that this is a proper 4-face-coloring, as illustrated above. Faces F, F' sharing an edge e are distinct faces, since G is 2-edge-connected. Edge e belongs to a cycle C in at least one of H 1 , H2 (in both if e has color b). By the Jordan Curve Theorem, one of F, F' is inside C and the other is outside. All other cydes in H 1 and H2 fail to separate F and F', leaving them on the same side. Hence if e has color a, c, orb, then the parity of the number of cycles containing F and F' is different in H1, in H2, or in both, respectively. Thus F and F' receive different colors in the face-coloring we have constructed. Due to this theorem, a proper 3-edge-coloring of a 3-regular graph is called a Tait coloring. The problem of showing that every 2-edge-connected 3-regular planar graph is 3-edge-colorable reduces to showing that every 3-connected 3-regular planar graph is 3-edge-colorable. 7.3.3.* Lemma. If G is a 3-regular graph with edge-connectivity 2, then G has subgraphs Gi, G2 and vertices u1, v1 E V(G1) and u2, v2 E V(G2) such that u1 ~ v1, also u2 ~ v2, and G consists of G1, G 2 and a ladder of some length joining Gi, G2 at u1, v1, u2, v2 as shown below.
Proof: If G has an edge cut of size 2 in which the two edges are incident, then the third edge incident to their common vertex is a cut-edge, contradicting K' = 2. Hence we may assume that the four endpoints in our minimum edge cut xy, uv are distinct. If x ~ y and u ~ v, then these are the four desired vertices and the ladder has only these two edges. When x ~ y, we extend the ladder (a similar argument applies when u ~ v). Let w be the third neighbor of x and z the third neighbor of y. If w = z, then the third edge incident to this vertex is a cut-edge. Hence w =f. z and the ladder extends. If w ~ z, then we are finished in this direction; otherwise, we repeat the argument till we obtain a nonadjacent pair at the base of the ladder. 7.3.4.* Theorem. All 2-edge-connected 3-regular simple planar graphs are 3edge-colorable if and only if all 3-connected 3-regular simple planar graphs are 3-edge-colorable.
302
## Chapter 7: Edges and Cycles
Proof: The second family is contained in the first. Hence it suffices to show that 3-edge-colorability for all graphs in the smaller family implies it also for the larger family. We use induction on n(G). Basis step (n(G) = 4): The only 2-edge-connected 3-regular simple planar graph with at most 4 vertices is K 4 , which is 3-edge-colorable. Induction step (n(G) > 4): Since K(G) = K'(G) when G is 3-regular (Theorem 4.1.11), we may restrict our attention to 3-regular graphs with edgeconnectivity 2. Lemma 7.3.3 gives us a decomposition of G into Gi, G2, and a ladder joining them. The length of the ladder is the distance from G1 to G2. Both G1 +u1 v1 and G 2+u 2v2 are 2-edge-connected and 3-regular. By the induction hypothesis, they are 3-edge-colorable; let f; be a proper, 3-edge-coloring ofGi + uiv;. Permute names of colors so that / 1(u1v 1) = 1 and so that f2(u2v2) is chosen from {l, 2} to have the same parity as the length of the ladder. Returning to G, color each G; as in /;. Beginning from the end of the ladder at Gi, color the rungs of the ladder with 3, and color the paths forming the sides of the ladder alternately with 1 and 2. The edges of the ladder at ui and v; now have the color fi(uiv;). Thus we have assembled a proper 3-edge-coloring of G .
Thus thelFour Color Theorem reduces to finding Tait colorings of 3-edgeconnected 3-regular planar graphs. The statement of their existence was known as Tait's conjecture and is equivalent to the Four Color Theorem.
GRINBERG'S THEOREM
Every Hamiltonian 3-regular graph has a Tait coloring (Exercise 1). Tait believed that this completed a proof of the Four Color Theorem, because he assumed that every 3-connected 3-regular planar graph is Hamiltonian. Not until 1946 was an explicit counterexample found, although the gap in the proof was noticed earlier. Later, Grinberg [1968] discovered a simple necessary condition that led to many 3-regular 3-connected non-Hamiltonian planar graphs, including the Grinberg graph of Exercise 16.
7.3.5. Theorem. (Grinberg [1968]) If G is a loopless plane graph having a Hamiltonian cycle C, and G .has f( faces oflength i inside C and f(' faces oflength i outside C, then Li(i - 2)(// - !(') = 0. Proof: Considering the faces inside and outside C separately, we want to show that. L;U - 2)// = L;(i - 2)f('. No changes on one side affect the sum on the other side. Furthermore, we can switch inside and outside by projecting the embedding onto a sphere and puncturing a face inside C. Hence we need only show that L:;(i - 2) f( is constant. When there are no inside edges, the sum is n - 2. With this as the basis step, we prove by induction on the number of inside edges that the sum is always n - 2. Suppose that Li(i - 2)// = n - 2 when there are k edges inside C. We can obtain any graph with k + 1 edges inside C by adding an edge to such a graph.
## Section 7.3: Planarity, Coloring, and Cycles
303
The added edge cuts a face of some length r into two faces of lengths s and t. We have s + t = r + 2, because the new edge contributes to both new faces and each edge on the old face contributes to one new face. No other contribution to the sum changes. Since (s - 2) + (t - 2) = (r - 2), the contribution from these faces also remains the same. By the induction hypothesis, the sum is n - 2. Being a necessary condition, Grinberg's condition can be used to show that graphs are not Hamiltonian. The arguments can often be simplified using modular arithmetic. Two numbers that are not congruent mod k are not equal. We apply this to the first known non-Hamiltonian 3-connected 3-regular planar graph (Tutte [1946]). Tutte used an ad hoc argument to prove that this graph is not Hamiltonian. For many years it was the only known example (see . Exercise 17 for the smallest now known). 7.3.6. Example. Grinberg's condition and the Tutte graph. The Tutte graph G appears on the left below. Let H denote each component obtained by deleting the central vertex and the three long edges. Since a Hamiltonian cycl~ must visit the central vertex of G, it must traverse one copy of H along a Hamiltonian path joining the other entrances to H, which we call x and y. We therefore study a graph that has a Hamiltonian cycle if and only if H has a Hamiltonian x, y-path. Such a graph H' (on the right below) is obtained by adding an x, y-path oflength two through a new vertex.
H'
x '
The plane graph H' has five 5-faces, three 4-faces, and one 9-face. Grinberg's condition becomes 2a4 + 3a 5 + 7ag = 0, where a; = f/ - f/'. Since the unbounded face is always outside, the equation reduces mod 3 to 2a4 7 mod 3. Since f~ + f!; = 3, the possibilities for a4 are +3, + 1, -1, -3. The only choice satisfying 2a4 7 mod 3 is a4 = -1, which requires that two of the 4-faces lie outside the Hamiltonian cycle. However, the 4-faces having having a vertex of degree 2 cannot lie outside the cycle, since the edges incident to the vertex of degree 2 separate the face from the outside face. We can reach a contradiction faster by subdividing one edge incident to each vertex of degree 2. This does not change. the existence of a spanning cycle. The resulting graph has seven 5-faces, one 4-face, and one 11-face. The
304
## Chapter 7: Edges and Cycles
required equation becomes 2 (1) = 9 - 3a5, which has no solution since the left side is not a multiple of 3. We have not presented a systematic procedure for proving the nonexistence of solutions to equations with integer variables. Our arguments involving divisibility are merely tricks to avoid listing cases, but such tricks often work. High connectivity makes it harder to avoid spanning cycles. Tutte [1956] (extended by Thomassen [1983]) proved that every 4-connected planar graph is Hamiltonian. Barnette [1969] conjectured that every planar 3-connected 3-r gular bipartite graph is Hamiltonian.
0
SNARKS (optional)
Another approach to the Four Color Theorem is to study which 3-regular graphs are 3-edge-colorable. In a discussion focusing on 3-regular graphs and graphs without cut-edges, it is convenient to have simple adjectives to descrihP these properties.
7.3.7. Definition. A bridgeless graph is a graph without cut-edges. A cubic graph is a graph that is regular of degree 3. 7.3.8. Conjecture. (3-edge-coloring Conjecture-Tutte [1967]) Every bridgeless cubic non-3-edge-colorable graph contains a subdivision of the Petersen graph.
Conjecture 7.3.8 has been proved! Like the Four Color Theorem, its computer-assisted proof uses discharging methods. The proof will appear in a series of five papers by Robertson, Sanders, Seymour, and Thomas [2001]. Since every subdivision of the Petersen graph is nonplanar, Conjecture 7.3.8 implies Tait's Conjecture and hence the Four Color Theorem. One natural approach to the conjecture, like the idea of reducibility for the Four Color Theorem, is to derive properties that a minimal counterexample must have. In this language, Theorem 7.3.4 says that a minimal counterexample must be 3edge-connected. In the next lemma, we make this statement precise and obtain_ several other properties.
7.3.9. Definition. A trivial edge cut is an edge cut whose deletion isolates a single vertex. Other edge cuts are nontrivial. 7.3.10. Lemma. If a non-3-edge-colorable cubic graph G has connectivity 2 or girth less than 4 or a nontrivial 3-edge cut, then G contains a subdivision of a smaller non-3-edge-colorable cubic graph. Proof: Suppose first that G has an edge cut of size 2. As discussed in Lemma 7.3.3, these edges have no common vertices. Deleting the edge cut and adding one edge to each piece yields cubic graphs G 1 + u 1 v1 and G 2 + u 2 v 2 As argued
## Section 7.3: Planarity, Coloring, and Cycles
305
in Theorem 7.3.4, at least one of these graphs is not 3-edge-colorable. Since the added edge can be replaced by a path through the other piece, G contains a subdivision of this smaller non-3-edge-colorable graph. Next suppose that G contains a triangle. Let G' be the graph obtained from G by contracting the triangle to a single vertex. A proper 3-edge-coloring of G' could be expanded into a proper 3-edge-coloring of G as shown below. Also, G contains a subdivision of G', obtained by deleting one edge of the triangle. Suppose that G contains a 4-cycle but no triangle. Let G' be the cubic graph obtained from G by deleting two opposite edges of the 4-cycle and replacing the resulting paths of length 3 with single edges. Since G has no triangle, the new edges are not loops. A proper 3-edge-coloring of G' yields a proper 3-edgecoloring of G via the two cases shown below. Also G contains a subdivision of G', so G' is the desired smaller graph.
A~
+---a
Finally, suppose that G contains a nontrivial 3-edge cut [S, S]. Since we may assume that G is 3-edge-connected, the three edges of the cut are pairwise disjoint. The two graphs obtained by contracting G[S] or G[S] to a single vertex are also 3-regular. If both are 3-edge-colorable, then the colors can be renamed to agree on the edges of the cut, yielding a proper 3-edge-coloring of G. Thus at least one of these graphs is not 3-edge-colorable. It remains only to show that G contains a subdivision of G [S] (and similarly ofG[S]). Let a, b, c be the endpoints in S of the edges in the cut. Since G is 3edge-connected, the cut is a bond, and G[S] is connected (Proposition 4.i.15). Thus G[S] contains an a, b-path Panda path from c to P. Adding these paths and the edges of the cut to G[S] completes a subdivision of G[S].
7.3.11. Definition. A snark is a 2-edge-connected 3-regular graph that is not 3-colorable, has girth at least 5, and has no non-trivial 3-edge cut. A prime snark is one that contains no subdivision of a smaller snark.
In this language, we have reduced Tutte's 3-edge-coloring Conjecture to the statement that the Petersen graph is the only prime snark. Again, we note that the conjecture has been proved (Robertson-Sanders-Seymour-Thomas [2001]). After the Petersen graph in 1898, by 1975 only three more snarks had been found: the 18-vertex Blanusa [1946] snark, the 210-vertex Descartes [1948] snark, and the 50-vertex Szekeres [1973] snark. This prompted Martin Gardner [1976] to invent the term "snark", evoking the r&rity of the creature in Lewis Carroll's "The Hunting of the Snark".
306
## Chapter 7: Edges and Cycles
Isaacs [1975] then showed that the earlier snarks arise from the Petersen graph via an operation that generates infinite families of snarks.
7.3.12. Definition. The dot product of cubic graphs G and H is the cubic graph formed from G + H by deleting disjoint edges uv and wx from G, deleting adjacent vertices y and z from H, and adding edges from u and v to NH(Y) - {z} and from wand x to NH(Z) - {y}.
The dot product of two snarks is a snark (Exercise 23). Applying it to two copies of the Petersen graph yields the Blanusa snark shown below. This graph has a non-trivial 4-edge cut. Kochol [1996] introduced a more general operation that yields snarks with large girth and higher connectedness properties.
7.3.13. Example. The flower snarks. Isaacs also found an explicit infinite fam-
ily of snarks (Exercise 21) that don't arise via the dot product. Independently
discovered by Grinberg, they have 4k vertices, for odd k :::_ 5. Begin with three disjoint k-cycles. Let {x;}, {y; }, {z;} be the three vertex sets, indexed cyclically. For each i add a.vertex w; with N(w;) = {x;, y;, z;}. The resulting graph Gk is 3-edge-colorable. Let Hk be the graph obtained by replacing the edges xkx1 and YkYl with XkYl and YkX1. If k is odd and k :::_ 5, then Hk is a snark. If k is even, then Hk is 3-edge-colorable. The drawing of Hk in which {z;} is a central cycle suggests the name "flower snark".
307
## FLOWS AND CYCLE COVERS (optional)
Tait's Theorem (Theorem 7.3.2) states that 3-edge-colorability and 4-facecolorability are equivalent for plane triangulations. When extending this beyond planar graphs, we need a concept that makes sense for all graphs and is.equivalent to 4-face-coloring on plane graphs. Additional information about this topic (and about snark,s) appears in the monograph by Zhang [1997].
7.3.14. Definition. A fl.ow on a graph G is a pair (D, f) such that 1) D is an orientation of G, 2) f is a weight function on E(G), and 3) each v E V(G) satisfies LweNh"(v) f(vw) = LueN;;(v) f(uv). A k-:flow is an integer-valued flow such that I/ (e)I ~ k - 1 for all e E E(G). A flow is nowhere-zero or positive if f(e) is nonzero or positive, respectively, for all e E E(G).
The usage of "flow" here is somewhat different from that in Chapter 4. In both contexts, the word "flow" suggests the conservation constraints imposed at each vertex. The bound of k - 1 on flow value evokes the notion of capacity. We can alte:r: the orientation to make all weights positive.
7.3.15. Proposition. For a graph G, the following are equivalent: A) G has a positive k-flow. B) G has a nowhere-zero k-flow. C) G has a nowhere-zero k-flow for each orientation of G. Proof: Simultaneously changing the orientation of an edge and the sign of its weight does not affect the conservation constraints.
Thu!i! the existence of a nowhere-zero k-flow does not depend on the choice of the orientation. We can also take linear combinations of flows.
7.3.16. Proposition. If (D, Ji), ... , (D, f,) are flows on G, and g = I:;=l a; f;, then {D, g) is a flow on G. Proof: For each v E V ( G), the net flow out of v under each fi is zero, and hence it is also zero under g. 7.3.17. Proposition. For a flow on G, the net flow out of any set S ~ V(G) is zero. Thus a graph with a nowhere-zero flow has no cut-edge. Proof: We sum the net flows out of vertices of S. Edges leaving S contribute with positive weight, edges entering S contribute with negative weight, and edges within S contribute positively at their tails and negatively at their heads. The net flow out of Sis thus the sum of the net flowi;; out of the vertices of S, which is zero. This implies that the net flow across any edge cut is zero, so it cannot consist of a single edge with nonzero weight.
308
## Chapter 7: Edges and Cycles
Thus we restrict our attention to graphs without cut-edges (bridgeless graphs). What distinguishes flows here from circulations in Section 4.3 is that we forbid zero as a weight. Nowhere-zero flows enable us to extend Tait'i;; Theorem. We begin by interpreting Eulerian graphs in the context of nowhere-zero flows; connectedness is no longer important.
7.3.18. Definition. A graph is an even graph if every vertex has even degree. 7.3.19. Proposition. A graph has a nowhere-zero 2-flow if and only ifit is an even graph. Proof: Given a nowhere-zero 2-fiow, we obtain a positive 2-flow. Since this assigns weight 1 to every edge, the orientation must have as many edges entering each vertex as leaving it. Thus each vertex degree is even. Conversely, when each vertex degree is even, each component has an Eulerian circuit. Orienting the edges to follow such a circuit and assigning weight 1 to each edge yields a positive 2-flow.
Nowhere-zero 3-flows are more subtle, even for 3-regular graphs.
7.3.20. Proposition. (Tutte [1949]) A cubic graph has a nowhere-zero 3-flow if and only if it is bipartite. Proof: Let G be a cubic X, Y-bigraph. Every regular bipartite graph has a 1factor. Orient the edges of a I-factor from X to Y, and give them weight 2. Orient all other edges from Y to X, and give them weight 1. The il0-;, in and out of every vertex is 2, so this is a nowhere-zero 3-flow. Conversely, let G be a cubic graph with a nowhere-zero 3-flow. By Proposition 7.3.15, we may assume that the flow is 1 or 2 on each edge. Since the net flow is 0, there must be one edge with flow 2 and two edges with flow 1 at each vertex. Thus the edges with flow 2 form a matching. The X be the set of tails and Y the set of heads of these edges. Since the net flow is 0 at each vertex, each edge with flow 2 points from X to Y, and each edge with flow 1 points from Y to X. Thus X, Y is a bipartition of G. 7.3.21. Example. Since the Petersen graph is cubic and not bipartite, it has no nowhere-zero 3-flow. We will see that it also has no nowhere-zero 4-flow. Below we show a nowhere-zero 4-flow in the 3-regular simple graph C3 o K 2 .
1
1
3
To understand the duality between flows and colorings, we characterize the plane graphs with nowhere-zero k-flows.
## Section 7.3: Planarity, Coloring, and Cycles
309
7.3.22. Theorem. (Tutte [1954b]) A plane bridgeless graph is k-face-colorable if and only if it has a nowhere-zero k-flow. Proof: (Younger [1983], refined by Seymour) Let f be a flow on a plane graph G. We define a function g on the set of faces by letting g(F) be the net flow accumulated by traveling from face F out to the unbounded face. Each time we cross an edge ewe count+ f(e) if e is directed toward our right, - f(e) if e is directed toward our left. The value assigned to the outside face is 0. The function g is well-defined; that is, g(F) is independent of our route to the outside face. We can change a route into any other by a succession of changes where we go the "other way" around some vertex v (shown on the left below). The change increases or decreases our accumulation for this portion by the net flow out of v, which is 0. Note that the difference between the values on faces with a common edge e is f(e).
/
/ /
Conversely, given a function g defined on the faces, we can invert the process to obtain a flow (shown on the right .above). As we stand on face F and look and face F' across edge e, we let f (e) = g(F) - g(F') if e is directed toward our right, f (e) = g(F') - g(F) if e is directed toward our left. Thus flows correspond to face-colorings. The face-coloring is proper if and only if the flow is nowhere-zero. If the flow is a nowhere-zero k-flow, then reducing the labels in the coloring to congruence classes in {0, ... , k - 1} produces a proper k-coloring. Conversely, a proper k-face-coloring using these colors pro duces a nowhere-zero k-flow. The correspondence between face-labelings and flows in Theorem 7 .3.22 is valid when the labels come from any abelian group. Applied using the group of binary ordered pairs under addition ((0, 0) is the identity), the statement proved by this argument is precisely Tait's Theorem itself. Since we can study flows on all graphs, we can consider the flow problem as a general dual notion to vertex coloring. "Nowhere-zero" is the analogue of "proper". Since every nowhere-zero k-flow is a nowhere-zero k + 1-flow, the natural problem is to minimize k such that G has a nowhere-zero k-flow. This minimum is the flow number of G, by analogy with "chromatic number". Since we say "G is k-colorable" when G has a proper k-coloring, the natural analogue would be to say "G is k-flowable" instead of "G has a nowhere-zero k-flow". This language is not yet common, so we will use it sparingly. By Tait's Theorem, Theorem 7.3.22 states that a cubic bridgeless planar graph is 3-edge-colorable if and only if it has a nowhere-zero 4-flow. We want
310
## Chapter 7: Edges and Cycles
to extend this correspondence by dropping the condition on planarity. A simple observation about parity will be useful.
7.3.23. Lemma. In a nowhere-zero k-flow, every vertex is incident to an even number of edges of odd weight. Proof: Since at each vertex the total weight on entering edges equals the total weight on tixiting edges, the sum of the weights is even. 7.3.24. Theorem. Let G be a cubic graph. If G has a nowhere-zero 4-flow, then G is 3-edge-colorable. Proof: By Proposition 7.3.15, we may assume that G has a positive 4-flow (D, f), and thus f(e) E {l, 2, 3} for each edge e. By Lemma 7.3.23, each vertex is incident to exactly one edge of weight 2. Thus the edges of weight 2 form a 1-factor in G, and deleting them leaves a union of disjoint cycles. To complete a !-factorization, it suffices to show that each of these cycles has even length. Let C be such a cycle. The edges of weight 2 that are incident to vertices of C are chords or join V ( C) with V ( C). The chords occupy an even size subset of V ( C). Thus it suffices to show that the number of edges between V ( C) and V(C) is even. These edges all have weight 2. Since the net flow out of V(C) must be 0 and all edges between V ( C) and V ( C) h&ve flow 2, the number of edges leaving V ( C) must equal the number of edges entering it.
Since the Petersen graph is not 3-edge-colorable, Theorem 7 .3.24 implies that it is not 4-flowable. Existence of nowhere-zero k-flows is preserved by subdivision: when an edge e of weight j in a nowhere-zero k-flow is subdivided, replacing it with a path oflength 2 oriented in the same direction with weight j on both edges yields a nowhere-zero k-flow in the new graph. Thus subdivisions of the Petersen graph also have no nowhere-zero 4-flows. The converse of Theorem 7 .3.24 is true but not trivial, since it may not be possible to treat the color classes as edge sets of fixed weight and orient the graph to make this a 4-flow. In the graph C3 o K2 of Example 7.3.21, there is essentially only one proper 3-edge-coloring, and when the color classes are labeled 1, 2, 3 it is not possible to obtain a 4-flow. In the positive 4-flow in Example 7.3.21, the edges of weight 1 do not form a matching. Nevertheless, we can apply the next theorem to guarantee nowhere-zero 4-flows in cubic graphs. The characterization is more general, since it does not reqttj.re regularity.
7.3.25. Theorem. A graph has a nowhere-zero 4-flow if and only if it is the union of two even graphs.
## Section 7.3: Planarity, Coloring, and Cycles
311
Proof: Let G 1, G 2 be even graphs with G = G 1 U G 2 . Let D be an orientation of G, restricting to D; on G;. By Proposition 7.3.19 and Proposition 7.3.15, G; has a nowhere-zero 2-flow (D;, f;). Extend f; to E(G) by letting f;(e) = 0 for e E E(G) - E(G;). Let f = fi + 2/2. This weight function is odd on E(G 1) and is 2 on E(G) - E(G 1 ), so it is nowhere-zero. Its magnitude is; lways at most 3, and by Proposition 7.3.16 (D, f) is a flow; thus it is a nowhere-zero 4-flow. Conversely, let (D, f) be a nowhere-zero 4-flow on G. Let E 1 = {e E E(G): f(e) is odd}. By Lemma 7.3.23, E1 forms an even subgraph of G. Thus there is a nowhere-zero 2-flow (D1, /1) on Ei, where D 1 agrees with D. Extend fi to E(G) by letting fi(e) = 0 fore E E(G) - E 1 ; now (D, fi) is a 2-flow on G. Define hon E(G) by h = (f - fi)/2. By Proposition 7.3.16, (D, /2) is a flow on G. It is an integer flow, since f(e) - fi(e) is always even. By Lemma 7.3.23, the set E 2 = {e E E(G): f2(e) is odd} forms an even subgraph of G. For e E E(G) - Ei, we have f(e) = 2 and fi(e) == 0, which yields f2(e) = 1, so E(G) - E 1 -~ E 2 Now G is the union of two even subgraphs. 7 .3.26. Corollary. If G is a cubic graph, then G is 3-edge-colorable if and only if G has a nowhere-zero 4-flow. Proof: Every 3-edge-colorable cubic graph is the union of two even subgraphs: the edges of colors 1 and 2, and the edges of colors 1 and 3.
In light ofTheor~m 7.3.22, Corollary 7.3.26 generalizes Tait's Theorem. We have seen that subdivisions of the Petersen graph are not 4-flowable. Among bridgeless graphs, Tutte conjectured that excluding such subgraphs yields nowhere-zero 4-flows.
7.3.27. Conjecture. (Tutte's 4-flow Conjecture-Tutte [1966b]) Every bridgeless graph containing no subdivision of the Petersen graph is 4-flowable.
Since every graph containing a subdivision of the Petersen graph is nonplanar, Tutte's 4-flow Conjecture implies the Four Color Theorem. Since nowherezero 4-flows are equivalent to 3-edge-colorings on cubic graphs, the 4-flow Conjecture also implies the 3-edge-coloring Conjecture (which has been proved). Researchers have hoped for an elegant proof of Tutte's 4-flow Conjecture as a way of obtaining a shorter proof of the Four Color Theorem. We dose this section by describing of several other famous conjectures related to these. Every nowhere-zero k-flow is a nowhere-zero k + 1-flow, so conditions for nowhere-zero 3-flows or 5-flows should be more or less restrictive, respectively, than conditions for a nowhere-zero 4-flow. Statements of Tutte's 3-flow Conjecture appear in Steinberg [1976] and in Bondy-Murty [1976, Unsolved Problem 48].
7.3.28. Conjecture. (Tutte's 3-flow Conjecture) Every 4-edge-connected graph has a nowhere-zero 3-flow. 7.3.29. Conjecture. (Tutte's 5-flow Conject-q.re-Tutte [1954b]) Every bridge less graph has a nowhere-zero 5-flow.
312
## Chapter 7: Edges and Cycles
Kilpatrick [1975] and Jaeger [1979] proved that every bridgeless graph is 8-flowable. Seymour [1981] proved that these graphs are 6-flowable. We sketch the ideas of the 8-flow Theorem; details are requested in exercises. Both proofs reduce to the 3-edge-connected case, by showing that a smallest bridgeless graph without a nowhere-zero k-flow is simple, 2-connected, and 3-edge-connected (Exercise 26). The main step is then to express a 3-edgeconnecte::l graph as a union of subgraphs with good flows. A generalization of Theorem 7.3.25 then applies: If G 1 has a nowhere-zero k1 -flow and G2 has a nowhere-zero krflow, then G 1 U G 2 has a nowhere-zero k1 k2 -flow (Exercise 24). (The converse also holds but is not needed.) For the 8-flow Theorem, it then suffices to prove that a 3-edge-connected graph can be expressed as the union of three even subgraphs. First, adding an additional copy of each edge in G yields a 6-edge-connected graph G'. Then, the Tree-Packing Theorem of Nash-Williams (Corollary 8.2.59) yields three pairwise edge-disjoint spanning trees in G'. These correspond to three spanning trees in G. Since we obtained them as edge-disjoint trees in G', each edge of G appears in at most two of them. Within a spanning tree of G, one can find a parity subgraph of G, meaning a spanning subgraph H such that dH(v) dc(v) mod 2 for all v E V(G) (Exercise 25). The complement within E(G) of the edge set ofa parity subgraph is an even subgraph of G. Since our three spanning trees have no common edge, the complements of their parity subgraphs express Gas a union of three even subgraphs. By Proposition 7.3.19, each has a nowhere-zero 2-flow, and hence G has a nowhere-zero 8-flow. The approach in Seymour [1981] is similar; the task is to express a 3-edgeconnected graph as a union of an even graph and a 3-flowable graph. This uses more subtle concepts, including a notion of "modular" flows originally introduced by Tutte [1949]. Seymour's proof was refined by Younger [1983] and Jaeger [1988]. We refer the reader to Zhang [1997] for an exposition. Celmins [1984] proved that ifthe 5-flow Conjecture is false, then the smallest counterexample is a snark having girth at least 7 and no nontrivial edge cut with four edges.
We describe one additional conjecture and its relation to earlier topics. In a 2-edge-connected plane graph, all facial boundaries are cycles. Each edge lies in the boundary of two faces, so the facial cycles together cover every edge exactly twice. It is reasonable to ask whether such a covering can be obtained also for graphs that are not planar. 7.3.30. Definition. A cover of a graph G is a list of subgraphs whose union is G. A double cover is a cover with each edge appearing in exact)y two subgraphs in the list. A cycle double cover (CDC) is a double cover consisting of cycles. 7.3.31. Example. Together with the outer 5-cycle, the 5 rotations of the 5-cycle illustrated below form a CDC of the Petersen graph. The Petersen graph also has CDCs using cycles of other lengths (Exercise 36).
313
## Since cut-edges appear in no cycles, only bridgeless graphs have CDCs.
7.3.32. Conjecture. (Cycle Double Cover Conjecture-Szekeres [1973], Seymour [1979b]) Every bridgeless graph has a cycle double cover.
One might think that the CDC Conjecture follows immediately using embeddings on surfaces with handles, but such embeddings may have facial boundaries that traverse the same edge twice. The Strong Embedding Conjecture asserts that every 2-connected graph has an embedding (on some surface) in which the boundary of each face is a single cycle. Applying this to each block of a 2-edge-connected graph would yield the CDC Conjecture. In discussing the CDC, we must alert the reader to an unfortunate conflict in terminology. Throughout this book, we use the definition of cycle that is common in discussing connectivity, girth, circumference, planarity, etc. In this language, a circuit is an equivalence class of closed trails (ignoring the starting vertex), and an even graph is a graph whose vertex degrees are all even. A circuit traverses a connected even graph. The literature on cycle covers generally reverses this terminology, using "circuit" to mean what we call a cycle and "cycle" to mean what we call an even graph. Since the term "even graph" strongly evokes its definition, we hope that our usage will he clear. The alternative usage arises from other contexts. In a matroid (Section 8.2), the circuits are the minimal dependent sets, and in the cycle matroid of a graph these are the edge sets of the cycles. The cycle space ofa graph is a vector space (using scalars {O, l}) where the coordinates are indexed by the edges and the vectors correspond to the even subgraphs. The original CDC Conjecture states that every bridgeless graph has a double cover by even subgraphs. That phrasing is equivalent to ours, since every even graph is an edge-disjoint union of cycles. Thus we might seek a double cover by using a small number of even subgraphs. The cycles in a cycle double cover are even subgraphs; when cycles are pairwise edge-disjoint, they can be combined to forni a single even subgraph. This leads to the connection between integer flows and cycle double covers.
7.3.33. Proposition. A graph has a nowhere-zero 4-flow if and only if it has a cycle double cover forming three even subgraphs.
314
## Chapter 7: Edges and Cycles
Proof: Theorem 7.3.25 states that a graph has a nowhere-zero 4-flow if and only if it is the union of two even subgraphs Ei. E 2. Let 3 = E 1 t::.E2. At each vertex v the degree in 3 is the sum of the degrees in 1 and 2 minus twice the number of common incident edges; hence it is even. Hence E 3 is an even subgraph, and it contains precisely the edges that appear in just one of {Ei. 2 }. Cycle decompositions of Ei. E2, 3 thus combine to yield a CDC. Conversely, if a CDC forms three even subgraphs, then omitting one of them leaves the graph expressed as the union of two even subgraphs, and hence a nowhere-zero 4-flow exists.
Let P denote the family of graphs that do not contain a subdivision of the Petersen graph. By Proposition 7.3.33, Tutte's 4-flow Conjecture implies that every graph in P has a CDC. Alspach-Goddyn-Zhang [1994] proved a deep result that yields cycle double covers for graphs in P. (They proved that a stronger covering property holds for G ifand only ifG E P.) In light of Proposition 7.3.33, this is a partial result toward Tutte's 4-flow Conjecture. The CDC Conjecture is also related to snarks. Goddyn [1985] proved that if the CDC Conjecture is false, then the smallest counterexample is a snark with girth at least 8.
EXERCISES
7.3.1. (-)Prove that every Hamiltonian 3-regular graph has a Tait coloring. 7.3.2. (-)Exhibit 3-regular simple graphs with the following properties. a) Planar but not 3-edge-colorable. b) 2-connected but not 3-edge-colorable. c) Planar with connectivity 2, but not Hamiltonian .
7.3.3. Prove that every maximal plane graph other than K4 is 3-face-colorable. 7.3.4. Without using the Four Color Theorem, prove that every Hamiltonian plane graph is 4-face-colorable (nothing is assumed about the vertex degrees). 7.3.5. Prove that a 2-edge-connected plane graph is 2-face-colorable if and only if it is Eulerian. 7.3.6. Use Tait's Theorem (Theorem 7.3.2) to prove that x'(G)
## Section 7.3: Planarity, Coloring, and Cycles
315
7.3.7. (!)Let G be a plane triangulation. a) Prove that the dual G* has a 2-factor. b) Use part (a) to prove that the vertices of G can be 2-colored so that every face has vertices of both colors. (Hint: Use the idea in the proofofTheorem 7.3.2.) (Burstein (1974], Penaud [1975]) 7.3.8. (+)It has been conjectured that every planar triangulation has edge-chromatic number L'.l(G), and this has been proved when L'.l(G) is high enough. Show that x'(G) = Ll(G) for the graph of the icosahedron, illustrated below.
7.3.9. Prove that a proper 4-coloring of the icosahedron uses each color exactly 5 times. 7.3.10. Whitney [1931] proved that every 4-connected planar triangulation is Hamiltonian. Use this to reduce the Four Color Problem to the problem of proving that every Hamiltonian planar graph is 4-colorable. 7.3.11. Find a 5-connected planar graph. Does there exist a 6-connected planar graph? 7.3.12. Let G be a planar graph with at least three faces. Prove that G has a vertex partition into two sets whose induced subgraphs are trees if and only if G* is Hamiltonian. 7.3.13. (!) For each of the planar graphs below, present a Hamiltonian cycle or use planarity (Grinberg's condition) to prove that it is non-Hamiltonian.
7.3.14. Let G be the graph below. Prove that G has no Hamiltonian cycle. Explain why Grinberg's Theorem cannot be used directly to prove this.
316
## Chapter 7: Edges and Cycles
7.3.15. (!)Prove Grinberg's Theorem using Euler's Formula. 7.3.16. (!) Use Grinberg's condition to prove that the Grinberg graph (below) is not Hamiltonian.
7.3.17. (!)The smallest known 3-regular 3-connected planar graph that is not Hamiltonian has 38 vertices and appears below. Prove that this graph is not Hamiltonian. (Lederberg [1966], Bosak [1966], Barnette)
7.3.18. Let G be the grid graph Pm o P. Let Q be a Hamiltonian path from the upper left comer vertex to the lower right comer vertex, such as that shown in bold below. Note that Q partitions the grid into regions, of which some open to the left or downward and others open to the right or upward. Prove that the total area of the up-right regions (B) equals the total area of the down-left regions (A). (Fisher-Collins-Krompart [1994])
B B
A
A A A
B B
B B
7.3.19. (!) The generalized Petersen graph P(n, k) is the graph with vertices {u 1, ... , u.) and {v1, ... , v.) and edges {u;U;+i}, {u;v;), and {v;v;+d. where addition is modulo n. The Petersen graph itself is P.(5, 2). a) Prove that the subgraph of P(n, 2) induced by k consecutive pairs {u;, v;) has a
Section 7.3: Planarity, Coloring, and Cycles spanning cycle if k 1 mod 3 and k ::: 4. b) Use part (a) to prove that x'(P(n, 2)) = 3 ifn 2: 6.
317
7.3.20. (-) Let G be a 3-regular graph. Prove that if G is the union of three cycles, then G is 3-edge-colorable. 7.3.21. (+)"Flower snarks". Let Gk and Hk be as constructed in (Example 7.3.13). a) Prove that Gk is 3-edge-colorable. b) Prove that Hk is not 3-edge-colorable when k is odd. (Isaacs (1975)) 7.3.22. Prove that every edge cut of Kk DC, that does not isolate a vertex has at least 2k edges. 7.3.23. (*) Prove that applying the dot product operation (Definition 7.3.12) to two snarks yields a third snark. (Isaacs (1975)) 7.3.24. (*!)Let G 1 and G 2 be graphs. Prove that ifG 1 has a nowhere-zero k1 -flow and G 2 has a nowhere-zero k2-flow, then G 1 U G 2 has a nowhere-zero kik2-flow. 7.3.25. (!)A parity subgraph of G is a subgraph H such that dH(v) da(v) mod 2 for all v E V(G). Prove that every spanning tree of a connected graph G contains a partity subgraph of G. (Itai-Rodeh [1978]) 7.3.26. (*)Fork :;::, 3, prove that a smallest nontrivial 2-edge-connected graph G having no nowhere-zero k-flow must be simple, 2-connected, and 3-edge-connected. (Hint: First exclude loops and vertices of degree 2 and reduce to consideration of blocks. Then exclude multiple edges and finally edge cuts of size 2. In each case, compare G to a graph obtained from it by deleting or contracting edges.) 7.3.27. (*)Prove that every H'amiltonian graph has a nowhere-zero 4-flow. 7.3.28. (*)Prove that every bridgeless graph with a Hamiltonian path has a nowherezero 5-flow. (Jaeger (1978)) 7.3.29. 1 (*)Embed K6 on the torus, and let G be the dual graph. Find a nowhere-zero 5-flowion G. 7.3.30. (*)Prove that a graph G is the union of r even subgraphs if and only if G has a nowhere-zero 2' -flow. (Matthews (19781! 7.3.31. (*) Let G be a graph having a cycle double cover forming 2' even subgraphs. Prove that G has a nowhere-zero 2' -flow. (Jaege1 [1988))
dt(v)
7.3.32. (*!) A modular 3-orientation of a graph G is an orientation D such that di)(v) mod 3 for all v E V(G). Prove that a bridgeless graph has a nowhere-zero 3-flow if and only if it has a modular 3-orientation. (Steinberg-Younger (1989])
7.3.33. (*) Characterization of nowhere-zero k-flows. Let G be a bridgeiess graph, let D be an orientation of G, and let a and b be positive integers. Prove that the following statements are equivalent. (Hoffman [1958])
a)~
<
I?.
a
## for every nonempty proper vertex subset S.
318
Chapter 7: Edges and Cycles b) G has an integer fl.ow using weights in the interval [a, b). c) G has a real-valued fl.ow using weights in the interval [a, b].
7.3.34. (*)Find cycle double covers for the graphs C,. v Ki, C,. v 2Ki, and C,. v K2 7.3.35. (*) Find the cycle double covers with fewest cycles for every 3-regular simple graph with 6 vertices. 7.3.36. (*-) Let G be the Petersen graph. Find a cycle double cover of G whose elements are not all 5-cycles. Find a double cover of G consisting of 1-factors. (Hint: Consider the drawing of G having a 9-cycle on the "outside". Comment: Fulkerson [.1971) conjectured that every bridgeless cubic graph has a double cover consisting of 6 perfect matchings.) 7.3.37. (*) Prove that any two 6-cycles in the Petersen graph must have at least two common edges. Conclude that the Petersen graph has no CDC consisting of five 6-cycles. Use this and Exercise 7.3.20 to conclude that the Petersen graph has no CDC consisting of even cycles. (C.Q. Zhang) .7.3.38. (*!)A cycle double cover is orientable if its cycles can be oriented as directed cycles so that for each edge, the two cycles containing it traverse it in opposite directions. A digraph is even if d-(v) = d+(v) for each vertex v. a) Suppose that G has a nonnegative k-fl.ow (D, f). Prove that f can be expressed as I:;~:i fi, where each (D, fi) is a nonnegative 2-flow on G. (Hint: Use induction on k.) (Little-Tutte-Younger [1988)) b) Prove that a graph G has a positive k-fl.ow (D, f) if-and only if D is the union of k - 1 even digraphs such that each edge e in D appears in exactly f(e) of them. (LittleTutte-Younger [1988)) c) Prove that a graph G has a nowhere-zero 3-fl.ow if and only if it has an orientable cycle double cover forming three even subgraphs. (Tutte [1949]) 7.3.39. (*) Let G be a graph having a CDC formed from four even subgraphs. Prove that G also has a CPC formed from three even subgraphs. (Hint: Use symmetric differences.) 7.3:40. (*)In the Petersen graph, prove that the solution to the Chinese Postman Problem has total length 20, but the minimum total length of cycles covering the Petersen graph is 21. 7.3.41. (*) Let M be a perfect matching in the Petersen graph. Prove that there is no list of cycles in the Petersen graph that together cover every edge of M exactly twice and all other edges exactly once. (ltai-Rodeh [1978], Seymour [1979b]) 7.3.42. (*) Let G be a graph in which a shortest covering walk (that is, an optimal solution to the Chinese Postman Problem) decomposes into cycles. Prove that G has a cycle cover of total length at most e(G) + n(G) - 1. Determine the minimum length of a cycle cover of K3 ., in terms of the number of edges and vertices.
Chapters
In this chapter we explore more advanced or specialized material. Each section gives a glimpse of a topic that deserves its own chapter (or book). Several sections treat more difficult material near the end.
## 8.1. Perfect Graphs
We have discussed the lower bound x (G) 2: w ( G) for chromatic number; the vertices of a clique need different colors. In Section 5.3, we discussed graphs whose induced subgraphs all achieve equality in this bound. 8.1.1. Definition. A graph G is perfect if x (H) = w(H) for every induced subgraph H of G. When discussing perfect graphs, it is common ta use stable set to mean an independent set of vertices. As before, a clique is a set of pairwise adjacent vertices. As usual, maximum means maximum-sized. Since we focus on vertex coloring, again in this section we restrict our attention to simple graphs. Complementation converts cliques to stable sets and vice versa, so w(H) = a(H). Properly coloring H means expressing V(H) as a union of cliques in H; such a set of cliques in H is a clique covering of H. Thus for every graph G we have four optimization parameters of interest.
independence number a(G) clique number w( G) chromatic number x(G) clique covering number O(G)
max size of a stable set max size of a clique min size of a coloring min size of a clique covering
## Berge actually defined two types of perfection:
G is y-perfect if x(G[A]) = w(G[A]) for all A s; V(G). G is a-perfect if O(G[A]) = a(G[A]) for all As; V(G). 319
320
Our definition of perfect is the same as this definition of y-perfect (Berge used y(G) for chromatic number). Since G[A] is the complement of G[A], the definition of a-perfect can be stated in terms of Gas "x(G[A]) = w(G[A]) for all A 5; V (G)". Thus "G is a-perfect" has the same meaning as "G is y-perfect". We now use only one definition of perfection, because Lovasz [1972a] proved "G is y-perfect if and only if G is a-perfect". In terms of our original definition of perfection, this becomes "G is perfect if and only if G is perfect". This statement is the Perfect Graph Theorem (PGT). Always x(G) 2: w(G) and 8(G) 2: a(G), since a clique and a stable set share at most one vertex. A statement of perfection for a class of graphs is thus an integral min-max relation. We observed in Example 5.3.21 that several familiar min-max relations are statements that bipartite graphs, their line graphs, and the complements of such graphs are perfect. If k ::::. 2, then x(C2k+1) > w(C2k+1) and x(C2k+1) > w(C2k+1) (Exercise 1). Thus odd cycles and their complements (except C3 and C3 ) are imperfect. 8.1.2. Conjecture. (Strong Perfect Graph Conjecture (SPGC)-Berge (1960]) A graph G is perfect if and only if both G and G have no induced subgraph that is an odd cycle oflength at least 5. The SPGC remains open. Since the condition in the conjecture is selfcomplementary, the SPGC implies the PGT. Having presented several classical families of perfect graphs in Section 5.3, our goal now is to prove the Perfect Graph Theorem. Later we also study properties of minimal imperfect graphs and classes of perfect graphs. For further reading, Golumbic (1980] provides a thorough introduction to the subject. Berge-Chvatal (1984] collects and updates many of the classical papers.
## THE PERFECT GRAPH THEOREM
In 1960, Berge conjectured that y-perfection and a-perfection are equivalent (see Berge (1961]). Lovasz [1972a] stunned the world of coibinatorics by proving this important and well-known conjecture at the age of 22. FulJterson also studied it, reducing it to a statement he thought was too strong to be true. When Berge told him that Lovasz had proved it, within hours he proved the missing lemma (Lemma 8.1.4), thus illustrating that a theorem becomes easier to prove when known to be true (Fulkerson [1971]). We will prove the Perfect Graph Theorem using an operation that enlarges a graph without affecting the property of perfection. 8.1.3. Definition. Duplicating a vertex x of G produces a new graph G ox by adding a vertex x' with N(x') = N(x). The vertex multiplication of G by the nonnegative integer vector h = (hi. ... , hn) is the graph H = Goh whose vertex set consists of h; copies of each x; E V (G), with copies of x; and Xj adjacent in H if and only if x; # xj in G.
## Section B.l: Perfect Graphs
321
G ox1
(2, 1, 0, 3, 1)
8.1.4. Lemma. Vertex multiplication preserves y-perfection and a-perfection. Proof: We first observe that G o h can be obtained from an induced subgraph of G by successive vertex duplications. If every h; is 0 or 1, then Goh = G[A], where A = {i: h; > 0). Otherwise, start with G[A] and perform duplications until there are h; copies of x; (for each i). Each vertex duplication preserves the property that copies of x; and Xj are adjacent if and only if x;xj E E(G), so the resulting graph is Goh. If G is a-perfect but Goh is not, then some operation in the creation of Goh from G [A] produces a graph that is not a-perfect from an a-perfect graph. It thus suffices to prove that vertex duplication preserves a-perfection. The same reduction holds for y-perfection. Since every proper induced subgraph of Go x is an induced subgraph of G or a vertex duplication of an induced subgraph of G, we further reduce our claim to showing that x(G ox)= w(G ox) when G is y-perfect and that a(G ox)= fJ(G ox) when G is a-perfect. When G is y-perfect, we extend a proper coloring of G to a proper coloring of Go x by giving x' the same color as x. No clique contains both x and x', so w(G ox),= w(G). Hence x (Go x) = x (G) = w(G) = w(G ox). When G is a-perfect, we consider two cases. If x belongs to a maximum stable set in G, then adding x' to it yields a(G ox) = a(G) + 1. Since fJ(G) = a(G), we can obtain a clique covering of this size by adding x' as a 1-vertex clique to some set of fJ ( G) cliques covering G. If x belongs to no maximum stable set in G, then a(G ox)= a(G). Let Q be the clique containing x in a minimum clique cover of G. Since fJ(G) = a(G), Q intersects every maximum stable set in G. Since x belongs to no maximum stable set, Q' = Q - x also intersects every max~mum stable set. This yields a(G - Q') = a(G) - 1. Applying the a-perfection of G to the induced subgraph G- Q' (which containsx) yields fJ(G- Q') = a(G- Q'). Adding Q'U {x') to a set of a( G) - 1 cliques covering G - Q' yields a set of a( G) cliques covering Go x.
x'
~maximum
stable sets
## tl. t covermg t . t oft c 1que G
322
8.1.5. Lemma. In a minimal imperfect graph, no stable set intersects every maximum clique. Proof: If a stable set S in G intersects every w(G)-clique, then perfection of G - S yields x(G - S) = w(G - S) = w(G) - 1, and S completes a proper w(G)coloring of G. This makes G perfect. 8.1.6. Theorem. (The Perfect Graph Theorem (PGT) - Lovasz [1972a, 1972b]) A graph is perfect if and only if its complement is perfect. Proof: It suffices to show that a-perfection of G implies y-perfection of G; applying this to G yields the converse. If the claim fails, then we consider a minimal graph G that is a-perfect but not y-perfect. By Lemma 8.1.5, we may assume that every maximal stable set Sin G misses some maximum clique Q(S). We design a special vertex multiplication of G. Let S = {S;} be the list of maximal stable sets of G. We weight each vertex by its frequency in {Q(S;)}, letting hj be the number of stable sets S; ES such that Xj E Q(S;). By Lemma 8.1.4, H = Goh is a-perfect, yielding a(H) = ()(H). We use counting arguments for a(H) and ()(H) to obtain a contradiction. Let A be the 0,1-matrix of the ir1cidence relation between {Q(S;)} and V ( G); we have a;,j = 1 if and only if Xj E Q(S;). By construction, hj. is the number of ls in column j of A, and n(H) is the total number of l~ in A. Since each row has w(G) ls, also n(H) = w(G) ISi. Since vertex duplication cannot enlarge cliques, we have w(H) = w(G). Therefore, ()(H):::: n(H)/w(H) = ISi. We obtain a contradiction by proving that a(H) < ISi. Every stable set in H consists of copies of elements in some stable set of G, so a maximum stable set in H consists of all copies of all vertices in some maximal stable set of G. Hence a(H) = maxres L;: x;eT h;. The sum counts the ls in A that appea.r in the columns indexed by T. If we count these ls instead by rows, we obtain a(H) = maxres Lses IT n Q(S)I. Since T is a stable set, it has at most one vertex in each chosen clique Q(S). Furthermore, Tis disjoint from Q(T). With IT n Q(S)I ~ 1 for all SES, and IT n Q(T)I = 0, we have a(H) ~ ISi - 1.
V(G)
Q(T)
8.1.7.* Remark. Linear optimization and duality. Clique-vertex incidence matrices also arise in expressing a and () as integer optimizatfr1n problems. A linear (maximization) program can be written as "maximize c x over nonnegative vectors x such that Ax ~ b", where A is a matrix, b, care vectors, and each
## Section 8.1: Perfect Graphs
323
row of Ax s b is a linear constraint a; x s b; on the vector x of variables. A vector x satisfying all the constraints is a feasible solution. An integer linear program requires that each Xj also be an integer. Let A be the incidence matrix between maximal cliques and vertices in a graph G; we have a;.j = 1 when Vj E Q;. By definition, a(G) is the solutio1 to "max ln x such that Ax s lm" when the variables are required to be nonnegative integers. In the solution, xj is 1 or 0 depending on whether Vj is in the maximum stable set; the constraints prevent choosing adjacent vertices. Similarly, when B is the incidence matrix between maximal stable sets and vertices, w(G) is the solution to "max 111 x such that Bx :'.::: 1,," with integer variables. Every maximization program has a dual minimization program. When the max program is "maxc x such that Ax s b", the dual is "miny b such that yT A :::_ c". This program has a variable y; for each original constraint and a constraint for each original variable xj, and it interchanges c, max, s with b, min,::::. When stated in this form, the variables in both programs must be nonnegative. The integer programs dual to w and a seek the minimum number of stable set | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.85643470287323, "perplexity": 922.9027350208595}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250601628.36/warc/CC-MAIN-20200121074002-20200121103002-00245.warc.gz"} |
http://physics.stackexchange.com/questions/58164/net-work-output-of-an-engine-performing-carnot-cycle | # Net work output of an engine performing Carnot cycle?
My problem gives me a Carnot cycle heat engine with water as its working fluid, with $T_H$, $T_L$, and the fact that it starts from saturated liquid to saturated vapor in the heating process.
I need to find the net work output of this engine. So my solution goes like this:
$$\eta_{HE} = 1 - \frac {T_L}{T_H}$$
So I get $\eta_{HE}$. And then I also know that
$$\eta_{HE} = \frac {W_{HE}}{Q_H}$$
Since my goal is $W_{HE}$, I know I'll need to solve for $Q_H$ first. Since I have the state and temperatures of water at both states, I can get $h_H$ and $h_L$. I know that $h_L - h_H$ is some quantity of energy, and it must be either $Q_H$ or $Q_L$. The problem is, I don't know which it is. And I assume it is $h_L - h_H$ because it is heated first and then cooled. I hope this is correct.
I know that the change in enthalpy is the energy either released or absorbed by the system. In this case, since the temperature moves from higher to lower, it seems to me that the system is absorbing energy and hence this must be $Q_H$.
But it also makes sense to me to say that $h_L - h_H$ is the energy released by the system (as in an exothermic reaction), so it can be $Q_L$ too, because that is the heat rejected by the system.
So my question is, for the general case, which is it? Or is it situational? Does its being $Q_H$ or $Q_L$ depend on the resulting sign when I do my calculations? For this problem, the sign of $h_L - h_H$ that I computed is positive. Then this looks like heat entering the system and so it is $Q_H$. Am I correct?
Or, am I completely on the wrong track and should be chasing down another solution?
-
Also, I know I tag this as homework (as is the practice on this site for homework-y questions), but it's not actually graded homework. I'm just doing problems. – markovchain Mar 27 '13 at 4:35
Since the question was posted 10 months ago, I guess that the OP has already forgotten about it. Anyway, somebody might still be interested in it. So here's the explanation:
Your formula for efficiency:
$$\eta_{HE} = \frac {W_{HE}}{Q_H}$$
is correct. It is worth noting that $W_{HE}$ is Your net work output and $Q_H$ is the value of heat acquired from heat source (not the difference between the heat acquired and heat released - this one is exactly equal to $W_{HE}$).
Usually $h$ means specific enthalpy, so if $h_L$ and $h_H$ mean the values of specific enthalpy before and after heating respectively, $m(h_H-h_L)$ is Your value of heat acquired from the heat source $Q_H$, where $m$ is the mass of the working fluid that performs the Carnot cycle. Specific enthalpy at the end of heating process will always be higher than at the beginning, so $m(h_H-h_L)$ is always going to be positive.
If, as You said, You are able to obtain the values of $h_L$ and $h_H$ this solution will work regardless of the starting and end points parameters - specific enthalpy is a definite description of thermodynamic state.
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8823267817497253, "perplexity": 176.54139618369052}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394010354479/warc/CC-MAIN-20140305090554-00057-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/204804-simple-related-rates-prob-print.html | # Simple related rates prob
• October 7th 2012, 10:39 AM
AZach
Simple related rates prob
Helium is pumped into a spherical balloon at $2 \frac{ft^3}{s}$. How fast is the radius increasing after 2 minutes?
I've searched some of the other related rates probs posted on this forum, so I apologize if a question like this has been brought up already.
Basically, I understand I'm looking for $\frac{dV}{dt}$ , and that $\frac{dV}{dt} = \frac {dV}{dr} * \frac {dr}{dt}$.
Now, the rate is 2 $\frac{ft^3}{s}$ and the volume of a sphere is $V = \frac{4}{3} * {PI}r^3$.
Implicitly differentiating V with respect to T gets $\frac {dV}{dt} = 4 * {PI}r^2 \frac{dr}{dt}$
I know that the rate the volume increases with respect to time is $2 \frac{ft^3}{s}$, and I'm not sure what to do with my 2 minutes (or 120 seconds). Is the volume with respect to time 120 seconds and $2 \frac{ft^3}{s}$ is the rate with respect to time? That would make more sense I think because the dimensions of cubic feet with a factor of 2 is like a rate and seconds is most definitely a part of time.
However, when I try to solve $120 = 4PIr^2*2$ or $\frac{dV}{dt} = \frac {dV}{dr} * \frac {dr}{dt}$ and that's incorrect. Since I don't know the volume with respect to time, how can I properly go about solving for the rate with respect to time?
I get $r = sqrt(\frac{120}{8PI})$
• October 7th 2012, 10:59 AM
MarkFL
Re: Simple related rates prob
We are looking for $\frac{dr}{dt}$, and we know:
$dV=4\pi r^2\,dr$ hence:
$\frac{dV}{dt}=4\pi r^2\,\frac{dr}{dt}$
If we assume $V(0)=0$ then $V(t)=2t \text{ ft}^3$ and so:
$r(t)=\left(\frac{3t}{2\pi} \right)^{\frac{1}{3}}$
We are given:
$\frac{dV}{dt}=2\,\frac{\text{ft}^3}{\text{s}}$
$t=2\text{ min}=120\text{ s}$
Can you put this together to finish?
• October 7th 2012, 04:19 PM
AZach
Re: Simple related rates prob
Quote:
Originally Posted by MarkFL2
$r(t)=\left(\frac{3t}{2\pi} \right)^{\frac{1}{3}}$
Can you put this together to finish?
Do you mind if I ask where you derived that?
Well, I took the steps you laid out. since the problem asks for the rate of change with respect to time when t = 120 seconds, I plugged 120 into r(t) which == 3.86. Since we know the rate of change with respect to the volume we can plug that into dv/dr which gets 4pi(3.86)^2. And dV/dt is 2 cubic feet per second.
So the final equation came out to be $2\frac{ft^3}{s} = 186.8\frac{dr}{dt}$ which gave 0.01071 $\frac{ft^3}{s}$, and that means the growth of the balloon's volume has decreased significantly by the 2 minute mark. I'm just still unclear how you got the r(t) equation. (Shake)
Thanks much.
• October 7th 2012, 06:00 PM
skeeter
Re: Simple related rates prob
$V = \frac{4\pi}{3} r^3$
assuming $V(0) = 0$ ...
$2t = \frac{4\pi}{3} r^3$
solve for $r$ as a function of $t$
• October 7th 2012, 08:21 PM
AZach
Re: Simple related rates prob
Quote:
Originally Posted by skeeter
$V = \frac{4\pi}{3} r^3$
assuming $V(0) = 0$ ...
$2t = \frac{4\pi}{3} r^3$
solve for $r$ as a function of $t$
Oh! I kept looking at dV/dr. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 32, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9034978151321411, "perplexity": 572.9959920424883}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257827782.42/warc/CC-MAIN-20160723071027-00281-ip-10-185-27-174.ec2.internal.warc.gz"} |
http://mathhelpforum.com/discrete-math/185130-all-quantifier.html | # Math Help - for all quantifier
1. ## for all quantifier
Hello
A is a formula with variables x and y. What is the difference between
$\forall$ (x,y) A
$\forall$(x) $\forall$(y)A
Thank you...
2. ## Re: for all quantifier
Unless there's some special context in mind, I'd take them both as the same.
[Here using 'A' for the quantifier in ASCII:]
AxAyP
AxyP
some people write
(x)(y)P
They all mean the same.
3. ## Re: for all quantifier
What makes you think that there is a difference?
Can you come up with any examples where they wouldn't be the same?
4. ## Re: for all quantifier
If $\mathcal{L}$ is a first order language with variable letters $x,y,\ldots$ then, $(\forall x)(\forall y)A$ is a well formed formula and $(\forall (x,y))A$ it is not. We can of course use $(\forall (x,y))A$ as an alternative notation for $(\forall x)(\forall y)A$ .
5. ## Re: for all quantifier
I'd rather avoid the clutter of parenethess. It's enough to have
AxAyP
And then to allow that to be indicated by
AxyP
6. ## Re: for all quantifier
Thank you all for your replies . | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.918809175491333, "perplexity": 1495.1293061201088}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131297622.30/warc/CC-MAIN-20150323172137-00234-ip-10-168-14-71.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/108359/compactness-of-sobolev-embedding-for-domains-of-finite-measure/123494 | # Compactness of Sobolev embedding for domains of finite measure
Let $\Omega \subset \mathbb{R}^d$ be a domain of finite Lebesgue measure, not assumed to be smooth or bounded. Is it true that the embedding of, say, $W^{1,p}_0(\Omega)$ (Sobolev functions with zero boundary value) into $L^q(\Omega)$ is compact for $1/q > 1/p - 1/d$?
-
There are necessary and sufficient conditions on $\Omega$ for the compactness of such embedding. Check Mazy'a's book Sobolev Spaces, Springer Verlag, 2011, Section 5.5.2. – Liviu Nicolaescu Sep 28 '12 at 20:57
Since you want this only for functions with compact support, the embedding theorem follows directly from the one for functions on $\mathbb{R}^n$. – Deane Yang Mar 3 '13 at 23:04
@Deane, I am not sure I follow your argument here. There is no compact embedding for functions on $\mathbb{R}^n$. Could you axpand what do you mean here, I am sure I misunderstand something. – András Bátkai Mar 4 '13 at 9:01
@Andras There is no compact embedding on $\mathbb{R}^N$ but since the functions have zero boundary value they can be extended to a large ball where the embedding will hold, and therefore the restriction to $\Omega$ also embeds into $L^p$ compactly for $p<p^*$, as holds for smooth domains. – Daniel Spector Mar 4 '13 at 11:40
@Daniel, what disturbs me in this argument is only that the ball depends on the functions. – András Bátkai Mar 4 '13 at 16:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.918584942817688, "perplexity": 323.32451000926915}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657123274.33/warc/CC-MAIN-20140914011203-00333-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/viewtopic.php?p=90902 | ## Half life
$\frac{d[R]}{dt}=-k[R]^{2}; \frac{1}{[R]}=kt + \frac{1}{[R]_{0}}; t_{\frac{1}{2}}=\frac{1}{k[R]_{0}}$
DamianW
Posts: 35
Joined: Fri Sep 29, 2017 7:06 am
### Half life
for a second order reaction, when given something like 1/16 why do you flip it in the equation and write 16-1?
Alyssa Pelak 1J
Posts: 72
Joined: Fri Sep 29, 2017 7:04 am
Been upvoted: 1 time
### Re: Half life
you would use the integrated rate law for this question.
1/[A]=kt+1[A]0
1/[A]-1/[A]0=kt
You know that 1/[A] is 1/16[A]0 so you sub that into the equation:
1/(1[A]0/16)-1/[A]0=kt
In order to remove the fraction in the denominator you get
16/[A]0-1/[A]0=kt
Hope this helps!
Joanne Guan 1B
Posts: 30
Joined: Sat Jul 22, 2017 3:01 am
### Re: Half life
If you're talking about finding the time for when the concentration is 1/16 of the initial concentration, you would use the integrated second-order rate law which is:
1/{A] = 1/[A]0 + kt
[A] = (1/16)[A]0
Subbing that into the 1/[A] part would give you 1/((1/16)[A]0), and if you reorganize it, you have 16[A]0.
Salman Azfar 1K
Posts: 50
Joined: Thu Jul 13, 2017 3:00 am
### Re: Half life
The key is that when you put them into the formula (integrated second order rate law) I believe you end up using reciprocals due to the way the law is formatted. Just remember that half life problems for second order problems usually require use of the rate law; they aren't usually as simple as adding together multiple half lives for first order.
Timothy Kim 1B
Posts: 62
Joined: Fri Sep 29, 2017 7:04 am
### Re: Half life
Since it is a fraction, the value would appear flipped when it is used in the second-order reaction. take a look at the respective equations and you will see that the values correspond to each other.
Christina Bedrosian 1B
Posts: 33
Joined: Fri Sep 29, 2017 7:05 am
### Re: Half life
it is 1/(1/16) making it 16/1 (it flips); just plug it into the equation and you should be fine
Jennie Fox 1D
Posts: 66
Joined: Sat Jul 22, 2017 3:01 am
### Re: Half life
If you plug it in to the equation, you should get the correct answer. 1/(1/16) is 16. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8497632741928101, "perplexity": 3494.9129849252236}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496664752.70/warc/CC-MAIN-20191112051214-20191112075214-00526.warc.gz"} |
https://brilliant.org/problems/a-number-theory-problem-by-divyansh-tripathi-2/ | # A number theory problem by divyansh tripathi
Consider a 20-sided convex polygon K, with vertices A1, A2, . . . , A20 in that order. Find the number of ways in which three sides of K can be chosen so that every pair among them has at least two sides of K between them. (For example (A1A2, A4A5, A11A12) is an admissible triple while (A1A2, A4A5, A19A20) is not.)
× | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8944774270057678, "perplexity": 817.2993590994513}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218188213.41/warc/CC-MAIN-20170322212948-00229-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://mathhelpforum.com/geometry/122757-rhombus-print.html | # rhombus
• January 7th 2010, 04:21 AM
thereddevils
1 Attachment(s)
rhombus
A parallelogram ABCD with its diagonals meeting at the point O is shown in the diagram . AB is extendedto P such that BP=AB . THe line that passes through D and is parallel to AC meets PC produced at point R and angle CRD =90 .
(1) SHow that the triangles ABD and BPC are congruent .
I managed to prove this .
(2) Show that ABCD is a rhombus .
(3) CR:PC
Not sure bout (2) and (3)
• January 7th 2010, 06:37 AM
Hello thereddevils
Quote:
Originally Posted by thereddevils
A parallelogram ABCD with its diagonals meeting at the point O is shown in the diagram . AB is extendedto P such that BP=AB . THe line that passes through D and is parallel to AC meets PC produced at point R and angle CRD =90 .
(1) SHow that the triangles ABD and BPC are congruent .
I managed to prove this .
(2) Show that ABCD is a rhombus .
(3) CR:PC
Not sure bout (2) and (3)
For (1), I assume you used the facts that
$AD = BC$ (opp sides of a parallelogram)
$AB = BP$ (given)
$\angle DAB = \angle CBP$ (corresponding angles, $AD \| BC$)
So the triangles are congruent, SAS.
For (2), a parallelogram is a rhombus if its diagonals are perpendicular. Well, can you see how to use a pair of equal angles in the congruent triangles to prove $BD \| PR$? And you're given that $AC \| DR$, and $\angle DRC = 90^o$. So that will be enough to prove $DROC$ is a rectangle, and therefore $\angle DOC = 90^o$.
For (3), use the congruent triangles to show that $PC = BD$. Also, $RC=DO = \tfrac12DB$, to show that $CR:PC=1:2$.
Can you fill in all the details now?
• January 7th 2010, 07:39 AM
thereddevils
Quote:
Hello thereddevilsFor (1), I assume you used the facts that
$AD = BC$ (opp sides of a parallelogram)
$AB = BP$ (given)
$\angle DAB = \angle CBP$ (corresponding angles, $AD \| BC$)
So the triangles are congruent, SAS.
For (2), a parallelogram is a rhombus if its diagonals are perpendicular. Well, can you see how to use a pair of equal angles in the congruent triangles to prove $BD \| PR$? And you're given that $AC \| DR$, and $\angle DRC = 90^o$. So that will be enough to prove $DROC$ is a rectangle, and therefore $\angle DOC = 90^o$.
For (3), use the congruent triangles to show that $PC = BD$. Also, $RC=DO = \tfrac12DB$, to show that $CR:PC=1:2$.
Can you fill in all the details now?
Thanks Grandad , oh , so the diagonals of the parallelogram are NOT perpendicular to each other , so all these while i am wrong , i thought they are the same , since the diagonals of squares and rectangles are perpendicular to each other .
• January 7th 2010, 08:18 AM | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 24, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8868959546089172, "perplexity": 708.6291590908604}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422115867507.22/warc/CC-MAIN-20150124161107-00163-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://hal-insu.archives-ouvertes.fr/insu-03623402v1 | Long-term magnetic activity of a sample of M-dwarf stars from the HARPS program. I. Comparison of activity indices - Archive ouverte HAL Access content directly
Journal Articles Astronomy and Astrophysics - A&A Year : 2011
## Long-term magnetic activity of a sample of M-dwarf stars from the HARPS program. I. Comparison of activity indices
J. Gomes da Silva
• Function : Author
N. C. Santos
• Function : Author
X. Bonfils
• Function : Author
X. Delfosse
• Function : Author
T. Forveille
S. Udry
#### Abstract
Context. The search for extra-solar planets similar to Earth is becoming a reality, but as the level of the measured radial-velocity reaches the sub-m s-1, stellar intrinsic sources of noise capable of hiding the signal of these planets from scrutiny become more important.
Aims: Other stars are known to have magnetic cycles similar to that of the Sun. The relationship between these activity variations and the observed radial-velocity is still not satisfactorily understood. Following our previous work, which studied the correlation between activity cycles and long-term velocity variations for K dwarfs, we now expand it to the lower end of the main sequence. In this first paper our aim is to assess the long-term activity variations in the low end of the main sequence, having in mind a planetary search perspective.
Methods: We used a sample of 30 M0-M5.5 stars from the HARPS M-dwarf planet search program with a median timespan of observations of 5.2 years. We computed chromospheric activity indicators based on the Ca ii H and K, Hα, He i D3, and Na i D1 and D2 lines. All data were binned to average out undesired effects such as rotationally modulated atmospheric inhomogeneities. We searched for long-term variability of each index and determined the correlations between them.
Results: While the SCa II, Hα, and Na i indices showed significant variability for a fraction of our stellar sample (39%, 33%, and 37%, respectively), only 10% of our stars presented significant variability in the He i index. We therefore conclude that this index is a poor activity indicator at least for this type of stars. Although the Hα shows good correlation with SCa II for the most active stars, the correlation is lost when the activity level decreases. This result appears to indicate that the Ca ii - Hα correlation is dependent on the activity level of the star. The Na i lines correlate very well with the SCa II index for the stars with low activity levels we used, and are thus a good chromospheric activity proxy for early-M dwarfs. We therefore strongly recommend the use of the Na i activity index because the signal-to-noise ratio in the sodium lines spectral region is always higher than for the calcium lines.
Based on observations made with the HARPS instrument on the ESO 3.6-m telescope at La Silla Observatory under programme ID 072.C-0488(E).
### Dates and versions
insu-03623402 , version 1 (29-03-2022)
### Identifiers
• HAL Id : insu-03623402 , version 1
• ARXIV :
• BIBCODE :
• DOI :
### Cite
J. Gomes da Silva, N. C. Santos, X. Bonfils, X. Delfosse, T. Forveille, et al.. Long-term magnetic activity of a sample of M-dwarf stars from the HARPS program. I. Comparison of activity indices. Astronomy and Astrophysics - A&A, 2011, 534, ⟨10.1051/0004-6361/201116971⟩. ⟨insu-03623402⟩
### Export
BibTeX TEI Dublin Core DC Terms EndNote Datacite
5 View | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8309875726699829, "perplexity": 4335.675704909861}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948900.50/warc/CC-MAIN-20230328232645-20230329022645-00138.warc.gz"} |
https://math.stackexchange.com/questions/913078/eigenvalues-for-the-sturm-liouville-boundary-value-problem | # Eigenvalues for the Sturm-Liouville boundary value problem
Please show me how to calculate the eigenvalues for the following boundary value problem:
$$x''+\lambda x=0\\x(0)=0\\x(\pi)=0\\x'(\pi)=0$$
This is what I did: let $\lambda=\mu^2$
$$X(x)=A\cos\mu x+B\sin \mu x$$
$$X^{'}(x)=-A\mu \sin\mu x+B\mu \cos \mu x$$
Now applying the boundary conditions we get $$A=0$$
and using last two boundary conditions we get
$$A\cos\mu \pi+B\sin\mu\pi=0$$
$$-A\mu \cos\mu \pi+B\mu \sin \mu \pi=0$$
How to solve it further? Here the eigenvalue is $$\lambda=\mu^2=n^2$$
Does the eigenvalue satisfy this equation:
$$\sqrt{\lambda}+\tan \sqrt{\lambda}\pi=0 \textrm{ ?}$$
• From your equations you would find that $A=B=0$, except that you mixed up $\sin$, $\cos$ in the second system. That's because your problem is not well posed, it has too many boundary conditions. If you dropped one of the conditions at $x=\pi$, it would go smoothly. – user138530 Sep 1 '14 at 20:27
Hint:you have $Acos\mu \pi+Bsin\mu\pi=0$
$-A\mu cos\mu \pi+B\mu sin \mu \pi=0$. From these equation you have $2\mu Bsin\mu\pi=0$. Now when $\mu$ is not equal 0 then $sin\mu\pi=0$.As B can not be 0,otherwise X(x) will become zero. so you get $\mu\pi=n\pi$ where n is non-zero integer.That imply $\mu=n$. Now substitute this value you will get eigen value.
Look at the following solution :
Case 1: $\lambda=0$
In this case, we have $x''=0\implies x=Ct+D$. Now $x(0)=0\implies D=0$, so
$x=Ct$. Again $x(\pi)=0\implies C=0$.
Hence for $\lambda=0$, there are no eigen values.
Case 2: $\lambda >0$
Then, we let $\lambda=\alpha^2$. $0\ne\alpha\in \mathbb{R}$.
So we have $x''+\lambda x=x''+\alpha^2 x=0.$ $\to(1)$
Letting $x=e^{mt}$ and considering the auxiliary equation the solution for $(1)$ is
$x=A\cos \alpha t+B\sin \alpha t.$ Now by the boundary condition $x(0)=0,$ we get
$0=A$, so $x=B\sin \alpha t$.
By the second boundary condition $x(\pi)=0$, we see that :
$0=B\sin \alpha \pi$.
For non trivial eigen values we must have $\sin \alpha \pi=0\implies \alpha\pi=n\pi\implies\alpha=n, n\in \mathbb{N}$
Thus, $\lambda=n^2$ are the eigen values and correspondingly
$x=B\sin nt$ are the eigen vectors.
Case 3: $\lambda <0$
Assume $\lambda=-\beta^2, 0\ne \beta\in \mathbb{R}$ and proceed similarly to show that there are no real eigen values. $\square$
Please Note : $x'(\pi)=0$ is a redundant boundary condition, which doesn't do any good. Further it makes the problem look ill-posed. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9822059273719788, "perplexity": 259.849323353068}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257514.68/warc/CC-MAIN-20190524044320-20190524070320-00203.warc.gz"} |
https://brilliant.org/problems/conveyor-belt/ | # Conveyor belt
A 300 kg crate is dropped vertically onto a conveyor belt that is moving at 1.20 m/s. A motor maintains the belt’s constant speed. The belt initially slides under the crate, with a coefficient of friction of 0.400. After a short time, the crate is moving at the speed of the belt. During the period in which the crate is being accelerated, find the work done by the motor which drives the belt.
× | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9025845527648926, "perplexity": 535.0291965168733}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107880401.35/warc/CC-MAIN-20201022225046-20201023015046-00491.warc.gz"} |
http://mathhelpforum.com/algebra/53364-multiplying-dividing-rational-expressions.html | 1. ## Multiplying/Dividing Rational Expressions
Hey, this is actually a College class but im not very good at math so Im in an Intermediate Algebra, I still think it most likely goes here but, anyway... I missed a lecture on some of this and while I managed to learn a majority of the chapter myself im stuck on a few problems. I simply dont even understand how to start them.
Find the product - Rational Expressions/Equations
(Sofar I canceled ad, and 3b but thats about all I got)
ac + 3a +2c +6......ad - 5a + 2d - 10
--------------- X -----------------
ad + a + 2d + 2......bc + 3b - 4c - 12
Find the quotient - Rational Expressions/Equations
(I flipped the second fraction in both already, they were originally division but thats about all I got)
4a³ - 8a².....5a^4
--------- X -------
.....15a......3a² - 6a
y² + y - 20...........y
----------- X --------------
......7x²........3y² + 19y + 20
Those three I am really having trouble with. Sorry if I posted them wrong, I couldnt get the math code to work. I tried making them as readable as possible.
Thanks
2. Originally Posted by Shrinkwrap
...
Find the quotient - Rational Expressions/Equations[/B]
(I flipped the second fraction in both already, they were originally division but thats about all I got)
Those three I am really having trouble with. Sorry if I posted them wrong, I couldnt get the math code to work. I tried making them as readable as possible.[/INDENT]
Thanks
$\dfrac{ac + 3a +2c +6}{ad + a + 2d + 2} \cdot \dfrac{ad - 5a + 2d - 10}{bc + 3b - 4c - 12 } = \dfrac{a(c + 3) +2(c +3)}{a(d + 1) + 2(d + 1)} \cdot \dfrac{a(d - 5) + 2(d - 5)}{b(c + 3) - 4(c +3) } =$ $\dfrac{(a+2)(c + 3) }{(a+2)(d + 1) } \cdot \dfrac{(a+2)(d - 5)}{(b-4)(c + 3)} = \dfrac{(a+2)(d-5)}{(d+1)(b-4)}$
3. Originally Posted by Shrinkwrap
...
Find the quotient - Rational Expressions/Equations[/B]
(I flipped the second fraction in both already, they were originally division but thats about all I got)
....
$\dfrac{4a^3 - 8a^2}{15a} \cdot \dfrac{5a^4}{3a^2 - 6a }= \dfrac{4a^2(a-2)}{15a} \cdot \dfrac{5a^4}{3a(a-2)}= \dfrac{20a^6}{45a^2} = \dfrac49a^4$
I get you started with the last one: (I changed the $7x^2$ into $7y^2$ )
$\dfrac{y^2 + y - 20}{7y^2} \cdot \dfrac{y}{3y^2 + 19y + 20 } = \dfrac{(y-4)(y+5)}{7y^2} \cdot \dfrac{y}{(3y+4)(y+5)} =$
4. oh, very nice... thank you very much. Much easier now that you've shown me how its done. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9036152362823486, "perplexity": 1448.9753804509776}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345776257/warc/CC-MAIN-20131218054936-00031-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://www.emathematics.net/exponents.php?a=&pot=4 | User:
• Matrices
• Algebra
• Geometry
• Graphs and functions
• Trigonometry
• Coordinate geometry
• Combinatorics
Suma y resta Producto por escalar Producto Inversa
Monomials Polynomials Special products Equations Quadratic equations Radical expressions Systems of equations Sequences and series Inner product Exponential equations Matrices Determinants Inverse of a matrix Logarithmic equations Systems of 3 variables equations
2-D Shapes Areas Pythagorean Theorem Distances
Graphs Definition of slope Positive or negative slope Determine slope of a line Equation of a line Equation of a line (from graph) Quadratic function Parallel, coincident and intersecting lines Asymptotes Limits Distances Continuity and discontinuities
Sine Cosine Tangent Cosecant Secant Cotangent Trigonometric identities Law of cosines Law of sines
Equations of a straight line Parallel, coincident and intersecting lines Distances Angles in space Inner product
Exponents
Exponents with fractional exponents
When an exponent is expressed as a fraction, the numerator tells you the power the number is raised to, and the denominator tells you the root you take. The order in which you perform these operations does not matter.
Rules for working with fractional exponents Identify the power the number is raised to; identify the root you will find. Raise the number to the identified power. Take the identified root of the answer obtained.
$Solve\;7^{\frac{4}{3}}$
Step 1 Identify the power the number is raised to; identify the root you will find. Step 2 Raise the number to the identified power Raise 7 to the 4th power. 74=7·7·7·7=2401 Step 3 Take the identified root of the answer obtained. $\sqrt{2401}=13.4$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 2, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9420522451400757, "perplexity": 2353.6243106949432}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912207146.96/warc/CC-MAIN-20190327000624-20190327022624-00410.warc.gz"} |
http://downloadcenter.wimsedu.info/wims/en_H6~analysis~OEFCalcLimLnExp.en.html | # OEF Limit calculus with logarithms or exponentials --- Introduction ---
This module contains 7 exercises about the limit calculus of logarithm and exponential functions. The required and tested skills are:
• limits of polynoms and quotient of polynoms, of functions ln and exp ;
• computational properties of limits (theorems about the limits of sums, products, quotients, composed functions) ;
• indeterminate forms;
• compared growth properties between polynoms and the functions exp and ln.
The exercises are composed of several steps. An exercise goes on, even if a false reply has been given at the precedent step. The good answers are provided after each step, to enable further evaluations. NEW EXERCISES. PLEASE SIGNAL ANY BUG...
### Limit of u(x)*exp(kx)
We consider the function defined over .
The aim of the exercise is to compute step by step the limits of , at and at respectively.
• Let be the function defined over .
Evaluate the limits of at and at : ( )
= and =
• The limits of at and at are:
and
• Now evaluate the limits of at and at : ( )
= and =
• The limits of the exponential function at and at are:
and
• From the preceding results, one can deduce the limit of at by using =
• From the preceding results, and , one deduces that:
• From the preceding results, one can deduce the limit of at by using =
### Limit of u(x)*ln(kx)
Let us consider the function defined over .
The aim of the exercise is to evaluate step by step the limit of , at and at respectively.
• Let be the function defined over .
Evaluate the limits of at and at : ( )
= and =
• The limits of at and at are:
and
• Evaluate now the limits of at and at : ( )
= and =
• The limits of the logarithm at and at are:
and
• From the preceding results, one can deduce the limit of at by applying the =
• From the preceding results, by the , it comes that:
• From the preceding results, one can deduce the limit of at by applying the =
### Limit of k.ln(ax+b) or k/ln(ax+b)
Let be the function defined over by: .
The aim of the exercise is to evaluate step by step the limit of at .
• The function is of the form with:
= and =
• The function is of the form with and .
• Evaluate the limit of at : ( )
=
• The limit of at is:
• Evaluate the limit of ar )
=
• From the properties of the logarithm function, we know that:
• By variable renaming and by composition of limits, it comes that: ( )
=
• By composition, the limit of at is:
.
• Eventually, by the computational rules of the limits, it comes that: ( )
=
### Limit of k.exp(ax+b) or k/exp(ax+b)
Let be the function defined over by: .
The aim of the exercise is to evaluate step by step the limit of at .
• The function is of the form with:
= and =
• The function is of the form with and .
• Evaluate the limit of at : ( )
=
• The limit of at is:
• Evaluate the limit of at )
=
• From the properties of the exponential function, we know that:
• By variable renaming , and knowing that , it comes that: ( )
=
• The limit of at is:
.
• Eventually, by the computational rules of the limits, it comes that: ( )
=
### Compared growth : basic properties
The exercise deals with the basic rules of "compared growth" between on one hand logarithms or exponentials of a given variable and on the other hand powers of this variable.
• The sentence « » is:
• The sentence « » is .
The true sentence is: « ».
• Formally, this means that: =
### Indeterminate forms with ln or exp
Let be the function defined over by: .
So we have where, for any real in , and .
The aim of the exercise is to evaluate step by step the limit of at .
• Evaluate the limit of at :
=
• The limit of at is:
• Evaluate the limit of at :
=
• The limit of at is:
• Evaluate the limit of at =
• By variable renaming , knowing that , it comes that:
=
• The limit of at is:
.
• Can we deduce the limit at of by applying the computational rules of limits ?
• The computational rules of limits are valid, because there is no indeterminate form. The computational rules of limits are not valid, because there is an indeterminate form .
We use instead the properties of "compared growth": the exponential function dominates any polynom function any polynom function dominates the logarithm function . .
Then it comes:
=
### Basic limits (QUIZZ)
This exercise aims to test the knowledge of basic limits of logarithms and exponentials. Reply as quickly as possible !
= = = = = =
In order to access WIMS services, you need a browser supporting forms. In order to test the browser you are using, please type the word wims here: and press Enter''. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9887505173683167, "perplexity": 1579.3843609115804}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662587158.57/warc/CC-MAIN-20220525120449-20220525150449-00112.warc.gz"} |
https://vroomlab.wordpress.com/2020/09/28/13278/ | # Solving Kepler’s “Wine Barrel Problem” without Calculus
Shown below is a cylinder shaped wine barrel.
Fig. 1
From Fig. 1, we see that
$d^2 = (\frac{h}{2})^2 + (2r)^2\implies r^2=\frac{d^2}{4}-\frac{h^2}{16}\quad\quad\quad(1)$
and so,
$V=\pi r^2 h \overset{(1)}{\implies} V=\pi (\frac{d^2h}{4} - \frac{h^3}{16})\quad\quad\quad(2)$
Kepler’s “Wine Barrel Problem” can be stated as:
If $d$ is fixed, what value of $h$ gives the largest volume of $V$?
Kepler conducted extensive numerical studies on this problem. However, it was solved analytically only after the invention of calculus.
In the spring of 2012, while carrying out a research on solving maximization/minimization problems, I discovered the following theorem:
Theorem-1. For positive quantities $a_1, a_2, ..., a_n, c_1, c_2, ..., c_n$ and positive rational quantities $p_1, p_2, ..., p_n$, if $c_1a_1+c_2a_2+...+ c_na_n$ is a constant, then $a_1^{p_1}a_2^{p_2}...a_n^{p_n}$ attains its maximum if $\frac{c_1a_1}{p_1} = \frac{c_2a_2}{p_2} = ... = \frac{c_na_n}{p_n}$.
By applying Theorem-1, the “Wine Barrel Problem” can be solved analytically without calculus at all. It is as follows:
Rewrite (2) as
$V = \sqrt{16} \pi (\frac{h^2}{16})^{\frac{1}{2}}(\frac{d^2}{4} - \frac{h^2}{16})^1.\quad\quad\quad(3)$
Since
$\frac{h^2}{16} >0, \frac{d^2}{4} - \frac{h^2}{16} >0$ and $1\cdot \frac{h^2}{16} + 1\cdot(\frac{d^2}{4} -\frac{h^2}{16}) = \frac{d^2}{4}$, a constant,
by Theorem-1, when
$\frac{1\cdot\frac{h^2}{16}}{\frac{1}{2}} = \frac{1\cdot(\frac{d^2}{4}-\frac{h^2}{16})}{1},\quad\quad\quad$
or
$\frac{\frac{h^2}{16}}{\frac{1}{2}} = \frac{d^2}{4}-\frac{h^2}{16},\quad\quad\quad(4)$
$V$ (see (3)) attains its maximum.
Solving (4) for positive $h$, we have
$h = \frac{2}{\sqrt{3}}d.$
Discovered from the same research is another theorem for solving minimization problem without calculus:
Theorem-2. For positive quantities $a_1, a_2, ..., a_n, c_1, c_2, ..., c_n$ and positive rational quantities $p_1, p_2, ..., p_n$, if $a_1^{p_1}a_2^{p_2}...a_k^{p_k}$ is a constant, then $c_1a_1+c_2a_2+...+c_na_n$ attains its minimum if $\frac{c_1a_1}{p_1} = \frac{c_2a_2}{p_2} = ... = \frac{c_na_n}{p_n}$.
Let’s look at an example:
Problem: Find the minimum value of $\frac{1}{\sqrt[3]{x}} + 27x$ for $x>0$.
Since for $x >0, \frac{1}{\sqrt[3]{x}} >0, 27x >0$ and $(\frac{1}{\sqrt[3]{x}})^3\cdot 27x = 27$, a constant,
by Theorem-2, when
$\frac{\frac{1}{\sqrt[3]{x}}}{3} = 27x,\quad\quad\quad(5)$
$\frac{1}{\sqrt[3]{x}} + 27x$ attains its minimum.
Solving (5) for $x$ yields
$x = \frac{1}{27}$.
Therefore, at $x = \frac{1}{27}\approx 0.03703, \frac{1}{\sqrt[3]{x}} + 27x$ attains its minimum value $\sqrt[3]{27}+27\cdot\frac{1}{27}= 3+1=4$ (see Fig. 2).
Fig. 2
Nonetheless, neither Theorem-1 nor Theorem-2 is a silver bullet for solving max/min problems without calculus. For example,
Problem: Find the minimum value of $x^3-27x$ for $x>0$.
Theorem-2 is not applicable here (see Exercise-1). To solve this problem, we proceed as follows:
From $(x-3)^3 = x^3-9x^2+27x-27$, we have
$(x-3)^3+9x^2+27 = x^3+27x$
and so,
$(x-3)^3+9x^2+27-54x =x^3-27x$.
That is,
$(x-3)^3+9(x^2-6x+3)=x^3-27x$.
Or,
$x^3-27x= (x-3)^3+9(x^2-6x+9-6)$
$=(x-3)^3+9((x-3)^2-6)=(x-3)^3+9(x-3)^2-54$
$=-54+(x-3)^3+9(x-3)^2=-54+(x-3)^2(x-3+9)$
$=-54+(x-3)^2(x+6)$
i.e.
$x^3-27x = -54 +(x-3)^2(x+6)$.
Since $x>0, x+6>0, (x-3)^2 \ge 0 \implies (x-3)^2(x+6) \ge 0$,
$x^3-27x = -54 + (x-3)^3(x+6) \ge -54$,
with the “=” sign in “$\ge$” holds at $x=3$.
Therefore, $x^3-27x$ attains its minimum -54 at $x=3$ (see FIg. 3).
Fig. 3
Exercise-1 Explain why Theorem-2 is not applicable to finding the minimum of $x^3-27x$ for $x>0$.
## 1 thought on “Solving Kepler’s “Wine Barrel Problem” without Calculus”
1. Pingback: Fire & Water | Vroom | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 52, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9502460956573486, "perplexity": 1480.6352177198478}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038098638.52/warc/CC-MAIN-20210417011815-20210417041815-00239.warc.gz"} |
http://slave2.omega.jstor.org/stable/j.ctt7zv8k3 | # Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications
Bernd Heidergott
Geert Jan Olsder
Jacob van der Woude
Pages: 224
https://www.jstor.org/stable/j.ctt7zv8k3
1. Front Matter
(pp. i-iv)
(pp. v-viii)
3. Preface
(pp. ix-xii)
Bernd Heidergott, Geert Jan Olsder and Jacob van der Woude
4. Chapter Zero Prolegomenon
(pp. 1-10)
In this book we will model, analyze, and optimize phenomena in which the order of events is crucial. The timing of such events, subject to synchronization constraints, forms the core. This zeroth chapter can be viewed as an appetizer for the other chapters to come.
Consider a simple railway network between two cities, each with a station, as indicated in Figure 0.1. These stations are calledS1andS2, respectively, and are connected by two tracks. One track runs fromS1toS2, and the travel time for a train along this track is assumed to be 3...
5. ### PART I. MAX-PLUS ALGEBRA
• Chapter One Max-Plus Algebra
(pp. 13-27)
In the previous chapter we described max-plus algebra in an informal way. The present chapter contains a more rigorous treatment of max-plus algebra. In Section 1.1 basic concepts are introduced, and algebraic properties of max-plus algebra are studied. Matrices and vectors over max-plus algebra are introduced in Section 1.2, and an important model, calledheap of piecesorheap model, which can be described by means of max-plus algebra, is presented in Section 1.3. Finally, the projective space, a mathematical framework most convenient for studying limits, is introduced in Section 1.4.
Define$\varepsilon \overset{def}{\mathop{=}}\,-\infty$and$e\overset{def}{\mathop{=}}\,0$, and denoted by ℝmax...
• Chapter Two Spectral Theory
(pp. 28-46)
This chapter is devoted to spectral theory of matrices over the max-plus semiring. In Section 2.1 we will study the relation between graphs and matrices over the max-plus semiring. The basic observation is that any square matrix can be translated into a weighted graph (to be defined shortly) and that products and powers of matrices over the max-plus semiring have entries with a nice graph-theoretical interpretation. This interpretation will be further studied in Section 2.2. The key result will be that, under mild conditions, a square matrix over the max-plus semiring possesses a unique eigenvalue that equals the maximal average...
• Chapter Three Periodic Behavior and the Cycle-Time Vector
(pp. 47-71)
This chapter deals with sequences {x(k) :k∈ ℕ} generated by
x(k+ 1) =Ax(k),
fork≥ 0, whereA$\mathbb{R}_{\max }^{n\times n}$andx(0) =x0$\mathbb{R}_{\max }^{n}$is the initial condition. The sequences are then equivalently described by
x(k) =Akx0, (3.1)
for allk≥ 0.
Definition 3.1Let{x(k) :k∈ ℕ}be a sequence in$\mathbb{R}_{\max }^{n}$,and assume that for all jṉ the quantity ƞj, defined by$\underset{k\to \infty }{\mathop{\lim }}\,\frac{{{x}_{j}}(k)}{k}$,exists. The vector ƞ = (ƞ12,…,ƞn)is called the cycle-time vectorof the sequence x(k).If all ƞj’s...
• Chapter Four Asymptotic Qualitative Behavior
(pp. 72-84)
As in the previous chapter, we will study in this chapter sequences {x(k) :k∈ ℕ} given through
x(k+ 1) =Ax(k),k∈ ℕ, (4.1)
with initial vectorx(0) =x0$\mathbb{R}_{\max }^{n}$andA$\mathbb{R}_{\max }^{n\times n}$. Provided thatAis irreducible with unique eigenvalue λ and associated eigenvectorv, it follows forx(O) = v andk≥ 0 thatx(k) =A⊗kx(0) = λ⊗kv. In words, the vectorsx(k) are proportional tov, and we may therefore say that thequalitativeasymptotic behavior ofx(k) is completely characterized...
• Chapter Five Numerical Procedures for Eigenvalues of Irreducible Matrices
(pp. 85-94)
In this chapter we discuss two numerical procedures for irreducible matrices over max-plus algebra. The first one, calledKarp’s algorithm, will be presented in Section 5.1 and yields the eigenvalue of an irreducible matrix. The second one, called apower algorithm, to be presented in Section 5.2, yields the eigenvalue and a corresponding eigenvector. Notice that we have already encountered an algorithm for computing the eigenvalue in Chapter 2. Indeed, by Theorem 2.9 the eigenvalue of an irreducible matrixAis equal to the maximal average circuit weight of the communication graph ofA.
We start in this chapter from...
• Chapter Six A Numerical Procedure for Eigenvalues of Reducible Matrices
(pp. 95-112)
The generalized eigenmode of a square matrix has been introduced and studied in Chapter 3. More specifically, in Sections 3.2 and 3.3 the existence of a generalized eigenmode of a square regular matrix has been proved by making use of its normal form. As the proofs in Sections 3.2 and 3.3 are constructive, a conceptual algorithm has been obtained by which a generalized eigenmode in principle can be computed. See in particular the proof of Corollary 3.16. However, the obtained algorithm heavily relies on a normal form of the matrix involved.
In this chapter an alternative algorithm is presented.Howard’s...
6. ### PART II. TOOLS AND APPLICATIONS
• Chapter Seven Petri Nets
(pp. 115-125)
In this chapter we will give a brief introduction to Petri nets as a modeling tool. We will show that a subclass of Petri nets, the so-called event graphs, is a suitable modeling aid for the construction of max-plus linear systems (i.e., for the construction of equations like (0.9) or (4.7)). In Section 7.1, the definitions of a Petri net and a timed event graph will be given. The construction of max-plus linear systems, starting from an event graph description of a model, will be treated in Section 7.2 for the autonomous case (i.e., when no external inputs are considered),...
• Chapter Eight The Dutch Railway System Captured in a Max-Plus Model
(pp. 126-139)
This chapter and the next deal with the application of max-plus algebra in a study of the timetable of the Dutch railway system. The starting point is the railway track layout, consisting of a number of lines along which trains run up and down, and the requested synchronization data (i.e., which trains should wait for which other trains in order to allow passengers to transfer from one to the other). It will be assumed that this data is provided. In addition, it is assumed that a timetable is given with a period of one hour (or, a frequency of one...
• Chapter Nine Delays, Stability Measures, and Results for the Whole Network
(pp. 140-152)
This chapter is a follow-up to the previous one. Once a timetable is given, we are interested in its sensitivity with respect to disturbances in the system. A question that came up during one of the discussions at the Dutch railway headquarters was, how many minutes can all changeover times be increased such that a timetable with a period of sixty minutes still can be maintained? The underlying reason for this question was that an increase in age of the average passenger is expected during the coming years due to the baby boom after World War II. Older passengers walk...
• Chapter Ten Capacity Assessment
(pp. 153-160)
This chapter illustrates the application of max-plus algebra to models with sharing of and competition for resources. Section 10.1 will describe the occupation of a railway track (being the resource) by two types of trains. Slow and fast trains alternately use the track. The heaps of pieces approach, as introduced in Section 1.3, provides useful insights. Section 10.2 will deal with a real-life study in which the competition for resources stems from the situation in which a double-track railway line passes through three tunnels, each of them essentially functioning as a single-track section. Section 10.2.1 will deal with a stylized...
7. ### PART III. EXTENSIONS
• Chapter Eleven Stochastic Max-Plus Systems
(pp. 163-176)
This chapter is devoted to the study of sequences {x(k) :k∈ ℕ} satisfying the recurrence relation
x(k+ 1) =A(k) ⊗x(k),k≥0, (11.1)
wherex(0) =x0$\mathbb{R}_{\max }^{n}$is the initial value and {A(k) :k∈ ℕ} is a sequence ofnxnmatrices over$\mathbb{R}_{\max }$· In order to develop a meaningful mathematical theory, we need some additional assumptions on {A(k) :k∈ ℕ}. The approach presented in this chapter assumes that {A(k) :k∈ ℕ} is a sequence of random matrices$\mathbb{R}_{\max }^{n\times n}$in defined on a common probability space....
• Chapter Twelve Min-Max-Plus Systems and Beyond
(pp. 177-190)
In this chapter min-max-plus systems will be studied. Such systems can be viewed as an extension of max-plus systems in the sense that in addition to the max and plus operators, the min(imization) operator is now also allowed. This gives more flexibility with respect to modeling issues. At the end of this chapter, we will briefly discuss the imbedding of min-max-plus systems in the even more general class of nonexpansive systems.
Min-max-plus systems are described by expressions in which the three operations minimization, maximization, and addition appear. They can be viewed as an extension of max-plus expressions in the sense...
• Chapter Thirteen Continuous and Synchronized Flows on Networks
(pp. 191-200)
So far, we have formulated timed events as discrete flows on networks. In this section, we consider a continuous version of such flows.
One possible way to define, describe, and analyze such continuous flows is by limit arguments in timed event graphs (Chapter 7). In such an approach tokens are split up into mini-tokens (say, one original token consists ofNidentical minitokens); the original corresponding place is replaced byNplaces in series, with one mini-token in each of them and with transitions in between. The original holding times are divided byN(firing times remain zero). A transition...
8. Bibliography
(pp. 201-205)
9. List of Symbols
(pp. 206-208)
10. Index
(pp. 209-213) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.890625536441803, "perplexity": 1362.0553586974356}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400193391.9/warc/CC-MAIN-20200920031425-20200920061425-00082.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/183404-proving-row-space-column-space.html | Math Help - proving row space column space
1. proving row space column space
A , B are nXn matrices
and
AB=(A)^t
t-is transpose
prove that the space spanned by A's row equals the space spanned by A's columns
i know that there dimentions are equals
so in order to prove equality i need to prove that one is a part of the other
how to do it?
2. Re: proving row space column space
Originally Posted by transgalactic
A , B are nXn matrices
and
AB=(A)^t
t-is transpose
prove that the space spanned by A's row equals the space spanned by A's columns
i know that there dimentions are equals
so in order to prove equality i need to prove that one is a part of the other
how to do it?
an element in the row space of $A$ is in the form $A^tx$, where $x$ in a vector. an element in the column space of $A$ is in the form $Ay$, where $y$ is a vector. so if the vector $x$ is given and we let $Bx = y,$ then $A^tx = Ay$. this proves that the row space of $A$ is a subspace of the column space of $A$.
3. Re: proving row space column space
each column i of (AB)_i=A*B_i
i was told by my proff that that column i of AB is a member from the span of the columns of A
but i dont get this result
suppose the member of B_i column is (c1,c2,..,cn)
so the multiplication of A by the B_i column
we get then the first member is dot product from the first row with (c1,c2,..,cn)
i cant see how its a variation from the A columns?
4. Re: proving row space column space
the key is this simple fact that if $x$ is a vector with entries $x_1, \ldots, x_n$ and $v_1, \ldots v_n$ are the columns of an $n \times n$ matrix $C$, then $Cx = x_1v_1 + \ldots + x_n v_n$, which is an element of the column space of $C$. it should be clear now that $C^tx$ is an element of the row space of $C$.
5. Re: proving row space column space
i cant get the diagram
of a coefficient next to each of A's columns from this coulmn by matrix multiplication
6. Re: proving row space column space
Originally Posted by transgalactic
i cant get the diagram
of a coefficient next to each of A's columns from this coulmn by matrix multiplication
well, i suggest you take $2 \times 2$ matrices first to get an idea.
let $C = \begin{pmatrix}a & b \\ c & d \end{pmatrix}.$ so the columns are $v_1=\begin{pmatrix}a \\ c \end{pmatrix}$ and $v_2 = \begin{pmatrix}b \\ d \end{pmatrix}$. now let $x = \begin{pmatrix}x_1 \\ x_2 \end{pmatrix}$. show that $Cx=x_1v_1 + x_2v_2.$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 26, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9741125702857971, "perplexity": 457.84644304502837}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394678698575/warc/CC-MAIN-20140313024458-00016-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://cjtcs.cs.uchicago.edu/articles/2022/1/contents.html | ### Volume 2022
#### Complexity of counting feedback vertex sets
Kévin Perrot
Laboratoire d'Informatique et Systèmes, Aix-Marseille Univeristé,
France
kevin DOT perrot AT lis-lab DOT fr
March 26, 2022
#### Abstract
In this note we study the computational complexity of feedback arc set counting problems in directed graphs, highlighting some subtle yet common properties of counting classes. Counting the number of feedback arc sets of cardinality $k$ and the total number of feedback arc sets are $\#P$-complete problems, while counting the number of minimum feedback arc sets is only proven to be $\#P$-hard. Indeed, this latter problem is $\#OptP[\log n]$-complete, hence if it belongs to $\#P$ then $P=NP$.
• The article: PDF (836 KB)
• Source material: ZIP (68 KB) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9939048886299133, "perplexity": 2070.8981953130096}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104655865.86/warc/CC-MAIN-20220705235755-20220706025755-00268.warc.gz"} |
http://www.physicsforums.com/showpost.php?p=4196287&postcount=2 | Thread: Dissipative Lagrangian View Single Post
PF Gold
P: 1,164
I do not understand how the lagrangian L can correspond to two such oscillators.
You wrote two equations above, one is the equation of h.o. with friction, the other one has friction term with opposite sign, which leads to run-away.
The energy here is defined as
$$E = p_x \dot x + p_y \dot y - L = m \dot x\dot y.$$
and since the Lagrangian does not depend on time, it should be constant in time.
You can verify this by multiplying the solutions for $\dot x, \dot y$ - the exponentials will cancel out and the result does not depend on time.
However, all this seems very artificial - I would like to see some useful application of it. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9573965072631836, "perplexity": 330.3142368552454}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500825010.41/warc/CC-MAIN-20140820021345-00417-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://brilliant.org/problems/can-you-find-the-value-of-f20pi/ | # Can you find the value of $$f'(20\pi)$$?
Calculus Level 3
Let $$f(x)$$ be a function defined for all $$x\in\mathbb{R}$$ such that $$f(x+y)=f(x)\cdot f(y)$$ for all $$x,y\in\mathbb{R}$$. Let $$f'(0)=18$$ and $$f(20\pi)=\tfrac12$$. What is the value of $$f'(20\pi)$$?
× | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9113688468933105, "perplexity": 62.492608223047505}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719542.42/warc/CC-MAIN-20161020183839-00489-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://www.futuresmag.com/2011/01/01/when-options-volatility-distorts-probability | From the January 01, 2011 issue of Futures Magazine • Subscribe!
# When options volatility distorts probability
Modern option pricing theories — from the basic Black-Scholes model to advanced mathematical algorithms — are based on probability concepts. One of the most popular indicators used to evaluate the investment attractiveness of both single options and their combinations is the probability to earn profit. This gauge expresses the probability that at the time of options expiration (or any other time point prior to expiration) a given option will make money. Whether the payoff function is positive or negative is determined by the future price of the underlying asset.
Applying classic probability theory, the profit probability of an option, or combination of different options, can be calculated by integrating the product of the combination payoff function by the probability density function over the price range (or ranges) for which the payoff function is positive:
where x is the underlying asset price; PF(B,S,x) is the payoff function of combination S with the underlying asset B; LogN(Mean,σ,x) is the probability density function of lognormal distribution with parameters Mean (mathematical expectation of the price); and variance σ; θ(y) is the theta-function with argument y = PF(B,S,x), which has the following values: θ(y) = 1, if y > 0, and θ(y) = 0 in other cases.
The above formula applies if the price is considered as a continuous variable.
In the discrete case, a finite price series {xt,t = t1, t2, …, tn} replaces the continuous variable. These prices constitute a set of all possible future outcomes. The set of probabilities {p(xti,t),i = 1, 2, …, n}, assigned to the corresponding prices, replaces the probability density function. Price index i forms two subsets:
I+ = {i : PF(B, S, Xt1) > 0}, where the payoff function is positive, and I- = {i : PF(B, S, Xt1) ≤ 0}, where the payoff is negative or equal to zero. Profit probability can be estimated as a sum of probabilities of all elements forming the first subset:
Estimating profit
Profit probability (PP) is widely used by traders and is included in almost all software products developed for analyzing options. It represents one of the main criteria used in different market scanners and rankers designed for identifying potential trading opportunities appearing on options exchanges. The popularity of this indicator is accounted for by the relative simplicity of the PP calculation and by sufficiently high effectiveness of its practical application. Combined with another important indicator, expected profit, PP enables accurate estimation of option profitability.
At the same time, according to longstanding observations, PP of short option combinations may be overvalued significantly during highly volatile periods. It is common knowledge that shorting options is one of the riskiest option strategies. In periods of high market volatility, and especially during financial crises, risks inherent in shorting options increase manifold. Because overstated estimates of PP inevitably lead to risk underestimation, it is absolutely essential to determine the extent to which the volatility of the underlying asset affects the probability values.
To accomplish this task, statistical studies were conducted using a five-year database containing prices of options and their underlying assets (from 2005 to 2010). This period includes data pertaining to both calm and extreme market conditions (the last financial crisis). Using these data, horizontal and vertical analyses were performed involving relationships between profit probability and underlying asset volatility.
In the vertical analysis, 1,000 of the most liquid U.S. stocks were used as underlying assets. For each of them, we created three short straddle combinations (using strike prices that are closest to the current underlying price) for the first, second and third weeks before the expiration date. All combinations were constructed for September 2010 expiration. In total, we estimated 3,000 short straddles.
For the horizontal analysis, five stocks were selected with actively traded options: Apple, Boeing, Ford Motor, General Electric and IBM. For each of these stocks, one short straddle was created on each trading day during a five-year period. By analogy with the vertical analysis, all combinations were created using the most liquid option contracts (the nearest expiration date and strike prices that are closest to the current underlying asset price). For all combinations, profit probability was calculated and recorded for implied and historical volatility values. Altogether, about 6,000 combinations were evaluated.
“Profit probability vs. volatility” (below) illustrates the relationships between profit probability and implied volatility. In both vertical and horizontal analyses the probability to earn profit increases as volatility increases. Although these relationships are not strong (correlation coefficients range from 0.36 to 0.66), they are statistically significant in all cases. At extremely high volatility levels, corresponding to crisis periods, PP rises to 80%-90%, and in some cases to nearly 100%. Obviously, such estimates are biased. Intuition suggests — and the experience of the latest financial crises proves this — that in periods of extreme market fluctuations, the probability to gain profit from short option positions decreases rather than increases. If this is the case, how can the existence of the direct relationship between probability and volatility be explained?
Solving the puzzle
To find the solution, we compare the time dynamics of implied and historical volatilities. It is commonly believed that during crises, option premiums (and, therefore, implied volatility) increase sharply because of rising uncertainty of market participants. Although historical volatility increases as well, this happens slower and with an evident time lag. “Trends in volatilities” (below) illustrates this phenomenon using Boeing stock as an example.
The divergence in cycles of two volatilities is because during crises, option premiums increase sharply, while historical volatility rises more slowly. This is because it is calculated with historical data that include prices from less volatile periods. Increased premiums imply higher values of the payoff function; relatively low historical volatility implies lower variance used in the lognormal probability density function. As it follows from the previous formulas, both factors cause PP to increase at higher levels of implied volatility.
Unjustifiably inflated values of PP, obtained in periods of high volatility, are inappropriate for estimating the profitability of option portfolios containing short positions. It is essential to develop methods that enable us to adjust PP according to the current volatility level. There are several approaches to solve this problem:
• Calculation of historical volatility — used as variance to build the probability density function — is based on an historical price series of a given length. The longer the price series, the greater the influence of old data — data belonging to the calm period preceding an extreme market — and the greater the divergence in historical and implied volatility cycles. This leads to overvaluation of PP. The distortions in probability estimates may be reduced significantly by regulating the length of the price series parameter, according to the current level of implied volatility. The parameter should relate inversely to the volatility.
• The standard method used for calculating historical volatility is based on historical prices, each of which has equal weight relative to all other prices. Alternatively, we can consider differential application of weight coefficients similar to what you would do in calculating an exponential moving average as opposed to a simple moving average. Higher weights would be assigned to recent prices, while older prices receive lower weight coefficients. As a result, recent price fluctuations would exert greater influence on the variance as compared to older price changes. The function setting the weights can be of any form — linear, exponential, etc.
• One of the main factors determining the payoff function of an option combination is the premium obtained by the trader as proceeds from opening the short position. During crises, the premium grows faster than historical volatility, leading to the divergence of volatilities and distortion of probability. Thus, to obtain unbiased PP it is possible to reduce the divergence between two volatilities by artificially decreasing the payoff function profile. This can be achieved by introducing the adjusting coefficient that lowers premium values by some fixed amount or by a certain coefficient.
High volatility is a fact in today’s market. What’s more, it often manifests without warning. It is critical that traders recognize this and adjust their analysis techniques accordingly. The methods discussed here are viable solutions to the problem of high volatility distorting profit expectations in options positions.
Sergey Izraylevich, Ph.D., and Vadim Tsudikman are authors of “Systematic Options Trading” (Financial Times Press, 2010), where you can find an extended discussion of practical application of the option profit probability indicator and the methods used to determine the optimal values of its parameters. Contact the authors at [email protected].
Page 1 of 2 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8459658622741699, "perplexity": 1329.6941148097887}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368703592489/warc/CC-MAIN-20130516112632-00042-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/integrating-factor-confusion.91076/ | # Integrating factor confusion
1. Sep 27, 2005
### asdf1
for the question, siny+cosydy=0, i want to find an integrating factor.
my work:
(1/F)(dF/dx)=(1/cosy)(cosy+siny)=1+tany
=>lny=x +xtany +c`
=> y =ce^(x+xtany)
however, the question wants the integrating factor to be e^x...
why?
2. Sep 27, 2005
### HallsofIvy
Staff Emeritus
Can I assume you mean sinydx+ cosydy= 0? Without a dx in there, it doesn't make sense. If that's the case, then an obvious integrating factor is 1/siny since multiplying through by that gives dx+ (cosx/sinx)dy= 0 which is clearly exact.
I don't know what you mean by "the question wants the integrating factor to be e^x"!
I wasn't aware that questions wanted anything!
3. Sep 27, 2005
### Corneo
Isn't that equation seperable?
4. Sep 28, 2005
### asdf1
opps!!! i'm sorry for the mistype! :P
you're right, it's "sinydx+ cosydy= 0"
that question wanted to prove that the integrating factor is e^x, but the integrating factor that i found was y =ce^(x+xtany)...
5. Sep 28, 2005
### lurflurf
so you want an integrating factor u such that
$$\frac{\partial}{\partial y}u\sin(y)=\frac{\partial}{\partial x}u\cos(y)$$
or
$$\frac{\partial u}{\partial y}\sin(y)+u\cos(y)=\frac{\partial u}{\partial x}\cos(y)$$
integrating factors are not unique so assume
$$\frac{\partial u}{\partial y}=0$$
6. Sep 28, 2005
### HallsofIvy
Staff Emeritus
If the problem says "show that ex is an integrating factor", thenyou don't have to find the integrating factor yourself (as lurflurf said, integrating factors are not unique), just multiply the equation by ex and see if the result is exact.
If you got ce^(x+xtany) as an integrating factor, you sure like doing things the hard way! As I said earlier, 1/sin y is an obvious integrating factor (because, as Corneo said, the equation is separable. Multiplying by
1/sin y "separates" it)
Last edited: Sep 28, 2005
7. Sep 29, 2005
### asdf1
lol...
i didn't think of that...
thank you very much!!! :)
Similar Discussions: Integrating factor confusion | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9696745872497559, "perplexity": 4005.4954310103776}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218188891.62/warc/CC-MAIN-20170322212948-00480-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://www.physicsoverflow.org/user/leastaction/history?start=20 | # Recent history for leastaction
6
years
ago
received upvote on question What is meant by "smooth" instantons, and is there a constraint on the winding number of $E_8$ instantons?
6
years
ago
question commented on What is meant by "smooth" instantons, and is there a constraint on the winding number of $E_8$ instantons?
6
years
ago
received upvote on question What is meant by "smooth" instantons, and is there a constraint on the winding number of $E_8$ instantons?
6
years
ago
received upvote on question What is meant by "smooth" instantons, and is there a constraint on the winding number of $E_8$ instantons?
6
years
ago
question commented on What is meant by "smooth" instantons, and is there a constraint on the winding number of $E_8$ instantons?
6
years
ago
posted a comment What is meant by "smooth" instantons, and is there a constraint on the winding number of $E_8$ instantons?
6
years
ago
posted a question What is meant by "smooth" instantons, ...
6
years
ago
posted a comment $\mathcal{N} = 2^\star$ supersymmetry
6
years
ago
received upvote on comment Contour integrals in complex coordinates in 2D CFT
6
years
ago
received upvote on comment Contour integrals in complex coordinates in 2D CFT
6
years
ago
edited a comment Contour integrals in complex coordinates in 2D CFT
6
years
ago
edited a comment Contour integrals in complex coordinates in 2D CFT
6
years
ago
posted a comment Contour integrals in complex coordinates in 2D CFT
6
years
ago
edited a comment Contour integrals in complex coordinates in 2D CFT
6
years
ago
edited a comment Contour integrals in complex coordinates in 2D CFT
6
years
ago
posted a comment Contour integrals in complex coordinates in 2D CFT
6
years
ago
edited a comment Contour integrals in complex coordinates in 2D CFT
6
years
ago
posted a comment Contour integrals in complex coordinates in 2D CFT
6
years
ago
received downvote on comment Contour integrals in complex coordinates in 2D CFT
6
years
ago
posted a comment Contour integrals in complex coordinates in 2D CFT | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8147562146186829, "perplexity": 1087.8625540451744}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948684.19/warc/CC-MAIN-20230327185741-20230327215741-00257.warc.gz"} |
https://www.zora.uzh.ch/id/eprint/146493/ | # Search for Higgs boson pair production in the bbττ final state in proton-proton collisions at $\sqrt{s}$ = 8 TeV
## Abstract
Results are presented from a search for production of Higgs boson pairs (HH) where one boson decays to a pair of b quarks and the other to a $\tau$ lepton pair. This work is based on proton-proton collision data collected by the CMS experiment at $\sqrt{s}$ = 8 TeV, corresponding to an integrated luminosity of 18.3 $fb^{−1}$. Resonant and nonresonant modes of HH production have been probed and no significant excess relative to the background-only hypotheses has been found in either mode. Upper limits on cross sections of the two HH production modes have been set. The results have been combined with previously published searches at $\sqrt{s}$ = 8 TeV, in decay modes to two photons and two b quarks, as well as to four b quarks, which also show no evidence for a signal. Limits from the combination have been set on resonant HH production by an unknown particle X in the mass range $m_X = 300$ GeV to $m_X = 1000$ GeV. For resonant production of spin 0 (spin 2) particles, the observed 95% CL upper limit is 1.13 pb (1.09 pb) at $m_X = 300$ GeV and to 21 fb (18 fb) at $m_X = 1000$ GeV. For nonresonant HH production, a limit of 43 times the rate predicted by the standard model has been set.
## Abstract
Results are presented from a search for production of Higgs boson pairs (HH) where one boson decays to a pair of b quarks and the other to a $\tau$ lepton pair. This work is based on proton-proton collision data collected by the CMS experiment at $\sqrt{s}$ = 8 TeV, corresponding to an integrated luminosity of 18.3 $fb^{−1}$. Resonant and nonresonant modes of HH production have been probed and no significant excess relative to the background-only hypotheses has been found in either mode. Upper limits on cross sections of the two HH production modes have been set. The results have been combined with previously published searches at $\sqrt{s}$ = 8 TeV, in decay modes to two photons and two b quarks, as well as to four b quarks, which also show no evidence for a signal. Limits from the combination have been set on resonant HH production by an unknown particle X in the mass range $m_X = 300$ GeV to $m_X = 1000$ GeV. For resonant production of spin 0 (spin 2) particles, the observed 95% CL upper limit is 1.13 pb (1.09 pb) at $m_X = 300$ GeV and to 21 fb (18 fb) at $m_X = 1000$ GeV. For nonresonant HH production, a limit of 43 times the rate predicted by the standard model has been set.
## Statistics
### Citations
Dimensions.ai Metrics
6 citations in Web of Science®
20 citations in Scopus®
### Altmetrics
Detailed statistics | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9821574687957764, "perplexity": 1189.771602337277}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400219221.53/warc/CC-MAIN-20200924132241-20200924162241-00453.warc.gz"} |
http://mathhelpforum.com/number-theory/118423-sums-squares-putnam-problem.html | # Math Help - Sums of squares : putnam problem
1. ## Sums of squares : putnam problem
Hello! This is a nice problem from Putnam a few years ago, for fun.
Show that there are infinitely many triples of consecutive integers, each of which is the sum of two squares.
2. Originally Posted by Bruno J.
Hello! This is a nice problem from Putnam a few years ago, for fun.
Show that there are infinitely many triples of consecutive integers, each of which is the sum of two squares.
let $n=4k^2(k^2+1), \ k \in \mathbb{Z}.$ then $n=(2k^2)^2 + (2k)^2, \ n+1=(2k^2 + 1)^2, \ n+2=(2k^2+1)^2 + 1.$
3. Very nice! Your intuition is very good.
My solution is different. I show that given any triple $n-1, n, n+1$, you can construct another triple. Since sums of two squares are closed under product, $(n-1)(n+1)=n^2-1$ is a sum of two squares, and thus we obtain $n^2-1,n^2,n^2+1$. So it suffices to show that we have one such triple, and 8,9,10 does the job.
Yours is better though
4. Originally Posted by Bruno J.
Very nice! Your intuition is very good.
My solution is different. I show that given any triple $n-1, n, n+1$, you can construct another triple. Since sums of two squares are closed under product, $(n-1)(n+1)=n^2-1$ is a sum of two squares, and thus we obtain $n^2-1,n^2,n^2+1$. So it suffices to show that we have one such triple, and 8,9,10 does the job.
Yours is better though
your solution is pretty good too! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9839503765106201, "perplexity": 248.52078416622754}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422115861305.18/warc/CC-MAIN-20150124161101-00035-ip-10-180-212-252.ec2.internal.warc.gz"} |
http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=GRL_6_A | # Network Flow - Maximum Flow
Time Limit : 1 sec, Memory Limit : 65536 KB
Japanese version is here
# Maximum Flow
A flow network is a directed graph which has a $source$ and a $sink$. In a flow network, each edge $(u, v)$ has a capacity $c(u, v)$. Each edge receives a flow, but the amount of flow on the edge can not exceed the corresponding capacity. Find the maximum flow from the $source$ to the $sink$.
## Input
A flow network is given in the following format.
$|V|\;|E|$
$u_0\;v_0\;c_0$
$u_1\;v_1\;c_1$
:
$u_{|E|-1}\;v_{|E|-1}\;c_{|E|-1}$
$|V|$, $|E|$ is the number of vertices and edges of the flow network respectively. The vertices in $G$ are named with the numbers 0, 1,..., $|V|-1$. The source is 0 and the sink is $|V|-1$.
$u_i$, $v_i$, $c_i$ represent $i$-th edge of the flow network. A pair of $u_i$ and $v_i$ denotes that there is an edge from $u_i$ to $v_i$ and $c_i$ is the capacity of $i$-th edge.
## Output
Print the maximum flow.
## Constraints
• $2 \leq |V| \leq 100$
• $1 \leq |E| \leq 1000$
• $0 \leq c_i \leq 10000$
## Sample Input 1
4 5
0 1 2
0 2 1
1 2 1
1 3 1
2 3 2
## Sample Output 1
3 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9407417178153992, "perplexity": 318.4048491352857}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948599549.81/warc/CC-MAIN-20171218005540-20171218031540-00373.warc.gz"} |
http://www.openthesis.org/documents/Measurement-interference-structure-functions-in-370590.html | Details
# Measurement of interference structure functions in quasielastic proton knockout from carbon-12
Abstract (Summary)
A description of the measurement of quasielastic interference structure functions for the nucleus $\sp{12}$C is presented. Longitudinally polarized electrons with an average polarization of 39 $\pm$ 4% and an initial energy of 660.0 MeV were scattered through 33.4 degrees from a graphite target. The scattered electrons were detected with a large magnetic spectrometer in coincidence with the knockout protons which were detected simultaneously by either of two small magnetic spectrometers placed out of the electron scattering plane. The forward-backward asymmetry A$\sb{91}$ and the beam helicity induced asymmetry A$\sbsp{01}{\prime}$ were measured, and the longitudinal-transverse interference structure functions $f\sb{01}$ and $f\sbsp{01}{\prime}$ were extracted for the $\sp{12}$C p-shell knockout reaction at a missing momentum of 115.0 MeV/c and a Q$\sp2$ of 0.13 (GeV/c)$\sp2$. This experiment was the first attempt in a series of $(\vec e,e\sp{\prime}p)$ experiments using multiple out-of-plane spectrometers to detect hadrons in isolating several interference structure functions simultaneously through precise asymmetry and cross section measurements. The equipment for these measurements was installed from 1995 to 1996 at the Bates Linear Accelerator Center in Middleton, Massachusetts, and this experiment was carried out in September of 1996.
Bibliographical Information: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9746648073196411, "perplexity": 2492.469201487505}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084890514.66/warc/CC-MAIN-20180121100252-20180121120252-00613.warc.gz"} |
http://link.springer.com/article/10.1007/BF01586447 | pure and applied geophysics
, Volume 118, Issue 1, pp 86–127
# The stratospheric sulfate aerosol layer: Processes,models, observations, and simulations
• R. C. Whitten
• O. B. Toon
• R. P. Rurco
Article
DOI: 10.1007/BF01586447
Whitten, R.C., Toon, O.B. & Rurco, R.P. PAGEOPH (1980) 118: 86. doi:10.1007/BF01586447
## Abstract
After briefly reviewing the observational data on the stratospheric sulfate aerosol layer, the chemical and physical processes that are likely to fix the properties of the layer are discussed. We present appropriate continuity equations for aerosol particles, and show how to solve the equations on a digital computer. Simulations of the unperturbed aerosol layer by various published models are discussed and the sensitivity of layer characteristics to variations in several aerosol model parameters is studied. We discuss model applications to anthropogenic pollution problems and demonstrate that moderate levels of aerospace activity (supersonic transport and space shuttle operations) will probably have only a negligible effect on global climate. Finally, we evaluate the possible climatic effect of a ten-fold increase in the atmospheric abundance of carbonyl sulfide.
### Key words
StratosphereAerosol layerSulfates | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8531965613365173, "perplexity": 3825.8383378813555}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738661640.68/warc/CC-MAIN-20160924173741-00271-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://socratic.org/questions/how-do-you-solve-6x-2-x-3-0 | Algebra
Topics
# How do you solve 6x^2 - x -3 =0?
Nov 22, 2015
$x \approx - 0.63 , \textcolor{w h i t e}{\times} x \approx 0.79$
#### Explanation:
It is a matter of practice so that you get used to spotting the factors.
The only 2 factors you are going to get for 3 is 1 and 3. This is because 3 is a prime number. The thing is that we have -3 so one has to be positive and the other negative.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Factors of 3 are only {1,3} because 3 is a prime number
Factors of 6 are {1,6} , {2,3}
Looking for -1 in $- x$
These factors do not work so revert to the standard solution formula:
$a {x}^{2} + b x + c = 0$
where
$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$
$a = 6$
$b = - 1$
$c = - 3$
$x = \frac{- \left(- 1\right) \pm \sqrt{{\left(- 1\right)}^{2} - 4 \left(6\right) \left(- 3\right)}}{2 \left(6\right)}$
$\textcolor{g r e e n}{\text{The use of brackets round negative numbers reduce}}$
$\textcolor{g r e e n}{\text{the chance of making a mistake!}}$
$x = \frac{1 \pm \sqrt{1 + 72}}{12}$
$x = \frac{1}{12} \pm \frac{\sqrt{73}}{12}$
$x \approx 0.08 \dot{3} \pm 0.71$
$x \approx - 0.63 , x \approx 0.79$
##### Impact of this question
204 views around the world
You can reuse this answer | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 14, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.860508382320404, "perplexity": 677.6731609398578}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145966.48/warc/CC-MAIN-20200224163216-20200224193216-00249.warc.gz"} |
http://kgslab.org/papers/paper/local-modularity-global-pleiotropy | A genotype-phenotype-fitness map reveals local modularity and global pleiotropy
Kinsler G*, Geiler-Samerotte K*, Petrov D, bioRxiv (2020).
Full text
DOI
Share
tweet
# Abstract
Building a genotype-phenotype-fitness map of adaptation is a central goal in evolutionary biology. It is notoriously difficult even when the adaptive mutations are known because it is hard to enumerate which phenotypes make these mutations adaptive. We address this problem by first quantifying how the fitness of hundreds of adaptive yeast mutants responds to subtle environmental shifts and then modeling the number of phenotypes they must collectively influence by decomposing these patterns of fitness variation. We find that a small number of phenotypes predicts fitness of the adaptive mutations near their original glucose-limited evolution condition. Importantly, phenotypes that matter little to fitness at or near the evolution condition can matter strongly in distant environments. This suggests that adaptive mutations are locally modular—affecting a small number of phenotypes that matter to fitness in the environment where they evolved—yet globally pleiotropic—affecting additional phenotypes that may reduce or improve fitness in new environments. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8136549592018127, "perplexity": 3944.385656264802}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107890586.57/warc/CC-MAIN-20201026061044-20201026091044-00103.warc.gz"} |
http://math.stackexchange.com/questions/252810/logic-as-subset-of-mathematics-and-mathematics-as-subset-of-logic?answertab=oldest | # Logic as subset of mathematics and mathematics as subset of logic
Is logic a subset of mathematics or is mathematics a subset of logic? I have heard the former view, but is there any argument for the latter?
-
Logicism: en.wikipedia.org/wiki/Logicism – Doug Spoonwood Dec 7 '12 at 4:16
Perfect Thanks! – vanattab Dec 7 '12 at 4:18
Mathematical logic is a branch of mathematics. But mathematical logic is by no means all of logic.
There have been recurrent attempts, from Frege through Whitehead/Russell and others, to develop mathematics within what they thought of as logic. The attempts failed, we have moved on.
-
"The attempts failed"-really? – user 170039 Jan 15 at 15:00
[(logic) $\cap$ (math) $\neq \varnothing$] $\;\land\;$ [(logic)$\setminus$(math) $\neq \varnothing$] $\;\land\;$ [(math)$\setminus$ (logic) $\neq \varnothing$].
That is, the intersection of logic and math is clearly not empty, but I think it is also the case that neither one completely encompasses (contains) the other.
Also note:
Mathematical Logic is a branch of mathematics, and is also of interest to (some) philosophers.
Likewise,
Philosophy of Math is a branch of Philosophy, which is also of interest to (some) mathematicians.
-
Nice answer, but every theorem can be rewritten using propositional calculus, for example. We can, of course, think of rules of inference as mathematical operations. – glebovg Dec 7 '12 at 4:28
Thanks, @glebovg. Of course, I root for philosophy as overarching ALL. Every discipline is (or was at one time) the birthchild of Philosophy! – amWhy Dec 7 '12 at 4:34
It is actually the other way around. Philosophy is the root of all sciences, including mathematics. You can think of mathematics as an application of logic.
-
Also, you can think of both mathematics and logic as subsets of the Philosophy of Proof.
-
I am not a mathematician, but here's my 2 cents.
Any mathematical entity must be in accordance with logic. There's not even such a concept in mathematic as "illogical" theoremes, proofs, equations or whatever. Any attempt to think of what such a, say, theorem, might look like, fails. Thus, it is natural to accept that all of mathematics is built within the body of logic.
However, there's no way of defining logic without mathematics. I.e. the only proper way to designate logic, is through mathematics.
So, it happens to be that mathematics is simply a kind of logic, and that logic can be used to describe itself to some extent. This can especially well be observed in gourps and sets theory, where borders between logic and mathematics become less pronounced.
At this point one would naturally come to the conclusion that despite that the original question seems sound, one has to define what mathematics and logic are.
What could serve better for that, than a little historical excourse?
Math appeared out of natural needs of humans. Originally it was made to improve the quality of life and organization. It was our minds predestination to find patterns in the surrounding world, that played the key role, and we began to develop it. So it was the product of interaction of our mind with the world. The product which we turned into a tool.
Eventually, due to the nature of perception, reframing occured and instead of finding patterns in the surrounding world, we began noticing patterns in patterns, then patterns in patterns of the patterns, and so on.. mathematics was developing and it essentially was a product of interaction of our mind with its previous products of interaction, with itself.
Logic was never different from that. However the term "logic" is mostly used to describe human actions and perceptions, where quantity is of a lesser value. It is also thought of as more accessible to an average person than mathematics. The ability of "logical" thinking is an essential ability of a healthy mind. In other words, logic is generally thought of as being "grounded" to the world of material things and everyday perception.
All that being said, there are different ways of using the word "logic". For example, there's Boolean logic which has no big meaning away from mathematics. And actually it is an algebra. So there's more a play of words than an actual well-defined difference between mathematics and logic.
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8188808560371399, "perplexity": 857.5177886065958}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701146550.16/warc/CC-MAIN-20160205193906-00070-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/introductory-particle-physics.587339/ | Introductory Particle Physics
1. welatiger
83
Hi all;
I will work master's degree in particle physics and I want an experimental textbook for particle physics, and i have two choices
is "Introduction to high energy physics" by Perkins
or "particle physics" by B.R. Martin & G. Shaw
2. kloptok
188
I haven't used Perkins myself so I can't say anything about it, but Martin&Shaw I've used. It is quite nice and often comprehensive and contains most of what you need on an introductory level (basic QM is probably all you will need). It also has solutions to all problems which is nice if you use it for self study.
However, I would instead recommend Griffiths "Introduction to Elementary Particles". It is better than the other in my opinion, better explanations of the concepts in the comprehensive style of Griffiths. It is a bit more advanced than the previous two (don't know about Perkins as I said) since it also introduces a basic form of Quantum Field Theory to be able to calculate cross sections, but it covers the same concepts (at least as Martin&Shaw).
3. welatiger
83
thank you so much
i think that Martin's book advantage is it is up to date
246 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8052375912666321, "perplexity": 882.8127525789432}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042987628.47/warc/CC-MAIN-20150728002307-00192-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://www.maa.org/publications/periodicals/loci/joma/raindrops-summary?device=desktop | # Raindrops - Summary
Author(s):
David A. Smith and Lawrence C. Moore
1. Why is it important to consider air resistance when modeling raindrops as falling objects?
2. What important feature did you find in both resistance models that was lacking in the no-resistance model? How did the slope fields reveal this feature? How does it appear symbolically in the differential equations?
3. Explain in your own words how Euler's Method generates a solution of an initial value problem. In particular, explain how Euler's Method uses the same information that is used to generate a slope field.
4. Explain why
is an exact solution of the drizzle drop problem. How does this formula reveal the terminal velocity you know already? If you guessed the form of the solution on Page 3, compare this symbolic form with your guess. Are the two proposed solutions the same? If not, describe how they differ. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9346707463264465, "perplexity": 781.8201950618319}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422115865010.11/warc/CC-MAIN-20150124161105-00004-ip-10-180-212-252.ec2.internal.warc.gz"} |
http://cms.math.ca/cmb/kw/function%20space | location: Publications → journals
Search results
Search: All articles in the CMB digital archive with keyword function space
Expand all Collapse all Results 1 - 3 of 3
1. CMB 2015 (vol 58 pp. 757)
Han, Yanchang
Embedding Theorem for Inhomogeneous Besov and Triebel-Lizorkin Spaces on RD-spaces In this article we prove the embedding theorem for inhomogeneous Besov and Triebel-Lizorkin spaces on RD-spaces. The crucial idea is to use the geometric density condition on the measure. Keywords:spaces of homogeneous type, test function space, distributions, Calderón reproducing formula, Besov and Triebel-Lizorkin spaces, embeddingCategories:42B25, 46F05, 46E35
2. CMB 2008 (vol 51 pp. 570)
Lutzer, D. J.; Mill, J. van; Tkachuk, V. V.
Amsterdam Properties of $C_p(X)$ Imply Discreteness of $X$ We prove, among other things, that if $C_p(X)$ is subcompact in the sense of de Groot, then the space $X$ is discrete. This generalizes a series of previous results on completeness properties of function spaces. Keywords:regular filterbase, subcompact space, function space, discrete spaceCategories:54B10, 54C05, 54D30
3. CMB 1999 (vol 42 pp. 321)
Kikuchi, Masato
Averaging Operators and Martingale Inequalities in Rearrangement Invariant Function Spaces We shall study some connection between averaging operators and martingale inequalities in rearrangement invariant function spaces. In Section~2 the equivalence between Shimogaki's theorem and some martingale inequalities will be established, and in Section~3 the equivalence between Boyd's theorem and martingale inequalities with change of probability measure will be established. Keywords:martingale inequalities, rearrangement invariant function spacesCategories:60G44, 60G46, 46E30
top of page | contact us | privacy | site map | | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8433390259742737, "perplexity": 2964.758920107648}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560284429.99/warc/CC-MAIN-20170116095124-00290-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/454080/question-about-mode-expansion-of-free-compact-boson | # Question about Mode expansion of free compact boson
$$(1+1)$$-Dim free compact boson, Lagrangian is $$\mathcal{L}= \frac{1}{2}(\partial_\mu\phi(\sigma,t))^2$$ with $$\phi(x,t)\sim\phi(x,t)+2\pi r$$ and periodic boundary condition along $$x$$, i.e. $$\phi(\sigma,t)= \phi(\sigma+\beta, t)$$.
Equation of motion is $$(\partial_t^2 -\partial_\sigma^2) \phi(\sigma,t)=0$$
I saw the following saying but I don't understand:
The mode expansion of $$\phi(x,t)$$ is $$\begin{eqnarray} { \phi(\sigma,t) =} \nonumber \\ && \hat{x} + \frac{2 \pi}{\beta} r w \sigma + \frac{\pi}{\beta} \hat{p} t \nonumber \\ && + \frac{1}{2} \sum_{n = 1}^{\infty} [ \frac{a_n}{\sqrt{n}} e^{ - in( \sigma + t) \frac{2 \pi}{\beta}} + \frac{a^{\dagger}_n}{\sqrt{n}} e^{ in( \sigma + t) \frac{2 \pi}{\beta}} ] \nonumber \\ && + \frac{1}{2} \sum_{n = 1}^{\infty} [ \frac{\tilde{a}_n}{\sqrt{n}} e^{ in( \sigma - t) \frac{2 \pi}{\beta}} + \frac{\tilde{a}^{\dagger}_n}{\sqrt{n}} e^{ - in( \sigma - t) \frac{2 \pi}{\beta}} ] \end{eqnarray}$$ with $$w\in\mathbb{Z}$$ is winding number.
My questions:
1. Why are there second and third terms in compact free boson and no such terms in non-compact free boson?
I know there can exist wind solution for compact case since it's a map from $$S$$ to $$S$$. And we can check that $$\frac{2 \pi}{\beta} r w \sigma$$ satisfies the EOM. However why we don't include such a term $$\propto \sigma$$ or $$\propto t$$ in non-compact case. And $$\phi(\sigma, t) = \alpha \sigma$$ and $$\phi(\sigma,t)= \alpha t$$ are certainly solutions of EOM, why we don't include these solutions in mode expansion for non-compact free boson?
1. How to prove that in compact case, except above modes there are no other solutions which satisfy the EOM and PBC?
Because $$\phi(\sigma, t)= \alpha t + \alpha' x$$ is a solution of EOM and doesn't belong to plane wave. There are also many other solutions like $$t^2 +\sigma^2$$$$\sigma t$$ or $$\sigma^3 + 3 \sigma t^2$$, etc. Certainly these solutions can't satisfy PBC. But how to prove that in compact case, except plane wave and linear solutions there are no other solutions which can satisfy the EOM and PBC?
1. Those terms include momentum and winding instanton modes $$\phi_{n,m}$$ because the target space (TS) coordinate is in principle many-valued $$\phi(x,t)\sim\phi(x,t)+2\pi r$$ with infinitely many branches labelled by an integer. The momentum and winding number tell the monodromy in the $$t$$- and the $$x$$-direction on the world sheet (WS), i.e. the change in target space branch.
2. If we subtract these instanton contributions $$\phi_{n,m}$$, the remainder $$\phi-\phi_{n,m}$$ has TS coordinate within the same target space branch/can be treated as being single-valued on the whole WS, i.e. the boundary conditions (BCs) don't connect different branches, so that the remainder $$\phi-\phi_{n,m}$$ is just an ordinary Fourier series, i.e. the oscillator modes.
3. So the string $$\phi$$ is a sum of an affine instanton part $$\phi_{n,m}$$ and an oscillator part. In the uncompactified case, the instanton part has 1 instanton. (This terminology is used in the same semantic sense that, say, a bicycle without gears has 1 gear.)
• But why in non-compact case, we don't include the term like $\phi(\sigma,t)= \alpha \sigma+ \alpha' t$? And only take the Fourier mode into consideration? – maplemaple Jan 14 at 16:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 29, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8951554298400879, "perplexity": 491.37306832556123}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256887.36/warc/CC-MAIN-20190522163302-20190522185302-00223.warc.gz"} |
https://brilliant.org/problems/prismatic-package-nice-one/ | # Prismatic Package
Calculus Level 2
It is given a prismatic package which sides measure $$x,y,z$$ such that its volume is $$108$$. Its surface (without the cover) is given by the equation: $$S = xy + 2yz + 2xz$$
Setting $$x,y,z$$ as variables, what is the value of $$x+y+z$$ if we want to minimize the surface area of the package?
× | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9518567323684692, "perplexity": 331.186623450672}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187828411.81/warc/CC-MAIN-20171024105736-20171024125736-00754.warc.gz"} |
http://nrich.maths.org/1279 | ### Coordinate Tan
What are the coordinates of the coloured dots that mark out the tangram? Try changing the position of the origin. What happens to the coordinates now?
### A Cartesian Puzzle
Find the missing coordinates which will form these eight quadrilaterals. These coordinates themselves will then form a shape with rotational and line symmetry.
### Criss Cross Quiz
A game for 2 players. Practises subtraction or other maths operations knowledge.
# Coordinate Cunning
##### Stage: 2 Challenge Level:
This month's game is linked with our World of Tan. It is for 2 players. Little Mai and Little Fung explain how you play:
This text is usually replaced by the Flash movie.
LM: First you must decide where to place the origin, (0,0). It can be anywhere.
LF: Now you must decide who is blue and who is red.
LM: Blue goes first and chooses a point. Blue has to state the position of their chosen point in relation to the origin.
LF: BUT, if you get the coordinates of the dot wrong, you don't get that point.
LM: And you lose a go.
LF: Red now chooses a point and gives its coordinates.
LM: You then take it turns to go?
LF: Until one person has four dots in line of their colour. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8286018967628479, "perplexity": 1797.0547097568788}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657130272.73/warc/CC-MAIN-20140914011210-00196-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
http://math.stackexchange.com/questions/286452/how-to-solve-system-of-equations-with-multiple-constraints | # How to solve system of equations with multiple constraints?
I have a system of equations that looks like this:
$$\begin{array}{rl} a_1 b_1 c_1+a_2 b_2 c_2+a_3 b_3 c_3&=1000\\ a_1+a_2+a_3&=1\\ a_2&=0.6 \,a_1\\ b_1+b_2+b_3&=500 \end{array}$$
and $$a_1 a_2 a_3 b_1 b_2 b_3 c_1 c_2 c_3 > 0$$
$c_1,c_2,c_3$ are free.
I have no experience with linear programming, some in linear algebra. How do I go about finding the optimal solution so that $a_2b_2c_2$ is maximized?
-
The first equation is not linear. – copper.hat Jan 25 '13 at 6:17
My mistake, I've updated the title. – mirai Jan 25 '13 at 6:17
Optimal in what sense? – Max Jan 25 '13 at 6:20
@Max optimal in the sense that it maximizes $a_2$ – mirai Jan 25 '13 at 14:41
I formatted the equations, removed the redundant one, check them. I'm not sure if the positive condition is for the product or for each variable. – leonbloy Jan 25 '13 at 15:03
If the coefficients $a_i,b_i,c_i$ are not constrained to be non-negative, $a_2$ can be made as large as possible.
Consider $b_1 = b_2 = b_3 = \frac{500}{3}$. Let $a_2 =M$ be any positive number. $a_1 = \frac{5}{3}M$. Also, let $c_3 = -1$. Then, $c_1 = c_2 = \frac{18+5M}{8M}$ will satisfy all the equations. If $M$ is sufficiently large$(>\frac{3}{8}), a_3<0$. So, the product of all terms will be positive.
So, $a_2$ can be made as large as wanted. In other words, it is unbounded.
If you assume that all the $a_i,b_i,c_i$ are non-negative, the problem is straight-forward. Define new variables $x_1 = a_1b_1c_1,x_2 = a_2b_2c_2,x_3 = a_3b_3c_3$ in the first relation. Then, solve the simple LP that arises.
Try any online solver or matlab or mathematica for a sloution. You need not know how to solve a LP to get a solution.
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8496564626693726, "perplexity": 380.00766603457413}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375097475.46/warc/CC-MAIN-20150627031817-00179-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/2274788/error-analysis-of-a-non-restoring-division-algorithm-studying-the-iteration | # Error analysis of a non restoring division algorithm. Studying the iteration.
I'm not sure my derivation is correct, and I also need to find out the error of the iteration/sequence I'm about to derive, but I can't figure out the error once the iteration is finished.
Let $x,y,q,r$ integers (expressed in binary) such that
$$\begin{array}{l} x = x_{2k-1} \ldots x_0 = \sum_{j=0}^{2k-1} x_j 2^j\\ y = y_{k-1} \ldots y_0 = \sum_{j=0}^{k-1} y_j 2^j\\ q = q_{k-1} \ldots q_0 = \sum_{j=0}^{k-1} q_j 2^j\\ r = r_{k-1} \ldots r_0 = \sum_{j=0}^{k-1} r_j 2^j \end{array}$$
The idea would be write down an algorithm such that $$x = qy + r$$
To derive the iteration we define for $0 \leq j \leq k$
$$q^{(j)} = q_{k - 1}2^{k - 1} \ldots q_{k-j}2^{k-j} = \left\lfloor \frac{q}{2^{k-j}} \right\rfloor$$
We define $r^{(j)}$ as the solution of $$q^{(j)}y + r^{(j)} = x$$ It is in particular clear that $q^{(0)} = 0 \Rightarrow r^{(0)} = x$ For $0 \leq j \leq k-1$ taking the finite difference of the previous equation provides \begin{multline} \Delta_j (q^{(j)}y + r^{(j)}) = \Delta_j x \Rightarrow (\Delta_j q^{(j)})y + \Delta_j r^{(j)} = \Delta_jx \Rightarrow \\ (q^{(j+1)} - q^{(j)})y + r^{(j+1)} - r^{(j)} = 0 \Rightarrow q_{k-j-1} 2^{k-j-1}y + r^{(j+1)} - r^{(j)} = 0 \Rightarrow \\ r^{(j+1)} = r^{(j)} - q_{k-j-1}2^{k-j-1}y \end{multline} therefore I end up with the following sequences of partial reminder $$r^{(j+1)} = \left\{ \begin{array}{l} r^{(j)} - q_{k-j-1}2^{k-j-1}y & 0 \leq j \leq k - 1\\ x & j = - 1 \end{array} \right.$$
Choosing $q_{k-j-1} = \text{signum}(r^{(j)} - 2^{k-1-j}y)$ allow the sequence to converge. The question is... how do analyse the sequence in order to derive a bound on $r^(j)$?
I have easily derived
$$|r^{(j+1)} - r^{(j)}| = 2^{k-1-j}y$$
But this doesn't seem to lead at anything... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9995170831680298, "perplexity": 903.7049713258101}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579251779833.86/warc/CC-MAIN-20200128153713-20200128183713-00037.warc.gz"} |
http://math.stackexchange.com/questions/5/how-can-you-prove-that-the-square-root-of-two-is-irrational/16544 | # How can you prove that the square root of two is irrational?
I have read a few proofs that $\sqrt{2}$ is irrational.
I have never, however, been able to really grasp what they were talking about.
Is there a simplified proof that $\sqrt{2}$ is irrational?
-
I can't edit yet, could someone edit "grash" to "grasp"? Thanks. – Coltin Jul 20 '10 at 19:53
$\sqrt{2}$ isn't integer (it's strictly in between 1 and 2). So if it's rational, it's equal to an irreducible fraction $p/q$. Then the fraction $p^2 / q^2$ is also irreducible, but it is equal to 2, which is an integer! – Alexei Averchenko Apr 3 '11 at 1:16
Plato said I can't call myself human unless I can prove this. – Mike Jones May 7 '11 at 19:48
You use a proof by contradiction. Basically, you suppose that $\sqrt{2}$ can be written as $p/q$. Then you know that $2q^2 = p^2$. However, both $q^2$ and $p^2$ have an even number of factors of two, so $2q^2$ has an odd number of factors of 2, which means it can't be equal to $p^2$.
-
Thanks! This is quite clear. – John Gietzen Jul 22 '10 at 14:51
This is by far my favorite proof of $\sqrt{2}$ irrational. I believe it is due to Chaitin (at least I think it's his -- it's in his book Meta Math! (p. 98), and he does not attribute it to anyone else). Of course, this depends on unique prime factorization, but it's still quite elementary. The descent method in the standard proof is, of course, hidden in the prime factorization proof, but that's a fine place for it. Note that the original poster couldn't grasp the popular proof, and I bet the descent with contradiction is the obstacle -- I've seen that with many students. – David Lewis Apr 3 '11 at 10:53
This proof easily generalizes to any exponent k and ratio b >= 2 which is not a perfect power of k, as follows (not in Chaitin's book, but it ain't so hard)... Assume $m^k = b n^k$ Then the unique prime factorizations of $m^k$ and $n^k$ must have all exponents that are multiples of k, and that must also therefore be true of b. But that means b is a perfect k-th power, $b = c^k$ for some integer c. The case k = b = 2 is the classical theorem, with 2 not a perfect square. – David Lewis Apr 3 '11 at 10:54
It is interesting that the Greeks (and I guess most everybody since) missed this proof, because they had unique prime factorization (cf. Euclid's algorithm), and this proof makes clear that that the irrationality of non-perfect roots is intimately related to it. – David Lewis Apr 3 '11 at 10:55
@Bill -- thanks. Makes sense that this proof is not due to Chaitin. It's just that I had never seen it, and it is so attractive (IMHO) that I assumed it must be recent, or it would be better known. He does not attribute it, but I guess that is normal with ancient, folklore proofs. I wonder if anyone does know where or from whom it did originate -- did the Greeks know it? As for not requiring unique prime factorization, you are correct, mathematically. But I was thinking more pedagogically -- it's feasible to introduce this proof in school when prime factorization has been taught. – David Lewis Mar 14 '12 at 17:27
Assume that $\sqrt{2}$ is rational. Then there exists some rational $R=\sqrt{2}=\frac{Q}{D}$, where $Q$ and $D$ are positive integers and relatively prime (since $R$ can be expressed in simplified form).
Now consider $R^2 = 2 = \frac{Q^2}{D^2}$. Since $Q$ and $D$ are relatively prime, this means that only $Q^2$ can have $2$ in its prime decomposition, and the exponent must be one. Thus, $Q^2 = 2^1 x$, for some odd integer $x$. But $Q^2$ is a square, and thus the exponents for all of its prime factors must be even. Here we have a contradiction.
Thus, $\sqrt{2}$ must be irrational.
-
If you are implicitly using uniqueness of prime factorizations then you need to explicitly state that, and state how it applies to yields your deduction. This is essential for proofs at this level. – Bill Dubuque Apr 14 '12 at 0:19
Another method is to use continued fractions (which was used in one of the first proofs irrationality of $\displaystyle \pi$).
Instead of $\displaystyle \sqrt{2}$, we will consider $\displaystyle 1 + \sqrt{2}$.
Now $\displaystyle v = 1 + \sqrt{2}$ satisfies
$$v^2 - 2v - 1 = 0$$
i.e
$$v = 2 + \frac{1}{v}$$
This leads us to the following continued fraction representation
$$1 + \sqrt{2} = 2 + \cfrac{1}{2 + \cfrac{1}{2 + \dots}}$$
Any number with an infinite simple continued fraction is irrational and any number with a finite simple continued fraction is rational and has at most two such simple continued fraction representations.
Thus it follows that $\displaystyle 1 + \sqrt{2}$ is irrational, and so $\displaystyle \sqrt{2}$ is irrational.
Exercise: Show that the Golden Ratio is irrational.
-
The continued fraction proof in Aryabhata's answer can be recast into an elementary form that requires no knowledge of continued fractions. Below is a variant of such that John Conway (JHC) often mentions, followed by my (WGD) reinterpretation of it to highlight the key role played by the principality of (denominator) ideals in $\:\mathbb Z\:$ (which I call unique fractionization).
THEOREM (JHC) $\quad \rm r = \sqrt{n}\ \:$ is integral if rational,$\:$ for $\:\rm n\in\mathbb{N}$
Proof $\ \ \$ Put $\ \ \displaystyle\rm r = \frac{A}B ,\;$ least $\rm\; B>0\:.\;$ $\ \displaystyle\rm\sqrt{n}\; = \frac{n}{\sqrt{n}} \ \Rightarrow\ \frac{A}B = \frac{nB}A.\ \:$ Taking fractional parts yields $\rm\displaystyle\ \frac{b}B = \frac{a}A\$ for $\rm\ 0 \le b < B\:.\$ But $\rm\displaystyle\ B\nmid A\ \Rightarrow\:\ b\ne 0\ \:\Rightarrow\ \frac{A}B = \frac{a}b\$ contra $\rm B$ least. $\:$ QED
Abstracting out the Euclidean descent at the heart of the above proof yields the following
THEOREM (WGD) $\quad \rm r = \sqrt{n}\ \:$ is integral if rational,$\:$ for $\:\rm n\in\mathbb{N}$
Proof $\ \$ Put $\ \ \displaystyle\rm r = \frac{A}B ,\;$ least $\rm\; B>0\:.\;$ $\ \displaystyle\rm\sqrt{n}\; = \frac{n}{\sqrt{n}} \ \Rightarrow\ \frac{A}B = \frac{nB}A\ \Rightarrow\ B\:|\:A\$ by this key result:
Unique Fractionization $\$ The least denominator $\rm\:B\:$ of a fraction divides every denominator.
Proof $\rm\displaystyle\ \ \frac{A}B = \frac{C}D\ \Rightarrow\ \frac{D}B = \frac{C}A \:.\$ Taking fractional parts $\rm\displaystyle\ \frac{b}B = \frac{a}A\$ where $\rm\ 0 \le b < B\:.\$ But
$\rm\displaystyle\ \:B\nmid D\ \Rightarrow\ b\ne 0\ \Rightarrow\ \frac{A}B = \frac{a}b\ \$ contra leastness of $\rm\:B\:.\quad\quad$ QED
Thus JHC's proof essentially "inlines" the above proof - which is better viewed as principality of (denominator) ideals in $\mathbb Z\:,$ cf. my post here. See also this discussion between John Conway and I.
-
or...suppose gcd(a,b) = 1, and $\sqrt{n} = \frac{a}{b}$ for integers a and b. then $a^2 = b^2n$. since $b^2$ divides $b^2n$, it must be the case that $b^2|a^2$. if p is a prime that divides b then p divides $a^2$, hence p divides a contradicting gcd(a,b) = 1. thus, there can be no such prime that divides b, so b = -1 or b = 1, that is, a/b is an integer. – David Wheeler Mar 14 '12 at 9:36
If $\sqrt 2$ were rational, we could write it as a fraction $a/b$ in lowest terms. Then $$a^2 = 2 b^2.$$ Look at the last digit of $a^2$. It has to be $0$, $1$, $4$, $5$, $6$ or $9$. Now look at the last digit of $2b^2$. It has to be $0$, $2$ or $8$. As $a^2$ and $2b^2$ are the same number, its last digit must be $0$. But that's only possible if $a$ ends in $0$ and $b$ ends in $0$ or $5$. Either way both $a$ and $b$ are multiples of $5$ contradicting $a/b$ being in lowest terms.
-
Can you explain how you got the numbers, 0, 1, 4, 5, 6, or 9 – Tyler Hilton Aug 10 '10 at 19:59
The last digit of the square of a number depends only on the last digit of the number. To see this, just think about how you usually multiply two numbers (by hand) and focus on what can contribute to the 1's column. From here, you just compute 0^2, 1^2, 2^2,..., 9^2 and record the last digits to get 0,1,4,9,6,5,6,9,4,1, which, not counting multiples, is 0,1,4,5,6, or 9. – Jason DeVito Aug 16 '10 at 2:20
You can also use the rational root test on the polynomial equation $x^2-2=0$ (whose solutions are $\pm \sqrt{2}$). If this equation were to have a rational solution $\frac{a}{b}$, then $a \vert 2$ and $b \vert 1$, hence $\frac{a}{b}\in \{\pm 1, \pm 2\}$. However, it's straightforward to check that none of $1,-1,2,-2$ satisfy the equation $x^2-2=0$. Therefore the equation has no rational roots and $\sqrt{2}$ is irrational.
-
There is also a proof of this theorem that uses the well ordering property of the set of positive integers, that is in a non empty set of positive integers there is always a least element. The proof follows the approach of proof by contradiction but uses the well ordering principle to find the contradiction :) -
Let us assume $\sqrt{2}$ is rational, hence it can be written down in the form $\sqrt{2}=a/b$ assuming that both $a$ and $b$ are positive integers in that case if we look at the set $S = \{k\sqrt{2} \mid k, k\sqrt{2}\text{ are integers}\}$ we find that it's a non empty set of positive integers, it's non empty because $a = b\sqrt{2}$ is in the above set. Now using the Well ordering principle we know that every set of positive integers which is non-empty has a minimum element, we assume that smallest element to be $s$ and let it equal to $s =t\sqrt{2}$. Now an interesting thing happens if we take the difference between the following quantities $s\sqrt{2} - s = (s-t)\sqrt{2} = s(\sqrt{2} - 1)$ which is a smaller element than $s$ itself, hence contradicting the very existence of $s$ being the smallest element. Hence we find that $\sqrt{2}$ is irrational.
I know the proof but I am still amazed at how the author came up with the set assumption. Sometimes such assumptions make you feel kinda dumb :). If anyone has some insight regarding how to come up with such assumptions kindly post your answer in the comment, otherwise I would just assume that it was a workaround.
-
Here are some of my favorite (sketches) of proofs for the irrationality of $\sqrt{2}$.
• Using Newton's method to approximate roots of the polynomial $f(x) = x^2 - 2$, then showing that the sequence does not converge to a rational number.
• Proof by contradiction, assume that $\sqrt{2} = \frac{n}{m}$ for some $n,m \in \mathbb{Z}$ with $m \neq 0$, then $2m^{2} = n^2$, hence $n$ must be even and we can let $n = 2k$ for some $k \in \mathbb{Z}$, but then $m^2 = 2k^2$ will also be even, which is impossible if $\frac{n}{m}$ is reduced. Therefore, $\sqrt{2}$ cannot be expressed as a ratio of integers.
• Since $f(x) = x^2 -2$ is irreducible over $\mathbb{Q}[x]$, its roots must lie in some finite extension field $\mathbb{Q}(\sqrt{2})$ over the rationals.
[Reposted from closed topicProve the square root of 2 is irrational
-
It would help to give further details about the first and third methods. I know a lot about these topics yet I cannot be sure precisely what you have in mind. – Bill Dubuque Nov 7 '12 at 20:15
Another one that is understandable by high schoolers and below.
We will use the following lemma:
If $n$ is an integer, $n^2$ is even (resp. odd) iff $n$ is even (resp. odd).
For the high-schoolers, the proof is about writing $(2k)^2 = 2(2k^2)$ and $(2k+1)^2=2(2k^2+2k)+1$ ...
Now, assume $\sqrt 2 = \frac{a}{b}$ with $a$ and $b$ strictly positive integers.
Then $a^2=2b^2$, $\implies a^2$ is even ($=2b^2$), $\implies a$ is even (from the lemma), $\implies a=2a_1$ with $a_1 \in \mathbb N^*$, $\implies b^2=2a_1^2$.
Repeat with $b$ to find that $b=2b_1$ with $b_1 \in \mathbb N^*$ and $(a_1,b_1)$ verifies $a_1^2=2b_1^2$.
By repeating these two steps, we build two sequences $(a_n)_{n\in \mathbb N}$ and $(b_n)_{n\in \mathbb N}$ with values in $\mathbb N^*$ and strictly decreasing, which is impossible, ergo $\sqrt{2}$ is irrational.
(Here of course we use the well-ordering principle which most high schoolders would not know about, but the intuition that the sequence would hit $0$ after at most $a_0=a$ steps is easy to get).
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9562661647796631, "perplexity": 213.30953280293033}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802770815.80/warc/CC-MAIN-20141217075250-00006-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://mathhelpforum.com/pre-calculus/106104-really-hard-word-problem.html | # Math Help - Really hard word problem
1. ## Really hard word problem
In a financial arrangement, you are promised $900 the first day and each day after that you will receive 75% of the previous day's amount. When one day's amount drops below$1, you stop getting paid from that day on. What day is the first day you receive no payment and what is your total income? Use a formula for the nth partial sum of a geometric sequence.
2. You should know the formulas dealing with geometric sequences and series. First you need to find the term where a_n < 1. You have your initial value and common ratio so you can just plug the info into the sequence formula. Then once you know the ending term use the geometric series formula to solve for the sum.
You know what formulas I mean right?
3. Originally Posted by Jameson
You should know the formulas dealing with geometric sequences and series. First you need to find the term where a_n < 1. You have your initial value and common ratio so you can just plug the info into the sequence formula. Then once you know the ending term use the geometric series formula to solve for the sum.
You know what formulas I mean right?
yea i know what you mean. but it just really hard when i do word problems. like usually different problems ask me they already give me a1 and r and i just cant figure out the terms. i just need help. like i know the form is An=A1(r)^(n-1) but i just dont see what you see.
4. Originally Posted by thepride
yea i know what you mean. but it just really hard when i do word problems. like usually different problems ask me they already give me a1 and r and i just cant figure out the terms. i just need help. like i know the form is An=A1(r)^(n-1) but i just dont see what you see.
It's ok. I'll walk you through it.
You have the right formula. All you need to do is figure out a_1 and r. a_1 is the starting value so this should be easy to figure out since there are very few numbers. Now for the ratio, you know it's decreasing each time by 75%, but written in decimal form the ratio is .75. You want to find for what n you get a_n is 1 or less than one. So let a_n=1, plug in the other info I just wrote about and solve for n.
Does that make more sense?
5. Originally Posted by Jameson
It's ok. I'll walk you through it.
You have the right formula. All you need to do is figure out a_1 and r. a_1 is the starting value so this should be easy to figure out since there are very few numbers. Now for the ratio, you know it's decreasing each time by 75%, but written in decimal form the ratio is .75. You want to find for what n you get a_n is 1 or less than one. So let a_n=1, plug in the other info I just wrote about and solve for n.
Does that make more sense?
ok sorry it took so long for me to get back to ya....this website is godly slow.
ok this is what i got from what you explain don't laugh if im not anywhere close.
a1=900 because thats what he gets on the first day correct?
r= 75%=.75
an=900*.75^(n-1)
what do you think??
6. What more help do you need? Jameson told you, "You want to find for what n you get a_n is 1 or less than one". That is, you want to find n so that $900(.75)^n\le 1$ or, equivalently, $(.75)^n\le \frac{1}{900}= 0.00111...$. You can find n by using logarithms or just by calculating $(.75)^n$ for different n. (Start around n= 20.)
7. i dont know if i did this right but this is what i did but it not the correct answer as the back of the book.
900(.75)^20 less than or equal to 1
i got 2.85 less than or equal to 1
did i do this right.....what is the correct answer? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8903153538703918, "perplexity": 396.10098919569265}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375097710.59/warc/CC-MAIN-20150627031817-00232-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://mathematics.huji.ac.il/event/analysis-seminar-manuel-friedrich-m%C3%BCnster?delta=0 | # Analysis Seminar: Manuel Friedrich (Münster) — Emergence of rigid polycrystals from atomistic systems
We investigate the emergence of rigid polycrystalline structures from atomistic particle systems. The atomic interaction is governed by a suitably normalized pair interaction energy, where the `sticky disk’ interaction potential models the atoms as hard spheres that interact when they are tangential. The discrete energy is frame invariant and no underlying reference lattice on the atomistic configurations is assumed. By means of Gamma-convergence, we characterize the asymptotic behavior of configurations with finite surface energy scaling in the infinite particle limit. The effective continuum theory is described in terms of a piecewise constant field delineating the local orientation and micro-translation of the configuration. The limiting energy is local and concentrated on the grain boundaries, i.e., on the boundaries of the zones where the underlying microscopic configuration has constant parameters. The corresponding surface energy density depends on the relative orientation of the two grains, their microscopic translation misfit, and the normal to the interface. Joint work with Leonard Kreutz (Münster) and Bernd Schmidt (Augsburg).
## Date:
Wed, 07/04/2021 - 12:00 to 13:00 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8329155445098877, "perplexity": 1390.0442917489854}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989526.42/warc/CC-MAIN-20210514121902-20210514151902-00204.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/134352-iven-any-group-g-subset-w-let-w-smallest-subgroup-g-contai.html | # Math Help - iven any group G and a subset W, let < W > be the smallest subgroup of G which contai
1. ## iven any group G and a subset W, let < W > be the smallest subgroup of G which contai
Given any group G and a subset W, let < W > be the smallest subgroup of G which contains W?
A) prove that there is such a subgroup < W > in G. ( < W > is called the subgroup generated by W.)
B) If gwg ^ -1 € W for all g € G, w € W, prove that < W > is a normal subgroup of G.
C) Now, let U={xyx^ -1y^-1 | x, y € G}. Prove that < U > is normal in G.
2. Originally Posted by snick
Given any group G and a subset W, let < W > be the smallest subgroup of G which contains W?
A) prove that there is such a subgroup < W > in G. ( < W > is called the subgroup generated by W.)
B) If gwg ^ -1 € W for all g € G, w € W, prove that < W > is a normal subgroup of G.
C) Now, let U={xyx^ -1y^-1 | x, y € G}. Prove that < U > is normal in G.
A) Let $\mathcal{M}=\left\{H:W\subseteq H\leqslant G\right\}$. Define $K=\bigcap_{M\in\mathcal{M}}M$
B) What definition do you use?
C) What have you tried? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9752945899963379, "perplexity": 1740.6258991893803}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701158601.61/warc/CC-MAIN-20160205193918-00284-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://brilliant.org/problems/jump-frog-jump/ | # Jump frog, jump!
An infinite line of stepping stones stretches out into an infinitely large lake.
A frog starts on the second stone from the shore.
Every second, he takes a jump to a neighboring stone. He has a 60% chance of jumping one stone closer to the shore and a 40% chance of jumping one stone further away from the shore.
What is the expected value for the number of jumps he will take before reaching the first stone (the one closest to the shore)?
Other Expected Value Quizzes
× | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9626933336257935, "perplexity": 879.9827395296539}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156613.38/warc/CC-MAIN-20180920195131-20180920215531-00270.warc.gz"} |
http://math.stackexchange.com/questions/284074/hmm-as-special-case-of-mrf?answertab=votes | # HMM as special case of MRF
I have learned that any Hidden Markov Model (HMM) can be described as a special case of a Markov Random Field (MRF) model.
However, AFAIK, the dependencies in a HMM are directed, while the dependencies in a MRF are undirected.
So, how can I represent a HMM as an MRF? How can I represent the potential functions of the MRF, as a function of the transition probabilities of the HMM?
Suppose we have an HMM, where $x_t$ are the observations and $z_t$ are the latent states:
$p(x_1,\ldots,x_T,z_1,\ldots,z_T) = p(z_1)\prod_{t=1}^Tp(z_{t+1}|z_t)p(x_t|z_t)$
My conjecture was that the equivalent MRF is a linear graph, with variables $z_1 \ldots z_T$, and the following potential functions:
$\psi_{\{z_t\}} =$
• $p(z_1) * p(x_1 | z_1)$ $(t == 1)$
• $p(x_t | z_t)$ $(t>1)$
And:
$\psi_{\{z_t,z_{t+1}\}} = p(z_{t+1} | z_t)$
Is this correct?
-
Hmm is a Markov process with partial observations. Markov processes are special cases of mrf. – Ilya Jan 22 '13 at 7:22
In general, any directed graphical model can be converted into an undirected graphical model (though the converse is not true). They're just two different ways to express a joint probability distribution. An excellent review of graphical models is given here, and some discussion of converting directed to undirected models is given in section 2.5. The critical step involves connecting the parents of every child node together, which is cheekily referred to as "moralization". However, in an HMM, each node in the directed model has only one parent, so no moralization is necessary (in other words, statisticians consider single parents perfectly moral).
The actual probabilities in an HMM are given by the following, where $x_t$ are the observations and $z_t$ are the latent states:
$p(x_1,\ldots,x_T,z_1,\ldots,z_T) = p(z_1)\prod_{t=1}^Tp(z_{t+1}|z_t)p(x_t|z_t)$
So if we write down an MRF with a single-node clique for $z_1$ with $\psi_{\{z_1\}} = p(z_1)$, cliques for $\{z_{t},z_{t+1}\}$ with $\psi_{\{z_{t},z_{t+1}\}} = p(z_{t+1}|z_t)$ and cliques for $\{z_{t},x_{t}\}$ with $\psi_{\{z_{t},x_{t}\}} = p(x_{t}|z_t)$, this probability distribution is clearly the same as the one above for an HMM.
One advantage of directed graphical models over their undirected cousins is that they explicitly model how different variables change when manipulated. If we fix the value of one node in our directed graphical model, the graph structure tells us which downstream variables will be affected. This information is thrown out when converting to an undirected model. If, however, we are only interested in passive observation rather than manipulation, the two describe exactly the same distribution over data.
Another advantage is that directed graphical models are generally much easier to sample from, because there is a clear order to how data are generated. In undirected graphical models, one usually has to run a Markov chain to convergence, for instance by Gibbs sampling, which can be very slow.
-
Thank you, this makes sense, but can you please look at my conjecture, that I added to the question, and tell me if it is also a possible solution? – Erel Segal Halevi Jan 22 '13 at 10:42
What you wrote down is a perfectly well-defined MRF, but it's not the MRF that corresponds to that HMM. An MRF representation of an HMM should assign a probability to the same set of variables as the directed model, that is, $z_t$ and $x_t$. What you gave only defines a probability over $z_t$. – David Pfau Jan 22 '13 at 18:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.916643500328064, "perplexity": 441.20485593890305}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1412037663218.28/warc/CC-MAIN-20140930004103-00305-ip-10-234-18-248.ec2.internal.warc.gz"} |
https://eprints.utas.edu.au/3887/ | # Two implications of common models of microbial growth
Brown, SC 2007 , 'Two implications of common models of microbial growth' , ANZIAM Journal, vol. 49 , C230-C242 .
Preview
PDF
## Abstract
Analysis of a generalised growth equation shows that both the maximum growth rate of a microbial culture and the duration of the lag phase are related to each other and to the maximum growth. Similar relationships apply to growth expressions, such as the logistic and Gompertz models, that are special cases of the generalised model. Moreover, the same relationships are observed qualitatively in measurements of the growth of Salmonella species. These results may allow the characterisation of microbial growth with fewer parameters than is usually the case and imply the likelihood of a fundamental physiological interdependence between maximum growth rate, the duration of the lag time and the maximum growth.
Item Type: Article Brown, SC ANZIAM Journal Cambridge University Press 1446-1811 Open Journal Systemshttp://pkp.sfu.ca/?q=ojs View statistics for this item | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9317789077758789, "perplexity": 1508.6290030212838}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178375439.77/warc/CC-MAIN-20210308112849-20210308142849-00626.warc.gz"} |
http://math.stackexchange.com/questions/454157/what-is-this-expression | # What is this expression?
I saw this nice relief on the University of Warsaw's library building, but I'm left wondering what the line beneath $\pi$ and above Collatz is saying. I'm not familiar with the arrow notation.
Can anyone identify it or give a reference for it?
-
That is the long exact sequence of homology groups of a pair $(X,A)$ of topological spaces. – Stefan Hamcke Jul 28 '13 at 15:54
Looks like Mayer-Vietoris en.wikipedia.org/wiki/Mayer%E2%80%93Vietoris_sequence to me – BlackAdder Jul 28 '13 at 15:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8779045343399048, "perplexity": 936.9004321020282}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931008520.8/warc/CC-MAIN-20141125155648-00198-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/proving-a-function-is-differetiable-in-r.183747/ | # Proving a function is differetiable in R
1. Sep 9, 2007
### Simfish
f(x) is a function with...
$$| f(x) - f(y) | \leq |x-y|^2 \forall x,y \in \Re$$
(a) prove differentiability in R, find f'
(b) prove f continuous
====
my steps;
(a) $$\frac{| f(x) - f(y) |}{|x-y|} \leq |x-y|$$
Then by the definition of differentiability as stated in Apostol "Mathematica Analysis pg. 104, f is differentiable if the limit of the function as x -> y exists.
So by the inequality, as x -> y, we know that the limit is bounded and therefore must exist. The value of the limit is simply |x-y|, so is that always the derivative of the function? (or could it be |x|)? Since the derivative, after all, must always be positive?
(b) differentiability implies continuity
2. Sep 10, 2007
### SiddharthM
Using epsilon-delta definition of limit, let epsilon=delta to show the limit of f(x)-f(y)/(x-y) is zero. that is as x goes to y.
derivatives aren't always positive. consider y=-x.
(-1)^n n a natural number is bounded but has no limit.
Last edited: Sep 10, 2007
Similar Discussions: Proving a function is differetiable in R | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9879316687583923, "perplexity": 1293.4538226417612}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463610374.3/warc/CC-MAIN-20170528181608-20170528201608-00230.warc.gz"} |
https://mathhelpboards.com/threads/positive-definite-matrices.5857/#post-26675 | # Positive definite matrices
#### Fernando Revilla
##### Well-known member
MHB Math Helper
Jan 29, 2012
661
I quote a question from Yahoo! Answers
1. Prove that if X ∈ R^(d×n) then XX^T and X^TX are both positive semidefinite.
6. Prove that if X ∈ R^(d×n) has rank d, then XX^T is positive definite (invertible).
I have given a link to the topic there so the OP can see my response.
#### Fernando Revilla
##### Well-known member
MHB Math Helper
Jan 29, 2012
661
For all $x\in\mathbb{R}^{d\times 1}:$
$$x^T(XX^T)x=(X^Tx)^T(X^Tx)=(y_1,\ldots,y_n) \begin{pmatrix}{y_1}\\{\vdots}\\{y_n}\end{pmatrix}=y_1^2+\ldots+y_n^2\geq 0$$
which implies $XX^T$ is positive semidefinite (or positive definite). Similar arguments for $X^TX$.
If $\text{rank }X=d$, then $\text{rank }(XX^T)=\text{rank }X=d$, which implies $XX^T$ is invertible. This means that $XX^T$ is congruent to a matrix $\text{diag }(\alpha_1,\ldots,\alpha_d)$ with $\alpha_i>0$ for all $i$, as a consequence $XX^T$ is positive definite
#### MrJava
##### New member
Aug 3, 2013
7
When I try to solve the case for XTX I get stuck at the following:
xT(XTX)x = xTXTXx = (Xx)TXx
Please kindly guide me next step.
#### Fernando Revilla
##### Well-known member
MHB Math Helper
Jan 29, 2012
661
When I try to solve the case for XTX I get stuck at the following: xT(XTX)x = xTXTXx = (Xx)TXx
Right. Now, the difference is that $x\in \mathbb{R}^{n\times 1}$ instead of $\mathbb{R}^{d\times 1}.$ So, for all $x\in \mathbb{R}^{n\times 1}$
$$(Xx)^T(Xx)=(w_1,\ldots,w_d) \begin{pmatrix}{w_1}\\{\vdots}\\{w_d}\end{pmatrix} =w_1^2+\ldots+w_d^2\geq 0$$
#### MrJava
##### New member
Aug 3, 2013
7
Right. Now, the difference is that $x\in \mathbb{R}^{n\times 1}$ instead of $\mathbb{R}^{d\times 1}.$ So, for all $x\in \mathbb{R}^{n\times 1}$
$$(Xx)^T(Xx)=(w_1,\ldots,w_d) \begin{pmatrix}{w_1}\\{\vdots}\\{w_d}\end{pmatrix} =w_1^2+\ldots+w_d^2\geq 0$$
Ok I get the point, thank you. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.951434314250946, "perplexity": 2321.480421365366}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104683683.99/warc/CC-MAIN-20220707033101-20220707063101-00619.warc.gz"} |
https://www.physicsforums.com/threads/sequence-satisfying-a-condition-for-all-n.339493/ | # Homework Help: Sequence satisfying a condition for all n
1. Sep 22, 2009
### StarTiger
1. The problem statement, all variables and given/known data
Suppose that a sequence {s_n} of positive numbers satisfies the condition s_(n+1) > αs_n for all n where α > 1. Show that s_n → ∞
My teacher mentioned something about making it into a geometric sequence and taking the log. I'm just confused.
2. Relevant equations
3. The attempt at a solution
2. Sep 22, 2009
### LCKurtz
You can start with s2 > as1. Now what about s3? Can you compare it to s2 and s1? Continue...
3. Sep 22, 2009
### HallsofIvy
So $s_2> a s_n$, $s_3> a s_2> a(a s_1)= a^2 s_1$, $s_4> a s_3> a(a^2 s_1)= a^3 s_1$. So $s_n>$ a to what power times $s_1$? What does that have to do with a "geometric sequence"?
4. Sep 22, 2009
### fmam3
If $$(s_n)$$ is a sequence and the limit $$\lim_{n \to \infty}|s_{n+1} / {s_n}| = L$$ exists and $$L < 1$$, then $$\lim s_n$$ converges. If not, what do you think happens? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9338323473930359, "perplexity": 1388.9412523130034}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676590901.10/warc/CC-MAIN-20180719125339-20180719145339-00186.warc.gz"} |
https://socratic.org/questions/5a40cfb97c014951ff09988f | Chemistry
Topics
# Question 9988f
Dec 26, 2017
Here's what I got.
#### Explanation:
The idea here is that the nitrite anion, ${\text{NO}}_{2}^{-}$, acts as a weak base in aqueous solution, so right from the start, you should expect to find
$\text{pH" > 7" " and " " "pOH} < 7$
The nitrite anions are delivered to the solution by the soluble sodium nitrite in a $1 : 1$ mole ratio, so you know that you start with
["NO"_ 2^(-)]_0 = "0.125 M"
Now, when the nitrite anion is present in aqueous solution, the following equilibrium is established
${\text{NO"_ (2(aq))^(-) + "H"_ 2"O"_ ((l)) rightleftharpoons "HNO"_ (2(aq)) + "OH}}_{\left(a q\right)}^{-}$
The base dissociation constant, ${K}_{b}$, of the nitrite anion can be calculated by using the fact that an aqueous solution at ${25}^{\circ} \text{C}$ has
$\textcolor{b l u e}{\underline{\textcolor{b l a c k}{{K}_{a} \cdot {K}_{b} = 1 \cdot {10}^{- 14}}}}$
In your case, the acid dissociation of the nitrous acid, ${\text{HNO}}_{2}$, is equal to $6.5 \cdot {10}^{- 4}$, so you can say that the nitrite anion will have
${K}_{b} = \frac{1 \cdot {10}^{- 14}}{6.5 \cdot {10}^{- 4}} = 1.54 \cdot {10}^{- 11}$
Now, by definition, the base dissociation constant is equal to
${K}_{b} = \left(\left[{\text{HNO"_2] * ["OH"^(-)])/(["NO}}_{2}^{-}\right]\right)$
Notice that the nitrite anions react in $1 : 1$ mole ratios to produce nitrous acid and hydroxide anions, so if you say that $x$ $\text{M}$ represents the equilibrium concentrations of the two products, you can say that the equilibrium concentration of the nitrite anions will be equal to
${\left[{\text{NO"_ 2^(-)] = ["NO}}_{2}^{-}\right]}_{0} - x$
This basically means that in order for the reaction to produce $x$ $\text{M}$ of nitrous acid and hydroxide anions, it must consume $x$ $\text{M}$ of nitrite anions.
This means that you have
${K}_{b} = \frac{x \cdot x}{0.125 - x}$
Since the value of the base dissociation constant is so small compared with the initial concentration of the nitrite anions, which means that the equilibrium lies far to the left, you can use the approximation
$0.125 - x \approx 0.125$
You will thus have
${K}_{b} = {x}^{2} / 0.125$
which will get you
x = sqrt( (0.125 * 1.54 * 10^(-11))
$x = 1.39 \cdot {10}^{- 6}$
Since $x$ represents the equilibrium concentration of the hydroxide anions, you can say that you have
["OH"^(-)] = color(darkgreen)(ul(color(black)(1.39 * 10^(-6)color(white)(.)"M")))
Using the fact that an aqueous solution at ${25}^{\circ} \text{C}$ has
color(blue)(ul(color(black)(["H"_3"O"^(+)] * ["OH"^(-)] = 1 * 10^(-14)color(white)(.)"M"^2)))
you can say that this solution will have
["H"_3"O"^(+)] = (1 * 10^(-14) "M"^color(red)(cancel(color(black)(2))))/(1.39 * 10^(-6)color(red)(cancel(color(black)("M")))) = color(darkgreen)(ul(color(black)(7.19 * 10^(-9)color(white)(.)"M")))
Consequently, you will have
"pOH" = - log(["OH"^(-)])
$\text{pOH} = - \log \left(1.39 \cdot {10}^{- 6}\right) = \textcolor{\mathrm{da} r k g r e e n}{\underline{\textcolor{b l a c k}{5.86}}}$
and
"pH" = - log (["H"_3"O"^(+)])
$\text{pH} = - \log \left(7.19 \cdot {10}^{- 9}\right)$
$\text{pH} = \textcolor{\mathrm{da} r k g r e e n}{\underline{\textcolor{b l a c k}{8.14}}}$
Notice that you get the same result by using
$\textcolor{b l u e}{\underline{\textcolor{b l a c k}{\text{pH + pOH} = 14}}}$
The equilibrium concentrations are rounded to three sig figs, but I'll leave the $\text{pH}$ and the $\text{pOH}$ of the solution rounded to two decimal places instead of three.
As predicted, the $\text{pH}$ of the solution is indeed $> 7$, which is consistent with the fact that the nitrite anion acts as a weak base in aqueous solution. You can also say that sodium nitrite is a basic salt.
$\textcolor{w h i t e}{\frac{a}{a}}$
A MORE ACCURATE APPROACH
An interesting thing to keep in mind here is that pure water already contains hydronium and hydroxide anions $\to$ think the auto-ionization of water here.
At ${25}^{\circ} \text{C}$, pure water has
color(blue)(ul(color(black)(["H"_3"O"^(+)] = ["OH"^(-)] = 1 * 10^(-7)color(white)(.)"M")))
Now, notice that the equilibrium concentration of hydroxide anions that we calculated
["OH"^(-)] = 1.39 * 10^(-6)color(white)(.)"M"
is actually quite small and comparable to the initial concentration of hydroxide anions, i.e. $1 \cdot {10}^{- 7}$ $\text{M}$.
To account for this, you can use the fact that the solution already contains hydroxide anions when you dissolve the salt. So if you use $x$ $\text{M}$ to represent the equilibrium concentration of nitrous acid, you can say that the equilibrium concentration of hydroxide anions will be
$1 \cdot {10}^{- 7} + x$
This means that the concentration of hydroxide anions starts at $1 \cdot {10}^{- 7}$ in pure water and increases by $x$ $\text{M}$ when $x$ $\text{M}$ of nitrite anions react.
The base dissociation constant will now be
${K}_{b} = \frac{x \cdot \left(1 \cdot {10}^{- 7} + x\right)}{0.125 - x}$
Using the same approximation as before, you will end up with
${x}^{2} + 1 \cdot {10}^{- 7} \cdot x - 0.125 \cdot 1.54 \cdot {10}^{- 11} = 0$
This quadratic equation will produce two solutions, one positive and one negative. Since we're looking for concentration here, you can discard the negative solution to say that
$x = 1.34 \cdot {10}^{- 6} \textcolor{w h i t e}{.} \text{M}$
This time, you will have
["OH"^(-)] = 1.34 * 10^(-6)color(white)(.)"M"
["H"_3"O"^(+)] = 7.46 * 10^(-9)color(white)(.)"M"
$\text{pOH} = 5.87$
$\text{pH} = 8.13$
Now, because the percent error that you get by disregarding the initial concentration of hydroxide anions is < color(red)(5%)
"% error" = (1.39 * color(red)(cancel(color(black)(10^(-6)color(white)(.)"M"))) - 1.34 * color(red)(cancel(color(black)(10^(-6)color(white)(.)"M"))))/(1.34 * color(red)(cancel(color(black)(10^(-6)color(white)(.)"M")))) * 100% = 3.73% < color(red)(5%)#
you can use the values that you got without accounting for the initial concentration of the hydroxide anions.
##### Impact of this question
464 views around the world | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 64, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9511942863464355, "perplexity": 1260.7978266993068}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153860.57/warc/CC-MAIN-20210729140649-20210729170649-00585.warc.gz"} |
http://mathhelpforum.com/discrete-math/133029-another-derivation.html | # Math Help - another derivation
1. ## another derivation
The problem:
Show that the set $\{\neg(p\rightarrow q), \neg(q\rightarrow r)\}$ is inconsistent.
The hint is to show that $\neg(q\rightarrow r)\vdash q$ and $\neg(p \rightarrow q)\vdash \neg q$
And of course the hint totally makes sense but I just cannot do it! I feel like I've played around all day with the three axioms and MP and tried using the deduction theorem and deriving things from the empty set and trying to reverse engineer the derivation and UGH! I don't understand how to do these things, is there like a method to figuring it out or is it just reverse engineering and trial-and-error or what?
Any help or hints or anything will be MUCH appreciated!
2. Here is an outline that uses Deduction Theorem (DT). I assume that (not p) is an abbreviation for (p -> false). If negation is a primitive connective (i.e., not an abbreviation for anything), then some steps below need to be adjusted.
Code:
(1) not q assumption
(2) q assumption
(3) false (1), (2), MP
(4) r (3), EFQ
(5) q -> r (2), (4), DT; close assumption (2)
(6) not (q -> r) assumption
(7) false (5), (6), MP
(8) not (not q) (1), (7), DT; close assumption (1)
(9) q (8), DNE (double-negation elimination)
3. We aren't using truth values because with our notation, $\vdash$ means that there is a formal derivation only, while $\models$ means that it's a logical consequence (using truth values). So like the $\vdash$ means you can play around with the shape of the formula only. (At least, in my understanding but obviously I'm a n00b!)
Also how can you assume q and not q? I think the only thing we are allowed to assume is $\neg(q\rightarrow r)$ and then put that in the axioms somehow. Also we don't have the ability to get rid of double negations either.
I'm sorry! Is there something that I'm just not understanding?
4. You understand that we do not have your particular text material.
However, as I read the question here is my take.
$\neg \left( {p \to q} \right) \equiv p \wedge \neg q$
By simplification $\neg q$.
Likewise, $\neg \left( {q \to r} \right) \equiv q \wedge \neg r$
By simplification $q$.
5. We aren't using truth values
I am not talking about truth values either. Everything I wrote above is about derivations.
Since you mentioned the three axioms, MP and the Deduction Theorem, I assume you are using Hlbert-style derivations (see in particular the three axioms -- are they the same that you are using?). Hilbert system is easy to study but hard to use in practice. To construct a complete derivation of the required formula is pretty hard, so I am describing how to build it in theory without actually building it.
In particular, DT is very convenient, but note that this is not a primitive inference rule. It is an algorithm describing how to construct a derivation of $\Gamma\vdash p\to q$ from $\Gamma,p\vdash q$. By looking into the proof, one can actually build the required derivation in every concrete instance, but this is rarely done.
Similarly for DNE: this is not a primitive rule, but for each formula $p$ there is a derivation of $\neg\neg p\to p$. This derivation can be incorporated into bigger derivations.
Also how can you assume q and not q?
By using DT. By assuming $q$ and deriving $r$ I mean deriving $q\vdash r$. Then you apply the DT to obtain a derivation of $\vdash q\to r$, where the assumption $q$ is no longer present (it is said to be closed).
Applications of DT can be nested. You can have the following sequence: (1) assume $p$, (2) assume $q$, (3) derive $p, q\vdash r$, (4) apply DT to get $p\vdash q\to r$, (5) continue this derivation to get $p\vdash r'$, (6) apply DT to get $\vdash p\to r'$.
6. Okay, that makes a lot more sense. I was definitely confused when you said something about not p being the same as p --> false. Those are the axioms that we're using! It's so much easier to find other sources online now that I know the name. For some reason we haven't mentioned the name Hilbert at all in the book, but from what I've looked at so far, that's what it is. It's really helpful to know that you can use nested DTs too. Thank you so much! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 26, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8944306969642639, "perplexity": 423.6087366454497}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500836106.97/warc/CC-MAIN-20140820021356-00161-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://douknow000.com/2020/12/16/physics/ | # The great Maxwell’s equation.
Hey everyone, Welcome to DO u know 000. In this post we are going to see some new people and what they said about light in their laws. This is a joke down there.
Carl Friedrich Gauss : The net electric flux through any hypothetical closed surface is equal to
times the net electric charge within that closed surface”. So, We can see that the magnetic flux cannot be enclosed within a closed surface of any shape. {gauss law}
Michael Faraday : The induced voltage in a circuit is proportional to the rate of change over time of the magnetic flux through that circuit. {faraday’s law}
Emil Lenz : Induced electric current flows in a direction such that the current opposes the change that induced it. {lenz’s law}
André-Marie Ampère : A magnetic field induced by an electric current is, at any point, directly proportional to the product of the current intensity and the length of the current conductor, inversely proportional to the square of the distance between the point and the conductor, and perpendicular to the plane joining the point and the conductor. { ampere’s law }
With all these references A person named James Clerk Maxwell made few equations.
equations by Maxwell
These equations represent how the electromagnetic waves work, I mean the spectrum.
These were made to demonstrate how fluctuating, the electromagnetic is in the vacuum. at all wavelengths.
We can realize that the light we are seeing daily is concluded from those 4 equations.
Here the joke is:
## AND THERE WAS LIGHT…
.
I think you may got the joke!!! Please comment if you understand the joke.
If you got it Read this, If you didn’t get the joke Read this. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9510974287986755, "perplexity": 568.104549281893}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154432.2/warc/CC-MAIN-20210803061431-20210803091431-00390.warc.gz"} |
http://math.stackexchange.com/questions/17119/distinctness-is-maintained-after-adding-some-element-to-all-sets | # Distinctness is maintained after adding some element to all sets
Let $S=\{S_1,S_2,\ldots,S_n\}$ be a set of $n$ distinct subsets with $S_i \subseteq \{1,\ldots,n\}$ for $i=1,\ldots, n$ then $k \in \{1,\ldots,n\}$ exists with $S_i \cup \{k\}$ is distinct for $i=1,\ldots,n$.
I found this in a old problem sheet of mine about sets and graph theory. Is there a elegant solution to this problem?
-
Perhaps you mean $S_i \cup \{k\}$ are distinct? Curly brackets need to be escaped in laTex as otherwise they are interpreted as bracketing an arbitrary laTex expression. – hardmath Jan 11 '11 at 19:03
Thank you, I missed this mistake of mine. – Listing Jan 11 '11 at 19:04
This result follows from Bondy's theorem (in fact, it is equivalent) which states that,
Given $\displaystyle n$ distinct sets $\displaystyle S_1, S_2, \dots, S_n$, each a subset of $\displaystyle \{1,2, \dots, n\}$, there is a set $\displaystyle A \subset \{1,2, \dots, n\}$, with $\displaystyle |A| \leq n-1$ such that the sets $\displaystyle S_i \cap A$ are all distinct.
Pick a $\displaystyle k \notin A$. Then we have that if $\displaystyle S_i \cup \{k\} = S_j \cup \{k\}$, then $\displaystyle (S_i \cup \{k\}) \cap A = (S_j \cup \{k\}) \cap A$. Since $\displaystyle k \notin A$, it follows that $\displaystyle S_i \cap A = S_j \cap A$ contradicting the result of Bondy's theorem.
You can find a short proof and a sketch of an elegant linear algebra proof (originally due to Babai and Frankl) of Bondy's theorem, in the excellent book, Extremal Combinatorics by Stasys Jukna.
-
Good to know... Anyway, it doesn't make a big difference, that's why I deleted it. Your argument is really nice! – t.b. Jan 12 '11 at 5:30
@THeo: Yeah, I suppose it is minor. Thanks! – Aryabhata Jan 12 '11 at 5:36
Thank you for this find :-), Great – Listing Jan 12 '11 at 7:47
THIS IS INCORRECT
Suppose that this wasn't the case. Then for each $k \in \{1, \dots, n\}$ you can find $S_a \subset S_b$ such that $S_b \setminus S_a = \{k\}$.
Consider a directed graph where the $n$ vertices correspond to the sets and there is an edge from $S_a$ to $S_b$ if and only if $S_b \setminus S_a$ consists of exactly one element. Clearly there won't be any loops so it is a forest with at most $n-1$ edges. Now by pigeonhole principle one of the edges must get two $k$s which is impossible.
-
Thank you, now its also clear why the topic is graph theory. – Listing Jan 11 '11 at 19:55
This does not look right. $S = \{ \{1\}, \{1,2\}, \{1,3\}, \{1,2,3\} \}$. There are 4 edges here. Also, you are constructing a directed graph and then claiming no cycles implying $n-1$ edges, which is not true for directed acyclic graphs. If you are constructing an undirected graph, the description seems wrong and the example in this comment shows there can be loops. – Aryabhata Jan 11 '11 at 20:40
You are right, Moron. I was too hasty answering this. – J. J. Jan 12 '11 at 5:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8998963236808777, "perplexity": 347.997760600348}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783395548.53/warc/CC-MAIN-20160624154955-00040-ip-10-164-35-72.ec2.internal.warc.gz"} |
http://mathoverflow.net/revisions/111987/list | 2 made one statement more precise
May I add some information on this topic? Firstly, the space $C(X)$ is not usually a Frechet space---you need some countability condition on the compact subsets of $X$, e.g., it being $\sigma$-compact and locally compact. It is not even complete in the general case---for that you need the condition that it be a $k_R$-space. The dual of $C(X)$ can be identified, with the aid of some abstract locally convex theory and the RRT for compact spaces, with the space of measures on $K$ with compact support (i.e. those arising from measures on some compact subset in the natural way). If $X$ is locally compact, then Bourbaki used the dual of the space of continuous functions with compact support as the {\it definition} of the space of (unbounded) measures on $X$. One can then interpret its members as measures in the classical sense (i.e. as functions defined on a suitable class of sets) by the usual extension methods. I would suggest that the most useful extension of the Riesz representation theorem is the one for bounded, Radon measures on a (completely regular) space. For this one has to go beyond the more common classes of Banach or even locally convex spaces, something which was done by Buck in the 50's. He introduced a locally convex topology on $C^b(X)$ (the bounded, continuous functions) using weighted seminorms for which exactly the kind of representation theorem one would expect and hope for obtains. He did this for locally compact spaces but it was soon extended to the general case, using the methods of mixed topologies and Saks spaces of the polish school. There are many indications that this is the correct structure---the natural versions of the Stone-Weierstrass theorem hold for it and its spectrum (regarding $C^b(X)$ as an algebra) is identifiable with $X$ so that one has a form of the Gelfand-Naimark theory. Further indications of its suitability are that if one considers generalised spectra, i.e., continuous, algebraic homomorphisms into more general algebras then one obtains interesting results and concepts. The important case is where $A$ is $L(H)$ (or, more generally, a von Neumann algebra). One then gets spaces of observables (in the sense of quantum theory) in the case where the underlying topological space is the real line and this provides them in a natural way with a structure which opens a path to a natural and rigorous approach to analysis in the context of spaces of observables---distributions, analytic functions, ...).
1
May I add some information on this topic? Firstly, the space $C(X)$ is not usually a Frechet space---you need some countability condition on the compact subsets of $X$, e.g., it being $\sigma$-compact and locally compact. It is not even complete in the general case---for that you need the condition that it be a $k_R$-space. The dual of $C(X)$ can be identified, with the aid of some abstract locally convex theory and the RRT for compact spaces, with the space of measures on $K$ with compact support (i.e. those arising from measures on some compact subset in the natural way). If $X$ is locally compact, then Bourbaki used the dual of the space of continuous functions with compact support as the {\it definition} of the space of (unbounded) measures on $X$. One can then interpret its members as measures in the classical sense (i.e. as functions defined on a suitable class of sets) by the usual extension methods. I would suggest that the most useful extension of the Riesz representation theorem is the one for bounded, Radon measures on a (completely regular) space. For this one has to go beyond the more common classes of Banach or even locally convex spaces, something which was done by Buck in the 50's. He introduced a locally convex topology on $C^b(X)$ (the bounded, continuous functions) using weighted seminorms for which exactly the kind of representation theorem one would expect and hope for obtains. He did this for locally compact spaces but it was soon extended to the general case, using the methods of mixed topologies and Saks spaces of the polish school. There are many indications that this is the correct structure---the natural versions of the Stone-Weierstrass theorem hold for it and its spectrum (regarding $C^b(X)$ as an algebra) is identifiable with $X$ so that one has a form of the Gelfand-Naimark theory. Further indications of its suitability are that if one considers generalised spectra, i.e., continuous, algebraic homomorphisms into more general algebras then one obtains interesting results and concepts. The important case is where $A$ is $L(H)$ (or, more generally, a von Neumann algebra). One then gets spaces of observables (in the sense of quantum theory) and this provides them in a natural way with a structure which opens a path to a natural and rigorous approach to analysis in the context of spaces of observables---distributions, analytic functions, ...). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9285393357276917, "perplexity": 199.55984205667013}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368702444272/warc/CC-MAIN-20130516110724-00054-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://probabilityexam.wordpress.com/2013/05/18/exam-p-practice-problem-73-wait-time-at-a-busy-restaurant/ | # Exam P Practice Problem 73 – Wait Time at a Busy Restaurant
Both Problem 73-A and Problem 73-B use the following information.
A certain restaurant is very busy in the evening time during the weekend. When customers arrive, they typically have to wait for a table.
When a customer has to wait for a table, the wait time (in minutes) follows a distribution with the following density function.
$\displaystyle f(x)=\frac{1}{1800} \ x, \ \ \ \ \ \ \ \ \ 0
A customer plans to dine at this restaurant on five Saturday evenings during the next 3 months. Assume that the customer will have to wait for a table on each of these evenings.
______________________________________________________________________
Problem 73-A
What is the probability that the minimum wait time for a table during the next 3 months for this customer will be more than half an hour?
$\text{ }$
$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.24$
$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.25$
$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.36$
$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.42$
$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.75$
$\text{ }$
$\text{ }$
$\text{ }$
$\text{ }$
______________________________________________________________________
Problem 73-B
What is the mean of the maximum wait time (in minutes) for a table during the next 3 months for this customer?
$\text{ }$
$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 40.0$
$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 50.5$
$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 51.4$
$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 54.5$
$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 55.4$
______________________________________________________________________
$\text{ }$
$\text{ }$
$\text{ }$
$\text{ }$
$\text{ }$
$\text{ }$
$\text{ }$
$\text{ }$
______________________________________________________________________
______________________________________________________________________
$\copyright \ 2013 \ \ \text{Dan Ma}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 26, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9998151659965515, "perplexity": 39.64186513564028}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105195.16/warc/CC-MAIN-20170818233221-20170819013221-00576.warc.gz"} |
https://www.physicsforums.com/threads/work-energy-theorem-problem.155617/ | # Work-Energy Theorem Problem
1. Feb 10, 2007
### VinceStolen
1. The problem statement, all variables and given/known data
A driver in a car is on a level road traveling at a speed of "v". He puts on the brakes and they lock and skid rather than roll. I have to use the Work-Energy Theorem to give an equation for the stopping distance of the car in terms of "v". the acceleration of gravity "g" and the coefficient of kinetic friction "u(k)" between the tires and the road.
2. Relevant equations
W = EK(f) - EK(i)
3. The attempt at a solution
I attempted to use various formulas I have that use friction and gravity but came up to no success. I am hoping someone else knows what they are doing.
2. Feb 10, 2007
### PhanthomJay
You have identified the work done in your formula. What force does this work? How do you calculate it? What is the definition of work?
3. Feb 10, 2007
### VinceStolen
The frictional force is the force doing this work. So W = -F(friction)*x. And F(friction) = u(k)mg. So -u(k)mg*x = 0 - (1/2)mv^2 ... and solve for x?
4. Feb 10, 2007
### PhanthomJay
Looks good!
5. Feb 10, 2007
### VinceStolen
Thank you so much. You were extremely helpful.
Similar Discussions: Work-Energy Theorem Problem | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8699800968170166, "perplexity": 1174.7007697806355}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814140.9/warc/CC-MAIN-20180222160706-20180222180706-00523.warc.gz"} |
http://arxiv-export-lb.library.cornell.edu/abs/2108.05771 | math.CA
(what is this?)
# Title: Fourier dimension of the cone
Abstract: It is shown that the cone in $\mathbb{R}^{d+1}$ has Fourier dimension $d-1$. This verifies a conjecture of Fraser and Kroon. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9854665398597717, "perplexity": 1218.656194373283}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103947269.55/warc/CC-MAIN-20220701220150-20220702010150-00392.warc.gz"} |
https://brilliant.org/discussions/thread/1xn-in-fractions/ | # $1+x^n$ in Fractions?
Could you calculate
$\large \int \frac{1}{1+x^n} dx$
for every positive integer $n$?
Note by Pepper Mint
3 years, 8 months ago
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
When posting on Brilliant:
• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2
paragraph 1
paragraph 2
[example link](https://brilliant.org)example link
> This is a quote
This is a quote
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.
print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$
Sort by:
Ok......What if someone replaces the + sign with a - sign........Can we solve it and generalize it??
- 3 years, 4 months ago
That is a good point we can use its series expansion... WAIT THE SERIES EXPANSION!!! We can use it to maybe SOLVE THE INTEGRAL (in series form BUT WHO CARES)!!! $\frac { 1 }{ 1+x } =1-x+{ x }^{ 2 }-{ x }^{ 3 }+{ x }^{ 4 }-...=\sum _{ n=0 }^{ \infty }{ { (-1) }^{ n }{ x }^{ n } }$ $\frac { 1 }{ 1+{ x }^{ 2 } } =1-{ x }^{ 2 }+{ x }^{ 4 }-{ x }^{ 6 }+{ x }^{ 8 }-...=\sum _{ n=0 }^{ \infty }{ { (-1) }^{ n }{ x }^{ 2n } }$ $\frac { 1 }{ 1+{ x }^{ 3 } } =1-{ x }^{ 3 }+{ x }^{ 6 }-{ x }^{ 9 }+{ x }^{ 12 }-...=\sum _{ n=0 }^{ \infty }{ { (-1) }^{ n }{ x }^{ 3n } }$ So: $\frac { 1 }{ 1+{ x }^{ k } } =1-{ x }^{ k }+{ x }^{ 2k }-{ x }^{ 3k }+{ x }^{ 4k }-...=\sum _{ n=0 }^{ \infty }{ { (-1) }^{ n }{ x }^{ kn } }$ and...: $\int { \frac { 1 }{ 1+{ x }^{ k } } dx } =\int { (1-{ x }^{ k }+{ x }^{ 2k }-{ x }^{ 3k }+{ x }^{ 4k }-...)dx } =\int { \sum _{ n=0 }^{ \infty }{ { (-1) }^{ n }{ x }^{ kn } } dx }$
Simplifying the integral of the right: (Not including the +C in the integral) $\int { \sum _{ n=0 }^{ \infty }{ { (-1) }^{ n }{ x }^{ kn } } dx } =\sum _{ n=0 }^{ \infty }{ \int { { (-1) }^{ n }{ x }^{ kn }dx } } =\sum _{ n=0 }^{ \infty }{ \frac { { (-1) }^{ n }{ x }^{ kn+1 } }{ kn+1 } } =x-\frac { { x }^{ k+1 } }{ k+1 } +\frac { { x }^{ 2k+1 } }{ 2k+1 } -\frac { { x }^{ 3k+1 } }{ 3k+1 } +\frac { { x }^{ 4k+1 } }{ 4k+1 } -...$ ... which is so far all the work that can be done, without using any special functions.
Thus: $\int { \frac { 1 }{ 1+{ x }^{ k } } dx } =\sum _{ n=0 }^{ \infty }{ { \frac { { (-1) }^{ n }{ x }^{ kn+1 } }{ kn+1 } } }$ Done
- 3 years, 4 months ago
Just glancing at it and some solutions computed with WolframAlpha, it looks like you have to use partial fractions to decompose it and then integrate term-by-term, which makes me unsure about whether or not a closed-form solution exists...
- 3 years, 4 months ago
Integrate term by term... that is correct. In fact, the terms you need to integrate are actually very surprising.. x^n and -x^k! The answer is that $\int { \frac { 1 }{ 1+{ x }^{ k } } dx } =\sum _{ n=0 }^{ \infty }{ { \frac { { (-1) }^{ n }{ x }^{ kn+1 } }{ kn+1 } } }$!
- 3 years, 4 months ago
You can see the power rule in the summation!
- 3 years, 4 months ago
Anyway, we know that: $\int { \frac { 1 }{ 1+{ x } } } dx=\ln { (1+x) } +C$ and $\int { \frac { 1 }{ 1+{ x }^{ 2 } } } dx=\arctan { x } +C$ but $\int { \frac { 1 }{ 1+{ x }^{ 3 } } } dx$ is a mess...
(Just in case you don't believe me): $\int { \frac { 1 }{ 1+{ x }^{ 3 } } } dx=-\frac { \ln { |{ x }^{ 2 }-x+1| } -2(\ln { |x+1| } +\sqrt { 3 } \arctan { (\frac { 2x-1 }{ \sqrt { 3 } } ) } ) }{ 6 } +C$
- 3 years, 4 months ago
@Pepper Mint Well, we can solve this sort of definite integral ranging from 0 to infinity.........this is simply using Beta function....!!!
- 3 years, 1 month ago
But it won't help solve for the indefinite integral. hmm
- 3 years, 1 month ago
@Pepper Mint Have a look at this paper...
- 2 years, 4 months ago | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 23, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9940370321273804, "perplexity": 2680.7964125957405}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154878.27/warc/CC-MAIN-20210804142918-20210804172918-00673.warc.gz"} |
https://www.projecteuclid.org/euclid.aoms/1177692625 | ## The Annals of Mathematical Statistics
### Fixed Alternatives and Wald's Formulation of the Noncentral Asymptotic Behavior of the Likelihood Ratio Statistic
T. W. F. Stroud
#### Abstract
Let $X$ be a random vector, taking values in $p$-dimensional Euclidean space $\mathscr{E}^p$ with density $f(x; \theta)$. The parameter $\theta$ belongs to a subset $\Theta$ of a Euclidean space $\mathscr{E}^q$ and is unkown. Let $g$ be a function over the parameter space having continuous first partial derivatives and taking values in $\mathscr{E}^r (r \leqq q)$. To test the hypothesis $g(\theta) = 0$ against the alternative $g(\theta) \neq 0$ using a sample of $n$ independent observations of $X$, one frequently uses the Neyman-Pearson generalized likelihood ratio test statistic $\lambda_n$. The limiting distribution of $-2\ln\lambda_n$ under the null hypothesis, as $n \rightarrow \infty$, was shown by Wilks (1938) to be chi-square with $r$ degrees of freedom (assuming regularity conditions). If $\{\theta_n\}$ is a sequence of alternatives converging to a point of the null hypothesis at the rate $n^{\frac{1}{2}}$, the limiting distribution is noncentral chi-square with noncentrality parameter equal to the limit of $n\lbrack g(\theta_n)\rbrack' \sum^{-1}_g (\theta_n)\lbrack g(\theta_n)\rbrack$, where $\sum_g(\theta)$ is the asymptotic covariance matrix of the quantity $n^{\frac{1}{2}}\lbrack g(\hat{\theta}) - g(\theta)\rbrack$ as $n \rightarrow \infty$ with $\theta$ fixed ($\hat{\theta}$ denoting the maximum-likelihood estimator of $\theta$ based on sample size $n$). This noncentral convergence was first proved by Wald (1943), along with a number of other results, on the basis of some rather severe uniformity conditions. Davidson and Lever (1970) have proved the result using more intuitive assumptions. Feder (1968) has obtained asymptotic noncentral chi-square for the case where both the hypothesis and alternative regions are cones in $\Theta$; this is essentially a generalization of $g(\theta) = 0$ versus $g(\theta) \neq 0$, since the hypothesis $g(\theta) = 0$ is locally equivalent to a hyperplane and $g(\theta) \neq 0$ to its complement. Despite the generality, Feder's assumptions are quite mild compared with Wald's. The result appears in Wald's paper as a special case of a more general statement entitled "Theorem IX." This theorem states that for $\theta \in \Theta$ and $-\infty < t < \infty$ the relationship \begin{equation*}\tag{1.1}P_\theta\lbrack -2 \ln \lambda_n < t\rbrack - P_\theta\lbrack K_n < t\rbrack \rightarrow 0\end{equation*} holds uniformly in $t$ and $\theta$, where $K_n$ has a noncentral chi-square distribution with $r$ degrees of freedom and noncentrality parameter equal to $n\lbrack g(\theta)\rbrack' \sum^{-1}_g (\theta)\lbrack g(\theta)\rbrack$. This formulation of Wald is too strong. It will be shown by counterexample that, if $\theta$ is held fixed while $n \rightarrow \infty$, relationship (1.1) fails to hold uniformly in $t$. The counterexample is that of testing the value of the mean of a normal distribution with unknown mean and variance. Wald's proof of Theorem IX treats two cases separately, case (i) where $\theta_n$ approaches the null hypothesis set at the rate $n^{-\frac{1}{2}}$ or faster, and case (ii) where it does not. The proof of (1.1) in case (i) requires convergence of $\theta_n$ at the rate $n^{-\frac{1}{2}}$ in order that the Taylor series expansion of the logarithm behave nicely. In case (ii) there is no reason at all to believe the distribution of $K_n$ to be a good approximation to that of $-2\ln\lambda_n$. From Wald's paper (page 480, line following (212)) one gets the impression that Wald felt that the statement of uniform convergence of (1.1) in case (ii) was trivial, since pointwise convergence is trivial (because both terms tend to zero for fixed $t$). But, since $K_n$ does not converge in distribution to a random variable in case (ii), there is really no reason why pointwise convergence should imply uniform convergence. In the same paper, Wald (1943) also described a test procedure based only on the unrestricted maximum-likelihood estimator $\hat{\theta}_n$. This procedure rejects for large values of the statistic $Q_n = n\lbrack g(\hat{\theta}_n)\rbrack' \sum^{-1}_g (\hat{\theta}_n)\lbrack g(\hat{\theta}_n)\rbrack.$ Wald claimed in his paper that (1.1) again holds uniformly in $t$ and $\theta$ if $-2\ln\lambda_n$ is replaced by $Q_n$. This claim too is false, in the stated generality, as the same counterexample will demonstrate. Keeping $\theta$ as a fixed alternative while $n \rightarrow \infty$ has the disadvantage that the limiting behavior of each of the quantities $-2\ln\lambda_n, Q_n$ and $K_n$ is degenerate in the sense that the probability mass moves out to infinity with increasing $n$. However, statement (1.1), uniform in $t$ for fixed $\theta$, has meaning here since both $-2\ln\lambda_n$ (or $Q_n$) and $K_n$ may be related to quantities with genuine limiting normal distributions which must be identical or at least very similar in order for (1.1) to be uniform in $t$. The precise result is embodied in a theorem presented in Section 2 of this paper. In Sections 3 and 4 we consider the case of $X$ normally distributed with mean $\mu$ and variance $\sigma^2$, where $-\infty < \mu < \infty, 0 < \sigma_1 < \sigma < \sigma_2$, and the hypothesis to be tested is $\mu = 0$. It is shown in Sections 3 and 4, respectively, that for this problem the relationships $P_\theta\lbrack Q_n < t\rbrack - P_\theta\lbrack K_n < t\rbrack \rightarrow 0$ and $P_\theta\lbrack -2\ln\lambda_n < t\rbrack - P_\theta\lbrack K_n < t\rbrack \rightarrow 0$ fail to be uniform in $t$ when $\theta = (\mu, \sigma)$ is fixed and satisfies $\mu \neq 0, \sigma_1^2 < \sigma^2 < \sigma_2^2 - \mu^2$. The space of values of $\sigma$ has been truncated in order to satisfy Wald's regularity conditions. In the following section boldface letters denote vectors and matrices. The law of the random vector $\mathbf{x}$ is denoted throughout by $\mathscr{L}(\mathbf{x})$. In particular, $\mathscr{N}(\mathbf{\mu}, \mathbf{\Sigma})$ refers to a normal law with mean vector $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}$. By $\mathscr{L}(\mathbf{x}_n) \rightarrow \mathscr{L}(\mathbf{y})$ or $\mathscr{L}(\mathbf{x}_n) \rightarrow \mathscr{N}(\mathbf{\mu}, \mathbf{\Sigma})$ is meant, respectively, that the law of $\mathbf{x}_n$ converges to the law of $\mathbf{y}$ or to the stated normal law, as $n \rightarrow \infty$. The definitions of the Mann-Wald symbols $O_p$ and $o_p$ may be found in Chernoff ((1956), Section 2), as may the statements of some basic results of large-sample theory which are used freely in the proof of the theorem.
#### Article information
Source
Ann. Math. Statist., Volume 43, Number 2 (1972), 447-454.
Dates
First available in Project Euclid: 27 April 2007
https://projecteuclid.org/euclid.aoms/1177692625
Digital Object Identifier
doi:10.1214/aoms/1177692625
Mathematical Reviews number (MathSciNet)
MR307406
Zentralblatt MATH identifier
0238.62023
JSTOR | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9837723970413208, "perplexity": 156.42772557746636}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496664567.4/warc/CC-MAIN-20191112024224-20191112052224-00350.warc.gz"} |
http://orbit.dtu.dk/en/publications/bilinear-relative-equilibria-of-identical-point-vortices(0fd644eb-a474-4344-87c9-d1f7fea3cae5).html | ## Bilinear Relative Equilibria of Identical Point Vortices
Publication: Research - peer-reviewJournal article – Annual report year: 2012
### DOI
A new class of bilinear relative equilibria of identical point vortices in which the vortices are constrained to be on two perpendicular lines, conveniently taken to be the x- and y-axes of a Cartesian coordinate system, is introduced and studied. In the general problem we have m vortices on the y-axis and n on the x-axis. We define generating polynomials q(z) and p(z), respectively, for each set of vortices. A second-order, linear ODE for p(z) given q(z) is derived. Several results relating the general solution of the ODE to relative equilibrium configurations are established. Our strongest result, obtained using Sturm’s comparison theorem, is that if p(z) satisfies the ODE for a given q(z) with its imaginary zeros symmetric relative to the x-axis, then it must have at least n−m+2 simple, real zeros. For m=2 this provides a complete characterization of all zeros, and we study this case in some detail. In particular, we show that, given q(z)=z 2+η 2, where η is real, there is a unique p(z) of degree n, and a unique value of η 2=A n , such that the zeros of q(z) and p(z) form a relative equilibrium of n+2 point vortices. We show that $A_{n} \approx\frac{2}{3}n + \frac{1}{2}$, as n→∞, where the coefficient of n is determined analytically, the next-order term numerically. The paper includes extensive numerical documentation on this family of relative equilibria.
Original language English Journal of Nonlinear Science 22 5 849-885 0938-8974 http://dx.doi.org/10.1007/s00332-012-9129-2 Published - 2012
Citations Web of Science® Times Cited: No match on DOI | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8273418545722961, "perplexity": 1048.2118664596164}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783395679.92/warc/CC-MAIN-20160624154955-00178-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://warwick.ac.uk/fac/sci/masdoc/current/msc-modules/ma916/pg/abc/ | # Approximate Bayesian Calculation
## Brief on ABC
Approximate Bayesian Computation (ABC) aims to tackle problems with intractable likelihood by approximating the posterior distribution without explicit likelihood evaluation. The original ABC rejection sampling algorithm is as follows, as presented in Beaumont (2002).
• Propose $$\theta_0$$ from prior $$p(\theta)$$.
• Compute $$S_{\theta_0}$$
• For $$U \sim U(0,1)$$, accept $$\theta$$
Suppose $$\theta$$ is an unknown parameter of interest with prior $$\pi( \theta)$$ . The algorithm proposes values $$\theta_0$$ from the prior, and accepts them with some probability. Suppose $$S$$ is a summary statistic computed from a data set, and let $$D$$ be the observed value of this summary statistic given the value $$\theta_0$$.
Instead of considering the full likelihood, we now accept proposals with probability proportional to $$\mathbb{ P }( S = D | \theta )$$, that is the proposal $$\theta$$ is accepted if
\begin{equation*}
\mathbb{ P }\left( S = D | \theta \right) > \max_{ \theta' \in \Theta }\left\{ \mathbb{ P }\left( S = D | \theta' \right) \right\} U
\end{equation*}
where $$U$$ is an independent random variable with distribution $$U \sim U(0,1)$$. It can be shown that when $$S$$ is a sufficient statistic for $$\theta$$ the algorithm results in a sample from the posterior distribution.
The algorithm suffers from two main drawbacks:
• Firstly, it can be very difficult to evaluate $$\mathbb{ P }\left( S = D | \theta \right)$$ in practice.
• Secondly, $$\mathbb{ P }\left( S = D | \theta \right)$$ can be very small (even 0 in the case of continuous distributions), resulting in very few accepted proposals.
Instead of considering $$\mathbb{ P }\left( S = D | \theta \right)$$ directly, simulate a sample from the model under $$\theta$$, compute the summary statistic from the simulated data set (denoted $$D_S$$) and consider accept proposals when $$D_S = D$$. This approach overcomes the first problem by removing the need to calculate probabilities explicitly, but still suffers from the second problem since in most situations $$\mathbb{ P }\left( D_S = D | \theta \right)$$ will be very small even for plausible values of $$\theta$$.
The second problem problem can be overcome by introducing a tolerance $$\epsilon > 0$$ , and accepting a proposal whenever $$|| D_S - D || \leq \epsilon$$ in some appropriate norm.
The value of $$\epsilon$$ will have to be fine-tuned on a case-by-case basis, but if an appropriate tolerance can be found then the second problem is also resolved.
## ABC for the SLFV
Implementing ABC for the SLFV model requires specification of a test statistic. A sufficient statistic ensures that a sample generated by ABC is from the exact posterior distribution, provided $$\epsilon$$= 0 (see Beaumont 2002). Unfortunately, there are no known sufficient statistics for the Fleming-Viot process in general, but Tavare (1997) argues that many statistics are suitable for practical inference.
We use an approach which measures homogeneity in the lineages in the sample. To make this precise, we divide the sample of n lineages into $$\lfloor n/2 \rfloor =: n_0$$ disjoint pairs. We label these draws as $$(d_i, m_i )_{ i = 1 }^{ n_0 }$$, where $$d_i$$ denotes the euclidean distance between the $$i^{ \text{th} }$$ pair and similarity score $$m_i$$ given by for a pair $$(x_{i_1}, x_{i_2})$$:
\begin{equation*}
m_i :=
\begin{cases}
1 &\text{ if } \kappa^{ ( i_1 ) } = \kappa^{ ( i_2 ) } \\
0 &\text{ otherwise}
\end{cases}
\end{equation*}
It remains to specify how to pair the individuals. Choosing pairs to cover a wide range of distances is a good approach, which we achieve in a cheap way by pairing individuals uniformly at random.
In order to simulate similar binary vectors $$\mathbf{m}^{\theta}$$ from the SLFV model we derive the probability of identity, $$\psi_{ \theta }(x)$$ of two individuals sampled at separation $$x$$ as follows.
\begin{align*}
0 = &( 1 - \psi_{ \theta }( x ) ) \psi_{ \theta }( x ) \int_{ 0 }^{ \infty } | B_x( r ) \cap B_0( r ) | \int_0^1 u^2 \nu_r( du ) \mu(dr ) \\
&+ \int_{ \mathbb{ T } } \int_0^{ \infty } \int_0^1 \frac{ 2u }{ \pi r^2 } ( | B_0( r ) \cap B_y( r ) | - u | B_0( r ) \cap B_y( r ) \cap B_x( r ) | ) ( \psi_{ \theta }( x - y ) - \psi_{ \theta }( x ) ) \nu_r( du ) \mu( dr ) dy \\
&- 2m \psi_{ \theta }( x ) \left( \psi_{ \theta }( x ) - \psi_{ \theta }( x ) Q_1(d,P) + ( \psi_{ \theta }( x ) - 1 ) Q_2(d,P) \right)
\end{align*}
We then would numerically solve this non-linear equation in order to get numerical values for the probability of identity.
A degree of freedom to consider is how to pair the original individuals. It would seem that choosing pairs to cover a wide range of distances is a good approach, but for our initial trials we just choose the pairs uniformly at random over the spatial region.
The following algorithm will provide a value $$\theta$$, approximately from the desired distribution given the observed data $$\pi( \theta | \eta)$$.
## ABC Algorithm using Pairing Statistic
Require $$(\mathbf{d},\mathbf{m})=(d_i, m_i )_{ i = 1 }^{ n_0 }$$
1. Sample $$\theta \sim \pi( \theta )$$ (Uniform)
2. Draw vector of binary outcomes $$\mathbf{ m }^{ ( \theta ) } = ( m^{ ( \theta ) }_i )_{ i = 1 }^{ n_0 }$$ from $$\text{Ber}\psi_{ \theta } \left( \mathbf{d} \right)$$
3. Calculate the Hamming distance $$d_H( \mathbf{ m }, \mathbf{ m }^{ (\theta) } ):=$$# of entries $$\mathbf{ m }$$ and $$\mathbf{ m }^{ (\theta)}$$ differ
4. If $$d_H( \mathbf{ m }, \mathbf{ m }^{ (\theta) } )\leq \epsilon$$
Accept $$\theta$$
5. Else
Go to 1
## Inference Results
Using the previous algorithm, ABC was performed on a sample of size 10,000 of two different types, in a torus of side length $$L=10$$, assuming a mutation rate $$\theta=0.05$$ and using a uniform prior on the interval (0,0.1).
In addition, different values for $$\epsilon$$ were tested in order to have some idea of the sensitivity of the approximate likelihood to this variable. Similarly, different refinements for the mesh used in the numerical scheme were tested.
Below: Likelihood function using length interval h=2 for the mesh and $$\epsilon=47\%$$.
Below: Likelihood function using length interval h=2 for the mesh and $$\epsilon=48\%$$.Below: Likelihood function using length interval h=2 for the mesh and $$\epsilon=50\%$$.
Observe that the likelihood is actually "too" sensitive to the value of $$\epsilon$$. For $$\epsilon=47\%$$, most of values were rejected, for $$\epsilon=50\%$$ almost all values were accepted and for $$\epsilon=48\%$$ it seems to produce a feasible likelihood. In addition, the computing time is also quite dependent on this variable. For $$\epsilon=47\%$$, on average it took around 6.5 minutes for obtaining one accepted value, for $$\epsilon=48\%$$ around 1.1 minutes and for $$\epsilon=50\%$$ around 0.2 minutes. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9819071888923645, "perplexity": 653.0801527938189}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864110.40/warc/CC-MAIN-20180621075105-20180621095105-00546.warc.gz"} |
https://math.stackexchange.com/questions/228356/how-to-find-solutions-of-x2-3y2-2 | # How to find solutions of $x^2-3y^2=-2$?
According to MathWorld,
Pentagonal Triangular Number: A number which is simultaneously a pentagonal number $P_n$ and triangular number $T_m$. Such numbers exist when $$\frac{1}{2}n(3n-1)=\frac{1}{2}m(m+1).$$ Completing the square gives $$(6n-1)^2-3(2m+1)^2=-2.$$ Substituting $x=6n-1$ and $y=2m+1$ gives the Pell-like quadratic Diophantine equation $$x^2-3y^2=-2,$$ which has solutions $(x,y)=(5,3),(19,11),(71,41),(265,153), \ldots$.
However, it does not state how these solutions for $(x,y)$ were obtained.
I know that the solution $(5,3)$ can be obtained by observing that $1$ is both a pentagonal and a triangular number.
Does obtaining the other solutions simply involve trial-and-error? Or is there a way to obtain these solutions?
$$\left( \begin{array}{cc} 2 & 3 \\ 1 & 2 \end{array} \right) \left( \begin{array}{c} -1 \\ 1 \end{array} \right) \; = \; \left( \begin{array}{c} 1 \\ 1 \end{array} \right),$$
$$\left( \begin{array}{cc} 2 & 3 \\ 1 & 2 \end{array} \right) \left( \begin{array}{c} 1 \\ 1 \end{array} \right) \; = \; \left( \begin{array}{c} 5 \\ 3 \end{array} \right),$$
$$\left( \begin{array}{cc} 2 & 3 \\ 1 & 2 \end{array} \right) \left( \begin{array}{c} 5 \\ 3 \end{array} \right) \; = \; \left( \begin{array}{c} 19 \\ 11 \end{array} \right),$$
$$\left( \begin{array}{cc} 2 & 3 \\ 1 & 2 \end{array} \right) \left( \begin{array}{c} 19 \\ 11 \end{array} \right) \; = \; \left( \begin{array}{c} 71 \\ 41 \end{array} \right),$$
$$\left( \begin{array}{cc} 2 & 3 \\ 1 & 2 \end{array} \right) \left( \begin{array}{c} 71 \\ 41 \end{array} \right) \; = \; \left( \begin{array}{c} 265 \\ 153 \end{array} \right),$$
$$\left( \begin{array}{cc} 2 & 3 \\ 1 & 2 \end{array} \right) \left( \begin{array}{c} 265 \\ 153 \end{array} \right) \; = \; \left( \begin{array}{c} 989 \\ 571 \end{array} \right),$$
$$\left( \begin{array}{cc} 2 & 3 \\ 1 & 2 \end{array} \right) \left( \begin{array}{c} 989 \\ 571 \end{array} \right) \; = \; \left( \begin{array}{c} 3691 \\ 2131 \end{array} \right),$$
EDIT, March 2016: From the stuff with the matrix above, we can use the Cayley-Hamilton theorem to give separate linear recurrences for $x$ and for $y.$ Just these: $$x_{k+2} = 4 x_{k+1} - x_k,$$ $$y_{k+2} = 4 y_{k+1} - y_k.$$ The $x$ sequence is $$1, 5, 19, 71, 265, 989, 3691, 13775, 51409, 191861, \ldots$$ while the $y$ sequence is $$1, 3, 11, 41, 153, 571, 2131, 7953, 29681, 110771, \ldots$$
Well. The theorem of Lagrange is that all values of the quadratic form (that are primitively represented) occur as output of the neighboring forms method, the same as doing continued fractions, if they are below $\frac{1}{2} \; \sqrt \Delta$ in absolute value, where in this case $\Delta = 12.$ So half the square root of that is $\sqrt 3,$ and $2$ is larger than this. This means that, while $-2$ is permitted to show up by the continued fraction method, it is possible that unexpected representations may occur. However, one may check with Conway's topograph method from The Sensual Quadratic Form and confirm that all appearances of $-2$ are along the "river" itself, meaning the simplest possible collection, as I illustrate with the matrix multiplications above. For your viewing pleasure, the topograph for $x^2 - 3 y^2,$ with a fair amount of detail:
=-=-=-=-=-=-=-=-=-=-=
=-=-=-=-=-=-=-=-=-=-=
Oh, well. The $-2$ at coordinates $(5,3)$ goes in the lower right open space, while the $-2$ at coordinates $(-5,3)$ goes in the lower left open space. If you think about it long enough, each edge in the infinite tree, including the little blue numbered arrow and the value on either side, is an indefinite quadratic form equivalent to $\langle 1,0,-3 \rangle,$ but is also an element in $PSL_2 \mathbb Z$ given by a little 2 by 2 matrix using the two column vectors in green.
Note that the automorph $$\left( \begin{array}{cc} 2 & 3 \\ 1 & 2 \end{array} \right)$$ is visible as a pair of column vectors corresponding once again to $\langle 1,0,-3 \rangle,$ as, indeed, it must.
• Where did you obtain the matrix $\begin{bmatrix} 2 & 3\\ 1 & 2 \end{bmatrix}$ and how come you started off the multiplication with the matrix $\begin{bmatrix} -1 \\ 1 \end{bmatrix}$? Feb 4 '15 at 16:29
• @user130018, that matrix does appear by placing two of the green column vectors side by side. Or for a Pell form $x^2 - n y^2,$ find the fundamental solution $u^2 - n v^2 = 1$ with minimal positive entries, then the matrix is $$\left( \begin{array}{cc} u & nv \\ v & u \end{array} \right)$$ Feb 4 '15 at 18:40
• @user130018, if you do your diagram with that much detail (I'm not so sure they are asking for more than the values this time) you will eventually find that your square matrix is $$\left( \begin{array}{cc} 664 & 975 \\ 585 & 859 \end{array} \right)$$ for your form $3 x^2 + x y - 5 y^2.$ Getting that far with the (green) $x,y$ coordinates is a large effort but can be done by hand. I wrote software, also know many ways to find the matrix. Feb 4 '15 at 18:50
• @user130018, the other book I recommend is John Stillwell, Elements of Number Theory, pages 87-99. Excerpt, not necessarily all those pages, books.google.com/… Page 100 is also nice, it says Conway made a video on this, available from the AMS !! Feb 4 '15 at 19:38
• Feb 4 '15 at 19:45
Suppose that we have found a particular solution of $x^2-3y^2=-2$, say $(x_0,y_0)$. We can then write $$(x_0+y_0\sqrt{3})(x_0-y_0\sqrt{3})=-2.$$ Note that $2^2-3(1^2)=1$. Write this as $$(2+\sqrt{3})(2-\sqrt{3})=1.$$ Combining the two results above, we see that $$(x_0+\sqrt{3}y_0)(2+\sqrt{3})(x_0-\sqrt{3}y_0)(2-\sqrt{3})=-2.$$ Expanding, we get $$[2x_0+3y_0+\sqrt{3}(x_0+2y_0)] [2x_0+3y_0-\sqrt{3}(x_0+2y_0)]=-2.$$ This just says that $$(2x_0+3y_0)^2-3(x_0+2y_0)^3=-2.$$ Put $x_1=2x_0+3y_0$, and $y_1=x_0+2y_0$. We have shown that $x_1^2-3y_1^2=-2$.
In general, once we have found a solution $(x_n,y_n)$ we can find another solution $(x_{n+1},y_{n+1})$ where $$x_{n+1}=2x_n+ 3y_n \qquad\text{and}\qquad y_{n+1}=x_n+2y_n.$$
Remark: The above idea is very old. You may be interested in looking up the Brahmagupta Identity.
• Sorry, I did not notice you had the automorph approach, just starting a different way. Nov 4 '12 at 0:53
If you're a bit familiar with algebraic number theory:
$x^2 - 3y^2$ is the norm of the element $x + y\sqrt{3}$ in $\mathbb{Q}(\sqrt{3})$. Given the obvious element $1 + \sqrt{3}$ with norm $-2$, every other possibility differs by multiplication with an element of norm $1$. Dirichlet's unit theorem characterizes them: all powers of $2 + \sqrt{3}$ (up to $\pm 1$).
So the solutions are given by $\pm x \pm y\sqrt{3} = (1 + \sqrt{3})(2 + \sqrt{3})^n$ for $n \in \mathbb{Z}$.
This is an issue that comes up over and over again. The quadratic form $m^2-3n^2$ happens to be the norm form for the quadratic field $\mathbb{Q}(\sqrt3)$. That is, when you write $z=m+n\sqrt3$ and $\bar z=m-n\sqrt3$, you see that $z\mapsto\bar z$ preserves both multiplication and addition. So $z\mapsto z\bar z$ is also multiplicative, taking integral things in the field to ordinary integers. And it takes the value $\pm1$ on the group of units of the corresponding integer ring $\mathbb{Z}[\sqrt3]$. We know, from the study of Pell’s Equation, or from continued fractions, or from much more advanced methods, that every unit is plus-or-minus a power of the primitive unit $2+\sqrt3$.
So what? If you can only find one of these quadratic integers, $z_0$, whose “norm” $z\bar z$ is equal to $-2$, you can get all the others by multiplying by units. But of course the norm of $1+\sqrt3$ is $-2$, you’ve got your recipe for finding all. So: $(1+\sqrt3)(2+\sqrt3)=5+3\sqrt3$; $(1+\sqrt3)(2+\sqrt3)^2=19+11\sqrt3$, etc.
As an alternative approach which you might like to investigate:
If you write $\sqrt{3}$ as a continued fraction, you get
$$1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+ \cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\cdots}}}}}}}}$$
If you then calculate the partial convergents by stopping the continued fraction after a certain point, you will find that the solutions appear as the numerators and denominators of some of the convergents. It is an interesting exercise to decide which ones.
Here's another approach.
$$X^2-AY^2=B\tag1$$ $$x^2-Ay^2=1\tag2$$
If we know fundamental solution $(a,b/A)$ for $(2)$ and “trivial” solutions $(t,v)$ for $(1)$ then:
$$X_n = \sum_{k=0}^{n}\frac{a^{n-k}b^k\displaystyle\binom{n}{k}\left(\left(\left\lceil\frac{k}{2}\right\rceil -\left\lfloor\frac{k}{2}\right\rfloor\right)v + \left(\left\lceil\frac{k+1}{2}\right\rceil -\left\lfloor\frac{k+1}{2}\right\rfloor\right)t\right)}{A^{\left\lfloor\frac{k}{2}\right\rfloor}}$$
$$Y_n = \sum_{k=0}^n \frac{a^{n-k}b^k\displaystyle\binom{n}{k}\left(\left( \left\lceil\frac{k+1}{2}\right\rceil -\left\lfloor\frac{k+1}{2}\right\rfloor\right)v + \left(\left\lceil\frac{k}{2}\right\rceil - \left\lfloor\frac{k}{2}\right\rfloor\right)t\right)}{A^{\left\lceil\frac{k}{2}\right\rceil}}$$
For $X^2 -3Y^2 = -2$; $t = 1, v = 1, a = 2, b/A = 1$
$$X_n = \sum_{k=0}^n 3^{k-\left\lfloor\frac{k}{2}\right\rfloor}2^{n-k}\binom{n}{k}$$
$$Y_n = \sum_{k=0}^n 3^{k-\left\lceil\frac{k}{2}\right\rceil}2^{n-k}\binom{n}{k}$$
• I've replaced the words 'floor' and 'ceiling' with the corresponding symbols. If you didn't want the symbols, you can always revert back to your original answer. Jan 8 '16 at 5:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.939821183681488, "perplexity": 199.80242993336944}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300244.42/warc/CC-MAIN-20220116210734-20220117000734-00152.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/36125-bxc-not-equal-0-implies-linearly-independent.html | # Math Help - a.(bXc) not equal 0 implies linearly independent?
1. ## a.(bXc) not equal 0 implies linearly independent?
A.(BXC) /= 0, must a,b,c be linearly independent?
2. Originally Posted by szpengchao
A.(BXC) /= 0, must a,b,c be linearly independent?
Do you mean $a \cdot (b \times c) = 0$ or $a \cdot (b \times c) \neq 0$?
Either way if a,b,c lie in the same 2 dimensional plane, then $a \cdot (b \times c) = 0$. Can you see that? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9602168798446655, "perplexity": 4312.752896608462}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131300031.99/warc/CC-MAIN-20150323172140-00227-ip-10-168-14-71.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/141125/bounding-the-level-for-eigenforms-satisfying-a-deformation-condition | # Bounding the level for eigenforms satisfying a deformation condition
Let $k$ be a finite field of char $p \geq 3$. Given an absolutely irreducible, continuous, odd representation $\overline{\rho}: G_\mathbb{Q} \longrightarrow GL_2(k)$ and a deformation condition $D$ for $\overline{\rho}$, let $S(D)$ be the collection of all newforms with associated $p$-adic representation in $D$. If $f \in S(D)$ then is its level bounded? I remember reading somewhere that one might work out the level using local Langlands but do not recall the reference or the argument.
-
For an arbitrary $D$ the answer is "no". For some specific $D$'s the answer is "yes"; it mainly depends on what local condition at $p$ you impose. What sort of $D$ did you have in mind? – David Loeffler Sep 3 '13 at 15:39
Yes, I would require that $\overline\rho$ is ordinary at $p$ and $D$ consists of ordinary lifts with fixed determinant. – unramified Sep 3 '13 at 16:53
Do you also fix the Hodge--Tate weights at $p$? Are you imposing any conditions at primes away from $p$? – David Loeffler Sep 3 '13 at 18:05
The Hodge-Tate weights are $0$ and $k-1$ for some integer $k \geq 2$. Away from $p$ the lifts have fixed determinant (given by $k$.) – unramified Sep 3 '13 at 18:15
Let's bound the level of such an $f$ in two stages. Firstly, let's look at a prime $\ell \ne p$. Here there is a theorem of Livne and (independently) Carayol which says that if $\rho$ is a lifting of $\bar\rho$, the exponent of $\ell$ dividing the Artin conductor of $\rho$ is bounded (it's at most 2 more than the $\ell$-conductor of $\bar\rho$). That leaves just the power of $p$ dividing the level to be controlled; and it's easy to see that if you require $\rho|_{D_p}$ to be upper-triangular, with fixed determinant and Hodge--Tate weights, then the conductors of the characters occuring along the diagonal are bounded above and this gives you a bound on the level of $f$ at $p$.
Under these stipulations, couldn't $\rho|D_p$ be of the form $0 \to \psi \to \rho \to \psi^{-1} \det \rho \to 0$ for any $\psi$ lifting the unramified line in $\overline{\rho}$? Such things don't have bounded conductor. – David Hansen Sep 3 '13 at 20:39
Then $\rho$ won't be ordinary. Depending on your conventions, one of $\psi$ and $\psi^{-1} \det(\rho)$ needs to be an unramified character times a power of cyclotomic. – David Loeffler Sep 4 '13 at 6:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9530714154243469, "perplexity": 399.6792063201613}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131296951.54/warc/CC-MAIN-20150323172136-00186-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/time-and-motion.160250/ | # Time and motion
1. Mar 11, 2007
### petm1
Can we split time from motion? I would think not, Space as we measure it is still expanding at a rate that is faster than light; if this is correct then we need to at least have some understanding as to why and I would think that time in the sense of it being a potential for movement, is expanding ahead of all motion we measure as space. Would this be a fair statement?
Can we call our visible universe a single duration of time? I would think not, unless we take into account the part of time that motion has not yet filled.
2. Mar 12, 2007
### petm1
Would the concept of time being potential movement, remembering that motion already fills part of that potential, fit on the universal scale? Can, this concept be used to explain why the universe could be expanding faster than light? Could dark energy be the potential, both larger and smaller, between that of motion with its speed limit of light, and time's expanding "potential for motion” because relative motion has not reached it yet?
Similar Discussions: Time and motion | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.817709743976593, "perplexity": 665.9729932678974}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105922.73/warc/CC-MAIN-20170819201404-20170819221404-00379.warc.gz"} |
http://math.stackexchange.com/questions/91904/are-these-two-versions-of-the-dominated-convergence-theorem-equivalent?answertab=votes | # Are these two versions of the dominated convergence theorem equivalent?
## First version:
This is from an old script of my professor:
Let $f_n$ be a sequence of integrable functions. Let $f$ be a measurable function such that $$\lim_{n\to\infty} f_n(\omega) = f(\omega)$$ $P$-almost everywhere. Let $g\geq 0$ be a positive function with the property $$\int g\,dP<\infty$$ such that $$|f_n(\omega)|\leq g(\omega)$$ $P$-almost everywhere. Then $f$ is integrable and it is true that $$\lim_{n\to\infty}\int_\Omega f_n\,dP=\int_\Omega f\,dP$$
## Second version:
I read this one on Wikipedia:
Let $f_n$ denote a sequence of real-valued measurable functions on a measure space $(\Omega ,\mathcal{A},P)$. Assume that the sequence converges pointwise to a function $f$ and is dominated by some integrable function $g$ in the sense that $|f_n(x)| \leq g(x)$ for all $x\in \Omega$. Then the limiting function $f$ is integrable and $$\lim_{n\to\infty}\int_\Omega f_n\,dP=\int_\Omega f\,dP$$
The difference here is that $f_n$ are real-valued measurable functions (not integrable as in the version above). Are these versions still equivalent?
Thanks for anyone who enlightens me.
-
Since each $f_n$ satisfies $|f_n(\omega)| \leq g(\omega)$ almost everywhere and $g$ is integrable, it follows from the monotonicity of the integral that each $f_n$ is integrable, so the second version follows from the first.
The first version follows from the second version by throwing away a null-set $N$ containing all points where $f_n$ doesn't converge to $g$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9885143637657166, "perplexity": 72.60296578777444}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644068749.35/warc/CC-MAIN-20150827025428-00039-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/69720/proving-a-trigonometric-identity | # Proving a trigonometric identity
How can one prove the validity of this trigonometric identity? $$2\arccos\sqrt{x} = \frac{\pi }{2}-\arcsin(2x-1)$$
-
Let $\theta=\arccos(\sqrt{x})$. Now use a familiar trigonometric identity for $\cos(2\theta)$. – André Nicolas Oct 4 '11 at 5:51
So it does hold. Thanks! – liberias Oct 4 '11 at 6:08
EDITED in response to valdo's answer.
Your identity $$2\arccos \sqrt{x}=\frac{\pi }{2}-\arcsin (2x-1),\qquad 0\le x\le 1\tag{0},$$ may be rewritten as $$\arcsin (2x-1)=\frac{\pi }{2}-2\arccos \sqrt{x},\qquad 0\le x\le 1\tag{1}.$$
For identity $(1)$ to be valid$^1$ it is enough that
$$\sin \left( \arcsin (2x-1)\right) =\sin \left( \frac{\pi }{2}-2\arccos \sqrt{ x}\right).\tag{2}$$ The LHS of $(2)$ is $$\sin \left( \arcsin (2x-1)\right) =2x-1,\tag{3}$$ and the RHS, $$\sin \left( \frac{\pi }{2}-2\arccos \sqrt{x}\right) =\cos \left( 2\arccos \sqrt{x}\right) =2\cos ^{2}\left( \arccos \sqrt{x}\right) -1\tag{4}.$$ And so, it is enough that we have
$$2x-1 =2\cos ^{2}\left( \arccos \sqrt{x}\right) -1 =2\left( \sqrt{x}\right) ^{2}-1 =2x-1,\qquad x\ge 0,\tag{5}$$
which is indeed an identity. Consequently, all the previous identities are valid e so, also the given identity $(0)$.
--
$^1$ See valdo's detailed explanation in his answer.
-
Correct, but incomplete. Please see my post – valdo Oct 4 '11 at 9:56
@valdo: Thanks! I added a note pointing to your explanation. – Américo Tavares Oct 4 '11 at 10:52
Hints:
1. $\arcsin(x)+\arccos(x)=\frac{\pi}{2}$
2. $\cos\,2x=2\cos^2x-1$
-
I fully agree with Américo Tavares's solution, except one little moment.
If you prove that sin(a) = sin(b) this does not automatically mean that a = b. Strictly speaking the consequence of sin(a) = sin(b) is the following:
a = b + 2πn (n - any integer)
or
a = π - b + 2πn (n - any integer)
The proof would be complete if we prove that only (1) is possible, whereas n=0.
Let's start with LHS. We have arcsin(2x−1). The arcsin function's image is [-π/2, π/2]. It's defined only for x in [0, 1]. For x=0 we have -π/2, for x=1 we have π/2. And it's also easy to see that the function is ascending across all the defined range of x.
Now let's look at RHS. arccos(x^[1/2]) is defined for x>=0. Intersecting this with the domain of LHS we restrict the analysis for x in [0, 1]. The image is [0, π]. For x=0 we have π/2, and for x=1 we have 0. And it's also clear that the function is descending.
Taking into account the whole RHS we get -π/2 for x=0, and π/2 for x=1 (and it's ascending). Which equals to the LHS.
Worth to add that both sides are continuous and smooth functions inside the domain (excluding the endpoints).
From all that we can deduce that indeed LHS and RHS are equal on the defined domain
-
OK, just for fun let's try another method. If you know calculus, show that both sides have the same derivative with respect to $x$ (it's $-1/\sqrt{x-x^2}$) and also show that the two expressions are equal when $x=1$ (or when $x=0$ or whatever).
-
$\cos(2\arccos(\sqrt{x})) =\cos(\frac{\pi}{2}-\arcsin(2x-1))$
Since $\cos(\frac{\pi}{2}-\alpha)=\sin(\alpha)$ we may write:
$2(\cos(\arccos(\sqrt{x})))^2 -1= \sin(\arcsin(2x-1))$
$2(\sqrt{x})^2 -1=2x-1$
$2x-1=2x-1$
-
Your first line seems to assume the identity is true. – Michael Hardy Oct 4 '11 at 6:07
$\cos u =\cos v$ does not in general imply that $u=v$. Your "proof" would work if $\pi/2$ was replaced by $5\pi/2$, but the result would be false. But your idea can be made to work with an additional line or so. – André Nicolas Oct 4 '11 at 6:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9189260601997375, "perplexity": 629.0245996323148}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802770829.36/warc/CC-MAIN-20141217075250-00076-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/406938/has-anyone-studied-this-operator/407087 | # Has anyone studied this operator?
I've been studying a particular unary operator on the commutative ring $\mathbb{Z}/n\mathbb{Z}$. The operator is:
$\downarrow(x) = y\pmod{n}$ iff $n \equiv y \pmod{x}$, where $0< x,y \le n$.
The operator isn't distributive on addition or multiplication, so its use is probably fairly limited. I've proven a couple fairly trivial results, but I'm interested in seeing if there are more resources available.
One of the fairly trivial results: If $n$ is composite and $n=pq$ and WLOG $p \le q$, then if $q\ge kp+k$ I can show that
$(\downarrow(q-k))\mid p$.
-
This is not really an operator on $\mathbb{Z}/n\mathbb{Z}$, it's an operator on the numbers $\{1,2,\ldots, n\}$. The reason is that if $x\equiv x'\pmod{n}$, we would expect any such operator to satisfy $\downarrow (x)=\downarrow(x')$, but this one doesn't.
-
The definition of the operator can be extended trivially to satisfy this property, though. – Foo Barrigno May 30 at 18:59
Alas, it cannot. $n\equiv y\pmod{x}$ means $x|(n-y)$. $n\equiv y\pmod{x+n}$ means $(x+n)|(n-y)$. These two are rarely both true, and that's just for one choice of $x'$. Try some examples and you'll see. – vadim123 May 30 at 19:09
The "trivially" part was mapping x to the value I the range. In that sense, yes, it's an operator on the numbers {1,2,...,n}, but as long as you're rigorous about how the operator acts on each congruence class, it should be fine. – Foo Barrigno May 30 at 22:07
Yes. If the modulus is prime $\, n = p,\,$ then iterating the map $\ x\mapsto p\ {\rm mod}\ x\$ yields Gauss's algorithm for computing inverses mod $\,p.\,$ By this variant of the Euclidean algorithm, Gauss deduced the prime divisor property, i.e. if a prime divides a product then it divides some factor, which immediately yields uniqueness of prime factorizations (fundamental theorem of arithmetic). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.957770586013794, "perplexity": 407.20360907907957}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164921422/warc/CC-MAIN-20131204134841-00082-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://export.arxiv.org/abs/2106.01089 | astro-ph.HE
(what is this?)
# Title: Testing the magnetic flux paradigm for AGN radio loudness with a radio intermediate quasar
Abstract: For understanding the diversity of jetted active galactic nuclei (AGN) and especially the puzzling wide range in their radio-loudness, it is important to understand what role the magnetic fields play in setting the power of relativistic jets in AGN. We have performed multi-frequency (4-24 GHz) VLBA phase-referencing observations of the radio-intermediate quasar III Zw 2 using three nearby calibrators as reference sources to estimate jet magnetic flux by measuring the core-shift effect. By combining the self-referencing core-shift of each calibrator with the phase-referencing core-shifts, we obtained an upper limit of 0.16 mas for the core-shift between 4 and 24 GHz in III Zw 2. By assuming equipartition between magnetic and particle energy densities and adopting the flux-freezing approximation, we further estimated the upper limit for both magnetic field strength and poloidal magnetic flux threading the black hole. We find that the upper limit to the measured magnetic flux is smaller by at least a factor of five compared to the value predicted by the magnetically arrested disk (MAD) model. An alternative way to derive the jet magnetic field strength from the turnover of the synchrotron spectrum leads to an even smaller upper limit. Hence, the central engine of III Zw 2 has not reached the MAD state, which could explain why it has failed to develop a powerful jet, even though the source harbours a fast-spinning black hole. However, it generates an intermittent jet, which is possibly triggered by small scale magnetic field fluctuations as predicted by the magnetic flux paradigm of Sikora & Begelman (2013). We propose here that combining black hole spin measurements with magnetic field measurements from the VLBI core-shift observations of AGN over a range of jet powers could provide a strong test for the dominant factor setting the jet power relative to the accretion power available.
Comments: 29 pages, 22 figures, 3 Tables, accepted for publication with A&A Subjects: High Energy Astrophysical Phenomena (astro-ph.HE); Astrophysics of Galaxies (astro-ph.GA) Journal reference: A&A 652, A14 (2021) DOI: 10.1051/0004-6361/202140676 Cite as: arXiv:2106.01089 [astro-ph.HE] (or arXiv:2106.01089v1 [astro-ph.HE] for this version)
## Submission history
From: Wara Chamani [view email]
[v1] Wed, 2 Jun 2021 11:45:24 GMT (2582kb,D)
Link back to: arXiv, form interface, contact. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8506962656974792, "perplexity": 2834.5679728955056}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363290.39/warc/CC-MAIN-20211206042636-20211206072636-00307.warc.gz"} |
https://www.herongyang.com/Unicode/Font-Download-and-Install-GNU-Unifont.html | A tutorial example is provided on how to download and install GNU Unifont font family on Windows 7 systems.
GNU Unifont is a Unicode font family produced by Roman Czyborra and can be distributed under the GNU General Public License. The latest version of GNU Unifont covers 63,449 glyphs for large number of written languages.
Here is what I did to download and install it on my Windows 7 systems:
1. Go to http://unifoundry.com/unifont.html. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8995444178581238, "perplexity": 2243.7324074455764}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056974.30/warc/CC-MAIN-20210920010331-20210920040331-00043.warc.gz"} |
http://math.stackexchange.com/questions/53236/continuity-and-image-of-convergent-sequences | # Continuity and image of convergent sequences
Is it true that:
For a map $f:X\rightarrow Y$, between two topological spaces. If the image of every convergent sequence in $X$ is also convergent in $Y$. Then $f$ is continuous.
If it is true, how to prove it? Or if it is false, what is the counter-example? I guess it is false, because it is usually insufficient to characterize topological space with sequences. But I can't construct a counter-example. So I ask for help here.
Thanks for all the answers. Using nets or filters to characterize convergence seems to be a big topic such that I will spend some more time to digest. Before that, I seem to find an easy counter-example by myself.
Let $X=\{\{a\},\{a,b\},\emptyset\}$, every sequence in $X$ converges. The function $f$ from $X$ to $Y=\{\{f(a),f(b)\},\{f(b)\},\emptyset\}$. Then the image of every convergent sequence in $X$ is convergent in $Y$ but $f^{-1}(\{f(b)\})=\{b\}$ is not open in $X$, so $f$ is not continuous.
-
In certain kinds of spaces these things are the same; in general, you want to look at nets. See this discussion. – Dylan Moreland Jul 23 '11 at 5:03
@pluskid: Note that your counterexample does not work: The constant sequence $x_n=a$ converges to b (since every neighborhood of b contains a) but the constant sequence $f(x_n)=f(a)$ does not converge to $f(b)$ in Y (You are working with non-Hausdorff spaces - a sequence can have more than one limit.) – Martin Sleziak Jul 23 '11 at 7:21
As a side note: If you are trying to find a counterexample, it has to be infinite. Finite => first countable => Frechet-Urysohn => sequential. – Martin Sleziak Jul 23 '11 at 7:30
@Martin: The hypothesis is only that for every convergent sequence $x_n$ in $X$, $f(x_n)$ is a convergent sequence in $Y$. pluskid's counterexample does have this property, trivially, because every sequence in $Y$ converges. We are not assuming anything about what $f(x_n)$ converges to, and so this is not enough to force continuity. – Nate Eldredge Jul 23 '11 at 13:46
@Nate: You are right. I thought that we want $x_n\to x$ $\Rightarrow$ $f(x_n)\to f(x)$, since this is what is usually required. (This is what works for nets.) – Martin Sleziak Jul 23 '11 at 13:49
Construct a space which has a point which is not the limit of any sequence of points different from it, but which can be reached by a net (in other words, which is not isolated)
For example, let $\Omega$ be the smallest uncountable ordinal, let $X=\Omega+1$, and put the order topology on $X$. The biggest element $p$ in $X$ is such a point. Let $f:X\to X$ coincide with the identity map on $\Omega$ but with $f(p)=0$.
-
The characterization of continuous functions in terms of preservation of limits of sequences works well in a large class of topological spaces, but not all spaces. Better than restricting the class of spaces under consideration is to replace sequences with something better equipped to deal with the general case, namely nets or filters.
For this topic I recommend my notes on convergence in topological spaces. (Often I recommend my own notes just because I know what's in them and that they are available on the internet. This is an instance where I wouldn't really know where else to point to, as the standard texts all seem to tell different portions of the full story.)
Some specifics:
Proposition 2.3 shows that if $f: X \rightarrow Y$ is a map between topological spaces with $X$ first countable, then $f$ is continuous iff it preserves all limits of sequences.
In $\S 2.2$ I mention that the previous fact holds more generally when $X$ is sequential, which is the class of spaces studied in that section.
[This leaves open the question of giving a clean necessary and sufficient condition on a space $X$ such that for all spaces $Y$, a map $f: X \rightarrow Y$ is continuous iff all limits of sequences are preserved. I haven't really thought about this, but it would be interesting if there were a nice answer.]
In $\S 3$ I discuss nets. Proposition 3.2 shows that a map $f: X \rightarrow Y$ is continuous iff it preserves all limits of nets in $X$.
In $\S 5$ I discuss filters. Proposition 5.14 gives the analogous characterization of continuity in terms of preservation of convergent filters (actually "prefilters", which are more often called filter bases, but it is easy to see that this result remains true with all instances of "pre" omitted).
-
I think that being sequential is also necessary condition. For a space $(X,\tau)$ that is not sequential, it suffices to take $id:(X,\tau)\to(X,\tau')$, where $\tau'$ denotes the sequential coreflection (=closed subsets are precisely the sequentially closed subsets of the original topology). Since $X$ has sequentially closed subset that is not closed, this map is not continuous.\\I hope I did not miss something. – Martin Sleziak Jul 23 '11 at 6:18
I should have also written in my previous comment that $id: (X,\tau)\to(X,\tau')$ preserves convergent sequences. Hopefully, it was clear from the context. (Since I wanted to show that there is a map with domain $(X,\tau)$ that is sequentially continuous but not continuous.) – Martin Sleziak Jul 23 '11 at 9:33
EDIT: My post is about the spaces fulfilling the condition: If $x_n$ converges to $x$, then $f(x_n)$ converges to $f(x)$.
It seems, the you were asking about weaker condition: If $x_n$ is convergent, then $f(x_n)$ is convergent (as pointed out by Nate Eldredge).
I'll leave my answer, since it might be interesting for you anyway. (Any counterexample to the stronger condition is a counterexample to the weaker condition as well.)
This is true if $X$ is a sequential space. This paper gives a good introduction into the topic. Example 3.6 in this paper is an example of space that is not sequential. You can also find some useful information in this blog and its continuation. This question is also related to sequential spaces.
Your claim is true for arbitrary topological spaces if you replace sequences with nets.
There are many examples of spaces that are not sequential. Any non-discrete topological space with no non-trivial convergent sequences will do, such as cocountable topology on an uncountable set (see wikipedia article) or Stone-Čech compactification of countable discrete space (see this question).
For a space $X$ where every convergent sequence is eventually constant, you can take a discrete topological space $Y$ having at least 2 points. Then every function $f:X\to Y$ preserves convergence of sequences. But all such functions are continuous only if $X$ is discrete.
I will give the following counterexample (again, in this space all convergent sequences are eventually constant):
In this picture every arrow represents a convergent sequence (i.e. a topological space on a countable set, that has unique accumulation point and the neighborhoods of this point are precisely the cofinite sets; e.g. $\{0\}\cup\{1/n; n\in\mathbb N\}$ as the subspace of real line has this topology). If we make a quotient of a topological sum of such spaces (in the way given in this picture), we get a sequential space. The picture shows a subspace of a sequential space that is not sequential. (There is only one point that is not isolated. No sequence of isolated points converges to this point. Showing this is basically an exercise in working with quotient topology.)
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9710792899131775, "perplexity": 159.74677684279274}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375098072.11/warc/CC-MAIN-20150627031818-00181-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://mathhelpforum.com/algebra/166070-exponential-logarithmic-functions.html | # Thread: Exponential and Logarithmic Functions
1. ## Exponential and Logarithmic Functions
Kinda lost on this problem.
Would appreciate some help. Does this have to do with log properties?
-Auburn
2. No need for logarithms, just convert everything to the same base.
$\displaystyle \frac{25^{3x^2}}{5^{7x}} = \frac{1}{25}$
$\displaystyle \frac{(5^2)^{3x^2}}{5^{7x}} = \frac{1}{5^2}$
$\displaystyle \frac{5^{6x^2}}{5^{7x}} = 5^{-2}$
$\displaystyle 5^{6x^2 - 7x} = 5^{-2}$
$\displaystyle 6x^2 - 7x = -2$
$\displaystyle 6x^2 - 7x + 2 = 0$
$\displaystyle 6x^2 - 3x - 4x + 2 = 0$
$\displaystyle 3x(2x - 1) - 2(2x - 1) = 0$
$\displaystyle (2x-1)(3x-2) = 0$
$\displaystyle 2x-1 = 0$ or $\displaystyle 3x-2 = 0$
$\displaystyle x = \frac{1}{2}$ or $\displaystyle x = \frac{2}{3}$.
3. Right, I see now. Since it's the same base the exponents must be equal. Thank you kindly! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 13, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8079955577850342, "perplexity": 1109.7030850971287}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886117519.82/warc/CC-MAIN-20170823020201-20170823040201-00301.warc.gz"} |
https://learnzillion.com/lesson_plans/3917-8-interpret-quotients-of-rational-numbers-by-describing-real-world-contexts-c | # 8. Interpret quotients of rational numbers by describing real-world contexts (C)
teaches Common Core State Standards CCSS.Math.Content.7.NS.A.2c http://corestandards.org/Math/Content/7/NS/A/2/c
## You have saved this lesson!
Here's where you can access your saved items.
Dismiss
Card of
Lesson objective: Understand that real-world situations can be represented by quotients with negative numbers.
From 6.RP.3, students bring prior knowledge of unit rates. From 6.NS.5, students bring prior knowledge that positive and negative numbers can be used to represent quantities in real-world contexts. From 7.NS.1b, students bring prior knowledge that signs indicate direction in real-world contexts. This prior knowledge is extended as students represent real-world situations with quotients of negative numbers. A conceptual challenge students may encounter is that situations with different descriptions are represented with different but equivalent quotients.
The concept is developed through an extended example of temperature change, which leads students to write rates as fractions (or quotients) with negative numbers.
This work helps students deepen their understanding of equivalence because they discover that different placements of the negative sign in a fraction can result in equivalent quotients. Specifically, they learn that -p/q = p/-q = -(p/q).
Students engage in Mathematical Practice 2 (Reason abstractly and quantitatively) as they decontextualize and contextualize rational numbers by translating real-world temperature changes to numerical expressions and identifying equivalent representations.
Key vocabulary:
• ascended
• quotient
• rate
Special materials needed:
• none | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9146925806999207, "perplexity": 3499.2661985584527}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549423774.37/warc/CC-MAIN-20170721122327-20170721142327-00211.warc.gz"} |
https://ai.stackexchange.com/questions/1426/how-can-thousand-robot-swarm-coordinate-their-moves-without-bumping-into-each-ot/1443 | # How can thousand-robot swarm coordinate their moves without bumping into each other?
How can a swarm of small robots (like Kilobots) walking close to each other achieve collaboration without bumping into each other? For example, one study shows programmable self-assembly in a thousand-robot swarm (see article & video) which are moving without GPS-like system and by measuring distances to neighbours. This was achieved, because the robots were very slow.
Is there any way that similar robots can achieve much more efficient and quicker assembly by using more complex techniques of coordination? Not by walking around clock-wise (which I guess was the easiest way), but I mean using some more sophisticated way. Because waiting half a day (~11h) to create a simple star shape using a thousand-robot swarm is way too long! | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8120994567871094, "perplexity": 2206.7746251027856}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103573995.30/warc/CC-MAIN-20220628173131-20220628203131-00002.warc.gz"} |
https://www.cliffsnotes.com/study-guides/astronomy/the-science-of-astronomy/electromagnetic-radiation-light | The second great area of physics necessary to address the universe is the subject of light, or electromagnetic radiationVisible light is the relatively narrow frequency band of electromagnetic waves to which our eyes are sensitive. Wavelengths are usually measured in units of nanometers (1 nm = 10 −9m) or in units of angstroms (1 Å = 10 −10m). The colors of the visible spectrum stretch from violet with the shortest wavelength to red with the longest wavelength.
However, electromagnetic radiation consists of more than just visible light; it also includes (from short wavelength to long wavelength) gamma‐radiation, X‐radiation, ultraviolet, visible, infrared (heat), microwaves, and radio waves (see Figure 1). All of these forms of light have both electrical and magnetic characteristics. The properties of light (see the section, “Particle properties of light”) allow us to build devices to observe the universe and to deduce the physical nature of the sources that emit the radiation received during these observations. However, these same properties mean that light interacts with other matter before it reaches the observer and this often complicates our ability to observe other objects in the universe. Note that the word “radiation” can refer to any phenomena that radiates (moves) outwards from a source, here electromagnetic or light radiation. The term should not be confused with radiation associated with a radioactive source, i.e. nuclear radiation.
Figure 1
The electromagnetic spectrum. Visible light is only a small portion of the electromagnetic radiation that can be detected by various instruments.
## Particle properties of light
Light is such a complicated phenomena that no one model can be devised to explain its nature. Although light is generally thought of as acting like an electric wave oscillating in space accompanied by an oscillating magnetic wave, it can also act like a particle. A “particle” of light is called a photon, or a discrete packet of electromagnetic energy.
Most visible objects are seen by reflected light. There are a few natural sources of light, such as the Sun, stars, and a flame; other sources are man‐made, such as electrical lights. For an otherwise non‐luminous object to be visible, light from a source is reflected off the object into our eye. The property of reflection, that light can be reflected from appropriate surfaces, can most easily be understood in terms of a particle property, in the same sense that a ball bounces off a surface. A common example of reflection is mirrors, and in particular, telescope mirrors that use curved surfaces to redirect light received over a large area into a smaller area for detection and recording.
When reflection occurs in particle‐particle interactions (for example, colliding billiard balls), it's called scattering — light is scattered (reflected) off molecules and dust particles that have sizes comparable to the wavelengths of the radiation. As a consequence, light coming from an object seen behind dust is dimmer than it would be without the dust. This phenomena is termed extinction. Extinction can be seen in our own Sun when it becomes dimmer as its light passes through more of the dusty atmosphere as it sets. Similarly, stars seen from Earth seem fainter to the viewer than they would if there were no atmosphere. In addition, short wavelength blue light is preferentially scattered; thus objects look redder (astronomers refer to this as reddening); this occurs because the wavelength of blue light is very close to the size of the particles that cause the scattering. By analogy, consider ocean waves — a row boat whose length is close to the wavelength of the waves will bob up and down, whereas a long ocean liner will scarcely notice the waves. The Sun appears much redder at sunset. The light of stars also redden in passing through the atmosphere. You can see the scattered light by looking in directions away from the source of the light; hence the sky appears blue during the day.
Extinction and reddening of starlight are not caused by just the atmosphere. An exceedingly thin distribution of dust floats between the stars and affects the light that we receive as well. Astronomers must take into account the effect of dust on their observations to correctly describe the conditions of the objects that emit the light. Where interstellar dust is especially thick, no light passes through. Where dust clouds reflect starlight back in our direction, the observer may see blue interstellar wispiness like thin clouds surrounding some stars, or a nebula (to use the Latin word for cloud). A nebula formed by scattering of blue light is called a reflection nebulae.
## Wave properties of light
Most properties of light related to astronomical use and effects have the same properties as waves. Using an analogy to water waves, any wave can be characterized by two related factors. The first is a wavelength (λ) the distance (in meters) between similar positions on successive cycles of the wave, for example the crest‐to‐crest distance. The second is a frequency (f) representing the number of cycles that move by a fixed point each second. The fundamental characteristic of a wave is that multiplication of its wavelength by its frequency results in the speed with which the wave moves forward. For electromagnetic radiation this is the speed of light, c = 3 × 10 8 m/sec = 300,000 km/sec. The mid‐range of visible light has a wavelength of λ = 5500 Å = 5.5 × 10 −7 m, corresponding to a frequency f of 5.5 × 10 14 cycles/sec.
When light passes from one medium to another (for example, from water to air; from air to glass to air; from warmer, less dense regions of air to cooler, denser regions and vice‐versa) its direction of travel changes, a property termed refraction. The result is a visual distortion, as when a stick or an arm appears to “bend” when put into water. Refraction allowed nature to produce the lens of the eye to concentrate light passing through all parts of the pupil to be projected upon the retina. Refraction allows people to construct lenses to change the path of light in a desired fashion, for instance, to produce glasses to correct deficiencies in eyesight. And astronomers can build refracting telescopes to collect light over large surface areas, bringing it to a common focus. Refraction in the non‐uniform atmosphere is responsible for mirages, atmospheric shimmering, and the twinkling of stars. Images of objects seen through the atmosphere are blurred, with the atmospheric blurring or astronomical “seeing” generally about one second of arc at good observatory sites. Refraction also means that positions of stars in the sky may change if the stars are observed close to the horizon.
Related to refraction is dispersion, the effect of producing colors when white light is refracted. Because the amount of refraction is wavelength dependent, the amount of bending of red light is different than the amount of bending of blue light; refracted white light is thus dispersed into its component colors, such as by the prisms used in the first spectrographs (instruments specifically designed to disperse light into its component colors). Dispersion of the light forms a spectrum, the pattern of intensity of light as a function of its wave length, from which one can gain information about the physical nature of the source of light. On the other hand, dispersion of light in the atmosphere makes stars undesirably appear as little spectra near the horizon. Dispersion is also responsible for chromatic aberration in telescopes — light of different colors is not brought to the same focal point. If red light is properly focused, the blue will not be focused but will form a blue halo around a red image. To minimize chromatic aberration it is necessary to construct more costly multiple‐element telescope lenses.
When two waves intersect and thus interact with each other, interference occurs. Using water waves as an analogy, two crests (high points on the waves) or two troughs (low points) at the same place constructively interfere, adding together to produce a higher crest and a lower trough. Where a crest of one wave, however, meets a trough of another wave, there is a mutual cancellation or destructive interference. Natural interference occurs in oil slicks, producing colored patterns as the constructive interference of one wavelength occurs where other wavelengths destructively interfere. Astronomers make use of interference as another means of dispersing white light into its component colors. A transmission grating consisting of many slits (like a picket fence, but numbering in the thousands per centimeter of distance across the grating) produces constructive interference of the various colors as a function of angle. A reflection grating using multiple reflecting surfaces can do the same thing with the advantage that all light can be used and most of light energy can be thrown into a specific constructive interference region. Because of this higher efficiency, all modern astronomical spectrographs use reflection gratings.
A number of specialized observing techniques result from application of these phenomena, of which the most important is radio interferometry. The digital radio signals from arrays of telescopes can be combined (using a computer) to produce high‐resolution (down to 10 −3 second of arc resolution) “pictures” of astronomical objects. This resolution is far better than that achievable by any optical telescope, and thus, radio astronomy has become a major component in modern astronomical observation.
Diffraction is the property of waves that makes them seem to bend around corners, which is most apparent with water waves. Light waves are also affected by diffraction, which causes shadow edges to not be perfectly sharp, but fuzzy. The edges of all objects viewed with waves (light or otherwise) are blurred by diffraction. For a point source of light, a telescope behaves as a circular opening through which light passes and therefore produces an intrinsic diffraction pattern that consists of a central disk and a series of fainter diffraction rings. The amount of blurring as measured by the width of this central diffraction disk depends inversely on the size of the instrument viewing the source of light. The pupil of the human eye, about an eighth of an inch in diameter, produces a blurring greater than one arc minute in angular size; in other words, the human eye cannot resolve features smaller than this. The Hubble Space Telescope, a 90‐inch diameter instrument orbiting Earth above the atmosphere, has a diffraction disk of only 0.1 second of arc in diameter, allowing the achievement of well‐resolved detail in distant celestial objects.
The physical cause of diffraction is the fact that light passing through one part of an opening will interfere with light passing through all other parts of the opening. This self‐interference involves both constructive interference and destructive interference to produce the diffraction pattern.
## Kirchoff's three types of spectra
Both dispersive and interference properties of light are used to produce spectra from which information about the nature of the light‐emitting source can be gained. Over a century ago, the physicist Kirchoff recognized that three fundamental types of spectra (see Figure 2) are directly related to the circumstance that produces the light. These Kirchoff spectral types are comparable to Kepler's Laws in the sense that they are only a description of observable phenomena. Like Newton, who later was to mathematically explain the laws of Kepler, other researchers have since provided a sounder basis of theory to explain these readily observable spectral types.
Figure 2
Kirchhoff's three types of spectra. a) A continuous spectrum (blackbody spectrum) is radiation produced by warm, dense material; b) an emission line spectrum (bright line spectrum) is radiation created by a cloud of thin gas; and c) an absorption line spectrum (dark line spectrum) results from light passing through a cloud of thin gas.
Kirchoff's first type of spectrum is a continuous spectrum: Energy is emitted at all wavelengths by a luminous solid, liquid, or very dense gas — a very simple type of spectrum with a peak at some wavelength and little energy represented at short wavelengths and at long wavelengths of radiation. Incandescent lights, glowing coals in a fireplace, and the element of an electric heater are familiar examples of materials that produce a continuous spectrum. Because this type of spectrum is emitted by any warm, dense material, it is also called a thermal spectrum or thermal radiation. Other terms used to describe this type of spectrum are black body spectrum (since, for technical reasons, a perfect continuous spectrum is emitted by a material that is also a perfect absorber of radiation) and Planck radiation (the physicist Max Planck successfully devised a theory to describe such a spectrum). All these terminologies refer to the same pattern of emission from a warm dense material. In astronomy, warm interplanetary or interstellar dust produces a continuous spectrum. The spectra of stars are roughly approximated by a continuous spectrum.
Kirchoff's second type of spectrum is emission of radiation at a few discrete wavelengths by a tenuous (thin) gas, also known as an emission spectrum or a bright line spectrum. In other words, if an emission spectrum is observed, the source of the radiation must be a tenuous gas. The vapor in fluorescent tube lighting produces emission lines. Gaseous nebulae in the vicinity of hot stars also produce emission spectra.
Kirchoff's third type of spectrum refers not to the source of light, but to what might happen to light on its way to the observer: The effect of a thin gas on white light is that it removes energy at a few discrete wavelengths, known as an absorption spectrum or a dark line spectrum. The direct observational consequence is that if absorption lines are seen in the light coming from some celestial object, this light must have passed through a thin gas. Absorption lines are seen in the spectrum of sunlight. The overall continuous spectrum nature of the solar spectrum implies that the radiation is produced in a dense region in the Sun, then the light passes through a thinner gaseous region (the outer atmosphere of the Sun) on its way to Earth. Sunlight reflected from other planets shows additional absorption lines that must be produced in the atmospheres of those planets.
## Wien's and Stefan-Boltzman's Laws for Continuous Radiation
Kirchoff's three types of spectra give astronomers only a general idea of the state of the material that emits or affects the light. Other aspects of the spectra allow more of a quantitative definition of physical factors. Wien's Law says that in a continuous spectrum, the wavelength at which maximum energy is emitted is inversely proportional to temperature; that is, λ max = constant / T = 2.898 × 10‐3 K m / T where the temperature is measured in degrees Kelvin. Some examples of this are:
The Stefan‐Boltzman Law (sometimes called Stefan's Law) states that the total energy emitted at all wavelengths per second per unit surface area is proportional to the fourth power of temperature, or energy per second per square meter = σ T 4 = 5.67 × 10 8 watts/(m 2 K 4) T 4 (see Figure 3).
Figure 3
A graphical representation of continuous spectra for light sources of different temperature. Wien's Law is shown by the peak of radiation at shorter wavelengths for higher temperatures. The Stefan‐Boltzman Law is shown by the larger areas under each curve (representing the total energy emission at all wavelengths) for higher temperatures.
This simple principle produces a relationship between the total energy emitted by an object each second, the luminosity L, the radius r of a celestial object, assumed spherical, and the object's surface temperature. The total energy emitted per second = surface area × energy per second emitted by each unit area, or algebraically,
## Quantitative Analysis of Spectra
The development of the theory of quantum mechanics led to an understanding of the relationship between matter and its emission or absorption of radiation. If atoms are far enough apart that they do not affect each other, then each chemical element can emit or absorb light only at specific wavelengths. The energies of photons at these wavelengths correspond to the differences between the permitted energies of the electrons of the atoms. The negatively charged electrons can be considered to be in “orbits” about the positively charged protons in the nucleus of the atom, each orbit corresponding to a different energy. Quantum orbits can only be at certain energy levels, unlike orbits governed by gravity which can be at any energy. Emission of a photon of light occurs when an electron “drops” from a high energy state to a lower energy state. Absorption occurs when a photon of the right energy permits an electron to “jump” to a higher energy state. Most importantly, the pattern of absorption or emission lines is unique to each element. The strength of emission or absorption depends on how many atoms of the particular element are present as well as the temperature of the material, thus permitting both temperature and the chemical composition of the material producing the spectrum to be determined.
If atoms are progressively jammed closer together, the wavelengths of emission or absorption by any given atom will be slightly changed, thus some atoms will emit/absorb at slightly longer wavelengths and others will emit/absorb at slightly shorter wavelengths. The majority of atoms will emit/absorb at the same wavelengths that they would if unaffected by neighboring atoms. Astronomers therefore can differentiate between circumstances where the emitting or absorbing atoms are thinly dispersed (the spectral features will look very sharp) and where they are tightly packed together (the spectral features are broadened). In the extreme case of high density, the emission lines become completely blurred together and one observes a continuous spectrum.
## Doppler's Law
If a light source and observer are approaching each other, the observed wavelength of any spectral feature is shorter than what would be measured if the two were at rest with respect to each other. On the other hand, if the two are moving apart, the observed wavelength appears longer that the wavelength that would be measured at rest. The Doppler Shift or Doppler Effect is a recognition that the change of wavelength Øλ of a given spectral feature depends on the relative velocity of the source along the line of sight:
where Δ is the observed wavelength, Δ 0 is the rest or laboratory wavelength, v is the velocity toward (negative) or away (positive) from the observer, and c is the speed of light. Relative motion of a light source toward the observer results in a blueshift of the spectrum, as all wavelengths are measured shorter or bluer; relative motion of a light source away from the observer results in a redshift. This doesn't mean that the light literally turns blue or red, it means that the light has its color shifted toward the shorter or longer wavelength region of the spectrum, respectively (see Figure 4); in most situations the shift is very small because velocities are small compared to the speed of light. This simple form of the Doppler Law holds only if the velocity v is small with respect to the speed of light. A more complicated equation formulated by Einstein in his theory of relativity must be used if the source is moving near the speed of light. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8599416613578796, "perplexity": 564.9104411273946}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823895.25/warc/CC-MAIN-20181212134123-20181212155623-00603.warc.gz"} |
https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_7C_-_General_Physics/09%3A_Optics/9.4%3A_Summary | $$\require{cancel}$$
# 9.4: Summary
• Wendell Potter and David Webb et al.
• UC Davis
$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$
1. Become familiar with the idea of wavefronts and rays.
2. Geometric optics is the approximation that rays always travel in straight lines. This approximation is good provided that the wavelength is much smaller than anything it encounters (i.e. we are neglecting diffraction). The geometric optics approximation allows us to perform ray-tracings to locate images.
3. When a wave encounters an interface between two media, some of the wave's energy can reflect into the original medium while the rest can be transmitted into the new medium. Because the two media have different allowed wave speeds, the transmitted wave is typically deformed, a phenomenon called refraction.
4. The law of reflection states that $$\theta_{inc} = \theta_{ref}$$, where both angles are measured from the normal of the reflecting surface.
5. Objects with rough surfaces have normals that change over their surface. As a result, light incident on rough surfaces is reflected in all directions. This is calleddiffuse reflection.
6. Each non-absorbing material has a refractive index that describes how quickly light travels through it. The higher the refractive index, the slower light travels in that medium. The refractive index in a medium is defined as $$n_{medium} = c/v_{medium}$$, where $$v_{medium}$$ is the speed of the wave in the medium and $$c$$ is the speed of light in vacuum.
7. To find the direction that light bends when refracted, we use Snell’s law $$n_1 \sin \theta_1 = n_2 \sin \theta_2$$. Both $$\theta_1$$ and $$\theta_2$$ are measured from the normal of the interface between the two media.
8. If Snell’s law cannot be satisfied then none of the wave can be transmitted; instead it is all reflected. This phenomenon is called total internal reflection. Total internal reflection can only occur when the light is coming from a faster medium and reaches the boundary between media.
9. Our eyes can only track back the rays that reach our eyes, and so if rays appear to be coming from somewhere then our brain thinks there is an object there. If there is no object there, the object our brain thinks it sees is called an image.
10. Images come in two types: real and virtual.
• A real image is where the light rays actually come to a point and then spread out again. This sort of image can be placed on a screen.
• A virtual image is an image where the light rays do not cross, but our brain traces back the rays and is tricked into thinking that they cross.
11. For thin lenses or particular curved mirrors there is a focal length $$f$$. The relationship between the object distance $$o$$ and image distance $$i$$ is $\dfrac{1}{o} + \dfrac{1}{i} = \dfrac{1}{f}$which is known as the thin lens equation. If $$i$$ is positive,this is a real image; if $$i$$ is negative this is a virtual image.
12. The image of an object is typically a different size than the object. We use the magnification $$m = −i/o$$ to describe the change in size; for example $$m=2$$ means the image is twice as big as the original object. If the magnification is negative then this means the image is inverted.
# Derivations
In this chapter we simply presented Snell's Law, magnification, and the thin lens equation as true instead of deriving these results from what we know. We present the derivations here for the interested reader. While not necessary to learn and apply these equations, understanding these derivations will deepen your understanding of the concepts they reflect.
## Snell's Law
Let us draw both the peaks (as wavefronts) and the rays of light together for light that is traveling from air into water.
Look at the distance between the wavefronts on the boundary, shown as a bold line between the two indicated normals (dashed lines). Let us call this distance $$h$$ for hypotenuse, because it is the hypotenuse of both right-angled triangles indicated in the water and in the air. We know that the distance between the wavefronts (which makes up the opposite side of these triangles) is given by the wavelength in that medium. Writing this out for the triangle in the water we have
$\sin \theta_w = \dfrac{\lambda_w}{h} \implies h = \dfrac{\lambda_w}{\sin \theta_w}$
For the triangle in the air we have a similar relationship:
$\sin \theta_a = \dfrac{\lambda_a}{h} \implies h = \dfrac{\lambda_a}{\sin \theta_a}$
Because we know that the hypotenuse is the same in both of these equations, we are lead to conclude that
$\dfrac{\lambda_w}{\sin \theta_w} = \dfrac{\lambda_a}{\sin \theta_a}$
By multiplying this equation by the frequency $$f$$ and recalling that $$v_{wave} = f \lambda$$ we have
$\dfrac{f \lambda_w}{\sin \theta_w} = \dfrac{f \lambda_a}{\sin \theta_a}$
$\dfrac{v_w}{\sin \theta_w} = \dfrac{v_a}{\sin \theta_a}$
Finally we recall that $$v_a = c/n_a$$ (and a similar result for water) we have
$\dfrac{c}{n_w \sin \theta_w} = \dfrac{c}{n_a \sin \theta_a}$
This results holds if and only if
$n_w \sin \theta_w = n_a \sin \theta_a$
is true. This has been Snell's Law.
## Magnification
Consider an object with height $$h_o$$ and a converging lens that produces a real image with height $$h_i$$. Examine at the principal ray that goes through the center. Because it is a straight line, the gradient (slope) does not change.
We see that the gradient on the left hand side is
$\text{gradient }= \dfrac{\Delta y}{\Delta x} = \dfrac{-h_o}{o}$
We can use the information on the right-hand side to calculate the gradient we get
$\text{gradient }= \dfrac{\Delta y}{\Delta x} = \dfrac{h_i}{i}$
Because this ray does not bend, we know these gradients are the same. Therefore: $\dfrac{-h_o}{o} = \dfrac{h_i}{i}$ Rearranging this equation we have
$m \equiv \dfrac{h_i}{h_o} = -\dfrac{i}{o}$
## The Thin Lens Equation
We will prove the thin lens equation for a converging lens that produces a real image. The other cases can be shown in a similar manner. Examine the ray that enters the lens parallel to the optical axis and refracts through the focal point.
We know that there are two ways of calculating the gradient of the ray that passes through the focal point. The first manipulates the fact that the incoming ray has the same height as the object, and drops to the optical axis within a focal length
$\text{gradient} = \dfrac{\Delta y}{\Delta x} = -\dfrac{h_o}{f}$
The second way of calculating the gradient uses the fact that the height of the ray drops to the location of the image in the distance $$i$$. Because $$h_i < 0$$ we should be slightly careful with the sign of $$\Delta y$$
$\Delta y = y_f - y_i = h_i - h_o \implies \text{gradient} = \dfrac{h_i-h_o}{i}$
We can get rid of $$h_i$$ by applying our equation for the magnification:
$h_i = m h_o = -\dfrac{i}{o}h_o$
Writing out our gradient again we obtain
$\text{gradient} = \dfrac{-\frac{i}{o} h_o - h_o}{i} = - \left( \dfrac{1}{o} + \dfrac{1}{i} \right) h_o$
As a straight line has a constant gradient, the segment we use should not matter. Therefore these two expressions for the gradient must be equal:
$- \left( \dfrac{1}{o} + \dfrac{1}{i} \right) h_o = - \dfrac{h_o}{f}$
Cancelling the $$-h_o$$ from both sides leaves the thin lens equation
$\dfrac{1}{o} + \dfrac{1}{i} = \dfrac{1}{f}$
9.4: Summary is shared under a not declared license and was authored, remixed, and/or curated by Wendell Potter and David Webb et al.. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8457217812538147, "perplexity": 290.6289028855076}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103920118.49/warc/CC-MAIN-20220701034437-20220701064437-00415.warc.gz"} |
https://www.physicsforums.com/threads/mechanical-principles-rotating-systems.759681/ | # Homework Help: Mechanical Principles - Rotating Systems
1. Jun 27, 2014
### louise
A load of mass 250kg is required to be lifted by a means of a winding drum and cable. The mass will be initially at rest, accelerated uniformly upwards for 4 seconds and then decelerated uniformly for one second. such that a final height of 10.5 metres is gained.
The winding drum has a mass of 225kg, a diameter of 1.5m and a radius of gyration of 320mm. The bearings of the drum have a constant frictional torque of 5Nm.
Calculate:
a) The maximum velocity reached by the mass.
b) The maximum angular velocity of the winding drum.
c) The total work done whilst accelerating the load upwards.
d) The input torque to the driving motor whilst accelerating the load.
e) The average power required whilst accelerating the load.
f) The maximum power required from the drive motor.
My solutions so far:
a) Using: s=ut+1/2at^2 (1/2 being half and ^2 being squared)
a=2(s-ut)/t^
=2(10.5-0*4)/4^2
= 1.31m/s^2
Using: v=u+at
=0+1.13*4
=5.24m/s^2
b) Using: θ=((ω1+ω2)t)/2 UNSURE ON THIS ON FOR DEFINITE!
=((0+5.24)4)/2
θ/t=Angular Velocity
10.48/2∏ (∏ being used as Pi)
=16.46revs
c) NEED HELP I HAVE THIS EQUATION BUT I NEED TO FIGURE OUT ALL THE LETTERS FIRST
WD=mg(h2-h1)+1/2m(v^2-u^2)+1/2I(ω2^2-ω1^2)+(Fr*θ)
I know mK^2 =I m being the mass and K being the radius of gyration?
An to find h2 I use S=((u+v)t)/2
d)Ang Power =WD/t?
f)??
g) Max Power = T*ω1??
2. Jun 27, 2014
### Simon Bridge
Welcome to PF;
Your overall strategy appears to involve trying to find the right equations to use.
It is better to use your understanding of the physics.
(a) fair enough - you can also draw a v-t diagram.
(b) what is the relationship between angular velocity and linear velocity?
(c) ignore the formula - work = change in energy: where does the energy put into winding the drum go?
(d) W=Fd normally - what is the angular form?
(e&f) what is the definition of power?
note: if you know the linear equation, you can get the equivalent rotational one by substituting:
velocity -> angular velocity
acceleration -> angular acceleration
force -> torque
displacement -> angle
3. Jun 27, 2014
### louise
Yes, I am looking for help with the equations and advice as to whether the methods I've attempted to use are correct.
a) I have already done a velocity time graph to represent the two in relation with each other. Wasn't too sure that my working for the calculations were correct.
b) They are the same, but because the winding drum is the rotating object you can't use the same formula. That's where i was confused.
c) Into the load? I have no idea.
d) I needed the average power not angular sorry! W=Fd? We've never being taught that. What does the lettering stand for?
e&f) WD/time = Pavg Torque*?=Pmax
4. Jun 27, 2014
### louise
PS. I'm not a whizz at physics and the formulas involved with Mech Principles, its just one unit of many we have to do.
5. Jun 27, 2014
### Simon Bridge
Oh good - well from the v-t graph you just do a bit of geometry to check your result via equations.
No ... linear velocity and angular velocity are not the same - they have a reationship.
Have you see v=rω?
Come on - what kinds of energy are there?
Is the load moving vertically? What kind of energy changes?
Is the load changing speed? What kind of energy changes?
Is there any friction?
It is the definition of Work.
W = work
F = force
d = displacement
6. Jun 28, 2014
### dean barry
M = load mass = 250 kg
m = drum mass = 225 kg
k = radius of gyration = 0.32 metres
Im assuming the mass moment of inertia (i) = m * k ² = 225 * 0.32 ² = 23.04 kg-m²
----------------------------------------------------------------------------------------------------
There is a short cut that might help you cut out some work, you can give the load and drum a combined equivalent mass (EM) for use in this type of problem :
Give the load an example constant velocity, say 10 m/s and calculate the linear KE from :
KE linear = ½ * M * v ²
Calculate the rotation rate ω ( radians / second ) of the drum with the load at 10 m/s from v / r
Calculate the rotary KE of the drum from :
KE rotary = ½ * i * ω ²
Install in the following equation :
EM (in kgs) = ( 1 + ( KE rotary / KE linear ) ) * M
If you have a linear acceleration rate (a) for the load for instance, you can calculate the force (f) required to accelerate both the load and the drum from :
f = EM * a
im uncertain as to the performance you require under accelerating and braking, can you post the graph you have ?
thanks
dean
7. Jun 30, 2014
### dean barry
Sorry louise, led you up the wrong path there, disregard my input.
dean | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9048620462417603, "perplexity": 2093.3618143034687}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676591150.71/warc/CC-MAIN-20180719164439-20180719184439-00132.warc.gz"} |
https://www.ncbi.nlm.nih.gov/pubmed/?term=9390264 | Format
Choose Destination
Pac Symp Biocomput. 1996:638-52.
# Empirical free energy calculations of human immunodeficiency virus type 1 protease crystallographic complexes. II. Knowledge-based ligand-protein interaction potentials applied to thermodynamic analysis of hydrophobic mutations.
### Author information
1
Agouron Pharmaceuticals, Inc., San Diego, CA 92121-1121, USA.
### Abstract
Empirical free energy calculations of HIV-1 protease crystallographic complexes based on the developed knowledge-based ligand-protein interaction potentials have enabled a detailed thermodynamic analysis. Binding free energies are estimated within an empirical model that postulates that hydrophobic effect, mean field ligand-protein interaction potentials and conformational entropy changes are the dominant forces that determine complex formation. To provide a quantitative framework of the binding thermodynamics contributions the derived knowledge-based potentials have been linked with the hydrophobicity and conformational entropy scales originally developed to explain protein stability. The comparative analysis of studied inhibitors provides reasonable estimates of distinctions in their binding affinity with HIV-1 protease and gives insight into the nature of the binding determinants. The binding free energy changes upon a simple hydrophobic mutation Ile -> Val in the JG-365, MVT-101 and U75875 inhibitors of HIV-1 protease have been evaluated within a model that includes the effects of solvation, cavity formation, conformational entropy and mean field ligand-protein interactions. In general, free energy changes associated with a particular perturbation of a system can not be rigorously decomposed into separate terms from first principles. We explored the relationships between the changes in hydrophobic contributions and mean field ligand-protein interaction energies in the context of a totally buried and dense area of the binding site. We assume, therefore, that these simple hydrophobic deletions would not induce noticeable conformational changes in the enzyme and can be interpreted with some confidence in the framework of the model. The analysis has revealed the decisive effect of the energetics of ligand-protein interactions on the estimated free energy changes.
PMID:
9390264
[Indexed for MEDLINE]
Free full text | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.866100013256073, "perplexity": 2577.9295876535084}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267158001.44/warc/CC-MAIN-20180922005340-20180922025740-00297.warc.gz"} |
https://www.nbccomedyplayground.com/is-higher-insertion-loss-better/ | Is higher insertion loss better?
Is higher insertion loss better?
Insertion loss is expressed in decibels, or dBs, and should be a positive number as it indicates how much signal was lost by comparing input power to output power. The lower the number, the better the insertion loss performance – an insertion loss of 0.2dB is better than 0.4dB.
What is the difference between attenuation and insertion loss?
Attenuation, now referred to as Insertion loss, is the amount of loss incurred on a signal in a cable or link. It is measured in decibels (dB) and is dependent on distance and frequency. The attenuation of a cable or link will define the maximum distance a signal can travel and still be picked up at receiver.
What is differential insertion loss?
Differential and single-ended insertion loss are used to describe the signal loss in a device, depending on the type of signaling used. Insertion loss in a differential interconnect or balanced network is called differential insertion loss.
Why return loss should be less than 10 dB?
The return loss measures the reflected wave to the incident wave, that is RL = -20 log(Γ). So, a return loss of -10 dB means that the reflected wave is 10 dB lower than the incident wave. This is approximately equal to a reflection coefficient of 0.3, so 30% of the incident wave is wasted.
Does insertion loss include return loss?
Microwave power is sent down a transmission line from the left and it reaches the component. The ratio of incident power to transmitted power, in dB terminology, is the insertion loss. The ratio of incident power to the reflected power, in dB terminology, is the return loss.
Is high return loss Good?
Return loss is a measure of how well devices or lines are matched. A match is good if the return loss is high. A high return loss is desirable and results in a lower insertion loss. From a certain perspective ‘Return Loss’ is a misnomer.
What is insertion loss vs return loss?
The ratio of incident power to transmitted power, in dB terminology, is the insertion loss. The ratio of incident power to the reflected power, in dB terminology, is the return loss.
Why is return loss important?
Return loss is a measurement parameter that expresses how well a device or line matches. A high return loss is advantageous as it will result in a lower insertion loss.
What is a good value for return loss?
The return loss scale is normally set up from 0 to 60 dB with 0 being an open or a short and 60 dB would be close to a perfect match. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8238300085067749, "perplexity": 1019.7467271806513}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662534773.36/warc/CC-MAIN-20220521014358-20220521044358-00782.warc.gz"} |
http://math.stackexchange.com/questions/152354/proof-of-int-limits-af-int-limits-mathbbrf1-a-for-the-lebesgue-inte | # Proof of $\int\limits_{A}f=\int\limits_{\mathbb{R}}f{1}_A$ for the Lebesgue integral
Let $f:\mathbb{R}\to [0,\infty]$ be a measurable function and $A\subset \mathbb{R}$. Then, show that
$$\int\limits_{A}f=\int\limits_{\mathbb{R}}f{1}_A \tag{1}$$
where ${1}_A$ is the characteristic function of $A$ defined as
$${1}_A(x)=\begin{cases}1 & \text{if x\in A,} \\ 0 &\text{if x\notin A.} \end{cases} \tag{2}$$
and $\int\limits_{A}f$ is the Lebesgue integral of $f$ on $A$ defined as:
$$\int\limits_{A}f=\sup\left\{\int\limits_{A}s:0\le s\le f\text{ and }s\text{ is simple}\right\} \tag{3}$$
I can easily prove this property for simple functions so take this for granted:
$$\int\limits_{A}s=\int\limits_{\mathbb{R}}s{1}_A \tag{4}$$
where $s:\mathbb{R}\to [0,\infty]$ is a simple function. Thus to prove (1) we need to show that:
\begin{gather} %omg wall of text code - mixedmath \sup\left\{\int\limits_{A}s:0\le s\le f\text{ and }s\text{ is simple}\right\}=\sup\left\{\int\limits_{\mathbb{R}}s:0\le s\le f{1}_A\text{ and }s\text{ is simple}\right\}\notag\\ \sup\left\{\int\limits_{\mathbb{R}}s{1}_A:0\le s\le f\text{ and }s\text{ is simple}\right\}=\sup\left\{\int\limits_{\mathbb{R}}s:0\le s\le f{1}_A\text{ and }s\text{ is simple}\right\} \tag{5} \end{gather}
My question is how do we prove (5)?
PROOF: It can be easily shown that $\int\limits_{A}f=\int\limits_{A}f1_A$ and since $A\subset \mathbb{R}$, $$\int\limits_{A}f=\int\limits_{A}f1_A\le \int\limits_{\mathbb{R}}f1_A \tag{6}$$ We just have to show that $$\int\limits_{A}f\ge\int\limits_{\mathbb{R}}f{1}_A$$ The last inequality is proven in the answer given by Thomas.E
For a completely different approach you can look at my answer
-
Hint: If $s\colon\mathbb R \to[0,\infty)$ is simple, so is $s|_A$, on the other side, if $s \colon A \to [0,\infty)$ is simple, extend $s$ by zero to a simple function $\sigma\colon \mathbb R \to [0,\infty)$... – martini Jun 1 '12 at 7:04
I edited it to number the equations as I gathered that you had intended. – mixedmath Jun 1 '12 at 7:49
I am back. Dear martini, I agree with your observation but how does it continue from there? – SomeoneContinuous Jun 1 '12 at 9:53
If martini isn't Someone, that Continuous ly monitors your question, you should write @martini instead... and +1, interesting question. – draks ... Jun 1 '12 at 11:00
Thx @draks, now if $f \colon \mathbb R \to [0,\infty]$ is given, suppose $s \le f1_A$ on $\mathbb R$, then $s|_A \le f|_A$, giving you one inequality, for the other suppose $s \le f|_A$, then $\sigma \le f1_A$, giving you the other one ... does this help? – martini Jun 1 '12 at 11:05
Let $\varepsilon>0$. By definition of the Lebesgue integral you find a simple function $0\leq s_{\varepsilon}\leq f 1_{A}$ with $\int_{\mathbb{R}}f 1_{A}\leq \varepsilon+\int_{\mathbb{R}} s_{\varepsilon}$. Note that this implies $s_{\varepsilon}=s_{\varepsilon}1_{A}$ since $0\leq s_{\varepsilon}(x)\leq f(x) 1_{A}(x)=0$ for all $x\in A^{c}$. Using what you have proven to apply for simple functions $(*)$ and the choice of $s_{\varepsilon}$ $(**)$ it follows that \begin{align*} \int_{\mathbb{R}} f1_{A}\leq \varepsilon+\int_{\mathbb{R}} s_{\varepsilon}=\varepsilon+\int_{\mathbb{R}}s_{\varepsilon}1_{A}\overset{(*)}{=} \varepsilon +\int_{\mathbb{A}} s_{\varepsilon} \overset{(**)}{\leq} \varepsilon+\int_{A}f 1_{A}=\varepsilon +\int_{A}f \end{align*} since $1_{A}(x)= 1$ for $x\in A$. Since the choice of $\varepsilon>0$ was arbitrary it follows that \begin{align*} \int_{\mathbb{R}} f1_{A}\leq \int_{A}f. \end{align*} And the other inequality you have already proven.
In the comments section ThomasE. proposed a completely different yet beautiful approach that I will now present here. First a lemma: $$\int\limits_{\mathbb{R}}f=\int\limits_{A}f+\int\limits_{A^c}f$$ Proof: Let $s_1,s_2$ be any simple functions on $A$ and $A^c$ respectively and define $s(x)=\begin{cases}s_1(x) & \text{if$x\in A$,} \\ s_2(x) &\text{if$x\in A^c$} \end{cases}$. Then, \begin{gather}\int\limits_{A}f+\int\limits_{A^c}f=\sup\left\{\int\limits_{A}s_1:0\le s_1\le f|_A\right\}+ \sup\left\{\int\limits_{A^c}s_2:0\le s_2\le f|_{A^c}\right\}\\ \int\limits_{A}f+\int\limits_{A^c}f=\sup\left\{\int\limits_{A}s_1+\int\limits_{A^c}s_2:0\le s_1\le f|_A\text{ and }0\le s_2\le f|_{A^c}\right\} =\sup\left\{\int\limits_{\mathbb{R}}s:0\le s\le f\right\}\\ \int\limits_{A}f+\int\limits_{A^c}f\le \int\limits_{\mathbb{R}}f \end{gather} and \begin{gather} \int\limits_{\mathbb{R}}f=\sup\left\{\int\limits_{\mathbb{R}}s:0\le s\le f\text{ and }s\text{ is simple}\right\}= \sup\left\{\int\limits_{A}s+\int\limits_{A^c}s:0\le s_A\le f\text{ and }0\le s|A^c\le f\right\}\\ \int\limits_{\mathbb{R}}f=\sup\left\{\int\limits_{A}s:0\le s|A\le f\right\}+\sup\left\{\int\limits_{A^c}s:0\le s\le f^c\right\}\le\int\limits_{A}f+\int\limits_{A^c}f \end{gather} Thus, the Lemma is proven. Now $$\int\limits_{A}f=\int\limits_{A}f+\int\limits_{A^c}0=\int\limits_{A}f1_A+\int\limits_{A^c}f1_A=\int\limits_{\mathbb{R}}f1_A$$ This all seems to be correct to my eyes, but is it?
With this lemma it works. If you want to avoid working suprumems over sets in equalities, you may do something like the following too (you already got the idea). $"\Rightarrow"$: Let $0\leq s\leq f$ be arbitrary simple function, and define $s_{1}=s 1_{A}$ and $s_{2}=s 1_{A^{c}}$, whence $s=s_{1}+s_{2}$ and $s_{1}(x)\leq f(x)$ for $x\in A$ and $s_{2}(x)\leq f(x)$ for $x\in A^{c}$. Now using what you had already proven $\int_{R} s=\int_{R} s 1_{A}+s 1_{A^{c}}=\int_{R}s 1_{A}+\int_{R}s 1_{A^{c}}=\int_{A} s_{1}+\int_{A^{c}} s_{2}\leq \int_{A}f+\int_{A^{c}}f$. (Continues below) – Thomas E. Jun 1 '12 at 12:55
... by taking sup over all such $s$ we obtain $\int_{\mathbb{R}} f\leq \int_{A}f+\int_{A^{c}}f$. Can you show the other direction similarly? By choosing arbitrary simple functions $0\leq s_{1}\leq f|_{A}$ and $0\leq s_{2}\leq f|_{A^{c}}$, and showing $\int_{A}s_{1}+\int_{A}s_{2}\leq \int_{\mathbb{R}}f$, and taking supremum over all such $s_{1}$ and $s_{2}$ obtaining the other inequality. – Thomas E. Jun 1 '12 at 12:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 8, "x-ck12": 0, "texerror": 0, "math_score": 0.984985888004303, "perplexity": 1028.4818115207843}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345774525/warc/CC-MAIN-20131218054934-00091-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://rd.springer.com/article/10.1007%2Fs10711-017-0251-z | Geometriae Dedicata
, Volume 191, Issue 1, pp 171–198
# Structure of attractors for boundary maps associated to Fuchsian groups
• Svetlana Katok
• Ilie Ugarcovici
Original Paper
## Abstract
We study dynamical properties of generalized Bowen–Series boundary maps associated to cocompact torsion-free Fuchsian groups. These maps are defined on the unit circle (the boundary of the Poincaré disk) by the generators of the group and have a finite set of discontinuities. We study the two forward orbits of each discontinuity point and show that for a family of such maps the cycle property holds: the orbits coincide after finitely many steps. We also show that for an open set of discontinuity points the associated two-dimensional natural extension maps possess global attractors with finite rectangular structure. These two properties belong to the list of “good” reduction algorithms, equivalence or implications between which were suggested by Zagier.
## Keywords
Fuchsian groups Reduction theory Boundary maps Attractor
37D40
## Notes
### Acknowledgements
We thank the anonymous referee for several useful comments and suggestions.
The second author was partially supported by the Simons Foundation (Grant No. 281407).
## References
1. 1.
Adler, R., Flatto, L.: Geodesic flows, interval maps, and symbolic dynamics. Bull. Am. Math. Soc. 25(2), 229–334 (1991)
2. 2.
Bowen, R., Series, C.: Markov maps associated with Fuchsian groups. Inst. Hautes Études Sci. Publ. Math. No. 50, 153–170 (1979)
3. 3.
Jones, G.A., Singerman, D.: Complex Functions. Cambridge University Press, Cambridge (1987)
4. 4.
Katok, S.: Fuchsian Groups. University of Chicago Press, Chicago (1992)
5. 5.
Katok, S., Ugarcovici, I.: Structure of attractors for $$(a, b)$$-continued fraction transformations. J. Mod. Dyn. 4, 637–691 (2010)
6. 6.
Katok, S., Ugarcovici, I.: Applications of $$(a,b)$$-continued fraction transformations. Ergod. Theory Dyn. Syst. 32, 755–777 (2012)Google Scholar
7. 7.
Maskit, B.: On Poincaré’s theorem for fundamental polygons. Adv. Math. 7, 219–230 (1971)
8. 8.
Series, C.: Symbolic dynamics for geodesic flows. Acta Math. 146, 103–128 (1981)
9. 9.
Smeets, I.: On continued fraction algorithms. Ph.D. thesis, Leiden, (2010)Google Scholar
10. 10.
Zagier, D.: Possible notions of “good” reduction algorithms. Personal communication (2007)Google Scholar | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9347124099731445, "perplexity": 2588.9443489651067}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514577478.95/warc/CC-MAIN-20190923172009-20190923194009-00007.warc.gz"} |
http://www.quantumdiaries.org/2011/03/14/tevatron-experiments-report-new-higgs-search-results | ## View Blog | Read Bio
### Tevatron experiments report new Higgs search results
Improved analysis techniques and more data have made the Tevatron experiments more sensitive to the Higgs boson. CDF and DZero exclude a significant portion of the high-Higgs-mass range.
The CDF and DZero experiments at Fermilab have reached new ground in their quest to find the Higgs boson, a key member of the particle zoo known as the Standard Model. For the first time, each experiment by itself excludes regions of the expected Higgs mass range as more sophisticated data analysis techniques and more data from the Tevatron particle collider have increased their sensitivity to the Higgs boson. This week, the two collaborations, each comprising about 500 scientists, will present details of their results at conferences and seminars around the world, including the Rencontres de Moriond in Italy.
“This makes the Tevatron the frontrunner in the hunt for the Standard Model Higgs boson,” said Fermilab physicist Rob Roser, co-spokesperson for the CDF experiment. “We are getting more mileage out of 10 years worth of Tevatron Run II data.”
The Tevatron, four miles in circumference, is the world’s highest-energy proton-antiproton collider.
“It is impressive to see the progress in the analysis of the Tevatron data from CDF and DZero,” said Fermilab Director Pier Oddone. “Step by step they are narrowing the space in which the Higgs could be hiding.”
Searches by previous experiments and constraints due to precision measurements of the Standard Model of Particles and Forces indicate that the Higgs particle should have a mass between 114 and 185 GeV/c2. (For comparison: 100 GeV/c2 is equivalent to 107 times the mass of a proton.) The CDF and DZero experiments are now sensitive to excluding Higgs bosons with masses from 153 to 179 GeV/c2. Statistical fluctuations in the number of observed particle collisions that mimic a Higgs signal, mixed with collisions that may have produced a Higgs boson, affect the actual range that can be excluded with 95 percent certainty. Combining their independent Higgs analyses, the two experiments now exclude a Higgs boson with a mass between 158 and 173 GeV/c2. The recording of additional collisions and further improved analysis of data will reduce the size of the statistical fluctuations and, over time, could reveal a signal from the Higgs boson.
“Fermilab plans to operate the Tevatron collider until September 2011,” said DZero co-spokesperson Stefan Söldner-Rembold, of the University of Manchester. “During this time, we will increase what is already the largest data set from a hadron collider at the Energy Frontier.”
For the present data analysis, CDF and DZero scientists concentrated on the search for a high-mass Higgs boson that has a mass heavier than 130 GeV/c2. But the Tevatron experiments also continue to look for a low-mass Higgs boson. In this case, the Higgs boson decays mainly into bottom quarks, which would create a different pattern in the CDF and DZero detectors than the decay products of a high-mass Higgs.
“The low-mass scenario now seems to be the more likely option,” said CDF co-spokesperson Giovanni Punzi, of the University of Pisa and the National Institute of Nuclear Physics (INFN) in Italy. “In the coming months, our collaborations will focus on both the high-mass and low-mass scenarios and optimize our analysis techniques for the entire Higgs mass range.”
Said DZero co-spokesperson Dmitri Denisov, of Fermilab, “If the Higgs boson exists, hints of its presence will emerge from the Tevatron data. If it does not exist, the CDF and DZero collaborations expect to rule out the remainder of the allowed mass range and scientists would have to wonder again: how do fundamental particles acquire mass?”
— Kurt Riesselmann
Combined the Tevatron experiments now are sensitive to a Higgs mass from 153 to 179 GeV/c2, but statistical fluctuation reduce the actual mass range that can be excluded so far. For the first time, the experiments now also exclude Higgs mass ranges individually (see CDF and DZero graphics).
Tags: , , , | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8251224756240845, "perplexity": 1753.9744265323043}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049276415.60/warc/CC-MAIN-20160524002116-00020-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/does-a-light-pulse-get-amplified-when-it-goes-to-a-less-dense-medium.370025/ | # Does a light pulse get amplified when it goes to a less dense medium?
1. Jan 16, 2010
### quantum123
Since a string pulse get amplified when it enters a less dense medium such as from the thick heavy rope to a lighter string, why does not light do the same?
2. Jan 16, 2010
### jety
3. Jan 16, 2010
### quantum123
Come to think of it, there is a medium and the medium is the electromagnetic field.
4. Jan 16, 2010
### jety
It never goes to a "different medium".
5. Jan 16, 2010
### Bob S
When light goes from a medium such as glass (index of refraction n=1.5) to air, about 4% of the energy is reflected, and 96% of the energy is transmitted. The ratio of the transverse E (electric) vector to the transverse H (magnetic) vector in light is proportional to 1/n, so the ratio E/H increases when the light goes from glass to air. But the transmitted energy decreases by 4%. So the amplitude E increases, but the transmitted energy, which is proportional to E times H (Poynting vector), decreases.
Bob S
6. Jan 16, 2010
Thanks, Bob! | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8733941316604614, "perplexity": 2292.2314948459343}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738661775.4/warc/CC-MAIN-20160924173741-00151-ip-10-143-35-109.ec2.internal.warc.gz"} |
http://clay6.com/qa/24227/which-of-the-following-can-act-as-sink-for-co-2-and-so-2 | # Which of the following can act as sink for $CO_2$ and $SO_2$
$(a)\;aq\;KOH\qquad(b)\;Plants\qquad(c)\;Seawater\qquad(d)\;Soil$
Seawater
Hence (c) is the correct answer. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9954137802124023, "perplexity": 460.52718065716635}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647600.49/warc/CC-MAIN-20180321082653-20180321102653-00283.warc.gz"} |
https://www.transtutors.com/questions/consider-the-classical-hagen-poiseuille-solution-for-fully-developed-laminar-flow-an-6380635.htm | # Consider the classical Hagen–Poiseuille solution for fully developed laminar flow and heat transfer.
Consider the classical Hagen–Poiseuille solution for fully developed laminar flow and heat transfer through a parallel-plate channel [11]. The plate-to-plate spacing D and the wall heat flux q'' are given. The wall heat flux is uniform; therefore, the longitudinal temperature gradient dT/dx is constant and fixed by design. Derive an expression for the volumetric rate of entropy generation S''' gen and plot the profile of entropy generation distribution over the channel cross section
## Plagiarism Checker
Submit your documents and get free Plagiarism report
Free Plagiarism Checker | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9244734644889832, "perplexity": 2721.149948275829}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243988793.99/warc/CC-MAIN-20210507120655-20210507150655-00332.warc.gz"} |
https://www.physicsforums.com/threads/minimum-polynomial-over-a-field.285603/ | Minimum polynomial over a field
1. Jan 17, 2009
math8
The question is to determine the dimension over Q(rationals)of the extension Q(sqrt(3+2sqrt2)).
So we need to find [Q(sqrt(3+2sqrt2)): Q].
All I can say is that (3+2sqrt2) = (1+sqrt2)^2.
I also know that we're trying to find the degree of the minimum polynomial over Q that has sqrt(3+2sqrt2) as a root.
But I don't know how to proceed.
2. Jan 17, 2009
Hurkyl
Staff Emeritus
I'm not really sure how to explain it, but it seems obvious to me how to go about constructing that particular field extension out of other field extensions that are very easy to understand. (And your observation makes this even more obvious)
You have a problem: you want to compute [Q(sqrt(3+2sqrt2)): Q].
You can simplify this problem.
Do so.
3. Jan 17, 2009
math8
I would say that this degree is 2, and the minimum polynomial is (x^2)-2x-1.
4. Jan 17, 2009
math8
My question is, are there some other details that need to be specified other than showing that (x^2)-2x-1 is irreducible?
5. Jan 17, 2009
Hurkyl
Staff Emeritus
You don't actually need to find the minimal polynomial of that element. You just have to find the degree of the field extension. And you already know the degree of Q(sqrt(3+2sqrt2)); it would be clear if you wrote the extension differently....
If you're not sure about the details, you can always look at the definitions and theorems! It would certainly be faster than asking for help over the internet.
But yes, the relevant theorem is:
Theorem: If f is an irreducible integer polynomial of degree d, and f(a) = 0, then [Q(a) : Q] = d.
Similar Discussions: Minimum polynomial over a field | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8081910610198975, "perplexity": 515.3274351931514}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218186608.9/warc/CC-MAIN-20170322212946-00004-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://encyclopedia.kids.net.au/page/ex/Exponential_function | Encyclopedia > Exponential function
Article Content
Exponential function
The exponential function is one of the most important functions in mathematics. It is written as exp(x) or $e^x$ (where e is the base of the natural logarithm) and can be defined in two equivalent ways, the first an infinite series, the second a limit:
The graph of ex does not ever touch the x axis, although it comes very close.
$\exp(x) = \sum_{n = 0}^{\infty} {x^n \over n!}$
$\exp(x) = \lim_{n \to \infty} \left( 1 + {x \over n} \right)^n$
Here $n!$ stands for the factorial of $n$ and $x$ can be any real or complex number, or even any element of a Banach algebra or the field of p-adic numbers.
If x is real, then exp(x) is positive and strictly increasing. Therefore its inverse function, the natural logarithm ln(x), is defined for all positive x. Using the natural logarithm, one can define more general exponential functions as follows:
$a^x = \exp(\ln(a) x)$
for all $a > 0$ and all real $x$.
The exponential function also gives rise to the trigonometric functions (as can be seen from Euler's formula) and to the hyperbolic functions. Thus we see that all elementary functions except for the polynomials spring from the exponential function in one way or another.
Exponential functions "translate between addition and multiplication" as is expressed in the following exponential laws:
$a^0 = 1$
$a^1 = a$
$a^{x + y} = a^x a^y$
$a^{x y} = \left( a^x \right)^y$
${1 \over a^x} = \left({1 \over a}\right)^x = a^{-x}$
$a^x b^x = (a b)^x$
These are valid for all positive real numbers a and b and all real numbers x. Expressions involving fractions and roots can often be simplified using exponential notation because
${1 \over a} = a^{-1}$
$\sqrt{a} = a^{1/2}$
$\sqrt[n]{a} = a^{1/n}$
Exponential function and differential equations
The major importance of the exponential functions in the sciences stems from the fact that they are constant multiples of their own derivatives:
${d \over dx} a^{bx} = \ln(a) b a^{bx}.$
If a variable's growth or decay rate is proportional to its size, as is the case in unlimited population growth, continuously compounded interest or radioactive decay, then the variable can be written as a constant times an exponential function of time.
The exponential function thus solves the basic differential equation
${dy \over dx} = y$
and it is for this reason commonly encountered in differential equations. In particular the solution of linear ordinary differential equations can frequently be written in terms of exponential functions. These equations include Schrödinger equation and the Laplace's equation as well as the equations for simple harmonic motion.
Exponential function on the complex plane
When considered as a function defined on the complex plane, the exponential function retains the important properties
$\exp(z + w) = \exp(z) \exp(w)$
$\exp(0) = 1$
$\exp(z) \ne 0$
$\exp'(z) = \exp(z)$
for all z and w. The exponential function on the complex plane is a holomorphic function which is periodic with imaginary period $2 \pi i$ which can be written as
$\exp(a + bi) = \exp(a) \cdot (\cos(b) + i * \sin(b))$
where $a$ and $b$ are real values. This formula connects the exponential function with the trigonometric functions, and this is the reason that extending the natural logarithm to complex arguments yields a multi-valued function ln(z). We can define a more general exponentiation:
$z^w = \exp(\ln(z) w)$
for all complex numbers z and w. This is then also a multi-valued function. The above stated exponential laws remain true if interpreted properly as statements about multi-valued functions.
It is easy to see, that the exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the centre at 0, noting that the case of a line parallel with the real or imaginary axis maps to a line or circle.
Exponential function for matrices and Banach algebras
The definition of the exponential function exp given above can be used verbatim for every Banach algebra, and in particular for square matrices. In this case we have
$\exp(x + y) = \exp(x) \exp(y)$
if $xy = yx$ (we should add the general formula involving commutators here.)
$\exp(0) = 1$
exp(x) is invertible with inverse exp(-x)
the derivative of exp at the point x is that linear map which sends u to exp(xu.
In the context of non-commutative Banach algebras, such as algebras of matrices or operators on Banach or Hilbert spaces, the exponential function is often considered as a function of a real argument:
$f(t) = \exp(t A)$
where $A$ is a fixed element of the algebra and $t$ is any real number. This function has the important properties
$f(s + t) = f(s) f(t)$
$f(0) = 1$
$f'(t) = A f(t)$
Exponential map on Lie algebras
The "exponential map" sending a Lie algebra to the Lie group that gave rise to it shares the above properties, which explains the terminology. In fact, since R is the Lie algebra of the Lie group of all positive real numbers with multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie algebra M(n, R) of all square real matrices belongs to the Lie group of all invertible square matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9677357077598572, "perplexity": 180.8575157253688}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507452681.5/warc/CC-MAIN-20141017005732-00368-ip-10-16-133-185.ec2.internal.warc.gz"} |
http://link.springer.com/article/10.1007/BF02746912 | , Volume 44, Issue 3, pp 140-144
# A Majorana-Oppenheimer formulation of quantum electrodynamics
Rent the article at a discount
Rent now
* Final gross prices may vary according to local VAT.
## Summary
It is shown that, according to suggestions by Majorana and Oppenheimer, it is possible to formulate quantum electrodynamics in terms of the same formalism in a Dirac’s electron theory, thus giving a more direct physical interpretation to the quantum field which describes the photon. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.813229501247406, "perplexity": 1092.868913892629}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802770399.32/warc/CC-MAIN-20141217075250-00105-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/114177/for-a-finite-flat-etale-morphism-fy-to-x-is-f-1-y-deg-f-1-x-nilpote | # For a finite flat (etale?) morphism $f:Y\to X$, is $f_*1_Y-\deg f . 1_X$ nilpotent in $A^0(X)$, where $A^*$ is the algebraic cobordism?
Let $A^\ast$ be an algebraic oriented cohomology theory (i.e. it is equipped with certain push-forwards for projective morphisms of smooth varieties over the base field $k$; see section 2 of http://www.math.uiuc.edu/K-theory/0535/orient.pdf for more detail); let $f:Y\to X$ be a finite morphism of smooth varieties whose degree is $d$. I would like to prove the following conjecture: if $f^\ast h=0$ for $h\in A^\ast (X)$, then $d^lh=0$ for some $l>0$ (one cannot take $l=1$ here when $A^*$ is the K-theory). To this end it suffices to verify that $f_\ast 1_Y-d$ is nilpotent in $A^0(X)$ (since $f_\ast f^\ast h=f_\ast 1_Y\cdot h$ by the property (v) in the reference cited). It seems sufficient to prove the latter for $A^\ast$ being the algebraic cobordism (as defined by Levine and Morel), since this is the universal algebraic oriented cohomology theory.
I would like to say that $f_\ast 1_Y-d$ is supported in codimension 1. Does $A^0(X)$ possess a multiplicative coniveau filtration? If $f$ is generically etale, then I can use the fact that $f'_{*}1_{Y'}=d$ for $f':Y'\to X'$ being the (etale) morphism of generic points; Levine proves this in his cobordism book.
Is there a better way to prove my conjecture (that would work even if $f$ is not generically etale)?
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9846911430358887, "perplexity": 72.41423724550145}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131299054.80/warc/CC-MAIN-20150323172139-00208-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://sciencing.com/convert-volume-percent-weight-percent-8740558.html | # How to Convert Gas From a Volume Percent to a Weight Percent
••• Bluberries/iStock/GettyImages
Print
Volume percents characterize the composition of gas mixtures. An example of a gas mixture is air that consists of primarily oxygen and nitrogen gasses. Gas mixtures obey the ideal gas law that sets the relation between the gas volume, temperature and pressure. According to this law, the volume is proportional to the number of moles of a gas, and therefore, the mole percentage is the same as the volume percents for gas mixtures. Weight percents refer to mass of gasses in the mixtures and are required for stoichiometry calculations in chemistry.
Write down the composition of the gas mixture. For example, the mixture consists of oxygen O2 and nitrogen N2, and their respective volume percents are 70 and 30.
Calculate the molar mass of the first gas in the mixture; in this example, the molar mass of oxygen, O2 is 2×16 = 32 grams per mole. Note that the atomic weight of oxygen is 16, and the number of the atoms in the molecule is 2.
Calculate the molar mass of the second gas in the mixture; in this example, the molar mass of nitrogen, N2 is 2× 4 = 28 grams per mole. Note that the atomic weight of nitrogen is 14, and the number of the atoms in the molecule is 2.
Divide the volume percent of the first gas by 100, and then multiply the respective molar mass to calculate the weight of the first gass in one mole of the mixture. In this example, the mass of the oxygen is:
\frac{70}{100}\times 32=22.4\text{ grams}
Divide the volume percent of the second gas by 100, and then multiply the respective molar mass to calculate the weight of the second gass in one mole of the mixture. In this example, the mass of the oxygen is:
\frac{30}{100}\times 28=8.4\text{ grams}
Add up the weights of the gasses to compute the mass of one mole of the mixture. In this example, the mass of the mixture is 22.4 + 8.4 = 30.8 grams.
Divide the weight of the first gas by the mass of the mixture, and then multiply by 100 to calculate the weight percent. In this example, the weight percent of oxygen is:
\frac{22.4}{30.8}\times 100 = 72.7
Divide the weight of the second gas by the mass of the mixture, and then multiply by 100 to calculate the weight percent. In this example, the weight percent of nitrogen is:
\frac{8.4}{30.8}\times 100 = 27.3
Dont Go!
We Have More Great Sciencing Articles! | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9433850646018982, "perplexity": 563.6423153351125}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500044.16/warc/CC-MAIN-20230203055519-20230203085519-00371.warc.gz"} |
http://mathhelpforum.com/algebra/203875-complex-numbers-set-points.html | # Math Help - Complex numbers - set of points
1. ## Complex numbers - set of points
{z ∈ C | Im(z) = 2 Re(z) − 2}
2. ## Re: Complex numbers - set of points
Originally Posted by Tala
{z ∈ C | Im(z) = 2 Re(z) − 2}
That is just the set $\{(x,y)|~y=2x-2\}$.
3. ## Re: Complex numbers - set of points
But how do I draw it in the complex plane?
4. ## Re: Complex numbers - set of points
Originally Posted by Tala
But how do I draw it in the complex plane?
The same way you graph it in $R^2$.
$\text{Re}(z)$ is the $x\text{-axis}$ and $\text{Im}(z)$ is the $y\text{-axis}$.
It is a line.
5. ## Re: Complex numbers - set of points
But how do I plot this in Re(z)-2 ?
6. ## Re: Complex numbers - set of points
Originally Posted by Tala
But how do I plot this in Re(z)-2 ?
Can you plot $y=2x-2~?$ If you can that is the same graph as $\text{Im}(z)=2\text{Re}(z)-2$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9257501363754272, "perplexity": 1903.4297553659974}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802765610.7/warc/CC-MAIN-20141217075245-00043-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://deepai.org/publication/correlated-bandits-or-how-to-minimize-mean-squared-error-online | # Correlated bandits or: How to minimize mean-squared error online
While the objective in traditional multi-armed bandit problems is to find the arm with the highest mean, in many settings, finding an arm that best captures information about other arms is of interest. This objective, however, requires learning the underlying correlation structure and not just the means. Sensors placement for industrial surveillance and cellular network monitoring are a few applications, where the underlying correlation structure plays an important role. Motivated by such applications, we formulate the correlated bandit problem, where the objective is to find the arm with the lowest mean-squared error (MSE) in estimating all the arms. To this end, we derive first an MSE estimator based on sample variances/covariances and show that our estimator exponentially concentrates around the true MSE. Under a best-arm identification framework, we propose a successive rejects type algorithm and provide bounds on the probability of error in identifying the best arm. Using minimax theory, we also derive fundamental performance limits for the correlated bandit problem.
## Authors
• 3 publications
• 12 publications
• ### Multiple Identifications in Multi-Armed Bandits
We study the problem of identifying the top m arms in a multi-armed band...
05/14/2012 ∙ by Sébastien Bubeck, et al. ∙ 0
• ### Adaptive Monte Carlo via Bandit Allocation
We consider the problem of sequentially choosing between a set of unbias...
05/13/2014 ∙ by James Neufeld, et al. ∙ 0
• ### Best Arm Identification for Contaminated Bandits
We propose the Contaminated Best Arm Identification variant of the Multi...
02/26/2018 ∙ by Jason Altschuler, et al. ∙ 0
• ### Risk-aware Multi-armed Bandits Using Conditional Value-at-Risk
Traditional multi-armed bandit problems are geared towards finding the a...
01/04/2019 ∙ by Ravi Kumar Kolla, et al. ∙ 0
• ### Gaussian Data-aided Sensing with Multichannel Random Access and Model Selection
In this paper, we study data-aided sensing (DAS) for a system consisting...
12/04/2019 ∙ by Jinho Choi, et al. ∙ 0
• ### On the bias, risk and consistency of sample means in multi-armed bandits
In the classic stochastic multi-armed bandit problem, it is well known t...
02/02/2019 ∙ by Jaehyeok Shin, et al. ∙ 0
• ### Robust Interference Management for SISO Systems with Multiple Over-the-Air Computations
In this paper, we consider the over-the-air computation of sums. Specifi...
04/21/2020 ∙ by Jaber Kakar, et al. ∙ 0
##### This week in AI
Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.
## 1 Introduction
The traditional multi-armed bandit problem aims to find the arm with the highest payoff. This is often motivated by practical applications such as to identify an ad with highest payoff in showing to users, or identifying a strategy with maximum payoff. In this work, we consider a setting with the objective being the identification of an arm/node which best captures the entire information of a system, i.e., the identification of arm which can best estimate all the other arms. In contrast to the traditional multi-armed bandit problem, this objective involves an estimation of the correlation structure among the various arms. This is motivated by several practical applications. For instance, in internet-of-things, sensors are used to take measurements from multiple locations with the objective of estimating the underlying parameter, e.g., temperature, over a region. Resource constraints mean that it might not possible to place sensors at the desired level of granularity. However, an estimate of the underlying distribution enables one to form an estimate of the parameter at points not measured. This estimate of the statistics of the underlying randomness is often formed using limited measurements from multiple points, before choosing the final location of the sensors. Another application of interest is in identifying members who can best approximate the social network. Instances include sensors used for measuring temperature in a region (Guestrin et al., 2005), thermal sensors on microprocessors (Long et al., 2008), optimizing queries over a sensornet (Deshpande et al., 2004) and placing sensors to detect contaminants in a water distribution network (Krause et al., 2008). In all these applications, the underlying correlation structure plays an important role. Problems of similar interest have also been studied in the realm of information theory in Boda and Narayan (2017), Boda (2018).
In this paper, we formulate a variant of the stochastic -armed bandit problem, where the objective is to identify the arm that best estimates all the other correlated arms. We measure how good an arm can estimate other arms using the mean-squared error (MSE) criterion, defined as follows:
−2ex\cEi≜K∑j=1E[(Xj−E[Xj|Xi])2]. (1)
We assume that the arms
are correlated sub-Gaussian random variables (r.v.s).
Paul et al. (2014) consider a celluar network application, where the goal is to to monitor large communication networks with huge traffic. Since observing every node is computationally intensive, companies such as AT&T use measurements from various nodes to identify a subset which best captures the average behavior of the network. The requirement is for an algorithm that reduces the data acquisition cost by identifying the most-correlated subset of nodes, while using a minimum number of sample measurements. The authors in (Paul et al., 2014) show that a model approximating the underlying nodes as Gaussian r.v.s is useful and reliable.
Closely related problems in other application contexts include (i) selecting a few blogs that capture the information cascade (Leskovec et al., 2007); (ii) finding a subset of people that captures best the average behavior of a community; To put it differently, the notions of centrality in the context of document/news summarization (Erkan and Radev, 2004) and prestige in social networks (Heidemann et al., 2010) are closely related to the MSE objective in (1). In each of these applications, there is a cost associated with acquiring data and the challenge is to find the most correlated subset of blogs/people/etc using minimal observations about the community.
We study the basic problem of identifying the arm which has the best MSE in estimating the remaining arms in a multi-armed bandit framework. We consider the best arm identification setting (Audibert et al., 2010; Kaufmann et al., 2015), where a bandit algorithm is given a fixed sampling budget, and is evaluated based on the probability of incorrect identification. Challenges encountered for such a setup include:
(i) Any estimate for the MSE requires estimation of the underlying correlations, without assuming knowledge of the variances.
(ii) Estimate of the MSE of an arm involves estimating the correlation of arm with the remaining arms. This requires samples from all pairs of arms associated with . In particular, sampling arm alone would be insufficient towards estimating arm ’s MSE; and hence
(iii) A bandit algorithm needs to optimize sampling across all pairs of arms and not just among arms. This requires intricate decisions over a larger set, in contrast to the classical mean-value optimizing algorithms in a best arm identification framework.
We summarize our contributions below.
• First, we introduce a new formulation to study the identification of arm which best estimates all arms. We first design an estimate and develop the concentration bound for the estimate of mean-squared error formed from available samples. Our estimator builds on the difference estimator introduced in (Liu and Bubeck, 2014), but estimation is technically more challenging in our setting as the underlying variances are not known and unlike Liu and Bubeck (2014), not necessarily assumed to be one.
• Second, we analyze a nonadaptive uniform sampling strategy (i.e., a strategy that pulls each pair of arms an equal number of times) and propose an algorithm inspired by popular successive rejects (SR) (Audibert et al., 2010) for best-arm identification, but more intricate due to the nonlinearity of the objective function, the MSE objective function (1). A naive SR strategy that operates over phases, discarding all arm pairs associated with the arm having lowest empirical MSE is suboptimal. Instead, our SR algorithm maintains active sets for arms as well as pairs and discards a pair only if both constituent arms are out of the active arms set. We provide an upper bound, on the probability of error in identifying the best arm, for our SR algorithm and the latter bound involves a hardness measure that factors in the gaps in MSEs as well as the correlations, which are specific to the correlated bandit problem. As in the classic bandit setup, the upper bound shows that SR algorithm requires fewer samples to find the best arm in comparison to a uniform sampling strategy, especially, when is large and the underlying gaps (difference between MSE of optimal and suboptimal arms) are uneven.
• Third, we prove a lower bound over all bandit problems with a certain hardness measure and to the best of our knowledge, this is the first lower bound for the correlated bandit problem that involves adaptive sampling strategies. The lower bound involves constructing problem transformations, where the optimal arm is “swapped” with one of the sub-optimal ones, resulting in problem instances. Unlike in the classic setup, any local change in the distribution of an arm impacts the MSE of all the other arms. Moreover, pulling arm pairs instead of individual arms makes the lower bound technically more challenging.
In (Liu and Bubeck, 2014), which is the closest related work, the authors consider a bandit problem, where the objective is to identify a subset of arms most correlated among themselves, i.e., to identify the local correlation structure within a subset of arms themselves. On the other hand, our problem is about forming global inference from samples of subsets of arms to identify the arm that is most correlated to the remaining arms. In Liu and Bubeck (2014), the authors consider a setting with positively correlated arms with unit variance, making the estimation task and hence, the overall best arm identification slightly easier. As we show later in Section 3, their estimation scheme does not extend to the more general non-unit variance setup that we consider. Finally, we also prove fundamental limits on the performance of any correlated bandit algorithm, through information-theoretic lower bounds, and to the best of our knowledge, no lower bounds exist for a correlated bandit problem.
The rest of the paper is organized as follows: In Section 2, we formalize the correlated bandit problem. In Section 3, we present the MSE estimation scheme and derive a concentration bound for our estimator. In Section 4, we examine uniform sampling strategy, while in Section 5, we present a successive-rejects type algorithm. In Section 6, we present a lower bound for the correlated bandit problem. We provide the convergence proofs in Section 7. While not the thrust of this work, we provide a few illustrative examples in Section 8 showing the performance of our successive-rejects type algorithm. Finally, in Section 9 we provide our concluding remarks.
## 2 Model
We consider a set of correlated arms , whose samples are i.i.d. in time. For each arm , let denote the minimum mean-squared error (MMSE) of estimating all the remaining arms, i.e.,
\cEi≜ming E[(X\cM−g(Xi))T(X\cM−g(Xi))]. (2)
Consider the special case of jointly Gaussian r.v.s
, whose joint probability distribution is characterized by the mean (taken to be zero for the sake of expository simplicity), and
covariance matrix :
Σ=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣σ21ρ12σ1σ2…ρ1Kσ1σKρ12σ1σ2σ22…ρ2Kσ2σK⋮⋮⋱⋮ρ1Kσ1σKρ2Kσ2σK…σ2K⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦. (3)
In the above, , is the variance of arm and , the correlation coefficient between arms and .
The best estimate , which achieves the minimum in (2), is known to be the MMSE estimate. For zero-mean jointly Gaussian r.v.s, this is given by (cf. Chapter 3 of (Hajek, 2009))
g∗(Xi)=E[X\cM|Xi]=[E[X1|Xi]…E[XK|Xi]]T, with E[Xj|Xi]=E[XjXi]E[X2i]Xi=ρijσjσiXi. (4)
The corresponding MMSE for arm is
(5)
Note that there is no error in arm estimating itself and the error in estimating the th arm is characterized by the correlation between and and the relevant variances. Further, the MMSE estimate for the case of Gaussian r.v.s is linear. In the more general case of non-Gaussian r.v.s, the MMSE estimate is typically nonlinear and any online computation is typically a computationally intense task. In such cases, we restrict ourselves to employing an optimal linear estimator which is still defined as the right side of (4). Thus, the right-side of (5) holds for all optimal linear estimators, with it being optimal for Gaussian r.v.s.
We consider a setting where the arms are sub-Gaussian, and focus on linear estimators. We recall the definition of sub-Gaussianity below. A r.v. is said to be -sub-Gaussian if For equivalent characterizations of sub-Gaussianity, the reader is referred to Theorem 2.1 of Wainwright (2015).
Clean this part We consider a fixed budget best-arm identification framework, and the interaction of our (bandit) algorithm with the environment is given below.
Notice that, in each round, the algorithm above pulls a pair of arms, and this is necessary to learn the underlying correlation structure.
In our setting, the performance metric associated with each arm is its MSE , and the optimal arm, say , has the lowest MSE, i.e.,
The objective is to minimize the probability of error in identifying the best arm, i.e.,
\Prob^in≠i∗, (6)
where is the estimate of the best arm based on samples.
For the suboptimality of the arm is quantified by its gap in its MSE with respect to the optimal arm, i.e., The notation is used to refer to the best arm (with ties broken arbitrarily), i.e., s are ordered gaps of the arms:
Note that the problem with reduces to identifying the arm with higher variance and has no dependence on the correlation between the arms. The analysis of this case would be similar (estimate variance instead of mean) to the classical bandit problems and differs considerably from the setting with arms, which is the setting assumed hereafter.
## 3 MSE Estimation
Let denote the set of
i.i.d. samples obtained from the bivariate Gaussian distribution corresponding to the pair of arms
. To identify the optimal arm, we form an estimate of to which end we form estimates for the variances and the correlation coefficient . We employ the following estimators for the aforementioned quantities: For any ,
^ρij≜1−12⎛⎜⎝¯¯¯¯¯X2i^σ2i+¯¯¯¯¯X2j^σ2j−2¯¯¯¯¯¯¯¯¯¯¯¯XiXj^σi^σj⎞⎟⎠, (7) ^σ2i=¯¯¯¯¯X2i, ^σ2j=¯¯¯¯¯X2j,~{} where ~{} ¯¯¯¯¯X2i=1nn∑i=1X2it, and ¯¯¯¯¯¯¯¯¯¯¯¯XiXj=1nn∑i=1XitXjt.
The estimate for in (7) is akin to that proposed in Liu and Bubeck (2014), which considers a simpler setting where all the arms are known to have unit variance, i.e., For the unit variance setup, Liu and Bubeck (2014) establish via a likelihood ratio test that the difference based estimator for
1−12(¯¯¯¯¯X2i+¯¯¯¯¯X2j−2¯¯¯¯¯¯¯¯¯¯¯¯XiXj) (8)
is advantageous over the natural estimator for : . This superiority depends explicitly on the a priori knowledge of the variances being one, which is not applicable to the general setting considered here, i.e., a setting where the variances are not necessarily one. However, to exploit the optimality of the likelihood ratio test, we express the estimator above in the spirit of (8) which depend on the estimates of the variances to scale the samples to obtain
Unlike the unit variance setup of Liu and Bubeck (2014), it is not possible to obtain a difference based estimator in our setting. Nevertheless, concentrates faster as approaches and this can be argued as follows: On the high probability event , we have
\lx@paragraphsign((1−^ρij)−(1−ρij)≥ϵ,C) =\lx@paragraphsign(Yijn2n−1≥ϵ(1−ρij),C) ≤\lx@paragraphsign(¯Yijn2n−1≥ϵ2(1−ρij)) ≤\concsubexpTwotermsnϵ2(1−ρij), where Yijn≜1(1−ρij)⎛⎜⎝¯¯¯¯¯X2i^σ2i+¯¯¯¯¯X2j^σ2j−2¯¯¯¯¯¯¯¯¯¯¯¯XiXj^σi^σj⎞⎟⎠, and ¯Yijn≜1(1−ρij)⎛⎜⎝¯¯¯¯¯X2iσ2i+¯¯¯¯¯X2jσ2j−2¯¯¯¯¯¯¯¯¯¯¯¯XiXjσiσj⎞⎟⎠.
For any arm , the corresponding MSE is estimated using the quantities defined in (7) as follows:
^\cEi≜^σ2j(1−^ρ2ij)+∑p≠i,j^σ2p(1−^ρ2ip). (9)
The main result concerning the exponential concentration of the estimate around the true MSE is presented below. (MSE Concentration) Assume . Let be the MSE estimate given in (9), for . Then, for any , and for any , we have
\lx@paragraphsign(∣∣^\cEi−\cEi∣∣>ϵ)≤14Kexp(−nl2ϵ2cK3),
where is a universal constant, and . In the above, it suffices to look at , since is less than , owing to the assumption that .
###### Proof.
See Section 7.2. ∎
The claim in Proposition 3 holds for the more general case of sub-Gaussian r.v.s . However, in this case, the MSE is best in the class of linear estimators, and is not necessarily the minimum MSE estimator.
## 4 Uniform Sampling
A simple approach towards identifying the best arm is to select each pair equal number of times, estimate the MSE errors and recommend the arm with the lowest MSE estimate to be optimal, i.e., the samples used for estimation are For uniform sampling, the probability of error in identifying the optimal arm is
\lx@paragraphsign(^An≠i∗) ≤84K2exp⎛⎝−nl2Δ2(1)cK7⎞⎠,
where is a universal constant.
###### Proof.
Proof uses Proposition 3 along with an union bound. See Section 7.3. ∎
If the correlations between all pairs of arms and the variances of all the arms are similar, then in the absence of this prior knowledge, the optimal strategy would involve sampling all pairs of arms an equal number of times. However, when this is not the case, uniform sampling might be a strictly inferior strategy because it fails to gather more samples which can enable a better estimation of MSE of arms with MSE close to the optimal arm. We present below a strategy which tries to sequentially zone in on a reduced set of possible candidates for the optimal arm and then sample the pairs of arms involved in the MSE estimation of these arms approximately equal number of times to get a better probability of error in identifying the best arm.
## 5 Successive Rejects
The successive rejects (SR) algorithm, which pulls pairs of arms111With abuse of notation, is used to denote the (unordered) pair of arms . to identify the arm which minimizes MSE, operates over phases as described in Figure 1 . The idea is to maintain a set of active arms and pairs of arms (for phase , these are denoted by and ) and eliminate arms (and some of their corresponding pairs) that have high MSE. The elimination scheme employed in Figure 1 departs significantly from the approach adopted in the classic SR algorithm for finding the arm with highest mean. To illustrate this, consider a setting with arms. If arms are out of contention after phase , . In the second phase, all the pairs in are pulled number of times. Now, if arm is out of contention at the end of this phase, the pairs and will be removed from and no longer be pulled in the later phases.
Notice that a strategy that finds the worst arm according to empirical MSE estimates and discards all pairs associated with that arm is clearly suboptimal, because samples from some of the discarded pairs of arms are essential to form estimate of MSE of arms which remain in contention. For e.g., in a -armed bandit setting, suppose that we discard all pairs associated with arm in some round. This would impact the quality of MSE estimate of arm , since the pair would be useful in training a better estimate of via .
Before presenting the main result that bounds the probability of error in identifying the best arm of the algorithm in Figure 1, we present the following problem complexities that capture the hardness of the learning task at hand (i.e., the order of number of samples required to find the best arm with reasonable probability):
H2=maxiiΔ2(i) and ¯¯¯¯¯H=∑i≠i∗1Δ2i. (10)
The quantities and , have a connotation similar to that in the classical bandit setup and satisfy222The proof follows in a similar fashion as in the classic bandit setup (Audibert et al., 2010) and is given in Appendix 7.1 for the sake of completeness.
H2≤¯¯¯¯¯H≤¯¯¯¯¯¯¯log(K)H2,
where is as defined in Figure 1.
Observe that the problem complexities depend both on the variances of the arms and the correlation between the arms through the gaps. The probability of error in identifying the best arm of SR satisfies
P(^An≠i∗)≤84K3exp(−l2cK5(n−K)¯¯¯¯¯¯¯log(K)H2),
where is a universal constant.
###### Proof.
See Section 7.4. ∎
From Theorem 4, it is apparent that an uniform sampling strategy would require samples to achieve a certain accuracy, while our SR variant for correlated bandits would require number of samples. SR scores over uniform sampling w.r.t. dependence on the number of arms because in our SR algorithm an increasing number of pairs of arms are removed from contention in successive phases. More importantly, SR has better dependence on the underlying gaps when compared to uniform sampling. In problem instances where the gaps are uneven, SR finds the best arm much faster than uniform sampling.
## 6 Lower Bound
To obtain the lower bound, we consider a -armed Gaussian bandit problem with the underlying joint probability distribution governed by the following covariance matrix:
Σ=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣1ρρρ…ρρ1ρ2ρ2…ρ2ρρ21ρ3…ρ3⋮⋮⋮⋮⋱⋮ρρ2ρ3…ρK−11⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦. (11)
Observe that is a valid covariance matrix and is positive semi-definite. The MSEs corresponding to arms are and more generally
\cEi=(i−1)−i−1∑i=1ρ2i+(K−i)(1−ρ2i),i=1,…,K.
Hence, we have the following order on the MSEs:
An approach in recent papers, cf. (Audibert et al., 2010; Kaufmann et al., 2015), for establishing lower bound for best-arm identification is to transform the bandit problem so that one of the sub-optimal arm is turned into an optimal one, while not affecting the rest of the arms. However, our setting involves correlated arms, with the correlation factors appearing in the mean-squared error objective and hence, one cannot make a sub-optimal arm optimal in a standalone fashion. We swap pairs of arms to interchange the MSE of a sub-optimal arm with that of the optimal arm and this introduces major deviations in the proof as compared classic -armed case, as we shall soon see. We describe our problem transformations next.
We form transformations of the bandit problem formulated at the beginning of this section. For “problem ,” arm is the best and for achieving this, we swap the first and th rows in . Let be the pdf associated with the given problem as in (11), and represent the pdf of the transformed bandit problem, where represents the
th transformation. Since we consider arms whose samples are i.i.d. in time, the joint distribution of
samples is a product distribution of the underlying random variables and for the transformed problem by . For compactness, we use , and , .
For any problem with , we define and and the min-max probability of error in identifying the optimal arm is given by the theorem below. For any bandit strategy that returns the arm after rounds, there exists a transformation of the covariance matrix such that the probability of error on the transformed problem satisfies
max1≤m≤K\lx@paragraphsignm(^An≠m)≥16exp(−6nKHlb−(K2)n~ϵn),
where is the problem complexity term,
, and .
###### Proof.
See Section 7.5. ∎
Note the gap between the upper and lower bounds on the probability of error in Theorems 1 and 6
. The problem complexity term in the upper bound involved the square of the gaps, whereas the lower bound involves just the gaps. We believe the upper bound for SR algorithm is optimal in terms of gap dependence and it would be interesting future work to establish a lower bound that involves squares of the gaps. In the lower bound proof, the Kullback-Leibler divergence terms for the transformed problems were bounded above by the gaps (for e.g., see (
LABEL:eq:kl-bound-lb) in Section LABEL:sec:proof-sketch), leading to an overall lower bound with complexity . Nevertheless, the current proof is challenging owing to (i) pairs of arms being pulled in each round; (ii) the covariance matrix in (11) is non-trivial and its problem transformations are novel and finally, (iii) arriving at the bound for the aforementioned KL-divergence terms requires non-trivial algebraic effort.
## 7 Convergence Proofs
### 7.1 Problem complexities
We begin by showing the relation between the different problem complexities defined in the Section 2.
¯¯¯¯¯H =∑i1(Δi)2=∑i1i(i1(Δi)2)≤¯¯¯¯¯¯¯log(K) maxi i(Δ(i))2=¯¯¯¯¯¯¯log(K)H2 ¯¯¯¯¯H =∑i1(Δi)2≥~i(Δ(~i))2=H2, ~i is the optimizer of H2 and lastly ¯¯¯¯¯H ≥1KuHlb, since Δi≤Ku, i=1,…,K.
### 7.2 Proof of Proposition 3
For establishing the main claim in Proposition 3, we shall use two well-known sub-exponential concentration bounds, which we are given below. (Concentration of sample variance) Let be independent sub-Gaussian r.v.s with common parameter . Let . Then, we have the following bound for any :
P(^σ2n>σ2+ϵ)≤exp(−n8min(ϵ2σ4,ϵσ2)), and P(^σ2n<σ2−ϵ)≤exp(−n8min(ϵ2σ4,ϵσ2)).
###### Proof.
By definition, it follows that the square of a sub-Gaussian r.v. is sub-exponential. The main claim now follows from the concentration bound for sub-exponential r.v.s in Proposition 2.2 of (Wainwright, 2015). ∎
(Concentration of sample standard deviation)
Under conditions of Lemma 7.2, letting , we have
P(^σn>σ+ϵ)≤exp(−nϵ28σ4), and P(^σn<σ−ϵ)≤exp(−nϵ28σ4), for any ϵ≥0.
###### Proof.
Consider
a vector of i.i.d. standard Gaussian r.v.s and a
-Lipschitz function . Then, using Gaussian concentration for Lipschitz functions (cf. Section 2.3 of Wainwright (2015))
\Probf(Z)−\Ef(Z)>ϵ≤exp(−nϵ22L2). (12)
For with i.i.d. Gaussian r.v.s , consider and . Observing that is -Lipschitz, changing the variable from to and using (12), we obtain
P(^σn>σ+ϵ)≤exp(−nϵ22σ2).
The other inequality bounding the left tail follows by an argument similar to above.
For the case of sub-Gaussian r.v.s, the main claim can be inferred from Theorem 3.1.1 in Vershynin (2016) and we provide the proof details below for the sake of completeness. Observe that
\Prob ⎷1nn∑i=1Z2i−1>ϵ ≤\Prob1nn∑i=1Z2i−1>max(ϵ,ϵ2) ≤exp(−nϵ28)
The first inequality above holds because implies , for any , while the final inequality follows from Lemma 7.2 after observing that is sub-exponential since is sub-Gaussian. The main claim follows by changing the variable to from . As before, the other inequality bounding the left tail follows by a completely parallel argument. ∎
Next, we state and prove a result that establishes exponential concentration of the sample correlation coefficient. Notice that the MSE estimate in (9) is comprised of sample variances and sample correlation coefficients. To prove that the MSE estimate concentrates, we shall use Lemma 7.2 for terms involving sample variances, and the lemma below for terms involving sample correlation coefficients.
(Concentration of sample correlation coefficient) For independent Gaussian rvs , with mean zero and covariance matrix as defined in (3) and with , formed from samples using (7), for any , and for any , we have
\lx@paragraphsign(∣∣^ρij−ρij∣∣>ϵ)≤26exp(−n8136(1+η)min(lϵ3,(lϵ3)2)),
where is a positive constant satisfying , .
###### Proof.
We bound for and . The analysis below holds in general.
Consider the following event:
\B={σ21−ϵ≤^σ21≤σ21+ϵ, σ22−ϵ≤^σ22≤σ22+ϵ,σ1−ϵ≤^σ1≤σ1+ϵ, σ2−ϵ≤^σ2≤σ2+ϵ}.
Using Lemmas 7.27.2,
\lx@paragraphsign(\Bc) ≤\lx@paragraphsign(^σ21>σ21+ϵ)+\lx@paragraphsign(^σ21<σ21−ϵ)+\lx@paragraphsign(^σ22>σ22+ϵ)+\lx@paragraphsign(^σ22<σ22−ϵ) \lx@paragraphsign(^σ1>σ1+ϵ)+\lx@paragraphsign(^σ1<σ1−ϵ)+\lx@paragraphsign(^σ2>σ2+ϵ)+\lx@paragraphsign(^σ2<σ2−ϵ) (13)
where the penultimate inequality relies on the assumption that . inlineinlinetodo: inline Why is the assumption required? It seems clean without out. Still haven’t seen a strong reason yet except that it will appear as throughout. As discussed, its easier to manage the constants this way. Feel free to genearlize.
Let and . Then, on the event , we have
\lx@paragraphsign(^ρ12<ρ12−ϵ,\B)=\lx@paragraphsign((1−^ρ12)−(1−ρ12)>ϵ,\B) =\lx@paragraphsign((1−ρ12)Y12n2−(1−ρ12)>ϵ,\B) ≤\lx@paragraphsign((1−ρ12)(¯Y12n2−1)>ϵ2,\B) ≤\lx@paragraphsign(¯Y12n2−1≥ϵ2(1−ρ12))+\lx@paragraphsign⎛⎜⎝¯¯¯¯¯X212(1^σ21−1σ21)>ϵ6,\B⎞⎟⎠ (14)
We now bound each term on the RHS above. The first term in (14) is bounded by an application of Lemma 7.2 as follows:
\lx@paragraphsign(¯Y12n2−1≥ϵ2(1−ρ12)) ≤\lx@paragraphsign(¯Y12n2−1≥ϵ4)(Since |ρij|≤1) ≤\concsubexpTwotermsnϵ2
inlineinlinetodo: inline Why is the equation above true? It is and not here. Also is not a centered -squared rv. has the true variances, and so it is sub-exponential, since it is like .
The second term in (14) is bounded as follows:
\lx@paragraphsign⎛⎜⎝¯¯¯¯¯X212(1^σ21−1σ21)>ϵ6,\B⎞⎟⎠≤\lx@paragraphsign(σ21−^σ21σ21≥ϵ3)≤\lx@paragraphsign(σ21−^σ21>lϵ3) ≤ \concsubexpTwotermsnlϵ3
where the penultimate inequality follows from an application of Lemma 7.2 and the last inequality uses the fact that . inlineinlinetodo: inlineShould this be ? Why do we need a condition on here? Doesn’t look like its used is a typo inlineinlinetodo: inline Also even if variances are less than , gaps can be as large as , so in general can be . The third term in (14) can be bounded in a similar fashion. The last term in (14) is bounded as follows:
\lx@paragraphsign(¯¯¯¯¯¯¯¯¯¯¯¯¯X1X2(−1^σ1^σ2+1σ1σ2)>ϵ3,\B) ≤\lx@paragraphsign(^σ1(^σ2−σ2)+σ2(^σ1−σ1)>ϵ^σ1σ1^σ2σ23¯¯¯¯¯¯¯¯¯¯¯¯¯X1X2,\B) ≤\lx@parag</ | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9555753469467163, "perplexity": 671.0665718258317}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738015.38/warc/CC-MAIN-20200808165417-20200808195417-00180.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/198014-how-rewrite-logarithm-print.html | # how to rewrite logarithm?
• Apr 27th 2012, 12:09 PM
kukid13
how to rewrite logarithm?
given the logarithm 7^(log7(x^2)-8log7(y)) , how can you rewrite as a single term that doesnt contain a logarithm?
• Apr 27th 2012, 12:29 PM
SpringFan25
Re: how to rewrite logarithm?
$\displaystyle 7^{log_7 x^2 - log_7 y^8}$
$\displaystyle 7^{log_7 \frac{x^2}{y^8}}$
can you finish?
• Apr 27th 2012, 12:43 PM
kukid13
Re: how to rewrite logarithm?
no i can get it there, but thats where i get stuck and cant figure out how to get it into a single term without a log
• Apr 27th 2012, 01:39 PM
Ivanator27
Re: how to rewrite logarithm? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9490689039230347, "perplexity": 2606.2494701768314}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794872766.96/warc/CC-MAIN-20180528091637-20180528111637-00031.warc.gz"} |
https://www.physicsforums.com/threads/whered-the-energy-come-from.352710/ | # Where'd the energy come from?
1. Nov 7, 2009
### trivia1
A rocket in space, of mass 1kg, accelerates at 2m/s squared. Between t=0 and t=1 it's change in KE is 0.5j, between t=999 and t=1000 it's change in KE is 999.5j. Yet the rocket motor power output hadn't changed. What explains the massive difference in KE transferred to the rocket?
2. Nov 7, 2009
### Danger
Welcome to PF, Trivia1.
I know nothing of math, so I'm just going to assume that your numbers are accurate. Consider that the speed, and thus KE, of the rocket is not increasing linearly. Speed builds up to some pretty ferocious levels given enough constant acceleration.
3. Nov 7, 2009
### Staff: Mentor
Hi trivia1, welcome to PF
D H's excellent rocket tutorial page explains it in detail: https://www.physicsforums.com/showthread.php?t=199087
The broad outline is that you need to consider the KE of the exhaust also. When you do that you find that the KE of the system (rocket + exhaust) changes at a constant rate equal to the power of the rocket motor.
4. Nov 7, 2009
### Sillyboy
from your data, I calculated it. It's reasonable, we can know that it is an accelaration process. You know, as the rocket accelarates, the traction decreased, so the resistence should decrease too if the mass of the rocket is constant. but as we know when the velocity increases, the resistance should increase too.so I think the mass of th rocket has decreased! It is just the KE of the decreased exghaust that contributes to the increased KE of the rocket!
I am sorry for my poor English!
Last edited: Nov 7, 2009
5. Nov 8, 2009
### trivia1
Thanks for the welcome Danger and DaleSpam
My difficulty with comprehending why there is such a large difference in KE growth over the two periods lies with the relationship between KE and Velocity squared. I've done a google search on this dynamic and the rocket model is the one generally used to try and explain it. I still don't get it. Here's the same problem, put in a different way. A rocket has a mass of 1kg and a velocity of 999m/s to observer A, and has a velocity of 0 to observer B. It accelerates at 2m/s squared. In the following second it gains 999.5j of KE in ref to A, and 0.5j in reference to observer B. It's the same rocket, same power. Obviously I'm missing something basic here.
6. Nov 8, 2009
### Danger
I see what you mean now. This is a problem of relativity. For the conditions that you've given, observer B must be moving in the same direction and at initially the same speed as the rocket.
Think of it as a gun problem as opposed to a rocket one (although the acceleration is eliminated). If you are standing still in relation to a shooter, and he shoots you, that bullet will impact you at 1,000 ft/sec (as an example; there's a vast range of ammo). If, on the other hand, you are running away from the shooter at 995 ft/sec, the bullet will eventually catch you and impact at 5 ft/sec. It wouldn't even leave a bruise.
7. Nov 8, 2009
### trivia1
In the examples above the acceleration should be 1m/s squared.
Danger, I understand that KE is relative to an observer, but what I don’t get is how the magnitude of the change in KE is related to the initial velocity. At extreme initial velocities the gain in energy for even slight increases in velocity is huge. A lkg rocket with initial V of 100000m/s has an engine applying a force of 1N for 1s. It gains 100000j of KE. That’s in addition to the KE the rocket already possessed. If it’s initial V is 0, change in KE is 0.5j. Yet the rocket engine converted the same amount of chemical energy in both cases. That’s the bit that confuses me. In one frame of reference huge gain, another frame of reference a tiny gain.
8. Nov 8, 2009
### rcgldr
You seemed to have missed DaleSpams point. The rocket engine is increasing the KE of both fuel and rocket, and the rate of increase in KE is constant if the rocket engine is producing constant thrust at the same mass flow rate of spent fuel.
Rocket propulsion relies on ejecting a part of its own internal mass (spend fuel) for propulsion. If no external forces are invovled, then note that the center of mass of the rocket and it's spent fuel never moves (regardless of the frame of reference).
If the frame of reference is the rockets initial velocity, then all of the starting increase in KE is going into the fuel. As the rockets speed increases, the KE of both the rocket and it's remaining fuel are increased, as well as the spent fuel. Eventually the rocket can reach a speed where it's moving faster than the terminal exhaust veolicity of the spent fuel, in which case the KE of the fuel being ejected is being decreased by the engine, relative to that original frame of reference where the rocket wasn't moving.
9. Nov 8, 2009
### qraal
You need to account for the kinetic energy of the exhaust required to produce the thrust. You'll find the kinetic energy all balances then. For example, say your rocket engine has an exhaust velocity, u. For a given thrust, T, the mass-flow rate μ = T/u. What's the kinetic energy of the exhaust? Starting at rest it's obvious, 1/2.μ.u2. But what about when you're at a speed v? The rocket, mass m, moves forward at v+T/m, while the exhaust jet goes backwards at (v - u) because it's pointed in the opposite direction to which the rocket is being propelled forward. Thus the exhaust's kinetic energy is 1/2.μ.(v-u)2 and the rocket's is 1/2.m.(v+T/m)2.
In sum: The mass ejected backwards loses kinetic energy while the mass moving forwards gains it. Jet power thus can rise even when the jet's exhaust velocity remains the same, relative to the rocket, the whole time.
Last edited: Nov 8, 2009
10. Nov 8, 2009
To get a constant acceleration you need a constant resultant force not a constant power.If the force remained constant then as the velocity increases the power (force times velocity) must increase also.
Last edited: Nov 8, 2009
11. Nov 8, 2009
### trivia1
Is the following correct?
At extreme initial velocities the gain in kinetic energy for even slight increases in velocity is huge. A l kg rocket with initial V of 100000m/s has an engine applying a force of 1N for 1s. It gains 100000j of KE. If it’s initial V is 0, and an engine applies a force of 1N for 1s the change in KE is 0.5j. Yet the rocket engine converted the same amount of chemical energy in both cases.
12. Nov 8, 2009
### qraal
I'm not so happy with my explanation, so I'll do a couple of expansions to illustrate what's going on a bit better.
Before an impulse the rocket + fuel's kinetic energy is 1/2(m+μ).v2, relative to a stationary observer. The potential energy of the fuel becomes kinetic energy and the tiny mass of fuel is propelled rearwards at speed (v-u), relative to the stationary observer.
But ask yourself: what is the rocket's speed relative to the rocket?
Prior to the impulse from the exhaust, by Galilean relativity, the speed is zero, then after the impulse, relative to that initial state, it gains by some small acceleration equal to the thrust/rocket-mass. And that's always true.
The confusion comes from comparing what a co-moving observer sees (constant jet-power) in the rocket's frame, and what a stationary observer sees the kinetic energy of the rocket to be. You just can't compare the two frames like that without confusing yourself.
So what does a stationary observer observe the jet-power to be when a rocket is in motion? Well the fuel packet starts with a kinetic energy of 1/2.μ.v2. It burns, expands in the combustion chamber and then exists at a speed (v-u) relative to the stationary observer. Thus the difference between before and after is 1/2.μ.(v-u)2 - 1/2.μ.v2 = 1/2.μ.u2 - μ.v.u.
Now 1/2.μ.v2 is obviously the initial kinetic energy of the propellant in the rocket's frame, but what is - μ.v.u? Oddly enough it's mirrored when we expand out the kinetic energy of the rocket, before and after...
the speed increment is μ.u/m, so after the impulse the rocket's KE is 1/2.m.(v + μ.u/m)2. Then the difference before and after is μ.v.u + (μ.u)2/2m. The second term (μ.u)2/2m is (Thrust)2/2m, which is the jet-power.
So what is μ.v.u? Well KE = 1/2.m.v2, thus d(KE)/dt is m.v.(dv/dt)... and in this case (dv/dt) is μ.u/m. That means μ.v.u is your "extra" kinetic energy and it was hiding in the maths the whole time.
13. Nov 8, 2009
### qraal
The propellant has gained kinetic energy along with the rocket that contains it. Its change in energy when it's burnt has to take that into account else you'll end up with this apparent paradox. When you do the maths it all adds up.
That being said it does tell you why ion rockets have such pitiful thrust levels for seemingly quite high power levels. For an exhaust velocity of 100,000 m/s you need 50 kW of power for every measly N of thrust, with perfectly efficient power conversion. Inefficiencies in power generation and powering the jet means an ion-rocket can't lift off from a planet with decent gravity. Sufficient power would melt the rocket from waste heat alone.
14. Nov 8, 2009
### Staff: Mentor
Yes, you are still neglecting the KE of the exhaust. Please read the tutorial and always do your analysis including the KE of the exhaust.
15. Nov 8, 2009
### D H
Staff Emeritus
The exhaust is important to understand real rockets. However, this apparent paradox is not limited to rockets. Cars can easily accelerate at 2 m/s2. The Bugatti Veyron, for example, accelerates from 0 to 100 km/h in 2.5 seconds (13.89 m/s2 average acceleration) and has a top speed of over 400 km/h. To a person standing on the ground, a Veyron starting from rest gains a specific kinetic energy of 385.8 joules/kg ($(100\,\text{km/s})^2/2$) in 2.5 seconds. From the perspective of another Veyron racing at a constant 400 km/h toward the accelerating Veyron, the accelerating Veyron gains a specific kinetic energy of 3472.2 joules/kg ($\left((500\,\text{km/s})^2-(400\,\text{km/s})^2\right)/2$) in that same 2.5 second interval.
So how does the exact same car gain 385.8 joules/kg in one frame and 3472.2 joules/kg in another? Where does that extra 3086.4 joules/kg come from? The car has to burn some fuel to accelerate. To the stationary observer, the energy of that fuel is (initially) purely potential energy. To the moving car, that same fuel has a lot of kinetic energy in addition to its potential energy. That extra 3086.4 joules/kg is, as qraal put it, "hiding in the maths the whole time."
16. Nov 8, 2009
### Bob_for_short
The work done by the engine is the force ma multiplied by the distance. That is the key to the answer. In the first case the distance is much smaller than in the second case. Than means, in order to keep the same acceleration the engine should do much more work in one second. So the engine output is in fact much larger in the second case. Surprise! The power is a frame-dependent thing. Fortunately a fast engine has some extra energy to spent.
Last edited: Nov 8, 2009
17. Nov 8, 2009
### D H
Staff Emeritus
This can lead you down a dangerous path, which is to conclude that a rocket's acceleration must decrease as it gains speed. This is after all exactly what happens with an automobile. This is not what happens with a rocket. In fact, the exact opposite is the case: The acceleration of a rocket with a constant thrust increases as fuel is burnt. This increased acceleration can be harmful to occupants of the rocket. For example, the Space Shuttle commences a "3-g throttle down" at about 7 minutes and 40 seconds into launch to compensate for this tendency of acceleration to increase as rocket mass decreases.
What is happening here is that you are ignoring the energy of the exhaust, Bob. If you take the energy transferred from the rocket proper to the exhaust it is clear that the engine's energy output can indeed be constant as posited in the original post.
18. Nov 8, 2009
Agreed and this is the point I was making in post number ten.Take a simple example where it is not necessary to consider efficiencies of engines,exhaust gases and so on,a mass being pulled across a smooth table by a string attached to a falling mass.The main energy conversion here is GPE to KE.The power input from the falling mass does not remain constant because it falls increasing distances in successive equal intervals of time.
Last edited: Nov 8, 2009
19. Nov 8, 2009
I think some of us are talking at cross purposes here. I agree with your analysis of a rockets motion with a constant thrust but the OP was referring to a constant power not thrust.
20. Nov 8, 2009
### D H
Staff Emeritus
For a rocket, constant power means constant thrust.
21. Nov 8, 2009
### Bob_for_short
Absolutely correct! I added a couple of phrases to my post.
22. Nov 8, 2009
Clearly they are not the same thing,thrust is a force and is measured in Newtons and power is rate of doing work and is measured in Watts
23. Nov 8, 2009
### rcgldr
When considering the power peformed by an engine of some type, the frame of reference should be related to the point of application of force to some external object. In the case of a car, the point of application of force is at the road surface, so the road the car drives on should be the frame of reference. If the car were to accelerate a short distance on a long flat bed, then the surface of that flat bed should be the frame of reference.
A rocket engine is a special case, because outside of the atmosphere, the rocket does not interact with any external objects, but instead relies on an internal interaction where part of it's own mass is accelerated and expelled at high speed.
24. Nov 8, 2009
### D H
Staff Emeritus
Clearly so. However in the case of a rocket, a rocket with constant power output will indeed have constant thrust.
25. Nov 8, 2009
### D H
Staff Emeritus
Only because the math is easiest in this frame. The only thing that prevents me from modeling the behavior of a car accelerating down a road on the surface of the Earth from the perspective of a Neptune-centered, Neptune-fixed frame of reference is that the math becomes ridiculously convoluted from this perspective. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9330083131790161, "perplexity": 969.0617327782406}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589902.8/warc/CC-MAIN-20180717203423-20180717223423-00069.warc.gz"} |
https://unapologetic.wordpress.com/2009/11/17/cramers-rule/?like=1&source=post_flair&_wpnonce=b0679ce6ea | # The Unapologetic Mathematician
## Cramer’s Rule
We’re trying to invert a function $f:X\rightarrow\mathbb{R}^n$ which is continuously differentiable on some region $X\subseteq\mathbb{R}^n$. That is we know that if $a$ is a point where $J_f(a)\neq0$, then there is a ball $N$ around $a$ where $f$ is one-to-one onto some neighborhood $f(N)$ around $f(a)$. Then if $y$ is a point in $f(N)$, we’ve got a system of equations
$\displaystyle f^j(x^1,\dots,x^n)=y^j$
that we want to solve for all the $x^i$.
We know how to handle this if $f$ is defined by a linear transformation, represented by a matrix $A=\left(a_i^j\right)$:
\displaystyle\begin{aligned}f^j(x^1,\dots,x^n)=a_i^jx^i&=y^j\\Ax&=y\end{aligned}
In this case, the Jacobian transformation is just the function $f$ itself, and so the Jacobian determinant $\det\left(a_i^j\right)$ is nonzero if and only if the matrix $A$ is invertible. And so our solution depends on finding the inverse $A^{-1}$ and solving
\displaystyle\begin{aligned}Ax&=y\\A^{-1}Ax&=A^{-1}y\\x&=A^{-1}y\end{aligned}
This is the approach we’d like to generalize. But to do so, we need a more specific method of finding the inverse.
This is where Cramer’s rule comes in, and it starts by analyzing the way we calculate the determinant of a matrix $A$. This formula
$\displaystyle\sum\limits_{\pi\in S_n}\mathrm{sgn}(\pi)a_1^{\pi(1)}\dots a_n^{\pi(n)}$
involves a sum over all the permutations $\pi\in S_n$, and we want to consider the order in which we add up these terms. If we fix an index $i$, we can factor out each matrix entry in the $i$th column:
$\displaystyle\sum\limits_{j=1}^na_i^j\sum\limits_{\substack{\pi\in S_n\\\pi(i)=j}}\mathrm{sgn}(\pi)a_1^{\pi(1)}\dots\widehat{a_i^j}\dots a_n^{\pi(n)}$
where the hat indicates that we omit the $i$th term in the product. For a given value of $j$, we can consider the restricted sum
$\displaystyle A_j^i=\sum\limits_{\substack{\pi\in S_n\\\pi(i)=j}}\mathrm{sgn}(\pi)a_1^{\pi(1)}\dots\widehat{a_i^j}\dots a_n^{\pi(n)}$
which is $(-1)^{i+j}$ times the determinant of the $i$$j$ “minor” of the matrix $A$. That is, if we strike out the row and column of $A$ which contain $a_i^j$ and take the determinant of the remaining $(n-1)\times(n-1)$ matrix, we multiply this by $(-1)^{i+j}$ to get $A_j^i$. These are the entries in the “adjugate” matrix $\mathrm{adj}(A)$.
What we’ve shown is that
$\displaystyle A_j^ia_i^j=\det(A)$
(no summation on $i$). It’s not hard to show, however, that if we use a different row from the adjugate matrix we find
$\displaystyle\sum\limits_{j=1}^nA_j^ka_i^j=\det(A)\delta_i^k$
That is, the adjugate times the original matrix is the determinant of $A$ times the identity matrix. And so if $\det(A)\neq0$ we find
$\displaystyle A^{-1}=\frac{1}{\det(A)}\mathrm{adj}(A)$
So what does this mean for our system of equations? We can write
\displaystyle\begin{aligned}x&=\frac{1}{\det(A)}\mathrm{adj}(A)y\\x^i&=\frac{1}{\det(A)}A_j^iy^j\end{aligned}
But how does this sum $A_j^iy^j$ differ from the one $A_j^ia_i^j$ we used before (without summing on $i$) to calculate the determinant of $A$? We’ve replaced the $i$th column of $A$ by the column vector $y$, and so this is just another determinant, taken after performing this replacement!
Here’s an example. Let’s say we’ve got a system written in matrix form
$\displaystyle\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}u\\v\end{pmatrix}$
The entry in the $i$th row and $j$th column of the adjugate matrix is calculated by striking out the $i$th column and $j$th row of our original matrix, taking the determinant of the remaining matrix, and multiplying by $(-1)^{i+j}$. We get
$\displaystyle\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$
and thus we find
$\displaystyle\begin{pmatrix}x\\y\end{pmatrix}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\begin{pmatrix}u\\v\end{pmatrix}=\frac{1}{ad-bc}\begin{pmatrix}ud-bv\\av-uc\end{pmatrix}$
where we note that
\displaystyle\begin{aligned}ud-bv&=\det\begin{pmatrix}u&b\\v&d\end{pmatrix}\\av-uc&=\det\begin{pmatrix}a&u\\c&v\end{pmatrix}\end{aligned}
In other words, our solution is given by ratios of determinants:
\displaystyle\begin{aligned}x&=\frac{\det\begin{pmatrix}u&b\\v&d\end{pmatrix}}{\det\begin{pmatrix}a&b\\c&d\end{pmatrix}}\\y&=\frac{\det\begin{pmatrix}a&u\\c&v\end{pmatrix}}{\det\begin{pmatrix}a&b\\c&d\end{pmatrix}}\end{aligned}
and similar formulae hold for larger systems of equations.
November 17, 2009 - Posted by | Algebra, Linear Algebra
1. From computational viewpoint, Cramer’s rule is nice for very small matrices, but in practice one should use Gaussian elimination or QR factorization to calculate the determinant and inverse of a matrix, or to solve a linear system.
Comment by timur | November 17, 2009 | Reply
• Yes, but Gaussian elimination doesn’t tell you about the analytic properties of the inverse.
Comment by Qiaochu Yuan | November 17, 2009 | Reply
• Going ogg on a tangent, whilst you’re right about computational requirements, Cramer’s rule is also worth knowing if you’re trying to simplify a linear system whose coefficients are themselves some functions of some other variables, since determinants have nice algebraic properties and interpretations as hyper-volumes. (This isn’t so much solving the linear system as simplifying it, say for large scale finite element models with structure. This is “algebraic form”, distinct from the “analytic properties” John Armstrong is using in his next post.)
It’s a shame that most mathematical degrees tend to briefly introduce Cramer’s rule almost as a historical artifact and then immediately introduce Gaussian elimination.
Comment by davetweed | November 19, 2009 | Reply
2. Related to this wedge product thread, and this post, is the nice way that one can achieve the result of cramer’s rule directly without cofactors, adjoints, or even determinants. Suppose you have
\begin{aligned}\begin{bmatrix}u \\ v\end{bmatrix}&=\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix} \\ &=\begin{bmatrix}a \\ c \end{bmatrix}x+\begin{bmatrix}b \\ d \end{bmatrix}y\end{aligned}
Now wedge both sides with one of the vectors, eliminating it. For example
\begin{aligned}\begin{bmatrix}u \\ v\end{bmatrix}\wedge\begin{bmatrix}a \\ c\end{bmatrix}=y\begin{bmatrix}b \\ d \end{bmatrix}\wedge\begin{bmatrix}a \\ c\end{bmatrix}\end{aligned}
If the system is can be solved for $y$ the bivectors on the left and right hand sides differ only by a constant, and one can solve by division.
I first saw this nicely illustrated in this online book by John Browne here:
Since the wedge calculation can be reduced to a determinant, at its core this is no different than Cramer’s rule, but I think it provides a nice conceptual clarity. It is also no more efficient, and no less numerically unstable, so you probably really want SVD for computational work.
Comment by peeterjoot | November 17, 2009 | Reply
3. Qiaochu is closest to the mark, as will become apparent later this week. I don’t just want the inverse, but I want to know something about the form of the inverse.
Comment by John Armstrong | November 18, 2009 | Reply
4. Oh, and I actually wanted to introduce the adjugate matrix for a completely unrelated(?) reason I’ll be getting to later. It’s something I only know a handful of people have thought about, and none very explicitly.
Comment by John Armstrong | November 18, 2009 | Reply
5. […] system of equations, which has a unique solution since the determinant of its matrix is . We use Cramer’s rule to solve it, and get an expression for our difference quotient as a quotient of two determinants. […]
Pingback by The Inverse Function Theorem « The Unapologetic Mathematician | November 18, 2009 | Reply
6. […] of a product matrix as a (quadratic) polynomial in the entries of and . As for inversion, Cramer’s rule expresses the entries of the inverse matrix as the quotient of a (degree ) polynomial in the […]
Pingback by General Linear Groups are Lie Groups « The Unapologetic Mathematician | June 9, 2011 | Reply | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 67, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9661592245101929, "perplexity": 221.54018966614078}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218187717.19/warc/CC-MAIN-20170322212947-00171-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://news.nationalgeographic.com/news/2010/06/100616-large-hadron-collider-lhc-higgs-boson-god-particle/ | # "God Particle" May Be Five Distinct Particles, New Evidence Shows
Standard physics can't explain "provocative" results, scientists say.
The "God particle" may actually be five distinct particles, evidence from a new atom-smashing experiment suggests.
Called the Higgs boson, the theoretical particle has been long sought by physicists who think it's responsible for all mass in the universe—hence the name God particle.
It's also one of the targets of experiments by the Large Hadron Collider (LHC), which began smashing subatomic particles together at half its maximum power in March.
According to the widely accepted standard model of physics, all particles acquire their mass by interacting with the Higgs boson.
But some theories say that the Higgs boson is not one, but multiple, particles with similar masses but different electrical charges.
Now, researchers at Fermilab in Batavia, Illinois, say they have found more evidence for this multiple-particle theory.
Single God-Particle Theory Challenged
In an experiment called DZero at the lab's Tevatron particle collider, scientists recently found that collisions of protons and antiprotons produced pairs of matter particles more often than pairs of antimatter particles.
The difference was tiny—less than one percent—but it can't be explained by a standard model that assumes the existence of a single Higgs boson, said study co-author Adam Martin, a theoretical physicist at Fermilab.
"It's a really small effect, but it's still much bigger than if you turn all the cranks with all the original rules in the standard model," Martin said.
"The standard model with just the one Higgs particle is too minimal to explain the DZero result."
The DZero results can, however, be explained if scientists assume the Higgs boson is actually five particles—an extension of the standard model called the two-Higgs doublet model.
"When we extend the standard model, we put in new particles and new interactions," said Martin, whose results were published recently on the physics-research website arXiv.org.
"These new interactions can even treat matter and antimatter differently and therefore lead to bigger effects in experiments."
Multiple God Particles "Quite Provocative"
If multiple Higgs bosons exist, they may interact with matter differently, which could in turn lead to new kinds of undiscovered physics beyond the standard model, scientists say.
"A lot of the schemes for extending the standard model include as a first step adding [more Higgs boson particles]," Martin said.
Chris Quigg is a theoretical physicist, also at Fermilab, but he was not involved in the study.
Though "quite provocative," the results are still preliminary, Quigg stressed.
"I know of nothing to make me explicitly doubt the result, but when something is so unexpected and yet so subtle it bears taking time and taking a deep breath," Quigg said. "It's important not to jump up and down too soon about this."
If Martin's team is correct and the Higgs boson is actually five different particles, then it should be detectable by the LHC in Switzerland. (See Large Hadron Collider pictures.)
"In our interpretation," study co-author Martin said, "these Higgses cannot be too heavy, so we should definitely see them during the LHC era."
David Evans, a physicist at the University of Birmingham and the head of the LHC’s ALICE project, added by email, "I personally think it is unlikely that we have five different Higgs particles.
"But if this was proved correct, it would make the LHC even more exciting." | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8796136975288391, "perplexity": 1647.9969411530649}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246635639.11/warc/CC-MAIN-20150417045715-00254-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://arxiv-export-lb.library.cornell.edu/abs/2108.02848 | math.NA
(what is this?)
# Title: Construction and application of provable positive and exact cubature formulas
Authors: Jan Glaubitz
Abstract: Many applications require multi-dimensional numerical integration, often in the form of a cubature formula. These cubature formulas are desired to be positive and exact for certain finite-dimensional function spaces (and weight functions). Although there are several efficient procedures to construct positive and exact cubature formulas for many standard cases, it remains a challenge to do so in a more general setting. Here, we show how the method of least squares can be used to derive provable positive and exact formulas in a general multi-dimensional setting. Thereby, the procedure only makes use of basic linear algebra operations, such as solving a least squares problem. In particular, it is proved that the resulting least squares cubature formulas are ensured to be positive and exact if a sufficiently large number of equidistributed data points is used. We also discuss the application of provable positive and exact least squares cubature formulas to construct nested stable high-order rules and positive interpolatory formulas. Finally, our findings shed new light on some existing methods for multivariate numerical integration and under which restrictions these are ensured to be successful.
Comments: 33 pages Subjects: Numerical Analysis (math.NA); Optimization and Control (math.OC) MSC classes: 65D30, 65D32, 41A55, 41A63, 42C05 DOI: 10.1093/imanum/drac017 Cite as: arXiv:2108.02848 [math.NA] (or arXiv:2108.02848v3 [math.NA] for this version)
## Submission history
From: Jan Glaubitz [view email]
[v1] Thu, 5 Aug 2021 21:03:12 GMT (2380kb)
[v2] Tue, 10 Aug 2021 20:30:31 GMT (2380kb)
[v3] Thu, 26 May 2022 16:41:23 GMT (1909kb)
Link back to: arXiv, form interface, contact. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8653859496116638, "perplexity": 1192.4874317297679}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103034930.3/warc/CC-MAIN-20220625095705-20220625125705-00694.warc.gz"} |
https://worldwidescience.org/topicpages/w/wave+acoustic+monitoring.html | #### Sample records for wave acoustic monitoring
1. Surface acoustic wave dust deposition monitor
Science.gov (United States)
Fasching, G.E.; Smith, N.S. Jr.
1988-02-12
A system is disclosed for using the attenuation of surface acoustic waves to monitor real time dust deposition rates on surfaces. The system includes a signal generator, a tone-burst generator/amplifier connected to a transmitting transducer for converting electrical signals into acoustic waves. These waves are transmitted through a path defining means adjacent to a layer of dust and then, in turn, transmitted to a receiving transducer for changing the attenuated acoustic wave to electrical signals. The signals representing the attenuated acoustic waves may be amplified and used in a means for analyzing the output signals to produce an output indicative of the dust deposition rates and/or values of dust in the layer. 8 figs.
2. Surface Acoustic Wave (SAW Resonators for Monitoring Conditioning Film Formation
Directory of Open Access Journals (Sweden)
Siegfried Hohmann
2015-05-01
Full Text Available We propose surface acoustic wave (SAW resonators as a complementary tool for conditioning film monitoring. Conditioning films are formed by adsorption of inorganic and organic substances on a substrate the moment this substrate comes into contact with a liquid phase. In the case of implant insertion, for instance, initial protein adsorption is required to start wound healing, but it will also trigger immune reactions leading to inflammatory responses. The control of the initial protein adsorption would allow to promote the healing process and to suppress adverse immune reactions. Methods to investigate these adsorption processes are available, but it remains difficult to translate measurement results into actual protein binding events. Biosensor transducers allow user-friendly investigation of protein adsorption on different surfaces. The combination of several transduction principles leads to complementary results, allowing a more comprehensive characterization of the adsorbing layer. We introduce SAW resonators as a novel complementary tool for time-resolved conditioning film monitoring. SAW resonators were coated with polymers. The adsorption of the plasma proteins human serum albumin (HSA and fibrinogen onto the polymer-coated surfaces were monitored. Frequency results were compared with quartz crystal microbalance (QCM sensor measurements, which confirmed the suitability of the SAW resonators for this application.
3. Monitoring polymer properties using shear horizontal surface acoustic waves.
Science.gov (United States)
Gallimore, Dana Y; Millard, Paul J; Pereira da Cunha, Mauricio
2009-10-01
Real-time, nondestructive methods for monitoring polymer film properties are increasingly important in the development and fabrication of modern polymer-containing products. Online testing of industrial polymer films during preparation and conditioning is required to minimize material and energy consumption, improve the product quality, increase the production rate, and reduce the number of product rejects. It is well-known that shear horizontal surface acoustic wave (SH-SAW) propagation is sensitive to mass changes as well as to the mechanical properties of attached materials. In this work, the SH-SAW was used to monitor polymer property changes primarily dictated by variations in the viscoelasticity. The viscoelastic properties of a negative photoresist film were monitored throughout the ultraviolet (UV) light-induced polymer cross-linking process using SH-SAW delay line devices. Changes in the polymer film mass and viscoelasticity caused by UV exposure produced variations in the phase velocity and attenuation of the SH-SAW propagating in the structure. Based on measured polymer-coated delay line scattering transmission responses (S(21)) and the measured polymer layer thickness and density, the viscoelastic constants c(44) and eta(44) were extracted. The polymer thickness was found to decrease 0.6% during UV curing, while variations in the polymer density were determined to be insignificant. Changes of 6% in c(44) and 22% in eta(44) during the cross-linking process were observed, showing the sensitivity of the SH-SAW phase velocity and attenuation to changes in the polymer film viscoelasticity. These results indicate the potential for SH-SAW devices as online monitoring sensors for polymer film processing.
4. Surface Acoustic Wave Monitor for Deposition and Analysis of Ultra-Thin Films
Science.gov (United States)
Hines, Jacqueline H. (Inventor)
2015-01-01
A surface acoustic wave (SAW) based thin film deposition monitor device and system for monitoring the deposition of ultra-thin films and nanomaterials and the analysis thereof is characterized by acoustic wave device embodiments that include differential delay line device designs, and which can optionally have integral reference devices fabricated on the same substrate as the sensing device, or on a separate device in thermal contact with the film monitoring/analysis device, in order to provide inherently temperature compensated measurements. These deposition monitor and analysis devices can include inherent temperature compensation, higher sensitivity to surface interactions than quartz crystal microbalance (QCM) devices, and the ability to operate at extreme temperatures.
5. Monitoring Gold Nanoparticle Growth in Situ via the Acoustic Vibrations Probed by Four-Wave Mixing.
Science.gov (United States)
Wu, Jian; Xiang, Dao; Gordon, Reuven
2017-02-21
We monitor in situ gold nanoparticle growth in aqueous solution by probing the acoustic vibrations with four-wave mixing. We observe two acoustic vibrational modes of gold nanoparticles from the nonlinear optical response: an extensional mode with longitudinal expansion and transverse contraction and a breathing mode with radial expansion and contraction. The mode frequencies, which show an inverse dependence on the nanoparticle diameter, allow one to monitor the nanoparticle size and size distribution during synthesis. The information about the nanoparticle size and size distribution calculated on the basis of the mode frequencies agrees well with the results obtained from the electron microscopy analysis, validating the four-wave mixing technique as an accurate and effective tool for in situ monitoring of colloidal growth.
6. Passive wireless surface acoustic wave sensors for monitoring sequestration sites CO2 emission
Energy Technology Data Exchange (ETDEWEB)
Wang, Yizhong [Univ. of Pittsburgh, PA (United States); Chyu, Minking [Univ. of Pittsburgh, PA (United States); Wang, Qing-Ming [Univ. of Pittsburgh, PA (United States)
2013-02-14
University of Pittsburgh’s Transducer lab has teamed with the U.S. Department of Energy’s National Energy Technology Laboratory (DOE NETL) to conduct a comprehensive study to develop/evaluate low-cost, efficient CO2 measuring technologies for geological sequestration sites leakage monitoring. A passive wireless CO2 sensing system based on surface acoustic wave technology and carbon nanotube nanocomposite was developed. Surface acoustic wave device was studied to determine the optimum parameters. Delay line structure was adopted as basic sensor structure. CNT polymer nanocomposite was fabricated and tested under different temperature and strain condition for natural environment impact evaluation. Nanocomposite resistance increased for 5 times under pure strain, while the temperature dependence of resistance for CNT solely was -1375ppm/°C. The overall effect of temperature on nanocomposite resistance was -1000ppm/°C. The gas response of the nanocomposite was about 10% resistance increase under pure CO2 . The sensor frequency change was around 300ppm for pure CO2 . With paralyne packaging, the sensor frequency change from relative humidity of 0% to 100% at room temperature decreased from over 1000ppm to less than 100ppm. The lowest detection limit of the sensor is 1% gas concentration, with 36ppm frequency change. Wireless module was tested and showed over one foot transmission distance at preferred parallel orientation.
7. Flexible surface acoustic wave respiration sensor for monitoring obstructive sleep apnea syndrome
Science.gov (United States)
Jin, Hao; Tao, Xiang; Dong, Shurong; Qin, Yiheng; Yu, Liyang; Luo, Jikui; Deen, M. Jamal
2017-11-01
Obstructive sleep apnea syndrome (OSAS) has received much attention in recent years due to its significant harm to human health and high morbidity rate. A respiration monitoring system is needed to detect OSAS, so that the patient can receive treatment in a timely manner. Wired and wireless OSAS monitoring systems have been developed, but they require a wire connection and batteries to operate, and they are bulky, heavy and not user-friendly. In this paper, we propose the use of a flexible surface acoustic wave (SAW) microsensor to detect and monitor OSAS by measuring the humidity change associated with the respiration of a person. SAW sensors on rigid 128° YX LiNbO3 substrate are also characterized for this application. Results show both types of SAW sensors are suitable for OSAS monitoring with good sensitivity, repeatability and reliability, and the response time and recovery time for the flexible SAW sensors are 1.125 and 0.75 s, respectively. Our work demonstrates the potential for an innovative flexible microsensor for the detection and monitoring of OSAS.
8. A Methodological Review of Piezoelectric Based Acoustic Wave Generation and Detection Techniques for Structural Health Monitoring
Directory of Open Access Journals (Sweden)
Zhigang Sun
2013-01-01
Full Text Available Piezoelectric transducers have a long history of applications in nondestructive evaluation of material and structure integrity owing to their ability of transforming mechanical energy to electrical energy and vice versa. As condition based maintenance has emerged as a valuable approach to enhancing continued aircraft airworthiness while reducing the life cycle cost, its enabling structural health monitoring (SHM technologies capable of providing on-demand diagnosis of the structure without interrupting the aircraft operation are attracting increasing R&D efforts. Piezoelectric transducers play an essential role in these endeavors. This paper is set forth to review a variety of ingenious ways in which piezoelectric transducers are used in today’s SHM technologies as a means of generation and/or detection of diagnostic acoustic waves.
9. Contactless Monitoring of Conductivity Changes in Vanadium Pentoxide Xerogel Layers Using Surface Acoustic Waves
Science.gov (United States)
Rimeika, Romualdas; Sereika, Raimundas; Čiplys, Daumantas; Bondarenka, Vladimiras; Sereika, Albertas; Shur, Michael
The hydrated form of the vanadium pentoxide (V2O5 ·nH2O) deposited by the sol-gel method on the piezoelectric YZ-LiNbO3 substrate has been studied using surface acoustic waves (SAWs). Brush-deposited and spin-coated layers, differing in thickness by an order of magnitude (∼1 μm and ∼0.1 μm, respectively) were studied. The variations with time in the transmitted SAW amplitude and phase during the gel-to-xerogel transition of V2O5 ·nH2O were observed and attributed to the acoustoelectric interaction. The possibilities of using the SAWs for contactless monitoring of the layer sheet conductivity have been demonstrated.
10. Acoustics waves and oscillations
CERN Document Server
Sen, S.N.
2013-01-01
Parameters of acoustics presented in a logical and lucid style Physical principles discussed with mathematical formulations Importance of ultrasonic waves highlighted Dispersion of ultrasonic waves in viscous liquids explained This book presents the theory of waves and oscillations and various applications of acoustics in a logical and simple form. The physical principles have been explained with necessary mathematical formulation and supported by experimental layout wherever possible. Incorporating the classical view point all aspects of acoustic waves and oscillations have been discussed together with detailed elaboration of modern technological applications of sound. A separate chapter on ultrasonics emphasizes the importance of this branch of science in fundamental and applied research. In this edition a new chapter ''Hypersonic Velocity in Viscous Liquids as revealed from Brillouin Spectra'' has been added. The book is expected to present to its readers a comprehensive presentation of the subject matter...
11. Quantitative Enhancement of Fatigue Crack Monitoring by Imaging Surface Acoustic Wave Reflection in a Space-Cycle Domain
Science.gov (United States)
Connolly, G. D.; Rokhlin, S. I.
2011-06-01
The surface wave acoustic method is applied to the in-situ monitoring of fatigue crack initiation and evolution on tension specimens. A small low-frequency periodic loading is also applied, resulting in a nonlinear modulation of reflected pulses. The acoustic wave reflections are collected for: each experimental cycle; a range of applied tension and modulation load levels; and a range of spatial propagation positions, and are presented in image form to aid pattern identification. Salient features of the image are then extracted and processed to evaluate the initiation time of the crack and its subsequent size evolution until sample failure. Additionally, a method for enhancing signal to noise ratio in Ti-6242 alloy samples is demonstrated.
12. Measuring soft tissue elasticity by monitoring surface acoustic waves using image plane digital holography
Science.gov (United States)
Li, Shiguang; Oldenburg, Amy L.
2011-03-01
The detection of tumors in soft tissues, such as breast cancer, is important to achieve at the earliest stages of the disease to improve patient outcome. Tumors often exhibit a greater elastic modulus compared to normal tissues. In this paper, we report our first study to measure elastic properties of soft tissues by mapping the surface acoustic waves (SAWs) with image plane digital holography. The experimental results show that the SAW velocity is proportional to the square root of elastic modulus over a range from 3.7-122kPa in homogeneous tissue phantoms, consistent with Rayleigh wave theory. This technique also permits detection of the interface of two-layer phantoms 10mm deep under surface and the interface depth by quantifying the SAW dispersion.
13. Tunable damper for an acoustic wave guide
Science.gov (United States)
Rogers, Samuel C.
1984-01-01
A damper for tunably damping acoustic waves in an ultrasonic waveguide is provided which may be used in a hostile environment such as a nuclear reactor. The area of the waveguide, which may be a selected size metal rod in which acoustic waves are to be damped, is wrapped, or surrounded, by a mass of stainless steel wool. The wool wrapped portion is then sandwiched between tuning plates, which may also be stainless steel, by means of clamping screws which may be adjusted to change the clamping force of the sandwiched assembly along the waveguide section. The plates are preformed along their length in a sinusoidally bent pattern with a period approximately equal to the acoustic wavelength which is to be damped. The bent pattern of the opposing plates are in phase along their length relative to their sinusoidal patterns so that as the clamping screws are tightened a bending stress is applied to the waveguide at 180.degree. intervals along the damping section to oppose the acoustic wave motions in the waveguide and provide good coupling of the wool to the guide. The damper is tuned by selectively tightening the clamping screws while monitoring the amplitude of the acoustic waves launched in the waveguide. It may be selectively tuned to damp particular acoustic wave modes (torsional or extensional, for example) and/or frequencies while allowing others to pass unattenuated.
14. Spin wave generation by surface acoustic waves
Science.gov (United States)
Li, Xu; Labanowski, Dominic; Salahuddin, Sayeef; Lynch, Christopher S.
2017-07-01
Surface acoustic waves (SAW) on piezoelectric substrates can excite spin wave resonance (SWR) in magnetostrictive films through magnetoelastic coupling. This acoustically driven SWR enables the excitation of a single spin wave mode with an in-plane wave vector k matched to the magnetoelastic wave vector. A 2D frequency domain finite element model is presented that fully couples elastodynamics, micromagnetics, and piezoelectricity with interface spin pumping effects taken into account. It is used to simulate SAW driven SWR on a ferromagnetic and piezoelectric heterostructure device with an interdigital transducer configuration. These results, for the first time, present the spatial distribution of magnetization components that, together with elastic wave, exponentially decays along the propagation direction due to magnetic damping. The results also show that the system transmission rate S21(dB) can be tuned by both an external bias field and the SAW wavevector. Acoustic spin pumping at magnetic film/normal metal interface leads to damping enhancement in magnetic films that decreases the energy absorption rate from elastic energy. This weakened interaction between the magnetic energy and elastic energy leads to a lower evanescence rate of the SAW that results in a longer distance propagation. With strong magnetoelastic coupling, the SAW driven spin wave is able to propagate up to 1200 μm. The results give a quantitative indication of the acoustic spin pumping contribution to linewidth broadening.
15. Oscillating nonlinear acoustic shock waves
DEFF Research Database (Denmark)
Gaididei, Yuri; Rasmussen, Anders Rønne; Christiansen, Peter Leth
2016-01-01
We investigate oscillating shock waves in a tube using a higher order weakly nonlinear acoustic model. The model includes thermoviscous effects and is non isentropic. The oscillating shock waves are generated at one end of the tube by a sinusoidal driver. Numerical simulations show...... that at resonance a stationary state arise consisting of multiple oscillating shock waves. Off resonance driving leads to a nearly linear oscillating ground state but superimposed by bursts of a fast oscillating shock wave. Based on a travelling wave ansatz for the fluid velocity potential with an added 2'nd order...... polynomial in the space and time variables, we find analytical approximations to the observed single shock waves in an infinitely long tube. Using perturbation theory for the driven acoustic system approximative analytical solutions for the off resonant case are determined....
16. Magnetic recording with acoustic waves
Energy Technology Data Exchange (ETDEWEB)
Li, Weiyang; Buford, Benjamin; Jander, Albrecht; Dhagat, Pallavi, E-mail: [email protected]
2014-09-01
We demonstrate acoustically assisted magnetic recording (AAMR), a new paradigm in magnetic data storage. In this concept, otherwise unwriteable high-coercivity media, requisite for thermally stable high-density data storage, are made amenable to recording by lowering their coercivity via strain induced by surface acoustic waves. The basic principles of AAMR are proven using galfenol, a low-coercivity magnetostrictive material, as the recording medium. It is shown that the writing field needed to record data in the presence of acoustic strain is lower than the coercivity of the unstrained galfenol film. Further, it is demonstrated that interference between acoustic waves can be tailored to selectively address a bit on the recording medium.
17. Monitoring of Soft Deposition Layers in Liquid-Filled Tubes with Guided Acoustic Waves Excited by Clamp-on Transducers.
Science.gov (United States)
Tietze, Sabrina; Singer, Ferdinand; Lasota, Sandra; Ebert, Sandra; Landskron, Johannes; Schwuchow, Katrin; Drese, Klaus Stefan; Lindner, Gerhard
2018-02-09
The monitoring of liquid-filled tubes with respect to the formation of soft deposition layers such as biofilms on the inner walls calls for non-invasive and long-term stable sensors, which can be attached to existing pipe structures. For this task a method is developed, which uses an ultrasonic clamp-on device. This method is based on the impact of such deposition layers on the propagation of circumferential guided waves on the pipe wall. Such waves are partly converted into longitudinal compressional waves in the liquid, which are back-converted to guided waves in a circular cross section of the pipe. Validating this approach, laboratory experiments with gelatin deposition layers on steel tubes exhibited a distinguishable sensitivity of both wave branches with respect to the thickness of such layers. This allows the monitoring of the layer growth.
18. Development of hydroacoustical techniques for the monitoring and classification of benthic habitats in Puck Bay: Modeling of acoustic waves scattering by seagrass
Science.gov (United States)
Raczkowska, A.; Gorska, N.
2012-12-01
Puck Bay is an area of high species biodiversity belonging to the Coastal Landscape Park of Baltic Sea Protected Areas (BSPA) and is also included in the list of the World Wide Fund for Nature (WWF) and covered by the protection program "Natura 2000". The underwater meadows of the Puck Bay are important for Europe's natural habitats due to their role in enhancing the productivity of marine ecosystems and providing shelter and optimal feeding conditions for many marine organisms. One of the dominant species comprising the underwater meadows of the Southern Baltic Sea is the seagrass Zostera marina. The spatial extent of underwater seagrass meadows is altered by pollution and eutrophication; therefore, to properly manage the area one must monitor its ecological state. Remote acoustic methods are useful tools for the monitoring of benthic habitats in many marine areas because they are non-invasive and allow researchers to obtain data from a large area in a short period of time. Currently there is a need to apply these methods in the Baltic Sea. Here we present an analysis of the mechanism of scattering of acoustic waves on seagrass in the Southern Baltic Sea based on the numerical modeling of acoustic wave scattering by the biological tissues of plants. The study was conducted by adapting a model developed on the basis of DWBA (Distorted Wave Born Approximation) developed by Stanton and Chu (2005) for fluid-like objects, including the characteristics of the Southern Baltic seagrass. Input data for the model, including the morphometry of seagrass leaves, their angle of inclination and the density plant cover, was obtained through the analysis of biological materials collected in the Puck Bay in the framework of a research project financed by the Polish Government (Development of hydroacoustic methods for studies of underwater meadows of Puck Bay, 6P04E 051 20). On the basis of the developed model, we have analyzed the dependence of the target strength of a single
19. Surface Acoustic Wave Devices
DEFF Research Database (Denmark)
Dühring, Maria Bayard
application is modulation of optical waves in waveguides. This presentation elaborates on how a SAW is generated by interdigital transducers using a 2D model of a piezoelectric, inhomogeneous material implemented in the high-level programming language Comsol Multiphysics. The SAW is send through a model...
20. Millimeter waves: acoustic and electromagnetic.
Science.gov (United States)
Ziskin, Marvin C
2013-01-01
This article is the presentation I gave at the D'Arsonval Award Ceremony on June 14, 2011 at the Bioelectromagnetics Society Annual Meeting in Halifax, Nova Scotia. It summarizes my research activities in acoustic and electromagnetic millimeter waves over the past 47 years. My earliest research involved acoustic millimeter waves, with a special interest in diagnostic ultrasound imaging and its safety. For the last 21 years my research expanded to include electromagnetic millimeter waves, with a special interest in the mechanisms underlying millimeter wave therapy. Millimeter wave therapy has been widely used in the former Soviet Union with great reported success for many diseases, but is virtually unknown to Western physicians. I and the very capable members of my laboratory were able to demonstrate that the local exposure of skin to low intensity millimeter waves caused the release of endogenous opioids, and the transport of these agents by blood flow to all parts of the body resulted in pain relief and other beneficial effects. Copyright © 2012 Wiley Periodicals, Inc.
1. Ion Acoustic Waves in the Presence of Electron Plasma Waves
DEFF Research Database (Denmark)
Michelsen, Poul; Pécseli, Hans; Juul Rasmussen, Jens
1977-01-01
Long-wavelength ion acoustic waves in the presence of propagating short-wavelength electron plasma waves are examined. The influence of the high frequency oscillations is to decrease the phase velocity and the damping distance of the ion wave.......Long-wavelength ion acoustic waves in the presence of propagating short-wavelength electron plasma waves are examined. The influence of the high frequency oscillations is to decrease the phase velocity and the damping distance of the ion wave....
2. Robust acoustic wave manipulation of bubbly liquids
Energy Technology Data Exchange (ETDEWEB)
Gumerov, N. A., E-mail: [email protected] [Institute for Advanced Computer Studies, University of Maryland, College Park, Maryland 20742 (United States); Center for Micro- and Nanoscale Dynamics of Dispersed Systems, Bashkir State University, Ufa 450076 (Russian Federation); Akhatov, I. S. [Center for Design, Manufacturing and Materials, Skolkovo Institute of Science and Technology, Moscow 143026 (Russian Federation); Ohl, C.-D. [Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371 (Singapore); Center for Micro- and Nanoscale Dynamics of Dispersed Systems, Bashkir State University, Ufa 450076 (Russian Federation); Sametov, S. P. [Center for Micro- and Nanoscale Dynamics of Dispersed Systems, Bashkir State University, Ufa 450076 (Russian Federation); Khazimullin, M. V. [Center for Micro- and Nanoscale Dynamics of Dispersed Systems, Bashkir State University, Ufa 450076 (Russian Federation); Institute of Molecule and Crystal Physics, Ufa Research Center of Russian Academy of Sciences, Ufa 450054 (Russian Federation); Gonzalez-Avila, S. R. [Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371 (Singapore)
2016-03-28
Experiments with water–air bubbly liquids when exposed to acoustic fields of frequency ∼100 kHz and intensity below the cavitation threshold demonstrate that bubbles ∼30 μm in diameter can be “pushed” away from acoustic sources by acoustic radiation independently from the direction of gravity. This manifests formation and propagation of acoustically induced transparency waves (waves of the bubble volume fraction). In fact, this is a collective effect of bubbles, which can be described by a mathematical model of bubble self-organization in acoustic fields that matches well with our experiments.
3. PORTABLE ACOUSTIC MONITORING PACKAGE (PAMP)
Energy Technology Data Exchange (ETDEWEB)
John L. Loth; Gary J. Morris; George M. Palmer; Richard Guiler; Patrick Browning
2004-07-20
The Portable Acoustic Monitoring Package (PAMP) has been designed to record and monitor the acoustic signal in natural gas transmission lines. In particular the three acoustic signals associated with a line leak. The system is portable ({approx}30 lbs) and is designed for line pressures up to 1000 psi. It has become apparent that cataloging of the various background acoustic signals in natural gas transmission line is very important if a system to identify leak signals is to be developed. The low-pressure (0-200 psig) laboratory test phase has been completed and a number of field trials have been conducted. Before the cataloging phase could begin, a few problems identified in field trials identified had to be corrected such as: (1) Decreased microphone sensitivity at line pressures above 250 psig. (2) The inability to deal with large data sets collected when cataloging the variety of signals in a transmission line. (3) The lack of an available online acoustic calibration system. These problems have been solved and the WVU PAMP is now fully functional over the entire pressure range found in the Natural Gas transmission lines in this region. Field portability and reliability have been greatly improved. Data collection and storage have also improved to the point were the full acoustic spectrum of acoustic signals can be accurately cataloged, recorded and described.
4. Wear monitoring of single point cutting tool using acoustic emission ...
was carried out to study the wear monitoring in single point cutting tool using acoustic emission techniques. 2. Propagation of stress wave due to crater wear and flank wear. Figure 1 show the crater wear occurred on the rake face of the tool. This crater wear emits stress wave, which propagates as spherical wave front and ...
5. On Collisionless Damping of Ion Acoustic Waves
DEFF Research Database (Denmark)
Jensen, Vagn Orla; Petersen, P.I.
1973-01-01
Exact theoretical treatments show that the damping of ion acoustic waves in collisionless plasmas does not vanish when the derivative of the undisturbed distribution function at the phase velocity equals zero.......Exact theoretical treatments show that the damping of ion acoustic waves in collisionless plasmas does not vanish when the derivative of the undisturbed distribution function at the phase velocity equals zero....
6. Dual-mode acoustic wave biosensors microarrays
Science.gov (United States)
Auner, Gregory W.; Shreve, Gina; Ying, Hao; Newaz, Golam; Hughes, Chantelle; Xu, Jianzeng
2003-04-01
We have develop highly sensitive and selective acoustic wave biosensor arrays with signal analysis systems to provide a fingerprint for the real-time identification and quantification of a wide array of bacterial pathogens and environmental health hazards. We have developed an unique highly sensitive dual mode acoustic wave platform prototype that, when combined with phage based selective detection elements, form a durable bacteria sensor. Arrays of these new real-time biosensors are integrated to form a biosensor array on a chip. This research and development program optimizes advanced piezoelectric aluminum nitride wide bandgap semiconductors, novel micromachining processes, advanced device structures, selective phage displays development and immobilization techniques, and system integration and signal analysis technology to develop the biosensor arrays. The dual sensor platform can be programmed to sense in a gas, vapor or liquid environment by switching between acoustic wave resonate modes. Such a dual mode sensor has tremendous implications for applications involving monitoring of pathogenic microorganisms in the clinical setting due to their ability to detect airborne pathogens. This provides a number of applications including hospital settings such as intensive care or other in-patient wards for the reduction of nosocomial infections and maintenance of sterile environments in surgical suites. Monitoring for airborn pathogen transmission in public transportation areas such as airplanes may be useful for implementation of strategies for redution of airborn transmission routes. The ability to use the same sensor in the liquid sensing mode is important for tracing the source of airborn pathogens to local liquid sources. Sensing of pathogens in saliva will be useful for sensing oral pathogens and support of decision-making strategies regarding prevention of transmission and support of treatment strategies.
7. Acoustic multivariate condition monitoring - AMCM
Energy Technology Data Exchange (ETDEWEB)
Rosenhave, P.E. [Vestfold College, Maritime Dept., Toensberg (Norway)
1997-12-31
In Norway, Vestfold College, Maritime Department presents new opportunities for non-invasive, on- or off-line acoustic monitoring of rotating machinery such as off-shore pumps and diesel engines. New developments within acoustic sensor technology coupled with chemometric data analysis of complex signals now allow condition monitoring of hitherto unavailable flexibility and diagnostic specificity. Chemometrics paired with existing knowledge yields a new and powerful tool for condition monitoring. By the use of multivariate techniques and acoustics it is possible to quantify wear and tear as well as predict the performance of working components in complex machinery. This presentation describes the AMCM method and one result of a feasibility study conducted onboard the LPG/C Norgas Mariner owned by Norwegian Gas Carriers as (NGC), Oslo. (orig.) 6 refs.
8. Swimming using surface acoustic waves.
Directory of Open Access Journals (Sweden)
Yannyk Bourquin
Full Text Available Microactuation of free standing objects in fluids is currently dominated by the rotary propeller, giving rise to a range of potential applications in the military, aeronautic and biomedical fields. Previously, surface acoustic waves (SAWs have been shown to be of increasing interest in the field of microfluidics, where the refraction of a SAW into a drop of fluid creates a convective flow, a phenomenon generally known as SAW streaming. We now show how SAWs, generated at microelectronic devices, can be used as an efficient method of propulsion actuated by localised fluid streaming. The direction of the force arising from such streaming is optimal when the devices are maintained at the Rayleigh angle. The technique provides propulsion without any moving parts, and, due to the inherent design of the SAW transducer, enables simple control of the direction of travel.
9. Acoustic wave coupled magnetoelectric effect
Energy Technology Data Exchange (ETDEWEB)
Gao, J.S. [Institute of information Engineering, Suqian College, Suqian 223800 (China); Magnetoelectronic Laboratory, Nanjing Normal University, Nanjing 210023 (China); Zhang, N., E-mail: [email protected] [Magnetoelectronic Laboratory, Nanjing Normal University, Nanjing 210023 (China)
2016-07-15
Magnetoelectric (ME) coupling by acoustic waveguide was developed. Longitudinal and transversal ME effects of larger than 44 and 6 (V cm{sup −1} Oe{sup −1}) were obtained with the waveguide-coupled ME device, respectively. Several resonant points were observed in the range of frequency lower than 47 kHz. Analysis showed that the standing waves in the waveguide were responsible for those resonances. The frequency and size dependence of the ME effects were investigated. A resonant condition about the geometrical size of the waveguide was obtained. Theory and experiments showed the resonant frequencies were closely influenced by the diameter and length of the waveguide. A series of double-peak curves of longitudinal magnetoelectric response were obtained, and their significance was discussed initially. - Highlights: • Magnetoelectric (ME) coupling by acoustic waveguide was developed. • The frequency and size dependence of the ME effects were investigated. • A resonant condition about the geometrical size of the waveguide was obtained. • A series of double-peak curves of longitudinal magnetoelectric response were obtained, and their significance was discussed initially.
10. Unidirectional propagation of designer surface acoustic waves
CERN Document Server
Lu, Jiuyang; Ke, Manzhu; Liu, Zhengyou
2014-01-01
We propose an efficient design route to generate unidirectional propagation of the designer surface acoustic waves. The whole system consists of a periodically corrugated rigid plate combining with a pair of asymmetric narrow slits. The directionality of the structure-induced surface waves stems from the destructive interference between the evanescent waves emitted from the double slits. The theoretical prediction is validated well by simulations and experiments. Promising applications can be anticipated, such as in designing compact acoustic circuits.
11. Acoustic emission monitoring of degradation of cross ply laminates.
Science.gov (United States)
Aggelis, D G; Barkoula, N M; Matikas, T E; Paipetis, A S
2010-06-01
The scope of this study is to relate the acoustic activity of damage in composites to the failure mechanisms associated with these materials. Cross ply fiber reinforced composites were subjected to tensile loading with recording of their acoustic activity. Acoustic emission (AE) parameters were employed to monitor the transition of the damage mechanism from transverse cracking (mode I) to delamination (mode II). Wave propagation measurements in between loading steps revealed an increase in the relative amplitude of the propagated wave, which was attributed to the development of delamination that confined the wave to the top longitudinal plies of the composite.
12. Microfabricated bulk wave acoustic bandgap device
Science.gov (United States)
Olsson, Roy H.; El-Kady, Ihab F.; McCormick, Frederick; Fleming, James G.; Fleming, Carol
2010-06-08
A microfabricated bulk wave acoustic bandgap device comprises a periodic two-dimensional array of scatterers embedded within the matrix material membrane, wherein the scatterer material has a density and/or elastic constant that is different than the matrix material and wherein the periodicity of the array causes destructive interference of the acoustic wave within an acoustic bandgap. The membrane can be suspended above a substrate by an air or vacuum gap to provide acoustic isolation from the substrate. The device can be fabricated using microelectromechanical systems (MEMS) technologies. Such microfabricated bulk wave phononic bandgap devices are useful for acoustic isolation in the ultrasonic, VHF, or UHF regime (i.e., frequencies of order 1 MHz to 10 GHz and higher, and lattice constants of order 100 .mu.m or less).
13. S wave propagation in acoustic anisotropic media
Science.gov (United States)
Stovas, Alexey
2017-04-01
The acoustic anisotropic medium can be defined in two ways. The first one is known as a pseudo-acoustic approximation (Alkhalifah, 1998) that is based on the fact that in TI media, P wave propagation is weakly dependent on parameter known as "vertical S-wave velocity" (Thomsen, 1986). The standard way to define the pseudo-acoustic approximation is to set this parameter to zero. However, as it was shown later (Grechka et al., 2004), there is "S wave artifact" in such a medium. Another way is to define the stack of horizontal solid-fluid layers and perform an upscaling based on the Backus (1962) averaging. The stiffness coefficient that responds to "vertical S wave velocity" turns to zero if any of layers has zero vertical S wave velocity. In this abstract, I analyze the S wave propagation is acoustic anisotropic medium and define important kinematic properties such as the group velocity surface and Dix-type equations. The kinematic properties can easily be defined from the slowness surface. In elastic transversely isotropic medium, the equations for P and SV wave slowness surfaces are coupled. Setting "vertical S wave velocity" to zero, results in decoupling of equations. I show that the S wave group velocity surface is given by quasi-astroidal form with the reference astroid defined by vertical and horizontal projections of group velocity. I show that there are cusps attached to both vertical and horizontal symmetry axes. The new S wave parameters include vertical, horizontal and normal moveout velocities. With the help of new parameterization, suitable for S wave, I also derived the Dix-type of equations to define the effective kinematical properties of S waves in multi-layered acoustic anisotropic medium. I have shown that effective media defined from P and S waves have different parameters. I also show that there are certain symmetries between P and S waves parameters and equations. The proposed method can be used for analysis of S waves in acoustic anisotropic
14. Real-time monitoring of methanol concentration using a shear horizontal surface acoustic wave sensor for direct methanol fuel cell without reference liquid measurement
Science.gov (United States)
Tada, Kyosuke; Nozawa, Takuya; Kondoh, Jun
2017-07-01
In recent years, there has been an increasing demand for sensors that continuously measure liquid concentrations and detect abnormalities in liquid environments. In this study, a shear horizontal surface acoustic wave (SH-SAW) sensor is applied for the continuous monitoring of liquid concentrations. As the SH-SAW sensor functions using the relative measurement method, it normally needs a reference at each measurement. However, if the sensor is installed in a liquid flow cell, it is difficult to measure a reference liquid. Therefore, it is important to establish an estimation method for liquid concentrations using the SH-SAW sensor without requiring a reference measurement. In this study, the SH-SAW sensor is installed in a direct methanol fuel cell to monitor the methanol concentration. The estimated concentration is compared with a conventional density meter. Moreover, the effect of formic acid is examined. When the fuel temperature is higher than 70 °C, it is necessary to consider the influence of liquid conductivity. Here, an estimation method for these cases is also proposed.
15. Writing magnetic patterns with surface acoustic waves
Energy Technology Data Exchange (ETDEWEB)
Li, Weiyang; Buford, Benjamin; Jander, Albrecht; Dhagat, Pallavi, E-mail: [email protected] [School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, Oregon 97331 (United States)
2014-05-07
A novel patterning technique that creates magnetization patterns in a continuous magnetostrictive film with surface acoustic waves is demonstrated. Patterns of 10 μm wide stripes of alternating magnetization and a 3 μm dot of reversed magnetization are written using standing and focusing acoustic waves, respectively. The magnetization pattern is size-tunable, erasable, and rewritable by changing the magnetic field and acoustic power. This versatility, along with its solid-state implementation (no moving parts) and electronic control, renders it as a promising technique for application in magnetic recording, magnonic signal processing, magnetic particle manipulation, and spatial magneto-optical modulation.
16. Focusing of Acoustic Waves through Acoustic Materials with Subwavelength Structures
KAUST Repository
Xiao, Bingmu
2013-05-01
In this thesis, wave propagation through acoustic materials with subwavelength slits structures is studied. Guided by the findings, acoustic wave focusing is achieved with a specific material design. By using a parameter retrieving method, an effective medium theory for a slab with periodic subwavelength cut-through slits is successfully derived. The theory is based on eigenfunction solutions to the acoustic wave equation. Numerical simulations are implemented by the finite-difference time-domain (FDTD) method for the two-dimensional acoustic wave equation. The theory provides the effective impedance and refractive index functions for the equivalent medium, which can reproduce the transmission and reflection spectral responses of the original structure. I analytically and numerically investigate both the validity and limitations of the theory, and the influences of material and geometry on the effective spectral responses are studied. Results show that large contrasts in impedance and density are conditions that validate the effective medium theory, and this approximation displays a better accuracy for a thick slab with narrow slits in it. Based on the effective medium theory developed, a design of a at slab with a snake shaped" subwavelength structure is proposed as a means of achieving acoustic focusing. The property of focusing is demonstrated by FDTD simulations. Good agreement is observed between the proposed structure and the equivalent lens pre- dicted by the theory, which leads to robust broadband focusing by a thin at slab.
17. Imaging of Acoustic Waves in Piezoelectric Ceramics by Coulomb Coupling
Science.gov (United States)
Habib, Anowarul; Shelke, Amit; Pluta, Mieczyslaw; Kundu, Tribikram; Pietsch, Ullrich; Grill, Wolfgang
2012-07-01
The transport properties of bulk and guided acoustic waves travelling in a lead zirconate titanate (PZT) disc, originally manufactured to serve as ultrasonic transducer, have been monitored by scanned Coulomb coupling. The images are recorded by excitation and detection of ultrasound with local electric field probes via piezoelectric coupling. A narrow pulse has been used for excitation. Broadband coupling is achieved since neither mechanical nor electrical resonances are involved. The velocities of the traveling acoustic waves determined from the images are compared with characteristic velocities calculated from material properties listed by the manufacturer of the PZT plate.
18. Studying materials using acoustic waves
Science.gov (United States)
Apfel, R. E.
1988-03-01
This final report summarizes the activity of the contractor in meeting the objectives of the contract. A comprehensive bibliography and list of participants on the contract work are included along with a discussion including: microcavitation, microparticle characterization, interfacial characterization using acoustic levitation, measurements of the acoustic nonlinear parameter for determining the composition of mixtures.
19. Classical acoustic waves in damped media.
Science.gov (United States)
Albuquerque, E L; Mauriz, P W
2003-05-01
A Green function technique is employed to investigate the propagation of classical damped acoustic waves in complex media. The calculations are based on the linear response function approach, which is very convenient to deal with this kind of problem. Both the displacement and the gradient displacement Green functions are determined. All deformations in the media are supposed to be negligible, so the motions considered here are purely acoustic waves. The damping term gamma is included in a phenomenological way into the wave vector expression. By using the fluctuation-dissipation theorem, the power spectrum of the acoustic waves is also derived and has interesting properties, the most important of them being a possible relation with the analysis of seismic reflection data.
20. Modeling acoustic wave propagation in isotropic medium
Science.gov (United States)
Krasnoveikin, V. A.; Druzhinin, N. V.; Derusova, D. A.; Tarasov, S. Yu.
2017-12-01
The paper carries out the graphical analysis of acoustic wave propagation in plates of different thickness to reveal the surface wave patterns formed on the plate surfaces. The results of the analysis allowed explaining the non-uniform distribution of the surface wave pattern nodes formed on the PMMA plate by a point oscillator. The wave pattern reconstruction made it possible to reveal fundamental and reflected waves as well as their interference patterns with node distributions on the surfaces of the plate. These results may be useful for defect detection in composite materials such as delamination, impact damage, gaps, etc.
1. Acoustic metasurface for refracted wave manipulation
Science.gov (United States)
Han, Li-Xiang; Yao, Yuan-Wei; Zhang, Xin; Wu, Fu-Gen; Dong, Hua-Feng; Mu, Zhong-Fei; Li, Jing-bo
2018-02-01
Here we present a design of a transmitted acoustic metasurface based on a single row of Helmholtz resonators with varying geometric parameters. The proposed metasurface can not only steer an acoustic beam as expected from the generalized Snell's law of refraction, but also exhibits various interesting properties and potential applications such as insulation of two quasi-intersecting transmitted sound waves, ultrasonic Bessel beam generator, frequency broadening effect of anomalous refraction and focusing.
2. Active micromixer using surface acoustic wave streaming
Science.gov (United States)
Branch,; Darren W. , Meyer; Grant D. , Craighead; Harold, G [Ithaca, NY
2011-05-17
An active micromixer uses a surface acoustic wave, preferably a Rayleigh wave, propagating on a piezoelectric substrate to induce acoustic streaming in a fluid in a microfluidic channel. The surface acoustic wave can be generated by applying an RF excitation signal to at least one interdigital transducer on the piezoelectric substrate. The active micromixer can rapidly mix quiescent fluids or laminar streams in low Reynolds number flows. The active micromixer has no moving parts (other than the SAW transducer) and is, therefore, more reliable, less damaging to sensitive fluids, and less susceptible to fouling and channel clogging than other types of active and passive micromixers. The active micromixer is adaptable to a wide range of geometries, can be easily fabricated, and can be integrated in a microfluidic system, reducing dead volume. Finally, the active micromixer has on-demand on/off mixing capability and can be operated at low power.
3. Circuit quantum acoustodynamics with surface acoustic waves.
Science.gov (United States)
Manenti, Riccardo; Kockum, Anton F; Patterson, Andrew; Behrle, Tanja; Rahamim, Joseph; Tancredi, Giovanna; Nori, Franco; Leek, Peter J
2017-10-17
The experimental investigation of quantum devices incorporating mechanical resonators has opened up new frontiers in the study of quantum mechanics at a macroscopic level. It has recently been shown that surface acoustic waves (SAWs) can be piezoelectrically coupled to superconducting qubits, and confined in high-quality Fabry-Perot cavities in the quantum regime. Here we present measurements of a device in which a superconducting qubit is coupled to a SAW cavity, realising a surface acoustic version of cavity quantum electrodynamics. We use measurements of the AC Stark shift between the two systems to determine the coupling strength, which is in agreement with a theoretical model. This quantum acoustodynamics architecture may be used to develop new quantum acoustic devices in which quantum information is stored in trapped on-chip acoustic wavepackets, and manipulated in ways that are impossible with purely electromagnetic signals, due to the 10(5) times slower mechanical waves.In this work, Manenti et al. present measurements of a device in which a tuneable transmon qubit is piezoelectrically coupled to a surface acoustic wave cavity, realising circuit quantum acoustodynamic architecture. This may be used to develop new quantum acoustic devices.
4. Acoustic-Emission Weld-Penetration Monitor
Science.gov (United States)
Maram, J.; Collins, J.
1986-01-01
Weld penetration monitored by detection of high-frequency acoustic emissions produced by advancing weld pool as it melts and solidifies in workpiece. Acoustic emission from TIG butt weld measured with 300-kHz resonant transducer. Rise in emission level coincides with cessation of weld penetration due to sudden reduction in welding current. Such monitoring applied to control of automated and robotic welders.
5. Broadband Acoustic Cloak for Ultrasound Waves
Science.gov (United States)
Zhang, Shu; Xia, Chunguang; Fang, Nicholas
2011-01-01
Invisibility devices based on coordinate transformation have opened up a new field of considerable interest. We present here the first practical realization of a low-loss and broadband acoustic cloak for underwater ultrasound. This metamaterial cloak is constructed with a network of acoustic circuit elements, namely, serial inductors and shunt capacitors. Our experiment clearly shows that the acoustic cloak can effectively bend the ultrasound waves around the hidden object, with reduced scattering and shadow. Because of the nonresonant nature of the building elements, this low-loss (˜6dB/m) cylindrical cloak exhibits invisibility over a broad frequency range from 52 to 64 kHz. Furthermore, our experimental study indicates that this design approach should be scalable to different acoustic frequencies and offers the possibility for a variety of devices based on coordinate transformation.
6. Nonlinear interaction between acoustic gravity waves
Directory of Open Access Journals (Sweden)
P. Axelsson
1996-03-01
Full Text Available The resonant interaction between three acoustic gravity waves is considered. We improve on the results of previous authors and write the new coupling coefficients in a symmetric form. Particular attention is paid to the low-frequency limit.
7. Nonlinear interaction between acoustic gravity waves
Directory of Open Access Journals (Sweden)
P. Axelsson
Full Text Available The resonant interaction between three acoustic gravity waves is considered. We improve on the results of previous authors and write the new coupling coefficients in a symmetric form. Particular attention is paid to the low-frequency limit.
8. Surface acoustic wave probe implant for predicting epileptic seizures
Energy Technology Data Exchange (ETDEWEB)
Gopalsami, Nachappa [Naperville, IL; Kulikov, Stanislav [Sarov, RU; Osorio, Ivan [Leawood, KS; Raptis, Apostolos C [Downers Grove, IL
2012-04-24
A system and method for predicting and avoiding a seizure in a patient. The system and method includes use of an implanted surface acoustic wave probe and coupled RF antenna to monitor temperature of the patient's brain, critical changes in the temperature characteristic of a precursor to the seizure. The system can activate an implanted cooling unit which can avoid or minimize a seizure in the patient.
9. Energy scavenging system by acoustic wave and integrated wireless communication
Science.gov (United States)
Kim, Albert
The purpose of the project was developing an energy-scavenging device for other bio implantable devices. Researchers and scientist have studied energy scavenging method because of the limitation of traditional power source, especially for bio-implantable devices. In this research, piezoelectric power generator that activates by acoustic wave, or music was developed. Follow by power generator, a wireless communication also integrated with the device for monitoring the power generation. The Lead Zirconate Titanate (PZT) bimorph cantilever with a proof mass at the free end tip was studied to convert acoustic wave to power. The music or acoustic wave played through a speaker to vibrate piezoelectric power generator. The LC circuit integrated with the piezoelectric material for purpose of wireless monitoring power generation. However, wireless monitoring can be used as wireless power transmission, which means the signal received via wireless communication also can be used for power for other devices. Size of 74 by 7 by 7cm device could generate and transmit 100mVp from 70 mm distance away with electrical resonant frequency at 420.2 kHz..
10. Acoustic waves in granular materials
NARCIS (Netherlands)
Mouraille, O.J.P.; Luding, Stefan
2008-01-01
Dynamic simulations with discrete elements are used to obtain more insight into the wave propagation in dense granular media. A small perturbation is created on one side of a dense, static packing and examined during its propagation until it arrives at the opposite side. The influence of
11. Acoustic spin pumping in magnetoelectric bulk acoustic wave resonator
Energy Technology Data Exchange (ETDEWEB)
Polzikova, N. I., E-mail: [email protected]; Alekseev, S. G.; Pyataikin, I. I.; Kotelyanskii, I. M.; Luzanov, V. A.; Orlov, A. P. [Kotel’nikov Institute of Radio Engineering and Electronics of Russian Academy of Sciences, Mokhovaya 11, building 7, Moscow, 125009 (Russian Federation)
2016-05-15
We present the generation and detection of spin currents by using magnetoelastic resonance excitation in a magnetoelectric composite high overtone bulk acoustic wave (BAW) resonator (HBAR) formed by a Al-ZnO-Al-GGG-YIG-Pt structure. Transversal BAW drives magnetization oscillations in YIG film at a given resonant magnetic field, and the resonant magneto-elastic coupling establishes the spin-current generation at the Pt/YIG interface. Due to the inverse spin Hall effect (ISHE) this BAW-driven spin current is converted to a dc voltage in the Pt layer. The dependence of the measured voltage both on magnetic field and frequency has a resonant character. The voltage is determined by the acoustic power in HBAR and changes its sign upon magnetic field reversal. We compare the experimentally observed amplitudes of the ISHE electrical field achieved by our method and other approaches to spin current generation that use surface acoustic waves and microwave resonators for ferromagnetic resonance excitation, with the theoretically expected values.
12. Study of Ion Acoustic Wave Damping through Green's Functions
DEFF Research Database (Denmark)
Hsuan, H.C.S.; Jensen, Vagn Orla
1973-01-01
Green's function analyses of ion acoustic waves in streaming plasmas show that, in general, the waves damp algebraically rather than exponentially with distance from exciter.......Green's function analyses of ion acoustic waves in streaming plasmas show that, in general, the waves damp algebraically rather than exponentially with distance from exciter....
13. Acoustical Wave Propagation in Sonic Composites
Directory of Open Access Journals (Sweden)
Iulian Girip
2015-09-01
Full Text Available The goal of this paper is to discuss the technique of controlling the mechanical properties of sonic composites. The idea is to architecture the scatterers and material from which they are made, their number and geometry in order to obtain special features in their response to external waves. We refer to perfectly reflecting of acoustical waves over a desired range of frequencies or to prohibit their propagation in certain directions, or confining the waves within specified volumes. The internal structure of the material has to be chosen in such a way that to avoid the scattering of acoustical waves inside the material. This is possible if certain band-gaps of frequencies can be generated for which the waves are forbidden to propagate in certain directions. These bandgaps can be extended to cover all possible directions of propagation by resulting a full band-gap. If the band-gaps are not wide enough, their frequency ranges do not overlap. These band-gaps can overlap due to reflections on the surface of thick scatterers, as well as due to wave propagation inside them. growth.
14. Resonant surface acoustic wave chemical detector
Energy Technology Data Exchange (ETDEWEB)
Brocato, Robert W.; Brocato, Terisse; Stotts, Larry G.
2017-08-08
Apparatus for chemical detection includes a pair of interdigitated transducers (IDTs) formed on a piezoelectric substrate. The apparatus includes a layer of adsorptive material deposited on a surface of the piezoelectric substrate between the IDTs, where each IDT is conformed, and is dimensioned in relation to an operating frequency and an acoustic velocity of the piezoelectric substrate, so as to function as a single-phase uni-directional transducer (SPUDT) at the operating frequency. Additionally, the apparatus includes the pair of IDTs is spaced apart along a propagation axis and mutually aligned relative to said propagation axis so as to define an acoustic cavity that is resonant to surface acoustic waves (SAWs) at the operating frequency, where a distance between each IDT of the pair of IDTs ranges from 100 wavelength of the operating frequency to 400 wavelength of the operating frequency.
15. An acoustical model based monitoring network
NARCIS (Netherlands)
Wessels, P.W.; Basten, T.G.H.; Eerden, F.J.M. van der
2010-01-01
In this paper the approach for an acoustical model based monitoring network is demonstrated. This network is capable of reconstructing a noise map, based on the combination of measured sound levels and an acoustic model of the area. By pre-calculating the sound attenuation within the network the
16. Optimized reflector stacks for solidly mounted bulk acoustic wave resonators
NARCIS (Netherlands)
Jose, Sumy; Jansman, André B.M.; Hueting, Raymond Josephus Engelbart; Schmitz, Jurriaan
2010-01-01
The quality factor (Q) of a solidly mounted bulk acoustic wave resonator (SMR) is limited by substrate losses, because the acoustic mirror is traditionally optimized to reflect longitudinal waves only. We propose two different design approaches derived from optics to tailor the acoustic mirror for
17. Surface Acoustic Wave Strain Sensor Model
OpenAIRE
Wilson, William; Gary ATKINSON
2011-01-01
NASA Langley Research Center is investigating Surface Acoustic Wave (SAW) sensor technology for harsh environments aimed at aerospace applications. To aid in development of sensors a model of a SAW strain sensor has been developed. The new model extends the modified matrix method to include the response of Orthogonal Frequency Coded (OFC) reflectors and the response of SAW devices to strain. These results show that the model accurately captures the strain response of a SAW sensor on a Langasi...
18. Comparison of Transmission Line Methods for Surface Acoustic Wave Modeling
Science.gov (United States)
Wilson, William; Atkinson, Gary
2009-01-01
Surface Acoustic Wave (SAW) technology is low cost, rugged, lightweight, extremely low power and can be used to develop passive wireless sensors. For these reasons, NASA is investigating the use of SAW technology for Integrated Vehicle Health Monitoring (IVHM) of aerospace structures. To facilitate rapid prototyping of passive SAW sensors for aerospace applications, SAW models have been developed. This paper reports on the comparison of three methods of modeling SAWs. The three models are the Impulse Response Method (a first order model), and two second order matrix methods; the conventional matrix approach, and a modified matrix approach that is extended to include internal finger reflections. The second order models are based upon matrices that were originally developed for analyzing microwave circuits using transmission line theory. Results from the models are presented with measured data from devices. Keywords: Surface Acoustic Wave, SAW, transmission line models, Impulse Response Method.
19. Non-Linear Excitation of Ion Acoustic Waves
DEFF Research Database (Denmark)
Michelsen, Poul; Hirsfield, J. L.
1974-01-01
The excitation of ion acoustic waves by nonlinear coupling of two transverse magnetic waves generated in a microwave cavity was investigated. Measurements of the wave amplitude showed good agreement with calculations based on the Vlasov equation.......The excitation of ion acoustic waves by nonlinear coupling of two transverse magnetic waves generated in a microwave cavity was investigated. Measurements of the wave amplitude showed good agreement with calculations based on the Vlasov equation....
20. Deconvolution of acoustically detected bubble-collapse shock waves.
Science.gov (United States)
Johansen, Kristoffer; Song, Jae Hee; Johnston, Keith; Prentice, Paul
2017-01-01
The shock wave emitted by the collapse of a laser-induced bubble is detected at propagation distances of 30, 40and50mm, using a PVdF needle hydrophone, with a non-flat end-of-cable frequency response, calibrated for magnitude and phase, from 125kHz to 20MHz. High-speed shadowgraphic imaging at 5×106 frames per second, 10nstemporal resolution and 256 frames per sequence, records the bubble deflation from maximum to minimum radius, the collapse and shock wave generation, and the subsequent rebound in unprecedented detail, for a single sequence of an individual bubble. The Gilmore equation for bubble oscillation is solved according to the resolved bubble collapse, and simulated shock wave profiles deduced from the acoustic emissions, for comparison to the hydrophone recordings. The effects of single-frequency calibration, magnitude-only and full waveform deconvolution of the experimental data are presented, in both time and frequency domains. Magnitude-only deconvolution increases the peak pressure amplitude of the measured shock wave by approximately 9%, from single-frequency calibration, with full waveform deconvolution increasing it by a further 3%. Full waveform deconvolution generates a shock wave profile that is in agreement with the simulated profile, filtered according to the calibration bandwidth. Implications for the detection and monitoring of acoustic cavitation, where the role of periodic bubble collapse shock waves has recently been realised, are discussed. Copyright © 2016 Elsevier B.V. All rights reserved.
1. Acoustic Monitoring for Spaceflight Vehicle Applications Project
Data.gov (United States)
National Aeronautics and Space Administration — This SBIR will develop and demonstrate acoustic sensor technology enabling real-time, remotely performed measuring and monitoring of sound pressure levels and noise...
2. Cavitation controlled acoustic probe for fabric spot cleaning and moisture monitoring
Science.gov (United States)
Sheen, Shuh-Haw; Chien, Hual-Te; Raptis, Apostolos C.
1997-01-01
A method and apparatus are provided for monitoring a fabric. An acoustic probe generates acoustic waves relative to the fabric. An acoustic sensor, such as an accelerometer is coupled to the acoustic probe for generating a signal representative of cavitation activity in the fabric. The generated cavitation activity representative signal is processed to indicate moisture content of the fabric. A feature of the invention is a feedback control signal is generated responsive to the generated cavitation activity representative signal. The feedback control signal can be used to control the energy level of the generated acoustic waves and to control the application of a cleaning solution to the fabric.
3. Twisted electron-acoustic waves in plasmas
Energy Technology Data Exchange (ETDEWEB)
Aman-ur-Rehman, E-mail: [email protected] [Department of Nuclear Engineering, Pakistan Institute of Engineering and Applied Sciences (PIEAS), P. O. Nilore, Islamabad 45650 (Pakistan); Department of Physics and Applied Mathematics (DPAM), Pakistan Institute of Engineering and Applied Sciences (PIEAS), P.O. Nilore, Islamabad 45650 (Pakistan); Ali, S.; Khan, S. A. [National Centre for Physics at Quaid-e-Azam University Campus, Shahdra Valley Road, Islamabad 44000 (Pakistan); Shahzad, K. [Department of Physics and Applied Mathematics (DPAM), Pakistan Institute of Engineering and Applied Sciences (PIEAS), P.O. Nilore, Islamabad 45650 (Pakistan)
2016-08-15
In the paraxial limit, a twisted electron-acoustic (EA) wave is studied in a collisionless unmagnetized plasma, whose constituents are the dynamical cold electrons and Boltzmannian hot electrons in the background of static positive ions. The analytical and numerical solutions of the plasma kinetic equation suggest that EA waves with finite amount of orbital angular momentum exhibit a twist in its behavior. The twisted wave particle resonance is also taken into consideration that has been appeared through the effective wave number q{sub eff} accounting for Laguerre-Gaussian mode profiles attributed to helical phase structures. Consequently, the dispersion relation and the damping rate of the EA waves are significantly modified with the twisted parameter η, and for η → ∞, the results coincide with the straight propagating plane EA waves. Numerically, new features of twisted EA waves are identified by considering various regimes of wavelength and the results might be useful for transport and trapping of plasma particles in a two-electron component plasma.
4. Acoustic evaluation of wood quality in standing trees. Part I, Acoustic wave behavior
Science.gov (United States)
Xiping Wang; Robert J. Ross; Peter Carter
2007-01-01
Acoustic wave velocities in standing trees or live softwood species were measured by the time-of-flight (TOF) method. Tree velocities were compared with acoustic velocities measured in corresponding butt logs through a resonance acoustic method. The experimental data showed a skewed relationship between tree and log acoustic measurements. For most trees tested,...
5. Identification of rocket-induced acoustic waves in the ionosphere
Science.gov (United States)
Mabie, Justin; Bullett, Terence; Moore, Prentiss; Vieira, Gerald
2016-10-01
Acoustic waves can create plasma disturbances in the ionosphere, but the number of observations is limited. Large-amplitude acoustic waves generated by energetic sources like large earthquakes and tsunamis are more readily observed than acoustic waves generated by weaker sources. New observations of plasma displacements caused by rocket-generated acoustic waves were made using the Vertically Incident Pulsed Ionospheric Radar (VIPIR), an advanced high-frequency radar. Rocket-induced acoustic waves which are characterized by low amplitudes relative to those induced by more energetic sources can be detected in the ionosphere using the phase data from fixed frequency radar observations of a plasma layer. This work is important for increasing the number and quality of observations of acoustic waves in the ionosphere and could help improve the understanding of energy transport from the lower atmosphere to the thermosphere.
6. Acoustic field distribution of sawtooth wave with nonlinear SBE model
Energy Technology Data Exchange (ETDEWEB)
Liu, Xiaozhou, E-mail: [email protected]; Zhang, Lue; Wang, Xiangda; Gong, Xiufen [Key Laboratory of Modern Acoustics, Ministry of Education, Institute of Acoustics, Nanjing University, Nanjing 210093 (China)
2015-10-28
For precise prediction of the acoustic field distribution of extracorporeal shock wave lithotripsy with an ellipsoid transducer, the nonlinear spheroidal beam equations (SBE) are employed to model acoustic wave propagation in medium. To solve the SBE model with frequency domain algorithm, boundary conditions are obtained for monochromatic and sawtooth waves based on the phase compensation. In numerical analysis, the influence of sinusoidal wave and sawtooth wave on axial pressure distributions are investigated.
7. Wave-Flow Interactions and Acoustic Streaming
CERN Document Server
Chafin, Clifford E
2016-01-01
The interaction of waves and flows is a challenging topic where a complete resolution has been frustrated by the essential nonlinear features in the hydrodynamic case. Even in the case of EM waves in flowing media, the results are subtle. For a simple shear flow of constant n fluid, incident radiation is shown to be reflected and refracted in an analogous manner to Snell's law. However, the beam intensities differ and the system has an asymmetry in that an internal reflection gap opens at steep incident angles nearly oriented with the shear. For EM waves these effects are generally negligible in real systems but they introduce the topic at a reduced level of complexity of the more interesting acoustic case. Acoustic streaming is suggested, both from theory and experimental data, to be associated with vorticity generation at the driver itself. Bounds on the vorticity in bulk and nonlinear effects demonstrate that the bulk sources, even with attenuation, cannot drive such a strong flow. A review of the velocity...
8. Surface Acoustic Wave Strain Sensor Model
Directory of Open Access Journals (Sweden)
William WILSON
2011-04-01
Full Text Available NASA Langley Research Center is investigating Surface Acoustic Wave (SAW sensor technology for harsh environments aimed at aerospace applications. To aid in development of sensors a model of a SAW strain sensor has been developed. The new model extends the modified matrix method to include the response of Orthogonal Frequency Coded (OFC reflectors and the response of SAW devices to strain. These results show that the model accurately captures the strain response of a SAW sensor on a Langasite substrate. The results of the model of a SAW Strain Sensor on Langasite are presented.
9. Surface acoustic wave propagation in graphene film
Energy Technology Data Exchange (ETDEWEB)
Roshchupkin, Dmitry, E-mail: [email protected]; Plotitcyna, Olga; Matveev, Viktor; Kononenko, Oleg; Emelin, Evgenii; Irzhak, Dmitry [Institute of Microelectronics Technology and High-Purity Materials Russian Academy of Sciences, Chernogolovka 142432 (Russian Federation); Ortega, Luc [Laboratoire de Physique des Solides, Univ. Paris-Sud, CNRS, UMR 8502, 91405 Orsay Cedex (France); Zizak, Ivo; Erko, Alexei [Institute for Nanometre Optics and Technology, Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Albert-Einstein Strasse 15, 12489 Berlin (Germany); Tynyshtykbayev, Kurbangali; Insepov, Zinetula [Nazarbayev University Research and Innovation System, 53 Kabanbay Batyr St., Astana 010000 (Kazakhstan)
2015-09-14
Surface acoustic wave (SAW) propagation in a graphene film on the surface of piezoelectric crystals was studied at the BESSY II synchrotron radiation source. Talbot effect enabled the visualization of the SAW propagation on the crystal surface with the graphene film in a real time mode, and high-resolution x-ray diffraction permitted the determination of the SAW amplitude in the graphene/piezoelectric crystal system. The influence of the SAW on the electrical properties of the graphene film was examined. It was shown that the changing of the SAW amplitude enables controlling the magnitude and direction of current in graphene film on the surface of piezoelectric crystals.
10. Nonlinear Acoustic Wave Interactions in Layered Media.
Science.gov (United States)
1980-03-06
entire complex plane. The residue of GX at all of its poles must be determined in order to evaluate Equation (2-22) via Cauchy’s Residue Theorem ...4 ,0 0 Q) N +1 - 0 L I. 35 zero. Then, by Cauchy’s Residue Theorem , the Green’s function G [Equation (2-27)] for forward-guided modes is given by a...34Connection Between the Fay and Fubini Solutions for Plane Sound Waves of Finite-Amplitude." Journal of the Acoustical Society of America, 39 (1966
11. Electron Acoustic Waves in Pure Ion Plasmas
Science.gov (United States)
Anderegg, F.; Affolter, M.; Driscoll, C. F.; O'Neil, T. M.; Valentini, F.
2012-10-01
Electron Acoustic Waves (EAWs) are the low-frequency branch of near-linear Langmuir (plasma) waves: the frequency is such that the complex dielectric function (Dr, Di) has Dr= 0; and flattening'' of f(v) near the wave phase velocity vph gives Di=0 and eliminates Landau damping. Here, we observe standing axisymmetric EAWs in a pure ion column.footnotetextF. Anderegg, et al., Phys. Rev. Lett. 102, 095001 (2009). At low excitation amplitudes, the EAWs have vph˜1.4 v, in close agreement with near-linear theory. At moderate excitation strengths, EAW waves are observed over a range of frequencies, with 1.3 v vphvph.footnotetextF. Valentini et al., arXiv:1206.3500v1. Large amplitude EAWs have strong phase-locked harmonic content, and experiments will be compared to same-geometry simulations, and to simulations of KEENfootnotetextB. Afeyan et al., Proc. Inertial Fusion Sci. and Applications 2003, A.N.S. Monterey (2004), p. 213. waves in HEDLP geometries.
12. Nonlinear propagation of weakly relativistic ion-acoustic waves in ...
2016-10-06
4], in the polar ... and potentiality in investigating various types of collec- tive processes in astrophysical, space as well as ... Different types of ion-acoustic, dust-acoustic or elec- tron-acoustic waves have been studied [27–31] ...
13. Spherical dust acoustic solitary waves with two temperature ions
CERN Document Server
Eslami, Esmaeil
2014-01-01
The nonlinear dust acoustic waves in unmagnetized dusty plasma which consists of two temperature Boltzmann distributed ions and Boltzmann distributed electrons in spherical dimension investigated and obtained spherical Kadomtsev Petviashvili (SKP) equation and shown that the dust acoustic solitary wave can exist in the SKP equation.
14. Influence of acoustic waves on TEA CO 2 laser performance
CSIR Research Space (South Africa)
Von Bergmann, H
2007-01-01
Full Text Available In this paper the author’s present results on the influence of acoustic waves on the output laser beam from high repetition rate TEA CO 2 lasers. The authors show that acoustic waves generated inside the cavity lead to deterioration in beam quality...
15. Wave-wave interactions and deep ocean acoustics.
Science.gov (United States)
Guralnik, Z; Bourdelais, J; Zabalgogeazcoa, X; Farrell, W E
2013-10-01
Deep ocean acoustics, in the absence of shipping and wildlife, is driven by surface processes. Best understood is the signal generated by non-linear surface wave interactions, the Longuet-Higgins mechanism, which dominates from 0.1 to 10 Hz, and may be significant for another octave. For this source, the spectral matrix of pressure and vector velocity is derived for points near the bottom of a deep ocean resting on an elastic half-space. In the absence of a bottom, the ratios of matrix elements are universal constants. Bottom effects vitiate the usual "standing wave approximation," but a weaker form of the approximation is shown to hold, and this is used for numerical calculations. In the weak standing wave approximation, the ratios of matrix elements are independent of the surface wave spectrum, but depend on frequency and the propagation environment. Data from the Hawaii-2 Observatory are in excellent accord with the theory for frequencies between 0.1 and 1 Hz, less so at higher frequencies. Insensitivity of the spectral ratios to wind, and presumably waves, is indeed observed in the data.
16. On Mass Loading and Dissipation Measured with Acoustic Wave Sensors: A Review
Directory of Open Access Journals (Sweden)
Marina V. Voinova
2009-01-01
Full Text Available We summarize current trends in the analysis of physical properties (surface mass density, viscosity, elasticity, friction, and charge of various thin films measured with a solid-state sensor oscillating in a gaseous or liquid environment. We cover three different types of mechanically oscillating sensors: the quartz crystal microbalance with dissipation (QCM-D monitoring, surface acoustic wave (SAW, resonators and magnetoelastic sensors (MESs. The fourth class of novel acoustic wave (AW mass sensors, namely thin-film bulk acoustic resonators (TFBARs on vibrating membranes is discussed in brief. The paper contains a survey of theoretical results and practical applications of the sensors and includes a comprehensive bibliography.
17. Acoustic wave characterization of silicon phononic crystal plate
Science.gov (United States)
Feng, Duan; Jiang, Wanli; Xu, Dehui; Xiong, Bin; Wang, Yuelin
2015-08-01
In this paper, characterization of megahertz Lamb waves in a silicon phononic crystal based asymmetry filter by laser Doppler vibrometer is demonstrated. The acoustic power from a piezoelectric substrate was transmitted into the silicon superstrate by fluid coupling method, and measured results show that the displacement amplitude of the acoustic wave in the superstrate was approximately one fifth of that in the piezoelectric substrate. Effect of the phononic bandgap on the propagation of Lamb wave in the silicon superstrate is also measured, and the result shows that the phononic crystal structure could reflect part of the acoustic waves back.
18. Vehicle exhaust gas chemical sensors using acoustic wave resonators
Energy Technology Data Exchange (ETDEWEB)
Cernosek, R.W.; Small, J.H.; Sawyer, P.S.; Bigbie, J.R. [Sandia National Labs., Albuquerque, NM (United States); Anderson, M.T. [3M Industrial and Consumer Sector Research Lab., St. Paul, MN (United States)
1998-03-01
Under Sandias Laboratory Directed Research and Development (LDRD) program, novel acoustic wave-based sensors were explored for detecting gaseous chemical species in vehicle exhaust streams. The need exists for on-line, real-time monitors to continuously analyze the toxic exhaust gases -- nitrogen oxides (NOx), carbon monoxide (CO), and hydrocarbons (HC) -- for determining catalytic converter efficiency, documenting compliance to emission regulations, and optimizing engine performance through feedback control. In this project, the authors adapted existing acoustic wave chemical sensor technology to the high temperature environment and investigated new robust sensor materials for improving gas detection sensitivity and selectivity. This report describes one new sensor that has potential use as an exhaust stream residual hydrocarbon monitor. The sensor consists of a thickness shear mode (TSM) quartz resonator coated with a thin mesoporous silica layer ion-exchanged with palladium ions. When operated at temperatures above 300 C, the high surface area film catalyzes the combustion of the hydrocarbon vapors in the presence of oxygen. The sensor acts as a calorimeter as the exothermic reaction slightly increases the temperature, stressing the sensor surface, and producing a measurable deviation in the resonator frequency. Sensitivities as high as 0.44 (ppm-{Delta}f) and (ppm-gas) have been measured for propylene gas, with minimum detectable signals of < 50 ppm of propylene at 500 C.
19. Acoustic emission health monitoring of steel bridges
NARCIS (Netherlands)
Pahlavan, P.L.; Paulissen, J.H.; Pijpers, R.J.M.; Hakkesteegt, H.C.; Jansen, T.H.
2014-01-01
Despite extensive developments in the field of Acoustic Emission (AE) for monitoring fatigue cracks in steel structures, the implementation of AE systems for large-scale bridges is hindered by limitations associated with instrumentation costs and signal processing complexities. This paper sheds
20. Wind, waves, and acoustic background levels at Station ALOHA
Science.gov (United States)
Duennebier, Fred K.; Lukas, Roger; Nosal, Eva-Marie; Aucan, JéRome; Weller, Robert A.
2012-03-01
Frequency spectra from deep-ocean near-bottom acoustic measurements obtained contemporaneously with wind, wave, and seismic data are described and used to determine the correlations among these data and to discuss possible causal relationships. Microseism energy appears to originate in four distinct regions relative to the hydrophone: wind waves above the sensors contribute microseism energy observed on the ocean floor; a fraction of this local wave energy propagates as seismic waves laterally, and provides a spatially integrated contribution to microseisms observed both in the ocean and on land; waves in storms generate microseism energy in deep water that travels as seismic waves to the sensor; and waves reflected from shorelines provide opposing waves that add to the microseism energy. Correlations of local wind speed with acoustic and seismic spectral time series suggest that the local Longuet-Higgins mechanism is visible in the acoustic spectrum from about 0.4 Hz to 80 Hz. Wind speed and acoustic levels at the hydrophone are poorly correlated below 0.4 Hz, implying that the microseism energy below 0.4 Hz is not typically generated by local winds. Correlation of ocean floor acoustic energy with seismic spectra from Oahu and with wave spectra near Oahu imply that wave reflections from Hawaiian coasts, wave interactions in the deep ocean near Hawaii, and storms far from Hawaii contribute energy to the seismic and acoustic spectra below 0.4 Hz. Wavefield directionality strongly influences the acoustic spectrum at frequencies below about 2 Hz, above which the acoustic levels imply near-isotropic surface wave directionality.
1. Simulation and Optimization of Surface Acoustic Wave Devises
DEFF Research Database (Denmark)
Dühring, Maria Bayard
2007-01-01
In this paper a method to model the interaction of the mechanical field from a surface acoustic wave and the optical field in the waveguides of a Mach-Zehnder interferometer is presented. The surface acoustic waves are generated by interdigital transducers using a plane strain model...... of a piezoelectric, inhomogeneous material and reflections from the boundaries are avoided by applying perfectly matched layers. The optical modes in the waveguides are modeled by the time-harmonic wave equation for the magnetic field. The two models are coupled using the stress-optical relation and the change...... in effective refractive index introduced in the Mach-Zehnder interferometer arms by the stresses from the surface acoustic wave is calculated. It is shown that the effective refractive index of the fundamental optical mode increases at a surface acoustic wave crest and decreases at a trough. The height...
2. Acoustic-gravity waves in the atmosphere generated by infragravity waves in the ocean
National Research Council Canada - National Science Library
Godin, Oleg A; Zabotin, Nikolay A; Bullett, Terence W
2015-01-01
.... We show that, at frequencies below a certain transition frequency of about 3 mHz, infragravity waves continuously radiate their energy into the upper atmosphere in the form of acoustic-gravity waves...
3. Modulation of cavity-polaritons by surface acoustic waves
DEFF Research Database (Denmark)
de Lima, M. M.; Poel, Mike van der; Hey, R.
2006-01-01
We modulate cavity-polaritons using surface acoustic waves. The corresponding formation of a mini-Brillouin zone and band folding of the polariton dispersion is demonstrated for the first time. Results are in good agreement with model calculations.......We modulate cavity-polaritons using surface acoustic waves. The corresponding formation of a mini-Brillouin zone and band folding of the polariton dispersion is demonstrated for the first time. Results are in good agreement with model calculations....
4. Surface acoustic wave devices for sensor applications
Science.gov (United States)
Bo, Liu; Xiao, Chen; Hualin, Cai; Mohammad, Mohammad Ali; Xiangguang, Tian; Luqi, Tao; Yi, Yang; Tianling, Ren
2016-02-01
Surface acoustic wave (SAW) devices have been widely used in different fields and will continue to be of great importance in the foreseeable future. These devices are compact, cost efficient, easy to fabricate, and have a high performance, among other advantages. SAW devices can work as filters, signal processing units, sensors and actuators. They can even work without batteries and operate under harsh environments. In this review, the operating principles of SAW sensors, including temperature sensors, pressure sensors, humidity sensors and biosensors, will be discussed. Several examples and related issues will be presented. Technological trends and future developments will also be discussed. Project supported by the National Natural Science Foundation of China (Nos. 60936002, 61025021, 61434001, 61574083), the State Key Development Program for Basic Research of China (No. 2015CB352100), the National Key Project of Science and Technology (No. 2011ZX02403-002) and the Special Fund for Agroscientific Research in the Public Interest of China (No. 201303107). M.A.M is additionally supported by the Postdoctoral Fellowship (PDF) program of the Natural Sciences and Engineering Research Council (NSERC) of Canada and the China Postdoctoral Science Foundation (CPSF).
5. Observation of terahertz radiation coherently generated by acoustic waves
Science.gov (United States)
Armstrong, Michael R.; Reed, Evan J.; Kim, Ki-Yong; Glownia, James H.; Howard, William M.; Piner, Edwin L.; Roberts, John C.
2009-04-01
Over the past decade, pioneering and innovative experiments using subpicosecond lasers have demonstrated the generation and detection of acoustic and shock waves in materials with terahertz frequencies, the highest possible frequency acoustic waves. In addition to groundbreaking demonstrations of acoustic solitons, these experiments have led to new techniques for probing the structure of thin films. Terahertz-frequency electromagnetic radiation has been used in applications as diverse as molecular and material excitations, charge transfer, imaging and plasma dynamics. However, at present, existing approaches to detect and measure the time dependence of terahertz-frequency strain waves in materials use direct optical probes-time-resolved interferometry or reflectrometry. Piezoelectric-based strain gauges have been used in acoustic shock and strain wave experiments for decades, but the time resolution of such devices is limited to ~100ps and slower, the timescale of electronic recording technology. We have recently predicted that terahertz-frequency acoustic waves can be detected by observing terahertz radiation emitted when the acoustic wave propagates past an interface between materials of differing piezoelectric coefficients. Here, we report the first experimental observation of this fundamentally new phenomenon and demonstrate that it can be used to probe structural properties of thin films.
6. Positive amplitude electron acoustic solitary waves in auroral plasma
Science.gov (United States)
Ghosh, S. S.; Lakhina, G. S.
Rapidly moving positive potential pulses have been observed by FAST and POLAR satellites in downward current region of auroral plasma. They are characterized by their high velocities (> 1000 km/s) which are of the order of the electron drift velocities and are found to be associated with electron beams. Interestingly, it is observed that the width of such electron mode solitary waves increases with the amplitude [Ergun et al. (1998)]. Theoretically, they are interpreted as BGK electron phase space holes. However, Berthomier et al. (2000) have shown that a positive amplitude solitary wave may well exist for an electron acoustic mode. According to a weakly nonlinear theory, the width of such an electron acoustic solitary wave is expected to decrease with increasing amplitude which contradicts the observation. On the other hand, in our previous work, we have shown that the width of a large amplitude rarefactive ion acoustic solitary wave increases with an increasing amplitude [Ghosh et al. (2004)]. In the present work, we have extended our analysis to an electron acoustic solitary wave. A fully nonlinear solution of positive amplitude electron acoustic solitary waves (electron acoustic solitary holes) has been obtained by adopting the Sagdeev pseudopotetial technique. The plasma is assumed to be magnetized and traversed by the electron beam. The existence domain of such electron acoustic solitary holes is studied in detail. It is found that the width of electron acoustic solitary holes increases with increasing amplitude. Theoretically estimated width-amplitude variation profiles have been compared with recent satellite observations. It is proposed that a model based on electron acoustic mode may well interpret the fast moving solitary holes for an appropriate parameter space. References:Berthomier et al., Phys. Plasma, 7, 2987 (2000).Ergun et al., Phys. Rev. Lett., 81, 826, (1998).Ghosh and Lakhina,, Nonlin. Process. Geophys, (2004), (to be appeared).
7. Distributed acoustic sensing for pipeline monitoring
Energy Technology Data Exchange (ETDEWEB)
Hill, David; McEwen-King, Magnus [OptaSense, QinetiQ Ltd., London (United Kingdom)
2009-07-01
Optical fibre is deployed widely across the oil and gas industry. As well as being deployed regularly to provide high bandwidth telecommunications and infrastructure for SCADA it is increasingly being used to sense pressure, temperature and strain along buried pipelines, on subsea pipelines and downhole. In this paper we present results from the latest sensing capability using standard optical fibre to detect acoustic signals along the entire length of a pipeline. In Distributed Acoustic Sensing (DAS) an optical fibre is used for both sensing and telemetry. In this paper we present results from the OptaSense{sup TM} system which has been used to detect third party intervention (TPI) along buried pipelines. In a typical deployment the system is connected to an existing standard single-mode fibre, up to 50km in length, and was used to independently listen to the acoustic / seismic activity at every 10 meter interval. We will show that through the use of advanced array processing of the independent, simultaneously sampled channels it is possible to detect and locate activity within the vicinity of the pipeline and through sophisticated acoustic signal processing to obtain the acoustic signature to classify the type of activity. By combining spare fibre capacity in existing buried fibre optic cables; processing and display techniques commonly found in sonar; and state-of-the-art in fibre-optic distributed acoustic sensing, we will describe the new monitoring capabilities that are available to the pipeline operator. Without the expense of retrofitting sensors to the pipeline, this technology can provide a high performance, rapidly deployable and cost effective method of providing gapless and persistent monitoring of a pipeline. We will show how this approach can be used to detect, classify and locate activity such as; third party interference (including activity indicative of illegal hot tapping); real time tracking of pigs; and leak detection. We will also show how an
8. Nonlinear acoustic waves in micro-inhomogeneous solids
CERN Document Server
Nazarov, Veniamin
2014-01-01
Nonlinear Acoustic Waves in Micro-inhomogeneous Solids covers the broad and dynamic branch of nonlinear acoustics, presenting a wide variety of different phenomena from both experimental and theoretical perspectives. The introductory chapters, written in the style of graduate-level textbook, present a review of the main achievements of classic nonlinear acoustics of homogeneous media. This enables readers to gain insight into nonlinear wave processes in homogeneous and micro-inhomogeneous solids and compare it within the framework of the book. The subsequent eight chapters covering: Physical m
9. Capacitive acoustic wave detector and method of using same
Science.gov (United States)
Yost, William T. (Inventor)
1994-01-01
A capacitor having two substantially parallel conductive faces is acoustically coupled to a conductive sample end such that the sample face is one end of the capacitor. A non-contacting dielectric may serve as a spacer between the two conductive plates. The formed capacitor is connected to an LC oscillator circuit such as a Hartley oscillator circuit producing an output frequency which is a function of the capacitor spacing. This capacitance oscillates as the sample end coating is oscillated by an acoustic wave generated in the sample by a transmitting transducer. The electrical output can serve as an absolute indicator of acoustic wave displacement.
10. Passive acoustic monitoring of human physiology during activity indicates health and performance of soldiers and firefighters
Science.gov (United States)
Scanlon, Michael V.
2003-04-01
The Army Research Laboratory has developed a unique gel-coupled acoustic physiological monitoring sensor that has acoustic impedance properties similar to the skin. This facilitates the transmission of body sounds into the sensor pad, yet significantly repels ambient airborne noises due to an impedance mismatch. The sensor's sensitivity and bandwidth produce excellent signatures for detection and spectral analysis of diverse physiological events. Acoustic signal processing detects heartbeats, breaths, wheezes, coughs, blood pressure, activity, motion, and voice for communication and automatic speech recognition. The health and performance of soldiers, firefighters, and other first responders in strenuous and hazardous environments can be continuously and remotely monitored with body-worn acoustic sensors. Comfortable acoustic sensors can be in a helmet or in a strap around the neck, chest, and wrist. Noise-canceling sensor arrays help remove out-of-phase motion noise and enhance covariant physiology by using two acoustic sensors on the front sides of the neck and two additional acoustic sensors on each wrist. Pulse wave transit time between neck and wrist acoustic sensors will indicate systolic blood pressure. Larger torso-sized arrays can be used to acoustically inspect the lungs and heart, or built into beds for sleep monitoring. Acoustics is an excellent input for sensor fusion.
11. Dromion solutions for an electron acoustic wave and its application ...
Abstract. The nonlinear evolution of an electron acoustic wave is shown to obey the Davey–. Stewartson I equation which admits so called dromion solutions. The importance of these two dimensional localized solutions for recent satellite observations of wave structures in the day side polar cap regions is discussed and the ...
12. Cylindrical and spherical dust-acoustic wave modulations in dusty ...
The nonlinear wave modulation of planar and non-planar (cylindrical and spherical) dust-acoustic waves (DAW) propagating in dusty plasmas, in the presence of non-extensive distributions for ions and electrons is investigated. By employing multiple scales technique, a cylindrically and spherically modified nonlinear ...
13. Quantum ion-acoustic solitary waves in weak relativistic plasma
ion-acoustic waves. Recently, Stenflo et al [24] observed two new low-frequency elec- trostatic modes in ultra-cold unmagnetized quantum dusty plasmas. Ali and Shukla ... waves in a nonuniform ultra-cold Fermi dusty gas composed of inertialess electrons, and ions as well ... the Van Allen radiation belts [34] etc. Streaming ...
14. Cylindrical and spherical dust-acoustic wave modulations in dusty ...
Abstract. The nonlinear wave modulation of planar and non-planar (cylindrical and spherical) dust-acoustic waves (DAW) propagating in dusty plasmas, in the presence of non-extensive distribu- tions for ions and electrons is investigated. By employing multiple scales technique, a cylindrically and spherically modified ...
15. Quantum ion-acoustic wave oscillations in metallic nanowires
Energy Technology Data Exchange (ETDEWEB)
Moradi, Afshin, E-mail: [email protected] [Department of Engineering Physics, Kermanshah University of Technology, Kermanshah, Iran and Department of Nano Sciences, Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran (Iran, Islamic Republic of)
2015-05-15
The low-frequency electrostatic waves in metallic nanowires are studied using the quantum hydrodynamic model, in which the electron and ion components of the system are regarded as a two-species quantum plasma system. The Poisson equation as well as appropriate quantum boundary conditions give the analytical expressions of dispersion relations of the surface and bulk quantum ion-acoustic wave oscillations.
16. Acoustic tweezing of particles using decaying opposing travelling surface acoustic waves (DOTSAW).
Science.gov (United States)
Ng, Jia Wei; Devendran, Citsabehsan; Neild, Adrian
2017-10-11
Surface acoustic waves offer a versatile and biocompatible method of manipulating the location of suspended particles or cells within microfluidic systems. The most common approach uses the interference of identical frequency, counter propagating travelling waves to generate a standing surface acoustic wave, in which particles migrate a distance less than half the acoustic wavelength to their nearest pressure node. The result is the formation of a periodic pattern of particles. Subsequent displacement of this pattern, the prerequisite for tweezing, can be achieved by translation of the standing wave, and with it the pressure nodes; this requires changing either the frequency of the pair of waves, or their relative phase. Here, in contrast, we examine the use of two counterpropagating traveling waves of different frequency. The non-linearity of the acoustic forces used to manipulate particles, means that a small frequency difference between the two waves creates a substantially different force field, which offers significant advantages. Firstly, this approach creates a much longer range force field, in which migration takes place across multiple wavelengths, and causes particles to be gathered together in a single trapping site. Secondly, the location of this single trapping site can be controlled by the relative amplitude of the two waves, requiring simply an attenuation of one of the electrical drive signals. Using this approach, we show that by controlling the powers of the opposing incoherent waves, 5 μm particles can be migrated laterally across a fluid flow to defined locations with an accuracy of ±10 μm.
17. Condition Monitoring and Management from Acoustic Emissions
DEFF Research Database (Denmark)
Pontoppidan, Niels Henrik Bohl
2005-01-01
In the following, I will use technical terms without explanation as it gives the freedom to describe the project in a shorter form for those who already know. The thesis is about condition monitoring of large diesel engines from acoustic emission signals. The experiments have been focused...... is the analysis of the angular position changes of the engine related events such as fuel injection and valve openings, caused by operational load changes. With inspiration from speech recognition and voice effects the angular timing changes have been inverted with the event alignment framework. With the event...
18. Ionospheric effects of magneto-acoustic-gravity waves: Dispersion relation
Science.gov (United States)
Jones, R. Michael; Ostrovsky, Lev A.; Bedard, Alfred J.
2017-06-01
There is extensive evidence for ionospheric effects associated with earthquake-related atmospheric disturbances. Although the existence of earthquake precursors is controversial, one suggested method of detecting possible earthquake precursors and tsunamis is by observing possible ionospheric effects of atmospheric waves generated by such events. To study magneto-acoustic-gravity waves in the atmosphere, we have derived a general dispersion relation including the effects of the Earth's magnetic field. This dispersion relation can be used in a general atmospheric ray tracing program to calculate the propagation of magneto-acoustic-gravity waves from the ground to the ionosphere. The presence of the Earth's magnetic field in the ionosphere can radically change the dispersion properties of the wave. The general dispersion relation obtained here reduces to the known dispersion relations for magnetoacoustic waves and acoustic-gravity waves in the corresponding particular cases. The work described here is the first step in achieving a generalized ray tracing program permitting propagation studies of magneto-acoustic-gravity waves.
19. Magneto-acoustic imaging by continuous-wave excitation.
Science.gov (United States)
Shunqi, Zhang; Zhou, Xiaoqing; Tao, Yin; Zhipeng, Liu
2017-04-01
The electrical characteristics of tissue yield valuable information for early diagnosis of pathological changes. Magneto-acoustic imaging is a functional approach for imaging of electrical conductivity. This study proposes a continuous-wave magneto-acoustic imaging method. A kHz-range continuous signal with an amplitude range of several volts is used to excite the magneto-acoustic signal and improve the signal-to-noise ratio. The magneto-acoustic signal amplitude and phase are measured to locate the acoustic source via lock-in technology. An optimisation algorithm incorporating nonlinear equations is used to reconstruct the magneto-acoustic source distribution based on the measured amplitude and phase at various frequencies. Validation simulations and experiments were performed in pork samples. The experimental and simulation results agreed well. While the excitation current was reduced to 10 mA, the acoustic signal magnitude increased up to 10-7 Pa. Experimental reconstruction of the pork tissue showed that the image resolution reached mm levels when the excitation signal was in the kHz range. The signal-to-noise ratio of the detected magneto-acoustic signal was improved by more than 25 dB at 5 kHz when compared to classical 1 MHz pulse excitation. The results reported here will aid further research into magneto-acoustic generation mechanisms and internal tissue conductivity imaging.
20. Individually Identifiable Surface Acoustic Wave Sensors, Tags and Systems
Science.gov (United States)
Hines, Jacqueline H. (Inventor); Solie, Leland P. (Inventor); Tucker, Dana Y. G. (Inventor); Hines, Andrew T. (Inventor)
2017-01-01
A surface-launched acoustic wave sensor tag system for remotely sensing and/or providing identification information using sets of surface acoustic wave (SAW) sensor tag devices is characterized by acoustic wave device embodiments that include coding and other diversity techniques to produce groups of sensors that interact minimally, reducing or alleviating code collision problems typical of prior art coded SAW sensors and tags, and specific device embodiments of said coded SAW sensor tags and systems. These sensor/tag devices operate in a system which consists of one or more uniquely identifiable sensor/tag devices and a wireless interrogator. The sensor device incorporates an antenna for receiving incident RF energy and re-radiating the tag identification information and the sensor measured parameter(s). Since there is no power source in or connected to the sensor, it is a passive sensor. The device is wirelessly interrogated by the interrogator.
1. Propagation characteristics of acoustic emission wave in reinforced concrete
Directory of Open Access Journals (Sweden)
Haoxiong Feng
Full Text Available Due to the complexity of components and damage mechanism of reinforced concrete, the wave propagation characteristics in reinforced concrete are always complicated and difficult to determine. The objective of this article is to study the failure process of reinforced concrete structure under the damage caused by pencil-broken. A new method on the basis of the acoustic emission technique and the Hilbert-Huang transform theory is proposed in this work. By using acoustic emission technique, the acoustic emission wave signal is generating while the real-time damage information and the strain field of the reinforced concrete structure is receiving simultaneously. Based on the Hilbert-Huang transform (HHT theory, the peak frequency characteristics of the acoustic emission signals were extracted to identify the damage modes of the reinforced concrete structure. The results demonstrate that this method can quantitatively investigate the acoustic emission wave propagation characteristic in reinforced concrete structures and might also be promising in other civil constructions. Keywords: Acoustic emission, Reinforced concrete structure, Hilbert-Huang transform (HHT, Propagation characteristics
2. Acoustic gravity wave growth and damping in convecting plasma
Directory of Open Access Journals (Sweden)
T. R. Robinson
Full Text Available The propagation of acoustic gravity waves through steadily convecting plasma in the thermosphere has been analysed theoretically. The growth and damping rates of internal gravity waves due to the feedback effects of wave-modulated Joule heating and Laplace forcing have been calculated. It is found that large convection flow velocities lead to the growth of large-scale internal gravity waves, whilst small- and medium-scale waves are heavily damped, under similar conditions. It has also been shown that wave growth is favoured for waves travelling against the plasma flow direction. The effects of critical coupling when wave phase speeds match the plasma flow speed have also been investigated. The results of these calculations are discussed in the context of the atmospheric energy budget and thermosphere-ionosphere coupling.
3. Acoustic gravity wave growth and damping in convecting plasma
Directory of Open Access Journals (Sweden)
T. R. Robinson
1994-01-01
Full Text Available The propagation of acoustic gravity waves through steadily convecting plasma in the thermosphere has been analysed theoretically. The growth and damping rates of internal gravity waves due to the feedback effects of wave-modulated Joule heating and Laplace forcing have been calculated. It is found that large convection flow velocities lead to the growth of large-scale internal gravity waves, whilst small- and medium-scale waves are heavily damped, under similar conditions. It has also been shown that wave growth is favoured for waves travelling against the plasma flow direction. The effects of critical coupling when wave phase speeds match the plasma flow speed have also been investigated. The results of these calculations are discussed in the context of the atmospheric energy budget and thermosphere-ionosphere coupling.
4. A metasurface carpet cloak for electromagnetic, acoustic and water waves.
Science.gov (United States)
Yang, Yihao; Wang, Huaping; Yu, Faxin; Xu, Zhiwei; Chen, Hongsheng
2016-01-29
We propose a single low-profile skin metasurface carpet cloak to hide objects with arbitrary shape and size under three different waves, i.e., electromagnetic (EM) waves, acoustic waves and water waves. We first present a metasurface which can control the local reflection phase of these three waves. By taking advantage of this metasurface, we then design a metasurface carpet cloak which provides an additional phase to compensate the phase distortion introduced by a bump, thus restoring the reflection waves as if the incident waves impinge onto a flat mirror. The finite element simulation results demonstrate that an object can be hidden under these three kinds of waves with a single metasurface cloak.
5. Introducing passive acoustic filter in acoustic based condition monitoring: Motor bike piston-bore fault identification
Science.gov (United States)
Jena, D. P.; Panigrahi, S. N.
2016-03-01
Requirement of designing a sophisticated digital band-pass filter in acoustic based condition monitoring has been eliminated by introducing a passive acoustic filter in the present work. So far, no one has attempted to explore the possibility of implementing passive acoustic filters in acoustic based condition monitoring as a pre-conditioner. In order to enhance the acoustic based condition monitoring, a passive acoustic band-pass filter has been designed and deployed. Towards achieving an efficient band-pass acoustic filter, a generalized design methodology has been proposed to design and optimize the desired acoustic filter using multiple filter components in series. An appropriate objective function has been identified for genetic algorithm (GA) based optimization technique with multiple design constraints. In addition, the sturdiness of the proposed method has been demonstrated in designing a band-pass filter by using an n-branch Quincke tube, a high pass filter and multiple Helmholtz resonators. The performance of the designed acoustic band-pass filter has been shown by investigating the piston-bore defect of a motor-bike using engine noise signature. On the introducing a passive acoustic filter in acoustic based condition monitoring reveals the enhancement in machine learning based fault identification practice significantly. This is also a first attempt of its own kind.
6. An Unconditionally Stable Method for Solving the Acoustic Wave Equation
Directory of Open Access Journals (Sweden)
Zhi-Kai Fu
2015-01-01
Full Text Available An unconditionally stable method for solving the time-domain acoustic wave equation using Associated Hermit orthogonal functions is proposed. The second-order time derivatives in acoustic wave equation are expanded by these orthogonal basis functions. By applying Galerkin temporal testing procedure, the time variable can be eliminated from the calculations. The restriction of Courant-Friedrichs-Levy (CFL condition in selecting time step for analyzing thin layer can be avoided. Numerical results show the accuracy and the efficiency of the proposed method.
7. Acoustic-Gravity Waves Interacting with a Rectangular Trench
Directory of Open Access Journals (Sweden)
2015-01-01
Full Text Available A mathematical solution of the two-dimensional linear problem of an acoustic-gravity wave interacting with a rectangular trench, in a compressible ocean, is presented. Expressions for the flow field on both sides of the trench are derived. The dynamic bottom pressure produced by the acoustic-gravity waves on both sides of the trench is measurable, though on the transmission side it decreases with the trench depth. A successful recording of the bottom pressures could assist in the early detection of tsunami.
8. Broadband aberration-free focusing reflector for acoustic waves
Science.gov (United States)
Wang, Aixia; Qu, Shaobo; Ma, Hua; Wang, Jiafu; Jiang, Wei; Feng, Mingde
2017-11-01
An aberration-free focusing reflector (AFR) for acoustic waves is proposed with the aim to eliminate spherical aberration and coma simultaneously. Meanwhile, the AFR can focus acoustic waves with low dispersion in a wide frequency range of 14-50 kHz. The broadband aberration-free focusing effect is originated from an elliptical reflection phase gradient profile, which is achieved by milling different depths of axisymmetric grooves on a planoconcave-like brass plate using the ray theory. Theoretical and numerical results are in good agreement. The designed AFR can find broad engineering, industrial and medical applications.
9. Strong wave/mean-flow coupling in baroclinic acoustic streaming
Science.gov (United States)
Chini, Greg; Michel, Guillaume
2017-11-01
Recently, Chini et al. demonstrated the potential for large-amplitude acoustic streaming in compressible channel flows subjected to strong background cross-channel density variations. In contrast with classic Rayleigh streaming, standing acoustic waves of O (ɛ) amplitude acquire vorticity owing to baroclinic torques acting throughout the domain rather than via viscous torques acting in Stokes boundary layers. More significantly, these baroclinically-driven streaming flows have a magnitude that also is O (ɛ) , i.e. comparable to that of the sound waves. In the present study, the consequent potential for fully two-way coupling between the waves and streaming flows is investigated using a novel WKBJ analysis. The analysis confirms that the wave-driven streaming flows are sufficiently strong to modify the background density gradient, thereby modifying the leading-order acoustic wave structure. Simulations of the wave/mean-flow system enabled by the WKBJ analysis are performed to illustrate the nature of the two-way coupling, which contrasts sharply with classic Rayleigh streaming, for which the waves can first be determined and the streaming flows subsequently computed.
10. Nanoliter-droplet acoustic streaming via ultra high frequency surface acoustic waves.
Science.gov (United States)
Shilton, Richie J; Travagliati, Marco; Beltram, Fabio; Cecchini, Marco
2014-08-06
The relevant length scales in sub-nanometer amplitude surface acoustic wave-driven acoustic streaming are demonstrated. We demonstrate the absence of any physical limitations preventing the downscaling of SAW-driven internal streaming to nanoliter microreactors and beyond by extending SAW microfluidics up to operating frequencies in the GHz range. This method is applied to nanoliter scale fluid mixing. © 2014 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
11. Acoustic Gravity Waves Generated by an Oscillating Ice Sheet in Arctic Zone
Science.gov (United States)
Abdolali, A.; Kadri, U.; Kirby, J. T., Jr.
2016-12-01
We investigate the formation of acoustic-gravity waves due to oscillations of large ice blocks, possibly triggered by atmospheric and ocean currents, ice block shrinkage or storms and ice-quakes.For the idealized case of a homogeneous weakly compressible water bounded at the surface by ice sheet and a rigid bed, the description of the infinite family of acoustic modes is characterized by the water depth h and angular frequency of oscillating ice sheet ω ; The acoustic wave field is governed by the leading mode given by: Nmax=\\floor {(ω h)/(π c)} where c is the sound speed in water and the special brackets represent the floor function (Fig1). Unlike the free-surface setting, the higher acoustic modes might exhibit a larger contribution and therefore all progressive acoustic modes have to be considered.This study focuses on the characteristics of acoustic-gravity waves generated by an oscillating elastic ice sheet in a weakly compressible fluid coupled with a free surface model [Abdolali et al. 2015] representing shrinking ice blocks in realistic sea state, where the randomly oriented ice sheets cause inter modal transition and multidirectional reflections. A theoretical solution and a 3D numerical model have been developed for the study purposes. The model is first validated against the theoretical solution [Kadri, 2016]. To overcome the computational difficulties of 3D models, we derive a depth-integrated equation valid for spatially varying ice sheet thickness and water depth. We show that the generated acoustic-gravity waves contribute significantly to deep ocean currents compared to other mechanisms. In addition, these waves travel at the sound speed in water carrying information on ice sheet motion, providing various implications for ocean monitoring and detection of ice-quakes. Fig1:Snapshots of dynamic pressure given by an oscillating ice sheet; h=4500m, c=1500m/s, semi-length b=10km, ζ =1m, omega=π rad/s. Abdolali, A., Kirby, J. T. and Bellotti, G
12. Surface spin-electron acoustic waves in magnetically ordered metals
CERN Document Server
Andreev, Pavel A
2015-01-01
Degenerate plasmas with motionless ions show existence of three surface waves: the Langmuir wave, the electromagnetic wave, and the zeroth sound. Applying the separated spin evolution quantum hydrodynamics to half-space plasma we demonstrate the existence of the surface spin-electron acoustic wave (SSEAW). We study dispersion of the SSEAW. We show that there is hybridization between the surface Langmuir wave and the SSEAW at rather small spin polarization. In the hybridization area the dispersion branches are located close to each other. In this area there is a strong interaction between these waves leading to the energy exchange. Consequently, generating the Langmuir waves with the frequencies close to hybridization area we can generate the SSEAWs. Thus, we report a method of creation of the SEAWs.
13. Characteristics of acoustic wave from atmospheric nuclear explosions conducted at the USSR Test Sites
Science.gov (United States)
Sokolova, Inna
2015-04-01
Availability of the acoustic wave on the record of microbarograph is one of discriminate signs of atmospheric (surface layer of atmosphere) and contact explosions. Nowadays there is large number of air wave records from chemical explosions recorded by the IMS infrasound stations installed during recent decade. But there is small number of air wave records from nuclear explosions as air and contact nuclear explosions had been conducted since 1945 to 1962, before the Limited Test Ban Treaty was signed in 1963 (the treaty banning nuclear weapon tests in the atmosphere, in outer space and under water) by the Great Britain, USSR and USA. That time there was small number of installed microbarographs. First infrasound stations in the USSR appeared in 1954, and by the moment of the USSR collapse the network consisted of 25 infrasound stations, 3 of which were located on Kazakhstan territory - in Kurchatov (East Kazakhstan), in Borovoye Observatory (North Kazakhstan) and Talgar Observatory (Northern Tien Shan). The microbarograph of Talgar Observatory was installed in 1962 and recorded large number of air nuclear explosions conducted at Semipalatinsk Test Site and Novaya Zemlya Test Site. The epicentral distance to the STS was ~700 km, and to Novaya Zemlya Test Site ~3500 km. The historical analog records of the microbarograph were analyzed on the availability of the acoustic wave. The selected records were digitized, the database of acoustic signals from nuclear explosions was created. In addition, acoustic signals from atmospheric nuclear explosions conducted at the USSR Test Sites were recorded by analogue broadband seismic stations at wide range of epicentral distances, 300-3600 km. These signals coincide well by its form and spectral content with records of microbarographs and can be used for monitoring tasks and discrimination in places where infrasound observations are absent. Nuclear explosions which records contained acoustic wave were from 0.03 to 30 kt yield for
14. Propagation of acoustic-gravity waves in arctic zones with elastic ice-sheets
Science.gov (United States)
Kadri, Usama; Abdolali, Ali; Kirby, James T.
2017-04-01
We present an analytical solution of the boundary value problem of propagating acoustic-gravity waves generated in the ocean by earthquakes or ice-quakes in arctic zones. At the surface, we assume elastic ice-sheets of a variable thickness, and show that the propagating acoustic-gravity modes have different mode shape than originally derived by Ref. [1] for a rigid ice-sheet settings. Computationally, we couple the ice-sheet problem with the free surface model by Ref. [2] representing shrinking ice blocks in realistic sea state, where the randomly oriented ice-sheets cause inter modal transition at the edges and multidirectional reflections. We then derive a depth-integrated equation valid for spatially slowly varying thickness of ice-sheet and water depth. Surprisingly, and unlike the free-surface setting, here it is found that the higher acoustic-gravity modes exhibit a larger contribution. These modes travel at the speed of sound in water carrying information on their source, e.g. ice-sheet motion or submarine earthquake, providing various implications for ocean monitoring and detection of quakes. In addition, we found that the propagating acoustic-gravity modes can result in orbital displacements of fluid parcels sufficiently high that may contribute to deep ocean currents and circulation, as postulated by Refs. [1, 3]. References [1] U. Kadri, 2016. Generation of Hydroacoustic Waves by an Oscillating Ice Block in Arctic Zones. Advances in Acoustics and Vibration, 2016, Article ID 8076108, 7 pages http://dx.doi.org/10.1155/2016/8076108 [2] A. Abdolali, J. T. Kirby and G. Bellotti, 2015, Depth-integrated equation for hydro-acoustic waves with bottom damping, J. Fluid Mech., 766, R1 doi:10.1017/jfm.2015.37 [3] U. Kadri, 2014. Deep ocean water transportation by acoustic?gravity waves. J. Geophys. Res. Oceans, 119, doi:10.1002/ 2014JC010234
15. Reflection and Transmission of Acoustic Waves at Semiconductor - Liquid Interface
Directory of Open Access Journals (Sweden)
J. N. Sharma
2011-09-01
Full Text Available The study of reflection and transmission characteristics of acoustic waves at the interface of a semiconductor halfspace underlying an inviscid liquid has been carried out. The reflection and transmission coefficients of reflected and transmitted waves have been obtained for quasi-longitudinal (qP wave incident at the interface from fluid to semiconductor. The numerical computations of reflection and transmission coefficients have been carried out with the help of Gauss elimination method by using MATLAB programming for silicon (Si, germanium (Ge and silicon nitride (Si3N4 semiconductors. In order to interpret and compare, the computer simulated results are plotted graphically. The study may be useful in semiconductors, seismology and surface acoustic wave (SAW devices in addition to engines of the space shuttles.
16. Monitoring power breakers using vibro acoustic techniques
Directory of Open Access Journals (Sweden)
Horia Balan
2017-09-01
Full Text Available Speaking about the commutation’s equipment, it can be said that the best solution in increasing reliability and lowering the maintenance costs is a continuous monitoring of the equipment. However, if the price/quality ratio is considered, it is obvious that, for the moment, the diagnosis can be also an acceptable solution. Nowadays the predictive maintenance for equipment’s diagnosis is currently replacing the preventive diagnosis. An efficient modality of lowering the maintenance costs is to online monitoring the power breakers, during their operation in the power systems. Consequently any connecting/disconnecting operations may be used in diagnosing a power breaker. Thus any supplementary and superfluous tests and/or maintenance maneuvers are avoided. The paper presents the operational maintenance in a power station with three high voltage active breakers, Areva type. The method of establishing the state of a breaker consists in the comparison between the signature of the acoustic signal provided by the manufacturer and the signal issued from the testing operation of the breaker’s state. The software processing procedure and the methodology of determining the faults of the monitored equipment are also developed. All the tests on the circuit breaker are made according the prescriptions of normative.
17. Operational monitoring of acoustic sensor networks
Directory of Open Access Journals (Sweden)
Boltenkov V.A.
2015-06-01
Full Text Available Acoustic sensor networks (ASN are widely used to monitor water leaks in the power generating systems. Since the ASN are used in harsh climatic conditions the failures of microphone elements of ASN are inevitable. That's why the failure detection of ASN elements is a problem of current interest. Two techniques of operational monitoring ASN are developed. Both of them are based on the placement of the test sound source within a network. The signal processing for ASN sensors had to detect the failed element. Techniques are based time difference of arrival (TDOA estimating at the each pair of ASN elements. TDOA estimates as argmaximum of cross-correlation function (CCF for signals on each microphone sensors pair. The M-sequence phase-shift keyed signal is applied as a test acoustic signal to ensure high accuracy of the CCF maximum estimation at low signal/noise ratio (SNR. The first technique is based on the isolation principle for TDOA sum at three points. It require to locate the test sound source in the far field. This is not always possible due to technological reasons. For the second proposed technique test sound source can be located near the ASN. It is based on a system of hyperbolic equations solving for each of the four elements of the ASN. Both techniques has been tested in the computer imitation experiment. It was found that for the SNR to –5 dB both techniques show unmistakable indicators of control quality. The second method requires significantly more time control.
18. Wave Monitoring with Wireless Sensor Networks
NARCIS (Netherlands)
Marin Perianu, Mihai; Chatterjea, Supriyo; Marin Perianu, Raluca; Bosch, S.; Dulman, S.O.; Kininmonth, Stuart; Havinga, Paul J.M.
2008-01-01
Real-time collection of wave information is required for short and long term investigations of natural coastal processes. Current wave monitoring techniques use only point-measurements, which are practical where the bathymetry is relatively uniform. We propose a wave monitoring method that is
19. Wireless Passive Strain Sensor Based on Surface Acoustic Wave Devices
OpenAIRE
Nomura, T.; Kawasaki, K.; Saitoh, A
2008-01-01
Surface acoustic wave (SAW) devices offer many attractive features for applications as chemical and physical sensors. In this paper, a novel SAW strain sensor that employs SAW delay lines has been designed. Two crossed delay lines were used to measure the two-dimensional strain. A wireless sensing system is also proposed for effective operation of the strain sensor. In addition, an electronic system for accurately measuring the phase characteristics of the signal wave from the passive strain ...
20. Acoustic Wave Dispersion and Scattering in Complex Marine Sediment Structures
Science.gov (United States)
2015-09-30
Acoustic wave dispersion and scattering in complex marine sediment structures Charles W. Holland The Pennsylvania State University Applied...shear waves on dispersion in marine sediments . The first step will be development of the theory. WORK COMPLETED A brief summary of the work...propagation and scattering in the seabed. OBJECTIVES The objectives are to advance understanding of 1) the nature and mechanisms leading to sediment
1. Theoretical Model of Acoustic Wave Propagation in Shallow Water
Directory of Open Access Journals (Sweden)
Kozaczka Eugeniusz
2017-06-01
Full Text Available The work is devoted to the propagation of low frequency waves in a shallow sea. As a source of acoustic waves, underwater disturbances generated by ships were adopted. A specific feature of the propagation of acoustic waves in shallow water is the proximity of boundaries of the limiting media characterised by different impedance properties, which affects the acoustic field coming from a source situated in the water layer “deformed” by different phenomena. The acoustic field distribution in the real shallow sea is affected not only by multiple reflections, but also by stochastic changes in the free surface shape, and statistical changes in the seabed shape and impedance. The paper discusses fundamental problems of modal sound propagation in the water layer over different types of bottom sediments. The basic task in this case was to determine the acoustic pressure level as a function of distance and depth. The results of the conducted investigation can be useful in indirect determination of the type of bottom.
2. Lithium niobate phononic crystal for surface acoustic waves
Science.gov (United States)
Benchabane, S.; Khelif, A.; Rauch, J. Y.; Robert, L.; Laude, V.
2006-02-01
The recent theoretical and experimental demonstrations of stop bands for surface acoustic waves have greatly enlarged the potential application field for phononic crystals. The possibility of a direct excitation of these surface waves on a piezoelectric material, and their already extensive use in ultrasonics make them an interesting basis for phononic crystal based, acoustic signal processing devices. In this paper, we report on the demonstration of the existence of an absolute band gap for surface waves in a piezoelectric phononic crystal. The Surface Acoustic Wave propagation in a square lattice, two-dimensional lithium niobate phononic crystal is both theoretically and experimentally studied. A plane wave expansion method is used to predict the band gap position and width. The crystal was then fabricated by reactive ion etching of a bulk lithium niobate substrate. Standard interdigital transducers were used to characterize the phononic structure by direct electrical generation and detection of surface waves. A full band gap around 200 MHz was experimentally demonstrated, and close agreement is found with theoretical predictions.
3. Numerical modelling of nonlinear full-wave acoustic propagation
Energy Technology Data Exchange (ETDEWEB)
Velasco-Segura, Roberto, E-mail: [email protected]; Rendón, Pablo L., E-mail: [email protected] [Grupo de Acústica y Vibraciones, Centro de Ciencias Aplicadas y Desarrollo Tecnológico, Universidad Nacional Autónoma de México, Ciudad Universitaria, Apartado Postal 70-186, C.P. 04510, México D.F., México (Mexico)
2015-10-28
The various model equations of nonlinear acoustics are arrived at by making assumptions which permit the observation of the interaction with propagation of either single or joint effects. We present here a form of the conservation equations of fluid dynamics which are deduced using slightly less restrictive hypothesis than those necessary to obtain the well known Westervelt equation. This formulation accounts for full wave diffraction, nonlinearity, and thermoviscous dissipative effects. A two-dimensional, finite-volume method using Roe’s linearisation has been implemented to obtain numerically the solution of the proposed equations. This code, which has been written for parallel execution on a GPU, can be used to describe moderate nonlinear phenomena, at low Mach numbers, in domains as large as 100 wave lengths. Applications range from models of diagnostic and therapeutic HIFU, to parametric acoustic arrays and nonlinear propagation in acoustic waveguides. Examples related to these applications are shown and discussed.
4. Enhancing Plasma Surface Modification using high Intensity and high Power Ultrasonic Acoustic Waves
DEFF Research Database (Denmark)
2010-01-01
high intensity and high power acoustic waves (102) by at least one ultrasonic high intensity and high power acoustic wave generator (101 ), wherein the ultrasonic acoustic waves are directed to propagate towards said surface (314) of the object (100) so that a laminar boundary layer (313) of a gas...
5. Ion Acoustic Waves in the Presence of Langmuir Oscillations
DEFF Research Database (Denmark)
Pécseli, Hans
1976-01-01
The dielectric function for long-wavelength, low-frequency ion acoustic waves in the presence of short-wavelength, high-frequency electron oscillations is presented, where the ions are described by the collision-free Vlasov equation. The effect of the electron oscillations can be appropriately de...
6. Modelling of bulk acoustic wave resonators for microwave filters
NARCIS (Netherlands)
Jose, Sumy; Hueting, Raymond Josephus Engelbart; Jansman, Andreas
2008-01-01
Modelling and development of high Q thin-film bulk acoustic wave (BAW) devices is a topic of research gaining attention due to good selectivity and steep transition band offered by these devices used for cellular applications. A preliminary survey of various modeling approaches of these devices and
7. A frequency selective acoustic transducer for directional Lamb wave sensing.
Science.gov (United States)
Senesi, Matteo; Ruzzene, Massimo
2011-10-01
A frequency selective acoustic transducer (FSAT) is proposed for directional sensing of guided waves. The considered FSAT design is characterized by a spiral configuration in wavenumber domain, which leads to a spatial arrangement of the sensing material producing output signals whose dominant frequency component is uniquely associated with the direction of incoming waves. The resulting spiral FSAT can be employed both for directional sensing and generation of guided waves, without relying on phasing and control of a large number of channels. The analytical expression of the shape of the spiral FSAT is obtained through the theoretical formulation for continuously distributed active material as part of a shaped piezoelectric device. Testing is performed by forming a discrete array through the points of the measurement grid of a scanning laser Doppler vibrometer. The discrete array approximates the continuous spiral FSAT geometry, and provides the flexibility to test several configurations. The experimental results demonstrate the strong frequency dependent directionality of the spiral FSAT and suggest its application for frequency selective acoustic sensors, to be employed for the localization of broadband acoustic events, or for the directional generation of Lamb waves for active interrogation of structural health. © 2011 Acoustical Society of America
8. Reflector stack optimization for Bulk Acoustic Wave resonators
NARCIS (Netherlands)
Jose, Sumy
2011-01-01
Thin-film bulk-acoustic-wave (BAW) devices are used for RF selectivity in mobile communication system and other wireless applications. Currently, the conventional RF filters are getting replaced by BAW filters in all major cell phone standards. In this thesis, we study solidly mounted BAW resonators
9. Quantum ion-acoustic solitary waves in weak relativistic plasma
A linear dispersion relation is also obtained taking into account the relativistic effect. The properties of quantum ion-acoustic solitary waves, obtained from the deformed KdV equation, are studied taking into account the quantum mechanical effects in the weak relativistic limit. It is found that relativistic effects significantly ...
10. Acoustic wave propagation in layered spherical structures
Science.gov (United States)
Yun-Tuan, Fang; Ting-Gen, Shen; Xilin, Tan
2004-03-01
Radiation from acoustic sources located inside quasi-periodically layered structures is studied using the transfer matrix method. In contrast to the periodically layered cases the transmission in the quasi-periodic systems has different band structures and decreases more slowly with the number of layers than in the periodic systems. The transmission periodically changes with the variation of media thickness in both types of systems, which may be useful in designing phonic devices. (
11. Acoustic precursor wave propagation in viscoelastic media.
Science.gov (United States)
Zhu, Guangran Kevin; Mojahedi, Mohammad; Sarris, Costas D
2014-03-01
Precursor field theory has been developed to describe the dynamics of electromagnetic field evolution in causally attenuative and dispersive media. In Debye dielectrics, the so-called Brillouin precursor exhibits an algebraic attenuation rate that makes it an ideal pulse waveform for communication, sensing, and imaging applications. Inspired by these studies in the electromagnetic domain, the present paper explores the propagation of acoustic precursors in dispersive media, with emphasis on biological media. To this end, a recently proposed causal dispersive model is employed, based on its interpretation as the acoustic counterpart of the Cole¿Cole model for dielectrics. The model stems from the fractional stress¿strain relation, which is consistent with the empirically known frequency power-law attenuation in viscoelastic media. It is shown that viscoelastic media described by this model, including human blood, support the formation and propagation of Brillouin precursors. The amplitude of these precursors exhibits a sub-exponential attenuation rate as a function of distance, actually being proportional to z(-p), where z is the distance traveled within the medium and 0.5
acoustic-pulse-based communication and imaging systems.
12. Acoustic Monitoring of the Arctic Ice Cap
Science.gov (United States)
Porter, D. L.; Goemmer, S. A.; Chayes, D. N.
2012-12-01
Introduction The monitoring of the Arctic Ice Cap is important economically, tactically, and strategically. In the scenario of ice cap retreat, new paths of commerce open, e.g. waterways from Northern Europe to the Far East. Where ship-going commerce is conducted, the U.S. Navy and U.S. Coast Guard have always stood guard and been prepared to assist from acts of nature and of man. It is imperative that in addition to measuring the ice from satellites, e.g. Icesat, that we have an ability to measure the ice extent, its thickness, and roughness. These parameters play an important part in the modeling of the ice and the processes that control its growth or shrinking and its thickness. The proposed system consists of three subsystems. The first subsystem is an acoustic source, the second is an array of geophones and the third is a system to supply energy and transmit the results back to the analysis laboratory. The subsystems are described below. We conclude with a plan on how to tackle this project and the payoff to the ice cap modeler and hence the users, i.e. commerce and defense. System Two historically tested methods to generate a large amplitude multi-frequency sound source include explosives and air guns. A new method developed and tested by the University of Texas, ARL is a combustive Sound Source [Wilson, et al., 1995]. The combustive sound source is a submerged combustion chamber that is filled with the byproducts of the electrolysis of sea water, i.e. Hydrogen and Oxygen, an explosive mixture which is ignited via a spark. Thus, no additional compressors, gases, or explosives need to be transported to the Arctic to generate an acoustic pulse capable of the sediment and the ice. The second subsystem would be geophones capable of listening in the O(10 Hz) range and transmitting that data back to the laboratory. Thus two single arrays of geophones arranged orthogonal to each other with a range of 1000's of kilometers and a combustive sound source where the two
13. Theory of Guided Acoustic Waves in Piezoelectric Solids.
Science.gov (United States)
1979-07-01
LABORATORY LEYE * THEORY OF GUIDED ACOUSTIC 0 WAVES IN PIEZOELECTRIC SOLIDS SUPRIYO DATTA 1I 717 I APPROVED FOR PUBLIC NLEASE. DISTRIBUTION UNLIMITD...Waves Line Acnus tic Waves Transducers and Reflectors 20 az. ACT -der’u, or’’’rqsJ !r~o~a rAf.mt , !ck rn , A non-itarative varia -;..onal cechnique...following chapters deal with a variety of interesting acous; tic field problems. Most of these results have already been published and the puroose of this
14. Acoustic wave propagation in an axisymmetric swirling jet.
Science.gov (United States)
Yu, J. C.; Mungur, P.
1973-01-01
An analysis has been developed to study the acoustic wave propagation in an axisymmetric swirling subsonic jet flow. The governing convected wave equation derived in the spherical coordinates includes mean shears, shear gradients and pressure gradients. The directivity patterns for various spinning and non-spinning modes due to the influence of the mean jet swirl were obtained by numerically integrating the governing wave equation. The mean flow field used in the computation was that obtained semiempirically for subsonic swirling turbulent jet and is completely specified once the degree of swirl is known. The dependence of sound directivity on jet Mach number, swirl ratio and frequency are discussed.
15. Langasite Surface Acoustic Wave Sensors: Fabrication and Testing
Energy Technology Data Exchange (ETDEWEB)
Zheng, Peng; Greve, David W.; Oppenheim, Irving J.; Chin, Tao-Lun; Malone, Vanessa
2012-02-01
We report on the development of harsh-environment surface acoustic wave sensors for wired and wireless operation. Surface acoustic wave devices with an interdigitated transducer emitter and multiple reflectors were fabricated on langasite substrates. Both wired and wireless temperature sensing was demonstrated using radar-mode (pulse) detection. Temperature resolution of better than ±0.5°C was achieved between 200°C and 600°C. Oxygen sensing was achieved by depositing a layer of ZnO on the propagation path. Although the ZnO layer caused additional attenuation of the surface wave, oxygen sensing was accomplished at temperatures up to 700°C. The results indicate that langasite SAW devices are a potential solution for harsh-environment gas and temperature sensing.
16. Optimization of Surface Acoustic Wave-Based Rate Sensors
Directory of Open Access Journals (Sweden)
Fangqian Xu
2015-10-01
Full Text Available The optimization of an surface acoustic wave (SAW-based rate sensor incorporating metallic dot arrays was performed by using the approach of partial-wave analysis in layered media. The optimal sensor chip designs, including the material choice of piezoelectric crystals and metallic dots, dot thickness, and sensor operation frequency were determined theoretically. The theoretical predictions were confirmed experimentally by using the developed SAW sensor composed of differential delay line-oscillators and a metallic dot array deposited along the acoustic wave propagation path of the SAW delay lines. A significant improvement in sensor sensitivity was achieved in the case of 128° YX LiNbO3, and a thicker Au dot array, and low operation frequency were used to structure the sensor.
17. Impact of Acoustic Standing Waves on Structural Responses: Reverberant Acoustic Testing (RAT) vs. Direct Field Acoustic Testing (DFAT)
Science.gov (United States)
Kolaini, Ali R.; Doty, Benjamin; Chang, Zensheu
2012-01-01
Loudspeakers have been used for acoustic qualification of spacecraft, reflectors, solar panels, and other acoustically responsive structures for more than a decade. Limited measurements from some of the recent speaker tests used to qualify flight hardware have indicated significant spatial variation of the acoustic field within the test volume. Also structural responses have been reported to differ when similar tests were performed using reverberant chambers. To address the impact of non-uniform acoustic field on structural responses, a series of acoustic tests were performed using a flat panel and a 3-ft cylinder exposed to the field controlled by speakers and repeated in a reverberant chamber. The speaker testing was performed using multi-input-single-output (MISO) and multi-input-multi-output (MIMO) control schemes with and without the test articles. In this paper the spatial variation of the acoustic field due to acoustic standing waves and their impacts on the structural responses in RAT and DFAT (both using MISO and MIMO controls for DFAT) are discussed in some detail.
18. Ultrahigh-frequency surface acoustic wave generation for acoustic charge transport in silicon
NARCIS (Netherlands)
Büyükköse, S.; Vratzov, B.; van der Veen, Johan (CTIT); Santos, P.V.; van der Wiel, Wilfred Gerard
2013-01-01
We demonstrate piezo-electrical generation of ultrahigh-frequency surface acoustic waves on silicon substrates, using high-resolution UV-based nanoimprint lithography, hydrogen silsequioxane planarization, and metal lift-off. Interdigital transducers were fabricated on a ZnO layer sandwiched between
19. Acoustic emission monitoring of the bending under tension test
DEFF Research Database (Denmark)
Moghadam, Marcel; Sulaiman, Mohd Hafis Bin; Christiansen, Peter
2017-01-01
Preliminary investigations have shown that acoustic emission has promising aspects as an online monitoring technique for assessment of tribological conditions during metal forming as regards to determination of the onset of galling. In the present study the acoustic emission measuring technique h...
20. Acoustic Signature Monitoring and Management of Naval Platforms
NARCIS (Netherlands)
Basten, T.G.H.; Jong, C.A.F. de; Graafland, F.; Hof, J. van 't
2015-01-01
Acoustic signatures make naval platforms susceptible to detection by threat sensors. The variable operational conditions and lifespan of a platform cause variations in the acoustic signature. To deal with these variations, a real time signature monitoring capability is being developed, with advisory
1. Tuneable film bulk acoustic wave resonators
CERN Document Server
Gevorgian, Spartak Sh; Vorobiev, Andrei K
2013-01-01
To handle many standards and ever increasing bandwidth requirements, large number of filters and switches are used in transceivers of modern wireless communications systems. It makes the cost, performance, form factor, and power consumption of these systems, including cellular phones, critical issues. At present, the fixed frequency filter banks based on Film Bulk Acoustic Resonators (FBAR) are regarded as one of the most promising technologies to address performance -form factor-cost issues. Even though the FBARs improve the overall performances the complexity of these systems remains high. Attempts are being made to exclude some of the filters by bringing the digital signal processing (including channel selection) as close to the antennas as possible. However handling the increased interference levels is unrealistic for low-cost battery operated radios. Replacing fixed frequency filter banks by one tuneable filter is the most desired and widely considered scenario. As an example, development of the softwa...
2. A three-microphone acoustic reflection technique using transmitted acoustic waves in the airway.
Science.gov (United States)
Fujimoto, Yuki; Huang, Jyongsu; Fukunaga, Toshiharu; Kato, Ryo; Higashino, Mari; Shinomiya, Shohei; Kitadate, Shoko; Takahara, Yutaka; Yamaya, Atsuyo; Saito, Masatoshi; Kobayashi, Makoto; Kojima, Koji; Oikawa, Taku; Nakagawa, Ken; Tsuchihara, Katsuma; Iguchi, Masaharu; Takahashi, Masakatsu; Mizuno, Shiro; Osanai, Kazuhiro; Toga, Hirohisa
2013-10-15
The acoustic reflection technique noninvasively measures airway cross-sectional area vs. distance functions and uses a wave tube with a constant cross-sectional area to separate incidental and reflected waves introduced into the mouth or nostril. The accuracy of estimated cross-sectional areas gets worse in the deeper distances due to the nature of marching algorithms, i.e., errors of the estimated areas in the closer distances accumulate to those in the further distances. Here we present a new technique of acoustic reflection from measuring transmitted acoustic waves in the airway with three microphones and without employing a wave tube. Using miniaturized microphones mounted on a catheter, we estimated reflection coefficients among the microphones and separated incidental and reflected waves. A model study showed that the estimated cross-sectional area vs. distance function was coincident with the conventional two-microphone method, and it did not change with altered cross-sectional areas at the microphone position, although the estimated cross-sectional areas are relative values to that at the microphone position. The pharyngeal cross-sectional areas including retropalatal and retroglossal regions and the closing site during sleep was visualized in patients with obstructive sleep apnea. The method can be applicable to larger or smaller bronchi to evaluate the airspace and function in these localized airways.
3. SILICON COMPATIBLE ACOUSTIC WAVE RESONATORS: DESIGN, FABRICATION AND PERFORMANCE
Directory of Open Access Journals (Sweden)
Aliza Aini Md Ralib
2014-12-01
Full Text Available ABSTRACT: Continuous advancement in wireless technology and silicon microfabrication has fueled exciting growth in wireless products. The bulky size of discrete vibrating mechanical devices such as quartz crystals and surface acoustic wave resonators impedes the ultimate miniaturization of single-chip transceivers. Fabrication of acoustic wave resonators on silicon allows complete integration of a resonator with its accompanying circuitry. Integration leads to enhanced performance, better functionality with reduced cost at large volume production. This paper compiles the state-of-the-art technology of silicon compatible acoustic resonators, which can be integrated with interface circuitry. Typical acoustic wave resonators are surface acoustic wave (SAW and bulk acoustic wave (BAW resonators. Performance of the resonator is measured in terms of quality factor, resonance frequency and insertion loss. Selection of appropriate piezoelectric material is significant to ensure sufficient electromechanical coupling coefficient is produced to reduce the insertion loss. The insulating passive SiO2 layer acts as a low loss material and aims to increase the quality factor and temperature stability of the design. The integration technique also is influenced by the fabrication process and packaging. Packageless structure using AlN as the additional isolation layer is proposed to protect the SAW device from the environment for high reliability. Advancement in miniaturization technology of silicon compatible acoustic wave resonators to realize a single chip transceiver system is still needed. ABSTRAK: Kemajuan yang berterusan dalam teknologi tanpa wayar dan silikon telah menguatkan pertumbuhan yang menarik dalam produk tanpa wayar. Saiz yang besar bagi peralatan mekanikal bergetar seperti kristal kuarza menghalang pengecilan untuk merealisasikan peranti cip. Silikon serasi gelombang akustik resonator mempunyai potensi yang besar untuk menggantikan unsur
4. Investigation into the Effect of Acoustic Radiation Force and Acoustic Streaming on Particle Patterning in Acoustic Standing Wave Fields
Science.gov (United States)
Yang, Yanye; Ni, Zhengyang; Guo, Xiasheng; Luo, Linjiao; Tu, Juan; Zhang, Dong
2017-01-01
Acoustic standing waves have been widely used in trapping, patterning, and manipulating particles, whereas one barrier remains: the lack of understanding of force conditions on particles which mainly include acoustic radiation force (ARF) and acoustic streaming (AS). In this paper, force conditions on micrometer size polystyrene microspheres in acoustic standing wave fields were investigated. The COMSOL® Mutiphysics particle tracing module was used to numerically simulate force conditions on various particles as a function of time. The velocity of particle movement was experimentally measured using particle imaging velocimetry (PIV). Through experimental and numerical simulation, the functions of ARF and AS in trapping and patterning were analyzed. It is shown that ARF is dominant in trapping and patterning large particles while the impact of AS increases rapidly with decreasing particle size. The combination of using both ARF and AS for medium size particles can obtain different patterns with only using ARF. Findings of the present study will aid the design of acoustic-driven microfluidic devices to increase the diversity of particle patterning. PMID:28753955
5. Investigation into the Effect of Acoustic Radiation Force and Acoustic Streaming on Particle Patterning in Acoustic Standing Wave Fields
Directory of Open Access Journals (Sweden)
Shilei Liu
2017-07-01
Full Text Available Acoustic standing waves have been widely used in trapping, patterning, and manipulating particles, whereas one barrier remains: the lack of understanding of force conditions on particles which mainly include acoustic radiation force (ARF and acoustic streaming (AS. In this paper, force conditions on micrometer size polystyrene microspheres in acoustic standing wave fields were investigated. The COMSOL® Mutiphysics particle tracing module was used to numerically simulate force conditions on various particles as a function of time. The velocity of particle movement was experimentally measured using particle imaging velocimetry (PIV. Through experimental and numerical simulation, the functions of ARF and AS in trapping and patterning were analyzed. It is shown that ARF is dominant in trapping and patterning large particles while the impact of AS increases rapidly with decreasing particle size. The combination of using both ARF and AS for medium size particles can obtain different patterns with only using ARF. Findings of the present study will aid the design of acoustic-driven microfluidic devices to increase the diversity of particle patterning.
6. Acoustic solitons: A robust tool to investigate the generation and detection of ultrafast acoustic waves
Science.gov (United States)
Péronne, Emmanuel; Chuecos, Nicolas; Thevenard, Laura; Perrin, Bernard
2017-02-01
Solitons are self-preserving traveling waves of great interest in nonlinear physics but hard to observe experimentally. In this report an experimental setup is designed to observe and characterize acoustic solitons in a GaAs(001) substrate. It is based on careful temperature control of the sample and an interferometric detection scheme. Ultrashort acoustic solitons, such as the one predicted by the Korteweg-de Vries equation, are observed and fully characterized. Their particlelike nature is clearly evidenced and their unique properties are thoroughly checked. The spatial averaging of the soliton wave front is shown to account for the differences between the theoretical and experimental soliton profile. It appears that ultrafast acoustic experiments provide a precise measurement of the soliton velocity. It allows for absolute calibration of the setup as well as the response function analysis of the detection layer. Moreover, the temporal distribution of the solitons is also analyzed with the help of the inverse scattering method. It shows how the initial acoustic pulse profile which gives birth to solitons after nonlinear propagation can be retrieved. Such investigations provide a new tool to probe transient properties of highly excited matter through the study of the emitted acoustic pulse after laser excitation.
7. Superresolution through the topological shaping of sound with an acoustic vortex wave antenna
CERN Document Server
Guild, Matthew D; Martin, Theodore P; Rohde, Charles A; Orris, Gregory J
2016-01-01
In this paper, we demonstrate far-field acoustic superresolution using shaped acoustic vortices. Compared with previously proposed near-field methods of acoustic superresolution, in this work we describe how far-field superresolution can be obtained using an acoustic vortex wave antenna. This is accomplished by leveraging the recent advances in optical vortices in conjunction with the topological diversity of a leaky wave antenna design. In particular, the use of an acoustic vortex wave antenna eliminates the need for a complicated phased array consisting of multiple active elements, and enables a superresolving aperture to be achieved with a single simple acoustic source and total aperture size less than a wavelength in diameter. A theoretical formulation is presented for the design of an acoustic vortex wave antenna with arbitrary planar arrangement, and explicit expressions are developed for the radiated acoustic pressure field. This geometric versatility enables variously-shaped acoustic vortex patterns t...
8. Acoustic myography: a noninvasive monitor of motor unit fatigue.
Science.gov (United States)
Barry, D T; Geiringer, S R; Ball, R D
1985-01-01
Acoustic myography is the recording of sounds produced by contracting muscle. These sounds become louder with increasing force of contraction. We have compared muscle sounds with surface EMG to monitor the dissociation of electrical from mechanical events (presumably, the loss of excitation-contraction coupling) which occur with motor unit fatigue. Acoustic signals were amplified using a standard phonocardiograph, recorded on FM magnetic tape, and digitally analyzed. Muscles were examined at rest, with intermittent contractions, and with sustained contractions. We found that with fatigue, the acoustic amplitude decayed, but the surface EMG amplitude did not. With decreased effort, however, the acoustic and the surface EMG amplitudes declined simultaneously. By simultaneously recording acoustic signals and needle EMG, individual motor units were resolved acoustically in two muscles with decreased numbers of motor units and increased motor unit size. Fasciculations also produced acoustic signals, although no acoustic signal has yet been found that correlates with fibrillations. Analysis of acoustic signals from muscle provides a noninvasive method for monitoring motor unit fatigue in vivo. It may also be useful in distinguishing muscle fatigue from decreased volition.
9. Study on Acoustic Catheter of Boiler Tube Leakage Monitoring Systems
Science.gov (United States)
Lv, Yongxing; Feng, Qiang
Boiler tube leakage is the major reason of affecting the safe operation of the unit now, there are 3 methods of the "four tube" leakage detection: Traditional method, filtering method and acoustic spectrum analysis, acoustic spectrum analysis is the common method, but this method have low sensitivity and the sensor damage easily. Therewith, designed the special acoustic catheter with acoustic resonance cavity type, proved by experiments, the acoustic catheter with acoustic resonance cavity type can enhance leakage sound, can accurately extract leakage signals, has high sensitivity, and can avoid the effect of sensor by fire and hot-gas when the furnace is in positive pressure situation, reduce the installation and maintenance costs of the boiler tube leakage monitor system.
10. Acoustic Guided Wave Testing of Pipes of Small Diameters
Science.gov (United States)
Muravev, V. V.; Muraveva, O. V.; Strizhak, V. A.; Myshkin, Y. V.
2017-10-01
Acoustic path is analyzed and main parameters of guided wave testing are substanti- ated applied to pipes of small diameters. The method is implemented using longitudinal L(0,1) and torsional T(0,1) waves based on electromagnetic-acoustic (EMA) transducers. The method of multiple reflections (MMR) combines echo-through, amplitude-shadow and time-shadow methods. Due to the effect of coherent amplification of echo-pulses from defects the sensitivity to the defects of small sizes at the signal analysis on the far reflections is increased. An oppor- tunity of detection of both local defects (dents, corrosion damages, rolling features, pitting, cracks) and defects extended along the pipe is shown.
11. Laser-generated acoustic wave studies on tattoo pigment
Science.gov (United States)
Paterson, Lorna M.; Dickinson, Mark R.; King, Terence A.
1996-01-01
A Q-switched alexandrite laser (180 ns at 755 nm) was used to irradiate samples of agar embedded with red, black and green tattoo dyes. The acoustic waves generated in the samples were detected using a PVDF membrane hydrophone and compared to theoretical expectations. The laser pulses were found to generate acoustic waves in the black and green samples but not in the red pigment. Pressures of up to 1.4 MPa were produced with irradiances of up to 96 MWcm-2 which is comparable to the irradiances used to clear pigment embedded in skin. The pressure gradient generated across pigment particles was approximately 1.09 X 1010 Pam-1 giving a pressure difference of 1.09 +/- 0.17 MPa over a particle with mean diameter 100 micrometers . This is not sufficient to permanently damage skin which has a tensile strength of 7.4 MPa.
12. The development of a surface acoustic wave strain sensor
OpenAIRE
Donohoe, Brian
2011-01-01
Multi sensors networks are becoming increasingly prevalent in modern engineering applications. In multi sensor networks, wireless sensors are preferred over traditional wired methods. Sensors based upon the surface acoustic wave resonators (SAWR) are often identified as a potential candidate to act as wireless and passive strain sensors. This thesis details the design, fabrication, modelling, calibration and packaging of SAW strain sensors as a general purpose modular strain sensor. The motiv...
13. Elastic Wave Propagation Mechanisms in Underwater Acoustic Environments
Science.gov (United States)
2015-09-30
Collis, and Robert I. Odom. Elastic parabolic equation solutions for oceanic T -wave generation and propagation from deep seismic sources. J. Acoust...navigation under Arctic ice. In Oceans , 2012, pages 1–8. IEEE, October 2012. 10.1109/ OCEANS .2012.6405005. PUBLICATIONS • Published in refereed journal...or elastic ice cover. OBJECTIVES To apply EPE solutions to scenarios that include fluid-elastic boundaries, either at the ocean floor, or at the
14. Wireless Passive Strain Sensor Based on Surface Acoustic Wave Devices
Directory of Open Access Journals (Sweden)
T. Nomura
2008-04-01
Full Text Available Surface acoustic wave (SAW devices offer many attractive features for applications as chemical and physical sensors. In this paper, a novel SAW strain sensor that employs SAW delay lines has been designed. Two crossed delay lines were used to measure the two-dimensional strain. A wireless sensing system is also proposed for effective operation of the strain sensor. In addition, an electronic system for accurately measuring the phase characteristics of the signal wave from the passive strain sensor is proposed.
15. Investigation of Ion Acoustic Waves in Collisionless Plasmas
DEFF Research Database (Denmark)
Christoffersen, G. B.; Jensen, Vagn Orla; Michelsen, Poul
1974-01-01
The Green's functions for the linearized ion Vlasov equation with a given boundary value are derived. The propagation properties of ion acoustic waves are calculated by performing convolution integrals over the Green's functions. For Te/Ti less than about 3 it is concluded that the collective...... interaction is very weak and that the propagation properties are determined almost completely by freely streaming ions. The wave damping, being due to phase mixing, is determined by the width of the perturbed distribution function rather than by the slope of the undisturbed distribution function at the phase...
16. Wave propagation in one-dimensional nonlinear acoustic metamaterials
Science.gov (United States)
Fang, Xin; Wen, Jihong; Bonello, Bernard; Yin, Jianfei; Yu, Dianlong
2017-05-01
The propagation of waves in nonlinear acoustic metamaterial (NAM) is fundamentally different from that in conventional linear ones. In this article we consider two one-dimensional (1D) NAM systems featuring respectively a diatomic and a tetratomic meta unit-cell. We investigate the attenuation of waves, band structures, and bifurcations to demonstrate novel nonlinear effects, which can significantly expand the bandwidth for elastic wave suppression and cause nonlinear wave phenomena. The harmonic averaging approach, continuation algorithm, and Lyapunov exponents (LEs) are combined to study the frequency responses, nonlinear modes, bifurcations of periodic solutions, and chaos. The nonlinear resonances are studied, and the influence of damping on hyperchaotic attractors is evaluated. Moreover, a ‘quantum’ behavior is found between the low-energy and high-energy orbits. This work provides a theoretical base for furthering understandings and applications of NAMs.
17. Impact of Acoustic Standing Waves on Structural Responses
Science.gov (United States)
Kolaini, Ali R.
2014-01-01
For several decades large reverberant chambers and most recently direct field acoustic testing have been used in the aerospace industry to test larger structures with low surface densities such as solar arrays and reflectors to qualify them and to detect faults in the design and fabrication. It has been reported that in reverberant chamber and direct acoustic testing, standing acoustic modes may strongly couple with the fundamental structural modes of the test hardware (Reference 1). In this paper results from a recent reverberant chamber acoustic test of a composite reflector are discussed. These results provide further convincing evidence of the acoustic standing wave and structural modes coupling phenomenon. The purpose of this paper is to alert test organizations to this phenomenon so that they can account for the potential increase in structural responses and ensure that flight hardware undergoes safe testing. An understanding of the coupling phenomenon may also help minimize the over and/or under testing that could pose un-anticipated structural and flight qualification issues.
18. Generation of ion-acoustic waves in an inductively coupled, low-pressure discharge lamp
Science.gov (United States)
Camparo, J. C.; Klimcak, C. M.
2006-04-01
For a number of years it has been known that the alkali rf-discharge lamps used in atomic clocks can exhibit large amplitude intensity oscillations. These oscillations arise from ion-acoustic plasma waves and have typically been associated with erratic clock behavior. Though large amplitude ion-acoustic plasma waves are clearly deleterious for atomic clock operation, it does not follow that small amplitude oscillations have no utility. Here, we demonstrate two easily implemented methods for generating small amplitude ion-acoustic plasma waves in alkali rf-discharge lamps. Furthermore, we demonstrate that the frequency of these waves is proportional to the square root of the rf power driving the lamp and therefore that their examination can provide an easily accessible parameter for monitoring and controlling the lamp's plasma conditions. This has important consequences for precise timekeeping, since the atomic ground-state hyperfine transition, which is the heart of the atomic clock signal, can be significantly perturbed by changes in the lamp's output via the ac-Stark shift.
19. Acoustic waves in tilted fiber Bragg gratings for sensing applications
Science.gov (United States)
Marques, Carlos A. F.; Alberto, Nélia J.; Domingues, Fátima; Leitão, Cátia; Antunes, Paulo; Pinto, João. L.; André, Paulo
2017-05-01
Tilted fiber Bragg gratings (TFBGs) are one of the most attractive kind of optical fiber sensor technology due to their intrinsic properties. On the other hand, the acousto-optic effect is an important, fast and accurate mechanism that can be used to change and control several properties of fiber gratings in silica and polymer optical fiber. Several all-optical devices for optical communications and sensing have been successfully designed and constructed using this effect. In this work, we present the recent results regarding the production of optical sensors, through the acousto-optic effect in TFBGs. The cladding and core modes amplitude of a TFBG can be controlled by means of the power levels from acoustic wave source. Also, the cladding modes of a TFBG can be coupled back to the core mode by launching acoustic waves. Induced bands are created on the left side of the original Bragg wavelength due to phase matching to be satisfied. The refractive index (RI) is analyzed in detail when acoustic waves are turned on using saccharose solutions with different RI from 1.33 to 1.43.
20. Guided wave opto-acoustic device
Energy Technology Data Exchange (ETDEWEB)
Jarecki, Jr., Robert L.; Rakich, Peter Thomas; Camacho, Ryan; Shin, Heedeuk; Cox, Jonathan Albert; Qiu, Wenjun; Wang, Zheng
2016-02-23
The various technologies presented herein relate to various hybrid phononic-photonic waveguide structures that can exhibit nonlinear behavior associated with traveling-wave forward stimulated Brillouin scattering (forward-SBS). The various structures can simultaneously guide photons and phonons in a suspended membrane. By utilizing a suspended membrane, a substrate pathway can be eliminated for loss of phonons that suppresses SBS in conventional silicon-on-insulator (SOI) waveguides. Consequently, forward-SBS nonlinear susceptibilities are achievable at about 3000 times greater than achievable with a conventional waveguide system. Owing to the strong phonon-photon coupling achievable with the various embodiments, potential application for the various embodiments presented herein cover a range of radiofrequency (RF) and photonic signal processing applications. Further, the various embodiments presented herein are applicable to applications operating over a wide bandwidth, e.g. 100 MHz to 50 GHz or more.
1. Surface acoustic wave devices as passive buried sensors
Science.gov (United States)
Friedt, J.-M.; Rétornaz, T.; Alzuaga, S.; Baron, T.; Martin, G.; Laroche, T.; Ballandras, S.; Griselin, M.; Simonnet, J.-P.
2011-02-01
Surface acoustic wave (SAW) devices are currently used as passive remote-controlled sensors for measuring various physical quantities through a wireless link. Among the two main classes of designs—resonator and delay line—the former has the advantage of providing narrow-band spectrum informations and hence appears compatible with an interrogation strategy complying with Industry-Scientific-Medical regulations in radio-frequency (rf) bands centered around 434, 866, or 915 MHz. Delay-line based sensors require larger bandwidths as they consists of a few interdigitated electrodes excited by short rf pulses with large instantaneous energy and short response delays but is compatible with existing equipment such as ground penetrating radar (GPR). We here demonstrate the measurement of temperature using the two configurations, particularly for long term monitoring using sensors buried in soil. Although we have demonstrated long term stability and robustness of packaged resonators and signal to noise ratio compatible with the expected application, the interrogation range (maximum 80 cm) is insufficient for most geology or geophysical purposes. We then focus on the use of delay lines, as the corresponding interrogation method is similar to the one used by GPR which allows for rf penetration distances ranging from a few meters to tens of meters and which operates in the lower rf range, depending on soil water content, permittivity, and conductivity. Assuming propagation losses in a pure dielectric medium with negligible conductivity (snow or ice), an interrogation distance of about 40 m is predicted, which overcomes the observed limits met when using interrogation methods specifically developed for wireless SAW sensors, and could partly comply with the above-mentioned applications. Although quite optimistic, this estimate is consistent with the signal to noise ratio observed during an experimental demonstration of the interrogation of a delay line buried at a depth of 5
2. Big Bend National Park: Acoustical Monitoring 2010
Science.gov (United States)
2013-06-01
During the summer of 2010 (September October 2010), the Volpe Center collected baseline acoustical data at Big Bend National Park (BIBE) at four sites deployed for approximately 30 days each. The baseline data collected during this period will he...
3. Wright Brothers National Memorial : acoustical monitoring 2011
Science.gov (United States)
2014-11-01
During the winter of 2011(September - November) baseline acoustical data were collected at Wright Brothers National Memorial (WRBR) at two sites deployed for approximately 30 days each. The baseline data collected during these periods will help park ...
4. Influence of surface acoustic waves induced acoustic streaming on the kinetics of electrochemical reactions
Science.gov (United States)
Tietze, Sabrina; Schlemmer, Josefine; Lindner, Gerhard
2013-12-01
The kinetics of electrochemical reactions is controlled by diffusion processes of charge carriers across a boundary layer between the electrode and the electrolyte, which result in a shielding of the electric field inside the electrolyte and a concentration gradient across this boundary layer. In accumulators the diffusion rate determines the rather long time needed for charging, which is a major drawback for electric mobility. This diffusion boundary can be removed by acoustic streaming in the electrolyte induced by surface acoustic waves propagating of the electrode, which results in an increase of the charging current and thus in a reduction of the time needed for charging. For a quantitative study of the influence of acoustic streaming on the charge transport an electropolishing cell with vertically oriented copper electrodes and diluted H3PO4-Propanol electrolytes were used. Lamb waves with various excitation frequencies were exited on the anode with different piezoelectric transducers, which induced acoustic streaming in the overlaying electrolytic liquid. An increase of the polishing current of up to approximately 100 % has been obtained with such a set-up.
5. Flow velocity measurement with the nonlinear acoustic wave scattering
Energy Technology Data Exchange (ETDEWEB)
Didenkulov, Igor, E-mail: [email protected] [Institute of Applied Physics, 46 Ulyanov str., Nizhny Novgorod, 603950 (Russian Federation); Lobachevsky State University of Nizhny Novgorod, 23 Gagarin ave., Nizhny Novgorod, 603950 (Russian Federation); Pronchatov-Rubtsov, Nikolay, E-mail: [email protected] [Lobachevsky State University of Nizhny Novgorod, 23 Gagarin ave., Nizhny Novgorod, 603950 (Russian Federation)
2015-10-28
A problem of noninvasive measurement of liquid flow velocity arises in many practical applications. To this end the most often approach is the use of the linear Doppler technique. The Doppler frequency shift of signal scattered from the inhomogeneities distributed in a liquid relatively to the emitted frequency is proportional to the sound frequency and velocities of inhomogeneities. In the case of very slow flow one needs to use very high frequency sound. This approach fails in media with strong sound attenuation because acoustic wave attenuation increases with frequency and there is limit in increasing sound intensity, i.e. the cavitation threshold. Another approach which is considered in this paper is based on the method using the difference frequency Doppler Effect for flows with bubbles. This method is based on simultaneous action of two high-frequency primary acoustic waves with closed frequencies on bubbles and registration of the scattered by bubbles acoustic field at the difference frequency. The use of this method is interesting since the scattered difference frequency wave has much lower attenuation in a liquid. The theoretical consideration of the method is given in the paper. The experimental examples confirming the theoretical equations, as well as the ability of the method to be applied in medical diagnostics and in technical applications on measurement of flow velocities in liquids with strong sound attenuation is described. It is shown that the Doppler spectrum form depends on bubble concentration velocity distribution in the primary acoustic beams crossing zone that allows one to measure the flow velocity distribution.
6. Microscale anechoic architecture: acoustic diffusers for ultra low power microparticle separation via traveling surface acoustic waves.
Science.gov (United States)
Behrens, Jan; Langelier, Sean; Rezk, Amgad R; Lindner, Gerhard; Yeo, Leslie Y; Friend, James R
2015-01-07
We present a versatile and very low-power traveling SAW microfluidic sorting device able to displace and separate particles of different diameter in aqueous suspension; the travelling wave propagates through the fluid bulk and diffuses via a Schröder diffuser, adapted from its typical use in concert hall acoustics to be the smallest such diffuser to be suitable for microfluidics. The effective operating power range is two to three orders of magnitude less than current SAW devices, uniquely eliminating the need for amplifiers, and by using traveling waves to impart forces directly upon suspended microparticles, they can be separated by size.
7. Langasite surface acoustic wave gas sensors: modeling and verification
Energy Technology Data Exchange (ETDEWEB)
Peng Zheng,; Greve, D. W.; Oppenheim, I. J.
2013-03-01
We report finite element simulations of the effect of conductive sensing layers on the surface wave velocity of langasite substrates. The simulations include both the mechanical and electrical influences of the conducting sensing layer. We show that three-dimensional simulations are necessary because of the out-of-plane displacements of the commonly used (0, 138.5, 26.7) Euler angle. Measurements of the transducer input admittance in reflective delay-line devices yield a value for the electromechanical coupling coefficient that is in good agreement with the three-dimensional simulations on bare langasite substrate. The input admittance measurements also show evidence of excitation of an additional wave mode and excess loss due to the finger resistance. The results of these simulations and measurements will be useful in the design of surface acoustic wave gas sensors.
8. Asymmetric wave transmission in a diatomic acoustic/elastic metamaterial
Energy Technology Data Exchange (ETDEWEB)
Li, Bing; Tan, K. T., E-mail: [email protected] [Department of Mechanical Engineering, The University of Akron, Akron, Ohio 44325-3903 (United States)
2016-08-21
Asymmetric acoustic/elastic wave transmission has recently been realized using nonlinearity, wave diffraction, or bias effects, but always at the cost of frequency distortion, direction shift, large volumes, or external energy. Based on the self-coupling of dual resonators, we propose a linear diatomic metamaterial, consisting of several small-sized unit cells, to realize large asymmetric wave transmission in low frequency domain (below 1 kHz). The asymmetric transmission mechanism is theoretically investigated, and numerically verified by both mass-spring and continuum models. This passive system does not require any frequency conversion or external energy, and the asymmetric transmission band can be theoretically predicted and mathematically controlled, which extends the design concept of unidirectional transmission devices.
9. Geodesic acoustic modes excited by finite beta drift waves
DEFF Research Database (Denmark)
Chakrabarti, Nikhil Kumar; Guzdar, P.N.; Kleva, R.G.
2008-01-01
Presented in this paper is a mode-coupling analysis for the nonlinear excitation of the geodesic acoustic modes (GAMs) in tokamak plasmas by finite beta drift waves. The finite beta effects give rise to a strong stabilizing influence on the parametric excitation process. The dominant finite beta...... effect is the combination of the Maxwell stress, which has a tendency to cancel the primary drive from the Reynolds stress, and the finite beta modification of the drift waves. The zonal magnetic field is also excited at the GAM frequency. However, it does not contribute to the overall stability...... of the three-wave process for parameters of relevance to the edge region of tokamaks....
10. Ion acoustic solitary waves in magneto-rotating plasmas
Energy Technology Data Exchange (ETDEWEB)
Mushtaq, A, E-mail: [email protected] [School of Physics, University of Sydney, New South Wales 2006 (Australia); Theoretical Plasma Physics Division, PINSTECH, Nilore, Islamabad (Pakistan)
2010-08-06
Propagation of an ion acoustic wave (IAW) in a magnetized electron-ion plasma, which is rotating around an axis at an angle {theta} with the direction of magnetic field, is studied by incorporating the effects of trapped and untrapped electron distributions. Employing the perturbation scheme, Korteweg-deVries and Schamel's modified KdV equations are derived for the small angle {theta} which may support the nonlinear IAW on a slow time scale of ion motion. The amplitude and width of the solitary wave in both cases (trapped and untrapped electrons) have been discussed with the effects of oblique rotation and external magnetic field. It is shown that the nonlinear effects considerably influence the propagation of waves in rotating plasmas.
11. Harmonic Wave Generated by Contact Acoustic Nonlinearity in Obliquely Incident Ultrasonic Wave
Energy Technology Data Exchange (ETDEWEB)
Yun, Dong Seok; Choi, Sung Ho; Kim, Chung Seok; Jhang, Kyung Young [Hangyang University, Seoul (Korea, Republic of)
2012-08-15
The objective of this study is to image the harmonic wave generated by contact acoustic nonlinearity in obliquely incident ultrasonic wave for early detection of closed cracks. A closed crack has been simulated by contacting two aluminum block specimens producing solid-solid contact interfaces and then acoustic nonlinearity has been imaged with contact pressure. Sampling phased array(SPA) and synthetic aperture focusing technique(SAFT) are used for imaging techniques. The amplitude of the fundamental frequency decreased with applying pressure. But, the amplitude of second harmonic increased with pressure and was a maximum amplitude at the simulation point of closed crack. Then, the amplitude of second harmonic decreased. As a result, harmonic imaging of contact acoustic nonlinearity is possible and it is expected to be apply for early detection of initial cracks.
12. Surface Modification on Acoustic Wave Biosensors for Enhanced Specificity
Directory of Open Access Journals (Sweden)
Nathan D. Gallant
2012-09-01
Full Text Available Changes in mass loading on the surface of acoustic biosensors result in output frequency shifts which provide precise measurements of analytes. Therefore, to detect a particular biomarker, the sensor delay path must be judiciously designed to maximize sensitivity and specificity. B-cell lymphoma 2 protein (Bcl-2 found in urine is under investigation as a biomarker for non-invasive early detection of ovarian cancer. In this study, surface chemistry and biofunctionalization approaches were evaluated for their effectiveness in presenting antibodies for Bcl-2 capture while minimizing non-specific protein adsorption. The optimal combination of sequentially adsorbing protein A/G, anti-Bcl-2 IgG and Pluronic F127 onto a hydrophobic surface provided the greatest signal-to-noise ratio and enabled the reliable detection of Bcl-2 concentrations below that previously identified for early stage ovarian cancer as characterized by a modified ELISA method. Finally, the optimal surface modification was applied to a prototype acoustic device and the frequency shift for a range of Bcl-2 concentration was quantified to demonstrate the effectiveness in surface acoustic wave (SAW-based detection applications. The surface functionalization approaches demonstrated here to specifically and sensitively detect Bcl-2 in a working ultrasonic MEMS biosensor prototype can easily be modified to detect additional biomarkers and enhance other acoustic biosensors.
13. Passive acoustic monitoring of bed load for fluvial applications
Science.gov (United States)
The sediment transported as bed load in streams and rivers is notoriously difficult to monitor cheaply and accurately. Passive acoustic methods are relatively simple, inexpensive, and provide spatial integration along with high temporal resolution. In 1963 work began on monitoring emissions from par...
14. Visualization of stress wave propagation via air-coupled acoustic emission sensors
Science.gov (United States)
Rivey, Joshua C.; Lee, Gil-Yong; Yang, Jinkyu; Kim, Youngkey; Kim, Sungchan
2017-02-01
We experimentally demonstrate the feasibility of visualizing stress waves propagating in plates using air-coupled acoustic emission sensors. Specifically, we employ a device that embeds arrays of microphones around an optical lens in a helical pattern. By implementing a beamforming technique, this remote sensing system allows us to record wave propagation events in situ via a single-shot and full-field measurement. This is a significant improvement over the conventional wave propagation tracking approaches based on laser doppler vibrometry or digital image correlation techniques. In this paper, we focus on demonstrating the feasibility and efficacy of this air-coupled acoustic emission technique by using large metallic plates exposed to external impacts. The visualization results of stress wave propagation will be shown under various impact scenarios. The proposed technique can be used to characterize and localize damage by detecting the attenuation, reflection, and scattering of stress waves that occurs at damage locations. This can ultimately lead to the development of new structural health monitoring and nondestructive evaluation methods for identifying hidden cracks or delaminations in metallic or composite plate structures, simultaneously negating the need for mounted contact sensors.
15. Examination of nanosecond laser melting thresholds in refractory metals by shear wave acoustics
Directory of Open Access Journals (Sweden)
A. Abdullaev
2017-07-01
Full Text Available Nanosecond laser pulse-induced melting thresholds in refractory (Nb, Mo, Ta and W metals are measured using detected laser-generated acoustic shear waves. Obtained melting threshold values were found to be scaled with corresponding melting point temperatures of investigated materials displaying dissimilar shearing behavior. The experiments were conducted with motorized control of the incident laser pulse energies with small and uniform energy increments to reach high measurement accuracy and real-time monitoring of the epicentral acoustic waveforms from the opposite side of irradiated sample plates. Measured results were found to be in good agreement with numerical finite element model solving coupled elastodynamic and thermal conduction governing equations on structured quadrilateral mesh. Solid-melt phase transition was handled by means of apparent heat capacity method. The onset of melting was attributed to vanished shear modulus and rapid radial molten pool propagation within laser-heated metal leading to preferential generation of transverse acoustic waves from sources surrounding the molten mass resulting in the delay of shear wave transit times. Developed laser-based technique aims for applications involving remote examination of rapid melting processes of materials present in harsh environment (e.g. spent nuclear fuels with high spatio-temporal resolution.
16. High-sensitivity open-loop electronics for gravimetric acoustic-wave-based sensors.
Science.gov (United States)
Rabus, David; Friedt, Jean-Michel; Ballandras, Sylvain; Martin, Gilles; Carry, Emile; Blondeau-Patissier, Virginie
2013-06-01
Detecting chemical species in gas phase has recently received an increasing interest mainly for security control, trying to implement new systems allowing for extended dynamics and reactivity. In this work, an open-loop interrogation strategy is proposed to use radio-frequency acoustic transducers as micro-balances for that purpose. The resulting system is dedicated to the monitoring of chemical compounds in gaseous or liquid-phase state. A 16 Hz standard deviation is demonstrated at 125 MHz, with a working frequency band in the 60 to 133 MHz range, answering the requirements for using Rayleigh- and Love-wave-based delay lines operating with 40-μm acoustic wavelength transducers. Moreover, this electronic setup was used to interrogate a high-overtone bulk acoustic wave resonator (HBAR) microbalance, a new sensor class allowing for multi-mode interrogation for gravimetric measurement improvement. The noise source still limiting the system performance is due to the analog-to-digital converter of the microcontroller, thus leaving open degrees-of-freedom for improving the obtained results by optimizing the voltage reference and board layout. The operation of the system is illustrated using a calibrated galvanic deposition at the surface of Love-wave delay lines to assess theoretical predictions of their gravimetric sensitivity and to compare them with HBAR-based sensor sensitivity.
17. Examination of nanosecond laser melting thresholds in refractory metals by shear wave acoustics
Science.gov (United States)
Abdullaev, A.; Muminov, B.; Rakhymzhanov, A.; Mynbayev, N.; Utegulov, Z. N.
2017-07-01
Nanosecond laser pulse-induced melting thresholds in refractory (Nb, Mo, Ta and W) metals are measured using detected laser-generated acoustic shear waves. Obtained melting threshold values were found to be scaled with corresponding melting point temperatures of investigated materials displaying dissimilar shearing behavior. The experiments were conducted with motorized control of the incident laser pulse energies with small and uniform energy increments to reach high measurement accuracy and real-time monitoring of the epicentral acoustic waveforms from the opposite side of irradiated sample plates. Measured results were found to be in good agreement with numerical finite element model solving coupled elastodynamic and thermal conduction governing equations on structured quadrilateral mesh. Solid-melt phase transition was handled by means of apparent heat capacity method. The onset of melting was attributed to vanished shear modulus and rapid radial molten pool propagation within laser-heated metal leading to preferential generation of transverse acoustic waves from sources surrounding the molten mass resulting in the delay of shear wave transit times. Developed laser-based technique aims for applications involving remote examination of rapid melting processes of materials present in harsh environment (e.g. spent nuclear fuels) with high spatio-temporal resolution.
18. Molding acoustic, electromagnetic and water waves with a single cloak.
Science.gov (United States)
Xu, Jun; Jiang, Xu; Fang, Nicholas; Georget, Elodie; Abdeddaim, Redha; Geffrin, Jean-Michel; Farhat, Mohamed; Sabouroux, Pierre; Enoch, Stefan; Guenneau, Sébastien
2015-06-09
We describe two experiments demonstrating that a cylindrical cloak formerly introduced for linear surface liquid waves works equally well for sound and electromagnetic waves. This structured cloak behaves like an acoustic cloak with an effective anisotropic density and an electromagnetic cloak with an effective anisotropic permittivity, respectively. Measured forward scattering for pressure and magnetic fields are in good agreement and provide first evidence of broadband cloaking. Microwave experiments and 3D electromagnetic wave simulations further confirm reduced forward and backscattering when a rectangular metallic obstacle is surrounded by the structured cloak for cloaking frequencies between 2.6 and 7.0 GHz. This suggests, as supported by 2D finite element simulations, sound waves are cloaked between 3 and 8 KHz and linear surface liquid waves between 5 and 16 Hz. Moreover, microwave experiments show the field is reduced by 10 to 30 dB inside the invisibility region, which suggests the multi-wave cloak could be used as a protection against water, sonic or microwaves.
19. Molding acoustic, electromagnetic and water waves with a single cloak
KAUST Repository
Xu, Jun
2015-06-09
We describe two experiments demonstrating that a cylindrical cloak formerly introduced for linear surface liquid waves works equally well for sound and electromagnetic waves. This structured cloak behaves like an acoustic cloak with an effective anisotropic density and an electromagnetic cloak with an effective anisotropic permittivity, respectively. Measured forward scattering for pressure and magnetic fields are in good agreement and provide first evidence of broadband cloaking. Microwave experiments and 3D electromagnetic wave simulations further confirm reduced forward and backscattering when a rectangular metallic obstacle is surrounded by the structured cloak for cloaking frequencies between 2.6 and 7.0 GHz. This suggests, as supported by 2D finite element simulations, sound waves are cloaked between 3 and 8 KHz and linear surface liquid waves between 5 and 16 Hz. Moreover, microwave experiments show the field is reduced by 10 to 30 dB inside the invisibility region, which suggests the multi-wave cloak could be used as a protection against water, sonic or microwaves. © 2015, Nature Publishing Group. All rights reserved.
20. OPERATING PROCEDURE FOR THE PORTABLE ACOUSTIC MONITORING PACKAGE (PAMP)
Energy Technology Data Exchange (ETDEWEB)
John L. Loth; Gary J. Morris; George M. Palmer; Richard Guiler; Patrick Browning
2004-08-29
The Portable Acoustic Monitoring Package (PAMP) has been designed to record and monitor acoustic signals in high-pressure natural gas (NG) transmission lines. Of particular interest are the three acoustic signals associated with a pipeline fracture. The system is portable (less than 30 lbm) and can be used at all line pressures up to 1000 psig. The PAMP requires a shut-off valve equipped 1/2 inch NPT access port in the pipeline. It is fully functional over the typical pressure range found in the natural gas transmission pipelines in the West Virginia, Virginia, Pennsylvania, and Ohio areas. With the use of the PAMP, a full spectrum of acoustic signals can be recorded and defined in terms of acoustic energy in decibels. To detect natural gas pipeline infringements and leaks, the acoustic energy generated inside the line is monitored with a sensitive pressure-equalized microphone and a step function type {Delta}p transducer. The assembly is mounted on a 1000 psig pipe fitting-tree called the PAMP. The electronics required to record, store and analyze the data are described within this report in the format of an operating manual.
1. Passive Wireless Hydrogen Sensors Using Orthogonal Frequency Coded Acoustic Wave Devices Project
Data.gov (United States)
National Aeronautics and Space Administration — This proposal describes the development of passive surface acoustic wave (SAW) based hydrogen sensors for NASA application to distributed wireless hydrogen leak...
2. Passive Wireless Hydrogen Sensors Using Orthogonal Frequency Coded Acoustic Wave Devices Project
Data.gov (United States)
National Aeronautics and Space Administration — This proposal describes the continued development of passive orthogonal frequency coded (OFC) surface acoustic wave (SAW) based hydrogen sensors for NASA application...
3. Radial wave crystals: radially periodic structures from anisotropic metamaterials for engineering acoustic or electromagnetic waves.
Science.gov (United States)
Torrent, Daniel; Sánchez-Dehesa, José
2009-08-07
We demonstrate that metamaterials with anisotropic properties can be used to develop a new class of periodic structures that has been named radial wave crystals. They can be sonic or photonic, and wave propagation along the radial directions is obtained through Bloch states like in usual sonic or photonic crystals. The band structure of the proposed structures can be tailored in a large amount to get exciting novel wave phenomena. For example, it is shown that acoustical cavities based on radial sonic crystals can be employed as passive devices for beam forming or dynamically orientated antennas for sound localization.
4. Lateral field excitation (LFE) of thickness shear mode (TSM) acoustic waves in thin film bulk acoustic resonators (FBAR) as a potential biosensor.
Science.gov (United States)
Dickherber, Anthony; Corso, Christopher D; Hunt, William
2006-01-01
Lateral field excitation (LFE) of a thin film bulk acoustic resonator (FBAR) is an ideal platform for biomedical sensors. A thickness shear mode (TSM) acoustic wave in a piezoelectric thin film is desirable for probing liquid samples because of the poor coupling of shear waves into the liquid. The resonator becomes an effective sensor by coating the surface with a bio- or chemi-specific layer. Perturbations of the surface can be detected by monitoring the resonance condition. Furthermore, FBARs can be easily fabricated to operate at higher frequencies, yielding greater sensitivity. An array of sensors offers the possibility of redundancy, allowing for statistical decision making as well as immediate corroboration of results. Array structures also offer the possibility of signature detection, by monitoring multiple targets in a sample simultaneously. This technology has immediate application to cancer and infectious disease diagnostics and also could serve as a tool for general proteomic research.
5. GPS-Acoustic Seafloor Geodesy using a Wave Glider
Science.gov (United States)
2013-12-01
The conventional approach to implement the GPS-Acoustic technique uses a ship or buoy for the interface between GPS and Acoustics. The high cost and limited availability of ships restricts occupations to infrequent campaign-style measurements. A new approach to address this problem uses a remote controlled, wave-powered sea surface vehicle, the Wave Glider. The Wave Glider uses sea-surface wave action for forward propulsion with both upward and downward motions producing forward thrust. It uses solar energy for power with solar panels charging the onboard 660 W-h battery for near continuous operation. It uses Iridium for communication providing command and control from shore plus status and user data via the satellite link. Given both the sea-surface wave action and solar energy are renewable, the vehicle can operate for extended periods (months) remotely. The vehicle can be launched from a small boat and can travel at ~ 1 kt to locations offshore. We have adapted a Wave Glider for seafloor geodesy by adding a dual frequency GPS receiver embedded in an Inertial Navigation Unit, a second GPS antenna/receiver to align the INU, and a high precision acoustic ranging system. We will report results of initial testing of the system conducted at SIO. In 2014, the new approach will be used for seafloor geodetic measurements of plate motion in the Cascadia Subduction Zone. The project is for a three-year effort to measure plate motion at three sites along an East-West profile at latitude 44.6 N, offshore Newport Oregon. One site will be located on the incoming plate to measure the present day convergence between the Juan de Fuca and North American plates and two additional sites will be located on the continental slope of NA to measure the elastic deformation due to stick-slip behavior on the mega-thrust fault. These new seafloor data will constrain existing models of slip behavior that presently are poorly constrained by land geodetic data 100 km from the deformation front.
6. Modeling of a Surface Acoustic Wave Strain Sensor
Science.gov (United States)
Wilson, W. C.; Atkinson, Gary M.
2010-01-01
NASA Langley Research Center is investigating Surface Acoustic Wave (SAW) sensor technology for harsh environments aimed at aerospace applications. To aid in development of sensors a model of a SAW strain sensor has been developed. The new model extends the modified matrix method to include the response of Orthogonal Frequency Coded (OFC) reflectors and the response of SAW devices to strain. These results show that the model accurately captures the strain response of a SAW sensor on a Langasite substrate. The results of the model of a SAW Strain Sensor on Langasite are presented
7. Non-Destructive Testing of Semiconductors Using Surface Acoustic Wave.
Science.gov (United States)
1983-12-31
the wafer can be evaluated with respect to lifetime and surface gen- Aeration velocity. The results can be shown in the form of images in pseudocolor...The mportant features of the plot are: enhaced by r-o beam spectroscopy as compared to 1) n te surface conductivicy euhibits a positive one beam, by... image scanning and signal processing is presented. PACS numbers: 72.50. + b The interaction of a surface acoustic wave (SAW) with METAL COW-,CT$* vaC 8. Longitudinal and Transverse Instability of Ion Acoustic Waves. Science.gov (United States) Chapman, T; Berger, R L; Cohen, B I; Banks, J W; Brunner, S 2017-08-04 Ion acoustic waves are found to be susceptible to at least two distinct decay processes. Which process dominates depends on the parameters. In the cases examined, the decay channel where daughter modes propagate parallel to the mother mode is found to dominate at larger amplitudes, while the decay channel where the daughter modes propagate at angles to the mother mode dominates at smaller amplitudes. Both decay processes may occur simultaneously and with onset thresholds below those suggested by fluid theory, resulting in the eventual multidimensional collapse of the mother mode to a turbulent state. 9. Location Dependence of Mass Sensitivity for Acoustic Wave Devices Directory of Open Access Journals (Sweden) Kewei Zhang 2015-09-01 Full Text Available It is introduced that the mass sensitivity (Sm of an acoustic wave (AW device with a concentrated mass can be simply determined using its mode shape function: the Sm is proportional to the square of its mode shape. By using the Sm of an AW device with a uniform mass, which is known for almost all AW devices, the Sm of an AW device with a concentrated mass at different locations can be determined. The method is confirmed by numerical simulation for one type of AW device and the results from two other types of AW devices. 10. Circuit Design of Surface Acoustic Wave Based Micro Force Sensor Directory of Open Access Journals (Sweden) Yuanyuan Li 2014-01-01 Full Text Available Pressure sensors are commonly used in industrial production and mechanical system. However, resistance strain, piezoresistive sensor, and ceramic capacitive pressure sensors possess limitations, especially in micro force measurement. A surface acoustic wave (SAW based micro force sensor is designed in this paper, which is based on the theories of wavelet transform, SAW detection, and pierce oscillator circuits. Using lithium niobate as the basal material, a mathematical model is established to analyze the frequency, and a peripheral circuit is designed to measure the micro force. The SAW based micro force sensor is tested to show the reasonable design of detection circuit and the stability of frequency and amplitude. 11. Solar wind implication on dust ion acoustic rogue waves Energy Technology Data Exchange (ETDEWEB) Abdelghany, A. M., E-mail: [email protected]; Abd El-Razek, H. N., E-mail: [email protected]; El-Labany, S. K., E-mail: [email protected] [Theoretical Physics Group, Department of Physics, Faculty of Science, Damietta University, New Damietta 34517 (Egypt); Moslem, W. M., E-mail: [email protected] [Department of Physics, Faculty of Science, Port Said University, Port Said 42521 (Egypt); Centre for Theoretical Physics, The British University in Egypt (BUE), El-Shorouk City, Cairo (Egypt) 2016-06-15 The relevance of the solar wind with the magnetosphere of Jupiter that contains positively charged dust grains is investigated. The perturbation/excitation caused by streaming ions and electron beams from the solar wind could form different nonlinear structures such as rogue waves, depending on the dominant role of the plasma parameters. Using the reductive perturbation method, the basic set of fluid equations is reduced to modified Korteweg-de Vries (KdV) and further modified (KdV) equation. Assuming that the frequency of the carrier wave is much smaller than the ion plasma frequency, these equations are transformed into nonlinear Schrödinger equations with appropriate coefficients. Rational solution of the nonlinear Schrödinger equation shows that rogue wave envelopes are supported by the present plasma model. It is found that the existence region of rogue waves depends on the dust-acoustic speed and the streaming temperatures for both the ions and electrons. The dependence of the maximum rogue wave envelope amplitude on the system parameters has been investigated. 12. Surface Acoustic Wave Vibration Sensors for Measuring Aircraft Flutter Science.gov (United States) Wilson, William C.; Moore, Jason P.; Juarez, Peter D. 2016-01-01 Under NASA's Advanced Air Vehicles Program the Advanced Air Transport Technology (AATT) Project is investigating flutter effects on aeroelastic wings. To support that work a new method for measuring vibrations due to flutter has been developed. The method employs low power Surface Acoustic Wave (SAW) sensors. To demonstrate the ability of the SAW sensor to detect flutter vibrations the sensors were attached to a Carbon fiber-reinforced polymer (CFRP) composite panel which was vibrated at six frequencies from 1Hz to 50Hz. The SAW data was compared to accelerometer data and was found to resemble sine waves and match each other closely. The SAW module design and results from the tests are presented here. 13. Mechanical Seal Opening Condition Monitoring Based on Acoustic Emission Technology Directory of Open Access Journals (Sweden) Erqing Zhang 2014-06-01 Full Text Available Since the measurement of mechanical sealing film thickness and just-lift-off time is very difficult, the sealing film condition monitoring method based on acoustic emission signal is proposed. The mechanical seal acoustic emission signal present obvious characteristics of time-varying nonlinear and pulsating. In this paper, the acoustic emission signal is used to monitor the seal end faces just-lift-off time and friction condition. The acoustic emission signal is decomposed by empirical mode decomposition into a series of intrinsic mode function with independent characteristics of different time scales and different frequency band. The acoustic emission signal only generated by end faces friction is obtained by eliminating the false intrinsic mode function components. The correlation coefficient of acoustic emission signal and Multi-scale Laplace Wavelet is calculated. It is proved that the maximum frequency (8000 Hz of the correlation coefficient is appeared at the spindle speed of 300 rpm. And at this time (300 rpm the end faces have just lifted off. By a set of mechanical oil seal running test, it is demonstrated that this method could accurately identify mechanical seal end faces just-lift-off time and friction condition. 14. A Finite Element Model of a MEMS-based Surface Acoustic Wave Hydrogen Sensor Directory of Open Access Journals (Sweden) Walied A. Moussa 2010-02-01 Full Text Available Hydrogen plays a significant role in various industrial applications, but careful handling and continuous monitoring are crucial since it is explosive when mixed with air. Surface Acoustic Wave (SAW sensors provide desirable characteristics for hydrogen detection due to their small size, low fabrication cost, ease of integration and high sensitivity. In this paper a finite element model of a Surface Acoustic Wave sensor is developed using ANSYS12© and tested for hydrogen detection. The sensor consists of a YZ-lithium niobate substrate with interdigital electrodes (IDT patterned on the surface. A thin palladium (Pd film is added on the surface of the sensor due to its high affinity for hydrogen. With increased hydrogen absorption the palladium hydride structure undergoes a phase change due to the formation of the β-phase, which deteriorates the crystal structure. Therefore with increasing hydrogen concentration the stiffness and the density are significantly reduced. The values of the modulus of elasticity and the density at different hydrogen concentrations in palladium are utilized in the finite element model to determine the corresponding SAW sensor response. Results indicate that with increasing the hydrogen concentration the wave velocity decreases and the attenuation of the wave is reduced. 15. A Surface Acoustic Wave Ethanol Sensor with Zinc Oxide Nanorods Directory of Open Access Journals (Sweden) Timothy J. Giffney 2012-01-01 Full Text Available Surface acoustic wave (SAW sensors are a class of piezoelectric MEMS sensors which can achieve high sensitivity and excellent robustness. A surface acoustic wave ethanol sensor using ZnO nanorods has been developed and tested. Vertically oriented ZnO nanorods were produced on a ZnO/128∘ rotated Y-cut LiNbO3 layered SAW device using a solution growth method with zinc nitrate, hexamethylenetriamine, and polyethyleneimine. The nanorods have average diameter of 45 nm and height of 1 μm. The SAW device has a wavelength of 60 um and a center frequency of 66 MHz at room temperature. In testing at an operating temperature of 270∘C with an ethanol concentration of 2300 ppm, the sensor exhibited a 24 KHz frequency shift. This represents a significant improvement in comparison to an otherwise identical sensor using a ZnO thin film without nanorods, which had a frequency shift of 9 KHz. 16. Contrast investigations of surface acoustic waves by stroboscopic topography. 1. Orientation contrast Energy Technology Data Exchange (ETDEWEB) Cerva, H.; Graeff, W. 1984-03-16 Surface acoustic waves are investigated by stroboscopic topography using synchrotron radiation from the storage ring DORIS. The observed contrast of the acoustic displacements of the lattice planes has the same periods as the acoustic wave. It is demonstrated that the major part of the contrast is due to orientation contrast of the curved net planes. Intensity maxima correspond to troughs of the acoustic wave, minima to crests. A numerical treatment yielding ray tracing maps, intensity curves as well as focusing conditions which are in quantitative agreement with the experimental data is presented. 17. Controlling an acoustic wave with a cylindrically-symmetric gradient-index system Science.gov (United States) Zhang, Zhe; Li, Rui-Qi; Liang, Bin; Zou, Xin-Ye; Cheng, Jian-Chun 2015-02-01 We present a detailed theoretical description of wave propagation in an acoustic gradient-index system with cylindrical symmetry and demonstrate its potential to numerically control acoustic waves in different ways. The trajectory of an acoustic wave within the system is derived by employing the theory of geometric acoustics, and the validity of the theoretical descriptions is verified numerically by using the finite element method simulation. The results show that by tailoring the distribution function of the refractive index, the proposed system can yield a tunable manipulation of acoustic waves, such as acoustic bending, trapping, and absorbing. Project supported by the National Basic Research Program of China (Grant Nos. 2010CB327803 and 2012CB921504), the National Natural Science Foundation of China (Grant Nos. 11174138, 11174139, 11222442, 81127901, and 11274168), NCET-12-0254, and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, China. 18. Analytical Solution for Waves in Planets with Atmospheric Superrotation. I. Acoustic and Inertia-Gravity Waves Science.gov (United States) Peralta, J.; Imamura, T.; Read, P. L.; Luz, D.; Piccialli, A.; López-Valverde, M. A. 2014-07-01 This paper is the first of a two-part study devoted to developing tools for a systematic classification of the wide variety of atmospheric waves expected on slowly rotating planets with atmospheric superrotation. Starting with the primitive equations for a cyclostrophic regime, we have deduced the analytical solution for the possible waves, simultaneously including the effect of the metric terms for the centrifugal force and the meridional shear of the background wind. In those cases when the conditions for the method of the multiple scales in height are met, these wave solutions are also valid when vertical shear of the background wind is present. A total of six types of waves have been found and their properties were characterized in terms of the corresponding dispersion relations and wave structures. In this first part, only waves that are direct solutions of the generic dispersion relation are studied—acoustic and inertia-gravity waves. Concerning inertia-gravity waves, we found that in the cases of short horizontal wavelengths, null background wind, or propagation in the equatorial region, only pure gravity waves are possible, while for the limit of large horizontal wavelengths and/or null static stability, the waves are inertial. The correspondence between classical atmospheric approximations and wave filtering has been examined too, and we carried out a classification of the mesoscale waves found in the clouds of Venus at different vertical levels of its atmosphere. Finally, the classification of waves in exoplanets is discussed and we provide a list of possible candidates with cyclostrophic regimes. 19. ANALYTICAL SOLUTION FOR WAVES IN PLANETS WITH ATMOSPHERIC SUPERROTATION. I. ACOUSTIC AND INERTIA-GRAVITY WAVES Energy Technology Data Exchange (ETDEWEB) Peralta, J.; López-Valverde, M. A. [Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía, 18008 Granada (Spain); Imamura, T. [Institute of Space and Astronautical Science-Japan Aerospace Exploration Agency 3-1-1, Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210 (Japan); Read, P. L. [Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford (United Kingdom); Luz, D. [Centro de Astronomia e Astrofísica da Universidade de Lisboa (CAAUL), Observatório Astronómico de Lisboa, Tapada da Ajuda, 1349-018 Lisboa (Portugal); Piccialli, A., E-mail: [email protected] [LATMOS, UVSQ, 11 bd dAlembert, 78280 Guyancourt (France) 2014-07-01 This paper is the first of a two-part study devoted to developing tools for a systematic classification of the wide variety of atmospheric waves expected on slowly rotating planets with atmospheric superrotation. Starting with the primitive equations for a cyclostrophic regime, we have deduced the analytical solution for the possible waves, simultaneously including the effect of the metric terms for the centrifugal force and the meridional shear of the background wind. In those cases when the conditions for the method of the multiple scales in height are met, these wave solutions are also valid when vertical shear of the background wind is present. A total of six types of waves have been found and their properties were characterized in terms of the corresponding dispersion relations and wave structures. In this first part, only waves that are direct solutions of the generic dispersion relation are studied—acoustic and inertia-gravity waves. Concerning inertia-gravity waves, we found that in the cases of short horizontal wavelengths, null background wind, or propagation in the equatorial region, only pure gravity waves are possible, while for the limit of large horizontal wavelengths and/or null static stability, the waves are inertial. The correspondence between classical atmospheric approximations and wave filtering has been examined too, and we carried out a classification of the mesoscale waves found in the clouds of Venus at different vertical levels of its atmosphere. Finally, the classification of waves in exoplanets is discussed and we provide a list of possible candidates with cyclostrophic regimes. 20. Acoustic waves in transversely excited atmospheric CO2 laser discharges: effect on performance and reduction techniques CSIR Research Space (South Africa) von Bergmann, HM 2008-08-01 Full Text Available Results are presented on the influence of acoustic waves on the performance of high-repetition-rate TEA CO2 lasers. It is shown that acoustic waves generated inside the laser cavity lead to nonuniform discharges, resulting in a deterioration... 1. Spectrum of the seismic-electromagnetic and acoustic waves caused by seismic and volcano activity Directory of Open Access Journals (Sweden) S. Koshevaya 2005-01-01 Full Text Available Modeling of the spectrum of the seismo-electromagnetic and acoustic waves, caused by seismic and volcanic activity, has been done. This spectrum includes the Electromagnetic Emission (EME, due to fracturing piezoelectrics in rocks and the Acoustic Emission (AE, caused by the excitation and the nonlinear passage of acoustic waves through the Earth's crust, the atmosphere, and the ionosphere. The investigated mechanism of the EME uses the model of fracturing and the crack motion. For its analysis, we consider a piezoelectric crystal under mechanical stresses, which cause the uniform crack motion, and, consequently, in the vicinity of the moving crack also cause non-stationary polarization currents. A possible spectrum of EME has been estimated. The underground fractures produce Very Low (VLF and Extremely Low Frequency (ELF acoustic waves, while the acoustic waves at higher frequencies present high losses and, on the Earth's surface, they are quite small and are not registered. The VLF acoustic wave is subject to nonlinearity under passage through the lithosphere that leads to the generation of higher harmonics and also frequency down-conversion, namely, increasing the ELF acoustic component on the Earth's surface. In turn, a nonlinear propagation of ELF acoustic wave in the atmosphere and the ionosphere leads to emerging the ultra low frequency (ULF acousto-gravity waves in the ionosphere and possible local excitation of plasma waves. 2. Damping-Growth Transition for Ion-Acoustic Waves in a Density Gradient DEFF Research Database (Denmark) D'Angelo, N.; Michelsen, Poul; Pécseli, Hans 1975-01-01 A damping-growth transition for ion-acoustic waves propagating in a nonuniform plasma (e-folding length for the density ln) is observed at a wavelength λ∼2πln. This result supports calculations performed in connection with the problem of heating of the solar corona by ion-acoustic waves generated... 3. Wear monitoring of single point cutting tool using acoustic emission ... Indian Academy of Sciences (India) However, the extent of improvement brought about by the coatings depends strongly on the cutting conditions, with the greatest benefits being seen at higher cutting speeds and feed rates. Among these methods, tool condition monitoring using Acoustic Techniques (AET) is an emerging one. Hence, the present work was ... 4. Monitoring of rapid sand filters using an acoustic imaging technique NARCIS (Netherlands) Allouche, N.; Simons, D.G.; Rietveld, L.C. 2012-01-01 A novel instrument is developed to acoustically image sand filters used for water treatment and monitor their performance. The instrument consists of an omnidirectional transmitter that generates a chirp with a frequency range between 10 and 110 kHz, and an array of hydrophones. The instrument was 5. Dynamic acoustics for the STAR-100. [computer algorithms for time dependent sound waves in jet Science.gov (United States) Bayliss, A.; Turkel, E. 1979-01-01 An algorithm is described to compute time dependent acoustic waves in a jet. The method differs from previous methods in that no harmonic time dependence is assumed, thus permitting the study of nonharmonic acoustical behavior. Large grids are required to resolve the acoustic waves. Since the problem is nonstiff, explicit high order schemes can be used. These have been adapted to the STAR-100 with great efficiencies and permitted the efficient solution of problems which would not be feasible on a scalar machine. 6. An acoustic wave equation for pure P wave in 2D TTI media KAUST Repository Zhan, Ge 2011-01-01 In this paper, a pure P wave equation for an acoustic 2D TTI media is derived. Compared with conventional TTI coupled equations, the resulting equation is unconditionally stable due to the complete isolation of the SV wave mode. To avoid numerical dispersion and produce high quality images, the rapid expansion method REM is employed for numerical implementation. Synthetic results validate the proposed equation and show that it is a stable algorithm for modeling and reverse time migration RTM in a TTI media for any anisotropic parameter values. © 2011 Society of Exploration Geophysicists. 7. An Investigation of Acoustic Wave Propagation in Mach 2 Flow Science.gov (United States) Nieberding, Zachary J. Hypersonic technology is the next advancement to enter the aerospace community; it is defined as the study of flight at speeds Mach 5 and higher where intense aerodynamic heating is prevalent. Hypersonic flight is achieved through use of scramjet engines, which intake air and compress it by means of shock waves and geometry design. The airflow is then directed through an isolator where it is further compressed, it is then delivered to the combustor at supersonic speeds. The combusted airflow and fuel mixture is then accelerated through a nozzle to achieve the hypersonic speeds. Unfortunately, scramjet engines can experience a phenomenon known as an inlet unstart, where the combustor produces pressures large enough to force the incoming airflow out of the inlet of the engine, resulting in a loss of acceleration and power. There have been several government-funded programs that look to prove the concept of the scramjet engine and also tackle this inlet unstart issue. The research conducted in this thesis is a fundamental approach towards controlling the unstart problem: it looks at the basic concept of sending a signal upstream through the boundary layer of a supersonic flow and being able to detect a characterizeable signal. Since conditions within and near the combustor are very harsh, hardware is unable to be installed in that area, so this testing will determine if a signal can be sent and if so, how far upstream can the signal be detected. This experimental approach utilizes several acoustic and mass injection sources to be evaluated over three test series in a Mach 2 continuous flow wind tunnel that will determine the success of the objective. The test series vary in that the conditions of the flow and the test objectives change. The research shows that a characterizeable signal can be transmitted upstream roughly 12 inches through the subsonic boundary layer of a supersonic cross flow. It is also shown that the signal attenuates as the distance between the 8. Acoustic Wave Treatment For Cellulite—A New Approach Science.gov (United States) Russe-Wilflingseder, Katharina; Russe, Elisabeth 2010-05-01 Background and Objectives: Cellulite is a biological caused modification of the female connective tissue. In extracorporeal shockwave therapy (ESWT) pulses are penetrating into the tissue without causing a thermal effect or micro lesions, but leading to a stimulation of tissue metabolism and blood circulation, inducing a natural repair process with cell activation and stem cells proliferation. Recently ESWT treatment showed evidence of remodelling collagen within the dermis and of stimulating microcirculation in fatty tissue. Study Design and Methods: The study was designed to assess acoustic wave treatment for cellulite by comparison treated vs. untreated side (upper-leg and buttock). Each individual served as its own control. 11 females with a BMI less then 30 and an age over 18 years were included. 6 treatments were given weekly with radial acoustic waves. Documentation was done before and 1, 4, 12 weeks after last treatment by standardized photo documentation, relaxed and with muscle contraction, measurement of body weight and circumference of the thigh, pinch test, and evaluation of hormonal status and lifestyle. The efficacy of AWT/EPAT was evaluated before and 1, 4, 12 weeks after last treatment. Patients rated the improvement of cellulite, overall satisfaction and acceptance. The therapist assessed improvement of cellulite, side effects and photo documentation treated vs. untreated side, before vs. after treatment. The blinded investigator evaluated the results using photo documentation right vs. left leg, before vs. after treatment in a frontal, lateral and dorsal view, relaxed and with muscle contraction. Results: The improvement of cellulite at the treated side was rated by patients with 27,3% at week 4 and 12, by the therapist with 34,1% at week 4 and 31,2% at week 12 after the last treatment The blinded investigator could verify an improvement of cellulite in an increasing number of patients with increasing time interval after treatment. No side 9. Surface acoustic waves in acoustic superlattice lithium niobate coated with a waveguide layer Science.gov (United States) Yang, G. Y.; Du, J. K.; Huang, B.; Jin, Y. A.; Xu, M. H. 2017-04-01 The effects of the waveguide layer on the band structure of Rayleigh waves are studied in this work based on a one-dimensional acoustic superlattice lithium niobate substrate coated with a waveguide layer. The present phononic structure is formed by the periodic domain-inverted single crystal that is the Z-cut lithium niobate substrate with a waveguide layer on the upper surface. The plane wave expansion method (PWE) is adopted to determine the band gap behavior of the phononic structure and validated by the finite element method (FEM). The FEM is also used to investigate the transmission of Rayleigh waves in the phononic structure with the interdigital transducers by means of the commercial package COMSOL. The results show that, although there is a homogeneous waveguide layer on the surface, the band gap of Rayleigh waves still exist. It is also found that increasing the thickness of the waveguide layer, the band width narrows and the band structure shifts to lower frequency. The present approach can be taken as an efficient tool in designing of phononic structures with waveguide layer. 10. Intraoperative monitoring during surgery for acoustic neuroma: benefits of an extratympanic intrameatal electrode Science.gov (United States) Mullatti, N; Coakham, H; Maw, A; Butler, S; Morgan, M 1999-01-01 OBJECTIVES—To assess the utility of an extratympanic intrameatal electrode for intraoperative monitoring during acoustic neuroma and other cerebellopontine angle tumour surgery and to define the neurophysiological and surgical factors which influence hearing preservation. METHODS—Twenty two patients, 18 with acoustic neuromas and four with other cerebellopontine angle tumours, underwent intraoperative monitoring during tumour excision. The extratympanic intrameatal electrode (IME) was used to record the electrocochleogram (ECoG) and surface electrodes to record the brainstem auditory evoked response (ABR). RESULTS—The compound action potential (CAP) of the ECoG was two and a half times greater in amplitude than wave I of the ABR and was easily monitored. Virtually instant information was available as minimal averaging was required. Continuous monitoring was possible from the commencement of anaesthesia to skin closure. The IME was easy to place, non-invasive, and did not interfere with the operative field. Operative procedures which affected CAP or wave V latency or amplitude were drilling around the internal auditory meatus, tumour dissection, nerve section, and brainstem and cerebellar retraction. Hearing was achieved in 59% of patients. CONCLUSIONS—The IME had significant benefits in comparison with other methods of monitoring. The technique provided information beneficial to preservation of hearing. PMID:10209169 11. Ultrasonic phased array with surface acoustic wave for imaging cracks Directory of Open Access Journals (Sweden) Yoshikazu Ohara 2017-06-01 Full Text Available To accurately measure crack lengths, we developed a real-time surface imaging method (SAW PA combining an ultrasonic phased array (PA with a surface acoustic wave (SAW. SAW PA using a Rayleigh wave with a high sensitivity to surface defects was implemented for contact testing using a wedge with the third critical angle that allows the Rayleigh wave to be generated. Here, to realize high sensitivity imaging, SAW PA was optimized in terms of the wedge and the imaging area. The improved SAW PA was experimentally demonstrated using a fatigue crack specimen made of an aluminum alloy. For further verification in more realistic specimens, SAW PA was applied to stainless-steel specimens with a fatigue crack and stress corrosion cracks (SCCs. The fatigue crack was visualized with a high signal-to-noise ratio (SNR and its length was measured with a high accuracy of better than 1 mm. The SCCs generated in the heat-affected zones (HAZs of a weld were successfully visualized with a satisfactory SNR, although responses at coarse grains appeared throughout the imaging area. The SCC lengths were accurately measured. The imaging results also precisely showed complicated distributions of SCCs, which were in excellent agreement with the optically observed distributions. 12. Crack propagation analysis using acoustic emission sensors for structural health monitoring systems. Science.gov (United States) Kral, Zachary; Horn, Walter; Steck, James 2013-01-01 Aerospace systems are expected to remain in service well beyond their designed life. Consequently, maintenance is an important issue. A novel method of implementing artificial neural networks and acoustic emission sensors to form a structural health monitoring (SHM) system for aerospace inspection routines was the focus of this research. Simple structural elements, consisting of flat aluminum plates of AL 2024-T3, were subjected to increasing static tensile loading. As the loading increased, designed cracks extended in length, releasing strain waves in the process. Strain wave signals, measured by acoustic emission sensors, were further analyzed in post-processing by artificial neural networks (ANN). Several experiments were performed to determine the severity and location of the crack extensions in the structure. ANNs were trained on a portion of the data acquired by the sensors and the ANNs were then validated with the remaining data. The combination of a system of acoustic emission sensors, and an ANN could determine crack extension accurately. The difference between predicted and actual crack extensions was determined to be between 0.004 in. and 0.015 in. with 95% confidence. These ANNs, coupled with acoustic emission sensors, showed promise for the creation of an SHM system for aerospace systems. 13. Acoustic monitoring of a fluidized bed coating process DEFF Research Database (Denmark) Naelapaa, Kaisa; Veski, Peep; Pedersen, Joan G. 2007-01-01 The aim of the study was to investigate the potential of acoustic monitoring of a production scale fluidized bed coating process. The correlation between sensor signals and the estimated amount of film applied and percentage release, respectively, were investigated in coating potassium chloride...... (KCl) crystals with ethylcellulose (EC). Vibrations were measured with two different types of accelerometers. Different positions for placing the accelerometers and two different product containers were included in the study. Top spray coating of KCl was chosen as a ‘worst case' scenario from a coating...... point perspective. The acoustic monitoring has the potential of summarising the commonly used means to monitor the coating process. The best partial least squares (PLS) regressions, obtained by the high frequency accelerometer, showed for the release a correlation coefficient of 0.92 and a root mean... 14. Universal instability of dust ion-sound waves and dust-acoustic waves Energy Technology Data Exchange (ETDEWEB) Tsytovich, V.N. [General Physics Institute, Russian Academy of Science Moscow, Moscow (Russian Federation); Watanabe, K. [National Inst. for Fusion Science, Toki, Gifu (Japan) 2002-01-01 It is shown that the dust ion-sound waves (DISW) and the dust-acoustic waves (DAW) are universally unstable for wave numbers less than some critical wave number. The basic dusty plasma state is assumed to be quasi-neutral with balance of the plasma particle absorption on the dust particles and the ionization with the rate proportional to the electron density. An analytical expression for the critical wave numbers, for the frequencies and for the growth rates of DISW and DAW are found using the hydrodynamic description of dusty plasma components with self-consistent treatment of the dust charge variations and by taking into account the change of the ion and electron distributions in the dust charging process. Most of the previous treatment do not take into account the latter process and do not treat the basic state self-consistently. The critical lengths corresponding to these critical wave numbers can be easily achieved in the existing experiments. It is shown that at the wave numbers larger than the critical ones DISW and DAW have a large damping which was not treated previously and which can be also measured. The instabilities found in the present work on their non linear stage can lead to formation of different types of dust self-organized structures. (author) 15. Generation of surface acoustic waves on doped semiconductor substrates Science.gov (United States) Yuan, M.; Hubert, C.; Rauwerdink, S.; Tahraoui, A.; van Someren, B.; Biermann, K.; Santos, P. V. 2017-12-01 We report on the electrical generation of surface acoustic waves (SAWs) on doped semiconductor substrates. This is implemented by using interdigital transducers (IDTs) placed on piezoelectric ZnO films sputtered onto evaporated thin metal layers. Two material systems are investigated, namely ZnO/Au/GaAs and ZnO/Ni/InP. The rf-field applied to the transducer is electrically screened by the highly conductive metal film underneath the ZnO film without any extra ohmic losses. As a result, absorption of the rf-field by the mobile carriers in the lossy doped region underneath the IDT is avoided, ensuring efficient SAW generation. We find that the growth temperature of the ZnO film on the metal layer affects its structure and, thus, the efficiency of SAW generation. With this technique, the SAW active layers can be placed close to doped layers, expanding the application range of SAWs in semiconductor devices. 16. Surface Acoustic Wave Tag-Based Coherence Multiplexing Science.gov (United States) Youngquist, Robert C. (Inventor); Malocha, Donald (Inventor); Saldanha, Nancy (Inventor) 2016-01-01 A surface acoustic wave (SAW)-based coherence multiplexing system includes SAW tags each including a SAW transducer, a first SAW reflector positioned a first distance from the SAW transducer and a second SAW reflector positioned a second distance from the SAW transducer. A transceiver including a wireless transmitter has a signal source providing a source signal and circuitry for transmitting interrogation pulses including a first and a second interrogation pulse toward the SAW tags, and a wireless receiver for receiving and processing response signals from the SAW tags. The receiver receives scrambled signals including a convolution of the wideband interrogation pulses with response signals from the SAW tags and includes a computing device which implements an algorithm that correlates the interrogation pulses or the source signal before transmitting against the scrambled signals to generate tag responses for each of the SAW tags. 17. Acoustic wave therapy for cellulite, body shaping and fat reduction. Science.gov (United States) Hexsel, Doris; Camozzato, Fernanda Oliveira; Silva, Aline Flor; Siega, Carolina 2017-06-01 Cellulite is a common aesthetic condition that affects almost every woman. To evaluate the efficacy of acoustic wave therapy (AWT) for cellulite and body shaping. In this open-label, single-centre trial, 30 women presenting moderate or severe cellulite underwent 12 sessions of AWT on the gluteus and back of the thighs, over six weeks. The following assessments were performed at baseline, and up to 12 weeks after treatment: Cellulite Severity Scale (CSS), body circumference measurements, subcutaneous fat thickness by magnetic resonance imaging (MRI), quality of life related by Celluqol ® and a satisfaction questionnaire. The treatment reduced cellulite severity from baseline up to 12 weeks after the last treatment session (subjects presenting severe cellulite: 60% to 38%). The mean CSS shifted from 11.1 to 9.5 (p cellulite appearance and reduce body circumferences. 18. Multilayer-graphene-based amplifier of surface acoustic waves Directory of Open Access Journals (Sweden) Stanislav O. Yurchenko 2015-05-01 Full Text Available The amplification of surface acoustic waves (SAWs by a multilayer graphene (MLG-based amplifier is studied. The conductivity of massless carriers (electrons or holes in graphene in an external drift electric field is calculated using Boltzmann’s equation. At some carrier drift velocities, the real part of the variable conductivity becomes negative and MLG can be employed in SAW amplifiers. Amplification of Blustein’s and Rayleigh’s SAWs in CdS, a piezoelectric hexagonal crystal of the symmetry group C6v, is considered. The corresponding equations for SAW propagation in the device are derived and can be applied to other substrate crystals of the same symmetry. The results of the paper indicate that MLG can be considered as a perspective material for SAW amplification and related applications. 19. Ultrafast high strain rate acoustic wave measurements at high static pressure in a diamond anvil cell Science.gov (United States) Armstrong, Michael R.; Crowhurst, Jonathan C.; Reed, Evan J.; Zaug, Joseph M. 2009-02-01 We describe experiments demonstrating the generation of ultrafast, high strain rate acoustic waves in a precompressed transparent medium at static pressure up to 24 GPa. We also observe shock waves in precompressed aluminum with transient pressures above 40 GPa under precompression. Using ultrafast interferometry, we determine parameters such as the shock pressure and acoustic wave velocity using multiple and single shot methods. These methods form the basis for material experiments under extreme conditions which are challenging to access using other techniques. 20. Study of nonlinear ion- and electron-acoustic waves in multi-component space plasmas Directory of Open Access Journals (Sweden) G. S. Lakhina 2008-11-01 Full Text Available Large amplitude ion-acoustic and electron-acoustic waves in an unmagnetized multi-component plasma system consisting of cold background electrons and ions, a hot electron beam and a hot ion beam are studied using Sagdeev pseudo-potential technique. Three types of solitary waves, namely, slow ion-acoustic, ion-acoustic and electron-acoustic solitons are found provided the Mach numbers exceed the critical values. The slow ion-acoustic solitons have the smallest critical Mach numbers, whereas the electron-acoustic solitons have the largest critical Mach numbers. For the plasma parameters considered here, both type of ion-acoustic solitons have positive potential whereas the electron-acoustic solitons can have either positive or negative potential depending on the fractional number density of the cold electrons relative to that of the ions (or total electrons number density. For a fixed Mach number, increases in the beam speeds of either hot electrons or hot ions can lead to reduction in the amplitudes of the ion-and electron-acoustic solitons. However, the presence of hot electron and hot ion beams have no effect on the amplitudes of slow ion-acoustic modes. Possible application of this model to the electrostatic solitary waves (ESWs observed in the plasma sheet boundary layer is discussed. 1. Acoustic wave emission for enhanced oil recovery (WAVE.O.R.) Energy Technology Data Exchange (ETDEWEB) Reichmann, S.; Amro, M. [TU Bergakademie, Freiberg (Germany); Giese, R.; Jaksch, K.; Krauss, F.; Krueger, K.; Jurczyk, A. [Helmholtz-Zentrum Potsdam - Deutsches GeoForschungsZentrum GFZ, Potsdam (Germany) 2016-09-15 In the project WAVE.O.R the potential of acoustic waves to enhance oil recovery was reviewed. The project focused on laboratory experiments of the oil displacement in sandstone cores under acoustic stimulation. Additionally, the Seismic Prediction While Drilling (SPWD) borehole device prototype was set up for a feasibility field test. The laboratory experiments showed that, depending on the stimulation frequency, acoustic stimulation allows for an enhanced oil recovery. For single frequency stimulation a mean increase of 3 % pore volumes was observed at distinguished frequencies. A cyclic stimulation, where two of these frequencies were combined, an increase of 5% pore volume was observed. The SPWD borehole device was tested and adjusted during feasibility tests in the GFZ underground laboratory in the research and education mine ''Reiche Zeche'' of the TU Bergakademie Freiberg and in the GFZ KTB-Deep Laboratory in Windischeschenbach. The first successful test of the device under realistic conditions was performed at the test site ''Piana di Toppo'' of the OGS Trieste, Italy. 2. Acoustic Float for Marine Mammal Monitoring Science.gov (United States) 2011-09-30 development by several teams, is to equip several ocean gliders with hydrophones and marine mammal call-detection software and send them out to monitor in...real time, but the gliders are relatively expensive, at upwards of$100,000 each [Rogers et al., 2004]. A vertical profiler float has been in...inexpensive and reliable tool for oceanogrphers [Kobayashi et al., 2006]. They also can dive to 2000 m whereass Slocum gilder and Seaglider are rated
3. The dust acoustic waves in three dimensional scalable complex plasma
CERN Document Server
Zhukhovitskii, D I
2015-01-01
Dust acoustic waves in the bulk of a dust cloud in complex plasma of low pressure gas discharge under microgravity conditions are considered. The dust component of complex plasma is assumed a scalable system that conforms to the ionization equation of state (IEOS) developed in our previous study. We find singular points of this IEOS that determine the behavior of the sound velocity in different regions of the cloud. The fluid approach is utilized to deduce the wave equation that includes the neutral drag term. It is shown that the sound velocity is fully defined by the particle compressibility, which is calculated on the basis of the scalable IEOS. The sound velocities and damping rates calculated for different 3D complex plasmas both in ac and dc discharges demonstrate a good correlation with experimental data that are within the limits of validity of the theory. The theory provides interpretation for the observed independence of the sound velocity on the coordinate and for a weak dependence on the particle ...
4. Single-electron transport driven by surface acoustic waves: Moving quantum dots versus short barriers
DEFF Research Database (Denmark)
Utko, Pawel; Hansen, Jørn Bindslev; Lindelof, Poul Erik
2007-01-01
We have investigated the response of the acoustoelectric-current driven by a surface-acoustic wave through a quantum point contact in the closed-channel regime. Under proper conditions, the current develops plateaus at integer multiples of ef when the frequency f of the surface-acoustic wave...... or the gate voltage V-g of the point contact is varied. A pronounced 1.1 MHz beat period of the current indicates that the interference of the surface-acoustic wave with reflected waves matters. This is supported by the results obtained after a second independent beam of surface-acoustic wave was added...... to an additional quantization mechanism, independent from those described in the standard model of 'moving quantum dots....
5. Ion acoustic wave generation by a standing electromagnetic field in a subcritical plasma
OpenAIRE
P. Fischer; Gauthereau, C.; Godiot, J.; G. Matthieussent
1987-01-01
An electromagnetic wave ( f = 9 GHz, Pi = 150 kW, τ = 1.5 μs) is launched into a subcritical argon plasma (n e ≃1011 cm-3, P0 ≃ 5 × 10-4 Torr), resulting in a standing wave. The associated ponderomotive force generates an ion acoustic wave with a wave vector equal to twice the electromagnetic one and with a frequency satisfying the usual dispersion relation (fA ≃ 150 kHz). The main features of the ion acoustic wave, as measured in this 3D experiment, agree with a simple theory. However, varyi...
6. Scattering of X-rays on the surface acoustic wave in the case of grazing geometry
CERN Document Server
Mkrtchyan, A R; Petrosian, A
2000-01-01
The scattering of X-rays on a crystal is considered in grazing geometry when a surface acoustic wave is excited normal to the diffraction vector. The intensity of wave field at finite distance from crystal to detector is obtained. It is shown that in the presence of surface acoustic wave the magnitude of the main peak of specular reflected diffracted wave intensity decreases and intensity of satellites increases. The main peak of specular reflected diffracted wave intensity is split up as the grazing observation angle increases.
7. Parallel electric field in the auroral ionosphere: excitation of acoustic waves by Alfvén waves
Directory of Open Access Journals (Sweden)
P. L. Israelevich
2004-09-01
Full Text Available We investigate a new mechanism for the formation of a parallel electric field observed in the auroral ionosphere. For this purpose, the excitation of acoustic waves by propagating Alfvén waves was studied numerically. We find that the magnetic pressure perturbation due to finite amplitude Alfvén waves causes the perturbation of the plasma pressure that propagates in the form of acoustic waves, and gives rise to a parallel electric field. This mechanism explains the observations of the strong parallel electric field in the small-scale electromagnetic perturbations of the auroral ionosphere. For the cases when the parallel electric current in the small-scale auroral perturbations is so strong that the velocity of current carriers exceeds the threshold of the ion sound instability, the excited ion acoustic waves may account for the parallel electric fields as strong as tens of mV/m.
8. Signal processing methodologies for an acoustic fetal heart rate monitor
Science.gov (United States)
Pretlow, Robert A., III; Stoughton, John W.
1992-01-01
Research and development is presented of real time signal processing methodologies for the detection of fetal heart tones within a noise-contaminated signal from a passive acoustic sensor. A linear predictor algorithm is utilized for detection of the heart tone event and additional processing derives heart rate. The linear predictor is adaptively 'trained' in a least mean square error sense on generic fetal heart tones recorded from patients. A real time monitor system is described which outputs to a strip chart recorder for plotting the time history of the fetal heart rate. The system is validated in the context of the fetal nonstress test. Comparisons are made with ultrasonic nonstress tests on a series of patients. Comparative data provides favorable indications of the feasibility of the acoustic monitor for clinical use.
9. Attenuation compensation in least-squares reverse time migration using the visco-acoustic wave equation
KAUST Repository
Dutta, Gaurav
2013-08-20
Attenuation leads to distortion of amplitude and phase of seismic waves propagating inside the earth. Conventional acoustic and least-squares reverse time migration do not account for this distortion which leads to defocusing of migration images in highly attenuative geological environments. To account for this distortion, we propose to use the visco-acoustic wave equation for least-squares reverse time migration. Numerical tests on synthetic data show that least-squares reverse time migration with the visco-acoustic wave equation corrects for this distortion and produces images with better balanced amplitudes compared to the conventional approach. © 2013 SEG.
10. Surface and quasi-longitudinal acoustic waves in KTiOAsO₄ single crystals.
Science.gov (United States)
Taziev, Rinat M
2014-02-01
Surface and quasi-longitudinal acoustic wave properties have been investigated in potassium titanyl arsenate (KTiOAsO₄, KTA) single crystals for the first time. Surface acoustic wave (SAW) velocity, electromechanical coupling coefficient and power flow angle characteristics have been obtained in rotated Y-cut of KTA crystals. High SAW electromechanical coupling coefficient (0.4%) is found in Z-cut of KTA crystals. For high-frequency devices it is promising the resonators on quasi-longitudinal acoustic wave in X-cut of KTA crystals with sharp response in interdigital transducer conductance at resonance frequency. Copyright © 2013 Elsevier B.V. All rights reserved.
11. Contrast investigations of surface acoustic waves by stroboscopic topography. 2. Wavefield deviation contrast
Energy Technology Data Exchange (ETDEWEB)
Cerva, H.; Graeff, W. (Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany, F.R.))
1985-02-16
When imaging a surface acoustic wave by stroboscopic X-ray topography a contrast contribution exists which can be related to the deviation of X-ray beams in the deformation field of the acoustic wave. With narrow entrance slits this contribution can be separated from the surface reflected waves. Using a beam path theory of Bonse the beam trajectories inside the crystal and the intensity profiles at the surface are calculated. It is also demonstrated that this contrast which has two nearly equal peaks within the acoustic period turns to the orientation contrast with one peak per period when increasing the distance between sample and film.
12. Generation of thermo-acoustic waves from pulsed solar/IR radiation
Science.gov (United States)
Rahman, Aowabin
Acoustic waves could potentially be used in a wide range of engineering applications; however, the high energy consumption in generating acoustic waves from electrical energy and the cost associated with the process limit the use of acoustic waves in industrial processes. Acoustic waves converted from solar radiation provide a feasible way of obtaining acoustic energy, without relying on conventional nonrenewable energy sources. One of the goals of this thesis project was to experimentally study the conversion of thermal to acoustic energy using pulsed radiation. The experiments were categorized into "indoor" and "outdoor" experiments, each with a separate experimental setup. The indoor experiments used an IR heater to power the thermo-acoustic lasers and were primarily aimed at studying the effect of various experimental parameters on the amplitude of sound waves in the low frequency range (below 130 Hz). The IR radiation was modulated externally using a chopper wheel and then impinged on a porous solid, which was housed inside a thermo-acoustic (TA) converter. A microphone located at a certain distance from the porous solid inside the TA converter detected the acoustic signals. The "outdoor" experiments, which were targeted at TA conversion at comparatively higher frequencies (in 200 Hz-3 kHz range) used solar energy to power the thermo-acoustic laser. The amplitudes (in RMS) of thermo-acoustic signals obtained in experiments using IR heater as radiation source were in the 80-100 dB range. The frequency of acoustic waves corresponded to the frequency of interceptions of the radiation beam by the chopper. The amplitudes of acoustic waves were influenced by several factors, including the chopping frequency, magnitude of radiation flux, type of porous material, length of porous material, external heating of the TA converter housing, location of microphone within the air column, and design of the TA converter. The time-dependent profile of the thermo-acoustic signals
13. On the local plane wave methods for in situ measurement of acoustic absorption
NARCIS (Netherlands)
Wijnant, Ysbrand H.
2015-01-01
In this paper we address a series of so-called local plane wave methods (LPW) to measure acoustic absorption. As opposed to other methods, these methods do not rely on assumptions of the global sound field, like e.g. a plane wave or diffuse field, but are based on a local plane wave assumption.
14. Plasma–maser instability of the ion acoustics wave in the presence ...
A theoretical study is made on the generation mechanism of ion acoustics wave in the presence of lower hybrid wave turbulence field in inhomogeneous plasma on the basis of plasma-maser interaction. The lower hybrid wave turbulence field is taken as the low-frequency turbulence field. The growth rate of test high ...
15. Acoustic module of the Acquabona (Italy debris flow monitoring system
Directory of Open Access Journals (Sweden)
A. Galgaro
2005-01-01
Full Text Available Monitoring of debris flows aimed to the assessment of their physical parameters is very important both for theoretical and practical purposes. Peak discharge and total volume of debris flows are crucial for designing effective countermeasures in many populated mountain areas where losses of lives and property damage could be avoided. This study quantifies the relationship between flow depth, acoustic amplitude of debris flow induced ground vibrations and front velocity in the experimental catchment of Acquabona, Eastern Dolomites, Italy. The analysis of data brought about the results described in the following. Debris flow depth and amplitude of the flow-induced ground vibrations show a good positive correlation. Estimation of both mean front velocity and peak discharge can be simply obtained monitoring the ground vibrations, through geophones installed close to the flow channel; the total volume of debris flow can be so directly estimated from the integral of the ground vibrations using a regression line. The application of acoustic technique to debris flow monitoring seems to be of the outmost relevance in risk reduction policies and in the correct management of the territory. Moreover this estimation is possible in other catchments producing debris flows of similar characteristics by means of their acoustic characterisation through quick and simple field tests (Standard Penetration Tests and seismic refraction surveys.
16. An Aquatic Acoustic Metrics Interface Utility for Underwater Sound Monitoring and Analysis
Energy Technology Data Exchange (ETDEWEB)
Ren, Huiying; Halvorsen, Michele B.; Deng, Zhiqun; Carlson, Thomas J.
2012-05-31
Fishes and other marine mammals suffer a range of potential effects from intense sound sources generated by anthropogenic underwater processes such as pile driving, shipping, sonars, and underwater blasting. Several underwater sound recording devices (USR) were built to monitor the acoustic sound pressure waves generated by those anthropogenic underwater activities, so the relevant processing software becomes indispensable for analyzing the audio files recorded by these USRs. However, existing software packages did not meet performance and flexibility requirements. In this paper, we provide a detailed description of a new software package, named Aquatic Acoustic Metrics Interface (AAMI), which is a Graphical User Interface (GUI) designed for underwater sound monitoring and analysis. In addition to the general functions, such as loading and editing audio files recorded by USRs, the software can compute a series of acoustic metrics in physical units, monitor the sound's influence on fish hearing according to audiograms from different species of fishes and marine mammals, and batch process the sound files. The detailed applications of the software AAMI will be discussed along with several test case scenarios to illustrate its functionality.
17. DNA-duplexes containing abasic sites: correlation between thermostability and acoustic wave properties.
Science.gov (United States)
Hianik, T; Wang, X; Andreev, S; Dolinnaya, N; Oretskaya, T; Thompson, M
2006-10-01
Aldehydic apurinic or apyrimidinic sites that lack a nucleobase moiety are one of the most common forms of toxic lesions in DNA. In the present study, a close structural analog of such a site, the 2-(hydroxymethyl) tetrahydrofuranyl residue, was synthesized in order to act as a model for damaged nucleic acid probes. Prepared oligodeoxyribonucleotides containing one, two or three abasic sites were hybridized to complementary sequences immobilized on a gold surface using the neutravidin-biotin interaction for study by thickness shear mode acoustic wave detector. Measurement of the complex electrical impedance of an AT-cut quartz device with immobilized biotinylated nucleotide allowed the detection of changes of series resonance frequency, Deltafs, and motional resistance, Rm, associated with duplex formation. The changes as detected by the acoustic wave method correlated well with the thermostability of DNA duplexes in solution. With respect to the latter, UV-monitored melting curves indicate that both the number of sites and their localization in the double-stranded structure influence the amount by which a 19 b.p. duplex is destabilized. The presence of 3 abasic sites completely destabilized the DNA duplex.
18. Vertically propagating acoustic waves launched by seismic waves visualized in ionograms
Science.gov (United States)
Maruyama, Takashi; Shinagawa, Hiroyuki
2013-04-01
After the magnitude 9.0 earthquake off the Pacific coast of Tohoku (near the east coast of Honshu, Japan), which occurred on 11 March 2011, an unusual multiple-cusp signature (MCS) was observed in ionograms at three ionosonde stations across Japan. Similar MCSs in ionograms were identified in 8 of 43 earthquakes with a seismic magnitude of 8.0 or greater for the period from 1957 to 2011. The appearance of MCSs at different epicentral distances exhibited traveling characteristics at a velocity of ~4.0 km/s, which is in the range of Rayleigh waves. There was a ~7 min offset in delay time at each epicentral distance in the travel-time diagram. This offset is consistent with the propagation time of acoustic waves from the ground to the ionosphere. We analyzed vertical structure of electron density perturbation that caused MCSs. The ionosonde technique is essentially radar-based measurement of a reflection at a height where the plasma frequency is equal to the sounding radio frequency and it is possible to obtain an electron density profile by sweeping the frequency. However, this measured height is not a true height because radio waves do not propagate at the speed of light in the ionosphere. The group velocity of radio waves decreases just below the reflection height where the sounding frequency approaches the plasma frequency. The amount of delay is larger when this region is thicker. The vertically propagating acoustic waves modulate the electron density. The radio wave speed greatly delays and a cusp signature appears in the echo trace at a phase of the periodic perturbation of electron density where the density gradient is most gradual. Simulations were conducted how large amplitude of density perturbation produces cusp signatures as observed. First, the real height density profile was obtained by converting the ionogram trace just before the arrival of coseismic disturbances. The electron density profile was then modified by adding a periodic perturbation and the
19. Fatigue crack sizing in rail steel using crack closure-induced acoustic emission waves
Science.gov (United States)
Li, Dan; Kuang, Kevin Sze Chiang; Ghee Koh, Chan
2017-06-01
The acoustic emission (AE) technique is a promising approach for detecting and locating fatigue cracks in metallic structures such as rail tracks. However, it is still a challenge to quantify the crack size accurately using this technique. AE waves can be generated by either crack propagation (CP) or crack closure (CC) processes and classification of these two types of AE waves is necessary to obtain more reliable crack sizing results. As the pre-processing step, an index based on wavelet power (WP) of AE signal is initially established in this paper in order to distinguish between the CC-induced AE waves and their CP-induced counterparts. Here, information embedded within the AE signal was used to perform the AE wave classification, which is preferred to the use of real-time load information, typically adopted in other studies. With the proposed approach, it renders the AE technique more amenable to practical implementation. Following the AE wave classification, a novel method to quantify the fatigue crack length was developed by taking advantage of the CC-induced AE waves, the count rate of which was observed to be positively correlated with the crack length. The crack length was subsequently determined using an empirical model derived from the AE data acquired during the fatigue tests of the rail steel specimens. The performance of the proposed method was validated by experimental data and compared with that of the traditional crack sizing method, which is based on CP-induced AE waves. As a significant advantage over other AE crack sizing methods, the proposed novel method is able to estimate the crack length without prior knowledge of the initial crack length, integration of AE data or real-time load amplitude. It is thus applicable to the health monitoring of both new and existing structures.
20. Ion-acoustic waves in ultracold neutral plasmas: Modulational instability and dissipative rogue waves
Energy Technology Data Exchange (ETDEWEB)
El-Tantawy, S.A., E-mail: [email protected]
2017-02-26
Progress is reported on the modulational instability (MI) of ion-acoustic waves (IAWs) and dissipative rogue waves (RWs) in ultracold neutral plasmas (UNPs). The UNPs consist of inertial ions fluid and Maxwellian inertialess hot electrons, and the presence of an ion kinematic viscosity is allowed. For this purpose, a modified nonlinear Schrödinger equation (NLSE) is derived and then solved analytically to show the occurrence of MI. It is found that the (in)stability regions of the wavepacks are dependent on time due to of the existence of the dissipative term. The existing regions of the MI of the IAWs are inventoried precisely. After that, we use a suitable transformation to convert the modified NLSE into the normal NLSE whose analytical solutions for rogue waves are known. The rogue wave propagation condition and its behavior are discussed. The impact of the relevant physical parameters on the profile of the RWs is examined. - Highlights: • UNPs are modeled by the phenomenological generalized hydrodynamic equations. • The derivative expansion method has been employed in order to derive a modified-NLSE. • A suitable transformation is used to transform the modified-NLSE into the standard NLSE. • The effect of the ion viscosity on the modulational instability and rogue waves is investigated.
1. Monitoring of drilling process with the application of acoustic signal
Directory of Open Access Journals (Sweden)
Laba Milan
2000-09-01
Full Text Available Monitoring of rock disintegration process at drilling, scanning of input quantities: thrust F, revolution n and the course of some output quantities: the drilling rate v and the power input P are needed for the control of this process. We can calculate the specific volume work of rock disintegration w and ϕ - quotient of drilling rate v and the specific volume work of disintegration w from the presented quantities.Works on an expertimental stand showed that the correlation relationships between the input and output quantities can be found by scanning the accompanying sound of the drilling proces.Research of the rock disintegration with small-diameter diamond drill tools and different rock types is done at the Institute of Geotechnics. The aim of this research is the possibility of monitoring and controlling the rock disintegration process with the application of acoustic signal. The acoustic vibrations accompanying the drilling process are recorded by a microphone placed in a defined position in the acoustic space. The drilling device (drilling stand, the drilling tool and the rock are the source of sound. Two basic sound states exist in the drilling stand research : the noise at no-load running and the noise at the rotary drilling of rock. Suitable quantities for optimizing the rock disintegration process are searched by the study of the acoustic signal. The dominant frequencies that characterize the disintegration process for the given rock and tool are searched by the analysis of the acoustic signal. The analysis of dominant frequencies indicates the possibility of determining an optimal regime for the maximal drilling rate. Extreme of the specific disintegration energy is determinated by the dispersion of the dominant frequency.The scanned acoustic signal is processed by the Fourier transformation. The Fourier transformation facilitates the distribution of the general non-harmonic periodic process into harmonic components. The harmonic
2. KAJIAN SIFAT AKUSTIK BUAH MANGGIS(Gracinia mangostana L) DENGAN MENGGUNAKAN GELOMBANG ULTRASONIK [Acoustic Study Of Mangosteene (Gracinia mangostana L) By Using Ultrasonic Wave
OpenAIRE
Jajang juansah 1); I Wayan Budiastra 2); Suroso 2)
2007-01-01
The wave used to study the acoustic properties of mangosteen is ultrasonic wave. Ultrasonic wave with frequency of 50 KHz was used to determine acoustic properties of mangosteen. The main wave properties were the attenuation, impedance of acoustic and acoustic velocity at mangosteen. Others have been evaluated were the correlation of attenuation and acoustic velocity at parts of mangosteen with its intact mangosteen. The acoustic parameters were related to the physic-chemical parameters of th...
3. Use of information system data of jet crushing acoustic monitoring for the process management
Directory of Open Access Journals (Sweden)
T.M. Bulanaya
2012-12-01
Full Text Available The graphic interpretation of amplitude and frequency of acoustic signals of loose material jet grinding process are resulted. Criteria of process management is determined on the basis of the acoustic monitoring data of jet mill acting.
4. Passive Wireless Cryogenic Liquid Level Sensors Using Orthogonal Frequency Coded Acoustic Wave Devices Project
Data.gov (United States)
National Aeronautics and Space Administration — This proposal describes the continued development of passive wireless surface acoustic wave (SAW) based liquid level sensors for NASA application to cryogenic liquid...
5. PASSIVE WIRELESS MULTI-SENSOR TEMPERATURE AND PRESSURE SENSING SYSTEM USING ACOUSTIC WAVE DEVICES Project
Data.gov (United States)
National Aeronautics and Space Administration — This proposal describes the development of passive surface acoustic wave (SAW) sensors and multi-sensor systems for NASA application to remote wireless sensing of...
6. Passive Wireless Multi-Sensor Temperature and Pressure Sensing System Using Acoustic Wave Devices Project
Data.gov (United States)
National Aeronautics and Space Administration — This proposal describes the continued development of passive, orthogonal frequency coded (OFC) surface acoustic wave (SAW) sensors and multi-sensor systems, an...
7. Passive Wireless Cryogenic Liquid Level Sensors Using Orthogonal Frequency Coded Acoustic Wave Devices Project
Data.gov (United States)
National Aeronautics and Space Administration — This proposal describes the development of passive wireless surface acoustic wave (SAW) based liquid level sensors for NASA application to cryogenic liquid level...
8. Visualization of GHz Acoustic Wave in LiNbO3 by Microwave Impedance Microscopy
Science.gov (United States)
Zheng, Lu; Dong, Hui; Wu, Xiaoyu; Huang, Yen-Lin; Wu, Weida; Wang, Zheng; Lai, Keji
Acoustic wave devices based on piezoelectric materials play a key role in the modern information technology and the research field of phononic metamaterials. High-resolution real-space mapping of the phononic modes is therefore of fundamental importance for the understanding of scattering, diffraction, and localization of the acoustic waves. To date, however, it has been challenging to directly image the GHz-range acoustic properties in piezoelectrics. Using a microwave impedance microscope (MIM), we demonstrate the ability to visualize the interference pattern of GHz acoustic waves in periodically poled lithium niobate (PPLN) samples, where the domain walls serve as good reflectors of the elastic deformation. The constructive and destructive interference regions exhibit different loss in the microwave images, which can be simulated by finite-element analysis of the PPLN samples. Our results pave the way to locally probe various phenomena of sound waves in phononic materials by nanoscale electromagnetic imaging.
9. A sound idea: Manipulating domain walls in magnetic nanowires using surface acoustic waves
Science.gov (United States)
Dean, J.; Bryan, M. T.; Cooper, J. D.; Virbule, A.; Cunningham, J. E.; Hayward, T. J.
2015-10-01
We propose a method of pinning and propagating domain walls in artificial multiferroic nanowires using electrically induced surface acoustic waves. Using finite-element micromagnetic simulations and 1D semi-analytical modelling, we demonstrate how a pair of interdigitated acoustic transducers can remotely induce an array of attractive domain wall pinning sites by forming a standing stress/strain wave along a nanowire's length. Shifts in the frequencies of the surface acoustic waves allow multiple domain walls to be synchronously transported at speeds up to 50 ms-1. Our study lays the foundation for energy-efficient domain wall devices that exploit the low propagation losses of surface acoustic waves to precisely manipulate large numbers of data bits.
10. Interaction of Acoustic Waves with a Cryogenic Nitrogen Jet at Sub- and Supercritical Pressures
National Research Council Canada - National Science Library
Chehroudi, B
2001-01-01
To better understand the nature of the interaction between acoustic waves and liquid fuel jets in rocket engines, cryogenic liquid nitrogen is injected into a room temperature high-pressure chamber...
11. Propagation of dust-acoustic waves in weakly ionized plasmas with ...
For an unmagnetized partially ionized dusty plasma containing electrons, singly charged positive ions, micron-sized massive negatively charged dust grains and a fraction of neutral atoms, dispersion relations for both the dust-ion-acoustic and the dust-acoustic waves have been derived, incorporating dust charge ...
12. Measurements of shock-induced guided and surface acoustic waves along boreholes in poroelastic materials
NARCIS (Netherlands)
Chao, G.; Smeulders, D.M.J.; Van Dongen, M.E.H.
2006-01-01
Acoustic experiments on the propagation of guided waves along water-filled boreholes in water-saturated porous materials are reported. The experiments were conducted using a shock tube technique. An acoustic funnel structure was placed inside the tube just above the sample in order to enhance the
13. Simulation of an Underwater Acoustic Communication Channel Characterized by Wind-Generated Surface Waves and Bubbles
NARCIS (Netherlands)
Dol, H.S.; Ainslie, M.A.; Colin, M.E.G.D.; Janmaat, J.
2012-01-01
Sea surface scattering by wind-generated waves and bubbles is regarded to be the main nonplatform-related cause of the time variability of shallow acoustic communication channels. Simulations for predicting the quality of acoustic communication links in such channels thus require adequate modelling
14. Piezoelectric thin films for bulk acoustic wave resonator applications: from processing to microwave filters
OpenAIRE
Lanz, Roman; Setter, Nava
2005-01-01
Bandpass filters for microwave frequencies realized with thin film bulk acoustic wave resonators (FBAR) are a promising alternative to current dielectric or surface acoustic wave filters for use in mobile telecommunication applications. With equivalent performance, FBAR filters are significantly smaller than dielectric filters and allow for a larger power operation than SAW filters. In addition, FBARs offer the possibility of on-chip integration, which will result in substantial volume and co...
15. Quantum ion acoustic solitary waves in electron-ion plasmas: A Sagdeev potential approach
Energy Technology Data Exchange (ETDEWEB)
Mahmood, S. [Theoretical Plasma Physics Division, PINSTECH, P.O. Nilore, Islamabad (Pakistan)], E-mail: [email protected]; Mushtaq, A. [Theoretical Plasma Physics Division, PINSTECH, P.O. Nilore, Islamabad (Pakistan)
2008-05-05
Linear and nonlinear ion acoustic waves are studied in unmagnetized electron-ion quantum plasmas. Sagdeev potential approach is employed to describe the nonlinear quantum ion acoustic waves. It is found that density dips structures are formed in the subsonic region in a electron-ion quantum plasma case. The amplitude of the nonlinear structures remains constant and the width is broadened with the increase in the quantization of the system. However, the nonlinear wave amplitude is reduced with the increase in the wave Mach number. The numerical results are also presented.
16. Propagation and localization of acoustic waves in Fibonacci phononic circuits
Energy Technology Data Exchange (ETDEWEB)
Aynaou, H [Laboratoire de Dynamique et d' Optique des Materiaux, Departement de Physique, Faculte des Sciences, Universite Mohamed Premier, 60000 Oujda (Morocco); Boudouti, E H El [Laboratoire de Dynamique et d' Optique des Materiaux, Departement de Physique, Faculte des Sciences, Universite Mohamed Premier, 60000 Oujda (Morocco); Djafari-Rouhani, B [Laboratoire de Dynamique et Structure des Materiaux Moleculaires, UMR CNRS 8024, UFR de Physique, Universite de Lille 1, F-59655 Villeneuve d' Ascq (France); Akjouj, A [Laboratoire de Dynamique et Structure des Materiaux Moleculaires, UMR CNRS 8024, UFR de Physique, Universite de Lille 1, F-59655 Villeneuve d' Ascq (France); Velasco, V R [Instituto de Ciencia de Materiales de Madrid, CSIC, Sor Juana Ines de la Cruz 3, 28049 Madrid (Spain)
2005-07-13
A theoretical investigation is made of acoustic wave propagation in one-dimensional phononic bandgap structures made of slender tube loops pasted together with slender tubes of finite length according to a Fibonacci sequence. The band structure and transmission spectrum is studied for two particular cases. (i) Symmetric loop structures, which are shown to be equivalent to diameter-modulated slender tubes. In this case, it is found that besides the existence of extended and forbidden modes, some narrow frequency bands appear in the transmission spectra inside the gaps as defect modes. The spatial localization of the modes lying in the middle of the bands and at their edges is examined by means of the local density of states. The dependence of the bandgap structure on the slender tube diameters is presented. An analysis of the transmission phase time enables us to derive the group velocity as well as the density of states in these structures. In particular, the stop bands (localized modes) may give rise to unusual (strong normal) dispersion in the gaps, yielding fast (slow) group velocities above (below) the speed of sound. (ii) Asymmetric tube loop structures, where the loops play the role of resonators that may introduce transmission zeros and hence new gaps unnoticed in the case of simple diameter-modulated slender tubes. The Fibonacci scaling property has been checked for both cases (i) and (ii), and it holds for a periodicity of three or six depending on the nature of the substrates surrounding the structure.
17. Surface acoustic wave nebulization facilitating lipid mass spectrometric analysis.
Science.gov (United States)
Yoon, Sung Hwan; Huang, Yue; Edgar, J Scott; Ting, Ying S; Heron, Scott R; Kao, Yuchieh; Li, Yanyan; Masselon, Christophe D; Ernst, Robert K; Goodlett, David R
2012-08-07
Surface acoustic wave nebulization (SAWN) is a novel method to transfer nonvolatile analytes directly from the aqueous phase to the gas phase for mass spectrometric analysis. The lower ion energetics of SAWN and its planar nature make it appealing for analytically challenging lipid samples. This challenge is a result of their amphipathic nature, labile nature, and tendency to form aggregates, which readily precipitate clogging capillaries used for electrospray ionization (ESI). Here, we report the use of SAWN to characterize the complex glycolipid, lipid A, which serves as the membrane anchor component of lipopolysaccharide (LPS) and has a pronounced tendency to clog nano-ESI capillaries. We also show that unlike ESI SAWN is capable of ionizing labile phospholipids without fragmentation. Lastly, we compare the ease of use of SAWN to the more conventional infusion-based ESI methods and demonstrate the ability to generate higher order tandem mass spectral data of lipid A for automated structure assignment using our previously reported hierarchical tandem mass spectrometry (HiTMS) algorithm. The ease of generating SAWN-MS(n) data combined with HiTMS interpretation offers the potential for high throughput lipid A structure analysis.
18. On the wave equation with semilinear porous acoustic boundary conditions
KAUST Repository
Graber, Philip Jameson
2012-05-01
The goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary/interior sources and a nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. The main difficulty in proving the local existence result is that the Neumann boundary conditions experience loss of regularity due to boundary sources. Using an approximation method involving truncated sources and adapting the ideas in Lasiecka and Tataru (1993) [28], we show that the existence of solutions can still be obtained. Second, we prove that under some restrictions on the source terms, then the local solution can be extended to be global in time. In addition, it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ODE and depends on the behavior of the damping terms. In several situations, the obtained ODE can be easily solved and the decay rates can be given explicitly. Third, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution ceases to exists and blows up in finite time. Moreover, in either the absence of the interior source or the boundary source, then we prove that the solution is unbounded and grows as an exponential function. © 2012 Elsevier Inc.
19. Following butter flavour deterioration with an acoustic wave sensor.
Science.gov (United States)
Gaspar, Cláudia R B S; Gomes, M Teresa S R
2012-09-15
Off-flavours develop naturally in butter and the process is accelerated by heat. An acoustic wave sensor was used to detect the aroma compounds evolved from heated butter and the results have shown that registered marked changes were coincident to odour changes detected by sensory analysis. The flavour compounds have also been analysed by GC/MS for identification. The response of the sensor was fully characterized in terms of the sensitivity to each of the identified compounds, and sensitivities of the system SPME/sensor were compared with the sensitivities of the system SPME/GC/MS. It was found that the sensor analytical system was more sensitive to methylketones than to fatty acids. The SPME/GC/MS system also showed the highest sensitivity to 2-heptanone, followed by 2-nonanone, but third place was occupied by undecanone and butanoic acid, to which the sensor showed moderate sensitivity. 2-heptanone was found to be an appropriate model compound to follow odour changes till the 500 h, and the lower sensitivity of the sensor to butanoic acid showed to be a positive characteristic, as saturation was prevented, and other more subtle changes in the flavour could be perceived. Copyright © 2012 Elsevier B.V. All rights reserved.
20. Acoustic Wave Velocity as a Selection Trait in Eucalyptus nitens
Directory of Open Access Journals (Sweden)
David Blackburn
2014-04-01
Full Text Available Previous studies in Eucalyptus nitens have revealed favourable genetic correlations exist between acoustic wave velocity (AWV in standing trees and modulus of elasticity (MOE, which can determine the suitability of trees for structural timber and/or engineered wood products. This study investigates the strength and stability of genetic variation in standing tree AWV across a range of environments in Tasmania, where there are a number of large plantation estates and breeding trials. Trees under study were from open-pollinated progeny trials established in 1993. Across sites, for standing tree AWV the ranking of E. nitens races did not change and within-race additive genetic correlations were strong (0.61 to 0.99. Heritabilities (0.16 to 0.74 and coefficients of additive genetic variation (2.6 to 4.8 were moderate for this trait. Correlations between standing tree AWV and both basic density and diameter at breast height (DBH were favourable. Results indicate that there is potential to improve MOE in E. nitens through the exploitation of genetic variation in AWV among and within races, the expression of genetic variation in AWV is relatively stable across different growing environments, and past selection for basic density and growth in pulpwood breeding programs is unlikely to have adversely affected MOE.
1. Employing Acoustic Emission for Monitoring Oil Film Regimes
Directory of Open Access Journals (Sweden)
David Mba
2013-07-01
Full Text Available The major purpose of a gear lubricant is to provide adequate oil film thickness to reduce and prevent gear tooth surface failures. Real time monitoring for gear failures is important in order to predict and prevent unexpected failures which would have a negative impact on the efficiency, performance and safety of the gearbox. This paper presents experimental results on the influence of specific oil film thickness on Acoustic Emission (AE activity for operational helical gears. Variation in film thickness during operations was achieved by spraying liquid nitrogen onto the rotating gear wheel. The experimental results demonstrated a clear relationship between the root mean square (r.m.s value of the AE signal and the specific film thickness. The findings demonstrate the potential of Acoustic Emission technology to quantify lubrication regimes on operational gears.
2. Controlling the transmission of ultrahigh frequency bulk acoustic waves in silicon by 45° mirrors.
Science.gov (United States)
Wang, Shengxiang; Gao, Jiaming; Carlier, Julien; Campistron, Pierre; NDieguene, Assane; Guo, Shishang; Matar, Olivier Bou; Dorothee, Debavelaere-Callens; Nongaillard, Bertrand
2011-07-01
In this paper, we present a feasible microsystem in which the direction of localized ultrahigh frequency (∼1GHz) bulk acoustic wave can be controlled in a silicon wafer. Deep etching technology on the silicon wafer makes it possible to achieve high aspect ratio etching patterns which can be used to control bulk acoustic wave to transmit in the directions parallel to the surface of the silicon wafer. Passive 45° mirror planes obtained by wet chemical etching were employed to reflect the bulk acoustic wave. Zinc oxide (ZnO) thin film transducers were deposited by radio frequency sputtering with a thickness of about 1μm on the other side of the wafer, which act as emitter/receptor after aligned with the mirrors. Two opponent vertical mirrors were inserted between the 45° mirrors to guide the transmission of the acoustic waves. The propagation of the bulk acoustic wave was studied with simulations and the characterization of S(21) scattering parameters, indicating that the mirrors were efficient to guide bulk acoustic waves in the silicon wafer. Copyright © 2011 Elsevier B.V. All rights reserved.
3. Effect of magnetic quantization on ion acoustic waves ultra-relativistic dense plasma
Science.gov (United States)
Javed, Asif; Rasheed, A.; Jamil, M.; Siddique, M.; Tsintsadze, N. L.
2017-11-01
In this paper, we have studied the influence of magnetic quantization of orbital motion of the electrons on the profile of linear and nonlinear ion-acoustic waves, which are propagating in the ultra-relativistic dense magneto quantum plasmas. We have employed both Thomas Fermi and Quantum Magneto Hydrodynamic models (along with the Poisson equation) of quantum plasmas. To investigate the large amplitude nonlinear structure of the acoustic wave, Sagdeev-Pseudo-Potential approach has been adopted. The numerical analysis of the linear dispersion relation and the nonlinear acoustic waves has been presented by drawing their graphs that highlight the effects of plasma parameters on these waves in both the linear and the nonlinear regimes. It has been noticed that only supersonic ion acoustic solitary waves can be excited in the above mentioned quantum plasma even when the value of the critical Mach number is less than unity. Both width and depth of Sagdeev potential reduces on increasing the magnetic quantization parameter η. Whereas the amplitude of the ion acoustic soliton reduces on increasing η, its width appears to be directly proportional to η. The present work would be helpful to understand the excitation of nonlinear ion-acoustic waves in the dense astrophysical environments such as magnetars and in intense-laser plasma interactions.
4. Snow Slab Failure Due to Biot-Type Acoustic Wave Propagation
OpenAIRE
Sidler Rolf
2014-01-01
Even though seismic methods are among the most used geophysical methods today their application in snow has been sparse. This might be related to the fact that commonly observed wave velocity attenuation and reflection coefficients can not be well explained by the widely used elastic or visco elastic models for wave propagation. Biot's well established model of wave propagation in porous media instead is much better suited to describe acoustic wave propagation in snow. This model predicts als...
5. Integration of thin film giant magnetoimpedance sensor and surface acoustic wave transponder
KAUST Repository
Li, Bodong
2012-03-09
Passive and remote sensing technology has many potential applications in implantable devices, automation, or structural monitoring. In this paper, a tri-layer thin film giant magnetoimpedance (GMI) sensor with the maximum sensitivity of 16%/Oe and GMI ratio of 44% was combined with a two-port surface acoustic wave(SAW) transponder on a common substrate using standard microfabrication technology resulting in a fully integrated sensor for passive and remote operation. The implementation of the two devices has been optimized by on-chip matching circuits. The measurement results clearly show a magnetic field response at the input port of the SAW transponder that reflects the impedance change of the GMI sensor.
6. Temporal window system: A new approach for dynamic detection application to surface acoustic wave gas sensors
Energy Technology Data Exchange (ETDEWEB)
Bordieu, C.; Rebiere, D.; Pistre, J. [and others
1996-12-31
Pattern recognition techniques based on artificial neural networks are now frequently used with good results for gas sensor signal processing (this includes the detection, the identification and the quantification of gases). In the literature, data sets needed for neural networks are practically always built with steady state sensor responses. This situation prevents these techniques from being used in real time applications. Nevertheless, for example in the case of surface acoustic wave (SAW) gas sensors, because of quite long response times due to kinetic factors concerning the gas adsorption and because gases are sometimes extremely dangerous and/or toxic (NO{sub x}, SO{sub 2}, organophosphorus compounds,...), the detection speed is an essential parameter and hence must be monitored in a real time mode. The purpose of this paper is to propose a new dynamic approach and to illustrate it with SAW sensor responses.
7. Surface Acoustic Wave (SAW-Enhanced Chemical Functionalization of Gold Films
Directory of Open Access Journals (Sweden)
Gina Greco
2017-10-01
Full Text Available Surface chemical and biochemical functionalization is a fundamental process that is widely applied in many fields to add new functions, features, or capabilities to a material’s surface. Here, we demonstrate that surface acoustic waves (SAWs can enhance the chemical functionalization of gold films. This is shown by using an integrated biochip composed by a microfluidic channel coupled to a surface plasmon resonance (SPR readout system and by monitoring the adhesion of biotin-thiol on the gold SPR areas in different conditions. In the case of SAW-induced streaming, the functionalization efficiency is improved ≈ 5 times with respect to the case without SAWs. The technology here proposed can be easily applied to a wide variety of biological systems (e.g., proteins, nucleic acids and devices (e.g., sensors, devices for cell cultures.
8. Coupling of an acoustic wave to shear motion due to viscous heating
Energy Technology Data Exchange (ETDEWEB)
Liu, Bin; Goree, J. [Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242 (United States)
2016-07-15
Viscous heating due to shear motion in a plasma can result in the excitation of a longitudinal acoustic wave, if the shear motion is modulated in time. The coupling mechanism is a thermal effect: time-dependent shear motion causes viscous heating, which leads to a rarefaction that can couple into a longitudinal wave, such as an acoustic wave. This coupling mechanism is demonstrated in an electrostatic three-dimensional (3D) simulation of a dusty plasma, in which a localized shear flow is initiated as a pulse, resulting in a delayed outward propagation of a longitudinal acoustic wave. This coupling effect can be profound in plasmas that exhibit localized viscous heating, such as the dusty plasma we simulated using parameters typical of the PK-4 experiment. We expect that a similar phenomenon can occur with other kinds of plasma waves.
9. Observation of dust acoustic shock wave in a strongly coupled dusty plasma
Energy Technology Data Exchange (ETDEWEB)
Sharma, Sumita K., E-mail: [email protected]; Boruah, A.; Nakamura, Y.; Bailung, H., E-mail: [email protected] [Physical Sciences Division, Institute of Advanced Study in Science and Technology, Guwahati 781035 (India)
2016-05-15
Dust acoustic shock wave is observed in a strongly coupled laboratory dusty plasma. A supersonic flow of charged microparticles is allowed to perturb a stationary dust fluid to excite dust acoustic shock wave. The evolution process beginning with steepening of initial wave front and then formation of a stable shock structure is similar to the numerical results of the Korteweg-de Vries-Burgers equation. The measured Mach number of the observed shock wave agrees with the theoretical results. Reduction of shock amplitude at large distances is also observed due to the dust neutral collision and viscosity effects. The dispersion relation and the spatial damping of a linear dust acoustic wave are also measured and compared with the relevant theory.
10. High-frequency programmable acoustic wave device realized through ferroelectric domain engineering
Science.gov (United States)
Ivry, Yachin; Wang, Nan; Durkan, Colm
2014-03-01
Surface acoustic wave devices are extensively used in contemporary wireless communication devices. We used atomic force microscopy to form periodic macroscopic ferroelectric domains in sol-gel deposited lead zirconate titanate, where each ferroelectric domain is composed of many crystallites, each of which contains many microscopic ferroelastic domains. We examined the electro-acoustic characteristics of the apparatus and found a resonator behavior similar to that of an equivalent surface or bulk acoustic wave device. We show that the operational frequency of the device can be tailored by altering the periodicity of the engineered domains and demonstrate high-frequency filter behavior (>8 GHz), allowing low-cost programmable high-frequency resonators.
11. Experimental verification of theoretical equations for acoustic radiation force on compressible spherical particles in traveling waves
Science.gov (United States)
Johnson, Kennita A.; Vormohr, Hannah R.; Doinikov, Alexander A.; Bouakaz, Ayache; Shields, C. Wyatt; López, Gabriel P.; Dayton, Paul A.
2016-05-01
Acoustophoresis uses acoustic radiation force to remotely manipulate particles suspended in a host fluid for many scientific, technological, and medical applications, such as acoustic levitation, acoustic coagulation, contrast ultrasound imaging, ultrasound-assisted drug delivery, etc. To estimate the magnitude of acoustic radiation forces, equations derived for an inviscid host fluid are commonly used. However, there are theoretical predictions that, in the case of a traveling wave, viscous effects can dramatically change the magnitude of acoustic radiation forces, which make the equations obtained for an inviscid host fluid invalid for proper estimation of acoustic radiation forces. To date, experimental verification of these predictions has not been published. Experimental measurements of viscous effects on acoustic radiation forces in a traveling wave were conducted using a confocal optical and acoustic system and values were compared with available theories. Our results show that, even in a low-viscosity fluid such as water, the magnitude of acoustic radiation forces is increased manyfold by viscous effects in comparison with what follows from the equations derived for an inviscid fluid.
12. Analysis of the effect of a rectangular cavity resonator on acoustic wave transmission in a waveguide
Science.gov (United States)
Porter, R.; Evans, D. V.
2017-11-01
The transmission of acoustic waves along a two-dimensional waveguide which is coupled through an opening in its wall to a rectangular cavity resonator is considered. The resonator acts as a classical band-stop filter, significantly reducing acoustic transmission across a range of frequencies. Assuming wave frequencies below the first waveguide cut-off, the solution for the reflected and transmitted wave amplitudes is formulated exactly within the framework of inviscid linear acoustics. The main aim of the paper is to develop an approximation in closed form for reflected and transmitted amplitudes when the gap in the thin wall separating the waveguide and the cavity resonator is assumed to be small. This approximation is shown to accurately capture the effect of all cavities resonances, not just the fundamental Helmholtz resonance. It is envisaged this formula (and more generally the mathematical approach adopted) could be used in the development of acoustic metamaterial devices containing resonator arrays.
13. Ultrafast high strain rate acoustic wave measurements at high static pressure in a diamond anvil cell
Energy Technology Data Exchange (ETDEWEB)
Armstrong, M; Crowhurst, J; Reed, E; Zaug, J
2008-02-04
We have used sub-picosecond laser pulses to launch ultra-high strain rate ({approx} 10{sup 9} s{sup -1}) nonlinear acoustic waves into a 4:1 methanol-ethanol pressure medium which has been precompressed in a standard diamond anvil cell. Using ultrafast interferometry, we have characterized acoustic wave propagation into the pressure medium at static compression up to 24 GPa. We find that the velocity is dependent on the incident laser fluence, demonstrating a nonlinear acoustic response which may result in shock wave behavior. We compare our results with low strain, low strain-rate acoustic data. This technique provides controlled access to regions of thermodynamic phase space that are otherwise difficult to obtain.
14. Proportional monitoring of the acoustic emission in crypto-conditions
Directory of Open Access Journals (Sweden)
Petr Dostál
2011-01-01
15. An Adaptive Framework for Acoustic Monitoring of Potential Hazards
Directory of Open Access Journals (Sweden)
Potamitis Ilyas
2009-01-01
Full Text Available Robust recognition of general audio events constitutes a topic of intensive research in the signal processing community. This work presents an efficient methodology for acoustic surveillance of atypical situations which can find use under different acoustic backgrounds. The primary goal is the continuous acoustic monitoring of a scene for potentially hazardous events in order to help an authorized officer to take the appropriate actions towards preventing human loss and/or property damage. A probabilistic hierarchical scheme is designed based on Gaussian mixture models and state-of-the-art sound parameters selected through extensive experimentation. A feature of the proposed system is its model adaptation loop that provides adaptability to different sound environments. We report extensive experimental results including installation in a real environment and operational detection rates for three days of function on a 24 hour basis. Moreover, we adopt a reliable testing procedure that demonstrates high detection rates as regards average recognition, miss probability, and false alarm rates.
16. Characterization of ablated porcine bone and muscle using laser-induced acoustic wave method for tissue differentiation
Science.gov (United States)
Nguendon, Hervé K.; Faivre, Neige; Meylan, Bastian; Shevchik, Sergey; Rauter, Georg; Guzman, Raphael; Cattin, Philippe C.; Wasmer, Kilian; Zam, Azhar
2017-07-01
A high power pulsed laser with millisecond pulse was used to interact with a bone and muscle of porcine, initiating an acoustic wave. We start to describe principle of laser ablation follows by the acoustic wave generation. Then, we present the characterization of these wave features for laser surgery applications.
17. Acoustic-wave generation in the process of CO2-TEA-laser-radiation interaction with metal targets in air
Science.gov (United States)
Apostol, Ileana; Teodorescu, G.; Serbanescu-Oasa, Anca; Dragulinescu, Dumitru; Chis, Ioan; Stoian, Razvan
1995-03-01
Laser radiation interaction with materials is a complex process in which creation of acoustic waves or stress waves is a part of it. As a function of the laser radiation energy and intensity incident on steel target surface ultrasound signals were registered and studied. Thermoelastic, ablation and breakdown mechanisms of generation of acoustic waves were analyzed.
18. Aero-acoustic Measurement and Monitoring of Dynamic Pressure Fields Project
Data.gov (United States)
National Aeronautics and Space Administration — This innovative and practical measurement and monitoring system optimally defines dynamic pressure fields, including sound fields. It is based on passive acoustic...
19. On an Acoustic Wave Equation Arising in Non-Equilibrium Gasdynamics. Classroom Notes
Science.gov (United States)
Chandran, Pallath
2004-01-01
The sixth-order wave equation governing the propagation of one-dimensional acoustic waves in a viscous, heat conducting gaseous medium subject to relaxation effects has been considered. It has been reduced to a system of lower order equations corresponding to the finite speeds occurring in the equation, following a method due to Whitham. The lower…
20. The first radial-mode Lorentzian Landau damping of dust acoustic space-charge waves
Energy Technology Data Exchange (ETDEWEB)
Lee, Myoung-Jae [Department of Physics and Research Institute for Natural Sciences, Hanyang University, Seoul 04763 (Korea, Republic of); Jung, Young-Dae, E-mail: [email protected] [Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, 110 8th Street, Troy, New York 12180-3590 (United States); Department of Applied Physics and Department of Bionanotechnology, Hanyang University, Ansan, Kyunggi-Do 15588 (Korea, Republic of)
2016-05-15
The dispersion properties and the first radial-mode Lorentzian Landau damping of a dust acoustic space-charge wave propagating in a cylindrical waveguide dusty plasma which contains nonthermal electrons and ions are investigated by employing the normal mode analysis and the method of separation of variables. It is found that the frequency of dust acoustic space-charge wave increases as the wave number increases as well as the radius of cylindrical plasma does. However, the nonthermal property of the Lorentzian plasma is found to suppress the wave frequency of the dust acoustic space-charge wave. The Landau damping rate of the dust acoustic space-charge wave is derived in a cylindrical waveguide dusty plasma. The damping of the space-charge wave is found to be enhanced as the radius of cylindrical plasma and the nonthermal property increase. The maximum Lorentzian Landau damping rate is also found in a cylindrical waveguide dusty plasma. The variation of the wave frequency and the Landau damping rate due to the nonthermal character and geometric effects are also discussed.
1. FDTD model of acoustic wave interaction with soft targets | Ikata ...
African Journals Online (AJOL)
Our interest has been on the character of the acoustic field inside the target and the interaction parameters which influence it. The numerical simulations suggest that for an acoustically denser target the interior field consist of alternate bands of high-(and low-) pressure, though in a narrow cylindrical target the interior is ...
2. Amplification of surface acoustic waves by transverse electric current in piezoelectric semiconductors
DEFF Research Database (Denmark)
Gulyaev, Yuri V.
1974-01-01
It is shown that the principal characteristic feature of the surface acoustic waves in piezoelectrics—the presence of an alternating electric field transverse to the surface, which can be of the same order of magnitude as the longitudinal field—may not only give rise to the known transverse...... acoustoelectric effect but also lead to amplification of surface acoustic waves by electron drift perpendicular to the surface. For Love waves in a piezoelectric semiconductor film on a highly conducting substrate, the amplification coefficient is found and the conditions necessary for amplification...
3. Energy Properties of Ion Acoustic Waves in Stable and Unstable Plasmas
DEFF Research Database (Denmark)
Jensen, Vagn Orla; Lynov, Jens-Peter
1979-01-01
acoustic waves that are growing or damped in space the time average of the sum of the potential and the kinetic energy density is independent of position. Energy absorption spectra in particle velocity space are calculated; they are relatively broad and complicated functions. This shows that plasma ions......Energy exchange between potential energy and ion kinetic energy in an ion acoustic wave is considered. In order to investigate the linear Landau damping or growth, the energy is calculated by use of first‐order quantities only so that nonlinear effects are not involved. It is found that for ion...... of all velocities exchange energy with the wave....
4. Finite element analysis of surface acoustic waves in high aspect ratio electrodes
DEFF Research Database (Denmark)
Dühring, Maria Bayard; Laude, Vincent; Khelif, Abdelkrim
2008-01-01
This paper elaborates on how the finite element method is employed to model surface acoustic waves generated by high aspect ratio electrodes and their interaction with optical waves in a waveguide. With a periodic model it is shown that these electrodes act as a mechanical resonator which slows...... down the SAWvelocity because of mechanical energy storage. A finite model is furthermore employed to study the acousto-optical interaction and shows that it is possible to get a bigger change in effective refractive index with these surface acoustic waves compared to using conventional interdigital...
5. Modeling and Analysis of Lateral Propagation of Surface Acoustic Waves Including Coupling Between Different Waves.
Science.gov (United States)
Zhang, Benfeng; Han, Tao; Tang, Gongbin; Zhang, Qiaozhen; Omori, Tatsuya; Hashimoto, Ken-Ya
2017-09-01
This paper discusses lateral propagation of surface acoustic waves (SAWs) in periodic grating structures when two types of SAWs exist simultaneously and are coupled. The thin plate model proposed by the authors is extended to include the coupling between two different SAW modes. First, lateral SAW propagation in an infinitely long periodic grating is modeled and discussed. Then, the model is applied to the Al-grating/42° YX-LiTaO3 (42-LT) substrate structure, and it is shown that the slowness curve shape changes from concave to convex with the Al grating thickness. The transverse responses are also analyzed on an infinitely long interdigital transducer on the structure, and good agreement is achieved between the present and the finite-element method analyses. Finally, SAW resonators are fabricated on the Cu grating/42-LT substrate structure, and it is experimentally verified that the slowness curve shape of the shear horizontal SAW changes with the Cu thickness.
6. The Effect of Dust Particles on Ion Acoustic Solitary Waves in a Dusty Plasma
Directory of Open Access Journals (Sweden)
Cheong Rim Choi
2004-09-01
Full Text Available In this paper we have examined the effect of dust charge density on nonlinear ion acoustic solitary wave which propagates obliquely with respect to the external magnetic field in a dusty plasma. For the dusty charge density below a critical value, the Sagdeev potential Ψ(n has a singular point in the region n<1, where n is the ion number density divided by its equilibrium number density. If there exists a dust charge density over the critical value, the Sagdeev potential becomes a finite function in the region n<1, which means that there may exist the rarefactive ion acoustic solitary wave. By expanding the Sagdeev potential in the small amplitude limit up to δ n4 near n=1, we find the solution of ion acoustic solitary wave. Therefore we suggest that the dust charge density plays an important role in generating the rarefactive solitary wave.
7. A Four-Quadrant PVDF Transducer for Surface Acoustic Wave Detection
Directory of Open Access Journals (Sweden)
Zhi Chen
2012-08-01
Full Text Available In this paper, a polyvinylidene fluoride (PVDF piezoelectric transducer was developed to detect laser-induced surface acoustic waves in a SiO2-thin film–Si-substrate structure. In order to solve the problems related to, firstly, the position of the probe, and secondly, the fact that signals at different points cannot be detected simultaneously during the detection process, a four-quadrant surface acoustic wave PVDF transducer was designed and constructed for the purpose of detecting surface acoustic waves excited by a pulse laser line source. The experimental results of the four-quadrant piezoelectric detection in comparison with the commercial nanoindentation technology were consistent, the relative error is 0.56%, and the system eliminates the piezoelectric surface wave detection direction deviation errors, improves the accuracy of the testing system by 1.30%, achieving the acquisition at the same time at different testing positions of the sample.
8. The viability of acoustic tomography in monitoring the circulation of Monterey Bay
OpenAIRE
James H Miller; Rowan, Theresa M.; Ehret, Laura L.; Dees, Robert C.
1990-01-01
This report presents the results of a fifteen month study on the viability of acoustic tomography in monitoring the circulation of Monterey Bay, California. The basis for ocean acoustic tomography is the measurement of travel times of coded acoustic signals between the transceivers. The sound speed field and current structure can be inferred from the fluctuations in the travel times. However, the extreme bathymetry of the Monterey Submarine Canyon complicates the acoustic transmissions in the...
9. Continuous micro-vortex-based nanoparticle manipulation via focused surface acoustic waves.
Science.gov (United States)
Collins, David J; Ma, Zhichao; Han, Jongyoon; Ai, Ye
2016-12-20
Despite increasing demand in the manipulation of nanoscale objects for next generation biological and industrial processes, there is a lack of methods for reliable separation, concentration and purification of nanoscale objects. Acoustic methods have proven their utility in contactless manipulation of microscale objects mainly relying on the acoustic radiation effect, though the influence of acoustic streaming has typically prevented manipulation at smaller length scales. In this work, however, we explicitly take advantage of the strong acoustic streaming in the vicinity of a highly focused, high frequency surface acoustic wave (SAW) beam emanating from a series of focused 6 μm substrate wavelength interdigital transducers patterned on a piezoelectric lithium niobate substrate and actuated with a 633 MHz sinusoidal signal. This streaming field serves to focus fluid streamlines such that incoming particles interact with the acoustic field similarly regardless of their initial starting positions, and results in particle displacements that would not be possible with a travelling acoustic wave force alone. This streaming-induced manipulation of nanoscale particles is maximized with the formation of micro-vortices that extend the width of the microfluidic channel even with the imposition of a lateral flow, occurring when the streaming-induced flow velocities are an order of magnitude larger than the lateral one. We make use of this acoustic streaming to demonstrate the continuous and differential focusing of 100 nm, 300 nm and 500 nm particles.
10. Time fractional effect on ion acoustic shock waves in ion-pair plasma
Science.gov (United States)
Abdelwahed, H. G.; El-Shewy, E. K.; Mahmoud, A. A.
2016-06-01
The nonlinear properties of ion acoustic shock waves are studied. The Burgers equation is derived and converted into the time fractional Burgers equation by Agrawal's method. Using the Adomian decomposition method, shock wave solutions of the time fractional Burgers equation are constructed. The effect of the time fractional parameter on the shock wave properties in ion-pair plasma is investigated. The results obtained may be important in investigating the broadband electrostatic shock noise in D- and F-regions of Earth's ionosphere.
11. Influence of high frequency electric field on the dispersion of ion-acoustic waves in plasma
Energy Technology Data Exchange (ETDEWEB)
Turky, A.; Cercek, M.; Tavzes, R.
1981-01-01
The modification of the ion-acoustic wave dispersion under the action of a high frequency electric field was studied experimentally, the wave propagating along and against the plasma stream. The frequency of the field amounted to approximately half the electron plasma frequency. It was found that the phase velocity of the ion wave and the plasma drift velocity decrease as the effective high frequency field power increases.
12. Effects of ion-atom collisions on the propagation and damping of ion-acoustic waves
DEFF Research Database (Denmark)
Andersen, H.K.; D'Angelo, N.; Jensen, Vagn Orla
1968-01-01
Experiments are described on ion-acoustic wave propagation and damping in alkali plasmas of various degrees of ionization. An increase of the ratio Te/Ti from 1 to approximately 3-4, caused by ion-atom collisions, results in a decrease of the (Landau) damping of the waves. At high gas pressure and....../or low wave frequency a "fluid" picture adequately describes the experimental results....
13. Volumetric nature of synchronization of the dust acoustic wave with an external modulation
Science.gov (United States)
Williams, Jeremiah
2017-10-01
The dust acoustic wave (also known as the dust density wave) is low-frequency, longitudinal mode that propagates through the dust component of the dusty plasma system and is self-excited by the free energy from the ion streaming through the dust component. In the laboratory setting, the majority of the self excited dust acoustic waves that are observed are nonlinear, which allows for detailed studies of the nonlinear properties of this wave mode at the kinetic level. One such nonlinear process is synchronization, which is observed when the self-excited dust acoustic wave mode couples with and adjusts to an externally applied modulation. In this poster, we will present volumetric measurements of naturally occurring dust acoustic waves in an rf discharge as it becomes synchronous with an externally applied modulation in the spatial and temporal domains by applying a time-resolved Hilbert Transform to high-speed video imaging of the wave mode over a range of experimental conditions. This work is supported by US National Science Foundation through Grant No. PHY-1615420.
14. Ionospheric acoustic and gravity wave activity above low-latitude thunderstorms
Energy Technology Data Exchange (ETDEWEB)
Lay, Erin Hoffmann [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2017-01-30
In this report, we study the correlation between thunderstorm activity and ionospheric gravity and acoustic waves in the low-latitude ionosphere. We use ionospheric total electron content (TEC) measurements from the Low Latitude Ionospheric Sensor Network (LISN) and lightning measurements from the World- Wide Lightning Location Network (WWLLN). We find that ionospheric acoustic waves show a strong diurnal pattern in summer, peaking in the pre-midnight time period. However, the peak magnitude does not correspond to thunderstorm area, and the peak time is significantly after the peak in thunderstorm activity. Wintertime acoustic wave activity has no discernable pattern in these data. The coverage area of ionospheric gravity waves in the summer was found to increase with increasing thunderstorm activity. Wintertime gravity wave activity has an observable diurnal pattern unrelated to thunderstorm activity. These findings show that while thunderstorms are not the only, or dominant source of ionospheric perturbations at low-latitudes, they do have an observable effect on gravity wave activity and could be influential in acoustic wave activity.
15. Acoustic monitoring method and system in laser-induced optical breakdown (LIOB)
Science.gov (United States)
O'Donnell, Matthew [Ann Arbor, MI; Ye, Jing Yong [Ann Arbor, MI; Norris, Theodore B [Dexter, MI; Baker, Jr., James R.; Balogh, Lajos P [Ann Arbor, MI; Milas, Susanne M [Ann Arbor, MI; Emelianov, Stanislav Y [Ann Arbor, MI; Hollman, Kyle W [Fenton, MI
2008-05-06
An acoustic monitoring method and system in laser-induced optical breakdown (LIOB) provides information which characterize material which is broken down, microbubbles in the material, and/or the microenvironment of the microbubbles. In one embodiment of the invention, femtosecond laser pulses are focused just inside the surface of a volume of aqueous solution which may include dendrimer nanocomposite (DNC) particles. A tightly focused, high frequency, single-element ultrasonic transducer is positioned such that its focus coincides axially and laterally with this laser focus. When optical breakdown occurs, a microbubble forms and a shock or pressure wave is emitted (i.e., acoustic emission). In addition to this acoustic signal, the microbubble may be actively probed with pulse-echo measurements from the same transducer. After the microbubble forms, received pulse-echo signals have an extra pulse, describing the microbubble location and providing a measure of axial microbubble size. Wavefield plots of successive recordings illustrate the generation, growth, and collapse of microbubbles due to optical breakdown. These same plots can also be used to quantify LIOB thresholds.
16. Health monitoring of Ceramic Matrix Composites from waveform-based analysis of Acoustic Emission
Directory of Open Access Journals (Sweden)
Maillet Emmanuel
2015-01-01
Full Text Available Ceramic Matrix Composites (CMCs are anticipated for use in the hot section of aircraft engines. Their implementation requires the understanding of the various damage modes that are involved and their relation to life expectancy. Acoustic Emission (AE has been shown to be an efficient technique for monitoring damage evolution in CMCs. However, only a waveform-based analysis of AE can offer the possibility to validate and precisely examine the recorded AE data with a view to damage localization and identification. The present work fully integrates wave initiation, propagation and acquisition in the analysis of Acoustic Emission waveforms recorded at various sensors, therefore providing more reliable information to assess the relation between Acoustic Emission and damage modes. The procedure allows selecting AE events originating from damage, accurate determination of their location as well as the characterization of effects of propagation on the recorded waveforms. This approach was developed using AE data recorded during tensile tests on carbon/carbon composites. It was then applied to melt-infiltrated SiC/SiC composites.
17. Nonlinear excitation of acoustic modes by large amplitude Alfv\\'en waves in a laboratory plasma
CERN Document Server
Dorfman, S
2013-01-01
The nonlinear three-wave interaction process at the heart of the parametric decay process is studied by launching counter-propagating Alfv\\'en waves from antennas placed at either end of the Large Plasma Device (LAPD). A resonance in the beat wave response produced by the two launched Alfv\\'en waves is observed and is identified as a damped ion acoustic mode based on the measured dispersion relation. Other properties of the interaction including the spatial profile of the beat mode and response amplitude are also consistent with theoretical predictions for a three-wave interaction driven by a non-linear pondermotive force.
18. Unattended Acoustic Sensor Systems for Noise Monitoring in National Parks
Science.gov (United States)
Detection and classification of transient acoustic signals is a difficult problem. The problem is often complicated by factors such as the variety of sources that may be encountered, the presence of strong interference and substantial variations in the acoustic environment. Furthermore, for most applications of transient detection and classification, such as speech recognition and environmental monitoring, online detection and classification of these transient events is required. This is even more crucial for applications such as environmental monitoring as it is often done at remote locations where it is unfeasible to set up a large, general-purpose processing system. Instead, some type of custom-designed system is needed which is power efficient yet able to run the necessary signal processing algorithms in near real-time. In this thesis, we describe a custom-designed environmental monitoring system (EMS) which was specifically designed for monitoring air traffic and other sources of interest in national parks. More specifically, this thesis focuses on the capabilities of the EMS and how transient detection, classification and tracking are implemented on it. The Sparse Coefficient State Tracking (SCST) transient detection and classification algorithm was implemented on the EMS board in order to detect and classify transient events. This algorithm was chosen because it was designed for this particular application and was shown to have superior performance compared to other algorithms commonly used for transient detection and classification. The SCST algorithm was implemented on an Artix 7 FPGA with parts of the algorithm running as dedicated custom logic and other parts running sequentially on a soft-core processor. In this thesis, the partitioning and pipelining of this algorithm is explained. Each of the partitions was tested independently to very their functionality with respect to the overall system. Furthermore, the entire SCST algorithm was tested in the
19. Tools for automated acoustic monitoring within the R package monitoR
Science.gov (United States)
Katz, Jonathan; Hafner, Sasha D.; Donovan, Therese
2016-01-01
The R package monitoR contains tools for managing an acoustic-monitoring program including survey metadata, template creation and manipulation, automated detection and results management. These tools are scalable for use with small projects as well as larger long-term projects and those with expansive spatial extents. Here, we describe typical workflow when using the tools in monitoR. Typical workflow utilizes a generic sequence of functions, with the option for either binary point matching or spectrogram cross-correlation detectors.
20. Tools for automated acoustic monitoring within the R package monitoR
DEFF Research Database (Denmark)
Katz, Jonathan; Hafner, Sasha D.; Donovan, Therese
2016-01-01
The R package monitoR contains tools for managing an acoustic-monitoring program including survey metadata, template creation and manipulation, automated detection and results management. These tools are scalable for use with small projects as well as larger long-term projects and those...... with expansive spatial extents. Here, we describe typical workflow when using the tools in monitoR. Typical workflow utilizes a generic sequence of functions, with the option for either binary point matching or spectrogram cross-correlation detectors....
1. Speech coding, reconstruction and recognition using acoustics and electromagnetic waves
Science.gov (United States)
Holzrichter, John F.; Ng, Lawrence C.
1998-01-01
The use of EM radiation in conjunction with simultaneously recorded acoustic speech information enables a complete mathematical coding of acoustic speech. The methods include the forming of a feature vector for each pitch period of voiced speech and the forming of feature vectors for each time frame of unvoiced, as well as for combined voiced and unvoiced speech. The methods include how to deconvolve the speech excitation function from the acoustic speech output to describe the transfer function each time frame. The formation of feature vectors defining all acoustic speech units over well defined time frames can be used for purposes of speech coding, speech compression, speaker identification, language-of-speech identification, speech recognition, speech synthesis, speech translation, speech telephony, and speech teaching.
2. Propagation of ion-acoustic waves in a warm dusty plasma with electron inertia
Science.gov (United States)
Barman, S. N.; Talukdar, A.
2011-08-01
The KdV equation is derived for weakly nonlinear ion-acoustic waves in an unmagnetized warm dusty plasma with electron inertia. It has been shown that the inclusion of electron inertia and pressure variation of the species not only significantly modifies the basic features (width and amplitude) of dust ion-acoustic solitions, but also introduces a new parametric regime for the existence of positive and negative solitons.
3. Detection of Metallic and Electronic Radar Targets by Acoustic Modulation of Electromagnetic Waves
Science.gov (United States)
2017-07-01
electronic targets within the near field of an ultra-wideband radar antenna operating in the ultra-high frequency band. 15. SUBJECT TERMS radar ...ARL-TR-8076● JULY 2017 US Army Research Laboratory Detection of Metallic and Electronic Radar Targets by Acoustic Modulation of...US Army Research Laboratory Detection of Metallic and Electronic Radar Targets by Acoustic Modulation of Electromagnetic Waves by Gregory
4. Novel Acoustic Wave Microsystems for Biophysical Studies of Cells
Science.gov (United States)
Senveli, Sukru Ufuk
Single cell analysis is an important topic for understanding of diseases. In this understanding, biomechanics approach serves as an important tool as it relates and connects the mechanical properties of biological cells with diseases such as cancer. In this context, analysis methods based on ultrasonics are promising owing to their non-invasive nature and ease of use. However, there is a lack of miniature systems that provide accurate ultrasonic measurements on single cancer cells for diagnostic purposes. The platform presented in this study exploits high frequency acoustic interaction and uses direct coupling of Rayleigh type SAWs with various samples placed inside microcavities to analyze their structural properties. The samples used are aqueous glycerin solutions and polystyrene microbeads for demonstrating proper system operation, and lead up to biological cells. The microcavity is instrumental in trapping a predetermined volume of sample inside and facilitating the interaction of the surface waves with the sample in question via a resonance condition. Ultimately, the resultant SAW reaching the output transducer incurs a phase delay due to its interaction with the sample in the microcavity. The system operates in a different manner compared to similar systems as a result of multiple wave reflections in the small volume and coupling back to the piezoelectric substrate. The proposed microsystem was first analyzed using finite element methods. Liquid and solid media were modeled by considering frequency dependent characteristics. Similarly, mechanical behavior of cells with respect to different conditions is considered, and biological cells are modeled accordingly. Prototype devices were fabricated on quartz and lithium niobate in a cleanroom environment. Process steps were optimized separately for devices with microcavities. Precise fabrication, alignment, and bonding of PDMS microchannels were carried out. Soft microprobes were fabricated out of SU-8, a
5. Computational simulation in architectural and environmental acoustics methods and applications of wave-based computation
CERN Document Server
Sakamoto, Shinichi; Otsuru, Toru
2014-01-01
This book reviews a variety of methods for wave-based acoustic simulation and recent applications to architectural and environmental acoustic problems. Following an introduction providing an overview of computational simulation of sound environment, the book is in two parts: four chapters on methods and four chapters on applications. The first part explains the fundamentals and advanced techniques for three popular methods, namely, the finite-difference time-domain method, the finite element method, and the boundary element method, as well as alternative time-domain methods. The second part demonstrates various applications to room acoustics simulation, noise propagation simulation, acoustic property simulation for building components, and auralization. This book is a valuable reference that covers the state of the art in computational simulation for architectural and environmental acoustics.
6. Solitonic, periodic and quasiperiodic behaviors of dust ion acoustic waves in superthermal plasmas
Energy Technology Data Exchange (ETDEWEB)
Saha, Asit, E-mail: [email protected] [Department of Mathematics, Sikkim Manipal Institute of Technology, Majitar, Rangpo, East-Sikkim (India); Chatterjee, Prasanta, E-mail: [email protected] [Department of Mathematics, Siksha Bhavana, Visva Bharati University, Santiniketan (India)
2015-08-15
The solitonic, periodic, and quasiperiodic behaviors of dust ion acoustic waves in superthermal plasmas with q-nonextensive electrons are studied using the bifurcation theory of planar dynamical systems through direct approach. Using a Galilean transformation, model equations are transformed to a Hamiltonian system involving electrostatic potential. The existence of solitary and periodic waves is shown for the unperturbed Hamiltonian system. Analytical forms of these waves are presented depending on physical parameters q and μ. The effects of q and μ are studied on characteristics of nonlinear dust ion acoustic solitary and periodic waves. It is observed that parameters q and μ significantly influence the characteristics of nonlinear dust ion acoustic solitary and periodic structures. Considering an external periodic perturbation, the quasiperiodic behavior of the perturbed Hamiltonian system for dust ion acoustic waves is studied. It is seen that the unperturbed Hamiltonian system has the solitary and periodic wave solutions whereas the perturbed Hamiltonian system has quasiperiodic motion for same values of parameters q,μ and v. (author)
7. Anomalous width variation of rarefactive ion acoustic solitary waves in the context of auroral plasmas
Directory of Open Access Journals (Sweden)
S. S. Ghosh
2004-01-01
Full Text Available The presence of dynamic, large amplitude solitary waves in the auroral regions of space is well known. Since their velocities are of the order of the ion acoustic speed, they may well be considered as being generated from the nonlinear evolution of ion acoustic waves. However, they do not show the expected width-amplitude correlation for K-dV solitons. Recent POLAR observations have actually revealed that the low altitude rarefactive ion acoustic solitary waves are associated with an increase in the width with increasing amplitude. This indicates that a weakly nonlinear theory is not appropriate to describe the solitary structures in the auroral regions. In the present work, a fully nonlinear analysis based on Sagdeev pseudopotential technique has been adopted for both parallel and oblique propagation of rarefactive solitary waves in a two electron temperature multi-ion plasma. The large amplitude solutions have consistently shown an increase in the width with increasing amplitude. The width-amplitude variation profile of obliquely propagating rarefactive solitary waves in a magnetized plasma have been compared with the recent POLAR observations. The width-amplitude variation pattern is found to fit well with the analytical results. It indicates that a fully nonlinear theory of ion acoustic solitary waves may well explain the observed anomalous width variations of large amplitude structures in the auroral region.
8. The Behavior of Multiple Acoustic Waves in the Lakes Bottom Sediments.
Science.gov (United States)
Krylov, P.; Nourgaliev, D. K.; Yasonov, P.
2016-12-01
Seismic studies are used for various tasks, such as the study of the bottom sediments properties, finding sunken objects, reconstruction the reservoir history, etc. Multiple acoustic waves are an enormous obstacle in obtaining full seismic record. Multiples from the bottom of a body of water (the surface of the base of water and the rock or sediment beneath it) and the air-water surface are common in lake seismic data. Multiple reflections on the seismic cross-sections are usually located on the double distance from the air/water surface. However, sometime multiple reflections from liquid deposits cannot be generated or they reflected from the deeper horizons. It is observed the phenomenon of changes in reflectance of the water/weakly consolidated sediments acoustic boundary under the influence of the acoustic wave. This phenomenon lies in the fact that after the first acoustic impact and reflection of acoustic wave for some time the reflectance of this boundary remains close to 0. This event on a cross-section can explain by the short-term changes in the properties of bottom sediments under the influence of shock? acoustic wave, with a further reduction of these properties to the next wave generation (generation period of 2 seconds). Perhaps in these deposits occurs thixotropic process. The paper presents the seismic acoustic cross-sections of Lake Balkhash (Kazakhstan), Turgoyak (Russia). The work was carried out according to the Russia Government's Program of Competitive Growth of Kazan Federal University, supported by the grant provided to the Kazan State University for performing the state program in the field of scientific research, and partially supported by the Russian Foundation for Basic research (grants ð 14-05-00785, 16-35-00452).
9. Lateral acoustic wave resonator comprising a suspended membrane of low damping resonator material
Science.gov (United States)
Olsson, Roy H.; El-Kady; , Ihab F.; Ziaei-Moayyed, Maryam; Branch; , Darren W.; Su; Mehmet F.,; Reinke; Charles M.,
2013-09-03
A very high-Q, low insertion loss resonator can be achieved by storing many overtone cycles of a lateral acoustic wave (i.e., Lamb wave) in a lithographically defined suspended membrane comprising a low damping resonator material, such as silicon carbide. The high-Q resonator can sets up a Fabry-Perot cavity in a low-damping resonator material using high-reflectivity acoustic end mirrors, which can comprise phononic crystals. The lateral overtone acoustic wave resonator can be electrically transduced by piezoelectric couplers. The resonator Q can be increased without increasing the impedance or insertion loss by storing many cycles or wavelengths in the high-Q resonator material, with much lower damping than the piezoelectric transducer material.
10. Deposition of aluminium nitride film by magnetron sputtering for diamond-based surface acoustic wave applications
Energy Technology Data Exchange (ETDEWEB)
Mortet, V.; Nesladek, M.; D' Haen, J.; Vanhoyland, G.; D' Olieslaeger, M. [IMO, Limburgs Universitair Centrum, Wetenschapspark 1, B-3590 Diepenbeek (Belgium); Elmazria, O.; Assouar, M.B.; Alnot, P. [LPMIA, Universite H. Poincare, Nancy I, F-54506 Vandoeuvre-les-Nancy Cedex (France)
2002-10-16
Diamond/piezoelectric material thin film layered structures are expected to be applied to high frequency surface acoustic wave (SAW) devices because of the high acoustic wave velocity of diamond. Aluminium nitride (AlN) has been chosen as piezoelectric material because of its both high phase velocity and high resistivity. AlN thin films have been deposited by DC pulsed magnetron sputtering on Si(100) substrates. Texture and structure of the films have been investigated by X-ray diffraction, cross-section and in-plane view scanning electronic microscopy observation, and atomic force microscopy. One-micron thick, smooth and (002) oriented AlN films have been successfully deposited on freestanding chemical vapour deposition (CVD) diamond layers. The surface acoustic wave characteristics of AlN/diamond structure were investigated. (Abstract Copyright [2002], Wiley Periodicals, Inc.)
11. One-way propagation of acoustic waves through a periodic structure
Science.gov (United States)
Xu, Zheng; Xu, Wei; Yan, Xu; Qian, Menglu; Cheng, Qian
2018-02-01
One-way acoustic transmission is achieved through a brass plate with a periodic grating on the surface. Using the Schlieren imaging technique, the positive and negative propagation processes of acoustic waves through the periodic structure were experimentally observed. Simulations were performed using the finite-element method. Both the experimental and simulation results revealed a very large transmission ratio between positive and negative incidence, thus demonstrating the feasibility of using this structure as an acoustic rectifier. The results indicate that the structure has a broadband working frequency. The structure has potential applications in ultrasonic medical devices and sonochemical reactors.
12. A multi-channel acoustics monitor for perioperative respiratory monitoring: preliminary data.
Science.gov (United States)
Jafarian, Kamal; Amineslami, Majid; Hassani, Kamran; Navidbakhsh, Mahdi; Lahiji, Mohammad Niakan; Doyle, D John
2016-02-01
This study pertains to a six-channel acoustic monitoring system for use in patient monitoring during or after surgery. The base hardware consists of a USB data acquisition system, a custom-built six-channel amplification system, and a series of microphones of various designs. The software is based on the MATLAB platform with data acquisition drivers installed. The displayed information includes: time domain signals, frequency domain signals, and tools to aid in the detection of endobronchial intubation. We hypothesize that the above mentioned arrangement may be helpful to the anesthesiologist in recognizing clinical conditions like wheezing, bronchospasm, endobronchial intubation, and apnea. The study also evaluated various types of microphone designs used to transduce breath sounds. The system also features selectable band-pass filtering using MATLAB algorithms as well as a collection of recordings obtained with the system to establish what respiratory acoustic signals look like under various conditions.
13. Anisotropic surface acoustic waves in tungsten/lithium niobate phononic crystals
Science.gov (United States)
Sun, Jia-Hong; Yu, Yuan-Hai
2018-02-01
Phononic crystals (PnC) were known for acoustic band gaps for different acoustic waves. PnCs were already applied in surface acoustic wave (SAW) devices as reflective gratings based on the band gaps. In this paper, another important property of PnCs, the anisotropic propagation, was studied. PnCs made of circular tungsten films on a lithium niobate substrate were analyzed by finite element method. Dispersion curves and equal frequency contours of surface acoustic waves in PnCs of various dimensions were calculated to study the anisotropy. The non-circular equal frequency contours and negative refraction of group velocity were observed. Then PnC was applied as an acoustic lens based on the anisotropic propagation. Trajectory of SAW passing PnC lens was calculated and transmission of SAW was optimized by selecting proper layers of lens and applying tapered PnC. The result showed that PnC lens can suppress diffraction of surface waves effectively and improve the performance of SAW devices.
14. Operational Performance Analysis of Passive Acoustic Monitoring for Killer Whales
Energy Technology Data Exchange (ETDEWEB)
Matzner, Shari; Fu, Tao; Ren, Huiying; Deng, Zhiqun; Sun, Yannan; Carlson, Thomas J.
2011-09-30
For the planned tidal turbine site in Puget Sound, WA, the main concern is to protect Southern Resident Killer Whales (SRKW) due to their Endangered Species Act status. A passive acoustic monitoring system is proposed because the whales emit vocalizations that can be detected by a passive system. The algorithm for detection is implemented in two stages. The first stage is an energy detector designed to detect candidate signals. The second stage is a spectral classifier that is designed to reduce false alarms. The evaluation presented here of the detection algorithm incorporates behavioral models of the species of interest, environmental models of noise levels and potential false alarm sources to provide a realistic characterization of expected operational performance.
15. Progress towards Acoustic Suspended Sediment Transport Monitoring: Fraser River, BC
Science.gov (United States)
Attard, M. E.; Venditti, J. G.; Church, M. A.; Kostaschuk, R. A.
2011-12-01
Our ability to predict the timing and quantity of suspended sediment transport is limited because fine sand, silt and clay delivery are supply limited, requiring empirical modeling approaches of limited temporal stability. A solution is the development of continuous monitoring techniques capable of tracking sediment concentrations and grain-size. Here we examine sediment delivery from upstream sources to the lower Fraser River. The sediment budget of the lower Fraser River provides a long-term perspective of the net changes in the channels and in sediment delivery to Fraser Delta. The budget is based on historical sediment rating curves developed from data collected from 1965-1986 by the Water Survey of Canada. We explore the possibility of re-establishing the sediment-monitoring program using hydro-acoustics by evaluating the use of a 300 kHz side-looking acoustic Doppler current profiler (aDcp), mounted just downstream of the sand-gravel transition at Mission, for continuous measurement of suspended sediment transport. Complementary field observations include conventional bottle sampling with a P-63 sampler, vertical profiles with a downward-looking 600 kHz aDcp, and 1200 kHz aDcp discharge measurements. We have successfully completed calibration of the downward-looking aDcp with the P-63 samples; the side-looking aDcp signals remain under investigation. A comparison of several methods for obtaining total sediment flux indicates that suspended sediment concentration (SSC) closely follows discharge through the freshet and peaks in total SSC and sand SSC coincide with peak measurements of discharge. Low flows are dominated by fine sediment and grain size increases with higher flows. This research assesses several techniques for obtaining sediment flux and contributes to the understanding of sediment delivery to sand-bedded portions of the river.
16. Integrated microfluidics system using surface acoustic wave and electrowetting on dielectrics technology.
Science.gov (United States)
Li, Y; Fu, Y Q; Brodie, S D; Alghane, M; Walton, A J
2012-03-01
This paper presents integrated microfluidic lab-on-a-chip technology combining surface acoustic wave (SAW) and electro-wetting on dielectric (EWOD). This combination has been designed to provide enhanced microfluidic functionality and the integrated devices have been fabricated using a single mask lithographic process. The integrated technology uses EWOD to guide and precisely position microdroplets which can then be actuated by SAW devices for particle concentration, acoustic streaming, mixing and ejection, as well as for sensing using a shear-horizontal wave SAW device. A SAW induced force has also been employed to enhance the EWOD droplet splitting function.
17. Different quantization mechanisms in single-electron pumps driven by surface acoustic waves
DEFF Research Database (Denmark)
Utko, P.; Gloos, K.; Hansen, Jørn Bindslev
2006-01-01
We have studied the acoustoelectric current in single-electron pumps driven by surface acoustic waves. We have found that in certain parameter ranges two different sets of quantized steps dominate the acoustoelectric current versus gate-voltage characteristics. In some cases, both types of quanti......We have studied the acoustoelectric current in single-electron pumps driven by surface acoustic waves. We have found that in certain parameter ranges two different sets of quantized steps dominate the acoustoelectric current versus gate-voltage characteristics. In some cases, both types...
18. Unipolar and Bipolar High-Magnetic-Field Sensors Based on Surface Acoustic Wave Resonators
Science.gov (United States)
Polewczyk, V.; Dumesnil, K.; Lacour, D.; Moutaouekkil, M.; Mjahed, H.; Tiercelin, N.; Petit Watelot, S.; Mishra, H.; Dusch, Y.; Hage-Ali, S.; Elmazria, O.; Montaigne, F.; Talbi, A.; Bou Matar, O.; Hehn, M.
2017-08-01
While surface acoustic wave (SAW) sensors have been used to measure temperature, pressure, strains, and low magnetic fields, the capability to measure bipolar fields and high fields is lacking. In this paper, we report magnetic surface acoustic wave sensors that consist of interdigital transducers made of a single magnetostrictive material, either Ni or TbFe2 , or based on exchange-biased (Co /IrMn ) multilayers. By controlling the ferromagnet magnetic properties, high-field sensors can be obtained with unipolar or bipolar responses. The issue of hysteretic response of the ferromagnetic material is especially addressed, and the control of the magnetic properties ensures the reversible behavior in the SAW response.
19. Experimental study of nonlinear dust acoustic solitary waves in a dusty plasma
CERN Document Server
Bandyopadhyay, P; Sen, A; Kaw, P K
2008-01-01
The excitation and propagation of finite amplitude low frequency solitary waves are investigated in an Argon plasma impregnated with kaolin dust particles. A nonlinear longitudinal dust acoustic solitary wave is excited by pulse modulating the discharge voltage with a negative potential. It is found that the velocity of the solitary wave increases and the width decreases with the increase of the modulating voltage, but the product of the solitary wave amplitude and the square of the width remains nearly constant. The experimental findings are compared with analytic soliton solutions of a model Kortweg-de Vries equation.
20. Dynamic motions of ion acoustic waves in plasmas with superthermal electrons
Energy Technology Data Exchange (ETDEWEB)
Saha, Asit, E-mail: [email protected] [Department of Mathematics, Sikkim Manipal Institute of Technology (India); Chatterjee, Prasanta [Department of Mathematics, Siksha Bhavana, Visva Bharati University (India); Wong, C.S. [Plasma Technology Research Centre, Department of Physics, University of Malaya, Kuala Lampur (Malaysia)
2015-12-15
The dynamic motions of ion acoustic waves an unmagnetized plasma with superthermal (q-non extensive) electrons are investigated employing the bifurcation theory of planar dynamical systems through direct approach. Using traveling wave transformation and initial conditions, basic equations are transformed to a planar dynamical system. Using numerical computations, all possible phase portraits of the dynamical system are presented. Corresponding to homoclinic and periodic orbits of the phase portraits, two new analytical forms of solitary and periodic wave solutions are derived depending on the non extensive parameter q and speed v of the traveling wave. Considering an external periodic perturbation, the quasiperiodic and chaotic motions of ion acoustic waves are presented. Depending upon different ranges of non extensive parameter q, the effect of q is shown on quasiperiodic and chaotic motions of ion acoustic waves with fixed value of v. It is seen that the unperturbed dynamical system has the solitary and periodic wave solutions, but the perturbed dynamical system has the quasiperiodic and chaotic motions with same values of parameters q and v. (author)
1. Surface Generated Acoustic Wave Biosensors for the Detection of Pathogens: A Review
Science.gov (United States)
Rocha-Gaso, María-Isabel; March-Iborra, Carmen; Montoya-Baides, Ángel; Arnau-Vives, Antonio
2009-01-01
This review presents a deep insight into the Surface Generated Acoustic Wave (SGAW) technology for biosensing applications, based on more than 40 years of technological and scientific developments. In the last 20 years, SGAWs have been attracting the attention of the biochemical scientific community, due to the fact that some of these devices - Shear Horizontal Surface Acoustic Wave (SH-SAW), Surface Transverse Wave (STW), Love Wave (LW), Flexural Plate Wave (FPW), Shear Horizontal Acoustic Plate Mode (SH-APM) and Layered Guided Acoustic Plate Mode (LG-APM) - have demonstrated a high sensitivity in the detection of biorelevant molecules in liquid media. In addition, complementary efforts to improve the sensing films have been done during these years. All these developments have been made with the aim of achieving, in a future, a highly sensitive, low cost, small size, multi-channel, portable, reliable and commercially established SGAW biosensor. A setup with these features could significantly contribute to future developments in the health, food and environmental industries. The second purpose of this work is to describe the state-of-the-art of SGAW biosensors for the detection of pathogens, being this topic an issue of extremely importance for the human health. Finally, the review discuses the commercial availability, trends and future challenges of the SGAW biosensors for such applications. PMID:22346725
2. The Ion Acoustic Solitary Waves and Double Layers in the Solar Wind Plasma
Directory of Open Access Journals (Sweden)
C. R. Choi
2006-09-01
Full Text Available Ion acoustic solitary wave in a plasma consisting of electrons and ions with an external magnetic field is reinvestigated using the Sagdeev's potential method. Although the Sagdeev potential has a singularity for n<1, where n is the ion number density, we obtain new solitary wave solutions by expanding the Sagdeev potential up to δ n^4 near n=1. They are compressiv (rarefactive waves and shock type solitary waves. These waves can exist all together as a superposed wave which may be used to explain what would be observed in the solar wind plasma. We compared our theoretical results with the data of the Freja satellite in the study of Wu et al.(1996. Also it is shown that these solitary waves propagate with a subsonic speed.
3. Acoustic Bloch Wave Propagation in a Periodic Waveguide
Science.gov (United States)
1991-07-24
matrix (Ramo, Whinnery, and Van Duzer , 1965). Given the amplitudes of the two travelling waves in a single cell, then, we can find the amplitudes of...harmonics (Ramo, Whinnery, and Van Duzer , 1965). ; is interesting to note that because the range of the sum index n in Eq. 2.53 includ negative integers...34backwar. wave structures" (Ramo, Whinnery, and Van Duzer , 1965). 2.4.3 The Convolution Representation The apparent simplicity of the Bloch wave function
4. Extracting the Green's function of attenuating heterogeneous acoustic media from uncorrelated waves.
Science.gov (United States)
Snieder, Roel
2007-05-01
The Green's function of acoustic or elastic wave propagation can, for loss-less media, be retrieved by correlating the wave field that is excited by random sources and is recorded at two locations. Here the generalization of this idea to attenuating acoustic waves in an inhomogeneous medium is addressed, and it is shown that the Green's function can be retrieved from waves that are excited throughout the volume by spatially uncorrelated injection sources with a power spectrum that is proportional to the local dissipation rate. For a finite volume, one needs both volume sources and sources at the bounding surface for the extraction of the Green's functions. For the special case of a homogeneous attenuating medium defined over a finite volume, the phase and geometrical spreading of the Green's function is correctly retrieved when the volume sources are ignored, but the attenuation is not.
5. Propagation of flexural waves in inhomogeneous plates exhibiting hysteretic nonlinearity: Nonlinear acoustic black holes.
Science.gov (United States)
Gusev, Vitalyi E; Ni, Chenyin; Lomonosov, Alexey; Shen, Zhonghua
2015-08-01
Theory accounting for the influence of hysteretic nonlinearity of micro-inhomogeneous material on flexural wave in the plates of continuously varying thickness is developed. For the wedges with thickness increasing as a power law of distance from its edge strong modifications of the wave dynamics with propagation distance are predicted. It is found that nonlinear absorption progressively disappearing with diminishing wave amplitude leads to complete attenuation of acoustic waves in most of the wedges exhibiting black hole phenomenon. It is also demonstrated that black holes exist beyond the geometrical acoustic approximation. Applications include nondestructive evaluation of micro-inhomogeneous materials and vibrations damping. Copyright © 2015 Elsevier B.V. All rights reserved.
6. Asymmetric transmission of acoustic waves in a layer thickness distribution gradient structure using metamaterials
Directory of Open Access Journals (Sweden)
Jung-San Chen
2016-09-01
Full Text Available This research presents an innovative asymmetric transmission design using alternate layers of water and metamaterial with complex mass density. The directional transmission behavior of acoustic waves is observed numerically inside the composite structure with gradient layer thickness distribution and the rectifying performance of the present design is evaluated. The layer thickness distributions with arithmetic and geometric gradients are considered and the effect of gradient thickness on asymmetric wave propagation is systematically investigated using finite element simulation. The numerical results indicate that the maximum pressure density and transmission through the proposed structure are significantly influenced by the wave propagation direction over a wide range of audible frequencies. Tailoring the thickness of the layered structure enables the manipulation of asymmetric wave propagation within the desired frequency range. In conclusion, the proposed design offers a new possibility for developing directional-dependent acoustic devices.
7. Real-time monitoring of controllable cavitation erosion in a vessel phantom with passive acoustic mapping.
Science.gov (United States)
Lu, Shukuan; Shi, Aiwei; Jing, Bowen; Du, Xuan; Wan, Mingxi
2017-11-01
Cavitation erosion in blood vessel plays an important role in ultrasound thrombolysis, drug delivery, and other clinical applications. The controllable superficial vessel erosion based on ultrasonic standing wave (USW) has been used to effectively prevent vessel ruptures and haemorrhages, and optical method is used to observe the experiments. But optical method can only work in transparent media. Compared with standard B-mode imaging, passive acoustic mapping (PAM) can monitor erosion in real time and has better sensitivity of cavitation detection. However, the conventionally used PAM has limitations in imaging resolution and artifacts. In this study, a unique PAM method that combined the robust Capon beamformer (RCB) with the sign coherence factor (SCF) was proposed to monitor the superficial vessel erosion in real time. The performance of the proposed method was validated by simulations. In vitro experiments showed that the lateral (axial) resolution of the proposed PAM was 2.31±0.51 (3.19±0.38) times higher than time exposure acoustics (TEA)-based PAM and 1.73±0.38 (1.76±0.48) times higher than RCB-based PAM, and the cavitation-to-artifact ratio (CAR) of the proposed PAM could be improved by 22.5±3.2dB and 7.1±1.2dB compared with TEA and RCB-based PAM. These results showed that the proposed PAM can precisely monitor the superficial vessel erosion and the erosion shift after USW modulation. This work may have the potential of developing a useful tool for precise spatial control and real-time monitoring of the superficial vessel erosion. Copyright © 2017 Elsevier B.V. All rights reserved.
8. Early corrosion monitoring of prestressed concrete piles using acoustic emission
Science.gov (United States)
Vélez, William; Matta, Fabio; Ziehl, Paul H.
2013-04-01
The depassivation and corrosion of bonded prestressing steel strands in concrete bridge members may lead to major damage or collapse before visual inspections uncover evident signs of damage, and well before the end of the design life. Recognizing corrosion in its early stage is desirable to plan and prioritize remediation strategies. The Acoustic Emission (AE) technique is a rational means to develop structural health monitoring and prognosis systems for the early detection and location of corrosion in concrete. Compelling features are the sensitivity to events related to micro- and macrodamage, non-intrusiveness, and suitability for remote and wireless applications. There is little understanding of the correlation between AE and the morphology and extent of early damage on the steel surface. In this paper, the evidence collected from prestressed concrete (PC) specimens that are exposed to salt water is discussed vis-à-vis AE data from continuous monitoring. The specimens consist of PC strips that are subjected to wet/dry salt water cycles, representing portions of bridge piles that are exposed to tidal action. Evidence collected from the specimens includes: (a) values of half-cell potential and linear polarization resistance to recognize active corrosion in its early stage; and (b) scanning electron microscopy micrographs of steel areas from two specimens that were decommissioned once the electrochemical measurements indicated a high probability of active corrosion. These results are used to evaluate the AE activity resulting from early corrosion.
9. Propagation of surface acoustic waves in n-type GaAs films
Science.gov (United States)
Wu, Chhi-Chong; Tsai, Jensan
1983-05-01
The effect of nonparabolicity on the amplification of surface acoustic waves in n-type GaAs films is investigated quantum mechanically in the GHz frequency region. Numerical results show that the amplification coefficient for the nonparabolic band structure is enhanced due to the nonlinear nature of the energy band in semiconductors. Moreover, the amplification coefficients in semiconductors depend on the temperature, the electronic screening effect, the frequency of sound waves, the applied electric field, and the thickness of the semiconductor film.
10. Universal morphologies of fluid interfaces deformed by the radiation pressure of acoustic or electromagnetic waves.
Science.gov (United States)
Bertin, N; Chraïbi, H; Wunenburger, R; Delville, J-P; Brasselet, E
2012-12-14
We unveil the generation of universal morphologies of fluid interfaces by radiation pressure regardless of the nature of the wave, whether acoustic or optical. Experimental observations reveal interface deformations endowed with steplike features that are shown to result from the interplay between the wave propagation and the shape of the interface. The results are supported by numerical simulations and a quantitative interpretation based on the waveguiding properties of the field is provided.
11. A porosity-based Biot model for acoustic waves in snow
OpenAIRE
Sidler, Rolf
2015-01-01
Phase velocities and attenuation in snow can not be explained by the widely used elastic or viscoelastic models for acoustic wave propagation. Instead, Biot's model of wave propagation in porous materials should be used. However, the application of Biot's model is complicated by the large property space of the underlying porous material. Here the properties of ice and air as well as empirical relationships are used to define the properties of snow as a function of porosity. Based on these rel...
12. Rapid Salmonella detection using an acoustic wave device combined with the RCA isothermal DNA amplification method
Directory of Open Access Journals (Sweden)
Antonis Kordas
2016-12-01
Full Text Available Salmonella enterica serovar Typhimurium is a major foodborne pathogen that causes Salmonellosis, posing a serious threat for public health and economy; thus, the development of fast and sensitive methods is of paramount importance for food quality control and safety management. In the current work, we are presenting a new approach where an isothermal amplification method is combined with an acoustic wave device for the development of a label free assay for bacteria detection. Specifically, our method utilizes a Love wave biosensor based on a Surface Acoustic Wave (SAW device combined with the isothermal Rolling Circle Amplification (RCA method; various protocols were tested regarding the DNA amplification and detection, including off-chip amplification at two different temperatures (30 °C and room temperature followed by acoustic detection and on-chip amplification and detection at room temperature, with the current detection limit being as little as 100 Bacteria Cell Equivalents (BCE/sample. Our acoustic results showed that the acoustic ratio, i.e., the amplitude over phase change observed during DNA binding, provided the only sensitive means for product detection while the measurement of amplitude or phase alone could not discriminate positive from negative samples. The method's fast analysis time together with other inherent advantages i.e., portability, potential for multi-analysis, lower sample volumes and reduced power consumption, hold great promise for employing the developed assay in a Lab on Chip (LoC platform for the integrated analysis of Salmonella in food samples.
13. Propagation of acoustic shock waves between parallel rigid boundaries and into shadow zones
Energy Technology Data Exchange (ETDEWEB)
Desjouy, C., E-mail: [email protected]; Ollivier, S.; Dragna, D.; Blanc-Benon, P. [Laboratoire de Mécanique des Fluides et d’Acoustique, UMR CNRS 5509, École Centrale de Lyon, Université de Lyon, 69134 Ecully Cedex (France); Marsden, O. [European Center For Medium Range Weather Forecasts, United Kingdom Shinfield (United Kingdom)
2015-10-28
The study of acoustic shock propagation in complex environments is of great interest for urban acoustics, but also for source localization, an underlying problematic in military applications. To give a better understanding of the phenomenon taking place during the propagation of acoustic shocks, laboratory-scale experiments and numerical simulations were performed to study the propagation of weak shock waves between parallel rigid boundaries, and into shadow zones created by corners. In particular, this work focuses on the study of the local interactions taking place between incident, reflected, and diffracted waves according to the geometry in both regular or irregular – also called Von Neumann – regimes of reflection. In this latter case, an irregular reflection can lead to the formation of a Mach stem that can modify the spatial distribution of the acoustic pressure. Short duration acoustic shock waves were produced by a 20 kilovolts electric spark source and a schlieren optical method was used to visualize the incident shockfront and the reflection/diffraction patterns. Experimental results are compared to numerical simulations based on the high-order finite difference solution of the two dimensional Navier-Stokes equations.
14. Dust acoustic solitary and shock waves in strongly coupled dusty ...
The Korteweg–de Vries–Burgers (KdV–Burgers) equation and modified Korteweg–de Vries–Burgers equation are derived in strongly coupled dusty plasmas containing nonthermal ions and Boltzmann distributed electrons. It is found that solitary waves and shock waves can be produced in this medium. The effects of ...
15. Dust acoustic solitary and shock waves in strongly coupled dusty ...
Abstract. The Korteweg–de Vries–Burgers (KdV–Burgers) equation and modified. Korteweg–de Vries–Burgers equation are derived in strongly coupled dusty plasmas con- taining nonthermal ions and Boltzmann distributed electrons. It is found that solitary waves and shock waves can be produced in this medium.
16. Analysis of ray trajectories of flexural waves propagating over generalized acoustic black hole indentations
Science.gov (United States)
Huang, Wei; Ji, Hongli; Qiu, Jinhao; Cheng, Li
2018-03-01
An Acoustic Black Hole (ABH) indentation embedded in thin-walled structures has been proved remarkably useful for broadband flexural wave focalization, in which the phase velocity of the flexural waves and the refractive index of the media undergo gradual changes from the outside towards the center of the indentation. A generalized two-dimensional ABH indentation can be defined by three geometric parameters: a power index, an extra thickness and a radius of a plateau at the indentation center. The dependence of the energy focalization on these parameters as well as the energy focalization process is of paramount importance for the understanding and design of effective ABH indentations. This work aims at investigating the energy focalization characteristics of flexural waves in such generalized ABH indentations. The calculation of the flexural ray trajectories is conducted to reveal and analyze the wave propagation features through numerical integration of the eikonal equation from the Geometric Acoustics Approximation (GAA). The theoretical results are verified by both experiment using wave visualization technique based on laser acoustic scanning method and finite element (FE) simulations. Finally, the influence of the geometric parameters on the flexural wave focalization characteristics in ABH indentations is discussed in detail.
17. Acoustic wave and eikonal equations in a transformed metric space for various types of anisotropy.
Science.gov (United States)
Noack, Marcus M; Clark, Stuart
2017-03-01
Acoustic waves propagating in anisotropic media are important for various applications. Even though these wave phenomena do not generally occur in nature, they can be used to approximate wave motion in various physical settings. We propose a method to derive wave equations for anisotropic wave propagation by adjusting the dispersion relation according to a selected type of anisotropy and transforming it into another metric space. The proposed method allows for the derivation of acoustic wave and eikonal equations for various types of anisotropy, and generalizes anisotropy by interpreting it as a change of the metric instead of a change of velocity with direction. The presented method reduces the scope of acoustic anisotropy to a selection of a velocity or slowness surface and a tensor that describes the transformation into a new metric space. Experiments are shown for spatially dependent ellipsoidal anisotropy in homogeneous and inhomogeneous media and sandstone, which shows vertical transverse isotropy. The results demonstrate the stability and simplicity of the solution process for certain types of anisotropy and the equivalency of the solutions.
18. Anomalous Refraction of Acoustic Guided Waves in Solids with Geometrically Tapered Metasurfaces.
Science.gov (United States)
Zhu, Hongfei; Semperlotti, Fabio
2016-07-15
The concept of a metasurface opens new exciting directions to engineer the refraction properties in both optical and acoustic media. Metasurfaces are typically designed by assembling arrays of subwavelength anisotropic scatterers able to mold incoming wave fronts in rather unconventional ways. The concept of a metasurface was pioneered in photonics and later extended to acoustics while its application to the propagation of elastic waves in solids is still relatively unexplored. We investigate the design of acoustic metasurfaces to control elastic guided waves in thin-walled structural elements. These engineered discontinuities enable the anomalous refraction of guided wave modes according to the generalized Snell's law. The metasurfaces are made out of locally resonant toruslike tapers enabling an accurate phase shift of the incoming wave, which ultimately affects the refraction properties. We show that anomalous refraction can be achieved on transmitted antisymmetric modes (A_{0}) either when using a symmetric (S_{0}) or antisymmetric (A_{0}) incident wave, the former clearly involving mode conversion. The same metasurface design also allows achieving structure embedded planar focal lenses and phase masks for nonparaxial propagation.
19. Horizontal Acoustic Barriers for Protection from Seismic Waves
Directory of Open Access Journals (Sweden)
Sergey V. Kuznetsov
2011-01-01
Full Text Available The basic idea of a seismic barrier is to protect an area occupied by a building or a group of buildings from seismic waves. Depending on nature of seismic waves that are most probable in a specific region, different kinds of seismic barriers can be suggested. Herein, we consider a kind of a seismic barrier that represents a relatively thin surface layer that prevents surface seismic waves from propagating. The ideas for these barriers are based on one Chadwick's result concerning nonpropagation condition for Rayleigh waves in a clamped half-space, and Love's theorem that describes condition of nonexistence for Love waves. The numerical simulations reveal that to be effective the length of the horizontal barriers should be comparable to the typical wavelength.
20. Measurement of Acoustic-to-Seismic Conversion Using T-wave Signals Recorded at Ascension Island and Diego Garcia
Science.gov (United States)
Pulli, J. J.; Kofford, A. S.; Newman, K. R.; Krumhansl, P. A.
2012-12-01
T-wave signals from sub-sea earthquakes are often recorded on coastal or island seismic stations (Linehan, 1940; Okal, 2008). The physical process of the acoustic-to-seismic conversion is poorly understood but likely depends on factors such as seafloor relief and sediment thickness at the location where the interaction occurs. Quantification of the conversion process is necessary to understand and interpret the seismic recordings, and allow for the calculation of in-water acoustic levels from these recordings where no in-water sensor recordings are available. Applications for this knowledge would include the calculation of in-water explosion yields and seismic airgun source levels. Here we present the measurement of the acoustic-to-seismic transfer functions at Ascension Island and Diego Garcia using hydroacoustic data from the International Monitoring System and broadband seismic data from the Global Seismic Network. For Ascension Island, a volcanic island formed above magmatic plumes, we used T-wave signals from earthquakes on the Central Mid-Atlantic Ridge and associated fracture zones. For Diego Garcia, an atoll of carbonate sequences and no volcanism, we used T-wave signals from earthquakes along the Sumatran Subduction Zone, the Indian Ocean Ridges, and the Chagos Arch. The methodology is based on the smoothed cross-spectra over a frequency band that is common to the acoustic and seismic recordings, typically 2-18 Hz. Preliminary results indicate that at 5 Hz the acoustic-to-seismic conversion is 2-4 times more efficient at Ascension Island than at Diego Garcia (124 nm/s/Pa vs. 51 nm/s/Pa, respectively), but nearly equal at 10 Hz (20 nm/s/Pa). At 15 Hz the conversion is more efficient at Diego Garcia (13 nm/s/Pa vs. 8 nm/s/Pa at Ascension). We also investigate the azimuthal variance of this transfer function, as well as the differences between the three components of seismic motion. As a verification of the methodology, we use the equivalent time domain
1. Development of a standing wave apparatus for calibrating acoustic vector sensors and hydrophones.
Science.gov (United States)
Lenhart, Richard D; Sagers, Jason D; Wilson, Preston S
2016-01-01
An apparatus was developed to calibrate acoustic hydrophones and vector sensors between 25 and 2000 Hz. A standing wave field is established inside a vertically oriented, water-filled, elastic-walled waveguide by a piston velocity source at the bottom and a pressure-release boundary condition at the air/water interface. A computer-controlled linear positioning system allows a device under test to be precisely located in the water column while the acoustic response is measured. Some of the challenges of calibrating hydrophones and vector sensors in such an apparatus are discussed, including designing the waveguide to mitigate dispersion, understanding the impact of waveguide structural resonances on the acoustic field, and developing algorithms to post-process calibration measurement data performed in a standing wave field. Data from waveguide characterization experiments and calibration measurements are presented and calibration uncertainty is reported.
2. Interaction acoustic waves with a layered structure containing layer of bubbly liquid
Directory of Open Access Journals (Sweden)
Gubaidullin Damir
2018-01-01
Full Text Available The results of a theoretical study of the effect of a bubble layer on the propagation of acoustic waves through a thin three-layered barrier at various angles of incidence are presented. The barrier consists of a layer of gel with polydisperse air bubbles bounded by layers of polycarbonate. It is shown that the presence of polydisperse air bubbles in the gel layer significantly changes the transmission and reflection of the acoustic signal when it interacts with such an obstacle for frequencies close to the resonant frequency of natural oscillations of the bubbles. The frequency range is identified where the angle of incidence has little effect on the reflection and transmission coefficients of acoustic waves.
3. Dependence of oscillational instabilities on the amplitude of the acoustic wave in single-axis levitators
DEFF Research Database (Denmark)
Orozco-Santillán, Arturo; Ruiz-Boullosa, Ricardo; Cutanda Henríquez, Vicente
2007-01-01
It is well known that acoustic waves exert forces on a boundary with which they interact; these forces can be so intense that they can compensate for the weight of small objects up to a few grams. In this way, it is possible to maintain solid or liquid samples levitating in a fluid, avoiding...... the use of containers, which may be undesirable for certain applications. Moreover, small samples can be manipulated by means of acoustic waves. In this paper, we report a study on the oscillational instabilities that can appear on a levitated solid sphere in single-axis acoustic devices. A theory...... published on the topic predicts that these instabilities appear when the levitator is driven with a frequency above the resonant frequency of the empty device. The theory also shows that the instabilities can either saturate to a state with constant amplitude, or they can grow without limit until the object...
4. High-frequency programmable acoustic wave device realized through ferroelectric domain engineering
Energy Technology Data Exchange (ETDEWEB)
Ivry, Yachin, E-mail: [email protected], E-mail: [email protected]; Wang, Nan; Durkan, Colm, E-mail: [email protected], E-mail: [email protected] [Nanoscience Centre, University of Cambridge, 11 JJ Thomson Avenue, Cambridge CB3 0FF (United Kingdom)
2014-03-31
Surface acoustic wave devices are extensively used in contemporary wireless communication devices. We used atomic force microscopy to form periodic macroscopic ferroelectric domains in sol-gel deposited lead zirconate titanate, where each ferroelectric domain is composed of many crystallites, each of which contains many microscopic ferroelastic domains. We examined the electro-acoustic characteristics of the apparatus and found a resonator behavior similar to that of an equivalent surface or bulk acoustic wave device. We show that the operational frequency of the device can be tailored by altering the periodicity of the engineered domains and demonstrate high-frequency filter behavior (>8 GHz), allowing low-cost programmable high-frequency resonators.
5. A Novel Bulk Acoustic Wave Resonator for Filters and Sensors Applications
Directory of Open Access Journals (Sweden)
Zhixin Zhang
2015-09-01
Full Text Available Bulk acoustic wave (BAW resonators are widely applied in filters and gravimetric sensors for physical or biochemical sensing. In this work, a new architecture of BAW resonator is demonstrated, which introduces a pair of reflection layers onto the top of a thin film bulk acoustic resonator (FBAR device. The new device can be transformed between type I and type II dispersions by varying the thicknesses of the reflection layers. A computational modeling is developed to fully investigate the acoustic waves and the dispersion types of the device theoretically. The novel structure makes it feasible to fabricate both type resonators in one filter, which offers an effective alternative to improve the pass band flatness in the filter. Additionally, this new device exhibits a high quality factor (Q in the liquid, which opens a possibility for real time measurement in solutions with a superior limitation of detection (LOD in sensor applications.
6. Time-domain imaging of gigahertz surface waves on an acoustic metamaterial
Science.gov (United States)
Otsuka, Paul H.; Mezil, Sylvain; Matsuda, Osamu; Tomoda, Motonobu; Maznev, Alexei A.; Gan, Tian; Fang, Nicholas; Boechler, Nicholas; Gusev, Vitalyi E.; Wright, Oliver B.
2018-01-01
We extend time-domain imaging in acoustic metamaterials to gigahertz frequencies. Using a sample consisting of a regular array of ∼1 μm diameter silica microspheres forming a two-dimensional triangular lattice on a substrate, we implement an ultrafast technique to probe surface acoustic wave propagation inside the metamaterial area and incident on the metamaterial from a region containing no microspheres, which reveals the acoustic metamaterial dispersion, the presence of band gaps and the acoustic transmission properties of the interface. A theoretical model of this locally resonant metamaterial based on normal and shear-rotational resonances of the spheres fits the data well. Using this model, we show analytically how the sphere elastic coupling parameters influence the gap widths.
7. Nonlinear propagation of dust-acoustic solitary waves in a dusty ...
... Lecture Workshops · Refresher Courses · Symposia · Live Streaming. Home; Journals; Pramana – Journal of Physics; Volume 80; Issue 6. Nonlinear propagation of dust-acoustic solitary waves in a dusty plasma with arbitrarily charged dust and trapped electrons. O Rahman A A Mamun. Volume 80 Issue 6 June 2013 pp ...
8. Studies on a surface acoustic wave (SAW) dosimeter sensor for organophosphorous nerve agents
NARCIS (Netherlands)
Nieuwenhuizen, M.S.; Harteveld, J.L.N.
1997-01-01
As a follow-up of previous work on a Surface Acoustic Wave (SAW) sensor for nerve agents, irreversible response effects have been studied in more detail. Surface analytical studies indicated that degradation products are responsible for the effects observed. In addition it was tried to explore these
9. Acoustic wave propagation in Ni3 R (R= Mo, Nb, Ta) compounds
Home; Journals; Pramana – Journal of Physics; Volume 76; Issue 4. Acoustic wave propagation in Ni3 ( = Mo, Nb, Ta) compounds. Pramod Kumar Yadawa ... Author Affiliations. Pramod Kumar Yadawa1. Department of Applied Physics, AMITY School of Engineering and Technology, Bijwasan, New Delhi 110 061, India ...
10. A filtered convolution method for the computation of acoustic wave fields in very large spatiotemporal domains
NARCIS (Netherlands)
Verweij, M.D.; Huijssen, J.
2009-01-01
The full-wave computation of transient acoustic fields with sizes in the order of 100x100x100 wavelengths by 100 periods requires a numerical method that is extremely efficient in terms of storage and computation. Iterative integral equation methods offer a good performance on these points, provided
11. Dust-acoustic solitary waves in a dusty plasma with two-temperature ...
acoustic waves in a dusty plasma (containing a negatively charged dust fluid, Boltzmann distributed electrons and two-temperature nonthermal ions) is investigated. The effects of two-temperature nonthermal ions on the basic properties of small but ...
12. Dust-acoustic solitary waves in a dusty plasma with two-temperature ...
Abstract. By using reductive perturbation method, the nonlinear propagation of dust-acoustic waves in a dusty plasma (containing a negatively charged dust fluid, Boltzmann distributed electrons and two-temperature nonthermal ions) is investigated. The effects of two-temperature nonthermal ions on the basic properties of ...
13. Effect of face fracturing on shear wave coda quality factor estimated from acoustic emission events
CSIR Research Space (South Africa)
Kgarume, T
2013-10-01
Full Text Available The dependency of the quality factor derived from S wave coda (Q(subc)) on frequency is analysed in order to understand the effect of fracturing ahead of a mining stope. Micro seismic events recorded using acoustic emission sensors in a mining...
14. First-principle simulation of the acoustic radiation force on microparticles in ultrasonic standing waves
DEFF Research Database (Denmark)
Jensen, Mads Jakob Herring; Bruus, Henrik
2013-01-01
The recent development in the field of microparticle acoutophoresis in microsystems has led to an increased need for more accurate theoretical predections for the acoustic radiation force on a single microparticle in an ultrasonic standing wave. Increasingly detailed analytical solutions of this ...
15. Observation of depolarized guided acoustic-wave Brillouin scattering in partially uncoated optical fibers
Science.gov (United States)
Hayashi, Neisei; Set, Sze Yun; Yamashita, Shinji
2018-02-01
We observed the depolarized guided acoustic-wave Brillouin scattering (GAWBS) spectrum in a highly nonlinear fiber with a partially side-stripped polymer coat. The linewidth of the GAWBS spectral line at 941 MHz was measured to be 6.4 MHz, which was 0.9 times that of a coated fiber.
16. Two-soliton and three-soliton interactions of electron acoustic waves ...
of the Kortweg–de Vries (KdV) equation [2]. On the other hand, electron acoustic waves (EAWs) do exhibit soliton solutions and have been investigated in detail both theoretically and experimentally by many resear- chers [3–5]. The evolution of small-amplitude EAWs in collisionless plasma is usually. Pramana – J. Phys.
17. Planar dust-acoustic waves in electron–positron–ion–dust plasmas ...
2014-09-19
Sep 19, 2014 ... Abstract. Propagation of small but finite nonlinear dust-acoustic solitary waves are investigated in a planar unmagnetized dusty plasma, which consists of electrons, positrons, ions and negatively charged dust particles with different sizes and masses. A Kadomtsev–Petviashvili (KP) equation is obtained by ...
18. Propagation of ion-acoustic waves in a dusty plasma with non ...
E-mail: kalyan [email protected]. MS received 30 April 2005; revised 18 April 2007; accepted 1 May 2007. Abstract. For an unmagnetised collisionless plasma .... constant. We further assume that the phase velocity of the ion-acoustic wave is low compared to the electron thermal velocity. The charge neutrality in the state.
19. Modeling nonlinear acoustic waves in media with inhomogeneities in the coefficient of nonlinearity
NARCIS (Netherlands)
Demi, L.; Verweij, M.D.; Van Dongen, K.W.A.
2010-01-01
The refraction and scattering of nonlinear acoustic waves play an important role in the realistic application of medical ultrasound. One cause of these effects is the tissue dependence of the nonlinear medium behavior. A method that is able to model those effects is essential for the design of
20. Planar dust-acoustic waves in electron–positron–ion–dust plasmas ...
Home; Journals; Pramana – Journal of Physics; Volume 84; Issue 1 ... Propagation of small but finite nonlinear dust-acoustic solitary waves are investigated in a planar unmagnetized dusty plasma, which consists of electrons, ... School of Science, Sichuan University of Science and Engineering, Zigong, 643000, China ...
1. Acoustic model of micro-pressure wave emission from a high-speed train tunnel
Science.gov (United States)
Miyachi, T.
2017-03-01
The micro-pressure wave (MPW) radiated from a tunnel portal can, if audible, cause serious problems around tunnel portals in high-speed railways. This has created a need to develop an acoustic model that considers the topography around a radiation portal in order to predict MPWs more accurately and allow for higher speed railways in the future. An acoustic model of MPWs based on linear acoustic theory is developed in this study. First, the directivity of sound sources and the acoustical effect of topography are investigated using a train launcher facility around a portal on infinitely flat ground and with an infinite vertical baffle plate. The validity of linear acoustic theory is then discussed through a comparison of numerical results obtained using the finite difference method (FDM) and experimental results. Finally, an acoustic model is derived that considers sound sources up to the second order and Green's function to represent the directivity and effect of topography, respectively. The results predicted by this acoustic model are shown to be in good agreement with both numerical and experimental results.
2. Lamb wave structural health monitoring using frequency-wavenumber analysis
Science.gov (United States)
Tian, Zhenhua; Yu, Lingyu
2013-01-01
Lamb waves have shown great potential for structural health monitoring (SHM) in plate-like structures. Their attractive features include sensitivity to a variety of damage types and the capability of traveling relatively long distance. However, Lamb waves are dispersive and multimodal. Moreover, the propagating Lamb waves may include incident, reflected and converted waves. Various wave modes make the interpretation of Lamb wave signal very difficult. This paper presents studies on Lamb wave propagation using frequency-wavenumber analysis. By using two-dimensional Fourier transform (2-D FT), the time-space wavefield can be transformed into frequency-wavenumber domain, where various wave modes and waves propagating in different directions can be clearly discerned. By a frequency-wavenumber filtering strategy, the desired wave modes or wave propagation at certain direction can be extracted and further utilized for the purpose of SHM. The frequency-wavenumber analysis and its applications to Lamb wave SHM are illustrated through two experimental investigations. One is Lamb wave propagation in a plate half immersed in water and the other is Lamb wave mode decomposition by using two-dimensional frequency-wavenumber filtering strategy. Lamb waves are excited by piezoelectric wafer sensor and measured by scanning laser Doppler vibrometer. Various wave modes were visualized and successfully decomposed.
3. Modeling of Acoustic Pressure Waves in Level-Dependent Earplugs
Science.gov (United States)
2008-09-01
of these boundary conditions were applied using user-defined functions ( UDFs ), which are user-written C- code that allows direct access to the FLUENT...ABSTRACT UNCLASSIFIED c. THIS PAGE UNCLASSIFIED 17. LIMITATION OF ABSTRACT UL 18. NUMBER OF PAGES 32 19b. TELEPHONE NUMBER (Include area code ...of interest in auditory acoustics (4). The commercial CFD code FLUENT (5) was used in this study. FLUENT is a general-purpose CFD package that
4. Diffraction of acoustic wave through a slit with a finitethickness
Science.gov (United States)
Burova, Marina; Andreeva, Andreana; Burov, Julian
2010-01-01
The acoustic field, transmitted through parallelepiped from fused silica (immersed in water) is registered, which parallelepiped acts the part of slit with a finite thickness. It is shown, that the finite thickness leads to a change of the acoustic wavelength and the distance up to a registration plane. The distance increases, if the acoustic velocity in the material is higher than that in water, independently of actual distance between the slit and data plane. It is shown, if the slit with finite thickness is placed, the Fresnel zone may be changed in comparison with that observed at the diffraction through infinite thin slit at the same distance (up to a registration plane) for given wavelength. Program module is created by MatLab program for simulation of the diffraction field trough an infinite slit. The experimental results are compared with the theoretical ones, obtained from the simulation. The simulation results shown, that at diffraction trough a slit with finite thickness, the registration distance of the far-field diffraction increases, i.e. the Fraunhofer zone may be drawn forward. The image of the object may be obtained in this case directly by inverse Fourier transformation from the registered data in the Fraunhofer zone.
5. The Influence of Trapped Particles on the Parametric Decay Instability of Near-Acoustic Waves
Science.gov (United States)
Affolter, M.; Anderegg, F.; Dubin, D. H. E.; Driscoll, C. F.
2017-10-01
We present quantitative measurements of a decay instability to lower frequencies of near-acoustic waves. These experiments are conducted on pure ion plasmas confined in a cylindrical Penning-Malmberg trap. The axisymmetric, standing plasma waves have near-acoustic dispersion, discretized by the axial wave number kz =mz(π /Lp) . The nonlinear coupling rates are measured between large amplitude mz = 2 (pump) waves and small amplitude mz = 1 (daughter) waves, which have a small frequency detuning Δω = 2ω1 -ω2 . Classical 3-wave parametric coupling rates are proportional to pump wave amplitude as Γ (δn2 /n0) , with oscillatory energy exchange for Γ Δω / 2 . Experiments on cold plasmas agree quantitatively for oscillatory energy exchange, and agree within a factor-of-two for decay instability rates. However, nascent theory suggest that this latter agreement is merely fortuitous, and that the instability mechanism is trapped particles. Experiments at higher temperatures show that trapped particles reduce the instability threshold below classical 3-wave theory predictions. Supported by NSF Grant PHY-1414570, and DOE Grants DE-SC0002451 and DE-SC0008693. M. Affolter is supported by the DOE FES Postdoctoral Research Program administered by ORISE for the DOE. ORISE is managed by ORAU under DOE Contract Number DE-SC0014664.
6. Enhancing the sensitivity of three-axis detectable surface acoustic wave gyroscope by using a floating thin piezoelectric membrane
Science.gov (United States)
Lee, Munhwan; Lee, Keekeun
2017-06-01
A new type of surface acoustic wave (SAW) gyroscope was developed on a floating thin piezoelectric membrane to enhance sensitivity and reliability by removing a bulk noise effect and by importing a higher amplitude of SAW. The developed device constitutes a two-port SAW resonator with a metallic dot array between two interdigital transducers (IDTs), and a one-port SAW delay line. The bulk silicon was completely etched away, leaving only a thin piezoelectric membrane with a thickness of one wavelength. A voltage controlled oscillator (VCO) was connected to a SAW resonator to activate the SAW resonator, while the SAW delay line was connected to the oscilloscope to monitor any variations caused by the Coriolis force. When the device was rotated, a secondary wave was generated, changing the amplitude of the SAW delay line. The highest sensitivity was observed in a device with a full acoustic wavelength thickness of the membrane because most of the acoustic field is confined within an acoustic wavelength thickness from the top surface; moreover, the thin-membrane-based gyroscope eliminates the bulk noise effect flowing along the bulk substrate. The obtained sensitivity and linearity of the SAW gyroscope were ˜27.5 µV deg-1 s-1 and ˜4.3%, respectively. Superior directivity was observed. The device surface was vacuum-sealed using poly(dimethylsiloxane) (PDMS) bonding to eliminate environmental interference. A three-axis detectable gyroscope was also implemented by placing three gyrosensors with the same configuration at right angles to each other on a printed circuit board.
7. A new mechanism for observation of THz acoustic waves: coherent THz radiation emission
Science.gov (United States)
Reed, Evan J.; Armstrong, Michael R.; Kim, Ki-Yong; Glownia, James M.; Howard, William M.; Piner, Edwin L.; Roberts, John C.
2009-02-01
Our simulations and experiments demonstrate a new physical mechanism for detecting acoustic waves of THz frequencies. We find that strain waves of THz frequencies can coherently generate radiation when they propagate past an interface between materials with different piezoelectric coefficients. By considering AlN/GaN heterostructures, we show that the radiation is of detectable amplitude and contains sufficient information to determine the time-dependence of the strain wave with potentially sub-picosecond, nearly atomic time and space resolution. This mechanism is distinct from optical approaches to strain wave measurement. We demonstrate this phenomenon within the context of high amplitude THz frequency strain waves that spontaneously form at the front of shock waves in GaN crystals. We also show how the mechanism can be utilized to determine the layer thicknesses in thin film GaN/AlN heterostructures.
8. A Novel Particulate Matter 2.5 Sensor Based on Surface Acoustic Wave Technology
Directory of Open Access Journals (Sweden)
Jiuling Liu
2018-01-01
Full Text Available Design, fabrication and experiments of a miniature particulate matter (PM 2.5 sensor based on the surface acoustic wave (SAW technology were proposed. The sensor contains a virtual impactor (VI for particle separation, a thermophoretic precipitator (TP for PM2.5 capture and a SAW sensor chip for PM2.5 mass detection. The separation performance of the VI was evaluated by using the finite element method (FEM model and the PM2.5 deposition characteristic in the TP was obtained by analyzing the thermophoretic theory. Employing the coupling-of-modes (COM model, a low loss and high-quality SAW resonator was designed. By virtue of the micro electro mechanical system (MEMS technology and semiconductor technology, the SAW based PM2.5 sensor detecting probe was fabricated. Then, combining a dual-port SAW oscillator and an air sampler, the experimental platform was set up. Exposing the PM2.5 sensor to the polystyrene latex (PSL particles in a chamber, the sensor performance was evaluated. The results show that by detecting the PSL particles with a certain diameter of 2 μm, the response of the SAW based PM2.5 sensor is linear, and in accordance with the response of the light scattering based PM2.5 monitor. The developed SAW based PM2.5 sensor has great potential for the application of airborne particle detection.
9. A surface acoustic wave passive and wireless sensor for magnetic fields, temperature, and humidity
KAUST Repository
Li, Bodong
2015-01-01
In this paper, we report an integrated single-chip surface acoustic wave sensor with the capability of measuring magnetic field, temperature, and humidity. The sensor is fabricated using a thermally sensitive LiNbO3 substrate, a humidity sensitive hydrogel coating, and a magnetic field sensitive impedance load. The sensor response to individually and simultaneously changing magnetic field, temperature and humidity is characterized by connecting a network analyzer directly to the sensor. Analytical models for each measurand are derived and used to compensate noise due to cross sensitivities. The results show that all three measurands can be monitored in parallel with sensitivities of 75 ppm/°C, 0.13 dB/%R.H. (at 50%R.H.), 0.18 dB/Oe and resolutions of 0.1 °C, 0.4%R.H., 1 Oe for temperature, humidity and magnetic field, respectively. A passive wireless measurement is also conducted on a current line using, which shows the sensors capability to measure both temperature and current signals simultaneously.
10. Acoustotaxis -in vitro stimulation in a wound healing assay employing surface acoustic waves.
Science.gov (United States)
Stamp, M E M; Brugger, M S; Wixforth, A; Westerhausen, C
2016-07-21
A novel, ultrasound based approach for the dynamic stimulation and promotion of tissue healing processes employing surface acoustic waves (SAW) on a chip is presented for the example of osteoblast-like SaOs-2 cells. In our investigations, we directly irradiate cells with SAW on a SiO2 covered piezoelectric LiNbO3 substrate. Observing the temporal evolution of cell growth and migration and comparing non-irradiated to irradiated areas on the chip, we find that the SAW-treated cells exhibit a significantly increased migration as compared to the control samples. Apart from quantifying our experimental findings on the cell migration stimulation, we also demonstrate the full bio compatibility and bio functionality of our SAW technique by using LDH assays. We safely exclude parasitic side effects such as a SAW related increased substrate temperature or nutrient flow by thoroughly monitoring the temperature and the flow field using infrared microscopy and micro particle image velocimetry. Our results show that the SAW induced dynamic mechanical and electrical stimulation obviously directly promotes the cell growth. We conclude that this stimulation method offers a powerful platform for future medical treatment, e.g. being implemented as a implantable biochip with wireless extra-corporal power supply to treat deeper tissue.
11. A method for crack sizing using Laser Doppler Vibrometer measurements of Surface Acoustic Waves.
Science.gov (United States)
Longo, Roberto; Vanlanduit, Steve; Vanherzeele, Joris; Guillaume, Patrick
2010-01-01
The goal of non-destructive testing (NDT) is to determine the position and size of structural defects, in order to measure the quality and evaluate the safety of building materials. Most NDT techniques are rather complex, however, requiring specialized knowledge. In this article, we introduce an experimental method for crack detection that uses Surface Acoustic Waves (SAWs) and optical measurements. The method is tested on a steel beam engraved with slots of known depth. A simple model to determine the cracks size is also proposed. At the end of the article, we describe a possible application: fatigue crack sizing on a damaged slat track. This technique represents a first step toward a better understanding of the crack growth, especially in its early stages (preferably when the cracks can still be repaired) and when it is possible to assume a linear propagation of the crack front. The ultimate goal of this research program is to develop a useful method of monitoring aircraft components during fatigue testing.
12. P Wave and S Wave Acoustic Velocities of Partial Molten Peridotite at Mantle P-T and MHz Frequencies
Science.gov (United States)
Weidner, D. J.; Li, L.; Whitaker, M. L.; Triplett, R.
2016-12-01
The speed that acoustic waves travel in a partially molten peridotite are crucial parameters to detect not only the presence of melt in the Earth's deep interior, but also understand many issues about the structure and dynamics of the mantle. Technical challenges have hindered such measurements in the laboratory. Here we report the experimental results on the ultrasonic acoustic wave velocities in a partial molten peridotite using multi-anvil high pressure apparatus located at beamline BM6 Advance Photon Source. We use the newly installed ultrasonic equipment using the pulse-echo-overlap method coupled with D-DIA device. X-ray radiography is used to measure sample length at high P-T. The X-ray diffraction spectrum is used to determine the pressure and sample conditions. Precise measurements of P and S wave velocities are obtained at 60 and 35 MHz respectively and are nearly simultaneous. We use a double reflector method to enable measurement of elastic wave velocities of cold-pressed polycrystalline sample which is sintered in situ at high P-T. Experiments were carried out up to 3 GPa and 1500 oC. Our preliminary results indicate that the KLB1 peridotite sample experienced a few percent decrease of both p and s wave velocities as partial melting occurs. The data define a small decrease in the bulk modulus as well as the shear modulus upon melting. This implies that dynamic melting is a significant process at megahertz frequencies.
13. Nonlocal analysis of the excitation of the geodesic acoustic mode by drift waves
DEFF Research Database (Denmark)
Guzdar, P.N.; Kleva, R.G.; Chakrabarti, N.
2009-01-01
The geodesic acoustic modes (GAMs) are typically observed in the edge region of toroidal plasmas. Drift waves have been identified as a possible cause of excitation of GAMs by a resonant three wave parametric process. A nonlocal theory of excitation of these modes in inhomogeneous plasmas typical...... of the edge region of tokamaks is presented in this paper. The continuum GAM modes with coupling to the drift waves can create discrete "global" unstable eigenmodes localized in the edge "pedestal" region of the plasma. Multiple resonantly driven unstable radial eigenmodes can coexist on the edge pedestal....
14. Acoustic Emissions (AE) Electrical Systems' Health Monitoring Project
Data.gov (United States)
National Aeronautics and Space Administration — Acoustic Emissions (AE) are associated with physical events, such as thermal activity, dielectric breakdown, discharge inception, as well as crack nucleation and...
15. Prediction and near-field observation of skull-guided acoustic waves
CERN Document Server
2016-01-01
Ultrasound waves propagating in water or soft biological tissue are strongly reflected when encountering the skull, which limits the use of ultrasound-based techniques in transcranial imaging and therapeutic applications. Current knowledge on the acoustic properties of the cranial bone is restricted to far-field observations, leaving its near-field properties unexplored. We report on the existence of skull-guided acoustic waves, which was herein confirmed by near-field measurements of optoacoustically-induced responses in ex-vivo murine skulls immersed in water. Dispersion of the guided waves was found to reasonably agree with the prediction of a multilayered flat plate model. It is generally anticipated that our findings may facilitate and broaden the application of ultrasound-mediated techniques in brain diagnostics and therapy.
16. Measurements of Finite Dust Temperature Effects in the Dispersion Relation of the Dust Acoustic Wave
Science.gov (United States)
Snipes, Erica; Williams, Jeremiah
2009-04-01
A dusty plasma is a four-component system composed of ions, electrons, neutral particles and charged microparticles. The presence of these charged microparticles gives rise to new plasma wave modes, including the dust acoustic wave. Recent measurements [1, 2] of the dispersion relationship for the dust acoustic wave in a glow discharge have shown that finite temperature effects are observed at higher values of neutral pressure. Other work [3] has shown that these effects are not observed at lower values of neutral pressure. We present the results of ongoing work examining finite temperature effects in the dispersion relation as a function of neutral pressure. [4pt] [1] E. Thomas, Jr., R. Fisher, and R. L. Merlino, Phys. Plasmas 14, 123701 (2007). [0pt] [2] J. D. Williams, E. Thomas Jr., and L. Marcus, Phys. Plasmas 15, 043704 (2008). [0pt] [3] T. Trottenberg, D. Block, and A. Piel, Phys. Plasmas 13, 042105 (2006).
17. Acoustic waves in the atmosphere and ground generated by volcanic activity
Energy Technology Data Exchange (ETDEWEB)
Ichihara, Mie; Lyons, John; Oikawa, Jun; Takeo, Minoru [Earthquake Research Institute, University of Tokyo, 1-1-1 Yayoi, Bunkyo-ku, Tokyo 113-0032 (Japan); Instituto Geofisico, Escuela Politecnica Nacional, Ladron de Guevara E11-253, Aptdo 2759, Quito (Ecuador); Earthquake Research Institute, University of Tokyo, 1-1-1 Yayoi, Bunkyo-ku, Tokyo 113-0032 (Japan)
2012-09-04
This paper reports an interesting sequence of harmonic tremor observed in the 2011 eruption of Shinmoe-dake volcano, southern Japan. The main eruptive activity started with ashcloud forming explosive eruptions, followed by lava effusion. Harmonic tremor was transmitted into the ground and observed as seismic waves at the last stage of the effusive eruption. The tremor observed at this stage had unclear and fluctuating harmonic modes. In the atmosphere, on the other hand, many impulsive acoustic waves indicating small surface explosions were observed. When the effusion stopped and the erupted lava began explosive degassing, harmonic tremor started to be transmitted also to the atmosphere and observed as acoustic waves. Then the harmonic modes became clearer and more stable. This sequence of harmonic tremor is interpreted as a process in which volcanic degassing generates an open connection between the volcanic conduit and the atmosphere. In order to test this hypothesis, a laboratory experiment was performed and the essential features were successfully reproduced.
18. Characterization of acoustic wave propagation in a concrete member after fire exposure
Science.gov (United States)
Chiang, Chih-Hung; Huang, Chin-Ting
2001-04-01
The acoustic wave propagation in a concrete member with embedded reinforcing bars was analyzed. Fire exposure was applied to two batches of concrete specimens prior to acoustic wave characterization. The fire duration and maximum temperature were simulated for experimental studies using a custom-built electric oven. A standard ultrasonic pulse velocity testing system for concrete was used to provide the through-transmission wave propagation. Multiple peaks were found in the frequency domain based on the fast Fourier transform of the waveform. This could be due to cracks induced by the incompatibility of thermal deformation of the constituents of concrete. Further study showed bond deterioration between reinforcing bars and concrete would also contribute to the variation in frequency content of the recorded waveform.
19. Comparison of artificial absorbing boundaries for acoustic wave equation modelling
Science.gov (United States)
Gao, Yingjie; Song, Hanjie; Zhang, Jinhai; Yao, Zhenxing
2017-12-01
Absorbing boundary conditions are necessary in numerical simulation for reducing the artificial reflections from model boundaries. In this paper, we overview the most important and typical absorbing boundary conditions developed throughout history. We first derive the wave equations of similar methods in unified forms; then, we compare their absorbing performance via theoretical analyses and numerical experiments. The Higdon boundary condition is shown to be the best one among the three main absorbing boundary conditions that are based on a one-way wave equation. The Clayton and Engquist boundary is a special case of the Higdon boundary but has difficulty in dealing with the corner points in implementaion. The Reynolds boundary does not have this problem but its absorbing performance is the poorest among these three methods. The sponge boundary has difficulties in determining the optimal parameters in advance and too many layers are required to achieve a good enough absorbing performance. The hybrid absorbing boundary condition (hybrid ABC) has a better absorbing performance than the Higdon boundary does; however, it is still less efficient for absorbing nearly grazing waves since it is based on the one-way wave equation. In contrast, the perfectly matched layer (PML) can perform much better using a few layers. For example, the 10-layer PML would perform well for absorbing most reflected waves except the nearly grazing incident waves. The 20-layer PML is suggested for most practical applications. For nearly grazing incident waves, convolutional PML shows superiority over the PML when the source is close to the boundary for large-scale models. The Higdon boundary and hybrid ABC are preferred when the computational cost is high and high-level absorbing performance is not required, such as migration and migration velocity analyses, since they are not as sensitive to the amplitude errors as the full waveform inversion.
20. Acoustical method of whole-body hydration status monitoring
Science.gov (United States)
Sarvazyan, A. P.; Tsyuryupa, S. N.; Calhoun, M.; Utter, A.
2016-07-01
An acoustical handheld hydration monitor (HM) for assessing the water balance of the human body was developed. Dehydration is a critical public health problem. Many elderly over age of 65 are particularly vulnerable as are infants and young children. Given that dehydration is both preventable and reversible, the need for an easy-to-perform method for the detection of water imbalance is of the utmost clinical importance. The HM is based on an experimental fact that ultrasound velocity in muscle is a linear function of water content and can be referenced to the hydration status of the body. Studies on the validity of HM for the assessment of whole-body hydration status were conducted in the Appalachian State University, USA, on healthy young adults and on elderly subjects residing at an assisted living facility. The HM was able to track changes in total body water during periods of acute dehydration and rehydration in athletes and day-to-day and diurnal variability of hydration in elderly. Results of human studies indicate that HM has a potential to become an efficient tool for detecting abnormal changes in the body hydration status.
1. Alfvén wave amplification as a result of nonlinear interaction with a magnetoacoustic wave in an acoustically active conducting medium
Science.gov (United States)
Zavershinsky, D. I.; Molevich, N. E.
2014-08-01
It is shown that Alfvén waves propagating parallel and antiparallel to a magnetic field can be generated and amplified in an acoustically active heat-releasing ionized medium. The amplification is due to parametric energy pumping from the unstable magnetoacoustic waves to the Alfvén waves.
2. Miniature acoustic wave lysis system and uses thereof
Energy Technology Data Exchange (ETDEWEB)
Branch, Darren W.; Vreeland, Erika Cooley; Smith, Gennifer Tanabe
2016-12-06
The present invention relates to an acoustic lysis system including a disposable cartridge that can be reversibly coupled to a platform having a small, high-frequency piezoelectric transducer array. In particular, the system releases viable DNA, RNA, and proteins from human or bacterial cells, without chemicals or additional processing, to enable high-speed sample preparation for clinical point-of-care medical diagnostics and use with nano/microfluidic cartridges. Also described herein are methods of making and using the system of the invention.
3. Magneto-acoustic ceramics for parametric sound wave phase conjugators
Science.gov (United States)
Brysev; Pernod; Preobrazhensky
2000-03-01
The paper reflects the recent experimental results on the elaboration and study of active materials for magneto-acoustic phase conjugators (MAPCs). The results of complex measurements of MAPC parameters are demonstrated on typical samples of NiFe2O4 magnetostrictive ceramics. The mechanism of strong dispersion of gain increments and output power of MAPCs is studied and explained by dispersion of critical values of the parametric modulation depth of sound velocity. A maximum output power 240 W at frequency 5 MHz is obtained for active element MAPC with critical current Ic = 9 A and electrical Q-factor equal to 80.
4. Coupling a Surface Acoustic Wave to an Electron Spin in Diamond via a Dark State
Directory of Open Access Journals (Sweden)
D. Andrew Golter
2016-12-01
Full Text Available The emerging field of quantum acoustics explores interactions between acoustic waves and artificial atoms and their applications in quantum information processing. In this experimental study, we demonstrate the coupling between a surface acoustic wave (SAW and an electron spin in diamond by taking advantage of the strong strain coupling of the excited states of a nitrogen vacancy center while avoiding the short lifetime of these states. The SAW-spin coupling takes place through a Λ-type three-level system where two ground spin states couple to a common excited state through a phonon-assisted as well as a direct dipole optical transition. Both coherent population trapping and optically driven spin transitions have been realized. The coherent population trapping demonstrates the coupling between a SAW and an electron spin coherence through a dark state. The optically driven spin transitions, which resemble the sideband transitions in a trapped-ion system, can enable the quantum control of both spin and mechanical degrees of freedom and potentially a trapped-ion-like solid-state system for applications in quantum computing. These results establish an experimental platform for spin-based quantum acoustics, bridging the gap between spintronics and quantum acoustics.
5. Coupling a Surface Acoustic Wave to an Electron Spin in Diamond via a Dark State
Science.gov (United States)
Golter, D. Andrew; Oo, Thein; Amezcua, Mayra; Lekavicius, Ignas; Stewart, Kevin A.; Wang, Hailin
2016-10-01
The emerging field of quantum acoustics explores interactions between acoustic waves and artificial atoms and their applications in quantum information processing. In this experimental study, we demonstrate the coupling between a surface acoustic wave (SAW) and an electron spin in diamond by taking advantage of the strong strain coupling of the excited states of a nitrogen vacancy center while avoiding the short lifetime of these states. The SAW-spin coupling takes place through a Λ -type three-level system where two ground spin states couple to a common excited state through a phonon-assisted as well as a direct dipole optical transition. Both coherent population trapping and optically driven spin transitions have been realized. The coherent population trapping demonstrates the coupling between a SAW and an electron spin coherence through a dark state. The optically driven spin transitions, which resemble the sideband transitions in a trapped-ion system, can enable the quantum control of both spin and mechanical degrees of freedom and potentially a trapped-ion-like solid-state system for applications in quantum computing. These results establish an experimental platform for spin-based quantum acoustics, bridging the gap between spintronics and quantum acoustics.
6. Multi-wavelength Observations of Solar Acoustic Waves Near Active Regions
Science.gov (United States)
Monsue, Teresa; Pesnell, Dean; Hill, Frank
2018-01-01
Active region areas on the Sun are abundant with a variety of waves that are both acoustically helioseismic and magnetohydrodynamic in nature. The occurrence of a solar flare can disrupt these waves, through MHD mode-mixing or scattering by the excitation of these waves. We take a multi-wavelength observational approach to understand the source of theses waves by studying active regions where flaring activity occurs. Our approach is to search for signals within a time series of images using a Fast Fourier Transform (FFT) algorithm, by producing multi-frequency power map movies. We study active regions both spatially and temporally and correlate this method over multiple wavelengths using data from NASA’s Solar Dynamics Observatory. By surveying the active regions on multiple wavelengths we are able to observe the behavior of these waves within the Solar atmosphere, from the photosphere up through the corona. We are able to detect enhancements of power around active regions, which could be acoustic power halos and of an MHD-wave propagating outward by the flaring event. We are in the initial stages of this study understanding the behaviors of these waves and could one day contribute to understanding the mechanism responsible for their formation; that has not yet been explained.
7. Phase control of electromagnetically induced acoustic wave transparency in a diamond nanomechanical resonator
Energy Technology Data Exchange (ETDEWEB)
Evangelou, Sofia, E-mail: [email protected]
2017-05-10
Highlights: • A high-Q single-crystal diamond nanomechanical resonator embedded with nitrogen-vacancy (NV) centers is studied. • A Δ-type coupling configuration is formed. • The spin states of the ground state triplet of the NV centers interact with a strain field and two microwave fields. • The absorption and dispersion properties of the acoustic wave field are controlled by the use of the relative phase of the fields. • Phase-dependent acoustic wave absorption, transparency, and gain are obtained. • “Slow sound” and negative group velocities are also possible. - Abstract: We consider a high-Q single-crystal diamond nanomechanical resonator embedded with nitrogen-vacancy (NV) centers. We study the interaction of the transitions of the spin states of the ground state triplet of the NV centers with a strain field and two microwave fields in a Δ-type coupling configuration. We use the relative phase of the fields for the control of the absorption and dispersion properties of the acoustic wave field. Specifically, we show that by changing the relative phase of the fields, the acoustic field may exhibit absorption, transparency, gain and very interesting dispersive properties.
8. Topology optimization applied to room acoustic problems and surface acoustic wave devices
DEFF Research Database (Denmark)
Dühring, Maria Bayard; Sigmund, Ole; Jensen, Jakob Søndergaard
and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: low-loss waveguide bends”, Applied Physics Letters, 84(12), 2022-2024 (2004) [3] M. B. Dühring, “Topology optimization for acoustic problems”, IUTAM Symposium on Topological Design Optimization of Structures......The work of this PhD-project is concerned with the method of topology optimization1, which has been developed and used since the late eighties to optimize the material distribution of structures in order to minimize static compliance. Since then it has successfully been applied to a range...... of engineering fields such as mechanism design, fluid problems and photonic and phononic band-gap materials and structures [1,2]. In this project topology optimization is first applied to control acoustic properties in a room [3]. It is shown how the squared sound pressure amplitude in a certain part of a room...
9. Quantitative acoustic emission monitoring of fatigue cracks in fracture critical steel bridges.
Science.gov (United States)
2014-01-01
The objective of this research is to evaluate the feasibility to employ quantitative acoustic : emission (AE) techniques for monitoring of fatigue crack initiation and propagation in steel : bridge members. Three A36 compact tension steel specimens w...
10. Time Reversal Acoustic Structural Health Monitoring Using Array of Embedded Sensors Project
Data.gov (United States)
National Aeronautics and Space Administration — Time Reversal Acoustic (TRA) structural health monitoring with an embedded sensor array represents a new approach to in-situ nondestructive evaluation of air-space...
11. Spatiotemporal monitoring of high-intensity focused ultrasound therapy with passive acoustic mapping
National Research Council Canada - National Science Library
Jensen, Carl R; Ritchie, Robert W; Gyöngy, Miklós; Collin, James R T; Leslie, Tom; Coussios, Constantin-C
2012-01-01
To demonstrate feasibility of monitoring high-intensity focused ultrasound (HIFU) treatment with passive acoustic mapping of broadband and harmonic emissions reconstructed from filtered-channel radiofrequency data in ex vivo bovine tissue...
12. Acoustic streaming in a microfluidic channel with a reflector: Case of a standing wave generated by two counterpropagating leaky surface waves.
Science.gov (United States)
Doinikov, Alexander A; Thibault, Pierre; Marmottant, Philippe
2017-07-01
A theory is developed for the modeling of acoustic streaming in a microfluidic channel confined between an elastic solid wall and a rigid reflector. A situation is studied where the acoustic streaming is produced by two leaky surface waves that propagate towards each other in the solid wall and thus form a combined standing wave in the fluid. Full analytical solutions are found for both the linear acoustic field and the field of the acoustic streaming. A dispersion equation is derived that allows one to calculate the wave speed in the system under study. The obtained solutions are used to consider particular numerical examples and to reveal the structure of the acoustic streaming. It is shown that two systems of vortices are established along the boundaries of the microfluidic channel.
13. Experimental observation of blood erythrocyte structure in the field of standing surface acoustic waves
Science.gov (United States)
Makalkin, D. I.; Korshak, B. A.; Brysev, A. P.
2017-09-01
The paper presents experimental results of observing the structurization effect for one of the formed elements of blood—erythrocytes—in the field of standing surface acoustic waves. Characteristic images of the striated structures formed by erythrocytes on the surface of lithium niobate as result of ultrasound action have been obtained. The results on the ultrasound structurization of erythrocytes in a blood sample and of calcium carbonate particles in an aqueous colloid solution have been comparatively analyzed. It has been noted that the achieved effect agrees qualitatively with the theoretical model of the behavior of colloid particle ensembles in an acoustic field developed by O.V. Rudenko et al.
14. Wavemaker theories for acoustic-gravity waves over a finite depth
CERN Document Server
Tian, Miao
2016-01-01
Acoustic-gravity waves (hereafter AGWs) in ocean have received much interest recently, mainly with respect to early detection of tsunamis as they travel at near the speed of sound in water which makes them ideal candidates for early detection of tsunamis. While the generation mechanisms of AGWs have been studied from the perspective of vertical oscillations of seafloor and triad wave-wave interaction, in the current study we are interested in their generation by wave-structure interaction with possible implication to the energy sector. Here, we develop two wavemaker theories to analyze different wave modes generated by impermeable (the classic Havelock's theory) and porous (porous wavemaker theory) plates in weakly compressible fluids. Slight modification has been made to the porous theory so that, unlike the previous theory, the new solution depends on the geometry of the plate. The expressions for three different types of plates (piston, flap, delta-function) are introduced. Analytical solutions are also de...
15. Theory of reflection reflection and transmission of electromagnetic, particle and acoustic waves
CERN Document Server
Lekner, John
2016-01-01
This book deals with the reflection of electromagnetic and particle waves by interfaces. The interfaces can be sharp or diffuse. The topics of the book contain absorption, inverse problems, anisotropy, pulses and finite beams, rough surfaces, matrix methods, numerical methods, reflection of particle waves and neutron reflection. Exact general results are presented, followed by long wave reflection, variational theory, reflection amplitude equations of the Riccati type, and reflection of short waves. The Second Edition of the Theory of Reflection is an updated and much enlarged revision of the 1987 monograph. There are new chapters on periodically stratified media, ellipsometry, chiral media, neutron reflection and reflection of acoustic waves. The chapter on anisotropy is much extended, with a complete treatment of the reflection and transmission properties of arbitrarily oriented uniaxial crystals. The book gives a systematic and unified treatment reflection and transmission of electromagnetic and particle...
16. Radiative transfer of acoustic waves in continuous complex media: Beyond the Helmholtz equation
CERN Document Server
Baydoun, Ibrahim; Pierrat, Romain; Derode, Arnaud
2016-01-01
Heterogeneity can be accounted for by a random potential in the wave equation. For acoustic waves in a fluid with fluctuations of both density and compressibility (as well as for electromagnetic waves in a medium with fluctuation of both permittivity and permeability) the random potential entails a scalar and an operator contribution. For simplicity, the latter is usually overlooked in multiple scattering theory: whatever the type of waves, this simplification amounts to considering the Helmholtz equation with a sound speed $c$ depending on position $\\mathbf{r}$. In this work, a radiative transfer equation is derived from the wave equation, in order to study energy transport through a multiple scattering medium. In particular, the influence of the operator term on various transport parameters is studied, based on the diagrammatic approach of multiple scattering. Analytical results are obtained for fundamental quantities of transport theory such as the transport mean-free path $\\ell^*$, scattering phase functi...
17. Nonlinear interaction between acoustic gravity waves in a rotating atmosphere
Directory of Open Access Journals (Sweden)
P. Axelsson
1996-01-01
Full Text Available The influence of the Earth's rotation on the resonant interaction of atmospheric waves is investigated. The explicit expressions for the coupling coefficients are presented. They are derived by means of two different techniques; first, by a direct expansion derivation from a set of reduced equations, and second, by a Hamiltonian method.
18. Modelling Acoustic Wave Propagation in Axisymmetric Varying-Radius Waveguides
DEFF Research Database (Denmark)
Bæk, David; Willatzen, Morten
2008-01-01
A computationally fast and accurate model (a set of coupled ordinary differential equations) for fluid sound-wave propagation in infinite axisymmetric waveguides of varying radius is proposed. The model accounts for fluid heat conduction and fluid irrotational viscosity. The model problem is solved...
19. Latest Trends in Acoustic Sensing
Directory of Open Access Journals (Sweden)
Cinzia Caliendo
2014-03-01
Full Text Available Acoustics-based methods offer a powerful tool for sensing applications. Acoustic sensors can be applied in many fields ranging from materials characterization, structural health monitoring, acoustic imaging, defect characterization, etc., to name just a few. A proper selection of the acoustic wave frequency over a wide spectrum that extends from infrasound (<20 Hz up to ultrasound (in the GHz–band, together with a number of different propagating modes, including bulk longitudinal and shear waves, surface waves, plate modes, etc., allow acoustic tools to be successfully applied to the characterization of gaseous, solid and liquid environments. The purpose of this special issue is to provide an overview of the research trends in acoustic wave sensing through some cases that are representative of specific applications in different sensing fields.
20. Auralization of concert hall acoustics using finite difference time domain methods and wave field synthesis
Science.gov (United States)
Hochgraf, Kelsey
Auralization methods have been used for a long time to simulate the acoustics of a concert hall for different seat positions. The goal of this thesis was to apply the concept of auralization to a larger audience area that the listener could walk through to compare differences in acoustics for a wide range of seat positions. For this purpose, the acoustics of Rensselaer's Experimental Media and Performing Arts Center (EMPAC) Concert Hall were simulated to create signals for a 136 channel wave field synthesis (WFS) system located at Rensselaer's Collaborative Research Augmented Immersive Virtual Environment (CRAIVE) Laboratory. By allowing multiple people to dynamically experience the concert hall's acoustics at the same time, this research gained perspective on what is important for achieving objective accuracy and subjective plausibility in an auralization. A finite difference time domain (FDTD) simulation on a three-dimensional face-centered cubic grid, combined at a crossover frequency of 800 Hz with a CATT-Acoustic(TM) simulation, was found to have a reverberation time, direct to reverberant sound energy ratio, and early reflection pattern that more closely matched measured data from the hall compared to a CATT-Acoustic(TM) simulation and other hybrid simulations. In the CRAIVE lab, nine experienced listeners found all hybrid auralizations (with varying source location, grid resolution, crossover frequency, and number of loudspeakers) to be more perceptually plausible than the CATT-Acoustic(TM) auralization. The FDTD simulation required two days to compute, while the CATT-Acoustic(TM) simulation required three separate TUCT(TM) computations, each taking four hours, to accommodate the large number of receivers. Given the perceptual advantages realized with WFS for auralization of a large, inhomogeneous sound field, it is recommended that hybrid simulations be used in the future to achieve more accurate and plausible auralizations. Predictions are made for a
1. Acoustic Pressure Waves in Vibrating 3-D Laminated Beam-Plate Enclosures
Directory of Open Access Journals (Sweden)
Charles A. Osheku
2009-01-01
Full Text Available The effect of structural vibration on the propagation of acoustic pressure waves through a cantilevered 3-D laminated beam-plate enclosure is investigated analytically. For this problem, a set of well-posed partial differential equations governing the vibroacoustic wave interaction phenomenon are formulated and matched for the various vibrating boundary surfaces. By employing integral transforms, a closed form analytical expression is computed suitable for vibroacoustic modeling, design analysis, and general aerospace defensive applications. The closed-form expression takes the form of a kernel of polynomials for acoustic pressure waves showing the influence of linear interface pressure variation across the axes of vibrating boundary surfaces. Simulated results demonstrate how the mode shapes and the associated natural frequencies can be easily computed. It is shown in this paper that acoustic pressure waves propagation are dynamically stable through laminated enclosures with progressive decrement in interfacial pressure distribution under the influence of high excitation frequencies irrespective of whether the induced flow is subsonic, sonic , supersonic, or hypersonic. Hence, in practice, dynamic stability of hypersonic aircrafts or jet airplanes can be further enhanced by replacing their noise transmission systems with laminated enclosures.
2. Measuring energy flux of magneto-acoustic wave in the magnetic elements by using IRIS
Science.gov (United States)
Kato, Yoshiaki; De Pontieu, Bart; Martinez-Sykora, Juan; Hansteen, Viggo; Pereira, Tiago; Leenaarts, Jorritt; Carlsson, Mats
NASA's Interface Region Imaging Spectrograph (IRIS) has opened a new window to explore the chromospheric/coronal waves that potentially energize the solar atmosphere. By using an imaging spectrograph covering the Si IV and Mg II h&k lines as well as a slit-jaw imager centered at Si IV and Mg II k onboard IRIS, we can determine the nature of propagating magneto-acoustic waves just below and in the transition region. In this study, we compute the vertically emergent intensity of the Si IV and Mg II h&k lines from a time series of snapshots of a magnetic element in a two-dimensional Radiative MHD simulation from the Bifrost code. We investigate the synthetic line profiles to detect the slow magneto-acoustic body wave (slow mode) which becomes a slow shock at the lower chromosphere in the magnetic element. We find that the Doppler shift of the line core gives the velocity amplitude of the longitudinal magneto-acoustic body wave. The contribution function of the line core indicates that the formation of Mg II h&k lines is associated with the propagating shocks and therefore the time evolution of the line core intensity represents the propagating shocks projected on the optical surface. We will report on measurement of the energy flux of slow modes in the magnetic elements by using IRIS observations.
3. Prediction and near-field observation of skull-guided acoustic waves
Science.gov (United States)
Estrada, Héctor; Rebling, Johannes; Razansky, Daniel
2017-06-01
Ultrasound waves propagating in water or soft biological tissue are strongly reflected when encountering the skull, which limits the use of ultrasound-based techniques in transcranial imaging and therapeutic applications. Current knowledge on the acoustic properties of the cranial bone is restricted to far-field observations, leaving its near-field unexplored. We report on the existence of skull-guided acoustic waves, which was herein confirmed by near-field measurements of optoacoustically-induced responses in ex-vivo murine skulls immersed in water. Dispersion of the guided waves was found to reasonably agree with the prediction of a multilayered flat plate model. We observed a skull-guided wave propagation over a lateral distance of at least 3 mm, with a half-decay length in the direction perpendicular to the skull ranging from 35 to 300 μm at 6 and 0.5 MHz, respectively. Propagation losses are mostly attributed to the heterogenous acoustic properties of the skull. It is generally anticipated that our findings may facilitate and broaden the application of ultrasound-mediated techniques in brain diagnostics and therapy.
4. Ionospheric Responses to Nonlinear Acoustic Waves Generated by Natural Hazard Events
Science.gov (United States)
Zettergren, M. D.; Snively, J. B.
2015-12-01
Ionospheric total electron content (TEC) fluctuations following large-magnitude earthquakes and resulting tsunamis, e.g. Tohoku in 2011, have been noted in many recent investigations [e.g., Galvan et al., Radio Science, 47(4), 2012]. Earthquakes impact the atmosphere through vertical displacements of the Earth's crust or ocean surfaces producing, as one effect, low-frequency acoustic waves. These waves can achieve significant amplitudes during propagation through the rarefied upper atmosphere, and are capable of driving sizable ionospheric electron density (TEC) fluctuations and electrical currents. Earthquake-generated acoustic waves are readily identifiable in GPS observations as 0.1-2 TECU, 3-5 mHz, oscillations, which are delayed from the quake occurrence by roughly the sound travel time between the ground and ionosphere. In some extreme cases, the onset of acoustic oscillations is concurrent with a persistent, sharp decrease in TEC (~5 TECU) above the epicenter [e.g., Kakinami et al., GRL, 39(13), 2012]. Ionospheric responses to large amplitude acoustic waves are investigated using a coupled atmosphere-ionosphere model [Zettergren and Snively, GRL, 40(20), 2013]. Of particular interest are effects of acoustic wave amplitude and nonlinearity on ionospheric responses, including production of detectable TEC oscillations and longer-lived responses like TEC depletions. The atmospheric dynamics model solves a Navier-Stokes' system of equations and incorporates generation of acoustic waves through acceleration source terms at ground-level. The ionospheric model solves a fluid system of equations for each of the major ionospheric species, and includes an electrostatic description of dynamo currents. The coupled model enables direct computation of observable quantities, such as vertical TEC and magnetic field fluctuations. Here we construct simulation case studies for realistic earthquake events and compare results against published TEC and magnetic field data. This
5. Effects of acoustic waves on stick-slip in granular media and implications for earthquakes
Science.gov (United States)
Johnson, P.A.; Savage, H.; Knuth, M.; Gomberg, J.; Marone, Chris
2008-01-01
It remains unknown how the small strains induced by seismic waves can trigger earthquakes at large distances, in some cases thousands of kilometres from the triggering earthquake, with failure often occurring long after the waves have passed. Earthquake nucleation is usually observed to take place at depths of 10-20 km, and so static overburden should be large enough to inhibit triggering by seismic-wave stress perturbations. To understand the physics of dynamic triggering better, as well as the influence of dynamic stressing on earthquake recurrence, we have conducted laboratory studies of stick-slip in granular media with and without applied acoustic vibration. Glass beads were used to simulate granular fault zone material, sheared under constant normal stress, and subject to transient or continuous perturbation by acoustic waves. Here we show that small-magnitude failure events, corresponding to triggered aftershocks, occur when applied sound-wave amplitudes exceed several microstrain. These events are frequently delayed or occur as part of a cascade of small events. Vibrations also cause large slip events to be disrupted in time relative to those without wave perturbation. The effects are observed for many large-event cycles after vibrations cease, indicating a strain memory in the granular material. Dynamic stressing of tectonic faults may play a similar role in determining the complexity of earthquake recurrence. ??2007 Nature Publishing Group.
6. Acoustic waves in the solar atmosphere. VII - Non-grey, non-LTE H(-) models
Science.gov (United States)
Schmitz, F.; Ulmschneider, P.; Kalkofen, W.
1985-01-01
The propagation and shock formation of radiatively damped acoustic waves in the solar chromosphere are studied under the assumption that H(-) is the only absorber; the opacity is non-grey. Deviations from local thermodynamic equilibrium (LTE) are permitted. The results of numerical simulations show the depth dependence of the heating by the acoustic waves to be insensitive to the mean state of the atmosphere. After the waves have developed into shocks, their energy flux decays exponentially with a constant damping length of about 1.4 times the pressure scale height, independent of initial flux and wave period. Departures from LTE have a strong influence on the mean temperature structure in dynamical chromosphere models; this is even more pronounced in models with reduced particle density - simulating conditions in magnetic flux tubes - which show significantly increased temperatures in response to mechanical heating. When the energy dissipation of the waves is sufficiently large to dissociate most of the H(-) ions, a strong temperature rise is found that is reminiscent of the temperature structure in the transition zone between chromosphere and corona; the energy flux remaining in the waves then drives mass motions.
7. Effects of acoustic waves on stick-slip in granular media and implications for earthquakes.
Science.gov (United States)
Johnson, Paul A; Savage, Heather; Knuth, Matt; Gomberg, Joan; Marone, Chris
2008-01-03
It remains unknown how the small strains induced by seismic waves can trigger earthquakes at large distances, in some cases thousands of kilometres from the triggering earthquake, with failure often occurring long after the waves have passed. Earthquake nucleation is usually observed to take place at depths of 10-20 km, and so static overburden should be large enough to inhibit triggering by seismic-wave stress perturbations. To understand the physics of dynamic triggering better, as well as the influence of dynamic stressing on earthquake recurrence, we have conducted laboratory studies of stick-slip in granular media with and without applied acoustic vibration. Glass beads were used to simulate granular fault zone material, sheared under constant normal stress, and subject to transient or continuous perturbation by acoustic waves. Here we show that small-magnitude failure events, corresponding to triggered aftershocks, occur when applied sound-wave amplitudes exceed several microstrain. These events are frequently delayed or occur as part of a cascade of small events. Vibrations also cause large slip events to be disrupted in time relative to those without wave perturbation. The effects are observed for many large-event cycles after vibrations cease, indicating a strain memory in the granular material. Dynamic stressing of tectonic faults may play a similar role in determining the complexity of earthquake recurrence.
8. Enhancing gas-phase reaction in a plasma using high intensity and high power ultrasonic acoustic waves
DEFF Research Database (Denmark)
2010-01-01
substantially 100 W. In this way, a high sound intensity and power are obtained that efficiently enhances a gas-phase reaction in the plasma, which enhances the plasma process, e.g. enabling more efficient ozone or hydrogen generation using plasma in relation to reaction speed and/or obtained concentration......This invention relates to enhancing a gas-phase reaction in a plasma comprising: creating plasma (104) by at least one plasma source (106), and wherein that the method further comprises: generating ultrasonic high intensity and high power acoustic waves (102) having a predetermined amount...... of acoustic energy by at least one ultrasonic high intensity and high power gas-jet acoustic wave generator (101), where said ultrasonic high intensity and high power acoustic waves are directed to propagate towards said plasma (104) so that at least a part of said predetermined amount of acoustic energy...
9. Monitoring of Global Acoustic Transmissions: Signal Processing and Preliminary Data Analysis
Science.gov (United States)
1991-09-01
signal level and 6.39x10- I volts2/Hz for the noise value. These values can be converted to acoustic power levels of 19 dB rel ipPa for the signal and...46.4 dB rel ipPa for the noise, as discussed below. Transmission loss over the mean acoustic path would then be about 213 dB - 19 dB = 194 dB. b. Signal...sensitivity of the hydrophones (-170 Db rel IpPa or 3.162 x 108 pPa/V). Acoustic power is the pressure force times the velocity of the pressure wave
10. Non-Invasive Acoustic-Based Monitoring of Heavy Water and Uranium Process Solutions
Energy Technology Data Exchange (ETDEWEB)
Pantea, Cristian [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Sinha, Dipen N. [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Lakis, Rollin Evan [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Beedle, Christopher Craig [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Davis, Eric Sean [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2017-10-20
This presentation includes slides on Project Goals; Heavy Water Production Monitoring: A New Challenge for the IAEA; Noninvasive Measurements in SFAI Cell; Large Scatter in Literature Values; Large Scatter in Literature Values; Highest Precision Sound Speed Data Available: New Standard in H/D; ~400 pts of data; Noninvasive Measurements in SFAI Cell; New funding from NA241 SGTech; Uranium Solution Monitoring: Inspired by IAEA Challenge in Kazakhstan; Non-Invasive Acoustic-Based Monitoring of Uranium in Solutions; Non-Invasive Acoustic-Based Monitoring of Uranium in Solutions; and finally a summary.
11. Nonlinear Scattering of Acoustic Waves by Vibrating Obstacles.
Science.gov (United States)
1983-06-01
completely neglected, the waveform becomes a triangular wave at a propagation distance equal to the discontinuity distance (see the discussion of this...the farfield (nearfield) is defined to be distances greater (lesser) than the distance to the last maximum in the on-axis diffraction pattern. This last...frequently referred to as the region in which Fraunhofer (Fresnel) diffractlion effects occur. 106 2. Electrical filtering problems due to experimental
12. Passive models of viscothermal wave propagation in acoustic tubes.
Science.gov (United States)
Bilbao, Stefan; Harrison, Reginald; Kergomard, Jean; Lombard, Bruno; Vergez, Christophe
2015-08-01
A continued fraction expansion to the immittances defining viscothermal wave propagation in a cylindrical tube has been presented recently in this journal, intended as a starting point for time domain numerical method design. Though the approximation has the great benefit of passivity, or positive realness under truncation, its convergence is slow leading to approximations of high order in practice. Other passive structures, when combined with optimisation methods, can lead to good accuracy over a wide frequency range, and for relatively low order.
13. Electron acoustic solitary waves with non-thermal distribution of electrons
Directory of Open Access Journals (Sweden)
S. V. Singh
2004-01-01
Full Text Available Electron-acoustic solitary waves are studied in an unmagnetized plasma consisting of non-thermally distributed electrons, fluid cold electrons and ions. The Sagdeev pseudo-potential technique is used to carry out the analysis. The presence of non-thermal electrons modifies the parametric region where electron acoustic solitons can exist. For parameters representative of auroral zone field lines, the electron acoustic solitons do not exist when either α > 0.225 or Tc/Th > 0.142, where α is the fractional non-thermal electron density, and Tc (Th represents the temperature of cold (hot electrons. Further, for these parameters, the simple model predicts negatively charged potential structures. Inclusion of an electron beam in the model may provide the positive potential solitary structures.
14. The frequency and damping of ion acoustic waves in collisional and collisionless two-species plasma
Energy Technology Data Exchange (ETDEWEB)
Berger, R L; Valeo, E J
2004-07-15
The dispersion properties of ion acoustic waves (IAW) are sensitive to the strength of ion-ion collisions in multi-species plasma in which the different species usually have differing charge-to-mass ratios. The modification of the frequency and damping of the fast and slow acoustic modes in a plasma composed of light (low Z) and heavy (high Z) ions is considered. In the fluid limit where the light ion scattering mean free path, {lambda}{sub th} is smaller than the acoustic wavelength, {lambda} = 2{pi}/k, the interspecies friction and heat flow carried by the light ions scattering from the heavy ions causes the damping. In the collisionless limit, k{lambda}{sub lh} >> 1, Landau damping by the light ions provides the dissipation. In the intermediate regime when k{lambda}{sub lh} {approx} 1, the damping is at least as large as the sum of the collisional and Landau damping.
15. The Frequency and Damping of Ion Acoustic Waves in Collisional and Collisionless Two-species Plasma
Energy Technology Data Exchange (ETDEWEB)
R.L. Berger; E.J. Valeo
2004-08-18
The dispersion properties of ion acoustic waves (IAW) are sensitive to the strength of ion-ion collisions in multi-species plasma in which the different species usually have differing charge-to-mass ratios. The modification of the frequency and damping of the fast and slow acoustic modes in a plasma composed of light (low Z) and heavy (high Z) ions is considered. In the fluid limit where the light ion scattering mean free path, {lambda}{sub th} is smaller than the acoustic wavelength, {lambda} = 2{pi}/k, the interspecies friction and heat flow carried by the light ions scattering from the heavy ions causes the damping. In the collisionless limit, k{lambda}{sub th} >> 1, Landau damping by the light ions provides the dissipation. In the intermediate regime when k{lambda}{sub th} {approx} 1, the damping is at least as large as the sum of the collisional and Landau damping.
16. Autonomous hydrophone array for long-term acoustic monitoring in the open ocean
Science.gov (United States)
D'Eu, J.-F.; Brachet, C.; Goslin, J.; Royer, J.-Y.; Ammann, J.
2009-04-01
We are developing an array of new autonomous hydrophones, benefiting from a long-lasting collaboration with the Pacific Marine Environmental Laboratory (NOAA and Oregon state University). The hydrophones are deployed on a mooring line anchored to the seafloor by an expendable anchor weight. The length of the line is adjusted so that the sensor (and buoy) lies in the middle of the SOFAR channel at about 1000m depth for mid-latitudes (depending on the speed-of-sound profile). The buoy at depth keeps the line under tension and prevents wave-motion noise from the sensor. The instrument continuously samples and records the acoustic signals at 240Hz for seismic studies, or 480Hz (or more) for marine mammal studies. The SOFAR channel acts as an acoustic wave-guide in the ocean so that acoustic waves can propagate with little attenuation over long distances. Autonomous hydrophones allow the detection and localization of the low-magnitude (Mw>2.5) seismic activity along oceanic ridges and in deformed intraplate areas, which remains generally undetected or poorly localized by land-based seismic networks. An array of hydrophones can monitor a much wider area (more than 1000 km across) than ocean-bottom seismometers, which suffer from the rapid attenuation of seismic waves in the crust and upper mantle. Arrays of autonomous hydrophones thus succeed in detecting and locating 30 to 50 times more earthquakes than those listed in the catalogs from land-based seismograph stations. Data are buffered on flash cards and then regularly stored on hard disks or on solid-state drives (e.g. 20Gb of data per year at 240Hz sampling rate). We use 24-bit sigma-delta converters with programmable gain amplifiers. As timing is a key issue for an accurate localisation of the seismic events, instruments are synchronized with GPS time and have a low-power, highly stable calibrated clock (10-8 drift). All electronics and batteries (Li or alcaline) are placed in titanium pressure cases for long
17. Remote acoustic monitoring of North Atlantic right whales (Eubalaena glacialis) reveals seasonal and diel variations in acoustic behavior.
Science.gov (United States)
Matthews, Leanna P; McCordic, Jessica A; Parks, Susan E
2014-01-01
Remote acoustic monitoring is a non-invasive tool that can be used to study the distribution, behavior, and habitat use of sound-producing species. The North Atlantic right whale (Eubalaena glacialis) is an endangered baleen whale species that produces a variety of stereotyped acoustic signals. One of these signals, the "gunshot" sound, has only been recorded from adult male North Atlantic right whales and is thought to function for reproduction, either as reproductive advertisement for females or as an agonistic signal toward other males. This study uses remote acoustic monitoring to analyze the presence of gunshots over a two-year period at two sites on the Scotian Shelf to determine if there is evidence that North Atlantic right whales may use these locations for breeding activities. Seasonal analyses at both locations indicate that gunshot sound production is highly seasonal, with an increase in the autumn. One site, Roseway West, had significantly more gunshot sounds overall and exhibited a clear diel trend in production of these signals at night. The other site, Emerald South, also showed a seasonal increase in gunshot production during the autumn, but did not show any significant diel trend. This difference in gunshot signal production at the two sites indicates variation either in the number or the behavior of whales at each location. The timing of the observed seasonal increase in gunshot sound production is consistent with the current understanding of the right whale breeding season, and our results demonstrate that detection of gunshots with remote acoustic monitoring can be a reliable way to track shifts in distribution and changes in acoustic behavior including possible mating activities.
18. Remote Acoustic Monitoring of North Atlantic Right Whales (Eubalaena glacialis) Reveals Seasonal and Diel Variations in Acoustic Behavior
Science.gov (United States)
Matthews, Leanna P.; McCordic, Jessica A.; Parks, Susan E.
2014-01-01
Remote acoustic monitoring is a non-invasive tool that can be used to study the distribution, behavior, and habitat use of sound-producing species. The North Atlantic right whale (Eubalaena glacialis) is an endangered baleen whale species that produces a variety of stereotyped acoustic signals. One of these signals, the “gunshot” sound, has only been recorded from adult male North Atlantic right whales and is thought to function for reproduction, either as reproductive advertisement for females or as an agonistic signal toward other males. This study uses remote acoustic monitoring to analyze the presence of gunshots over a two-year period at two sites on the Scotian Shelf to determine if there is evidence that North Atlantic right whales may use these locations for breeding activities. Seasonal analyses at both locations indicate that gunshot sound production is highly seasonal, with an increase in the autumn. One site, Roseway West, had significantly more gunshot sounds overall and exhibited a clear diel trend in production of these signals at night. The other site, Emerald South, also showed a seasonal increase in gunshot production during the autumn, but did not show any significant diel trend. This difference in gunshot signal production at the two sites indicates variation either in the number or the behavior of whales at each location. The timing of the observed seasonal increase in gunshot sound production is consistent with the current understanding of the right whale breeding season, and our results demonstrate that detection of gunshots with remote acoustic monitoring can be a reliable way to track shifts in distribution and changes in acoustic behavior including possible mating activities. PMID:24646524
19. Numerical study of the collar wave characteristics and the effects of grooves in acoustic logging while drilling
Science.gov (United States)
Yang, Yufeng; Guan, Wei; Hu, Hengshan; Xu, Minqiang
2017-05-01
Large-amplitude collar wave covering formation signals is still a tough problem in acoustic logging-while-drilling (LWD) measurements. In this study, we investigate the propagation and energy radiation characteristics of the monopole collar wave and the effects of grooves on reducing the interference to formation waves by finite-difference calculations. We found that the collar wave radiates significant energy into the formation by comparing the waveforms between a collar within an infinite fluid, and the acoustic LWD in different formations with either an intact or a truncated collar. The collar wave recorded on the outer surface of the collar consists of the outward-radiated energy direct from the collar (direct collar wave) and that reflected back from the borehole wall (reflected collar wave). All these indicate that the significant effects of the borehole-formation structure on collar wave were underestimated in previous studies. From the simulations of acoustic LWD with a grooved collar, we found that grooves broaden the frequency region of low collar-wave excitation and attenuate most of the energy of the interference waves by multireflections. However, grooves extend the duration of the collar wave and convert part of the collar-wave energy originally kept in the collar into long-duration Stoneley wave. Interior grooves are preferable to exterior ones because both the low-frequency and the high-frequency parts of the collar wave can be reduced and the converted inner Stoneley wave is relatively difficult to be recorded on the outer surface of the collar. Deeper grooves weaken the collar wave more greatly, but they result in larger converted Stoneley wave especially for the exterior ones. The interference waves, not only the direct collar wave but also the reflected collar wave and the converted Stoneley waves, should be overall considered for tool design.
20. Sunset Crater Volcano National Monument : Acoustical Monitoring 2010
Science.gov (United States)
2013-05-01
During the summer of 2010 (July - August), the Volpe Center collected baseline acoustical data at Sunset Crater Volcano National Monument (SUCR) at a site deployed for approximately 30 days. The baseline data collected during this period will help pa...
1. Acoustic detection and monitoring for transportation infrastructure security.
Science.gov (United States)
2009-09-01
Acoustical methods have been extensively used to locate, identify, and track objects underwater. Some of these applications include detecting and tracking submarines, marine mammal detection and identification, detection of mines and ship wrecks and ...
2. Casa Grande Ruins National Monument acoustical monitoring 2010
Science.gov (United States)
2014-11-01
During September 2010, The Volpe Center collected baseline acoustical data at Casa Grande National Monument (CAGR), at one site for 28 days. The baseline data collected during this period will help park managers and planners estimate the effects of f...
3. Acoustic monitoring of terrorist intrusion in a drinking water network
NARCIS (Netherlands)
Quesson, B.A.J.; Sheldon-Robert, M.K.; Vloerbergh, I.N.; Vreeburg, J.H.G.
2009-01-01
In collaboration with Kiwa Water Research, TNO (Netherlands Organisation for Applied Scientific Research) has investigated the possibilities to detect and classify aberrant sounds in water networks, using acoustic sensors. Amongst the sources of such sounds are pumps, drills, mechanical impacts,
4. Aespoe Pillar Stability Experiment. Acoustic emission and ultrasonic monitoring
Energy Technology Data Exchange (ETDEWEB)
Haycox, Jon; Pettitt, Will; Young, R. Paul [Applied Seismology Consultants Ltd., Shrewsbury (United Kingdom)
2005-12-15
This report describes the results from acoustic emission (AE) and ultrasonic monitoring of the Aespoe Pillar Stability Experiment (APSE) at SKB's Hard Rock Laboratory (HRL), Sweden. The APSE is being undertaken to demonstrate the current capability to predict spalling in a fractured rock mass using numerical modelling techniques, and to demonstrate the effect of backfill and confining pressure on the propagation of micro-cracks in rock adjacent to deposition holes within a repository. An ultrasonic acquisition system has provided acoustic emission and ultrasonic survey monitoring throughout the various phases of the experiment. Results from the entire data set are provided with this document so that they can be effectively compared to several numerical modelling studies, and to mechanical and thermal measurements conducted around the pillar volume, in an 'integrated analysis' performed by SKB staff. This document provides an in-depth summary of the AE and ultrasonic survey results for future reference. The pillar has been produced by excavating two 1.8 m diameter deposition holes 1 m apart. These were bored in 0.8 m steps using a Tunnel Boring Machine specially adapted for vertical drilling. The first deposition hole was drilled in December 2003. Preceding this a period of background monitoring was performed so as to obtain a datum for the results. The hole was then confined to 0.7 MPa internal over pressure using a specially designed water-filled bladder. The second deposition hole was excavated in March 2004. Heating of the pillar was performed over a two month period between ending in July 2004, when the confined deposition hole was slowly depressurised. Immediately after depressurisation the pillar was allowed to cool with cessation of monitoring occurring a month later. A total of 36,676 AE triggers were recorded over the reporting period between 13th October 2003 and 14th July 2004. Of these 15,198 have produced AE locations. The AE data set
5. A Comment on Interaction of Lower Hybrid Waves with the Current-Driven Ion-Acoustic Instability
DEFF Research Database (Denmark)
Schrittwieser, R.; Juul Rasmussen, Jens
1985-01-01
Majeski et al. (1984) have investigated the interaction between the current-driven 'ion-acoustic' instability and high frequency lower hybrid waves. The 'ion-acoustic' instability was excited by drawing an electron current through the plasma column of a single-ended Q-machine by means...
6. Validation of simulations of an underwater acoustic communication channel characterized by wind-generated surface waves and bubbles
NARCIS (Netherlands)
Dol, H.S.; Colin, M.E.G.D.; Ainlie, M.A.; Gerdes, F.; Schäfke, A.; Özkan Sertlekc, H.
2013-01-01
This paper shows that it is possible to simulate realistic shallow-water acoustic communication channels using available acoustic propagation models. Key factor is the incorporation of realistic time-dependent sea surface conditions, including both waves and bubbles due to wind.
7. Effects of aspect ratio on the mode couplings of thin-film bulk acoustic wave resonators
Science.gov (United States)
Li, Nian; Qian, Zhenghua; Yang, Jiashi
2017-05-01
We studied mode couplings in thin film bulk acoustic wave resonators of a piezoelectric film on a dielectric layer operating with the fundamental thickness-extensional mode. A system of plate equations derived in our previous paper was used which includes the couplings to the unwanted in-plane extension, flexure, fundamental and second-order thickness shear modes. It was shown that the couplings depend strongly on the plate length/thickness ratio. For a relatively clean operating mode with weak couplings to unwanted modes, a series of discrete values of the plate length/thickness ratio should be avoided and these values were determined in the present paper. The results can be of great significance to the design and optimization of film bulk acoustic wave resonators.
8. Rapid calculation of acoustic fields from arbitrary continuous-wave sources.
Science.gov (United States)
Treeby, Bradley E; Budisky, Jakub; Wise, Elliott S; Jaros, Jiri; Cox, B T
2018-01-01
A Green's function solution is derived for calculating the acoustic field generated by phased array transducers of arbitrary shape when driven by a single frequency continuous wave excitation with spatially varying amplitude and phase. The solution is based on the Green's function for the homogeneous wave equation expressed in the spatial frequency domain or k-space. The temporal convolution integral is solved analytically, and the remaining integrals are expressed in the form of the spatial Fourier transform. This allows the acoustic pressure for all spatial positions to be calculated in a single step using two fast Fourier transforms. The model is demonstrated through several numerical examples, including single element rectangular and spherically focused bowl transducers, and multi-element linear and hemispherical arrays.
9. Lattice Boltzmann approach for hydro-acoustic waves generated by tsunamigenic sea bottom displacement
Science.gov (United States)
Prestininzi, P.; Abdolali, A.; Montessori, A.; Kirby, J. T.; La Rocca, Michele
2016-11-01
Tsunami waves are generated by sea bottom failures, landslides and faults. The concurrent generation of hydro-acoustic waves (HAW), which travel much faster than the tsunami, has received much attention, motivated by their possible exploitation as precursors of tsunamis. This feature makes the detection of HAW particularly well-suited for building an early-warning system. Accuracy and efficiency of the modeling approaches for HAW thus play a pivotal role in the design of such systems. Here, we present a Lattice Boltzmann Method (LBM) for the generation and propagation of HAW resulting from tsunamigenic ground motions and verify it against commonly employed modeling solutions. LBM is well known for providing fast and accurate solutions to both hydrodynamics and acoustics problems, thus it naturally becomes a candidate as a comprehensive computational tool for modeling generation and propagation of HAW.
10. Laser photoacoustic technique for ultrasonic surface acoustic wave velocity evaluation on porcelain
Science.gov (United States)
Qian, K.; Tu, S. J.; Gao, L.; Xu, J.; Li, S. D.; Yu, W. C.; Liao, H. H.
2016-10-01
A laser photoacoustic technique has been developed to evaluate the surface acoustic wave (SAW) velocity of porcelain. A Q-switched Nd:YAG laser at 1064 nm was focused by a cylindrical lens to initiate broadband SAW impulses, which were detected by an optical fiber interferometer with high spatial resolution. Multiple near-field surface acoustic waves were observed on the sample surface at various locations along the axis perpendicular to the laser line source as the detector moved away from the source in the same increments. The frequency spectrum and dispersion curves were obtained by operating on the recorded waveforms with cross-correlation and FFT. The SAW phase velocities of the porcelain of the same source are similar while they are different from those of different sources. The marked differences of Rayleigh phase velocities in our experiment suggest that this technique has the potential for porcelain identification.
11. Spin dynamics in (110) GaAs quantum wells under surface acoustic waves
Science.gov (United States)
Couto, Odilon D. D., Jr.; Hey, R.; Santos, P. V.
2008-10-01
Long spin transport lengths (>60μm) independent of temperature up to approximately 80 K are demonstrated in (110) GaAs quantum wells using surface acoustic waves (SAWs). Study of the dynamics of spins aligned along the [110] direction shows that, in addition to the intrinsic absence of the D’yakonov-Perel’ spin-relaxation mechanism [Sov. Phys. Semicond.20, 110 (1986)], the Bir-Aronov-Pikus mechanism [Sov. Phys. JETP 42, 705 (1976)] is also suppressed due to the type-II carrier confinement imposed by the SAW piezoelectric potential. Experimental evidence is provided for suppression of the spin relaxation via motional narrowing effects induced by the mesoscopic carrier confinement in narrow stripes along the SAW wave front, thus demonstrating the tuning of the spin-relaxation rates with the acoustic power.
12. Surface acoustic wave regulated single photon emission from a coupled quantum dot-nanocavity system
CERN Document Server
Weiß, Matthias; Reichert, Thorsten; Finley, Jonathan J; Wixforth, Achim; Kaniber, Michael; Krenner, Hubert J
2016-01-01
A coupled quantum dot--nanocavity system in the weak coupling regime of cavity quantumelectrodynamics is dynamically tuned in and out of resonance by the coherent elastic field of a $f_{\\rm SAW}\\simeq800\\,\\mathrm{MHz}$ surface acoustic wave. When the system is brought to resonance by the sound wave, light-matter interaction is strongly increased by the Purcell effect. This leads to a precisely timed single photon emission as confirmed by the second order photon correlation function $g^{(2)}$. All relevant frequencies of our experiment are faithfully identified in the Fourier transform of $g^{(2)}$, demonstrating high fidelity regulation of the stream of single photons emitted by the system. The implemented scheme can be directly extended to strongly coupled systems and acoustically drives non-adiabatic entangling quantum gates based on Landau-Zener transitions.
13. Spectral element method for elastic and acoustic waves in frequency domain
Energy Technology Data Exchange (ETDEWEB)
Shi, Linlin; Zhou, Yuanguo; Wang, Jia-Min; Zhuang, Mingwei [Institute of Electromagnetics and Acoustics, and Department of Electronic Science, Xiamen, 361005 (China); Liu, Na, E-mail: [email protected] [Institute of Electromagnetics and Acoustics, and Department of Electronic Science, Xiamen, 361005 (China); Liu, Qing Huo, E-mail: [email protected] [Department of Electrical and Computer Engineering, Duke University, Durham, NC, 27708 (United States)
2016-12-15
Numerical techniques in time domain are widespread in seismic and acoustic modeling. In some applications, however, frequency-domain techniques can be advantageous over the time-domain approach when narrow band results are desired, especially if multiple sources can be handled more conveniently in the frequency domain. Moreover, the medium attenuation effects can be more accurately and conveniently modeled in the frequency domain. In this paper, we present a spectral-element method (SEM) in frequency domain to simulate elastic and acoustic waves in anisotropic, heterogeneous, and lossy media. The SEM is based upon the finite-element framework and has exponential convergence because of the use of GLL basis functions. The anisotropic perfectly matched layer is employed to truncate the boundary for unbounded problems. Compared with the conventional finite-element method, the number of unknowns in the SEM is significantly reduced, and higher order accuracy is obtained due to its spectral accuracy. To account for the acoustic-solid interaction, the domain decomposition method (DDM) based upon the discontinuous Galerkin spectral-element method is proposed. Numerical experiments show the proposed method can be an efficient alternative for accurate calculation of elastic and acoustic waves in frequency domain.
14. Spectral element method for elastic and acoustic waves in frequency domain
Science.gov (United States)
Shi, Linlin; Zhou, Yuanguo; Wang, Jia-Min; Zhuang, Mingwei; Liu, Na; Liu, Qing Huo
2016-12-01
Numerical techniques in time domain are widespread in seismic and acoustic modeling. In some applications, however, frequency-domain techniques can be advantageous over the time-domain approach when narrow band results are desired, especially if multiple sources can be handled more conveniently in the frequency domain. Moreover, the medium attenuation effects can be more accurately and conveniently modeled in the frequency domain. In this paper, we present a spectral-element method (SEM) in frequency domain to simulate elastic and acoustic waves in anisotropic, heterogeneous, and lossy media. The SEM is based upon the finite-element framework and has exponential convergence because of the use of GLL basis functions. The anisotropic perfectly matched layer is employed to truncate the boundary for unbounded problems. Compared with the conventional finite-element method, the number of unknowns in the SEM is significantly reduced, and higher order accuracy is obtained due to its spectral accuracy. To account for the acoustic-solid interaction, the domain decomposition method (DDM) based upon the discontinuous Galerkin spectral-element method is proposed. Numerical experiments show the proposed method can be an efficient alternative for accurate calculation of elastic and acoustic waves in frequency domain.
15. Acoustic Ecology and Remote Acoustic Monitoring of a Minke Whale Population
National Research Council Canada - National Science Library
Gedamke, Jason
2000-01-01
Sound is the most effective means of communication in the ocean. A uniquely inquisitive minke whale population on the northern Great Barrier Reef presents an unprecedented research opportunity to study minke acoustics...
16. Tunable arrayed waveguide grating driven by surface acoustic waves
Science.gov (United States)
Crespo-Poveda, Antonio; Hernández-Mínguez, Alberto; Biermann, Klaus; Tahraoui, Abbes; Gargallo, Bernardo; Muñoz, Pascual; Santos, Paulo V.; Cantarero, Andrés.; de Lima, Maurício M.
2016-03-01
We present a design approach for compact reconfigurable phased-array wavelength-division multiplexing (WDM) devices with N access waveguides (WGs) based on multimode interference (MMI) couplers. The proposed devices comprise two MMI couplers which are employed as power splitters and combiners, respectively, linked by an array of N single-mode WGs. First, passive devices are explored. Taking advantage of the transfer phases between the access ports of the MMI couplers, we derive very simple phase relations between the arms that provide wavelength dispersion at the output plane of the devices. When the effective refractive index of the WGs is modulated with the proper relative optical phase difference, each wavelength component can switch paths between the preset output channel and the remaining output WGs. Moreover, very simple phase relations between the modulated WGs that enable the reconfiguration of the output channel distribution when the appropriated coupling lengths of the MMI couplers are chosen are also derived. In this way, a very compact expression to calculate the channel assignment of the devices as a function of the applied phase shift is derived for the general case of N access WGs. Finally, the experimental results corresponding to an acoustically driven phased-array WDM device with five access WGs fabricated on (Al,Ga)As are shown.
17. Investigation of Ion Acoustic Wave Instabilities Near Positive Electrodes
Science.gov (United States)
Hood, Ryan; Chu, Feng; Baalrud, Scott; Merlino, Robert; Skiff, Fred
2017-10-01
Electron sheaths occur when an electrode is biased above the plasma potential, most often during the electron saturation portion of a Langmuir probe trace. Through the presheath, electrons are accelerated to velocities exceeding the electron thermal speed at the sheath edge, while ions do not develop any appreciable flow. PIC simulations have shown that ion acoustic instabilities are excited by the differential flow between ions and electrons in the presheath region of a low temperature plasma. We present the first experimental measurements investigating these instabilities using Laser-Induced Fluorescence diagnostics in a multidipole argon plasma. The plasma dispersion relation is measured from the power spectra of the imaged LIF signal and compared to the simulation results. In addition, optical pumping is measured using time-resolved LIF measurements and fit to a model in order to determine the diffusion rate, which may be enhanced due to the instability. This research was supported by the Office of Fusion Energy Sciences at the U.S. Department of Energy under contract DE-AC04-94SL85000.
18. Scattering of Evanescent Acoustic Waves by Regular and Irregular Objects
Science.gov (United States)
2006-12-01
simulated bottom. This system of liquids is more suitable for long-term indoor use than the vegetable-oil/ glycerin system used in related studies by a...published [ 18,19]. X. Reference List for the Main Report [1] C. F. Osterhoudt, Ph. D. Thesis (in preparation). [2] P. L. Marston, Annual Report for...evanescent waves incident on targets in a simulated sediment," to be presented at the December 2006 ASA meeting. [9] C. F. Osterhoudt, Ph.D. thesis in
19. Tuning Acoustic Wave Properties by Mechanical Resonators on a Surface
DEFF Research Database (Denmark)
Dühring, Maria Bayard; Laude, Vincent; Khelif, Abdelkrim
Vibrations generated by high aspects ratio electrodes are studied by the finite element method. It is found that the modes are combined of a surface wave and vibration in the electrodes. For increasing aspect ratio most of the mechanical energy is confined to the electrodes which act as mechanical...... resonators and slow down the velocity. It is furthermore found that the group delay can be increased compared to conventional thin electrodes. These results are interesting for filters and resonators as well as for delay lines....
20. A Possible Generalization of Acoustic Wave Equation Using the Concept of Perturbed Derivative Order
Directory of Open Access Journals (Sweden)
Abdon Atangana
2013-01-01
Full Text Available The standard version of acoustic wave equation is modified using the concept of the generalized Riemann-Liouville fractional order derivative. Some properties of the generalized Riemann-Liouville fractional derivative approximation are presented. Some theorems are generalized. The modified equation is approximately solved by using the variational iteration method and the Green function technique. The numerical simulation of solution of the modified equation gives a better prediction than the standard one.
1. Qubit-Assisted Transduction for a Detection of Surface Acoustic Waves near the Quantum Limit
Science.gov (United States)
Noguchi, Atsushi; Yamazaki, Rekishu; Tabuchi, Yutaka; Nakamura, Yasunobu
2017-11-01
We demonstrate ultrasensitive measurement of fluctuations in a surface-acoustic-wave (SAW) resonator using a hybrid quantum system consisting of the SAW resonator, a microwave (MW) resonator, and a superconducting qubit. The nonlinearity of the driven qubit induces parametric coupling, which up-converts the excitation in the SAW resonator to that in the MW resonator. Thermal fluctuations of the SAW resonator near the quantum limit are observed in the noise spectroscopy in the MW domain.
2. Experimental Study of Dust Acoustic Waves in the Strongly Correlated Regime
CERN Document Server
2016-01-01
Low frequency dust acoustic waves (DAW) were excited in a laboratory argon dusty plasma by modulating the discharge voltage with a low frequency AC signal. Metallic graphite particles were used as dust grains and a digital FFT technique was used to obtain dispersion characteristics. The experimental dispersion relation shows the reduction of phase velocity and a regime where $\\partial \\omega/\\partial k < 0$. A comparison is made with existing theoretical model.
3. Photonic integrated single-sideband modulator / frequency shifter based on surface acoustic waves
DEFF Research Database (Denmark)
Barretto, Elaine Cristina Saraiva; Hvam, Jørn Märcher
2010-01-01
Optical frequency shifters are essential components of many systems. In this paper, a compact integrated optical frequency shifter is designed making use of the combination of surface acoustic waves and Mach-Zehnder interferometers. It has a very simple operation setup and can be fabricated...... in standard semiconductor materials. The performance of the device is analyzed in detail, and by using multi-branch interferometers, the sensitivity of the device to fabrication tolerances can be drastically reduced....
4. Observation of self-excited dust acoustic wave in dusty plasma with nanometer size dust grains
Science.gov (United States)
Deka, Tonuj; Boruah, A.; Sharma, S. K.; Bailung, H.
2017-09-01
Dusty plasma with a nanometer size dust grain is produced by externally injecting carbon nanopowder into a radio frequency discharge argon plasma. A self-excited dust acoustic wave with a characteristic frequency of ˜100 Hz is observed in the dust cloud. The average dust charge is estimated from the Orbital Motion Limited theory using experimentally measured parameters. The measured wave parameters are used to determine dusty plasma parameters such as dust density and average inter particle distance. The screening parameter and the coupling strength of the dusty plasma indicate that the system is very close to the strongly coupled state.
5. Design of Passive Acoustic Wave Shaping Devices and Their Experimental Validation
DEFF Research Database (Denmark)
Christiansen, Rasmus Ellebæk; Sigmund, Ole; Fernandez Grande, Efren
We discuss a topology optimization based approach for designing passive acoustic wave shaping devices and demonstrate its application to; directional sound emission [1], sound focusing and wave splitting. Optimized devices, numerical and experimental results are presented and benchmarked against...... by the Helmholtz equation. An exterior 2D model domain is used and an array of point sources is considered as sound emitters. The optimization goal is to identify a distribution of solid material in a design sub-domain which produces a desired spatial sound feld pattern across a frequency band of interest...
6. Experimental observation of strong coupling effects on the dispersion of dust acoustic waves in a plasma
CERN Document Server
Bandyopadhyay, P; Sen, A; Kaw, P K
2016-01-01
The dispersion properties of low frequency dust acoustic waves in the strong coupling regime are investigated experimentally in an argon plasma embedded with a mixture of kaolin and $MnO_2$ dust particles. The neutral pressure is varied over a wide range to change the collisional properties of the dusty plasma. In the low collisional regime the turnover of the dispersion curve at higher wave numbers and the resultant region of $\\partial\\omega/\\partial k < 0$ are identified as signatures of dust-dust correlations. In the high collisional regime dust neutral collisions produce a similar effect and prevent an unambiguous identification of strong coupling effects.
7. Ion sound and dust acoustic waves at finite size of plasma particles
CERN Document Server
Andreev, Pavel A
2014-01-01
We consider influence of finite size of ions on properties of classic plasmas. We focus our attention on the ion sound for electron-ion plasmas. We also consider dusty plasmas, where we account finite size of ions and particles of dust and consider the dispersion of dust acoustic waves. Finite size of particles affects classical plasma properties. Finite size of particles gives considerable contribution for small wave lengths, which is area of appearing of quantum effects. Consequently, it is very important to consider finite size of ions in quantum plasmas as well.
8. Surface acoustic waves propagating over a rotating piezoelectric half-space.
Science.gov (United States)
Fang, H; Yang, J; Jiang, Q
2001-07-01
Surface acoustic waves (SAW) propagating over a piezoelectric half-space rotating at a constant angular rate about a fixed axis are analyzed using the linear theory of piezoelectricity, including Coriolis and centrifugal forces. Rotation sensitivity, the rotation induced change of wave speed, is studied. The dependence of the rotation sensitivity on the orientation of the rotation axis and the orientation of the material is examined. Numerical results for polarized ceramics PZT-5H are presented to show the detailed characteristics of the rotation sensitivity. The implications of the numerical results are discussed for different applications.
9. Calculation of an axial temperature distribution using the reflection coefficient of an acoustic wave.
Science.gov (United States)
Červenka, Milan; Bednařík, Michal
2015-10-01
This work verifies the idea that in principle it is possible to reconstruct axial temperature distribution of fluid employing reflection or transmission of acoustic waves. It is assumed that the fluid is dissipationless and its density and speed of sound vary along the wave propagation direction because of the fluid temperature distribution. A numerical algorithm is proposed allowing for calculation of the temperature distribution on the basis of known frequency characteristics of reflection coefficient modulus. Functionality of the algorithm is illustrated on a few examples, its properties are discussed.
10. Electron acoustic solitary waves in a magnetized plasma with nonthermal electrons and an electron beam
Energy Technology Data Exchange (ETDEWEB)
Singh, S. V., E-mail: [email protected]; Lakhina, G. S., E-mail: [email protected] [Indian Institute of Geomagnetism, New Panvel (W), Navi Mumbai (India); University of the Western Cape, Belville (South Africa); Devanandhan, S., E-mail: [email protected] [Indian Institute of Geomagnetism, New Panvel (W), Navi Mumbai (India); Bharuthram, R., E-mail: [email protected] [University of the Western Cape, Belville (South Africa)
2016-08-15
A theoretical investigation is carried out to study the obliquely propagating electron acoustic solitary waves having nonthermal hot electrons, cold and beam electrons, and ions in a magnetized plasma. We have employed reductive perturbation theory to derive the Korteweg-de-Vries-Zakharov-Kuznetsov (KdV-ZK) equation describing the nonlinear evolution of these waves. The two-dimensional plane wave solution of KdV-ZK equation is analyzed to study the effects of nonthermal and beam electrons on the characteristics of the solitons. Theoretical results predict negative potential solitary structures. We emphasize that the inclusion of finite temperature effects reduces the soliton amplitudes and the width of the solitons increases by an increase in the obliquity of the wave propagation. The numerical analysis is presented for the parameters corresponding to the observations of “burst a” event by Viking satellite on the auroral field lines.
11. Experimental observation of electron-acoustic wave propagation in laboratory plasma
Science.gov (United States)
Chowdhury, Satyajit; Biswas, Subir; Chakrabarti, Nikhil; Pal, Rabindranath
2017-06-01
In the field of fundamental plasma waves, the direct observation of electron-acoustic wave (EAW) propagation in laboratory plasmas remains a challenging problem, mainly because of heavy damping. In the Magnetized Plasma Linear Experimental device, the wave is observed and seen to propagate with the phase velocity ˜ 1.8 times the electron thermal velocity. A small amount of cold, drifting electrons, with the moderate bulk to cold temperature ratio ( ≈ 2 - 3), is present in the device. It plays a crucial role in reducing the damping. Our calculation reveals that the drift relaxes the stringent condition on the temperature ratio for wave destabilization. Growth rate becomes positive above a certain drift velocity even if the temperature ratio is moderate. The observed phase velocity agrees well with the theoretical estimate. Experimental realization of the mode may open up a new avenue in the EAW research.
12. Two-dimensional cylindrical ion-acoustic solitary and rogue waves in ultrarelativistic plasmas
Energy Technology Data Exchange (ETDEWEB)
Ata-ur-Rahman [Institute of Physics and Electronics, University of Peshawar, Peshawar 25000 (Pakistan); National Centre for Physics at QAU Campus, Shahdrah Valley Road, Islamabad 44000 (Pakistan); Ali, S. [National Centre for Physics at QAU Campus, Shahdrah Valley Road, Islamabad 44000 (Pakistan); Moslem, W. M. [Department of Physics, Faculty of Science, Port Said University, Port Said 42521 (Egypt); Mushtaq, A. [National Centre for Physics at QAU Campus, Shahdrah Valley Road, Islamabad 44000 (Pakistan); Department of Physics, Abdul Wali Khan University, Mardan 23200 (Pakistan)
2013-07-15
The propagation of ion-acoustic (IA) solitary and rogue waves is investigated in a two-dimensional ultrarelativistic degenerate warm dense plasma. By using the reductive perturbation technique, the cylindrical Kadomtsev–Petviashvili (KP) equation is derived, which can be further transformed into a Korteweg–de Vries (KdV) equation. The latter admits a solitary wave solution. However, when the frequency of the carrier wave is much smaller than the ion plasma frequency, the KdV equation can be transferred to a nonlinear Schrödinger equation to study the nonlinear evolution of modulationally unstable modified IA wavepackets. The propagation characteristics of the IA solitary and rogue waves are strongly influenced by the variation of different plasma parameters in an ultrarelativistic degenerate dense plasma. The present results might be helpful to understand the nonlinear electrostatic excitations in astrophysical degenerate dense plasmas.
13. Waves and Structures in Nonlinear Nondispersive Media General Theory and Applications to Nonlinear Acoustics
CERN Document Server
Gurbatov, S N; Saichev, A I
2012-01-01
"Waves and Structures in Nonlinear Nondispersive Media: General Theory and Applications to Nonlinear Acoustics” is devoted completely to nonlinear structures. The general theory is given here in parallel with mathematical models. Many concrete examples illustrate the general analysis of Part I. Part II is devoted to applications to nonlinear acoustics, including specific nonlinear models and exact solutions, physical mechanisms of nonlinearity, sawtooth-shaped wave propagation, self-action phenomena, nonlinear resonances and engineering application (medicine, nondestructive testing, geophysics, etc.). This book is designed for graduate and postgraduate students studying the theory of nonlinear waves of various physical nature. It may also be useful as a handbook for engineers and researchers who encounter the necessity of taking nonlinear wave effects into account of their work. Dr. Gurbatov S.N. is the head of Department, and Vice Rector for Research of Nizhny Novgorod State University. Dr. Rudenko O.V. is...
14. Measurement of Elastic Properties of Tissue by Shear Wave Propagation Generated by Acoustic Radiation Force
Science.gov (United States)
Tabaru, Marie; Azuma, Takashi; Hashiba, Kunio
2010-07-01
Acoustic radiation force (ARF) imaging has been developed as a novel elastography technology to diagnose hepatic disease and breast cancer. The accuracy of shear wave speed estimation, which is one of the applications of ARF elastography, is studied. The Young's moduli of pig liver and foie gras samples estimated from the shear wave speed were compared with those measured the static Young's modulus measurement. The difference in the two methods was 8%. Distance attenuation characteristics of the shear wave were also studied using finite element method (FEM) analysis. We found that the differences in the axial and lateral beam widths in pressure and ARF are 16 and 9% at F-number=0.9. We studied the relationship between two branch points in distance attenuation characteristics and the shape of ARF. We found that the maximum measurable length to estimate shear wave speed for one ARF excitation was 8 mm.
15. Control of single photon emitters in semiconductor nanowires by surface acoustic waves
Science.gov (United States)
Lazić, S.; Hernández-Mínguez, A.; Santos, P. V.
2017-08-01
We report on an experimental study into the effects of surface acoustic waves on the optical emission of dot-in-a-nanowire heterostructures in III-V material systems. Under direct optical excitation, the excitonic energy levels in III-nitride dot-in-a-nanowire heterostructures oscillate at the acoustic frequency, producing a characteristic splitting of the emission lines in the time-integrated photoluminescence spectra. This acoustically induced periodic tuning of the excitonic transition energies is combined with spectral detection filtering and employed as a tool to regulate the temporal output of anti-bunched photons emitted from these nanowire quantum dots. In addition, the acoustic transport of electrons and holes along a III-arsenide nanowire injects the electric charges into an ensemble of quantum dot-like recombination centers that are spatially separated from the optical excitation area. The acoustic population and depopulation mechanism determines the number of carrier recombination events taking place simultaneously in the ensemble, thus allowing control of the anti-bunching degree of the emitted photons. The results presented are relevant for the dynamic control of single photon emission in III-V semiconductor heterostructures.
16. Effect of particle-particle interactions on the acoustic radiation force in an ultrasonic standing wave
Energy Technology Data Exchange (ETDEWEB)
Lipkens, Bart, E-mail: [email protected] [Mechanical Engineering, Western New England University, Springfield, Massachusetts, 01119 (United States); Ilinskii, Yurii A., E-mail: [email protected]; Zabolotskaya, Evgenia A., E-mail: [email protected] [Applied Research Laboratories, The University of Texas at Austin, Austin, Texas 78713–8029 (United States)
2015-10-28
Ultrasonic standing waves are widely used for separation applications. In MEMS applications, a half wavelength standing wave field is generated perpendicular to a laminar flow. The acoustic radiation force exerted on the particle drives the particle to the center of the MEMS channel, where concentrated particles are harvested. In macro-scale applications, the ultrasonic standing wave spans multiple wavelengths. Examples of such applications are oil/water emulsion splitting [1], and blood/lipid separation [2]. In macro-scale applications, particles are typically trapped in the standing wave, resulting in clumping or coalescence of particles/droplets. Subsequent gravitational settling results in separation of the secondary phase. An often used expression for the radiation force on a particle is that derived by Gorkov [3]. The assumptions are that the particle size is small relative to the wavelength, and therefore, only monopole and dipole scattering contributions are used to calculate the radiation force. This framework seems satisfactory for MEMS scale applications where each particle is treated separately by the standing wave, and concentrations are typically low. In macro-scale applications, particle concentration is high, and particle clumping or droplet coalescence results in particle sizes not necessarily small relative to the wavelength. Ilinskii et al. developed a framework for calculation of the acoustic radiation force valid for any size particle [4]. However, this model does not take into account particle to particle effects, which can become important as particle concentration increases. It is known that an acoustic radiation force on a particle or a droplet is determined by the local field. An acoustic radiation force expression is developed that includes the effect of particle to particle interaction. The case of two neighboring particles is considered. The approach is based on sound scattering by the particles. The acoustic field at the location of
17. Bifurcation analysis for ion acoustic waves in a strongly coupled plasma including trapped electrons
Science.gov (United States)
El-Labany, S. K.; El-Taibany, W. F.; Atteya, A.
2018-02-01
The nonlinear ion acoustic wave propagation in a strongly coupled plasma composed of ions and trapped electrons has been investigated. The reductive perturbation method is employed to derive a modified Korteweg-de Vries-Burgers (mKdV-Burgers) equation. To solve this equation in case of dissipative system, the tangent hyperbolic method is used, and a shock wave solution is obtained. Numerical investigations show that, the ion acoustic waves are significantly modified by the effect of polarization force, the trapped electrons and the viscosity coefficients. Applying the bifurcation theory to the dynamical system of the derived mKdV-Burgers equation, the phase portraits of the traveling wave solutions of both of dissipative and non-dissipative systems are analyzed. The present results could be helpful for a better understanding of the waves nonlinear propagation in a strongly coupled plasma, which can be produced by photoionizing laser-cooled and trapped electrons [1], and also in neutron stars or white dwarfs interior.
18. Wave field characterization for non-destructive assessment of elastic properties using laser-acoustic sources in fluids and eye related tissues
Science.gov (United States)
Windisch, T.; Schubert, F.; Köhler, B.; Spörl, E.
2010-03-01
The age-related changes in the visco-elastic properties of the human lens are discussed with respect to presbyopia for a long time. All known measurement techniques are based on extracted lenses or are damaging the tissue. Hence, in vivo studies of lens hardness are not possible at the moment. To close this gap in lens diagnostics this project deals with an approach for a non-contact laser-acoustic characterization technique. Laser-generated wave fronts are reflected by the tissue interfaces and are also affected by the visco-elastic properties of the lens tissue. After propagating through the eye, these waves are recorded as corneal vibrations by laser vibrometry. A systematic analysis of amplitude and phase of these signals and the wave generation process shall give information about the interface locations and the tissues viscoelastic properties. Our recent studies on extracted porcine eyes proved that laser-acoustic sources can be systematically used for non-contacting generation and recording of ultrasound inside the human eye. Furthermore, a specific numerical model provides important contributions to the understanding of the complex wave propagation process. Measurements of the acoustic sources support this approach. Future investigations are scheduled to answer the question, whether this novel technique can be directly used during a laser surgery for monitoring purposes and if a purely diagnostic approach, e.g. by excitation in the aqueous humor, is also possible. In both cases, this technique offers a promising approach for non-contact ultrasound based eye diagnostics.
19. Plasma Heating During the Parametric Excitation of Acoustic Waves in Coronal Magnetic Loops
Science.gov (United States)
Zaitsev, V. V.; Kislyakova, K. G.
When studying microwave emission of active regions on the Sun, an effect of parametric resonance between 5-min velocity oscillations in the solar photosphere and sound oscillations of coronal magnetic loops modulating the microwave emission has been discovered for the first time. The effect shows itself as simultaneous excitation in coronal magnetic loop oscillations with periods 5, 10, and 3 min, which correspond to the pumping frequency, subharmonic, and the first upper frequency of parametric resonance. The parametric resonance can serve as an effective channel of transporting the energy of photospheric oscillations into the upper layers of the solar atmosphere. The energy of acoustic waves excited in a coronal magnetic loop, rate of dissipation of acoustic waves, and rate of heating of the coronal plasma are determined. The maximum temperature predicted for the apex of the loop is calculated as a function of velocity of photospheric oscillations, length of the loop, and electric current in the loop. It is shown that the mechanism proposed can explain the origin of quasi-stationary X-ray loops with temperatures of 3-6 MK. The lengths of these loops are resonant for acoustic waves excited by the 5-min photospheric oscillations. The use of the proposed mechanism to explain heating of the X-ray loops expected to be on stars of late spectral types is discussed.
20. Directional Acoustic Wave Manipulation by a Porpoise via Multiphase Forehead Structure
Science.gov (United States)
Zhang, Yu; Song, Zhongchang; Wang, Xianyan; Cao, Wenwu; Au, Whitlow W. L.
2017-12-01
Porpoises are small-toothed whales, and they can produce directional acoustic waves to detect and track prey with high resolution and a wide field of view. Their sound-source sizes are rather small in comparison with the wavelength so that beam control should be difficult according to textbook sonar theories. Here, we demonstrate that the multiphase material structure in a porpoise's forehead is the key to manipulating the directional acoustic field. Computed tomography (CT) derives the multiphase (bone-air-tissue) complex, tissue experiments obtain the density and sound-velocity multiphase gradient distributions, and acoustic fields and beam formation are numerically simulated. The results suggest the control of wave propagations and sound-beam formations is realized by cooperation of the whole forehead's tissues and structures. The melon size significantly impacts the side lobes of the beam and slightly influences the main beams, while the orientation of the vestibular sac mainly adjusts the main beams. By compressing the forehead complex, the sound beam can be expanded for near view. The porpoise's biosonar allows effective wave manipulations for its omnidirectional sound source, which can help the future development of miniaturized biomimetic projectors in underwater sonar, medical ultrasonography, and other ultrasonic imaging applications.
1. Detection of Volatile Organics Using a Surface Acoustic Wave Array System
Energy Technology Data Exchange (ETDEWEB)
ANDERSON, LAWRENCE F.; BARTHOLOMEW, JOHN W.; CERNOSEK, RICHARD W.; COLBURN, CHRISTOPHER W.; CROOKS, R.M.; MARTINEZ, R.F.; OSBOURN, GORDON C.; RICCO, A.J.; STATON, ALAN W.; YELTON, WILLIAM G.
1999-10-14
A chemical sensing system based on arrays of surface acoustic wave (SAW) delay lines has been developed for identification and quantification of volatile organic compounds (VOCs). The individual SAW chemical sensors consist of interdigital transducers patterned on the surface of an ST-cut quartz substrate to launch and detect the acoustic waves and a thin film coating in the SAW propagation path to perturb the acoustic wave velocity and attenuation during analyte sorption. A diverse set of material coatings gives the sensor arrays a degree of chemical sensitivity and selectivity. Materials examined for sensor application include the alkanethiol-based self-assembled monolayer, plasma-processed films, custom-synthesized conventional polymers, dendrimeric polymers, molecular recognition materials, electroplated metal thin films, and porous metal oxides. All of these materials target a specific chemical fi.mctionality and the enhancement of accessible film surface area. Since no one coating provides absolute analyte specificity, the array responses are further analyzed using a visual-empirical region-of-influence (VERI) pattern recognition algorithm. The chemical sensing system consists of a seven-element SAW array with accompanying drive and control electronics, sensor signal acquisition electronics, environmental vapor sampling hardware, and a notebook computer. Based on data gathered for individual sensor responses, greater than 93%-accurate identification can be achieved for any single analyte from a group of 17 VOCs and water.
2. The quality of our drinking water: aluminium determination with an acoustic wave sensor.
Science.gov (United States)
Veríssimo, Marta I S; Gomes, M Teresa S R
2008-06-09
A new methodology based on an inexpensive aluminium acoustic wave sensor is presented. Although the aluminium sensor has already been reported, and the composition of the selective membrane is known, the low detection limits required for the analysis of drinking water, demanded the inclusion of a preconcentration stage, as well as an optimization of the sensor. The necessary coating amount was established, as well as the best preconcentration protocol, in terms of oxidation of organic matter and aluminium elution from the Chelex-100. The methodology developed with the acoustic wave sensor allowed aluminium quantitation above 0.07 mg L(-1). Several water samples from Portugal were analysed using the acoustic wave sensor, as well as by UV-vis spectrophotometry. Results obtained with both methodologies were not statistically different (alpha=0.05), both in terms of accuracy and precision. This new methodology proved to be adequate for aluminium quantitation in drinking water and showed to be faster and less reagent consuming than the UV spectrophotometric methodology.
3. Acoustic Wave Monitoring of Biofilm Development in Porous Media
Science.gov (United States)
Biofilm development in porous media can result in significant changes to the hydrogeological properties of subsurface systems with implications for fluid flow and contaminant transport. As such, a number of numerical models and simulations have been developed in an attempt to qua...
4. A surface acoustic wave (SAW)-enhanced grating-coupling phase-interrogation surface plasmon resonance (SPR) microfluidic biosensor.
Science.gov (United States)
Sonato, A; Agostini, M; Ruffato, G; Gazzola, E; Liuni, D; Greco, G; Travagliati, M; Cecchini, M; Romanato, F
2016-04-07
A surface acoustic wave (SAW)-enhanced, surface plasmon resonance (SPR) microfluidic biosensor in which SAW-induced mixing and phase-interrogation grating-coupling SPR are combined in a single lithium niobate lab-on-a-chip is demonstrated. Thiol-polyethylene glycol adsorption and avidin/biotin binding kinetics were monitored by exploiting the high sensitivity of grating-coupling SPR under azimuthal control. A time saturation binding kinetics reduction of 82% and 24% for polyethylene and avidin adsorption was obtained, respectively, due to the fluid mixing enhancement by means of the SAW-generated chaotic advection. These results represent the first implementation of a nanostructured SAW-SPR microfluidic biochip capable of significantly improving the molecule binding kinetics on a single, portable device. In addition, the biochip here proposed is suitable for a great variety of biosensing applications.
5. Acoustic backscattering and radiation force on a rigid elliptical cylinder in plane progressive waves.
Science.gov (United States)
Mitri, F G
2016-03-01
6. Three dimensional full-wave nonlinear acoustic simulations: Applications to ultrasound imaging
Energy Technology Data Exchange (ETDEWEB)
Pinton, Gianmarco [Joint Department of Biomedical Engineering, University of North Carolina - North Carolina State University, 348 Taylor Hall, Chapel Hill, NC 27599, USA [email protected] (United States)
2015-10-28
Characterization of acoustic waves that propagate nonlinearly in an inhomogeneous medium has significant applications to diagnostic and therapeutic ultrasound. The generation of an ultrasound image of human tissue is based on the complex physics of acoustic wave propagation: diffraction, reflection, scattering, frequency dependent attenuation, and nonlinearity. The nonlinearity of wave propagation is used to the advantage of diagnostic scanners that use the harmonic components of the ultrasonic signal to improve the resolution and penetration of clinical scanners. One approach to simulating ultrasound images is to make approximations that can reduce the physics to systems that have a low computational cost. Here a maximalist approach is taken and the full three dimensional wave physics is simulated with finite differences. This paper demonstrates how finite difference simulations for the nonlinear acoustic wave equation can be used to generate physically realistic two and three dimensional ultrasound images anywhere in the body. A specific intercostal liver imaging scenario for two cases: with the ribs in place, and with the ribs removed. This configuration provides an imaging scenario that cannot be performed in vivo but that can test the influence of the ribs on image quality. Several imaging properties are studied, in particular the beamplots, the spatial coherence at the transducer surface, the distributed phase aberration, and the lesion detectability for imaging at the fundamental and harmonic frequencies. The results indicate, counterintuitively, that at the fundamental frequency the beamplot improves due to the apodization effect of the ribs but at the same time there is more degradation from reverberation clutter. At the harmonic frequency there is significantly less improvement in the beamplot and also significantly less degradation from reverberation. It is shown that even though simulating the full propagation physics is computationally challenging it
7. Existence domain of the compressive ion acoustic super solitary wave in a two electron temperature warm multi-ion plasma
Science.gov (United States)
Steffy, S. V.; Ghosh, S. S.
2017-10-01
The transition of an ion acoustic solitary wave into a "supersoliton," or a super solitary wave have been explored in a two electron temperature warm multi-ion plasma using the Sagdeev pseudopotential technique. It is generally believed that the ion acoustic solitary wave can be transformed to a super solitary wave only through a double layer. The present work shows that the transition route of an ion acoustic solitary wave to a super solitary wave is not unique. Depending on the electron temperature ratio, a regular solitary wave may transform to a super solitary wave either via the double layer, or through an extra-nonlinear solitary structure whose morphology differs from that of a regular one. These extra-nonlinear structures are associated with a fluctuation of the charge separation within the potential profile and are named as "variable solitary waves." Depending on these analyses, the upper and lower bounds of a super solitary wave have been deciphered and its existence domain has been delineated in the parametric space. It reveals that super solitary waves are a subset of a more generalized class of extra-nonlinear solitary structures called variable solitary waves.
8. Condition monitoring of industrial infrastructures using distributed fibre optic acoustic sensors
Science.gov (United States)
Hicke, Konstantin; Hussels, Maria-Teresa; Eisermann, René; Chruscicki, Sebastian; Krebber, Katerina
2017-04-01
Distributed fibre optic acoustic sensing (DAS) can serve as an excellent tool for real-time condition monitoring of a variety of industrial and civil infrastructures. In this paper, we portray a subset of our current research activities investigating the usability of DAS based on coherent optical time-domain reflectometry (C-OTDR) for innovative and demanding condition monitoring applications. Specifically, our application-oriented research presented here aims at acoustic and vibrational condition monitoring of pipelines and piping systems, of rollers in industrial heavy-duty conveyor belt systems and of extensive submarine power cable installations, respectively.
9. On the use of horizontal acoustic doppler profilers for continuous bed shear stress monitoring
NARCIS (Netherlands)
Vermeulen, B.; Hoitink, A.J.F.; Sassi, M.G.
2013-01-01
Continuous monitoring of bed shear stress in large river systems may serve to better estimate alluvial sediment transport to the coastal ocean. Here we explore the possibility of using a horizontally deployed acoustic Doppler current profiler (ADCP) to monitor bed shear stress, applying a prescribed
10. An effective sensor for tool wear monitoring in face milling: Acoustic ...
Abstract. Acoustic Emission (AE) has been widely used for monitoring man- ufacturing processes particularly those involving metal cutting. Monitoring the condition of the cutting tool in the machining process is very important since tool condition will affect the part size, quality and an unexpected tool failure may dam- age the ...
11. Acoustic isolation of a monopole logging while drilling tool by combining natural stopbands of pipe extensional waves
Science.gov (United States)
Su, Yuan-Da; Tang, Xiao-Ming; Xu, Song; Zhuang, Chun-Xi
2015-07-01
For extensional wave propagation along a cylindrical pipe, there exists a natural stopband in the frequency range between the first and second modes. This study explores the feasibility and practicality of building a drill collar acoustic extensional-wave isolator by combining the stopbands of pipes of different thicknesses. Numerical modelling shows that this is indeed possible and a stopband of designated width can be obtained using an optimization procedure. Laboratory measurement on an optimized design further verified this concept. The result provides a viable approach for the acoustic isolation design of a logging while drilling acoustic tool.
12. Remote Acoustic Emission Monitoring of Metal Ware and Welded Joints
Science.gov (United States)
Kapranov, Boris I.; Sutorikhin, Vladimir A.
2017-10-01
An unusual phenomenon was revealed in the metal-ultrasound interaction. Microwave sensor generates surface electric conductivity oscillations from exposure to elastic ultrasonic vibrations on regions of defects embracing micro-defects termed as “crack mouth.” They are known as the region of “acoustic activity,” method of Acoustic Emission (AE) method. It was established that the high phase-modulation coefficient of reflected field generates intentional Doppler radar signal with the following parameters: amplitude-1–5 nm, 6–30 dB adjusted to 70- 180 mm. This phenomenon is termed as “Gorbunov effect,” which is applied as a remote non-destructive testing method replacing ultrasonic flaw detection and acoustic emission methods.
13. Report on Non-invasive acoustic monitoring of D2O concentration Oct 31 2017
Energy Technology Data Exchange (ETDEWEB)
Pantea, Cristian [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Sinha, Dipen N. [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Lakis, Rollin Evan [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Beedle, Christopher Craig [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Davis, Eric Sean [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2017-11-06
There is an urgent need for real-time monitoring of the hydrogen /deuterium ratio (H/D) for heavy water production monitoring. Based upon published literature, sound speed is sensitive to the deuterium content of heavy water and can be measured using existing acoustic methods to determine the deuterium concentration in heavy water solutions. We plan to adapt existing non-invasive acoustic techniques (Swept-Frequency Acoustic Interferometry and Gaussian-pulse acoustic technique) for the purpose of quantifying H/D ratios in solution. A successful demonstration will provide an easily implemented, low cost, and non-invasive method for remote and unattended H/D ratio measurements with a resolution of less than 0.2% vol.
14. Computational study on full-wave inversion based on the acoustic wave-equation; Onkyoha hado hoteishiki full wave inversion no model keisan ni yoru kento
Energy Technology Data Exchange (ETDEWEB)
Watanabe, T.; Sassa, K. [Kyoto University, Kyoto (Japan); Uesaka, S. [Kyoto University, Kyoto (Japan). Faculty of Engineering
1996-10-01
The effect of initial models on full-wave inversion (FWI) analysis based on acoustic wave-equation was studied for elastic wave tomography of underground structures. At present, travel time inversion using initial motion travel time is generally used, and inverse analysis is conducted using the concept ray,` assuming very high wave frequency. Although this method can derive stable solutions relatively unaffected by initial model, it uses only the data of initial motion travel time. FWI calculates theoretical waveform at each receiver using all of observed waveforms as data by wave equation modeling where 2-D underground structure is calculated by difference calculus under the assumption that wave propagation is described by wave equation of P wave. Although it is a weak point that FWI is easily affected by noises in an initial model and data, it is featured by high resolution of solutions. This method offers very excellent convergence as a proper initial model is used, resulting in sufficient performance, however, it is strongly affected by initial model. 2 refs., 7 figs., 1 tab.
15. Spectral mass gauging of unsettled liquid with acoustic waves
Science.gov (United States)
Feller, Jeffrey; Kashani, Ali; Khasin, Michael; Muratov, Cyrill; Osipov, Viatcheslav; Sharma, Surendra
2017-12-01
Propellant mass gauging is one of the key technologies required to enable the next step in NASA’s space exploration program. At present, there is no reliable method to accurately measure the amount of unsettled liquid propellant in a large-scale propellant tank in micro- or zero gravity. Recently we proposed a new approach to use sound waves to probe the resonance frequencies of the two-phase liquid-gas mixture and take advantage of the mathematical properties of the high frequency spectral asymptotics to determine the volume fraction of the tank filled with liquid. We report the current progress in exploring the feasibility of this approach in the case of large propellant tanks, both experimental and theoretical. Excitation and detection procedures using solenoids for excitation and both hydrophones and accelerometers for detection have been developed. A 3% uncertainty for mass-gauging was demonstrated for a 200-liter tank partially filled with liquid for various unsettled configurations, such as tilts and artificial ullages.
16. Last call: Passive acoustic monitoring shows continued rapid decline of critically endangered vaquita.
Science.gov (United States)
Thomas, Len; Jaramillo-Legorreta, Armando; Cardenas-Hinojosa, Gustavo; Nieto-Garcia, Edwyna; Rojas-Bracho, Lorenzo; Ver Hoef, Jay M; Moore, Jeffrey; Taylor, Barbara; Barlow, Jay; Tregenza, Nicholas
2017-11-01
The vaquita is a critically endangered species of porpoise. It produces echolocation clicks, making it a good candidate for passive acoustic monitoring. A systematic grid of sensors has been deployed for 3 months annually since 2011; results from 2016 are reported here. Statistical models (to compensate for non-uniform data loss) show an overall decline in the acoustic detection rate between 2015 and 2016 of 49% (95% credible interval 82% decline to 8% increase), and total decline between 2011 and 2016 of over 90%. Assuming the acoustic detection rate is proportional to population size, approximately 30 vaquita (95% credible interval 8-96) remained in November 2016.
17. Electron-acoustic rogue waves in a plasma with Tribeche–Tsallis–Cairns distributed electrons
Energy Technology Data Exchange (ETDEWEB)
Merriche, Abderrzak [Faculty of Physics, Theoretical Physics Laboratory (TPL), Plasma Physics Group (PPG), University of Bab-Ezzouar, USTHB, B. P. 32, El Alia, Algiers 16111 (Algeria); Tribeche, Mouloud, E-mail: [email protected] [Faculty of Physics, Theoretical Physics Laboratory (TPL), Plasma Physics Group (PPG), University of Bab-Ezzouar, USTHB, B. P. 32, El Alia, Algiers 16111 (Algeria); Algerian Academy of Sciences and Technologies, Algiers (Algeria)
2017-01-15
The problem of electron-acoustic (EA) rogue waves in a plasma consisting of fluid cold electrons, nonthermal nonextensive electrons and stationary ions, is addressed. A standard multiple scale method has been carried out to derive a nonlinear Schrödinger-like equation. The coefficients of dispersion and nonlinearity depend on the nonextensive and nonthermal parameters. The EA wave stability is analyzed. Interestingly, it is found that the wave number threshold, above which the EA wave modulational instability (MI) sets in, increases as the nonextensive parameter increases. As the nonthermal character of the electrons increases, the MI occurs at large wavelength. Moreover, it is shown that as the nonextensive parameter increases, the EA rogue wave pulse grows while its width is narrowed. The amplitude of the EA rogue wave decreases with an increase of the number of energetic electrons. In the absence of nonthermal electrons, the nonextensive effects are more perceptible and more noticeable. In view of the crucial importance of rogue waves, our results can contribute to the understanding of localized electrostatic envelope excitations and underlying physical processes, that may occur in space as well as in laboratory plasmas.
18. Experimental Investigation on Acoustic Control Droplet Transfer in Ultrasonic-Wave-Assisted Gas Metal Arc Welding
Science.gov (United States)
Weifeng, Xie; Chenglei, Fan; Chunli, Yang; Sanbao, Lin
2018-02-01
Ultrasonic-wave-assisted gas metal arc welding (U-GMAW) is a new, advanced arc welding method that uses an ultrasonic wave emitted from an ultrasonic radiator above the arc. However, it remains unclear how the ultrasonic wave affects the metal droplet, hindering further application of U-GMAW. In this paper, an improved U-GMAW system was used and its superiority was experimentally demonstrated. Then a series of experiments were designed and performed to study how the ultrasonic wave affects droplet transfer, including droplet size, velocity, and motion trajectory. The behavior of droplet transfer was observed in high-speed images. The droplet transfer is closely related to the distribution of the acoustic field, determined by the ultrasonic current. Moreover, by analyzing the variably accelerated motion of the droplet, the acoustic control of the droplet transfer was intuitively demonstrated. Finally, U-GMAW was successfully used in vertical-up and overhead welding experiments, showing that U-GMAW is promising for use in welding in all positions.
19. The Propagation of Tsunami Generated Acoustic-Gravity Waves in the Atmosphere
Science.gov (United States)
Wu, Y.; Llewellyn Smith, S.; Rottman, J.; Broutman, D.; Minster, J. B. H.
2014-12-01
Tsunami-generated acoustic-gravity waves propagate in the atmosphere up to the ionosphere, where they have been observed to have an impact on the total electron content (TEC). We simulate the propagation of 2D&3D linearized acoustic-gravity waves in the atmosphere by Fourier transforming in the horizontal and solving the vertical structure with a tsunami-perturbed lower boundary and an upper radiation boundary conditions. Starting from the algorithm of Broutman (2013) and the atmospheric profile of the 2004 Sumatra Tsunami, we add compressibility to the atmosphere and extend the calculation to three dimensions. Compressibility is an important feature of the real atmosphere, and we investigate its effect on wave propagation. We obtain the vertical wavenumber as a function of buoyancy frequency, density scale height, sound speed, and background wind velocity. Results show that wind shear and compressibility have a significant impact on wave transmission and reflection. We also investigate the 3D problem to allow variations in the bottom boundary condition and in the background wind profiles. Results are quite similar to the 2D case.
20. Excitation of monochromatic and stable electron acoustic wave by two counter-propagating laser beams
Science.gov (United States)
Xiao, C. Z.; Liu, Z. J.; Zheng, C. Y.; He, X. T.
2017-07-01
The undamped electron acoustic wave is a newly-observed nonlinear electrostatic plasma wave and has potential applications in ion acceleration, laser amplification and diagnostics due to its unique frequency range. We propose to make the first attempt to excite a monochromatic and stable electron acoustic wave (EAW) by two counter-propagating laser beams. The matching conditions relevant to laser frequencies, plasma density, and electron thermal velocity are derived and the harmonic effects of the EAW are excluded. Single-beam instabilities, including stimulated Raman scattering and stimulated Brillouin scattering, on the excitation process are quantified by an interaction quantity, η =γ {τ }B, where γ is the growth rate of each instability and {τ }B is the characteristic time of the undamped EAW. The smaller the interaction quantity, the more successfully the monochromatic and stable EAW can be excited. Using one-dimensional Vlasov-Maxwell simulations, we excite EAW wave trains which are amplitude tunable, have a duration of thousands of laser periods, and are monochromatic and stable, by carefully controlling the parameters under the above conditions.
1. Plateau Waves of Intracranial Pressure and Multimodal Brain Monitoring.
Science.gov (United States)
Dias, Celeste; Maia, Isabel; Cerejo, Antonio; Smielewski, Peter; Paiva, José-Artur; Czosnyka, Marek
2016-01-01
The aim of this study was to describe multimodal brain monitoring characteristics during plateau waves of intracranial pressure (ICP) in patients with head injury, using ICM+ software for continuous recording. Plateau waves consist of an abrupt elevation of ICP above 40 mmHg for 5-20 min. This is a prospective observational study of patients with head injury who were admitted to a neurocritical care unit and who developed plateau waves. We analyzed 59 plateau waves that occurred in 8 of 18 patients (44 %). At the top of plateau waves arterial blood pressure remained almost constant, but cerebral perfusion pressure, cerebral blood flow, brain tissue oxygenation, and cerebral oximetry decreased. After plateau waves, patients with a previously better autoregulation status developed hyperemia, demonstrated by an increase in cerebral blood flow and brain oxygenation. Pressure and oxygen cerebrovascular reactivity indexes (pressure reactivity index and ORxshort) increased significantly during the plateau wave as a sign of disruption of autoregulation. Bedside multimodal brain monitoring is important to characterize increases in ICP and give differential diagnoses of plateau waves, as management of this phenomenon differs from that of regular ICP.
2. Cylindrical vector beam generation in fiber with mode selectivity and wavelength tunability over broadband by acoustic flexural wave.
Science.gov (United States)
Zhang, Wending; Huang, Ligang; Wei, Keyan; Li, Peng; Jiang, Biqiang; Mao, Dong; Gao, Feng; Mei, Ting; Zhang, Guoquan; Zhao, Jianlin
2016-05-16
Theoretical analysis and experimental demonstration are presented for the generation of cylindrical vector beams (CVBs) via mode conversion in fiber from HE11 mode to TM01 and TE01 modes, which have radial and azimuthal polarizations, respectively. Intermodal coupling is caused by an acoustic flexural wave applied on the fiber, whereas polarization control is necessary for the mode conversion, i.e. HE11x→TM01 and HE11y→TE01 for acoustic vibration along the x-axis. The frequency of the RF driving signal for actuating the acoustic wave is determined by the phase matching condition that the period of acoustic wave equals the beatlength of two coupled modes. With phase matching condition tunability, this approach can be used to generate different types of CVBs at the same wavelength over a broadband. Experimental demonstration was done in the visible and communication bands.
3. Ultrasonic guided wave mechanics for composite material structural health monitoring
Science.gov (United States)
Gao, Huidong
The ultrasonic guided wave based method is very promising for structural health monitoring of aging and modern aircraft. An understanding of wave mechanics becomes very critical for exploring the potential of this technology. However, the guided wave mechanics in complex structures, especially composite materials, are very challenging due to the nature of multi-layer, anisotropic, and viscoelastic behavior. The purpose of this thesis is to overcome the challenges and potentially take advantage of the complex wave mechanics for advanced sensor design and signal analysis. Guided wave mechanics is studied in three aspects, namely wave propagation, excitation, and damage sensing. A 16 layer quasi-isotropic composite with a [(0/45/90/-45)s]2 lay up sequence is used in our study. First, a hybrid semi-analytical finite element (SAFE) and global matrix method (GMM) is used to simulate guided wave propagation in composites. Fast and accurate simulation is achieved by using SAFE for dispersion curve generation and GMM for wave structure calculation. Secondly, the normal mode expansion (NME) technique is used for the first time to study the wave excitation characteristics in laminated composites. A clear and simple definition of wave excitability is put forward as a result of NME analysis. Source influence for guided wave excitation is plotted as amplitude on a frequency and phase velocity spectrum. This spectrum also provides a guideline for transducer design in guided wave excitation. The ultrasonic guided wave excitation characteristics in viscoelastic media are also studied for the first time using a modified normal mode expansion technique. Thirdly, a simple physically based feature is developed to estimate the guided wave sensitivity to damage in composites. Finally, a fuzzy logic decision program is developed to perform mode selection through a quantitative evaluation of the wave propagation, excitation and sensitivity features. Numerical simulation algorithms are
4. Design aspects of acoustic sensor networks for environmental noise monitoring
NARCIS (Netherlands)
Wessels, P.W.; Basten, T.G.H.
2016-01-01
An increase in public awareness of noise pollution and the impact of noise on human health has led to the need for enhanced insight in complex noise situations. This insight is commonly obtained either by brief measurements or by evaluation of a simplified acoustic model. Both of these approaches
5. Lathe stability charts via acoustic emission monitoring | Keraita ...
African Journals Online (AJOL)
Signal parameters characterizing acoustic emission (AE) detected during metal cutting have been theoretically correlated in a simple manner, to the work material properties, cutting conditions, and tool geometry. During chatter, the cutting conditions and the tool geometry change considerably. Self-exited chatter, an ...
6. Acoustic emission monitoring of crack formation during alkali silica\
Czech Academy of Sciences Publication Activity Database
Lokajíček, Tomáš; Přikryl, R.; Šachlová, Š.; Kuchařová, A.
2017-01-01
Roč. 220, MAR 30 (2017), s. 175-182 ISSN 0013-7952 R&D Projects: GA ČR(CZ) GAP104/12/0915 Keywords : Alkali-silica reaction * accelerated expansion test * ultrasonic sounding * acoustic emission * backscattered electron imaging Subject RIV: DB - Geology ; Mineralogy Impact factor: 2.569, year: 2016
7. Visco-acoustic wave-equation traveltime inversion and its sensitivity to attenuation errors
KAUST Repository
Yu, Han
2018-02-23
A visco-acoustic wave-equation traveltime inversion method is presented that inverts for the shallow subsurface velocity distribution. Similar to the classical wave equation traveltime inversion, this method finds the velocity model that minimizes the squared sum of the traveltime residuals. Even though, wave-equation traveltime inversion can partly avoid the cycle skipping problem, a good initial velocity model is required for the inversion to converge to a reasonable tomogram with different attenuation profiles. When Q model is far away from the real model, the final tomogram is very sensitive to the starting velocity model. Nevertheless, a minor or moderate perturbation of the Q model from the true one does not strongly affect the inversion if the low wavenumber information of the initial velocity model is mostly correct. These claims are validated with numerical tests on both the synthetic and field data sets.
8. Propagation of ion-acoustic solitary waves in a relativistic electron-positron-ion plasma
CERN Document Server
Saberian, E; Akbari-Moghanjoughi, M
2011-01-01
Propagation of large amplitude ion-acoustic solitary waves (IASWs) in a fully relativistic plasma consisting of cold ions and ultrarelativistic hot electrons and positrons is investigated using the Sagdeev's pseudopotential method in a relativistic hydrodynamics model. Effects of streaming speed of plasma fluid, thermal energy, positron density and positron temperature on large amplitude IASWs are studied by analysis of the pseudopotential structure. It is found that in regions that the streaming speed of plasma fluid is larger than that of solitary wave, by increasing the streaming speed of plasma fluid the depth and width of potential well increases and resulting in narrower solitons with larger amplitude. This behavior is opposite for the case where the streaming speed of plasma fluid is smaller than that of solitary wave. On the other hand, increase of the thermal energy results in wider solitons with smaller amplitude, because the depth and width of potential well decreases in that case. Additionally, th...
9. Landau damping of the dust-acoustic surface waves in a Lorentzian dusty plasma slab
Energy Technology Data Exchange (ETDEWEB)
Lee, Myoung-Jae [Department of Physics and Research Institute for Natural Sciences, Hanyang University, Seoul 04763 (Korea, Republic of); Jung, Young-Dae, E-mail: [email protected] [Department of Applied Physics and Department of Bionanotechnology, Hanyang University, Ansan, Kyunggi-Do 15588 (Korea, Republic of); Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, 110 8th Street, Troy, New York 12180-3590 (United States)
2016-01-15
Landau damping of a dust-acoustic surface wave propagating at the interfaces of generalized Lorentzian dusty plasma slab bounded by a vacuum is kinetically derived as the surface wave displays the symmetric and the anti-symmetric mode in a plasma slab. In the limiting case of small scaled wave number, we have found that Landau damping is enhanced as the slab thickness is increased. In particular, the damping of anti-symmetric mode is much stronger for a Lorentzian plasma than for a Maxwellian plasma. We have also found that the damping is more affected by superthermal particles in a Lorentzian plasma than by a Maxwellian plasma for both of the symmetric and the anti-symmetric cases. The variations of Landau damping with various parameters are also discussed.
10. Dust ion acoustic freak waves in a plasma with two temperature electrons featuring Tsallis distribution
Science.gov (United States)
Chahal, Balwinder Singh; Singh, Manpreet; Shalini; Saini, N. S.
2018-02-01
We present an investigation for the nonlinear dust ion acoustic wave modulation in a plasma composed of charged dust grains, two temperature (cold and hot) nonextensive electrons and ions. For this purpose, the multiscale reductive perturbation technique is used to obtain a nonlinear Schrödinger equation. The critical wave number, which indicates where the modulational instability sets in, has been determined precisely for various regimes. The influence of plasma background nonextensivity on the growth rate of modulational instability is discussed. The modulated wavepackets in the form of either bright or dark type envelope solitons may exist. Formation of rogue waves from bright envelope solitons is also discussed. The investigation indicates that the structural characteristics of these envelope excitations (width, amplitude) are significantly affected by nonextensivity, dust concentration, cold electron-ion density ratio and temperature ratio.
11. Symmetry and transformation of waves in optics and acoustics of crystals
CERN Document Server
Khatkevich, Anatol G
2016-01-01
It is show that in group representation by non-traditionally determining by the Maxwell equations, instead of wave, linear differential operator of momentous type from the common point of view the transformation of electromagnetic and ultrasonic radiation as well as the formation of caustics generation of solitons in crystals is represented. It is established that forming operator structural constants determine bias current with the connected charge and group velocity and also optical and acoustic axes of a crystal, which characterize its wave properties, moreover crystals are classified on common electromagnetic base. It is discovered that at change of crystal symmetry and representation of different wave process the problems also take place, which are similar to others spheres of physics and are constructed on the same aximatical base.
12. PIC simulation of compressive and rarefactive dust ion-acoustic solitary waves
Energy Technology Data Exchange (ETDEWEB)
Li, Zhong-Zheng; Zhang, Heng; Hong, Xue-Ren; Gao, Dong-Ning; Zhang, Jie; Duan, Wen-Shan, E-mail: [email protected] [College of Physics and Electronic Engineering and Joint Laboratory of Atomic and Molecular Physics of NWNU & IMP CAS, Northwest Normal University, Lanzhou 730070 (China); Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000 (China); Yang, Lei, E-mail: [email protected] [College of Physics and Electronic Engineering and Joint Laboratory of Atomic and Molecular Physics of NWNU & IMP CAS, Northwest Normal University, Lanzhou 730070 (China); Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000 (China); Department of Physics, Lanzhou University, Lanzhou 730000 (China)
2016-08-15
The nonlinear propagations of dust ion-acoustic solitary waves in a collisionless four-component unmagnetized dusty plasma system containing nonextensive electrons, inertial negative ions, Maxwellian positive ions, and negatively charged static dust grains have been investigated by the particle-in-cell method. By comparing the simulation results with those obtained from the traditional reductive perturbation method, it is observed that the rarefactive KdV solitons propagate stably at a low amplitude, and when the amplitude is increased, the prime wave form evolves and then gradually breaks into several small amplitude solitary waves near the tail of soliton structure. The compressive KdV solitons propagate unstably and oscillation arises near the tail of soliton structure. The finite amplitude rarefactive and compressive Gardner solitons seem to propagate stably.
13. Numerical simulations of three-dimensional nonlinear acoustic waves in bubbly liquids.
Science.gov (United States)
Vanhille, Christian; Campos-Pozuelo, Cleofé
2013-05-01
This paper presents three-dimensional simulations of nonlinear propagation of ultrasonic waves through bubbly liquids, which represent the continuity of our previous works included in the numerical tool SNOW-BL. The behavior of three-dimensional nonlinear acoustic waves in bubbly liquids is analyzed by means of numerical predictions. Nonlinearity, attenuation, and dispersion due to the presence of bubbles in the liquid are taken into account. The numerical solution to the differential problem is obtained by means of a finite-difference scheme. The simulations we present here consider a homogeneous distribution of bubbles in the liquid. Results compare high and low-amplitude waves to detect the nonlinear effects of the bubbles. Results are shown for radiation and enclosure problems. Copyright © 2012 Elsevier B.V. All rights reserved.
14. Paracousti-UQ: A Stochastic 3-D Acoustic Wave Propagation Algorithm.
Energy Technology Data Exchange (ETDEWEB)
Preston, Leiph [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
2017-09-01
Acoustic full waveform algorithms, such as Paracousti, provide deterministic solutions in complex, 3-D variable environments. In reality, environmental and source characteristics are often only known in a statistical sense. Thus, to fully characterize the expected sound levels within an environment, this uncertainty in environmental and source factors should be incorporated into the acoustic simulations. Performing Monte Carlo (MC) simulations is one method of assessing this uncertainty, but it can quickly become computationally intractable for realistic problems. An alternative method, using the technique of stochastic partial differential equations (SPDE), allows computation of the statistical properties of output signals at a fraction of the computational cost of MC. Paracousti-UQ solves the SPDE system of 3-D acoustic wave propagation equations and provides estimates of the uncertainty of the output simulated wave field (e.g., amplitudes, waveforms) based on estimated probability distributions of the input medium and source parameters. This report describes the derivation of the stochastic partial differential equations, their implementation, and comparison of Paracousti-UQ results with MC simulations using simple models.
15. Coupling of electrostatic ion cyclotron and ion acoustic waves in the solar wind
Energy Technology Data Exchange (ETDEWEB)
Sreeraj, T., E-mail: [email protected] [Indian Institute of Geomagnetism, Navi Mumbai (India); Singh, S. V., E-mail: [email protected]; Lakhina, G. S., E-mail: [email protected] [Indian Institute of Geomagnetism, Navi Mumbai (India); University of the Western Cape, Bellville 7535, Capetown (South Africa)
2016-08-15
The coupling of electrostatic ion cyclotron and ion acoustic waves is examined in three component magnetized plasma consisting of electrons, protons, and alpha particles. In the theoretical model relevant to solar wind plasma, electrons are assumed to be superthermal with kappa distribution and protons as well as alpha particles follow the fluid dynamical equations. A general linear dispersion relation is derived for such a plasma system which is analyzed both analytically and numerically. For parallel propagation, electrostatic ion cyclotron (proton and helium cyclotron) and ion acoustic (slow and fast) modes are decoupled. For oblique propagation, coupling between the cyclotron and acoustic modes occurs. Furthermore, when the angle of propagation is increased, the separation between acoustic and cyclotron modes increases which is an indication of weaker coupling at large angle of propagation. For perpendicular propagation, only cyclotron modes are observed. The effect of various parameters such as number density and temperature of alpha particles and superthermality on dispersion characteristics is examined in details. The coupling between various modes occurs for small values of wavenumber.
16. Finite-Difference Modeling of Acoustic and Gravity Wave Propagation in Mars Atmosphere: Application to Infrasounds Emitted by Meteor Impacts
Science.gov (United States)
Garcia, Raphael F.; Brissaud, Quentin; Rolland, Lucie; Martin, Roland; Komatitsch, Dimitri; Spiga, Aymeric; Lognonné, Philippe; Banerdt, Bruce
2017-10-01
The propagation of acoustic and gravity waves in planetary atmospheres is strongly dependent on both wind conditions and attenuation properties. This study presents a finite-difference modeling tool tailored for acoustic-gravity wave applications that takes into account the effect of background winds, attenuation phenomena (including relaxation effects specific to carbon dioxide atmospheres) and wave amplification by exponential density decrease with height. The simulation tool is implemented in 2D Cartesian coordinates and first validated by comparison with analytical solutions for benchmark problems. It is then applied to surface explosions simulating meteor impacts on Mars in various Martian atmospheric conditions inferred from global climate models. The acoustic wave travel times are validated by comparison with 2D ray tracing in a windy atmosphere. Our simulations predict that acoustic waves generated by impacts can refract back to the surface on wind ducts at high altitude. In addition, due to the strong nighttime near-surface temperature gradient on Mars, the acoustic waves are trapped in a waveguide close to the surface, which allows a night-side detection of impacts at large distances in Mars plains. Such theoretical predictions are directly applicable to future measurements by the INSIGHT NASA Discovery mission.
17. Nonlinear ion-acoustic structures in a nonextensive electron–positron–ion–dust plasma: Modulational instability and rogue waves
Energy Technology Data Exchange (ETDEWEB)
Guo, Shimin, E-mail: [email protected] [School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049 (China); Research Group MAC, Centrum Wiskunde and Informatica, Amsterdam, 1098XG (Netherlands); Mei, Liquan, E-mail: [email protected] [School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049 (China); Center for Computational Geosciences, Xi’an Jiaotong University, Xi’an, 710049 (China); Sun, Anbang [Research Group MAC, Centrum Wiskunde and Informatica, Amsterdam, 1098XG (Netherlands)
2013-05-15
The nonlinear propagation of planar and nonplanar (cylindrical and spherical) ion-acoustic waves in an unmagnetized electron–positron–ion–dust plasma with two-electron temperature distributions is investigated in the context of the nonextensive statistics. Using the reductive perturbation method, a modified nonlinear Schrödinger equation is derived for the potential wave amplitude. The effects of plasma parameters on the modulational instability of ion-acoustic waves are discussed in detail for planar as well as for cylindrical and spherical geometries. In addition, for the planar case, we analyze how the plasma parameters influence the nonlinear structures of the first- and second-order ion-acoustic rogue waves within the modulational instability region. The present results may be helpful in providing a good fit between the theoretical analysis and real applications in future spatial observations and laboratory plasma experiments. -- Highlights: ► Modulational instability of ion-acoustic waves in a new plasma model is discussed. ► Tsallis’s statistics is considered in the model. ► The second-order ion-acoustic rogue wave is studied for the first time.
18. A Catheter-Based Acoustic Interrogation Device for Monitoring Motility Dynamics of the Lower Esophageal Sphincter
Directory of Open Access Journals (Sweden)
Qian Lu
2014-08-01
Full Text Available This paper presents novel minimally-invasive, catheter-based acoustic interrogation device for monitoring motility dynamics of the lower esophageal sphincter (LES. A micro-oscillator actively emitting sound wave at 16 kHz is located at one side of the LES, and a miniature microphone is located at the other side of the sphincter to capture the sound generated from the oscillator. Thus, the dynamics of the opening and closing of the LES can be quantitatively assessed. In this paper, experiments are conducted utilizing an LES motility dynamics simulator. The sound strength is captured by the microphone and is correlated to the level of LES opening and closing controlled by the simulator. Measurements from the simulator model show statistically significant (p < 0.05 Pearson correlation coefficients (0.905 on the average in quiet environment and 0.736 on the average in noisy environment, D.O.F. = 9. Measuring the level of LES opening and closing has the potential to become a valuable diagnostic technique for understanding LES dysfunction and the disorders associated with it.
19. First in-situ passive acoustic monitoring for marine mammals during operation of a tidal turbine
DEFF Research Database (Denmark)
Malinka, Chloe Elizabeth; Gillespie, Douglas; MacAulay, Jamie
The development of marine renewables has raised concerns regarding impacts on marine wildlife, including collision risk. Here, we examine three months of passive acoustic monitoring (PAM) data collected at Tidal Energy Ltd.’s DeltaStream turbine deployment, in Ramsey Sound, Wales, UK. The PAM...... on animal presence and movement close to the turbine. This is the first time a tidal turbine has been equipped with such a PAM monitoring system, and is the first description of how marine mammals behave around an operational tidal turbine. The environmental monitoring methods presented here could be scaled...... in porpoise acoustic detection, with tidal cycle state (flood / ebb), time of day, low frequency noise levels and moon phase best explaining the acoustic presence of porpoise. There was a limited sample of turbine operation, meaning that there was insufficient data to understand the effect of turbine rotation...
20. 3D Ultrasonic Wave Simulations for Structural Health Monitoring
Science.gov (United States)
Campbell, Leckey Cara A/; Miler, Corey A.; Hinders, Mark K.
2011-01-01
Structural health monitoring (SHM) for the detection of damage in aerospace materials is an important area of research at NASA. Ultrasonic guided Lamb waves are a promising SHM damage detection technique since the waves can propagate long distances. For complicated flaw geometries experimental signals can be difficult to interpret. High performance computing can now handle full 3-dimensional (3D) simulations of elastic wave propagation in materials. We have developed and implemented parallel 3D elastodynamic finite integration technique (3D EFIT) code to investigate ultrasound scattering from flaws in materials. EFIT results have been compared to experimental data and the simulations provide unique insight into details of the wave behavior. This type of insight is useful for developing optimized experimental SHM techniques. 3D EFIT can also be expanded to model wave propagation and scattering in anisotropic composite materials. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8142998814582825, "perplexity": 3112.025391069715}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125948285.62/warc/CC-MAIN-20180426144615-20180426164615-00275.warc.gz"} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.